5. Relativistic Wave Equations and their Derivation · 5.2 The Klein–Gordon Equation 119 and the...

5. Relativistic Wave Equationsand their Derivation

5.1 Introduction

Quantum theory is based on the following axioms1:

1. The state of a system is described by a state vector |ψ⟩ in a linear space.2. The observables are represented by hermitian operators A..., and func-

tions of observables by the corresponding functions of the operators.3. The mean (expectation) value of an observable in the state |ψ⟩ is given

by ⟨A⟩ = ⟨ψ|A |ψ⟩.4. The time evolution is determined by the Schrodinger equation involving

the Hamiltonian H

i!∂ |ψ⟩∂t

= H |ψ⟩ . (5.1.1)

5. If, in a measurement of the observable A, the value an is found, then theoriginal state changes to the corresponding eigenstate |n⟩ of A.

We consider the Schrodinger equation for a free particle in the coordinaterepresentation

i!∂ψ

∂t= − !2

2m∇2ψ . (5.1.2)

It is evident from the differing orders of the time and the space derivativesthat this equation is not Lorentz covariant, i.e., that it changes its structureunder a transition from one inertial system to another.

Efforts to formulate a relativistic quantum mechanics began with at-tempts to use the correspondence principle in order to derive a relativis-tic wave equation intended to replace the Schrodinger equation. The firstsuch equation was due to Schrodinger (1926)2, Gordon (1926)3, and Klein(1927)4. This scalar wave equation of second order, which is now known asthe Klein–Gordon equation, was initially dismissed, since it led to negative1 See QM I, Sect. 8.3.2 E. Schrodinger, Ann. Physik 81, 109 (1926)3 W. Gordon, Z. Physik 40, 117 (1926)4 O. Klein, Z. Physik 41, 407 (1927)

116 5. Relativistic Wave Equations and their Derivation

probability densities. The year 1928 saw the publication of the Dirac equa-tion5. This equation pertains to particles with spin 1/2 and is able to de-scribe many of the single-particle properties of fermions. The Dirac equation,like the Klein–Gordon equation, possesses solutions with negative energy,which, in the framework of wave mechanics, leads to difficulties (see below).To prevent transitions of an electron into lower lying states of negative en-ergy, in 19306 Dirac postulated that the states of negative energy shouldall be occupied. Missing particles in these otherwise occupied states repre-sent particles with opposite charge (antiparticles). This necessarily leads toa many-particle theory, or to a quantum field theory. By reinterpreting theKlein–Gordon equation as the basis of a field theory, Pauli and Weisskopf7showed that this could describe mesons with spin zero, e.g., π mesons. Thefield theories based upon the Dirac and Klein–Gordon equations correspondto the Maxwell equations for the electromagnetic field, and the d’Alembertequation for the four-potential.

The Schrodinger equation, as well as the other axioms of quantum theory,remain unchanged. Only the Hamiltonian is changed and now represents aquantized field. The elementary particles are excitations of the fields (mesons,electrons, photons, etc.).

It will be instructive to now follow the historical development rather thanbegin immediately with quantum field theory. For one thing, it is concep-tually easier to investigate the properties of the Dirac equation in its inter-pretation as a single-particle wave equation. Furthermore, it is exactly thesesingle-particle solutions that are needed as basis states for expanding the fieldoperators. At low energies one can neglect decay processes and thus, here, thequantum field theory gives the same physical predictions as the elementarysingle-particle theory.

5.2 The Klein–Gordon Equation

5.2.1 Derivation by Means of the Correspondence Principle

In order to derive relativistic wave equations, we first recall the correspon-dence principle8. When classical quantities were replaced by the operators

energy E −→ i! ∂

∂t

and

5 P.A.M. Dirac, Proc. Roy. Soc. (London) A117, 610 (1928); ibid. A118, 351(1928)

6 P.A.M. Dirac, Proc. Roy. Soc. (London) A126, 360 (1930)7 W. Pauli and V. Weisskopf, Helv. Phys. Acta 7, 709 (1934)8 See, e.g., QM I, Sect. 2.5.1

5.2 The Klein–Gordon Equation 117

momentum p −→ !i∇ , (5.2.1)

we obtained from the nonrelativistic energy of a free particle

E =p2

2m, (5.2.2)

the free time-dependent Schrodinger equation

i! ∂

∂tψ = −!2∇2

2mψ . (5.2.3)

This equation is obviously not Lorentz covariant due to the different ordersof the time and space derivatives.

We now recall some relevant features of the special theory of relativity.9We will use the following conventions: The components of the space–timefour-vectors will be denoted by Greek indices, and the components of spa-tial three-vectors by Latin indices or the cartesian coordinates x, y, z. Inaddition, we will use Einstein’s summation convention: Greek indices thatappear twice, one contravariant and one covariant, are summed over, thesame applying to corresponding Latin indices.

Starting from xµ(s) = (ct,x), the contravariant four-vector representationof the world line as a function of the proper time s, one obtains the four-velocity xµ(s). The differential of the proper time is related to dx0 via ds =√

1 − (v/c)2 dx0, where

v = c (dx/dx0) (5.2.4a)

is the velocity. For the four-momentum this yields:

pµ = mcxµ(s) =1√

1 − (v/c)2

(mc

mv

)= four-momentum =

(E/c

p

).

(5.2.4b)

In the last expression we have used the fact that, according to relativisticdynamics, p0 = mc/

√1 − (v/c)2 represents the kinetic energy of the particle.

Therefore, according to the special theory of relativity, the energy E and themomentum px, py, pz transform as the components of a contravariant four-vector

pµ =(p0, p1, p2, p3

)=

(E

c, px, py, pz

). (5.2.5a)

9 The most important properties of the Lorentz group will be summarized in Sect.6.1.


The metric tensor

gµν =

⎛

⎜⎜⎝

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

⎞

⎟⎟⎠ (5.2.6)

yields the covariant components

pµ = gµνpν =(

E

c,−p

). (5.2.5b)

According to Eq. (5.2.4b), the invariant scalar product of the four-momentum is given by

pµpµ =E2

c2− p2 = m2c2 , (5.2.7)

with the rest mass m and the velocity of light c.From the energy–momentum relation following from (5.2.7),

E =√

p2c2 + m2c4 , (5.2.8)

one would, according to the correspondence principle (5.2.1), initially arriveat the following wave equation:

i! ∂

∂tψ =

√−!2c2∇2 + m2c4 ψ . (5.2.9)

An obvious difficulty with this equation lies in the square root of the spatialderivative; its Taylor expansion leads to infinitely high derivatives. Time andspace do not occur symmetrically.

Instead, we start from the squared relation:

E2 = p2c2 + m2c4 (5.2.10)

and obtain

−!2 ∂2

∂t2ψ = (−!2c2∇2 + m2c4)ψ . (5.2.11)

This equation can be written in the even more compact and clearly Lorentz-covariant form

(∂µ∂µ +

(mc

!

)2)

ψ = 0 . (5.2.11′)

Here xµ is the space–time position vector

xµ = (x0 = ct,x)

5.2 The Klein–Gordon Equation 119

and the covariant vector

∂µ =∂

∂xµ

is the four-dimensional generalization of the gradient vector. As is knownfrom electrodynamics, the d’Alembert operator ✷ ≡ ∂µ∂µ is invariant underLorentz transformations. Also appearing here is the Compton wavelength!/mc of a particle with mass m. Equation (5.2.11′) is known as the Klein–Gordon equation. It was originally introduced and studied by Schrodinger,and by Gordon and Klein.

We will now investigate the most important properties of the Klein–Gordon equation.

5.2.2 The Continuity Equation

To derive a continuity equation one takes ψ∗ times (5.2.11′)

ψ∗(

∂µ∂µ +(mc

!

)2)

ψ = 0

and subtracts the complex conjugate of this equation

ψ

(∂µ∂µ +

(mc

!

)2)

ψ∗ = 0 .

This yields

ψ∗∂µ∂µψ − ψ∂µ∂µψ∗ = 0∂µ(ψ∗∂µψ − ψ∂µψ∗) = 0 .

Multiplying by !2mi , so that the current density is equal to that in the non-

relativistic case, one obtains

∂

∂t

(i!

2mc2

(ψ∗ ∂ψ

∂t− ψ

∂ψ∗

∂t

))+ ∇ · !

2mi[ψ∗∇ψ − ψ∇ψ∗] = 0 .

(5.2.12)

This has the form of a continuity equation

ρ + div j = 0 , (5.2.12′)

with density

ρ =i!

2mc2

(ψ∗ ∂ψ

∂t− ψ

∂ψ∗

∂t

)(5.2.13a)

and current density

j =!

2mi(ψ∗∇ψ − ψ∇ψ∗) . (5.2.13b)


Here, ρ is not positive definite and thus cannot be directly interpreted as aprobability density, although eρ(x, t) can possibly be conceived as the corre-sponding charge density. The Klein–Gordon equation is a second-order dif-ferential equation in t and thus the initial values of ψ and ∂ψ

∂t can be chosenindependently, so that ρ as a function of x can be both positive and negative.

5.2.3 Free Solutions of the Klein–Gordon Equation

Equation (5.2.11) is known as the free Klein–Gordon equation in order todistinguish it from generalizations that additionally contain external poten-tials or electromagnetic fields (see Sect. 5.3.5). There are two free solutionsin the form of plane waves:

ψ(x, t) = ei(Et−p·x)/! (5.2.14)

with

E = ±√

p2c2 + m2c4 .

Both positive and negative energies occur here and the energy is not boundedfrom below. This scalar theory does not contain spin and could only describeparticles with zero spin.

Hence, the Klein–Gordon equation was rejected initially because the pri-mary aim was a theory for the electron. Dirac5had instead introduced a first-order differential equation with positive density, as already mentioned at thebeginning of this chapter. It will later emerge that this, too, has solutionswith negative energies. The unoccupied states of negative energy describe an-tiparticles. As a quantized field theory, the Klein–Gordon equation describesmesons7. The hermitian scalar Klein–Gordon field describes neutral mesonswith spin 0. The nonhermitian pseudoscalar Klein–Gordon field describescharged mesons with spin 0 and their antiparticles.

We shall therefore proceed by constructing a wave equation for spin-1/2fermions and only return to the Klein–Gordon equation in connection withmotion in a Coulomb potential (π−-mesons).

5.3 Dirac Equation

5.3.1 Derivation of the Dirac Equation

We will now attempt to find a wave equation of the form

i!∂ψ

∂t=

(!c

iαk∂k + βmc2

)ψ ≡ Hψ . (5.3.1)

Spatial components will be denoted by Latin indices, where repeated in-dices are to be summed over. The second derivative ∂2

∂t2 in the Klein–Gordon

5.3 Dirac Equation 121

equation leads to a density ρ =(ψ∗ ∂

∂tψ − c.c.). In order that the density be

positive, we postulate a differential equation of first order. The requirementof relativistic covariance demands that the spatial derivatives may only be offirst order, too. The Dirac Hamiltonian H is linear in the momentum operatorand in the rest energy. The coefficients in (5.3.1) cannot simply be numbers:if they were, the equation would not even be form invariant (having the samecoefficients) with respect to spatial rotations. αk and β must be hermitianmatrices in order for H to be hermitian, which is in turn necessary for apositive, conserved probability density to exist. Thus αk and β are N × Nmatrices and

ψ =

⎛

⎜⎝ψ1...

ψN

⎞

⎟⎠ an N -component column vector .

We shall impose the following requirements on equation (5.3.1):

(i) The components of ψ must satisfy the Klein–Gordon equation so thatplane waves fulfil the relativistic energy–momentum relation E2 = p2c2+m2c4.

(ii) There exists a conserved four-current whose zeroth component is a pos-itive density.

(iii) The equation must be Lorentz covariant. This means that it has thesame form in all reference frames that are connencted by a Poincaretransformation.

The resulting equation (5.3.1) is named, after its discoverer, the Dirac equa-tion. We must now look at the consequences that arise from the conditions(i)–(iii). Let us first consider condition (i). The two-fold application of Hyields

−!2 ∂2

∂t2ψ = −!2c2

∑

ij

12

(αiαj + αjαi

)∂i∂jψ

+!mc3

i

3∑

i=1

(αiβ + βαi

)∂iψ + β2m2c4ψ . (5.3.2)

Here, we have made use of ∂i∂j = ∂j∂i to symmetrize the first term on theright-hand side. Comparison with the Klein–Gordon equation (5.2.11′) leadsto the three conditions

αiαj + αjαi = 2δij 11 , (5.3.3a)

αiβ + βαi = 0 , (5.3.3b)

αi 2 = β2 = 11 . (5.3.3c)


5.3.2 The Continuity Equation

The row vectors adjoint to ψ are defined by

ψ† = (ψ∗1 , . . . , ψ∗

N ) .

Multiplying the Dirac equation from the left by ψ†, we obtain

i!ψ† ∂ψ

∂t=

!c

iψ†αi∂iψ + mc2ψ†βψ . (5.3.4a)

The complex conjugate relation reads:

−i!∂ψ†

∂tψ = −!c

i(∂iψ

†) αi†ψ + mc2ψ†β†ψ . (5.3.4b)

The difference of these two equations yields:

∂

∂t

(ψ†ψ

)= −c

((∂iψ

†) αi†ψ + ψ†αi∂iψ)

+imc2

!(ψ†β†ψ − ψ†βψ

).

(5.3.5)

In order for this to take the form of a continuity equation, the matrices αand β must be hermitian, i.e.,

αi† = αi , β† = β . (5.3.6)

Then the density

ρ ≡ ψ†ψ =N∑

α=1

ψ∗αψα (5.3.7a)

and the current density

jk ≡ cψ†αkψ (5.3.7b)

satisfy the continuity equation

∂

∂tρ + div j = 0 . (5.3.8)

With the zeroth component of jµ,

j0 ≡ cρ , (5.3.9)

we may define a four-current-density

jµ ≡ (j0, jk) (5.3.9′)

and write the continuity equation in the form

∂µjµ =1c

∂

∂tj0 +

∂

∂xkjk = 0 . (5.3.10)

The density defined in (5.3.7a) is positive definite and, within the frameworkof the single particle theory, can be given the preliminary interpretation of aprobability density.


5.3.3 Properties of the Dirac Matrices

The matrices αk, β anticommute and their square is equal to 1; see Eq.(5.3.3a–c). From (αk)2 = β2 = 11, it follows that the matrices αk and βpossess only the eigenvalues ±1.

We may now write (5.3.3b) in the form

αk = −βαkβ .

Using the cyclic invariance of the trace, we obtain

Trαk = −Trβαkβ = −Trαkβ2 = −Trαk .

From this, and from an equivalent calculation for β, one obtains

Trαk = Tr β = 0 . (5.3.11)

Hence, the number of positive and negative eigenvalues must be equal and,therefore, N is even. N = 2 is not sufficient since the 2 × 2 matrices11, σx, σy , σz contain only 3 mutually anticommuting matrices. N = 4 is thesmallest dimension in which it is possible to realize the algebraic structure(5.3.3a–c).

A particular representation of the matrices is

αi =(

0 σi

σi 0

), β =

(11 00 −11

), (5.3.12)

where the 4 × 4 matrices are constructed from the Pauli matrices

σ1 =(

0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)(5.3.13)

and the two-dimensional unit matrix. It is easy to see that the matrices(5.3.12) satisfy the conditions (5.3.3a–c):

e.g., αiβ + βαi =(

0 −σi

σi 0

)+

(0 σi

−σi 0

)= 0 .

The Dirac equation (5.3.1), in combination with the matrices (5.3.12), is re-ferred to as the “standard representation” of the Dirac equation. One calls ψa four-spinor or spinor for short (or sometimes a bispinor, in particular whenψ is represented by two two-component spinors). ψ† is called the hermitianadjoint spinor. It will be shown in Sect. 6.2.1 that under Lorentz transfor-mations spinors possess specific transformation properties.

5.3.4 The Dirac Equation in Covariant Form

In order to ensure that time and space derivatives are multiplied by matriceswith similar algebraic properties, we multiply the Dirac equation (5.3.1) byβ/c to obtain


−i!β∂0ψ − i!βαi∂iψ + mcψ = 0 . (5.3.14)

We now define new Dirac matrices

γ0 ≡ β

γi ≡ βαi .(5.3.15)

These possess the following properties:

γ0 is hermitian and (γ0)2 = 11. However, γk is antihermitian.

(γk)† = −γk and (γk)2 = −11.

Proof:(γk

)†= αkβ = −βαk = −γk ,

(γk

)2= βαkβαk = −11 .

These relations, together with

γ0γk + γkγ0 = ββαk + βαkβ = 0 and

γkγl + γlγk = βαkβαl + βαlβαk = 0 for k = l

lead to the fundamental algebraic structure of the Dirac matrices

γµγν + γνγµ = 2gµν11 . (5.3.16)

The Dirac equation (5.3.14) now assumes the form(−iγµ∂µ +

mc

!

)ψ = 0 . (5.3.17)

It will be convenient to use the shorthand notation originally introduced byFeynman:

v/ ≡ γ · v ≡ γµvµ = γµvµ = γ0v0 − γv . (5.3.18)

Here, vµ stands for any vector. The Feynman slash implies scalar multiplica-tion by γµ. In the fourth term we have introduced the covariant componentsof the γ matrices

γµ = gµνγν . (5.3.19)

In this notation the Dirac equation may be written in the compact form(−i∂/ +

mc

!

)ψ = 0 . (5.3.20)

Finally, we also give the γ matrices in the particular representation (5.3.12).From (5.3.12) and (5.3.15) it follows that


γ0 =(

11 00 −11

), γi =

(0 σi

−σi 0

). (5.3.21)

Remark. A representation of the γ matrices that is equivalent to (5.3.21) andwhich also satisfies the algebraic relations (5.3.16) is obtained by replacing

γ → MγM−1 ,

where M is an arbitrary nonsingular matrix. Other frequently encountered repre-sentations are the Majorana representation and the chiral representation (see Sect.

11.3, Remark (ii) and Eq. (11.6.12a–c)).

5.3.5 Nonrelativistic Limitand Coupling to the Electromagnetic Field

5.3.5.1 Particles at Rest

The form (5.3.1) is a particularly suitable starting point when dealing withthe nonrelativistic limit. We first consider a free particle at rest , i.e., withwave vector k = 0. The spatial derivatives in the Dirac equation then vanishand the equation then simplifies to

i!∂ψ

∂t= βmc2ψ . (5.3.17′)

This equation possesses the following four solutions

ψ(+)1 = e−

imc2! t

⎛

⎜⎜⎝

1000

⎞

⎟⎟⎠ , ψ(+)2 = e−

imc2! t

⎛

⎜⎜⎝

0100

⎞

⎟⎟⎠ ,

(5.3.22)

ψ(−)1 = e

imc2! t

⎛

⎜⎜⎝

0010

⎞

⎟⎟⎠ , ψ(−)2 = e

imc2! t

⎛

⎜⎜⎝

0001

⎞

⎟⎟⎠ .

The ψ(+)1 , ψ(+)

2 and ψ(−)1 , ψ(−)

2 correspond to positive- and negative-energysolutions, respectively. The interpretation of the negative-energy solutionsmust be postponed until later. For the moment we will confine ourselves tothe positive-energy solutions.

5.3.5.2 Coupling to the Electromagnetic Field

We shall immediately proceed one step further and consider the coupling toan electromagnetic field , which will allow us to derive the Pauli equation.


In analogy with the nonrelativistic theory, the canonical momentum p isreplaced by the kinetic momentum

(p− e

cA), and the rest energy in the

Dirac Hamiltonian is augmented by the scalar electrical potential eΦ,

i!∂ψ

∂t=

(cα ·

(p − e

cA

)+ βmc2 + eΦ

)ψ . (5.3.23)

Here, e is the charge of the particle, i.e., e = −e0 for the electron. At the endof this section we will arrive at (5.3.23), starting from (5.3.17).

5.3.5.3 Nonrelativistic Limit. The Pauli Equation

In order to discuss the nonrelativistic limit , we use the explicit representation(5.3.12) of the Dirac matrices and decompose the four-spinors into two two-component column vectors ϕ and χ

ψ ≡(

ϕ

χ

), (5.3.24)

with

i! ∂

∂t

(ϕ

χ

)= c

(σ · π χ

σ · π ϕ

)+ eΦ

(ϕ

χ

)+ mc2

(ϕ

−χ

), (5.3.25)

where

π = p − e

cA (5.3.26)

is the operator of the kinetic momentum.In the nonrelativistic limit, the rest energy mc2 is the largest energy

involved. Thus, to find solutions with positive energy, we write(

ϕ

χ

)= e−

imc2! t

(ϕ

χ

), (5.3.27)

where(ϕ

χ

)are considered to vary slowly with time and satisfy the equation

i! ∂

∂t

(ϕ

χ

)= c

(σ · π χ

σ · π ϕ

)+ eΦ

(ϕ

χ

)− 2mc2

(0χ

). (5.3.25′)

In the second equation, !χ and eΦχ may be neglected in comparison to2mc2χ, and the latter then solved approximately as

χ =σ · π2mc

ϕ . (5.3.28)

From this one sees that, in the nonrelativistic limit, χ is a factor of order∼ v/c smaller than ϕ. One thus refers to ϕ as the large, and χ as the small,component of the spinor.

Inserting (5.3.28) into the first of the two equations (5.3.25′) yields

i!∂ϕ

∂t=

(1

2m(σ · π)(σ · π) + eΦ

)ϕ . (5.3.29)

To proceed further we use the identity

σ · aσ · b = a · b + iσ · (a × b) ,

which follows from10,11 σiσj = δij + iϵijkσk, which in turn yields:

σ · π σ · π = π2 + iσ · π × π = π2 − e!c

σ · B .

Here, we have used12

(π × π)iϕ = −i!(−e

c

)εijk

(∂jA

k − Ak∂j

)ϕ

= i!e

cεijk

(∂jA

k)ϕ = i

!e

cBiϕ

with Bi = εijk∂jAk. This rearrangement can also be very easily carried out byapplication of the expression

∇ ×Aϕ + A × ∇ϕ = ∇ × Aϕ − ∇ϕ × A = (∇ × A) ϕ .

We thus finally obtain

i!∂ϕ

∂t=

[1

2m

(p − e

cA

)2− e!

2mcσ · B + eΦ

]ϕ . (5.3.29′)

This result is identical to the Pauli equation for the Pauli spinor ϕ, as is knownfrom nonrelativistic quantum mechanics13. The two components of ϕ describethe spin of the electron. In addition, one automatically obtains the correctgyromagnetic ratio g = 2 for the electron. In order to see this, we simplyneed to repeat the steps familiar to us from nonrelativistic wave mechanics.We assume a homogeneous magnetic field B that can be represented by thevector potential A:

10 Here, εijk is the totally antisymmetric tensor of third rank

εijk =

8<

:

1 for even permutations of (123)−1 for odd permutations of (123)

0 otherwise .

11 QM I, Eq.(9.18a)12 Vectors such as E, B and vector products that are only defined as three-vectors

are always written in component form with upper indices; likewise the ε tensor.Here, too, we sum over repeated indices.

13 See, e.g., QM I, Chap. 9.


B = curlA , A =12B × x . (5.3.30a)

Introducing the orbital angular momentum L and the spin S as

L = x × p , S =12

!σ , (5.3.30b)

then, for (5.3.30a), it follows14,15 that

i!∂ϕ

∂t=

(p2

2m− e

2mc(L + 2S) ·B +

e2

2mc2A2 + eΦ

)ϕ . (5.3.31)

The eigenvalues of the projection of the spin operator Se onto an arbitraryunit vector e are ±!/2. According to (5.3.31), the interaction with the elec-tromagnetic field is of the form

Hint = −µ · B +e2

2mc2A2 + eΦ , (5.3.32)

in which the magnetic moment

µ = µorbit + µspin =e

2mc(L + 2S) (5.3.33)

is a combination of orbital and spin contributions. The spin moment is ofmagnitude

µspin = ge

2mcS , (5.3.34)

with the gyromagnetic ratio (or Lande factor)

g = 2 . (5.3.35)

For the electron, e2mc = −µB

! can be expressed in terms of the Bohr magnetonµB = e0!

2mc = 0.927× 10−20erg/G.We are now in a position to justify the approximations made in this

section. The solution ϕ of (5.3.31) has a time behavior that is character-ized by the Larmor frequency or, for eΦ = −Ze2

0r , by the Rydberg energy

(Ry ∝ mc2α2, with the fine structure constant α = e20/!c). For the hydrogen

and other nonrelativistic atoms (small atomic numbers Z), mc2 is very muchlarger than either of these two energies, thus justifying for such atoms theapproximation introduced previously in the equation of motion for χ.

14 See, e.g., QM I, Chap. 9.15 One finds −p ·A−A ·p = −2A ·p = −2 1

2 (B × x) ·p = − (x × p) ·B = −L ·B,since (p · A) = !

i (∇ · A) = 0.


5.3.5.4 Supplement Concerning Couplingto an Electromagnetic Field

We wish now to use a different approach to derive the Dirac equation inan external field and, to facilitate this, we begin with a few remarks onrelativistic notation. The momentum operator in covariant and contravariantform reads:

pµ = i!∂µ and pµ = i!∂µ . (5.3.36)

Here, ∂µ = ∂∂xµ and ∂µ = ∂

∂xµ. For the time and space components, this

implies

p0 = p0 = i! ∂

∂ct, p1 = −p1 = i! ∂

∂x1=

!i

∂

∂x1. (5.3.37)

The coupling to the electromagnetic field is achieved by making the replace-ment

pµ → pµ − e

cAµ , (5.3.38)

where Aµ = (Φ,A) is the four-potential. The structure which arises here iswell known from electrodynamics and, since its generalization to other gaugetheories, is termed minimal coupling.

This implies

i! ∂

∂xµ→ i! ∂

∂xµ− e

cAµ (5.3.39)

which explicitly written in components reads:⎧⎪⎪⎪⎨

⎪⎪⎪⎩

i! ∂

∂t→ i! ∂

∂t− eΦ

!i

∂

∂xi→ !

i∂

∂xi+

e

cAi =

!i

∂

∂xi− e

cAi .

(5.3.39′)

For the spatial components this is identical to the replacement !i ∇ → !

i ∇−ecA or p → p− e

cA. In the noncovariant representation of the Dirac equation,the substitution (5.3.39′) immediately leads once again to (5.3.23).

If one inserts (5.3.39) into the Dirac equation (5.3.17), one obtains(−γµ

(i!∂µ − e

cAµ

)+ mc

)ψ = 0 , (5.3.40)

which is the Dirac equation in relativistic covariant form in the presence ofan electromagnetic field.


Remarks:

(i) Equation (5.3.23) follows directly when one multiplies (5.3.40), i.e.

γ0“i!∂0 −

ecA0

”ψ = −γi

“i!∂i −

ecAi

”ψ + mcψ

by γ0:

i!∂0ψ = αi“−i!∂i −

ecAi”

ψ +ecA0ψ + mcβψ

i! ∂∂t

ψ = c α ·“p − e

cA”

ψ + eΦψ + mc2βψ .

(ii) The minimal coupling, i.e., the replacement of derivatives by derivatives mi-nus four-potentials, has as a consequence the invariance of the Dirac equation(5.3.40) with respect to gauge transformations (of the first kind):

ψ(x) → e−i e!c α(x)ψ(x) , Aµ(x) → Aµ(x) + ∂µα(x) .

(iii) For electrons, m = me, and the characteristic length in the Dirac equationequals the Compton wavelength of the electron

λc =!

mec= 3.8 × 10−11cm .

Problems

5.1 Show that the matrices (5.3.12) obey the algebraic relations (5.3.3a–c).

5.2 Show that the representation (5.3.21) follows from (5.3.12).

5.3 Particles in a homogeneous magnetic field.Determine the energy levels that result from the Dirac equation for a (relativistic)particle of mass m and charge e in a homogeneous magnetic field B. Use the gaugeA0 = A1 = A3 = 0, A2 = Bx.

6. Lorentz Transformationsand Covariance of the Dirac Equation

In this chapter, we shall investigate how the Lorentz covariance of the Diracequation determines the transformation properties of spinors under Lorentztransformations. We begin by summarizing a few properties of Lorentz trans-formations, with which the reader is assumed to be familiar. The reader whois principally interested in the solution of specific problems may wish to omitthe next sections and proceed directly to Sect. 6.3 and the subsequent chap-ters.

6.1 Lorentz Transformations

The contravariant and covariant components of the position vector read:

xµ : x0 = ct , x1 = x , x2 = y , x3 = z contravariantxµ : x0 = ct , x1 = −x , x2 = −y , x3 = −z covariant .

(6.1.1)

The metric tensor is defined by

g = (gµν) = (gµν) =

⎛

⎜⎜⎝

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

⎞

⎟⎟⎠ (6.1.2a)

and relates covariant and contravariant components

xµ = gµνxν , xµ = gµνxν . (6.1.3)

Furthermore, we note that

gµν = gµσgσν ≡ δµ

ν , (6.1.2b)

i.e.,

(gµν) = (δµ

ν) =

⎛

⎜⎜⎝

1 0 0 00 1 0 00 0 1 00 0 0 1

⎞

⎟⎟⎠ .

132 6. Lorentz Transformations and Covariance of the Dirac Equation

The d’Alembert operator is defined by

✷ =1c2

∂2

∂t2−

3∑

i=1

∂2

∂xi 2= ∂µ∂µ = gµν∂µ∂ν . (6.1.4)

Inertial frames are frames of reference in which, in the absence of forces,particles move uniformly. Lorentz transformations tell us how the coordinatesof two inertial frames transform into one another.

The coordinates of two reference systems in uniform motion must berelated to one another by a linear transformation. Thus, the inhomogeneousLorentz transformations (also known as Poincare transformations) possessthe form

x′µ = Λµνxν + aµ , (6.1.5)

where Λµν and aµ are real.

Remarks:

(i) On the linearity of the Lorentz transformation:Suppose that x′ and x are the coordinates of an event in the inertialframes I ′ and I, respectively. For the transformation one could write

x′ = f(x) .

In the absence of forces, particles in I and I ′ move uniformly, i.e., theirworld lines are straight lines (this is actually the definition of an iner-tial frame). Transformations under which straight lines are mapped ontostraight lines are affinities, and thus of the form (6.1.5). The parametricrepresentation of the equation of a straight line xµ = eµs+dµ is mappedby such an affine transformation onto another equation for a straight line.

(ii) Principle of relativity: The laws of nature are the same in all inertialframes. There is no such thing as an “absolute” frame of reference. Therequirement that the d’Alembert operator be invariant (6.1.4) yields

ΛλµgµνΛρ

ν = gλρ , (6.1.6a)

or, in matrix form,

ΛgΛT = g . (6.1.6b)

Proof: ∂µ ≡ ∂∂xµ

=∂x′λ

∂xµ

∂∂x′λ = Λλ

µ∂′λ

∂µgµν∂ν = Λλµ∂′

λgµνΛρν∂′

ρ!= ∂′

λgλρ∂′ρ

⇒ ΛλµgµνΛρ

ν = gλρ .

6.1 Lorentz Transformations 133

The relations (6.1.6a,b) define the Lorentz transformations.Definition: Poincare group ≡ {inhomogeneous Lorentz transformation,aµ = 0}The group of homogeneous Lorentz transformations contains all elementswith aµ = 0.A homogeneous Lorentz transformation can be denoted by the shorthandform (Λ, a), e.g.,translation group (1, a)rotation group (D, 0) .

From the defining equation (6.1.6a,b) follow two important characteristics ofLorentz transformations:(i) From the definition (6.1.6a), it follows that (detΛ)2 = 1, thus

detΛ = ±1 . (6.1.7)

(ii) Consider now the matrix element λ = 0, ρ = 0 of the defining equation(6.1.6a)

Λ0µgµνΛ0

ν = 1 = (Λ00)

2 −∑

k

(Λ0k)2 = 1 .

This leads to

Λ00 ≥ 1 or Λ0

0 ≤ −1 . (6.1.8)

The sign of the determinant of Λ and the sign of Λ00 can be used to classify

the elements of the Lorentz group (Table 6.1). The Lorentz transformationscan be combined as follows into the Lorentz group L, and its subgroups orsubsets (e.g., L↓

+ means the set of all elements L↓+):

Table 6.1. Classification of the elements of the Lorentz group

sgn Λ00 det Λ

proper orthochronous L↑+ 1 1

improper orthochronous∗ L↑− 1 −1

time-reflection type∗∗ L↓− −1 −1

space–time inversion type∗∗∗ L↓+ −1 1

∗ spatial reflection ∗∗ time reflection ∗∗∗ space–time inversion

P =

0

B@

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

1

CA T =

0

B@

−1 0 0 00 1 0 00 0 1 00 0 0 1

1

CA PT =

0

B@

−1 0 0 00 −1 0 00 0 −1 00 0 0 −1

1

CA (6.1.9)


L Lorentz group (L.G.)

L↑+ restricted L.G. (is an invariant subgroup)

L↑ = L↑+ ∪ L↑

− orthochronous L.G.

L+ = L↑+ ∪ L↓

+ proper L.G.

L0 = L↑+ ∪ L↓

− orthochronous L.G.

L↑− = P · L↑

+

L↓− = T · L↑

+

L↓+ = P · T · L↑

+

The last three subsets of L do not constitute subgroups.

L = L↑ ∪ TL↑ = L↑+ ∪ PL↑

+ ∪ TL↑+ ∪ PTL↑

+ (6.1.10)

L↑ is an invariant subgroup of L; TL↑ is a coset to L↑.L↑

+ is an invariant subgroup of L; PL↑+, TL↑

+, PTL↑+ are cosets of L with

respect to L↑+. Furthermore, L↑, L+, and L0 are invariant subgroups of L

with the factor groups (E, P ), (E, P, T, PT ), and (E, T ).Every Lorentz transformation is either proper and orthochronous or can bewritten as the product of an element of the proper-orthochronous Lorentzgroup with one of the discrete transformations P , T , or PT .L↑

+, the restricted Lorentz group = the proper orthochronous L.G. consists ofall elements with detΛ = 1 and Λ0

0 ≥ 1; this includes:

(a) Rotations(b) Pure Lorentz transformations (= transformations under which space and

time are transformed). The prototype is a Lorentz transformation in thex1 direction

L1(η) =

⎛

⎜⎜⎝

L00 L0

1 0 0L1

0 L11 0 0

0 0 1 00 0 0 1

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

cosh η − sinh η 0 0− sinh η cosh η 0 0

0 0 1 00 0 0 1

⎞

⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎝

1√1−β2

− β√1−β2

0 0

− β√1−β2

1√1−β2

0 0

0 0 1 00 0 0 1

⎞

⎟⎟⎟⎟⎠, (6.1.11)

with tanh η = β. For this Lorentz transformation the inertial frame I ′

moves with respect to I with a velocity v = cβ in the x1 direction.

6.2 Lorentz Covariance of the Dirac Equation 135

6.2 Lorentz Covariance of the Dirac Equation

6.2.1 Lorentz Covariance and Transformation of Spinors

The principle of relativity states that the laws of nature are identical in everyinertial reference frame.

We consider two inertial frames I and I ′ with the space–time coordinatesx and x′. Let the wave function of a particle in these two frames be ψ andψ′, respectively. We write the Poincare transformation between I and I ′ as

x′ = Λx + a . (6.2.1)

It must be possible to construct the wave function ψ′ from ψ. This meansthat there must be a local relationship between ψ′ and ψ:

ψ′(x′) = F (ψ(x)) = F (ψ(Λ−1(x′ − a)) . (6.2.2)

The principle of relativity together with the functional relation (6.2.2) neces-sarily leads to the requirement of Lorentz covariance: The Dirac equation inI is transformed by (6.2.1) and (6.2.2) into a Dirac equation in I ′. (The Diracequation is form invariant with respect to Poincare transformations.) In orderthat both ψ and ψ′ may satisfy the linear Dirac equation, their functionalrelationship must be linear, i.e.,

ψ′(x′) = S(Λ)ψ(x) = S(Λ)ψ(Λ−1(x′ − a)) . (6.2.3)

Here, S(Λ) is a 4× 4 matrix, with which the spinor ψ is to be multiplied. Wewill determine S(Λ) below. In components, the transformation reads:

ψ′α(x′) =

4∑

β=1

Sαβ(Λ)ψβ(Λ−1(x′ − a)) . (6.2.3′)

The Lorentz covariance of the Dirac equation requires that ψ′ obey the equa-tion

(−iγµ∂′µ + m)ψ′(x′) = 0 , (c = 1, ! = 1) (6.2.4)

where

∂′µ =

∂

∂x′µ .

The γ matrices are unchanged under the Lorentz transformation. In orderto determine S, we need to convert the Dirac equation in the primed andunprimed coordinate systems into one another. The Dirac equation in theunprimed coordinate system

(−iγµ∂µ + m)ψ(x) = 0 (6.2.5)


can, by means of the relation

∂

∂xµ=

∂x′ν

∂xµ

∂

∂x′ν = Λνµ∂′

ν

and

S−1ψ′(x′) = ψ(x) ,

be brought into the form

(−iγµΛνµ∂′

ν + m)S−1(Λ)ψ′(x′) = 0 . (6.2.6)

After multiplying from the left by S, one obtains1

−iSΛνµγµS−1∂′

νψ′(x′) + mψ′(x′) = 0 . (6.2.6′)

From a comparison of (6.2.6′) with (6.2.4), it follows that the Dirac equationis form invariant under Lorentz transformations, provided S(Λ) satisfies thefollowing condition:

S(Λ)−1γνS(Λ) = Λνµγµ . (6.2.7)

It is possible to show (see next section) that this equation has nonsingu-lar solutions for S(Λ).2 A wave function that transforms under a Lorentztransformation according to ψ′ = Sψ is known as a four-component Lorentzspinor .

6.2.2 Determination of the Representation S(Λ)

6.2.2.1 Infinitesimal Lorentz Transformations

We first consider infinitesimal (proper, orthochronous) Lorentz transforma-tions

Λνµ = gν

µ + ∆ωνµ (6.2.8a)

with infinitesimal and antisymmetric ∆ωνµ

∆ωνµ = −∆ωµν . (6.2.8b)

This equation implies that ∆ωνµ can have only 6 independent nonvanishingelements.1 We recall here that the Λν

µ are matrix elements that, of course, commute withthe γ matrices.

2 The existence of such an S(Λ) follows from the fact that the matrices Λµν γν obey

the same anticommutation rules (5.3.16) as the γµ by virtue of (6.1.6a), and fromPauli’s fundamental theorem (property 7 on page 146). These transformationswill be determined explicitly below.


These transformations satisfy the defining relation for Lorentz transforma-tions

ΛλµgµνΛρ

ν = gλρ , (6.1.6a)

as can be seen by inserting (6.2.8) into this equation:

gλµgµνgρ

ν + ∆ωλρ + ∆ωρλ + O((∆ω)2

)= gλρ . (6.2.9)

Each of the 6 independent elements of ∆ωµν generates an infinitesimalLorentz transformation. We consider some typical special cases:

∆ω01 = −∆ω01 = −∆β : Transformation onto a coordinate

system moving with velocity c∆βin the x direction

(6.2.10)

∆ω12 = −∆ω12 = ∆ϕ : Transformation onto a coordinate

system that is rotated by an angle∆ϕ about the z axis. (See Fig. 6.1)

(6.2.11)

The spatial components are transformed under this passive transformation as fol-lows:

x′1 = x1 + ∆ϕx2

x′2 = −∆ϕx1 + x2

x′3 = x3or x′ = x +

0

@00

−∆ϕ

1

A× x = x +

˛˛˛e1 e2 e3

0 0 −∆ϕx1 x2 x3

˛˛˛

(6.2.12)

∆ϕ∆

x

x1

x′1

x2x′2

Fig. 6.1. Infinitesimal rotation, passive trans-formation

It must be possible to expand S as a power series in ∆ωνµ. We write

S = 11 + τ , S−1 = 11 − τ , (6.2.13)

where τ is likewise infinitesimal i.e. of order O(∆ωνµ). We insert (6.2.13) intothe equation for S, namely S−1γµS = Λµ

νγν , and get

(11 − τ)γµ(11 + τ) = γµ + γµτ − τγµ = γµ + ∆ωµνγν , (6.2.14)

from which the equation determining τ follows as


γµτ − τγµ = ∆ωµνγν . (6.2.14′)

To within an additive multiple of 11, this unambiguously determines τ . Giventwo solutions of (6.2.14′), the difference between them has to commute withall γµ and thus is proportional to 11 (see Sect. 6.2.5, Property 6). The nor-malization condition detS = 1 removes this ambiquity, since it implies tofirst order in ∆ωµν that

detS = det(11 + τ) = det 11 + Tr τ = 1 + Tr τ = 1 . (6.2.15)

It thus follows that

Tr τ = 0 . (6.2.16)

Equations (6.2.14′) and (6.2.16) have the solution

τ =18∆ωµν(γµγν − γνγµ) = − i

4∆ωµνσµν , (6.2.17)

where we have introduced the definition

σµν =i2

[γµ, γν ] . (6.2.18)

Equation (6.2.17) can be derived by calculating the commutator of τ withγµ; the vanishing of the trace is guaranteed by the general properties of theγ matrices (Property 3, Sect. 6.2.5).

6.2.2.2 Rotation About the z Axis

We first consider the rotation R3 about the z axis as given by (6.2.11). Ac-cording to (6.2.11) and (6.2.17),

τ(R3) =i2∆ϕσ12 ,

and with

σ12 = σ12 =i2

[γ1, γ2] = iγ1γ2 = i(

0 σ1

−σ1 0

) (0 σ2

−σ2 0

)=

(σ3 00 σ3

)

(6.2.19)

it follows that

S = 1 +i2∆ϕσ12 = 1 +

i2∆ϕ

(σ3 00 σ3

). (6.2.20)

By a succession of infinitestimal rotations we can construct the transfor-mation matrix S for a finite rotation through an angle ϑ. This is achieved bydecomposing the finite rotation into a sequence of N steps ϑ/N


ψ′(x′) = Sψ(x) = limN→∞

(1 +

i2N

ϑσ12

)N

ψ(x)

= ei2 ϑσ12

ψ

=(

cosϑ

2+ iσ12 sin

ϑ

2

)ψ(x) . (6.2.21)

For the coordinates and other four-vectors, this succession of transformationsimplies that

x′ = limN→∞

(11 +

ϑ

N

⎛

⎜⎜⎝

0 0 0 00 0 1 00 −1 0 00 0 0 0

⎞

⎟⎟⎠

)· · ·

(11 +

ϑ

N

⎛

⎜⎜⎝

0 0 0 00 0 1 00 −1 0 00 0 0 0

⎞

⎟⎟⎠

)x

= exp

{ϑ

⎛

⎜⎜⎝

0 0 0 00 0 1 00 −1 0 00 0 0 0

⎞

⎟⎟⎠

}x =

⎛

⎜⎜⎝

1 0 0 00 cosϑ sin ϑ 00 − sin ϑ cosϑ 00 0 0 1

⎞

⎟⎟⎠x , (6.2.22)

and is thus identical to the usual rotation matrix for rotation through anangle ϑ. The transformation S for rotations (6.2.21) is unitary (S−1 = S†).From (6.2.21), one sees that

S(2π) = −11 (6.2.23a)S(4π) = 11 . (6.2.23b)

This means that spinors do not regain their initial value after a rotationthrough 2π, but only after a rotation through 4π, a fact that is also confirmedby neutron scattering experiments3. We draw attention here to the analogywith the transformation of Pauli spinors with respect to rotations:

ϕ′(x′) = ei2 ω·σϕ(x) . (6.2.24)

6.2.2.3 Lorentz Transformation Along the x1 Direction

According to (6.2.10),

∆ω01 = ∆β (6.2.25)

and (6.2.17) becomes

τ(L1) =12∆βγ0γ1 =

12∆βα1 . (6.2.26)

We may now determine S for a finite Lorentz transformation along the x1

axis. For the velocity vc , we have tanh η = v

c .3 H. Rauch et al., Phys. Lett. 54A, 425 (1975); S.A. Werner et al., Phys. Rev. Lett.

35, 1053 (1975); also described in J.J. Sakurai, Modern Quantum Mechanics,p.162, Addison-Wesley, Red Wood City (1985).


The decomposition of η into N steps of ηN leads to the following transforma-

tion of the coordinates and other four-vectors:

x′µ = limN→∞

(g +

η

NI)µ

ν1

(g +

η

NI)ν1

ν2

· · ·(g +

η

NI)νN−1

νxν

gµν = δµ

ν ,

Iνµ =

⎛

⎜⎜⎝

0 −1 0 0−1 0 0 00 0 0 00 0 0 0

⎞

⎟⎟⎠ , I2 =

⎛

⎜⎜⎝

1 0 0 00 1 0 00 0 0 00 0 0 0

⎞

⎟⎟⎠ , I3 = I

x′ = eηIx =(

1 + ηI +12!

η2I2 +13!

η3I +14!

I2 . . .

)x

x′µ =(1 − I2 + I2 cosh η + I sinh η

)µ

νxν

=

⎛

⎜⎜⎝


0 0 1 00 0 0 1

⎞

⎟⎟⎠

⎛

⎜⎜⎝

x0

x1

x2

x3

⎞

⎟⎟⎠ . (6.2.27)

The N -fold application of the infinitesimal Lorentz transformation

L1

( η

N

)= 11 +

η

NI

then leads, in the limit of large N , to the Lorentz transformation (6.1.11)

L1(η) = eηI =

⎛

⎜⎜⎝


0 0 1 00 0 0 1

⎞

⎟⎟⎠ . (6.2.27′)

We note that the N infinitesimal steps of ηN add up to η. However, this does

not imply a simple addition of velocities.We now calculate the corresponding spinor transformation

S(L1) = limN→∞

(1 +

12

η

Nα1

)N

= eη2 α1

= 11 coshη

2+ α1 sinh

η

2.

(6.2.28)

For homogenous restricted Lorentz transformations, S is hermitian (S(L1)† =S(L1)).

For general infinitesimal transformations, characterized by infinitesimalantisymmetric ∆ωµν , equation (6.2.17) implies that

S(Λ) = 11 − i4σµν∆ωµν . (6.2.29a)


This yields the finite transformation

S(Λ) = e−i4 σµνωµν

(6.2.29b)

with ωµν = −ωνµ and the Lorentz transformation reads Λ = eω, where thematrix elements of ω are equal to ωµ

ν . For example, one can represent arotation through an angle ϑ about an arbitrary axis n as

S = ei2 ϑn·Σ , (6.2.29c)

where

Σ =(

σ 00 σ

). (6.2.29d)

6.2.2.4 Spatial Reflection, Parity

The Lorentz transformation corresponding to a spatial reflection is repre-sented by

Λµν =

⎛

⎜⎜⎝

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

⎞

⎟⎟⎠ . (6.2.30)

The associated S is determined, according to (6.2.7), from

S−1γµS = Λµνγν =

4∑

ν=1

gµνγν = gµµγµ , (6.2.31)

where no summation over µ is implied. One immediately sees that the solutionof (6.2.31), which we shall denote in this case by P , is given by

S = P ≡ eiϕγ0 . (6.2.32)

Here, eiϕ is an unobservable phase factor. This is conventionally taken tohave one of the four values ±1, ±i; four reflections then yield the identity 11.The spinors transform under a spatial reflection according to

ψ′(x′) ≡ ψ′(x′, t) = ψ′(−x, t) = eiϕγ0ψ(x) = eiϕγ0ψ(−x′, t) . (6.2.33)

The complete spatial reflection (parity) transformation for spinors is denotedby

P = eiϕγ0P(0) , (6.2.33′)

where P(0) causes the spatial reflection x → −x.

From the relationship γ0 ≡ β =(

11 00 −11

)one sees in the rest frame of

the particle, spinors of positive and negative energy (Eq. (5.3.22)) that areeigenstates of P – with opposite eigenvalues, i.e., opposite parity. This meansthat the intrinsic parities of particles and antiparticles are opposite.


6.2.3 Further Properties of S

For the calculation of the transformation of bilinear forms such as jµ(x), weneed to establish a relationship between the adjoint transformations S† andS−1.Assertion:

S†γ0 = bγ0S−1 , (6.2.34a)

where

b = ±1 for Λ00

{≥ +1≤ −1 . (6.2.34b)

Proof: We take as our starting point Eq. (6.2.7)

S−1γµS = Λµνγν , Λµ

ν real, (6.2.35)

and write down the adjoint relation

(Λµνγν)† = S†γµ†S†−1 . (6.2.36)

The hermitian adjoint matrix can be expressed most concisely as

γµ† = γ0γµγ0 . (6.2.37)

By means of the anticommutation relations, one easily checks that (6.2.37)is in accord with γ0† = γ0, γk† = −γk. We insert this into the left- andthe right-hand sides of (6.2.36) and then multiply by γ0 from the left- andright-hand side to gain

γ0Λµνγ0γνγ0γ0 = γ0S†γ0γµγ0S†−1γ0

Λµνγν = S−1γµS = γ0S†γ0γµ(γ0S†γ0)−1 ,

since (γ0)−1 = γ0. Furthermore, on the left-hand side we have made thesubstitution Λµ

νγν = S−1γµS. We now multiply by S and S−1:

γµ = Sγ0S†γ0γµ(γ0S†γ0)−1S−1 ≡ (Sγ0S†γ0)γµ(Sγ0S†γ0)−1 .

Thus, Sγ0S†γ0 commutes with all γµ and is therefore a multiple of the unitmatrix

Sγ0S†γ0 = b 11 , (6.2.38)

which also implies that

Sγ0S† = bγ0 (6.2.39)

and yields the relation we are seeking4

S†γ0 = b(Sγ0)−1 = bγ0S−1 . (6.2.34a)

Since (γ0)† = γ0 and Sγ0S† are hermitian, by taking the adjoint of (6.2.39)one obtains Sγ0S† = b∗γ0, from which it follows that

b∗ = b (6.2.40)

and thus b is real. Making use of the fact that the normalization of S is fixedby detS = 1, on calculating the determinant of (6.2.39), one obtains b4 = 1.This, together with (6.2.40), yields:

b = ±1 . (6.2.41)

The significance of the sign in (6.2.41) becomes apparent when one considers

S†S = S†γ0γ0S = bγ0S−1γ0S = bγ0Λ0νγν

= bΛ00 11 +

3∑

k=1

bΛ0k γ0γk

︸︷︷︸αk

.(6.2.42)

S†S has positive definite eigenvalues, as can be seen from the following.Firstly, detS†S = 1 is equal to the product of all the eigenvalues, and thesemust therefore all be nonzero. Furthermore, S†S is hermitian and its eigen-functions satisfy S†Sψa = aψa, whence

aψ†aψa = ψ†

aS†Sψa = (Sψa)†Sψa > 0

and thus a > 0. Since the trace of S†S is equal to the sum of all the eigen-values, we have, in view of (6.2.42) and using Tr αk = 0,

0 < Tr (S†S) = 4bΛ00 .

Thus, bΛ00 > 0. Hence, we have the following relationship between the signs

of Λ00 and b:

Λ00 ≥ 1 for b = 1

Λ00 ≤ −1 for b = −1 .(6.2.34b)

For Lorentz transformations that do not change the direction of time, wehave b = 1; while those that do cause time reversal have b = −1.

4 Note: For the Lorentz transformation L↑+ (restricted L.T. and rotations) and

for spatial reflections, one can derive this relation with b = 1 from the explicitrepresentations.


6.2.4 Transformation of Bilinear Forms

The adjoint spinor is defined by

ψ = ψ†γ0 . (6.2.43)

We recall that ψ† is referred to as a hermitian adjoint spinor. The additionalintroduction of ψ is useful because it allows quantities such as the currentdensity to be written in a concise form. We obtain the following transforma-tion behavior under a Lorentz transformation:

ψ′ = Sψ =⇒ ψ′† = ψ†S† =⇒ ψ′ = ψ†S†γ0 = b ψ†γ0S−1 ,

thus,

ψ′ = b ψS−1 . (6.2.44)

Given the above definition, the current density (5.3.7) reads:

jµ = c ψ†γ0γµψ = c ψγµψ (6.2.45)

and thus transforms as

jµ′ = c b ψS−1γµSψ = Λµνc b ψγνψ = bΛµ

νjν . (6.2.46)

Hence, jµ transforms in the same way as a vector for Lorentz transformationswithout time reflection. In the same way one immediately sees, using (6.2.3)and (6.2.44), that ψ(x)ψ(x) transforms as a scalar:

ψ′(x′)ψ′(x′) = bψ(x′)S−1Sψ(x′)= b ψ(x)ψ(x) .

(6.2.47a)

We now summarize the transformation behavior of the most important bi-linear quantities under orthochronous Lorentz transformations , i.e., transfor-mations that do not reverse the direction of time:

ψ′(x′)ψ′(x′) = ψ(x)ψ(x) scalar (6.2.47a)ψ′(x′)γµψ′(x′) = Λµ

νψ(x)γνψ(x) vector (6.2.47b)ψ′(x′)σµνψ′(x′) = Λµ

ρΛν

σψ(x)σρσψ(x) antisymmetric tensor

(6.2.47c)

ψ′(x′)γ5γµψ′(x′) = (det Λ)Λµ

ν ψ(x)γ5γνψ(x) pseudovector (6.2.47d)

ψ′(x′)γ5ψ′(x′) = (det Λ)ψ(x)γ5ψ(x) pseudoscalar, (6.2.47e)

where γ5 = iγ0γ1γ2γ3. We recall that det Λ = ±1; for spatial reflections thesign is −1.


6.2.5 Properties of the γ Matrices

We remind the reader of the definition of γ5 from the previous section:

γ5 ≡ γ5 ≡ iγ0γ1γ2γ3 (6.2.48)

and draw the reader’s attention to the fact that somewhat different definitionsmay also be encountered in the literature. In the standard representation(5.3.21) of the Dirac matrices, γ5 has the form

γ5 =(

0 1111 0

). (6.2.48′)

The matrix γ5 satisfies the relations{γ5, γµ

}= 0 (6.2.49a)

and

(γ5)2 = 11 . (6.2.49b)

By forming products of γµ, one can construct 16 linearly independent 4 × 4matrices. These are

Γ S = 11 (6.2.50a)Γ V

µ = γµ (6.2.50b)

Γ Tµν = σµν =

i2[γµ, γν ] (6.2.50c)

Γ Aµ = γ5γµ (6.2.50d)

Γ P = γ5 . (6.2.50e)

The upper indices indicate scalar, vector, tensor, axial vector (= pseudovec-tor), and pseudoscalar. These matrices have the following properties5:

1. (Γ a)2 = ±11 (6.2.51a)

2. For every Γ a except Γ S ≡ 11, there exists a Γ b, such that

Γ aΓ b = −Γ bΓ a . (6.2.51b)

3. For a = S we have TrΓ a = 0. (6.2.51c)Proof: Tr Γ a(Γ b)2 = −TrΓ bΓ aΓ b = −Tr Γ a(Γ b)2

Since (Γ b)2 = ±1, it follows that Tr Γ a = −Tr Γ a, thus proving the

assertion.

5 Only some of these properties will be proved here; other proofs are included asproblems.


4. For every pair Γ a, Γ b a = b there is a Γ c = 11, such that ΓaΓb = βΓc,β = ±1, ±i.Proof follows by considering the Γ .

5. The matrices Γ a are linearly independent.Suppose that

Pa

xaΓ a = 0 with complex coefficients xa. From property 3 above

one then has

TrX

a

xaΓ a = xS = 0 .

Multiplication by Γa and use of the properties 1 and 4 shows that subsequent

formation of the trace leads to xa = 0.6. If a 4 × 4 matrix X commutes with every γµ, then X ∝ 11.7. Given two sets of γ matrices, γ and γ′, both of which satisfy

{γµ, γν} = 2gµν ,

there must exist a nonsingular M

γ′µ = MγµM−1 . (6.2.51d)

This M is unique to within a constant factor (Pauli’s fundamental theo-rem).

6.3 Solutions of the Dirac Equation for Free Particles

6.3.1 Spinors with Finite Momentum

We now seek solutions of the free Dirac equation (5.3.1) or (5.3.17)

(−i∂/ + m)ψ(x) = 0 . (6.3.1)

Here, and below, we will set ! = c = 1.For particles at rest, these solutions [see (5.3.22)] read:

ψ(+)(x) = ur(m,0) e−imt r = 1, 2

ψ(−)(x) = vr(m,0) eimt ,(6.3.2)

for the positive and negative energy solutions respectively, with

u1(m,0) =

⎛

⎜⎜⎝

1000

⎞

⎟⎟⎠ , u2(m,0) =

⎛

⎜⎜⎝

0100

⎞

⎟⎟⎠ ,

v1(m,0) =

⎛

⎜⎜⎝

0010

⎞

⎟⎟⎠ , v2(m,0) =

⎛

⎜⎜⎝

0001

⎞

⎟⎟⎠ ,

(6.3.3)

6.3 Solutions of the Dirac Equation for Free Particles 147

and are normalized to unity. These solutions of the Dirac equation are eigen-functions of the Dirac Hamiltonian H with eigenvalues ±m, and also of theoperator (the matrix already introduced in (6.2.19))

σ12 =i2[γ1, γ2] =

(σ3 00 σ3

)(6.3.4)

with eigenvalues +1 (for r = 1) and −1 (for r = 2). Later we will show thatσ12 is related to the spin.

We now seek solutions of the Dirac equation for finite momentum in theform6

ψ(+)(x) = ur(k) e−ik·x positive energy (6.3.5a)ψ(−)(x) = vr(k) eik·x negative energy (6.3.5b)

with k0 > 0. Since (6.3.5a,b) must also satisfy the Klein–Gordon equation,we know from (5.2.14) that

kµkµ = m2 , (6.3.6)

or

E ≡ k0 =(k2 + m2

)1/2, (6.3.7)

where k0 is also written as E; i.e., k is the four-momentum of a particle withmass m.

The spinors ur(k) and vr(k) can be found by Lorentz transformation ofthe spinors (6.3.3) for particles at rest: We transform into a coordinate systemthat is moving with velocity −v with respect to the rest frame and then, fromthe rest-state solutions, we obtain the free wave functions for electrons withvelocity v. However, a more straightforward approach is to determine thesolutions directly from the Dirac equation. Inserting (6.3.5a,b) into the Diracequation (6.3.1) yields:

(k/ − m)ur(k) = 0 and (k/ + m)vr(k) = 0 . (6.3.8)

Furthermore, we have

k/k/ = kµγµkνγν = kµkν12{γµ, γν} = kµkνgµν . (6.3.9)

Thus, from (6.3.6), one obtains

(k/ − m)(k/ + m) = k2 − m2 = 0 . (6.3.10)

Hence one simply needs to apply (k/ + m) to the ur(m,0) and (k/ − m) tothe vr(m,0) in order to obtain the solutions ur(k) and vr(k) of (6.3.8). The6 We write the four-momentum as k, the four-coordinates as x, and their scalar

product as k · x.


normalization remains as yet unspecified; it must be chosen such that it iscompatible with the solution (6.3.3), and such that ψψ transforms as a scalar(Eq. (6.2.47a)). As we will see below, this is achieved by means of the factor1/

√2m(m + E):

ur(k) =k/ + m√

2m(m + E)ur(m,0) =

⎛

⎜⎜⎜⎜⎝

(E + m

2m

)1/2

χr

σ · k(2m(m + E))1/2

χr

⎞

⎟⎟⎟⎟⎠(6.3.11a)

vr(k) =−k/ + m√2m(m + E)

vr(m,0) =

⎛

⎜⎜⎜⎜⎝

σ · k(2m(m + E))1/2

χr

(E + m

2m

)1/2

χr

⎞

⎟⎟⎟⎟⎠. (6.3.11b)

Here, the solutions are represented by ur(m,0) =(χr

0

)and vr(m,0) =

(0

χr

)

with χ1 =(10

)and χ2 =

(01

).

In this calculculation we have made use of

k/

χr

0

!=

»k0

„11 00 −11

«− ki

„0 σi

−σi 0

«– χr

0

!

=

k0χr

0

!+

0

kiσiχr

!=

Eχr

k · σχr

!

and

−k/

0χr

!=

0

k0χr

!+

kiσiχr

0

!, r = 1, 2 .

From (6.3.11a,b) one finds for the adjoint spinors defined in (6.2.43)

ur(k) = ur(m,0)k/ + m√

2m(m + E)(6.3.12a)

vr(k) = vr(m,0)−k/ + m√2m(m + E)

. (6.3.12b)

Proof: ur(k) = u†r(k)γ0 = u†

r(m,0)(γµ†kµ+m)γ0√

2m(m+E)= u†

r(m,0)γ0(γµkµ+m)√

2m(m+E),

since γµ† = γ0γµγ0 and (γ0)2 = 11

Furthermore, the adjoint spinors satisfy the equations

ur(k) (k/ − m) = 0 (6.3.13a)


and

vr(k) (k/ + m) = 0 , (6.3.13b)

as can be seen from (6.3.10) and (6.3.12a,b) or (6.3.8).

6.3.2 Orthogonality Relations and Density

We shall need to know a number of formal properties of the solutions foundabove for later use. From (6.3.11) and (6.2.37) it follows that:

ur(k)us(k) = ur(m,0)(k/ + m)2

2m(m + E)us(m,0) . (6.3.14a)

With

ur(m,0)(k/ + m)2us(m,0) = ur(m,0)(k/2 + 2mk/ + m2)us(m,0)

= ur(m,0)(2m2 + 2mk/)us(m,0)

= ur(m,0)(2m2 + 2mk0γ0)us(m,0)

= 2m(m + E)ur(m,0)us(m,0)

= 2m(m + E)δrs ,

(6.3.14b)

ur(k)vs(k) = ur(m,0)k/2 − m2

2m(m + E)vs(m,0)

= ur(m,0) 0 vs(m,0) = 0

(6.3.14c)

and a similar calculation for vr(k), equations (6.3.14a,b) yield the

orthogonality relations

ur(k)us(k) = δrs ur(k) vs(k) = 0vr(k) vs(k) = −δrs vr(k)us(k) = 0.

(6.3.15)

Remarks:

(i) This normalization remains invariant under orthochronous Lorentz trans-formations:

u′r u′

s = u†r S† γ0 S us = u†

r γ0 S−1 S us = ur us = δrs . (6.3.16)

(ii) For these spinors, ψ(x)ψ(x) is a scalar,

ψ(+)(x)ψ(+)(x) = eik·xur(k)ur(k)e−ikx = 1 , (6.3.17)

is independent of k, and thus independent of the reference frame.In general, for a superposition of positive energy solutions, i.e., for


ψ(+)(x) =2∑

r=1

crur , with2∑

r=1

|cr|2 = 1 , (6.3.18a)

one has the relation

ψ(+)(x)ψ(+)(x) =∑

r,s

ur(k)us(k)c∗r cs =2∑

r=1

|cr|2 = 1 . (6.3.18b)

Analogous relationships hold for ψ(−).(iii) If one determines ur(k) through a Lorentz transformation corresponding

to −v, this yields exactly the above spinors. Viewed as an active trans-formation, this amounts to transforming ur(m,0) to the velocity v. Sucha transformation is known as a “boost”.

The density for a plane wave (c = 1) is ρ = j0 = ψγ0ψ . This is not aLorentz-invariant quantity since it is the zero-component of a four-vector:

ψ(+)r (x)γ0 ψ(+)

s (x) = ur(k)γ0 us(k)

= ur(k){k/, γ0}

2mus(k) =

E

mδrs (6.3.19a)

ψ(−)r (x)γ0 ψ(−)

s (x) = vr(k)γ0 vs(k)

= −vr(k){k/, γ0}

2mvs(k) =

E

mδrs . (6.3.19b)

In the intermediate steps here, we have used us(k) = (k//m)us(k), us(k) =us(k)(k//m) (Eqs. (6.3.8) and (6.3.13)) etc.

Note. The spinors are normalized such that the density in the rest frame is unity.

Under a Lorentz transformation, the density times the volume must remain con-stant. The volume is reduced by a factor

p1 − β2 and thus the density must increase

by the reciprocal factor 1√1−β2

= Em .

We now extend the sequence of equations (6.3.19).

For ψ(+)r (x) = e−i(k0x0−k·x)ur(k)

and ψ(−)s (x) = ei(k0x0+k·x)vs(k)

(6.3.20)

with the four-momentum k = (k0,−k), one obtains

ψ(−)r (x)γ0 ψ(+)

s (x) = e−2ik0x0vr(k)γ0 us(k)

=12e−2ik0x0

vr(k)

(− k/

mγ0 + γ0 k/

m

)us(k) (6.3.19c)

= 0

since the terms proportional to k0 cancel and since{kiγi, γ0

}= 0. In this

sense, positive and negative energy states are orthogonal for opposite energiesand equal momenta.


6.3.3 Projection Operators

The operators

Λ±(k) =±k/ + m

2m(6.3.21)

project onto the spinors of positive and negative energy, respectively:

Λ+ur(k) = ur(k) Λ−vr(k) = vr(k)Λ+vr(k) = 0 Λ−ur(k) = 0 .

(6.3.22)

Thus, the projection operators Λ±(k) can also be represented in the form

Λ+(k) =∑

r=1,2

ur(k) ⊗ ur(k)

Λ−(k) = −∑

r=1,2

vr(k) ⊗ vr(k) .(6.3.23)

The tensor product ⊗ is defined by

(a ⊗ b)αβ = aαbβ . (6.3.24)

In matrix form, the tensor product of a spinor a and an adjoint spinor b reads:

0

B@

a1

a2

a3

a4

1

CA`b1, b2, b3, b4

´=

0

BB@

a1b1 a1b2 a1b3 a1b4

a2b1 a2b2 a2b3 a2b4

a3b1 a3b2 a3b3 a3b4

a4b1 a4b2 a4b3 a4b4

1

CCA .

The projection operators have the following properties:

Λ2±(k) = Λ±(k) (6.3.25a)

Tr Λ±(k) = 2 (6.3.25b)Λ+(k) + Λ−(k) = 1 . (6.3.25c)

Proof:

Λ±(k)2 =(±k/ + m)2

4m2=

k/2 ± 2k/m + m2

4m2=

m2 ± 2k/m + m2

4m2

=2m(±k/ + m)

4m2= Λ±(k)

Tr Λ±(k) =4m2m

= 2

The validity of the assertion that Λ± projects onto positive and negative energystates can be seen in both of the representations, (6.3.21) and (6.3.22), by applying

them to the states ur(k) and vr(k). A further important projection operator, P (n),

which, in the rest frame projects onto the spin orientation n, will be discussed inProblem 6.15.


Problems

6.1 Prove the group property of the Poincare group.

6.2 Show, by using the transformation properties of xµ, that ∂µ ≡ ∂/∂xµ (∂µ ≡∂/∂xµ) transforms as a contravariant (covariant) vector.

6.3 Show that the N-fold application of the infinitesimal rotation in Minkowskispace (Eq. (6.2.22))

Λ = 1 +ϑN

0

B@

0 0 0 00 0 1 00 −1 0 00 0 0 0

1

CA

leads, in the limit N → ∞, to a rotation about the z axis through an angle ϑ (thelast step in (6.2.22)).

6.4 Derive the quadratic form of the Dirac equation»“

i!∂ − ecA”2

− i!ec

(αE + iΣB) − m2c2

–ψ = 0

for the case of external electromagnetic fields. Write the result using the electro-magnetic field tensor Fµν = Aµ,ν − Aν,µ, and also in a form explicitly dependenton E and B.

Hint: Multiply the Dirac equation from the left by γν`i!∂ν − e

cAν

´+ mc and, by

using the commutation relations for the γ matrices, bring the expression obtainedinto quadratic form in terms of the field tensor

»“i!∂ − e

cA”2

− !e2c

σµνFµν − m2c2

–ψ = 0 .

The assertion follows by evaluating the expression σµνFµν using the explicit formof the field tensor as a function of the fields E and B.

6.5 Consider the quadratic form of the Dirac equation from Problem 6.4 with thefields E = E0 (1, 0, 0) and B = B (0, 0, 1), where it is assumed that E0/Bc ≤ 1.Choose the gauge A = B (0, x, 0) and solve the equation with the ansatz

ψ(x) = e−iEt/!ei(kyy+kzz)ϕ(x)Φ ,

where Φ is a four-spinor that is independent of time and space coordinates. Cal-culate the energy eigenvalues for an electron. Show that the solution agrees withthat obtained from Problem 5.3 when one considers the nonrelativistic limit, i.e.,E0/Bc ≪ 1.

Hint: Given the above ansatz for ψ, one obtains the following form for the quadraticDirac equation:

[K(x, ∂x)11 + M ] ϕ(x)Φ = 0 ,

Problems 153

where K(x, ∂x) is an operator that contains constant contributions, ∂x and x. Thematrix M is independent of ∂x and x; it has the property M2 ∝ 11. This suggeststhat the bispinor Φ has the form Φ = (11 + λM)Φ0. Determine λ and the eigenval-ues of M . With these eigenvalues, the matrix differential equation reverts into anordinary differential equation of the oscillator type.

6.6 Show that equation (6.2.14′)

[γµ, τ ] = ∆ωµνγν

is satisfied by

τ =18∆ωµν(γµγν − γνγµ) .

6.7 Prove that γµ† = γ0γµγ0.

6.8 Show that the relation

S†γ0 = bγ0S−1

is satisfied with b = 1 by the explicit representations of the elements of the Poincaregroup found in the main text (rotation, pure Lorentz transformation, spatial reflec-tion).

6.9 Show that ψ(x)γ5ψ(x) is a pseudoscalar, ψ(x)γ5γµψ(x) a pseudovector, andψ(x)σµνψ(x) a tensor.

6.10 Properties of the matrices Γ a.Taking as your starting point the definitions (6.2.50a–e), derive the following prop-erties of these matrices:(i) For every Γ a (except Γ S) there exists a Γ b such that Γ aΓ b = −Γ bΓ a.(ii) For every pair Γ a, Γ b, (a = b) there exists a Γ c = 11 such that Γ aΓ b = βΓ c

with β = ±1, ±i.

6.11 Show that if a 4 × 4 matrix X commutes with all γµ, then this matrix X isproportional to the unit matrix.Hint : Every 4 × 4 matrix can, according to Problem 6.1, be written as a linearcombination of the 16 matrices Γ a (basis!).

6.12 Prove Pauli’s fundamental theorem for Dirac matrices: For any two four-dimensional representations γµ and γ′

µ of the Dirac algebra both of which satisfythe relation

{γµ, γν} = 2gµν

there exists a nonsingular transformation M such that

γ′µ = MγµM−1 .

M is uniquely determined to within a constant prefactor.


6.13 From the solution of the field-free Dirac equation in the rest frame, determinethe four-spinors ψ±(x) of a particle moving with the velocity v. Do this by applyinga Lorentz transformation (into a coordinate system moving with the velocity −v)to the solutions in the rest frame.

6.14 Starting from

Λ+(k) =X

r=1,2

ur(k) ⊗ ur(k) , Λ−(k) = −X

r=1,2

vr(k) ⊗ vr(k) ,

prove the validity of the representations for Λ±(k) given in (6.3.22).

6.15 (i) Given the definition P (n) = 12 (1+γ5n/), show that, under the assumptions

n2 = −1 and nµkµ = 0, the following relations are satisfied

(a) [Λ±(k), P (n)] = 0 ,

(b) Λ+(k)P (n) + Λ−(k)P (n) + Λ+(k)P (−n) + Λ−(k)P (−n) = 1 ,

(c) Tr [Λ±(k)P(±n)] = 1 ,

(d) P (n)2 = P (n)

(ii) Consider the special case n = (0, ez) where P (n) = 12

„1 + σ3 0

0 1 − σ3

«.

7. Orbital Angular Momentum and Spin

We have seen that, in nonrelativistic quantum mechanics, the angular mo-mentum operator is the generator of rotations and commutes with the Hamil-tonians of rotationally invariant (i.e., spherically symmetric) systems1. It thusplays a special role for such systems. For this reason, as a preliminary to thenext topic – the Coulomb potential – we present here a detailed investigationof angular momentum in relativistic quantum mechanics.

7.1 Passive and Active Transformations

For positive energy states, in the non-relativistic limit we derived the Pauliequation with the Lande factor g = 2 (Sect. 5.3.5). From this, we concludedthat the Dirac equation describes particles with spin S = 1/2. Following onfrom the transformation behavior of spinors, we shall now investigate angularmomentum in general.

In order to give the reader useful background information, we will startwith some remarks concerning active and passive transformations. Consider agiven state Z, which in the reference frame I is described by the spinor ψ(x).When regarded from the reference frame I ′, which results from I through theLorentz transformation

x′ = Λx , (7.1.1)

the spinor takes the form,

ψ′(x′) = Sψ(Λ−1x′) , passive with Λ . (7.1.2a)

A transformation of this type is known as a passive transformation. Oneand the same state is viewed from two different coordinate systems, which isindicated in Fig. 7.1 by ψ(x) = ψ′(x′).

On the other hand, one can also transform the state and then view theresulting state Z ′ exactly as the starting state Z from one and the samereference frame I. In this case one speaks of an active transformation. Forvectors and scalars, it is clear what is meant by their active transformation1 See QM I, Sect. 5.1

156 7. Orbital Angular Momentum and Spin

(rotation, Lorentz transformation). The active transformation of a vectorby the transformation Λ corresponds to the passive transformation of thecoordinate system by Λ−1. For spinors, the active transformation is definedin exactly this way (see Fig. 7.1).

The state Z ′, which arises through the transformation Λ−1, appears in Iexactly as Z in I ′, i.e.,

ψ′(x) = Sψ(Λ−1x) active with Λ−1 (7.1.2b)

Λ

Λ

ΛΛ−ΛΛ 11

I

I ′

ψ′′(x)

ψ(x) = ψ′(x′)

ψ′(x)

Z′′′

Z

Z′Fig. 7.1. Schematic representa-tion of the passive and activetransformation of a spinor; the en-closed area is intended to indicatethe region in which the spinor isfinite

The state Z ′′, which results from Z through the active transformation Λ,by definition appears the same in I ′ as does Z in I, i.e., it takes the formψ(x′). Since I is obtained from I ′ by the Lorentz transformation Λ−1, in Ithe spinor Z ′′ has the form

ψ′′(x) = S−1ψ(Λx) , active with Λ . (7.1.2c)

7.2 Rotations and Angular Momentum

Under the infinitesimal Lorentz transformation

Λµν = gµ

ν + ∆ωµν , (Λ−1)µ

ν = gµν − ∆ωµ

ν , (7.2.1)

a spinor ψ(x) transforms as

ψ′(x′) = Sψ(Λ−1x′) passive with Λ (7.2.2a)

or

ψ′(x) = Sψ(Λ−1x) active with Λ−1 . (7.2.2b)

7.2 Rotations and Angular Momentum 157

We now use the results gained in Sect. 6.2.2.1 (Eqs. (6.2.8) and (6.2.13)) toobtain

ψ′(x) = (11 − i4

∆ωµνσµν)ψ(xρ − ∆ωρνxν) . (7.2.3)

Taylor expansion of the spinor yields (1 − ∆ωµνxν∂µ)ψ(x) , so that

ψ′(x) = (11 + ∆ωµν(− i4

σµν + xµ∂ν))ψ(x) . (7.2.3′)

We now consider the special case of rotations through ∆ϕ, which are repre-sented by

∆ωij = −ϵijk∆ϕk (7.2.4)

(the direction of ∆ϕ specifies the rotation axis and |∆ϕ| the angle of rota-tion). If one also uses

σij = σij = ϵijkΣk , Σk =(

σk 00 σk

), (7.2.5)

(see Eq. (6.2.19)) one obtains for (7.2.3′)

ψ′(x) =(

1 + ∆ωij

(− i

4ϵijkΣk + xi∂j

))ψ(x)

=(

1 − ϵijk∆ϕk

(− i

4ϵijkΣk − xi∂j

))ψ(x)

=(

1 − ∆ϕk

(− i

42δkkΣk − ϵijkxi∂j

))ψ(x)

=(

1 + i∆ϕk

(12

Σk + ϵkijxi 1i

∂j

))ψ(x)

≡(1 + i∆ϕkJk

)ψ(x) .

(7.2.6)

Here, we have defined the total angular momentum

Jk = ϵkijxi 1i

∂j +12

Σk . (7.2.7)

With the inclusion of !, this operator reads:

J = x × !i

∇11 +!2

Σ , (7.2.7′)

and is thus the sum of the orbital angular momentum L = x × p and thespin !

2 Σ.The total angular momentum (= orbital angular momentum + spin) is

the generator of rotations: For a finite angle ϕk one obtains, by combining asuccession of infinitesimal rotations,

158 7. Orbital Angular Momentum and Spin

ψ′(x) = eiϕkJk

ψ(x) . (7.2.8)

The operator Jk commutes with the Hamiltonian of the Dirac equation con-taining a spherically symmetric potential Φ(x) = Φ(|x|)

[H, J i] = 0 . (7.2.9)

A straightforward way to verify (7.2.9) is by an explicit calculation of thecommutator (see Problem 7.1). Here, we consider general consequences re-sulting from the behavior, under rotation, of the structure of commutators ofthe angular momentum with other operators; Eq. (7.2.9) results as a specialcase. We consider an operator A, and let the result of its action on ψ1 be thespinor ψ2:

Aψ1(x) = ψ2(x) .

It follows that

eiϕkJk

A e−iϕkJk(eiϕkJk

ψ1(x))

=(eiϕkJk

ψ2(x))

or, alternatively,

eiϕkJk

A e−iϕkJk

ψ′1(x) = ψ′

2(x) .

Thus, in the rotated frame of reference the operator is

A′ = eiϕkJk

A e−iϕkJk

. (7.2.10)

Expanding this for infinitesimal rotations (ϕk → ∆ϕk) yields:

A′ = A − i∆ϕk[A, Jk] . (7.2.11)

The following special cases are of particular interest:

(i) A is a scalar (rotationally invariant) operator. Then, A′ = A and from(7.2.11) it follows that

[A, Jk] = 0 . (7.2.12)

The Hamiltonian of a rotationally invariant system (including a spheri-cally symmetric Φ(x) = Φ(|x|)) is a scalar; this leads to (7.2.9). Hence,in spherically symmetric problems the angular momentum is conserved.

(ii) For the operator A we take the components of a three-vector v . Asa vector, v transforms according to v′i = vi + ϵijk ∆ϕj vk. Equatingthis, component by component, with (7.2.11), vi + ϵijk∆ϕjvk = vi +i!∆ϕj

[Jj , vi

]which shows that

[J i, vj ] = i! ϵijk vk . (7.2.13)

Problems 159

The commutation relation (7.2.13) implies, among other things,[J i, Jj

]= i!ϵijkJk (7.2.14a)

[J i, Lj

]= i!ϵijkLk . (7.2.14b)

It is clear from the explicit representation Σk =(

σk 00 σk

)that the eigen-

values of the 4 × 4 matrices Σk are doubly degenerate and take the values±1. The angular momentum J is the sum of the orbital angular momentumL and the intrinsic angular momentum or spin S, the components of whichhave the eigenvalues ± 1

2 . Thus, particles that obey the Dirac equation havespin S = 1/2. The operator

(!2Σ

)2 = 34!211 has the eigenvalue 3!2

4 . Theeigenvalues of L2 and L3 are !2l(l + 1) and !ml, where l = 0, 1, 2, . . . andml takes the values −l,−l + 1, . . . , l − 1, l. The eigenvalues of J2 are thus!2j(j + 1), where j = l ± 1

2 for l = 0 and j = 12 for l = 0. The eigenvalues

of J3 are !mj , where mj ranges in integer steps between −j and j. The op-erators J2, L2, Σ2, and J3 can be simultaneously diagonalized. The orbitalangular momentum operators Li and the spin operators Σi themselves fulfillthe angular momentum commutation relations.Note: One is tempted to ask how it is that the Dirac Hamiltonian, a 4× 4 matrix,can be a scalar. In order to see this, one has to return to the transformation (6.2.6′).The transformed Hamiltonian including a central potential Φ(|x|)

(−iγν ∂′ν + m + eΦ(|x′|)) = S(−iγν ∂ν + m + eΦ(|x|))S−1

has, under rotations, the same form in both systems. The property “scalar” means

invariance under rotations, but is not necessarily limited to one-component spher-

ically symmetric functions.

Problems

7.1 Show, by explicit calculation of the commutator, that the total angular mo-mentum

J = x × p 11 +!2

Σ

commutes with the Dirac Hamiltonian for a central potential

H = c

3X

k=1

αkpk + βmc

!+ eΦ(|x|) .

CHAPTER 4

Boson Wave Equations

The dynamical quantum-mechanical wave equations of spin-0 pions, spin-1 vector particles, and massless spin-1 photons are formulated in a consistent one-particle fashion. For the spin-0 Klein-Gordon equation, the interpreta-tion of negative-energy states as describing antiparticles is stressed. The relativistic bound-state Coulomb problem is then solved for rc-mesic atoms. The parallel is made between the massive spin-1 and photon wave equations. The notion of currents, current conservation and gauge invariance for photon amplitudes is discussed in detail and linked to the principle of mini-mal replacement. Minimal coupling of photons to charged particles will be the basis of the general electromagnetic interaction to be considered in later chapters. Second-quantized field theories are briefly described, and an ana-logy is made between (relativistic) photons and nonrelativistic phonons.

4.A Spin-0 Klein-Gordon Equation

Derivation. For a particle moving at relativistic velocities (i.e., having a kinetic energy that is a substantial fraction of its rest mass), the nonrelati-vistic approximation for energy, E = m + p2/2m, is no longer valid and one must use instead the exact relation £ = (p2 + m2)*. The formal quantum-mechanical replacement p -» — iV would then result in the Hamiltonian

H = ( -V 2 + m2)* (4.1)

48

Spin-0 Klein-Gordon Equation 49

and a free particle Schrodinger-type equation id,il/ = ( -V 2 + m2)H. (4.2)

Since the square-root operation in (4.1) and (4.2) is difficult to interpret, it would seem more reasonable to construct a relativistic wave equation asso-ciated with the square of the Hamiltonian operator, (idt)2(p = H2<f>. Then defining the D'Alembertian operator as

n=d„d" = d2-\2=-p2, (4.3)

where pM = id^, one is naturally led to the free-particle Klein-Gordon equation

( • + m2)(t>(x) = 0, (p2 - m2)<f>(x) = 0. (4.4)

Such a particle is said to be "on its mass shell", p2 = m2.

Covariance. The manifest covariance of (4.4) for a quantum-mechanical (Lorentz) scalar wave function, </>'(x') = </>(x), i.e., for

|</.'> = l/A \<P), (4.5a)

4>'{x) = <x | <t>") = <x I UA| </>> = <A" lx 10> = </>(A" lx), (4.5b)

implies that

(a;3'" + m2)</)'(x') = (d^d" + m2)<t>{x) = 0. (4.6)

Due to these transformation properties, the wave function <p(x) must describe a spin-0 particle (e.g., a pion). Spin-0 solutions of the free-particle Klein-Gordon equation (4.4) are proportional to the invariant plane-wave functions

</>+(x)oce-ip'x = e'p"e-,'£', f ( x ) o c c i ' J = e " i p V £ ' . (4.7)

These equations are special cases of the general solutions of the Klein-Gordon wave equation containing a possible interaction term,

cP±(x) = (t>±(x)e+iEt. (4.8) It is therefore clear that the use of the operator H2 in forming a wave equation leads to seemingly unphysical negative-energy solutions e+,Et as well as physical positive-energy solutions e~'E' for the quantum mechanical state of the particle. While this problem led to a temporary discarding of the Klein-Gordon equation in the late 1920s and early 1930s, we have since learned to live with it, as will be discussed shortly.

Probability Current. Another problem which arises with solutions of (4.4) is the construction of a positive definite probability density. Paralleling postu-late v in Section LA, one searches for a covariant probability current density

50 Boson Wave Equations

h — (P> J) which obeys a continuity equation

d% = 8% + dlh = 8, j0 + V • j = 0, (4.9) where we have used (3.60), 3" = (d„ V). The obvious candidate for j^ is the hermitian form

7„(x) = 4>*iS,4> = 4>*tt>4> - ' K </>*)</>, (4-10) because this current is conserved (d • j = 0) for states </> and </»* obeying the Klein-Gordon equation (4.4):

d-j = id"(c/>*5M</>) - i5"(05>*) = i</>*D<£ - *'</>•</>* = 0. (4.11) Furthermore, since the spatial part of (4.10), j , is identical in form to the nonrelativistic current density (1.6) except for normalization, we are obliged by covariance arguments to accept the timelike component of (4.10), p, as the probability density. When combined with the general solutions (4.8), this probability density becomes

? + (x) = c/>*;30(K=2£|(/> + ( x ) | 2 > 0 (4.12a) for positive-energy solutions, and

p.(x) = <ptid0(t)_ = -2£|tf>_(x) |2<0 (4.12b) for negative-energy solutions, with E > 0. Clearly a negative probability density is unacceptable; we shall contend with (4.12b) shortly.

Wave Packets. Construct general spin-0, free-particle wave packets in the Hilbert space of positive and negative energy states as (normalized in a box—see Section l.B)

</, + (x) = I2£^a'e"""JC' *-W-J2§* *'*"'* (4-13) where the complex conjugation of bp follows the usual convention. The factor of 2£ (E > 0) in (4.13) is a manifestation of our covariant normaliza-tion of states. From (4.13) it is clear that the evolution of these packets does not alter their positive- and negative-energy character. Consider then a scalar product defined over the positive-energy states (4.13a) as (see Prob-lem 4.1)

<<*>'+, <P+> = \ d3x </>'*(xK<KM (4-14a)

= j ^ 4 % (4.14b)

= J d*p 8(p2 - m2)9(p0)a?ap/V, (4.14c)

where 8(p2 — m2)6(p0) in (4.14c) indicates that only positive-energy states with p2 = m2 are allowed. Note that this norm is time independent [differen-

tiate (4.14a) with respect to time and use the Klein-Gordon equation; the result is obvious for (4.14b) and (4.14c)]. Note too that the norm (4.14) is a Lorentz invariant. Also, the covariant normalization of states, <p' | p> = 2E83(p' — p), follows from (4.14b) with the replacement

a p K-*-*2£S 3 (p- P i ) (4.15)

for a plane wave of momentum p,.

Interpretation of Negative-Energy States. Historically, the resolution of the negative-energy and negative-probability-density problems led to a reformu-lation of the Klein-Gordon theory in a many-body context. It is possible, however, to stay within a single-particle framework (Stuckelberg 1941; Feynman 1949) by interpreting (4.12a) as the charge density of a positively charged, positive-energy particle "propagating" forward in time (t > 0, E > 0) via the plane-wave phase e~iEt. Similarly, one interprets (4.12b) as the charge density of a positively charged, negative-energy state propagating backward in time (t = — |r| < 0) via eiEt = e~im. Alternatively (4.12b) is the charge density of a negatively charged, positive-energy particle propagat-ing forward in time via the complex conjugate of the phase, e'E' = (e~'E')*. For neutral particles which are their own antiparticles (i.e., 7t" = n°, where c refers to the "charge conjugate" antiparticle—see Section 6.A), one can choose the wave function to be purely real or imaginary. In this case the probability density (4.12) vanishes, consistent with treating p as a charge density. Unfortunately, a thorough understanding of this interpretation must await a discussion of charge conjugation in Chapter 6 and "backward propagation in time" in Chapters 7 and 10.

The Stiickelberg-Feynman interpretation is ideally suited for scattering processes, where the particle is free and unlocalized before and after the collision. For bound-state wave packets, however, a particle constrained to Ax < m"l and Ap ~ 1/Ax ~ m demands a superposition of all Fourier com-ponents, negative as well as positive energy:

</>(*) = J ~ i K e - i r , + fcpV'-). (4-16)

Then (f)*4> contains interference terms e±2lEl which produce violent oscilla-tions, referred to as "Zitterbewegung". Since such "jittery-motion" plays a more significant role for spin4 particles, we postpone a detailed discussion of it until Chapter 5. Suffice it to say that as E -> m, such interference be-tween positive and negative energy components could have physical consequences.

Feshbach-Villars Formulation. It is possible to circumvent these interference terms by constructing a Klein-Gordon bound-state wave function which has no negative-energy component in the nonrelativistic limit (Feshbach and Villars 1958). For (p satisfying the Klein-Gordon equation with time

derivative fy, define

»"i(* + s 4 x-i(*-i*\ (417)

Because (j> = 0 + -+ e~iml as £ -*m implies q> -*• e~imt and x -* 0 by (4.17), (p is called the "large" and x the "small" positive energy component of <j>. More-over q> and x satisfy coupled first-order equations in time,

V2 V2

iq> = -— ((p + x) + m(p, ix = ^ , (•? + *) _ m*- ( 4 1 8 )

These equations can be unified into one matrix equation satisfied by the column vector

* = ( * ) ' ( 4 1 9 )

and resembling the Schrodinger form

ia,<D = H0O, (4.20)

where H0 is the 2 x 2 matrix "hamiltonian"

H0 = ^(1 + a)p2/2m + pm, (4.21) with

' -(1-tt —C i> While the positive- and negative-energy states are coupled in this scheme,

the problems associated with negative energy are not completely eliminated, because H0 as given by (4.21) is not hermitian in that

This links the large and small components q> and x together, implying as in (4.12) that the probability density p is not positive definite. If nothing else, however, this approach indicates that a second-order equation in time can be reduced to a first-order equation by doubling the Hilbert space of states via the column vector (4.19). [From an historical standpoint, Dirac learned this fact 30 years earlier when he discovered the first-order Dirac equation for spin-j particles (Dirac 1928).] Furthermore this formalism is ideally suited for nonrelativistic reductions. We will exploit a similar pattern in the case of the Dirac equation in Chapter 5.

External Fields. Next we consider the modification of the Klein-Gordon equation for spin-0 particles in the presence of an electromagnetic field, specified by the vector and scalar potentials A and A0 as the four-vector Ap = (A0, A). Following the minimal substitution procedure of classical

physics, we take

#„ = p„ - p„ - *Ki (4-23) where e is the electric charge of the particle, taken as positive unless other-wise specified. The Klein-Gordon equation then becomes

[(id - eAf - m2](f>(x) = 0, (4.24a)

or

(O + m2)(p(x) = -J(x), (4.24b)

where J(x) is a scalar current "source" density

J(x) = 2ieA • d<p + iecpd • A - e2A2(p, (4.25)

or J = V<p, V being an effective potential operator acting on <p. This scalar current density should be contrasted with the vector current

density j , , altered from the form (4.10) by the minimal substitution law (4-23): ^ ^

;„(x) = (f>*\id^ - eAjcp = i0*5„cp - 2e<j>*<l>A,. (4.26) The doubling of the A, term in (4.26) is a consequence of A^ being real. Following the procedure (4.11), one can demonstrate that (4.26) is also conserved, since in this case

d • j = i<p*UU<P* ~ 2ecp*<pd A~2eA- d((p*<p) = 0 (4.27) by use of (4.24b) and (4.25).

Bound-State Coulomb Atom. Finally we consider the specific bound-state Coulomb problem of a spin-0 n~ particle with charge —e bound to a heavy nucleus with charge Ze (n-mesic atom). The static potential for this configuration is (a = e2/4n)

eA, = Vd^o = - ^ 8,0. (4.28)

Writing the positive-energy wave function as (p(r, t) = cp(r)e~iE', the spatial part obeys a Klein-Gordon equation obtained from (4.24):

K) + V2 - m2 4>(r) = 0. (4.29)

This latter equation is solved by the standard method of separation of variables with 0(r) = <p(r)Y?(r) and V2 = r-l(d2/dr2)r - L2/r2, leading to the one-dimensional radial equation

ld2r / ( / + ! ) - ( Z « ) 2

{2EZ«1_ , p 2 ^ r dr2 r2 r J r _ - '_2 * ' + _ _ _ \^r) =-(E2- m2)<f>(r).

(4.30a)

In the limit £ -» m, E2 — m2 -»• 2m£NR, and Ztx <i\ (i.e., for small-Z atoms, Z <? 137), (4.30a) becomes the usual nonrelativistic Schrodinger radial equation

1 d2 1(1 + 1) 2mZ«" r drz r r <j>{r)= -2m£NR^(r). (4.30b)

We are familiar with the solutions of (4.30b) for integer values of n — / = 1, 2, ..., corresponding to the Bohr energy levels for n. = 1, 2, ...,

E - m = £NR = m(Za)2

(4.31)

Given (4.31), we can infer the Klein-Gordon energy levels by replacing m-> E, £NR -»(£2 - m2)/2E [by inspection of (4.30a)], and n -> n' [see Schiff (1968)]:

£ 2 - m 2 = -£2(Za)2/n'2. This relation can be solved explicitly for the relativistic energy as

(4.32a)

£ = m 1 + (Za)2

„'2 (4.32b)

Here ri is a new relativistic principal quantum number with ri — I' equal to the usual nonrelativistic quantum-number difference n — I, assuming only the integer values ri — l = 1,2, 3, . . . ; and /' is the effective relativistic orbital angular momentum, inferred from (4.30a) to be

/ ' ( / ' + l ) = / ( / + l ) - ( Z a ) 2 . (4.33) Solving (4.33) for /' gives

' ' = - i + [(' + i ) 2 - (Za) 2 ]* , (434a) „' = „ _ / + / - = „ _ ( / + -y + [(/ + 1)2 _ (Za)2]*, (4.34b)

where the positive sign of the square root has been chosen in order that /' may be nonnegative as Za -• 0, corresponding to bound radial solutions r' regular at the origin. This form of ri, (4.34b), is to be applied to the energy levels (4.32).

For Za < 1, both (4.32b) and (4.34) can be expanded in the form of (4.31) with n = 1, 2, ... (see problem 4.2),

£ — m JNR m(Za)2

~~2rF 1 + (Za)

n V + i An) (4.35)

This removes the / degeneracy of £ in the 0(a4) relativistic fine-structure term. For large Z, (4.34) provides the constraint that for /' and ri real, the discriminant of the square root must be positive (of course, it cannot vanish) and for s-waves (/ = 0) this means that

„, 1 137 Z < 2 a = l T

(4.36)

Spin-1 Wave Equation 55

If instead Z > l/2a, the centrifugal barrier term in (4.30) becomes attrac-tive for s-waves and the energy becomes imaginary, so that the wave func-tion has a damped exponential part and the particle orbits become unstable. The situation is similar to the classical relativistic situation for the Coloumb potential; for L2 < (Za)2 the centrifugal barrier also becomes attractive and the particle spirals in to the origin. Multiparticle quantum states then presumably play a role, with short-distance corrections such as vacuum polarization by pair creation (see Chapter 15) modifying the single-particle wave function. In the next chapter we shall again return to this strong-field limit for the case of bound electrons.

4.B Spin-1 Wave Equation

Derivation. Starting with a three-component wave function </>, describing a massive spin-1 free particle in its rest frame, two possible rest-frame covar-iant forms exist: a covariant four-vector ^ = (0, </>,) and a rank-two antisymmetric (field) tensor </>MV given by (recall the angular-momentum tensor operators LMV and J//v in Chapter 3) 0Oi = — c/>l0 = 5f < ; and cf)00 = (frjj = 0. In a general frame, the boosted form of </»/JV can be obtained from (/>M as

^ v = 3 ^ v - 5 v ^ . (4.37)

The free-particle dynamical relation between <\>^ and </>/lv is called the Proca equation:

<^„v = m2tfv (4.38)

Owing to the antisymmetric structure of (j)^, the derivative of (4.38) implies the subsidiary condition

5 " ^ = 0. (4.39) Since 0„ must transform according to the ( , j) representation of if, we know that (4.39) is required to rule out the spin-0 component in (/>,, (see Section 3.B). This in turn provides a group-theoretical justification for the dynamical equation (4.38). Moreover, using (4.37) to eliminate (j)^ in (4.38) and applying (4.39), we are led to a Klein-Gordon equation for the vector wave function,

( • + m2)<t>,(x) = 0, (4.40) which, as in the spin-0 case, guarantees the correct dynamical relation be-tween energy and momentum for a free particle.

Current Densities. Given the wave functions (^v and </>„, it is possible to construct a conserved, hermitian current density analogous to (4.10) but describing spin-1 probabilities:

The field equation (4.38) along with (4.37) then manifests 8 • j = 0. As was the case for the spin-0 probability density, j0 = p obtained from (4.41) is not a positive definite quantity. Likewise, negative energy and Zitterbewegung problems again arise, to be handled in a manner similar to spin-0 particles. We therefore proceed to indicate the new aspects associated with spin-1 wave functions.

In the presence of an electromagnetic field, the minimal replacement (4.23) converts the field equation (4.38) from the spin-1 free-particle Klein-Gordon equation (4.40) to a similar equation containing a vector source current (see Problem 4.3),

(a+m2)<t>fl(x) = Jtl(x). (4.42) The actual structure of J^ is not revealing, but if the subsidiary condition (4.39) is satisfied, then this source current is conserved. Consider too the modification of the spin one probability current density (4.41) in the presence of external electromagnetic fields. It is a straightforward task to show that the minimal replacement id^ -»id^ — eA^ converts (4.41) to

j„{x)=<l>*"['d*9<.fi + $fi9n* ~ 3*0,10 - eilA^p - Apg^ - A^g^W- (4-43)

It can be demonstrated that this current is conserved (8 • j = 0) in much the same manner as for the spin-0 case (see Problem 4.4). It turns out, however, that unlike the spin-0 and spin-j currents (the latter to be discussed in Chapter 5), the minimal spin-1 current is not unique (Lee 1965).

Free Particle Solutions. Finally we display the covariantly normalized free-particle, plane-wave solution of (4.40),

4>,(x) = e,(p)e-i"x, (4.44) with the polarization vector further specified by the helicity eigenvalues, E^\p) and k= ± 1, 0. The subsidiary condition d • (f> = 0 is therefore equiv-alent to p • e = 0, as derived earlier in (3.102). As in the spin-0 case, a factor of (2£)~* in (4.44) has been absorbed into the covariant normalization of the states, and a one-particle volume normalization factor of V-* has been set equal to unity. The orthogonality of the wave functions (4.44) then demands

e W')*(p^>(p)=-^ . A , (4.45) a result which can be verified from the specific forms (3.98) and (3.101). Likewise, in the rest frame of the particle, the completeness property of the polarization vectors implies

£#>(PMA)*(P) = <5.7. (4.46a) >.

and since <5y can be expressed in four-dimensional language as ~(9w — mllmv/m2), the boosted form of (4.46a) is

I ^ ) ( P K A ) * ( P ) = - ( ^ V - ^ ) . (4.46b)

Spin-1 Maxwell Equation 57

The p-meson (mp « 776 MeV) and the co-meson (m„, « 783 MeV) are examples of spin-1 particles occurring in nature. They are significantly heav-ier than the spin-0 7t-meson (m, x 140 MeV).

4.C Spin-1 Maxwell Equation

A massless spin-1 photon obeys a quantum-mechanical wave equation somewhat similar in structure to the massive one, with 0M and < /JV respec-tively replaced by the four-vector potential A^ and the antisymmetric field tensor F/lv, where

F„v = d„4v - M M - (4-47) The Maxwell free-field equation is then [setting m = 0 in (4.38)]

-d*F„v = nA„-dft(d-A) = 0. (4.48)

We take this as the dynamical wave equation for noninteracting photons.

Existence of Gauges. Note now that a subsidiary condition d A = 0 is no longer a direct consequence of the field equation itself, as was (4.39). Rather, an ambiguity now exists, and the value of d • A is correlated with the way we restrict the number of spin states of the photon to two, as required by the helicity constraint X = ± 1.

For plane waves, the representation for A^ satisfying (4.48) for a noninter-acting photon with k2 = co2 — k2 = 0 has the general form

Att(x) = sli(k)e-ikx. (4.49) In Section 3.D we chose the "transverse gauge" for ej,* 1}(k) in order to insure only two independent components in e :

£f)±1)(k) = 0, k-£( ± 1 )(k) = 0. (4.50)

But we cannot separately choose both conditions (4.50) in a frame-independent manner. A Lorentz-invariant choice is the combination k • s(k) = 0, and (4.49) then corresponds to the Lorentz gauge for (4.48):

d-A = 0, D ^ = 0. (4.51)

While (4.51) is not the only gauge decomposition of (4.48), it is the natural choice for free photons, paralleling massive spin-1 particles. The transverse or radiation gauge is d • A = 0. In any case, the orthogonality and com-pleteness relations for photon polarization vectors are analogous to (4.45) and (4.46):

e *w ' ) (k ) - e w>(k)=-^ , (4.52) valid in any gauge, while

X e^(k)£f*(k) = < 5 . v - ^ (4-53) x = ± i

is valid only in the transverse gauge (4.50).

Gauge Invariance. It is possible to transform A^ to a different gauge so that (4.50) and (4.51) no longer hold but (4.48) remains an identity. This is achieved by a gauge transformation

AÂ^ + Bîx), e„-> e„ + ?*„, (4.54) with £(x) an arbitrary scalar function becoming iê~'kx for a plane wave; thus d • A and k • e are transformed to

d-A-»nt(x), fc-e-»0. (4.55) Consequently, while k • e must always vanish for k1 = 0, d • A vanishes only in the Lorentz gauge (4.51). The connection between the minimal coupling replacement id^ -> id^ — eA^ and the gauge transformation (4.54) is the prin-ciple of "gauge invariance of the second kind". This is the coordinate-dependent phase transformation of a charged-particle wave function,

<f,(x)-+e-ie^<j)(x), (4.56a)

"where £(x) generates the gauge transformation (4.54), under which the can-onical momentum transforms simply:

(id„ - eA,M>{x) - g" «M(i3„ - d > ( x ) (4.56b) Observables such as the current densities (4.26) and (4.43) are built up

from bilinear products as </>*</> and </>*(«?,, — eAjcj), manifestly invariant under the phase transformations (4.56); thus current conservation is na-turally extended by this principle to include interactions in the presence of electromagnetic fields. From our viewpoint this is further justification for considering minimal replacement as a fundamental principle which gener-ates the only interaction between charged particles and photons. In the context of lagrangian field theory, the principle of gauge invariance of the second kind [the first kind corresponding to a constant phase in (4.56) and linked to charge conservation] plays the central role and is sometimes con-sidered the raison d'etre for the existence of A^ itself and its interaction with charged matter. Gauge principles recently have been used to generate other fundamental interactions (strong and weak), but such topics are beyond the scope of this book.

One final fact about gauge invariance of significant import for us later will be the manner in which the two physical spin states of the photon are realized for a general interaction with matter. This is most conveniently stated by the Lorentz-invariant S-matrix element, itself expressed in terms of the M-function of (3.87). Accordingly, we may write

Sw(k) = e^(k)M"{k), (4.57)

where Sw(k) and the polarization vector ef(k) represent a photon of momentum k and helicity L While M^ transforms like a simple four-vector under £?, we recall that e^ has a slightly modified transformation law (3.109) because a photon really behaves like an E(2) [and not an 0(3)] object under a little-group transformation. Applying (3.109) to (4.57) and using

Spin-1 Maxwell Equation 59

A / = (A"1)^ then leads to (see Problem 4.5)

Sw(k): = eikB

= eue

A / - A 0 ^ CO

eiX)(A~ ^M^k)

£W(A-1k)M"(A"1k) - A0velA)(A"1k)^M'i(k)

(4.58)

Aside from an overall phase factor (of no consequence for physical probabi-lities), the Lorentz-invariant S-matrix transforms as a simple scalar (Wein-berg 1964c)

Sw(k) = ea@Sw(\~1k). (4.59)

Then comparing (4.59) with (4.57) and (4.58), we see that MM must satisfy an additional constraint,

*"M„(k) = 0. (4.60) This condition is sometimes referred to as "on-shell" gauge invariance (on-shell refers to physical, as opposed to virtual or off-shell, energy or momentum—more about this later). It guarantees that e^l) transforms properly under little group transformations. The reference to gauge invar-iance means that (4.57) is invariant under the gauge transformation <V -»• en + ?kM' £ ' M - » £ ' M + ^ - M = £ ' M b y (4.60). We will take note of this fact many times in our later work.

Charged-Matter Currents. The probability current density j„ for charged particles plays a dual role in that ej^ is the charged source current density for the electromagnetic field. That is, the modification of the Maxwell field equation (4.48) in the presence of charged matter has the classical form

d*Fjx)=-ejtl(x), (4.61a) or in terms of the dynamical (Klein-Gordon-type) equation for /4„ in the Lorentz gauge (d • A = 0),

OA,(x) = ejll(x)^rir(x). (4.61b)

Note that the sign of the spin-1 source current in (4.61b) is opposite to that of the spin-0 source current in (4.24b). As a consequence we shall show later that this sign change is linked to a fundamentally attractive spin-0 force (e.g., the pion-exchange strong force) and a fundamentally repulsive spin-1 force (e.g., the photon-exchange electromagnetic force between particles of like charge).

It is clear from the divergenceless nature of the left-hand side of (4.61) that this charged-matter current must be conserved: d • j(x) = 0. As noted after (4.56), this result is also a consequence of the quantum-mechanical structure of the probability current density, as given for charged spinless particles in (4.26), coupled with the principle of gauge invariance of the second kind as

applied to such particles via (4.56). The connection between current conser-vation and gauge invariance takes on further significance when formulated in momentum space. For a free, spinless, charged particle absorbing the four-momentum q = p' — p from a photon, the current density constructed from positive-energy packets (4.13a) becomes, via (4.15),

= \ji\jf <vcpu+ipy«*- <PU>VVX, (4.62)

and the differential operator id^ in (4.62) requires the off-diagonal momentum-space current to be

<PU+IP> = (/>' + P i (4-63a)

for a free photon, q2 = (p' — p)2 = 0. On the other hand, in the presence of an external electromagnetic field, q1 = (p' — p)2 ± 0; and on grounds of Lorentz covariance and current conservation, the off-diagonal momentum-space electromagnetic current must have the form

e<p'\j;\p} = eF(q2)(p' + p))1. (4.63b)

Here the charged particle is still considered as free, as in initial- and final-state scattering configurations. Since d • j oc ^<p' \jll |p>, current conserva-tion implies, for p'2 = p2 = m2,

qf'<V'\j;\i>> = F(q2)(p'-p)-(p' + p) = F(q2)(m2 - m2) = 0, (4.64)

which demonstrates the absence of the only other possible covariant in (4.63b), q^ = (p' — p)^ The dimensionless Lorentz-invariant function F(q2) is called a form factor. It represents all possible interactions between the charged particle and photons, altered only by "strongly interacting" par-ticles also interacting with the spinless particle in question. At q2 = 0, (4.63a) demands that F(0) = 1, regardless of the type of interacting particles present.

The conversion of the free-particle current [(4.10) in coordinate space or (4.63a) in momentum space] to the interacting forms (4.26) or (4.63b) for q2 41 0 is a complicated dynamical process which we shall attempt to explain in the latter half of this book. At the present level of discussion the inter-esting connection is between the dynamical current-conservation statement (4.64) and the kinematical on-shell gauge-invariance constraint (4.60). For a free photon with momentum k and k2 = 0, (4.64) is a special case of (4.60) (with q replaced by k). An analogous situation exists for spin-1 matter cur-rents based upon (4.43) and also spin4 matter currents to be described in the next chapter.

Second Quantization: Photons and Phonons 61

4.D Second Quantization: Photons and Phonons

Thus far we have treated a photon as a "particle" having a quantum-mechanical wave function obeying the dynamical wave equation (4.48) for a free photon and (4.61) for a photon interacting with charged matter. Needless to say, these quantum field equations are identical in form to Maxwell's equations for the classical electromagnetic field tensor, related to physical electric and magnetic fields as

-Ei = F0i = dtAt-dtA0, (4.65a) Bt = hijkFjk = eiJkdjAk. (4.65b)

This complementarity relation between a quantum particle and a classical wave or field can be extended one step further, to a quantum field, then referred to as second quantization.

Noninteracting Photons. In such a quantum field theory, the vector potential of the radiation field is scaled to the space-time-averaged energy flux (Poynting vector), which in rationalized units (restoring h and c here) is

<S> = c<ExB> = £ — |Ak|2fc, (4.66) k c

where Ak is the Fourier component of the vector potential

A(r, t) = X (Ake*-re-to« + Aj<rft,reto«). (4.67) k

[Note that we have anticipated that A will be treated as an operator and have used Af instead of A* in (4.67).] If (4.66) is to represent a flux of photons of density N/V with energy hco and angular momentum h, then <S> = hcocN/V implies

|Ak|2 = ^ - (4.68)

In a second-quantized theory, (4.68) is interpreted as a matrix element <iVk|A^ • Ak|iVk> in a particle number "Fock space", with (restoring the polarization vector but suppressing its helicity components)

and <Nk\alak\Nk) = Nk. (4.70)

We see that the second-quantized field operator Ak has the same structure as the single-photon wave function (4.49), now noncovariantly normalized, except for the second-quantized operator ak, where a[ a^ is a number opera-tor according to (4.70).

To proceed more rigorously, one treats (4.67) as a normal-mode harmonic-oscillator expansion and expresses the field energy as

E = \ \d3x(E2 + B2y = YJh(o(Nk + $), (4.71) ^ J k

where use of (4.65), (4.67), and (4.69) leads to (4.71) with the identification (now being careful to respect the order of the operators A and Af)

<Nk\alak + akat\Nky = 2Nk + l. (4.72) Then abstracting (4.71) to a hamiltonian operator expressed in terms of canonical coordinates and momenta, the commutation relation [xk, pk] = ih leads to the second-quantized commutation relations (see Problem 4.6)

[ak, al] = (5k,k., K , Ok-] = 0. (4.73) It is therefore clear from (4.72) and (4.73) that ak is an annihilation operator and al a creation operator in Fock space satisfying

<Nk - 1 \ak\Nk) = <JVk|flJ|JVk - 1> = ^ (4.74) and consistent with a^ ak being the number operator as expected in (4.70).

Spin and Statistics. Second quantization is a natural way to build in the Bose statistics of the photon field from the outset via the commutation relations (4.73). Likewise the Bose statistics of spin-0 or spin-l massive particles can be built in by commutation relations such as (4.73) to form a similar free quantum field theory. A field theory of fermions also can be constructed in this way, with the commutation relations (4.73) replaced by anticommuta-tion relations in order to manifest the Fermi statistics. This connection between spin and statistics follows in a natural way from the requirement of positive definiteness for the free-field hamiltonian (Pauli 1940). Interacting quantum fields are described in terms of coordinate-space lagrangian densi-ties. While the methods of quantum field theory are elegant and powerful, we shall follow the simpler and intuitive (one-particle) methods of Feyman for the greater part of this book. This means we will be investigating the struc-ture of (one-particle) current and hamiltonian densities (usually in momen-tum space) rather than lagrangian densities (in coordinate space). The connection between spin and statistics will be invoked as a postulate, as elsewhere in quantum mechanics (postulate vi, Section l.A); no reference will be made to the second-quantized commutation relations (4.73).

Phonons. Before leaving this subject, it will prove useful to consider a non-relativistic quantum field construct, that of phonons, corresponding to boson quanta but associated with lattice vibrations in a solid. Like photons, phonons have no mass—but this is not dynamically relevant. Instead the relation between phonon energy (frequency) and momentum (wave number) depends upon a "dispersion law" resulting in cwq « constant at small q for "optical phonons" and coq = cs | q | at small | q | for "acoustical phonons",

Second Quantization: Photons and Phonons 63

where cs is the velocity of sound in a solid, cs ~ 105 cm/sec. Since phonons are induced by displacements of lattice-site ions, they can be associated with a vector "spin", having longitudinal as well as transverse components. The scale of the phonon displacement amplitude A with Fourier coefficients Aq and expansion (4.67) is set by the time-averaged displacement energy of a spring with spring constant k = Mco2, where M is the lattice ion mass,

<£> = 2xiMo;q2 |Aj2 . (4.75)

Since this classical energy represents only half the energy of the spring (the kinetic energy accounts for the other half, by the virial theorem), the quan-tum analog of (4.75) is \ha>q Nq for Nq phonons of angular frequency u>q, giving

l A ' l 2 = 2 * ^ ' < 4 J 6 )

Paralleling the photon case (4.68) and (4.69), it would appear that the second-quantized phonon displacement field operator is

with aq and aq satisfying the commutation relations (4.73) and a\ aq is the number operator similar to (4.70). In Chapter 9 we shall employ the one-particle phonon wave function (noncovariantly normalized)

A(x' l ) = V ipoTve{q)eiq' Xe_iM,'' ( 4 J 8 )

where the ion mass M in (4.77) is expressed as pV in (4.78) so that the amplitude displays the usual one-particle box normalization. The ion den-sity p varies from 2 to 20 g/cm3 in a solid, a typical value being 5 g/cm3.

General references for boson wave equations are Hamilton (1959), Roman (1960), Schweber (1961), Bjorken and Drell (1964), Bethe and Jackiw (1968), Schiff (1968), Baym (1969), and Berestetskii et al. (1971).

CHAPTER 5

Spin-i Dirac Equation

What the Schrodinger equation is to nonrelativistic physics, the Dirac equa-tion is to relativistic physics. We begin this chapter by describing three alternative ways of deriving this spin4 wave equation—the more the merrier, in order to develop as much intuition as possible about this fun-damental dynamical tool. Next we formulate the Dirac equation in a man-ifest covariant manner and emphasize the structure of y-matrix algebra and the positive and negative free-particle solutions. The Dirac equation in the presence of external fields is then generated by minimal replacement, and the resulting electron bound-state energies are obtained for the one-particle Coulomb atom and for a constant external magnetic field. We pay particular attention to the difference between the Dirac atom and the fine-structure level shifts in the Schrodinger atom. Finally, we develop free-particle Dirac equations for spin-j massless neutrinos and spin-f massive particles.

5.A Derivations of the Dirac Equation

The Dirac equation plays a fundamental role in any relativistic quantum theory, not only because it circumvents many of the problems arising from an unphysical interpretation of the Klein-Gordon equation, but also be-cause it naturally describes the basic spin4 constituents of matter at the atomic and nuclear level—electrons and nucleons (protons and neutrons). To appreciate the significance and beauty of the Dirac equation, it is well to describe three alternative derivations: the original relativistic approach of

64

Derivations of the Dirac Equation 65

Dirac, the ad hoc but elegant derivation via a Pauli spin-matrix replacement, and finally the group-theoretic derivation.

Dirac's Derivation. In order to obtain a positive definite probability density for spin4 particles, Dirac searched for a relativistic differential equation which was first order in time (Dirac 1928, 1958). To this end he interpreted the Schrodinger equation, (id, — HQ)^/ = 0, as generated for spin 4 particles by a free-particle matrix hamiltonian H0, first order in momentum for relati-vistic considerations:

H0 = a p + /?m. (5.1)

Here a; (i = 1, 2, 3) and /? are four matrices to be determined, and p is the usual quantum-mechanical momentum operator — iV for a particle of mass m. Since the single-component spin-0 Klein-Gordon equation leads to a rwo-component first order matrix (Feshbach-Villars) equation (4.20), it is natural to suppose that a two-component spin4 equation, second order in time, should be linked to a four-component matrix equation, first order in time. Such a "guess" was all the more impressive at the time because Dirac did not have the hindsight of the Feshbach-Villars equation. Thus one assumes that a, and ]8 are 4 x 4 matrices and that the Dirac wave function \// -»\j/„ is a four-component (Dirac) bispinor. The second-order equation in time for a free particle must, of course, be of the Klein-Gordon form, which in momentum space is obtained from the first-order equation via multi-plication by E + H0:

(E - H0)iP = 0 -> (E2 - H2)^ = 0. (5.2)

Dirac then demanded that the square of the 4 x 4 matrix hamiltonian (5.1) be constrained to E2 = p2 + m2 by (5.2), i.e.,

Hi = i(a,o, + ajOifapj + (/fat, + a,/*)p, + P2m2 = p2 + m2. (5.3)

This leads to the defining properties of the matrices a, and /?:

{a„a,} = 2Sy, {/J,al} = 0, a2 =/?2 = 1. (5.4)

Furthermore, since H0 must be an observable hermitian operator, so must a; and /? be hermitian:

a/ = «„ P = P. (5.5) The adjoint row bispinor i/ can be combined with the column bispinor xf/ to form a positive definite probability density

P = ^V= I Ma, (5-6) < r = l

now naturally linked with the hermitian probability current density

j = i/^ai/f. (5.7)

66 Spin-i Dirac Equation

a =

A justification for the form (5.7) is the resulting continuity equation 8,p + V • j = 0, which follows from (5.6), (5.7), and the Dirac equation

(id, - H0)i}> = 0, (5.8) where H0 obeys (5.1) and (5.3) (see Problem 5.1).

From the properties (5.4), it is clear that at = — /?<x,/? and /J = — a.l^a.l, which means that these matrices are traceless (Tr ft = YZ=i PooY-

Tr otj = Tr p = 0, (5.9) since Tr BA = Tr AB. It is then useful to describe at and /? by a specific representation, such as the Dirac-Pauli representation,

C »)• H i -")• [which, of course, satisfies the general properties (5.4), (5.5), and (5.9)], where the elements, a and 1, are themselves the 2 x 2 Pauli and identity matrices, obeying

oiOj = 8ul + isiJk(Tk. (5.11)

Dirac used this representation (5.10) to obtain the first (but by no means the last) profound prediction of the Dirac equation. Making a two-component reduction of (5.8) in the presence of an electromagnetic field via the Dirac-Pauli representation (5.10), he discovered that the electron must have the unique magnetic (dipole) moment

^ = 2X^neX2=2me°> <5"12>

i.e., the Lande g-factor is ge = 2, a result in almost perfect agreement with experiment and assumed ad hoc up to that time. We shall return to Dirac's method of predicting ge = 2 in Section 5.D and to the small but important corrections to this result in Chapter 15.

Derivation via Pauli-Spin-Matrix Replacement. This leads naturally to the second derivation of the Dirac equation with ge = 2 as input [see, e.g., van der Waerden (1932), Sakurai (1967)]. Recall from (5.11) that the square of any vector such as p2 is equivalent to the Pauli-spin-matrix replacement (a • p)2. The question then arises as to when this substitution is required, i.e., for all or only a few problems involving spin-^ particles. Furthermore, while (o • p)2 = p2, it is clear that in the presence of an electromagnetic field, the minimal replacement p -»• n = p — eA means (<r • Jt)2 =£ n2, but instead (5.11) implies, along with p x A = — i V x A — A x p , that

(a • n)2 — n2 = ia • n x n = —ea • B. (5.13)

Need nonrelativistic or relativistic momentum operators be subject to this Pauli-spinor replacement ? Consider first a nonrelativistic free-particle ham-iltonian with p2/2m -»• (<r • p)2/2m. In the presence of an electromagnetic

Derivations of the Dirac Equation 67

field, (5.13) then leads to the Pauli hamiltonian

H = ~(ap)2^^n2-~aB. (5.14) 2m 2m 2m

The spin-dependent term in (5.14) has the form — u • B, with \i = gnBJ where J = a/2 and g = 2, a very desirable result.

Given this success as input, we are encouraged to apply this mnemonic to all problems involving p2 and spin \. In particular we modify the spin4 Klein-Gordon equation to

[E2 - (a • p)2]</> = m2<$>, (5.15a) which then has the time-dependent factored form

(id, — a • p)(idt + a • p)</> = m24>. (5.15b) Now we parallel the discussion of the Feshbach-Villars version of the Klein-Gordon equation (Section 4.A) by defining

<PL = <t>> (PR = -(id, + o p)<t> (5.16)

and expressing the second-order equation (5.15) as two coupled first-order equations,

(id, + a • p)(pL = mcpR, (id,-o -p)(pR = m<pL. (5.17)

Further defining (p = <pR + <pL, X = <PR~ <Pu (5-18)

and using the Dirac-Pauli representation (5.10), the first-order equations (5.17) for these two-component spinors can be put into the four-component Dirac-equation form (id, — H0)\f/ = 0, with ty = (£) and

again the Dirac free-particle hamiltonian. Setting p = 0 in (5.17) means that <Pz.(0) = <PR(0) and x(0) = 0 for positive-energy states cpLR oc e~iEt, whereas <PL(®) = ~ <PR(0) and q>(0) = 0 for negative-energy states <pLR oc elEt. In four-component language with ^/+aze + ,Et, these p = 0 constraints can be ex-pressed as

#M0)=±M0), (5-20) stating that the Dirac-Pauli representation diagonalizes the bispinor energy states in the extreme nonrelativistic limit.

Alternatively we may use another representation for a, and /?, called the Weyl representation

68 Spin- Dirac Equation

also satisfying the defining relations (5.4), (5.5), and (5.9). Given (5.21), the two-component equations (5.17) again can be combined into the four-component Dirac form (5.8) provided that

in this representation (Problem 5.1). In Section 5.E we shall show that for m = 0, (5.17) indicates that <pR is polarized right-handed, corresponding to helicity X= + i , and q>L is polarized left-handed, with X = — \ [see also (3.94)]. Thus we see that the Weyl representation diagonalizes bispinor heli-city states in the extreme relativistic (m = 0) limit.

Group-Theory Derivation. The third derivation of the Dirac equation com-bines the Weyl representation for a; and fi with the group-theoretical formu-lation of relativistic spin-j states in Section 3.B. The motivation is to combine the (\, 0) and (0, \) irreducible representations of the homogeneous Lorentz group into a larger matrix (%, 0) + (0, \) representation 3), which is then equivalent to the adjoint representation S>(A) = / ^ (A - 1 ) / ? . Not by accident, this /? corresponds to the Dirac /? in the Weyl representation (5.21).

For the two-component ( , 0) boost D(LV) and the (0, ^) boost J5(L,,), the corresponding wave functions cpR and q>L satisfy the uncoupled equations

<pL(p) = D(Lp)(pL(0) = [2m(E + wi)]"*(£ + m - a • p)q>L(0), (5.22a)

Ml>) = D(Lp)(pR(0) = [2m(E + m)]-*(£ + m + a • v)<pR(0), (5.22b)

from which one can derive the coupled relations (5.17) (see Problem 5.1). Now use (5.22) along with the matrix boost obtained from (3.51) in the Weyl representation as

9{L>) = ( D ( ^ } 5 ( °^ } ) = [2m(E + m)]-*[(E + m)l + a • p],

(5.23) where 1 denotes the 4 x 4 unit matrix (sometimes deleted). Again expressing the Dirac bispinor in the Weyl representation as \j/ = (££), (5.22) and (5.23) boost i/f(0) from rest to

*(p) = ®(I*M0). (5.24) Finally, to recover the Dirac equation, operate on (5.24) with id, — H0, obtaining

(id, - H0)iA±(p) oc [±E - (a • p -I- j8m)][±£ + a p + m]il/±(0)

= ( £ 2 - p 2 - r o 2 ) > M 0 ) + m(±E + m-<x- p)(l + P)il/±(0).

(5.25)

Covariant Formulation 69

The first term on the right-hand side of (5.25) vanishes by the energy-momentum relation, and the second term is zero by (5.20). Thus the Dirac equation (id, — H0)i// = 0 is equivalent to the Klein-Gordon constraint plus (5.20), which projects out the positive- or negative-energy parts of \p. Put another way, the Dirac equation is the boosted form of the rest-frame energy-eigenvalue relation (5.20) subject to the energy-momentum (mass-shell) constraint E2 — p2 = m2 for p in an arbitrary direction.

In Chapter 6 we shall learn that (5.20) has invariant meaning as the spatial-inversion (parity) eigenvalue relation. Combining two-component spinors which transform into one another under spatial inversion as in (5.23) means that 3) is again equivalent to the parity transform via (3.47), P&(Lp)P = .@(L_P). This is sometimes considered the main motivation for the four-component Dirac formalism. Another is the fermion mass; if it is zero then the two-component equations (5.17) suffice.

5.B Covariant Formulation

Covariant y-Matrices. It is possible to formulate this relativistic four-component formalism in manifestly covariant language by expressing the Dirac matrices /} and a in terms of covariant ^-matrices y„ = (y0, y), defined as

y0 = fi, y = /fa. (5.26)

These matrices have the "lengths" (y? = y2 = y| = y2.)

y2o=-yf=l w' = r p 4 (5.27)

and satisfy fundamental anticommutation relations following from (5.4):

{y^y,} = y^v + y,y^ = ^9^- (5-28)

Note that (5.28) has the form of a covariant symmetric tensor equation; this implication will be discussed shortly. Next combine the hermitian and antihermitian relations yj = y0 and y/ = — y, into a "Dirac adjoint" matrix operation

iv = 7o7j7o = y„, (5.29)

where in general the Dirac adjoint (barred operation) of any 4 x 4 matrix is A = y0 A*y0 with AB = BA. That is, (5.29) implies that y„ is "self-barred". Then define the new y-matrices

7s = 70 7172 73 (5.30a) satisfying

75 = 7s, 7s = - 1 , 7s 7„ = -7„7s, (5.30b)

<V = ii[7„, 7v] = '(7„7v - 0„v) = -<rvtl (5.31a)

70 Spin4 Dirac Equation

satisfying a^ = a^, since i = —i and

Gij — £;jfc<T*' °i = kijk°jk> (5.31b)

ffoi = iyo y.- = '«,- = - 7s ffi. (5-3 lc) where a{ are now 4 x 4 matrices, built up from the 2 x 2 Pauli matrices as (" „). In the latter case, a = iy5o follows from a t = iyoTi^ysiyz73 — yo7i, etc.

Since 4 x 4 matrices have 16 possible elements, there must be 16 indepen-dent y-matrices. We may choose these independent matrices as the self-barred set

r , = 1, y„ <v, iy„y5, y5 = S(l), v(4), T(6), ,4(4), P(l), (5.32) where 1 is the one scalar (S) unit matrix, yM are the four vector (V) matrices, <r v the six antisymmetric tensor (T) matrices, iy^ y5 the four axial-vector (A) matrices, and y5 the one pseudoscalar (P) matrix, so that in all there are 1 + 4 + 6 + 4-1-1 = 16 independent matrices. The transformation laws for the various Tt verify their independence; this will be demonstrated later along with the implications of the "pseudo" property of A and P. Note the explicit factor of i in iyMy5. It preserves the relation that all 16 T, are self-barred, i.e., iy„y5 =ysyj= -iy5y^ = iy^s- Thus P, = r ; . Beware, however, of the other possible (non-self-barred) choices for y5 appearing in the literature, ±iy0yiy2y3- Finally, any general 4 x 4 matrix can be ex-panded in terms of these 16 Tj. In particular, one can verify that (see Prob-lem 5.2)

yMyv = 0„, - '<V> (5.33a)

y 5 ^ = i w ^ (5-33b) y„ yv yP = 0M. yP - gw y, + g,P y» - « w V"TS > (5.33c)

w y V Y = -3!y,y5, (5.33d) £,vpffy/'yYy''=-4!y5, (5.33e)

y^/vyp/ff CT/iv/p/ff y^pivia • yvpinto • yparptv yvo/n/p

+ y5 W " (5-33f) etc., where the y-matrix products on the left-hand sides have been expressed as linear combinations of the 16 Tt on the right-hand sides of (5.33), and we have used the fundamental properties of the Levi-Civita pseudotensor (e0123 = 1 ) >

(5.34a)

(5.34b)

F pnvpa — _ 4 |

F f '">" = — 3 I

P P P" — — "> 1

Jim'

9m

9»t>

9v« 9vp

(5.34c)

etc.

As regards the matrices /? and a, the y-matrices satisfying only (5.27) and (5.28) then have many possible representations. The representation-independence (Pauli-Good) theorem states that all representations of y-matrices are equivalent up to a similarity transformation, y'tl = S~lyflS. The Dirac-Pauli representation, diagonalizing energy via y0 in the extreme non-relativistic limit, is

1 0\ . 10 1)

(5.35) 7 o = 0 - 1 ' ^ 5 = 11 0

0 <T 0 / ' " \0

whereas the Weyl representation, diagonalizing helicity via y5 in the extreme relativistic limit, is

0 1\ / - l 0

(5.36) y° = ' i o ' iy5'~\ o i r

H- : ;> -[i:\ Other representations, including the Majorana and the light-plane represen-tation, are worked out in Problem 5.3.

Covariant Transformation Laws. To put the Dirac equation (id, — H0)i// = 0 on a covariant footing so as not to single out time via d„ multiply the equation on the left by y0—obtaining (iy0 d0 — y • p — m)i// = 0—and use p = — iV. This results in the covariant form of the Dirac equation,

(iy • d - m)\jj(x) = 0. (5.37)

Henceforth it will be convenient to define a "slash" operation 4 = y 'A so that (5.37) takes on the compact form (i</) — m)i// = 0. To verify that (5.37) is indeed covariant under homogeneous Lorentz transformations x' = Ax, we rely upon the discussion in Chapters 2 and 3 to obtain a 4 x 4 spinor matrix ^(A) which transforms the bispinor wave functions in a fashion analogous to (2.1c),

IA'(X') = ^(A)^(x). (5.38)

Since the Hilbert-space operator is (7A = exp( —jo»''vJ/lv/2), the obvious identification J/iV -» <J/1V /2 in the spin4 Dirac space means we can write

y(A) = exp(-ia/V„v/4). (5.39)

An infinitesimal transformation A„v -> g^ + OJ^ induces (5.39) to become y(A)-» 1 — ict/v<7„v /4, and the y-matrix identity (see Problem 5.2)

[ff„v, yP] = 2i(gpvyli-gPflyv) (5.40) then leads to the transformation law for y-matrices,

<f-i(A)ylty(V = \.f- (5-41)

72 Spin-j Dirac Equation

[Note that (5.41) bears the same relation to the Lorentz group as (2.15) and (2.52) do to the rotation group.] It is in the sense of (5.41) that yll transforms like a four-vector; since y^ is a constant matrix, yj, has no meaning other than (5.41). Then we establish the covariance of (5.37) by multiplying it on the left with y(A):

0 = y(A)(iy • d - m)iA(x)

= [ ^ ( A ^ y ' H A K - mMA)iA(x) (5-42) = (iA-Vd» - mty'(x') = (iy • d' - mW{x%

where we have used d'v = (dx^/dx'^d* = A'/S". Thus, (5.42) implies that the Dirac equation (5.37) is valid in any frame x' = Ax, an explicit demonstra-tion of the meaning of covariance.

Now we extend the Dirac adjoint or "bar" operation to bispinor column vectors. Define the row bispinor

$ = <AV (5-43) If if/ satisfies (5.37), then $/ satisfies the adjoint Dirac equation

(# - m)«A = ifr{ -1? - m) = 0, (5.44) since ytt = y^. Then using £?(A) = 5^_1(A), which follows from (5.39) and d^ = a^, we have from (5.38)

^'(x') = $(x)?(A) = $(x)&- X(A). (5.45)

Combining (5.38), (5.41), and (5.45), we can obtain the transformation laws for the bilinear covariants ^ r , \p (now "c-numbers" in the Dirac space, since a row times a column matrix is a pure number—see Problem 5.4),

S: ^'(x')iA'(x') = #(*M*) (5.46a) V: ^'(x')y^'(x') = A/.A(x)y>(x) (5.46b)

T: ^(x 'Kv^ ' (x ' ) = A/Av^(x)aa/!V(x) (5.46c)

A- ^'(x')iy,y5*'(*') = (det A)A„>(x)iyvy5iftx) (5.46d) P: ^(x')y5<A'(x') = (detA)^(x)y5^(x). (5.46e)

It is in this sense that the r ; transform like integral representations of if, indicating, for example, that the Dirac probability current (5.6) and (5.7), expressed in covariant form as

;„ = ^ ( 1 , atyr = ^(y0, y)* = fa^, (5.47) transforms as a four-vector, j'^x') = A/;'v(x), with a continuity equation d • j = 0 which is invariant from frame to frame.

Dirac Trace Algebra. The bilinear covariants just discussed can be used not only to construct the probability current;^, but also to form the S-matrix or transition-probability matrix elements for a process involving a spin-j parti-

cle. In analogy with (3.87) we write

<^'|S|A> = ^W'>M^W), (5.48) where i//A) and \j/(X"> are column and row bispinors which are eigenstates of helicity, and M is a scalar 4 x 4 matrix composed of invariant products of momenta and y-matrices. The complex conjugate (*) of (5.48) can be ex-pressed in terms of the Dirac adjoint matrix, M = y0 M^y0, according to

<A'|S|A>* = (^'>M^W))t = ^ttt>Mtj£ttt'>

= iAtU)yo7oMty0</'u') = VX)M\l/(X'\ The physical quantity of interest is the transition probability, which corre-sponds to the product of (5.48) and (5.49); summing over all possible helicity states (unpolarized spin sum), we write

X |a ' |S |A>|2 = X^u')M^A?iAu'), (5.50) k',k k'

where SP is a Dirac "projection" operator gp = £ ^wyw (5.51)

x obeying SP1 = SP for orthonormal bispinors ij/u 'tyU) = 5XX. Rearranging the bispinors in (5.50) to the form (M^,M)(7-<ri/'<r

A') (/#'>, a trace operation in the Dirac space (Tr A = ^ Aaa), and using (5.51) again, we obtain finally the unpolarized spin sum

£ |<A'|S|A>|2 = T r M ^ M ^ ' . (5.52)

Now we shall show shortly that the projection operators & and 3P' can be expressed in terms of y-matrices, so (5.52) represents in principle a trace over the product of many y-matrices. Consequently it will prove useful to describe the trace algebra of y-matrices. To begin with, in the Dirac space

Tr 1 = 4. (5.53a) Inspection of the Dirac-Pauli or Weyl representation [(5.35) and (5.36)] or application of the defining relations (5.28) and (5.4) along with the property Tr AB = Tr BA gives

Tr y„ = Tr y5 = 0, (5.53b)

Tr yMyv = i Tr{y„, yv} = 0(IV Tr 1 = 4^ v , (5.53c)

and from (5.30b) and (5.33b), Tr y5y„ = 0, Tr y5y„yv = -\iz^ Tr &" = 0. (5.53d)

Next, note that (5.33c) then implies Tr y yv yp = 0, which can be extended to a product of any odd number of y-matrices y ^ (excluding y5, which is an even product y0 yx y2 y3) via multiplication of (5.33c) by successive pairs of y-matrices, implying that

Tr yM = 0. (5.53e)


For even numbers of y-matrices, (5.53a) and (5.53c) are nonvanishing; for four y's, use of (5.33c) or (5.33f) leads to

Tr y»yvypy„ = Mg^Qpa - g^g*. + g^g,P\ (5.53f) Tr y5 y„ yv yp ya = - 4£„VP(T, (5.53g)

with £0123 = 1- The las t identity may be verified by choosing /i = 0, v = 1, p = 2, a = 3, and using (5.30a). Multiplication of (5.33f) by y„ yx leads to the trace of six y-matrices, but at this point and thereafter it is easier to apply repeatedly the anticommutation relation (5.28) to permute y„ from one side of an even number of y-matrices to the other and then use Tr Ay^ = Tr y„ A to find in general (see Problem 5.5)

Tr y„yvypyB • • • = g„v Tr ypyff • • • - g^ Tr yvyn • • • + 0^Tryvyp etc. (5.53h)

It is clear from the algebra (5.53) that including two additional y-matrices in a trace can lead to a considerable complication. Quite often, however, the indices of two of the y's are summed (contracted), in which case repeated applications of (5.28) results in (see Problem 5.5)

^ y „ f = - 2 y „ , (5.54a)

y*y„7vf = 40„v, (5.54b)

y. )V ?v yPf=- 2)V 7v y„, (5.54c) y*y*y*ypy.f = 2(yffy„yvyP + yPyvy„y,x), (5.54d)

etc. The identities (5.54) should be applied before computing y-traces accord-ing to (5.53).

The y-matrix traces can be used to verify the completeness of the 16 Dirac covariants r , of (5.32), with

Tr r , r , = 4 ^ , (5.55) where */, = +1 for T, = 1, y0, iy,y5, a, and r\{ = - 1 for T, = y5, yh iy0y5, o0i. Then any 4 x 4 matrix can be expanded as M = £ a, T; with the coefficients from (5.55) found to be 4a; = JJ, Tr Mr,. Also the completeness relation (5.55) is needed to develop Fierz "reshuffling" matrices (see Problem 5.6).

5.C Free-Particle Solutions of the Dirac Equation

Positive- and Negative-Energy Spinors. In the spirit of the free-particle spin-0 analysis, we now examine the positive- and negative-energy solutions of the covariant free-particle Dirac equation

( # - m t y ± ( x ) = 0. (5.56)

Free-Particle Solutions of the Dirac Equation 75

For the positive-energy solution corresponding to a particle of momentum p (with V = 1), we write

il/+(x) = u{p)eivxe-iEt, (5.57a)

while for the negative-energy solution with energy — E (E = ^/p2 + m2 > 0) associated with momentum — p,

^Mx) = w(p)e-'p•V£,. (5.57b)

In both cases we require the spin-^ particle of mass m to obey E2 — p2 = p2 = m2. Substituting (5.57) into (5.56) then leads to the momentum-space Dirac equations

{p - m)u(p) = 0, it(p)(p - m) = 0, (5.58a)

(p + m)v(p) = 0, v(p)(p + m) = 0. (5.58b)

It is convenient to normalize the positive-energy spinors u(p) and the negative-energy spinors v(p), choosing the signs according to (5.20), so that

u(p)w(p) = 2m, v(f)v (p) = — 2m, (559)

while i//± are normalized covariantly in analogy with the spin-0 Klein-Gordon solutions (4.7).

To obtain the free-particle solutions in the rest frame, we evaluate (5.58) in the limit p0 = m, p = 0, obtaining

yo«(0) = u(0), yov(0)=-v(0). (5.60) [Recall that the notation u(0) emphasizes the p = 0 aspect of the rest-frame wave functions; rest-frame two-component spinors will be written as </>(p) as a reference for rotations.] In the Dirac-Pauli representation with y0 diag-onal, (5.60) diagonalizes the energy (mass, in the rest frame) with the spinors expressed as

u(0) = y/2^fe\ r(0) = V2^(°J, (5.61)

where q> and i are two-component spinors, both normalized to unity in order that (5.59) may remain valid. In particular, we further specify these two component spinors as helicity eigenstates <pU)(p) and #<A)(p) = e+'*(pu,( — p) [so that y(p) represents a state with momentum —p = p^_e<n+4> in the latter case—see (6.46)], obeying the eigenvalue equation (3.89), i.e.,

V P ? W ( P ) = V 1 P ) (5-62) and represented by the rotated spinors (3.91). The reason for the choice of phase eT'* for k = ±\ in /(p) will be explained in Chapter 6.

We may also recognize (5.60) as equivalent to the boost constraint (5.20) in the Lorentz-group derivation of the Dirac equation. In this case, the

76 Spin-^ Dirac Equation

covariant version of the Dirac boost matrix (5.23) is, since E + a • p = 1>y0, ®(L,) = [2m(E + m)]-*(fiy0 + m)

, x (5-63)

so that @(L„) = y(Lp)- Application of this boost to (5.61) and using (5.60) with X = ±j then leads to

^)-»W^)--^i(,,7W). (5.64a)

^-*^>^>-7S^UA-J (564b) From the form of (5.64) it is clear that these bispinors satisfy the free-particle Dirac equations (5.58), since /*/> = \{p, p} = p2 = m2 implies that

[p - m)wU)(p) oc (p - m){p + m) = p2-m2 = 0,

{p + m)vw{p) az(p + m)(p - m) = p2 - m2 = 0. Further, the bispinors (5.64) are indeed helicity eigenstates, obeying by virtue of (5.62) and [a • p, p] = 0

& • p«(A)(p) = Xu^(j>), (5.65a)

-\e • pVA)(p) = *»w(p) (5.65b) for X = ±\, where a in (5.65) is now the four-component Dirac spin matrix.

Thus, the advantage of the explicit boost construction in the Dirac-Pauli representation (5.64) is that it manifests both the Dirac equation and helicity constraints satisfied by positive- and negative-energy bispinors. Moreover, the connection between u(p) and r(p) for momentum states — p in the latter case is made readily apparent by (5.64). There is another use for the boost operator, that of "de-boosting" u(p) and v(p) back to the rest frame, sometimes referred to as the "nonrelativistic reduction" procedure. Recalling that — Yy0 = fys <*> w e express the boost (5.63) and its Dirac adjoint in terms of the even (diagonal) operator 1 and the odd (off-diagonal) operator y5 a • p in the Dirac-Pauli representation, converting the bispinors to the form

^(p)=^,"::/s:p(<p7))- (5-66a) E + m + iy5 a y/E + m

« ( > ) = ( / ( 1 p ) , 0 ) £ ^ t o ) (5.66b) yJE + m

^-""^•"UA-J (Mfc) E + m + iy5a • p yjE + m

*«« = (0, e± Vu,(-P)) £ + m / F | y 5 < T ' P • (5.66d)


The nonrelativistic reduction of bilinear covariants such as ur ; u can be easily obtained using (5.66) to identify the net even operators (in the Dirac-Pauli representation) of ®r ; ®. For «r,H and vT^, only net even operators (two-component diagonal) contribute, while for ur ; v and vrt u only net odd operators (two-component off-diagonal) do (see Problem 5.7).

Free-Particle Projection Operators. Recalling that the two-component spinors satisfy the completeness relation

X > W W W ) ( * ) = 1 , (5.67) x

we expect the four-component bispinors to obey some sort of analogous relation. Given (5.64) and (5.67), we may compute

(E + m)i «w(p)iiw>(p) = {i> + m) I1 °) (P + m) * \U U / (5.68)

= i(/> + m){\ + y0)(p + m).

With the help of the anticommutation relations (5.28) and p2 = m2, we see that py0 i> + m2y0 = 2Ep, (y0 /> + py0) = IE, and (p + mf = 2m(p + m). Then the factor of (E + m) cancels from both sides of (5.68) and we obtain

2

X wa)(p)"U)(p) = /* + ™, (5.69a) x

and similarly 2

X vU){p)vw{p) = t-m. (5.69b) x

These useful relations can be verified by applying the Dirac operator (/> — m) to (5.69a) and (p + m) to (5.69b); the normalizations are easily confirmed in the rest frame via (5.61). The identities (5.69) define respectively positive and negative projection operators in the sense of (5.51). That is,

M r i - ^ ("0) are normalized energy projection operators obeying A+ = A+ and A+ A_ = 0 for p2 = m2. The additional identity A+ + A_ = 1, or equiv-alently from (5.69),

X («U)(p)tiU)(p) - «W,(P)»W,(P)) = 2w (5.71) x

is the Dirac analog of the two-component completeness relation (5.67). The usefulness of the energy projection operators (5.69) is that, for an

S-matrix specified by positive- or negative-energy free-particle bilinear co-

variants, the general unpolarized-spin sum (5.52) becomes in particular

X |uU)(p')Mua)(p)|2 = Tr M(p + m)M(p' + m), (5.71a) A ' , A

£ |DU)(p')Mi;U)(p)|2 = Tr M(p - m)M(p' - m), (5.71b) x'.x

and similarly for uMv and vMu. It is also possible to construct a covariant spin projection operator. In the

rest frame of the spin4 particle, such a spin projection operator is j( l + a • s), where s is the direction of spin polarization (s -» p corresponds to a helicity projection operator). Defining sM = (0, s) and m^ = (m, 0) in the rest frame, so that m • s = 0, it is clear that in a boosted frame p • s = 0. Then, replacing a by — iysyy0 and dropping the noncovariant factor y0 (since py0 does not change sign when applied to either u or i;), we obtain the covariant spin projection operator

E(s) = i(l + iy5/). (5.72) The invariant length s2 = — 1 means that £2(s) = Z(s), characteristic of any projection operator. We may then compute positive- or negative-energy polarized spin sums using the Dirac trace algebra by simply replacing, say, p ± m in (5.71) with Z(s)(p ± m) (or (p ± m)L(s), since [y5 £ p] = 0) for a particle whose spin is not unobserved, but pointing in the "direction" s. Specific helicity states can then be generated using s = + (p, Ep)/m in (5.72), whereas spin representations other than helicity follow from a different choice for sM.

Finally, a third set of Dirac projection operators

P ± = i ( l ± i y 5 ) (5.73)

will prove useful for our later work. They too obey P\ = P±, P+ P_ = 0, and P+ + P_ = 1. One immediate application of (5.73) stems from P + 7n = y^P-> which means that for free particle bispinors \jj = m'1^^ we have P± ip = m~ iii/)P+ i//, so that

ij, = P+il/ + P_il/ = m~l(i$ + m)P+il/. (5.74)

Thus for (j) = P+ i//,

( $ - w)i//= - m _ 1 ( n + m2)<f) = 0, (5.75)

which says that, given </> obeying the less restrictive Klein-Gordon equation (for bispinor wave functions), one may obtain \p via (5.74) which satisfies the more restrictive Dirac equation provided (j> = P+ if/. This is reminiscent of our second derivation of the Dirac equation built up from two-component spinors related by the differential operator in (5.16), here equivalent to (5.74) in the Weyl representation. In the above approach, the Dirac-Pauli re-presentation reduces (5.75) to two identical two-component spinor equa-tions. This is true even in the presence of electromagnetic fields, and we shall take advantage of this fact shortly.

Free-Particle Probability Current. We have noted in (5.6), (5.7), and (5.47) that the bilinear covariant i?y„i/' is the natural choice to represent a con-served probability current density. Consider the current density constructed from the free-particle positive-energy wave packet similar in structure to the Klein-Gordon form (4.13a), normalized in a box:

« A + W = J 2 ^ 4 « P « ( P K ' P J C (5-76)

(where we have deleted the helicity summation for clarity), giving

j ; (x ) = * + (x)y> + (x) = U l l \ w < ° P e*" '

(5.77)

with q = p' — p. The off-diagonal momentum-space current is then

<p' | j ; |p> = «(p')y,«(p). (5.78)

In this language, current conservation d • j(x) = 0 is equivalent to

<f <P' \ti I P> = «(P')<MP) = "(P')0*' - PHP) ={m- m)uu = 0, (5.79)

by the free-particle Dirac equations (5.58a). An analogous negative-energy packet with E = >/p2 + m2 > 0,

+-(*) = j^b*v(VW*, (5.80)

leads to a <p' |;'~ | p> which is also conserved in the sense of (5.79). As we have already noted in Section 4.C, such off-diagonal momentum-

space currents play an important role in the theory in their own right. Consider first a useful identity called the Gordon reduction. Defining P = \{p' + p),q = p' — P and using the y-matrix property (5.31a), it is easy to show (Problem 5.8) that

K*qv = P% + Y,P~2Pfl. (5.81)

Sandwiching this identity (5.81) between free-particle positive-energy bispinors, we note that p' operating to the left and f> to the right both become m in (5.81), again by the free-particle Dirac equations (5.58a). Rearranging this result, we obtain a form of the Gordon reduction,

U(P% u(p) = «(P') (~+ifff) «(P)> (5-82)

where the P^ and a^ qv terms in (5.82) are called the convection and spin currents, respectively. Clearly, relations similar to (5.82) can be generated by sandwiching (5.81) between v and v, or u and v, or v and u.

As a by-product of (5.82), the diagonal matrix element can be obtained with q = 0 and P = p:

"(PK «(p) = (P„H"(PMP) = 2p„. (5.83a) This result is "obvious" from covariance considerations alone, because the left-hand side of (5.83a) must be a simple four-vector and the only candidate depending upon p is p^, with a normalization factor following from the contraction with p^, using p2 = m2 and the Dirac equations (5.58). In a similar manner we deduce that

«(P))V«>(P) = (-Pjrn)v(p)v(p) = 2p„. (5.83b)

We may now calculate the expectation value of the Dirac velocity opera-tor a. Noting that the spatial integral of (5.77) generates #3(p' — p), which in turn picks out the diagonal matrix elements of j * , we use (5.83a) to write [a similar relation holds for the Klein-Gordon current (4.62)]

\d>xj;(x,t) = \^\ap\2pJE, (5.84)

with an analogous relation for j ~ , where ap is replaced by bp. For /j = 0we see that p0/E = 1 in (5.84) as well as for the jo integral. Since these jo integrals are j d3x \j/\{x)^i±(x), the latter integrals normalized to unity as the total probability, we learn that

V) 2£ K l 'V) ~2E | b p l ~L ( 5-8 5 )

Next we evaluate

d3x j±(x, t) = j d3x \j/i(x)aal/±(x) = <a>±, (5.86)

and from (5.84) conclude that

<«>± = + = <v.>±, (5-87)

where the latter two averages in (5.87) are weighted over the momentum-space probabilities (5.85). The important result (5.87) simply states that the positive- or negative-energy wave-packet expectation value of the current is just the relativistic group velocity vg, a conclusion paralleling the nonrelati-vistic situation (1.6), and expected in this case because in the Heisenberg picture,

^ = i[H, rj] = iak[pk, rj] = a,, (5.88)

Zitterbewegung. It turns out, however, that both positive- and negative-energy states must be included in the construction of the eigenfunctions of a.

Dirac Equation in an External Field 81

Consider then the general packet (again suppressing the helicity summation)

*(x) = I 2 K* k»(PK""x + b*vW<\, (5-89)

and computing <a> as before leads to

<«> = ( | ) + j ^ [a*b%u(P>tv(-v)e2iE' + a p b_ p D(-pMP)^ 2 ' £ ' ] -

(5.90) As was the case with the Klein-Gordon equation, the cross terms between the positive- and negative-energy states oscillate violently in time. Unlike the Klein-Gordon situation, however, the free-particle amplitudes multiplying the oscillating phases in (5.90) need not vanish as p -> 0. To see this, use a = iy5 a and the forms (5.66) to obtain the nonrelativistic reductions (Prob-lem 5.7)

i i ( p M - p ) -* 2m<p*{p)o(p{p), D(-P)XM(P) -• 2m<pt(p>(p(p). (5.91)

Thus Dirac Zitterbewegung is nonvanishing to zeroth order in momentum. While unphysical consequences can be avoided for free particles by choosing either a positive- or a negative-energy packet as in (5.76) or (5.80), for bound states this cannot be done. We return to this latter problem shortly.

5.D Dirac Equation in an External Field

Covariant Electromagnetic Interactions. Following the procedure used in the Klein-Gordon analysis of a particle in the presence of an electromagnetic field, we consider for spin-^ particles the minimal replacement of id^ by id^ — eA^ in the Dirac equation (5.37), giving

(# -eA- m)i//(x) = 0, or (# - m)tfr(x) = eA\p(x). (5.92) We will assume that this minimal coupling eA represents the fundamental interaction between the spin4 particles and photons. As in Dirac's original approach, the bispinor equation (5.92) obeys a Klein-Gordon constraint, obtained by multiplying (5.92) on the left by the operator i$ — eA + m:

[(id - eA)2 -m2- e\a^vF^(x) = 0, (5.93) where we have used the anticommutation relations of the y-matrices to write

Ui> + m = i[y„, y,]F"V + d • (Aij,). We see from (5.93) that the Dirac wave function also obeys the Klein-Gordon equation in the presence of an external field (4.24a) apart from the spin-dependent factor

i ^ v F ^ = i a - E - < r - B , (5.94)

which couples E and B respectively to the electric and magnetic dipole moments of the spin4 particle. In particular, an energy eigenstate of ^(x) in (5.93) obeys, for E2 — m2 = 2mENR + • • •,

£ N R " A = - ^ - B < A + ---, (5.95)

showing once again that \i = ea/2m and g = 2 are natural consequences of the Dirac equation.

It is also interesting to investigate the Dirac probability current, j^x), in the presence of an external field. In spite of the absence of a derivative term in ij/y i/f, one may consider the Gordon decomposition in coordinate space analogous to (5.82) and then make a minimal replacement to find (see Problem 5.8)

h(x) = ^ ^(x)a>(x) - ~ MxWWAAx) + ~ fl1f(XKi(4

(5.96)

It is clear that the first two terms in (5.96) have the same form as the nonrelativistic and Klein-Gordon current density (4.26), covariantly nor-malized in the latter case. The first and third terms in (5.96) correspond to the Gordon decomposition for A^ = 0. As in the spinless case (4.27), current conservation d • j(x) = 0 is an obvious consequence of (5.96).

Dirac Form Factors. Following now the discussion in Section 4.C, the prob-ability current density j„(x), whether due to nonrelativistic, Klein-Gordon, or Dirac particles, plays a dual role because ejj^x) represents a charged-matter current density which is a source for the electromagnetic field, \3A^ = ejp (in the Lorentz gauge). Thus the spin-j analog of the momentum-space current (4.63b) in the presence of an electromagnetic field for positive-energy scattered particles, obtained from (5.96), is

e<9'\tf \p> = eufa') 2m «(P). (5-97)

This is the most general form consistent with Lorentz covariance and the Dirac equation. The four-momentum transferred from the photon to the spin-j particles is q = p' — p, and the dimensionless invariant form factors Flt2(q2) parallel F(q2) in (4.63b) for spinless particles. For free photons, q2 = 0 and we must have Ft(0) = 1 in order that the coefficient of uy^u in (5.97) may be simply the charge e, consistent with the free-particle probabi-lity current density (5.78). Thus F^q2) is called the charge form factor. For q2 ± 0, the photon is not free, and the second term of (5.96) contributes to both Fx(q2) and F2(q2) of (5.97) in a complicated dynamical manner. We shall return to the q2 dependence of these form factors in later chapters.

Concerning the magnetic form factor F2(q2) in (5.97), since the y^ current already consists of a convection and magnetic moment part as given by

(5.82), with g = 2 in the latter case, the additional magnetic-moment term in (5.97) described by F2(q2) represents an anomalous correction to g = 2. That is, at q2 = 0,

F2(0) = ic = i(flr - 2) (5.98)

is the dimension less, anomalous magnetic moment of a spin4 particle. For the electron (e < 0),

or Ke ~ 0.001, due totally to electromagnetic interactions. For the proton (e > 0),

^ = ^ ( 1 + ^ ( 2 . 7 9 ) ^ , (5.99b)

or KP ~ 1.79, primarily due to strong interactions—as is the anomalous magnetic moment of the chargeless neutron,

/i„ = ^ - ( 0 + « „ ) * ( - 1 . 9 1 ) - ^ - , (5.99c) 2m„ 2m„

or K„ ~ — 1.91. In Chapter 15 we shall show that the minimal (QED) cou-pling in (5.92) ultimately generates Ke = cc/2n + 0(cc2), in perfect agreement with experiment. It will also turn out that we can estimate KP and K„ better than we have a right to expect. Lastly, the form factors Fl2{q2) a r e not a unique description of the dynamics of spin-| particles interacting with an external electromagnetic field. Two different but equivalent descriptions in terms of the Sachs form factors GE(q2) and GM(q2) are worked out in Prob-lem 5.8.

Constants of the Motion. For a free-particle hamiltonian a • p + fim, the fundamental commutation relations (5.4) lead to

[H, r x p] = — ia x p, [H, oj = 2ia x p, (5.100)

and it is therefore clear that the total angular momentum J = r x p + \a and helicity ^a • p are conserved operators. In a central, spherically symme-tric field V = V(r) = eA0 with A = 0, the Dirac hamiltonian becomes

H = a - p + j?m+ V(r), (5.101)

and while commutation relations similar to (5.100) become more cumber-some to apply (see, e.g., Sakurai 1967), group-theory arguments alone tell us that the total-angular-momentum operator must remain conserved.

To simplify the search for constants of the motion and their eigenvalues for spin-^ particles, it will prove convenient to reduce the four-component Dirac analysis back to two-component form. We consider then the less restrictive second-order equation (5.93) satisfied by a Dirac bispinor i/r for

positive-energy stationary states with n = p — eA and eE = — V V, B = 0,

(t + m){i - m)il/ = [(£ - V)2 + V2 - m2 + id • VFty = 0. (5.102)

Multiplying (5.102) on the left by the projection operators P± = i ( l ± iy5) of (5.73), we see that the four-component wave function </> = P+ ip satisfies (5.102), and the more restrictive Dirac wave function, obeying H\p = E\\i, is then recovered from (p by

i/> = m~ x{t + m)4> = m~ l[y0(E - V) - y • p + m]</>, (5.103) a situation similar to the free particle case (5.74). Next we work in the Dirac-Pauli representation with P+ = ^(} \), giving

<t> = P^=ft\ (5.104)

Then the four-component equation (5.102) reduces to two identical two-component equations [for V = V(r)],

2 1 3 2 L2 . ^dV r or r dr

( £ _ V{r))2 _ m2 + r - -j+ia • f <p = 0, (5.105)

equivalent to the Klein-Gordon form (4.29) except for the last spin-dependent term. If all we are interested in is the quantized energy levels, then we need only deal with (5.105), converting it to the form of the Schrodinger equation as in the Klein-Gordon case. We need not bother with recovering 4i from <t> via (5.103) and (5.104).

In order to separate variables in (5.105), we specialize further to a single-electron atom with V(r) = — Za/r and dV/dr = Z<x/r2. It is then natural to incorporate both V2 and the spin-dependent term in the angular part of (5.105) along with L2/r2, since these terms all fall off like r~2. In particular we make use of the two-component decomposition CT, OJ = du + ieijk ak to write

L2 = a • L(l + o • L) = (1 + a • L)2 - (1 + a • L). (5.106)

This suggests that we define the two-component spin operator

- A = (l + « - L ) + (Zo)w f, (5.107)

which has the square

A2 = (1 + a • L)2 - (Za)2, (5.108)

the cross terms in (5.108) dropping out because of the identity (see Problem 5.9)

{(1 + a • L), o f} = 0. (5.109)

Combining (5.106)-(5.108), we see that the coefficient of — r~2 in (5.105) has the simple form of an angular-momentum operator

L2 - (Za)ia • f - (Za)2 = A2 + A = A(A + 1). (5.110)

Next we obtain the eigenvalues of A2 and A by squaring J = L + ja, to write for states j = I ± \ =j±

(<,-L)j± = (J2-V-i)j± = {_{ll+{)\ (5.111a)

(l + c r - L ^ j ' + ^ t O + i), (5.111b)

where the upper (lower) cases in (5.111) correspond toj=l + \(j = l — j). Then we have

(A2)J± = () + i ) 2 - ( Z a ) 2 , (5.112a)

(A)J± = + [ 0 + i)2 - (Za)2]* - +A, (5.112b)

where the signs in (5.112b) are determined by (5.107) and (5.111b) with - A intrinsically positive for ;' = / + \ and Za -> 0. Following the Klein-Gordon analysis, we may now express the radial equation for the Dirac atom in the form of (4.30a),

1 d2 /'(/' + 1) 2EZa r dr r r q>(r) = -(E2 - m2)cp{r), (5.113)

with (5.110) and (5.112) giving, for) = / ± i

l'±(l'± + 1) = [A(A + l)]J± = A(A + 1). (5.114)

Solving (5.114) for the effective orbital-angular-momentum eigenvalue, we see that

r*-f-l\ (5.U5) As before, we have chosen the nonnegative solutions of /' in (5.114) as Za -»0, for otherwise the radial solution r' will not be regular at the origin.

The orbital constants of the motion now being understood for the Dirac atom, we proceed as in the Klein-Gordon atom to find the energy eigenvalues. Since the radial equations are identical in the two cases except for different values of /', we may conclude that the energy levels of the Dirac atom also are of the form (4.32b),

(Za)2

E = m 1 + n'2 (5.116)

with ri — I' = n— I constrained to the integer values 1, 2, ... as before. Sub-stituting /' given by (5.114) then leads to

n- = n - l + r ± = n - 0 - + i) + [(/ + W ~ (^ff (5.117) for both j = I ±\. [Note that this two-component analysis eliminates the need for an auxiliary quantum number k, appearing in the four-component analysis. See e.g. Sakurai (1967).] We see that the energy levels of a Dirac

86 Spin-5 Dirac Equation

^ N R — *• m — j 7

2 (5.118)

atom are identical in form to the Klein-Gordon levels, but with / in (4.34b) replaced by) in (5.117). Note too that for s-waves andy = \, the square root in (5.117) becomes imaginary unless Z < 1/a.

In the strong-field limit, Z > 1/a ~ 137 and the breakdown of the bound-state Dirac solutions is similar to the Klein-Gordon case with Z > l/2a. Once again, the breakdown has a classical analog for (Za)2 > L2 with multi-particle quantum states modifying the single particle orbit at short distances. These effects are not fully understood, nor have they been detected experimentally.

For Za < 1, one can expand (5.116) and (5.117) in powers of (Za) to find the fine-structure corrections to the Bohr formula for a single-electron atom ( n = l , 2 , ...),

m(Za)2 r | (Za)2 / 1 n V + \

now degenerate for a given value of j . We shall return to a detailed study of the hydrogen energy levels shortly. Finally, an analysis of the Dirac wave functions (proportional to associated Laguerre functions), based upon this two-component approach, depends upon (5.103). [See Auvil and Brown (1977).] Alternatively, the entire (and more complicated) four-component analysis of Hip = Exjj can be carried out in a straightforward manner [see e.g. Bethe and Salpeter (1957)]. For our purposes, the above analysis of the energy levels alone will suffice for our later work.

Before leaving this topic, it is worth mentioning that the two-component form of (5.93) also can be used to obtain the constants of the motion for spin4 problems other than the one-electron Coulomb atom. Consider, for example, an electron of momentum p moving through a region of constant magnetic field, B = Be3 and A = \E x r = \B( — ye1 + xe2) with A0 = 0. Applying the projection operator (5.104) in the Dirac-Pauli representation, the four-component equation (5.93) once more decouples into two identical two component equations:

[E2 - (p - eA)2 - m2 + eBo3]q> = 0. (5.119)

Since V • A = 0 and A • p = ^B • L, we may write (5.119) in the form

[p2 + p2 + (eB/2)2(x2 + y2)]q> = [E2 - p2 - m2 + eB(L3 + a3)]<p. (5.120)

Separating the z variable from the x and y variables in (5.120), we replace the operator p2 by the eigenvalue p2, and the magnetic-moment operator (pro-portional to L3 + a3) by the Zeeman-splitting eigenvalues /3 + 1 on the right-hand side of (5.120). The left-hand side of (5.120) has the structure of a nonrelativistic, two-dimensional harmonic-oscillator hamiltonian with a potential

(x2 + y2), (5.121)

corresponding to an angular frequency of

a) = y/k/^n= \eB\/2m. (5.122)

Since the eigenvalues of such an harmonic oscillator are known to be of the form

Eh.a. = (nx + ^,)(o + (ny + \)a> = nw, (5.123)

where n = nx + ny + 1 = 1, 2, 3, ..., we may identify the right-hand side of (5.120) as 2mEho. This leads to the constant of motion

E2 = pj + m2 + n\eB\ - (*3 ± \)eB, (5.124)

with E2 > pi + m2 implying n > | /3 | + 1. Thus we see that the energy and angular-momentum eigenvalues of a spin4 particle in a magnetic field can be found from (5.93) in a manner similar to those for the Klein-Gordon or Dirac atom. The two-component and four-component bispinor wave func-tions for this problem are proportional to Hermite polynomials, as might be expected from the structure of (5.120). (See Problem 5.10.)

Nonrelativistic Reduction. While the exact energy eigenvalues for a relati-vistic spin4 particle in an external field can always be obtained as in the last section, it is nonetheless of interest to expand the Dirac equation (5.92) in powers of ENK/m, where £NR = E — m. Starting with the Dirac hamiltonian (V = eA0, n = p - eA)

H = an + pm+V, (5.125)

we may express Hi]/ = Exjt in the Dirac- Pauli representation as

E - m - V - " ' * ) ( * ) = <>. (5.126) - a n E + m-V)\x)

This results in two coupled two-component spinor equations

(E — m — V)q> = a • nx, (E + m — V)% = a • nq>. (5.127)

For weak potentials V <g E ~ m, cp ex V lx is large while x<xm~l(p\s small. Identifying then q> as the relativistic extension of the nonrelativistic Schro-dinger wave function, we eliminate x in (5.127) to find (£NR = E — m)

HNR(p = ENR(p, (5.128a)

Assuming (£NR — V)/2m < 1, we expand

1 + ^ N L H Z r 1 * i _ f c L ^ . (5.129) 2m } 2m K '

88 Spin~j Dirac Equation

The leading term in (5.129) then converts (5.128b) to

where we have used (5.13) to obtain the Pauli magnetic-moment term with g = 2, a result now expected.

To next order in the expansion (5.129), we set A = 0 to isolate the correc-tion terms due to a weak electrostatic potential V:

Hi™ = hr+V + H'' (5131a)

4m2H' = -<f P(£NR - V)a p

= - (£NR - V)p2 + io • (pK) x p + (pK) • p,

(5.131b) where a^ = <50- + ieijk<jk and pt Vpt = Vp2 + (p; V)p{ have been used to obtain (5.131b). The first term in (5.131b) represents a relativistic momen-tum correction, since to leading order £NR — V ~ p2/2m means that

H^-iEn-V)^*-^. (5.132)

This potential then corresponds to the usual kinetic-energy correction

The second term in (5.131b) can be written for a spherically symmetric potential V = V(r) as

w " (fV) >< P) = ( l ^ y • L- (5134)

Consequently this term generates the familiar spin-o.bit interaction in (5.131),

Note that this relativistic derivation of H'so includes the troublesome factor of \ referred to as the Thomas precession, a factor which must be introduced in a subtle manner in the usual nonrelativistic derivation of (5.135) when transforming from the rest frame of the electron to the laboratory i.e. rest frame of the nucleus (Problem 5.9).

Finally, we interpret the third term in (5.131b), but first note that (pK) • p = — (VK) • V is not an hermitian operator. To convert it tohermi-tian form, we sandwich this term between a spatial integral over wave func-tions q>* and q> in a symmetric fashion and then integrate by parts.

Discarding the resulting surface term, we are led to the real integrand (pj/) . p_^ -i[jp*(VK) • V(\V) • \<p*]-^$(p*(V2V)(p. (5.136)

Alternatively, one can understand (5.136) as due to a "renormalization" of the large-component wave function <p, no longer the exact nonrelativistic Schrodinger solution to 0(p2/m2), but related to it by i//NR = (1 + p2/8m2)(p [see e.g. Sakurai (1967)]. In either case one is led to the hermitian Darwin interaction

H'Dar. = ^2(V2V). (5.137)

While this hamiltonian does not have an obvious analog in the usual non-relativistic formulation, it can be understood as a consequence of the relati-vistic effect of Zitterbewegung. To see this, account for the electron jittery motion to order of the Compton wavelength, Sr < m~\ by the expansion

Hzitt. = V(r + Sr) - V(r) = \ br{ 5r} + • • • * i(<5r)2V2F,

(5.138)

where symmetry dictates that the first-order term in (5.138) vanishes and the second-order derivatives are ^V2. Comparing (5.138) with (5.137), we see that (m 5r ~ 1)

HLr. *!(»t &r)-2Hziu. ~ Hzitl.. (5.139)

Fine-Structure Energy Levels of Hydrogen. To cap the discussion on bound states, we return to the Coulomb energy levels (5.118) for a Dirac atom, now specializing to the hydrogen atom with Z = 1:

ma2

" 2 M 2 1 + (5.140)

To understand how (5.140) arises in the context of Schrodinger theory, we write HNR = H0 + H', where H0 is the lowest-order Schrodinger hamilton-ian with V = — cc/r, which generates the Bohr energy levels for n = 1, 2, ... (structure):

.2 En=-^2=-^T2- (5-141) ma

In1 2a0n2

The small splittings of (5.141) to 0(<x2) are the fine-structure corrections caused by H' as given by (5.131b), viz.,

H' = H'nl + H;.0. + H'Dar, (5.142a)

generating the first-order perturbation-theory shifts

A£ft = (nl|H' |nV> = (nl\H'rel.|n/> + <n*|H'so.\n/> + <n/1H'D„.\n/>, (5.142b)

where | n/> refers to the usual unperturbed Laguerre eigenfunction solutions

To calculate the relativistic momentum-correction shift, note that

S J P V - - ( « + ?)« (5.143a)

(5.143b)

(5.144)

(5.145)

which breaks the /-degeneracy and is in fact the entire fine-structure shift for the Klein-Gordon atom (4.35).

The spin-orbit shift is found from (5.111a), (5.135) with dV/dr = a/r2 and

implies the square

^2<nl\p*\niy = (nl (*« + ? ) ' nl\

where one may easily verify that

<n/|r-1|n/> = ~ , <n/|r"2 |n/> = -n r

1 1 m or

>3(< + i ) ' Then combining (5.143) and (5.144) with (5.132) we find

1 4 <n / | # ; e l > /> = - ^ - 3 <n/|p*|n/> = - ~

1 3

giving

<«/|r"3|n/> =

<nl\H's.0.\nl) = ~2

ma

3 3

m a. n3/(/ + i ) ( /+ l ) '

(5.146)

nl <r L

1

nl

-4»3 C/ + i)(/ + i) for j = I ±\, but zero for / = 0.

Lastly, V2V = 47ta^3(r), and at the origin

n (1 - <5/,o) (5.147)

ll»)| m3a3

nn <5|,o>

which converts (5.137) to the Darwin shift

net <n/|H^ar.|n/> ma.

2m2 (nl\S3(r)\nl} = ^ dlt0.

(5.148)

(5.149)

While this Darwin shift is nonvanishing only for s-states, the important point is that it is precisely what the spin-orbit shift would be for / = 0 if 1 — <5, o were replaced by 1 in (5.147). In a sense then, the effect of Zitterbew-egung on the Dirac atom corresponds to a "continuation in /" of the spin-orbit energy shift down to / = 0.

Next, we combine (5.145), (5.147), and (5.149) according to (5.142b) to obtain the total fine-structure energy shift

1 _ 1 _3_" l + 12+2(j + m + l) 4n| / r i _

mot.4

= ~Jn* now breaking the) = I ±j degeneracy but preserving the / degeneracy. Note that (5.150) is identical to the relativistic momentum shift but with; replaced by /. Finally, adding (5.150) to £° as given by (5.141), we are led to the relativistic form (5.140)—a necessary but nonetheless satisfying conclusion. The Dirac theory therefore predicts excited states with fine structure split-ting for n = 2 and different j of the order \ x 10"4 eV, or equivalently, v = E/h ~ 104 Mc/sec.

Hyperfine Splitting. Aside from fine-structure splitting, one must account for hyperfine shifts due to the interaction of the proton and electron magnetic moments via the dipole magnetic field of the proton (in rationalized units):

V ' (5.151)

where the proton is assumed localized at the origin. Since iie\ip = — gegpe2/4memp with ge x 2, gp = 2(1 + KP) x 5.6, and since spherically symmetric s-states imply didjr'1 -+jdiJ'S71r~1, (5.151) becomes for s-states

H" = ^Lae-ap53(T). (5.152)

This splits every line of given j for / = 0 and S = \(ae + op) according to

("e • <*P)S=I ~ (««' °„)s=o = 1 - ( -3 ) = 4. (5.153) Then applying (5.148) and (5.153) to the matrix elements of (5.152), the net hyperfine shift for s-states is

A J ? h . f . _ 8 „ me me«* / S 1 s . x mp 2n

with the singlet spin state shifted down three times as much as the triplet state is shifted up from the fine-structure energy level. For the ground state, (5.154) corresponds to a 6 x 10"6-eV or 1420-Mc/sec shift. In wavelength language this is the 21-cm line for which astrophysicists search in order to detect the existence of neutral hydrogen in the intergalactic medium.

The story is not yet complete, however, because it was experimentally verified (Lamb and Retherford 1947) that, contrary to the Dirac theory, the

A£ft ma. " 2 ^

_J _3_ j + i 4n '

El n = 2

- v n>/z \ 2S| / 2 , 2 P | / 2

2 S i / 2

2P , / 2

Structure Fine Structure Lamb Shift m a 2 m a 4 m a 5

Figure 5.1 The n = 2 energy-level splittings in hydrogen.

2Si and 2P± energy levels are not degenerate. Rather, the 2Si line is shifted upward relative to the 2Pi state by some 1058 Mc/sec, as pictured in Figure 5.1. While the theory of this "Lamb shift" is now well understood, a detailed explanation must be postponed until Chapter 15. Qualitatively speaking, this shift is caused by the "cloud" of photons and virtual electron-positron pairs which surround the bound electron, modifying its form factors as given in (5.97) and the resulting Coulomb interaction with the proton nucleus. Suffice it to say now that theory and experiment agree exactly to five significant figures, a truly remarkable result rarely equaled in modern science.

5.E Wave Equations for Other Fermi Particles

Thus far we have explored the consequences of the Dirac equation for massive, spin4 fermions (e.g., electrons, protons). We now consider wave equations for massless spin-^ fermions (neutrinos) and also massive spin-f fermions (e.g., the A, 33 "resonance").

Massless Spin4 Particles. Recall from the discussion in Section 3.D that the two helicity states for a massive spin-^ particle, k = +-j, become reduced to just one helicity state as m -»• 0. The explicit structure of the two-component spinor boost operator in (3.94) shows that the 1 = \ right-handed massless state (with spin parallel to the momentum) survives for the (0, \) representa-tion, while X = — \ survives for the (j, 0) irreducible representation of the homogeneous Lorentz group. Experiment alone (see Chapter 13) has deter-mined that the massless neutrino is left handed and the antineutrino is right handed. Consequently the two-component (j, 0) free-particle wave function </>Lv for a neutrino satisfies a wave equation inferred from the boost in (3.94):

{id, + a • p)(pLv(x) = 0, (5.155a) whereas the (0, j) right-handed antineutrino wave equation is

{id, - <r • p)^ , (x) = 0. (5.155b) These equations also follow from the m = 0 limit of (5.17).

Wave Equations for Other Fermi Particles 93

While these wave equations were first obtained by Weyl (1929), they were rejected for 28 years because of noninvariance under spatial reflection {cpL -> cpR, q>R -»cpL). We shall return to this subject in Chapter 6. To link (5.155) with helicity eigenstates, note that the plane-wave solution of (5.155a) for positive-energy states, proportional to e~ipx with E= | p | , satisfies

<*P<PLV(P)= -<PLV(P) (5.156a) in accordance with a left-handed neutrino. Likewise, the plane-wave solu-tion of (5.155b) for positive-energy states, proportional to (e~ip '*)* with E = |p | , obeys

«" P<P*v(p) = <Pm(p), (5.156b) corresponding to a right-handed antineutrino. On the other hand, the plane-wave solution of (5.156b) for negative-energy neutrino states, also propor-tional to eip'x with E = | p |, satisfies

« - p > , - ( - p ) = ?v-(-p)- (5.156c) Comparing (5.156b) and (5.156c), we see that

<PKV(P)°C<PV-(-P)- (5.157) This is our first concrete example of the Feynman interpretation, identifying a negative-energy particle state with a positive-energy antiparticle state. More examples will be given in Chapter 6.

It is possible, and extremely useful, to couch these two-component neu-trino equations in four-component Dirac language. One way to proceed is to note that (5.155) corresponds to (5.17) with m = 0 along with the identification cpL-KpLv, (PR-+<PRV- We employ the (extreme relativistic) Weyl representation for the y-matrices, (5.36), and apply it to the projection operator P± = ^(1 ± iy5) of (5.73):

Then we define the Weyl-representation bispinors

uLv(P)-P-uLv(v)=(q>L^)\ (5.159a)

^ ( p ) = P + M p ) = ( g - i V ° _ ( _ p ) ) (5.159b)

with the positive-energy bispinor uLv(p) describing a left-handed neutrino and the negative-energy bispinor vm(p) describing a right-handed antineu-trino, and both having momentum p. As for the four-component dynamical equations that these bispinors satisfy, since P + projects out the lower negative-energy components in the Weyl representation and uLv has no such component, it is obvious that

P+»L v = i( l + iy5)uLv = 0, (5.160a)

94 Spin-i Dirac Equation

and similarly, P-vFf, = $(l-iys)v», = 0. (5.160b)

We may also proceed by starting with the free-particle Dirac equations (5.58) for m = 0,

M P ) = M P ) = 0, (5-161) then multiplying (5.161) on the left by iy5 y0 and using iy5 y0 y = a, E = | p | we obtain

<r • pu(p) = iy5 u(p), a • pu(p) = - iy5 v(p). (5.162)

Next observe that the bispinor helicity eigenstates (5.65) become in this case

« ' P"ZJP) = - « £ » » - 5 ' W P ) = ^V(P)- (5163)

Combining (5.162) with (5.163), we are once more led to (5.160). Another useful bispinor expression for massless fermions is the probabi-

lity current density j ^ — ^iy^- Applying (5.159), we may write the momentum-space current for Weyl neutrinos obeying (5.155) as (dropping the factor of \ in P_ by convention)

<P'I#1P> = tUp'Wi - iysWM (5-164) Since yM(l — iy5) = (1 + iys)yll, the latter y-matrix combinations used in (5.164) can also be expressed as uF- = uP+. For an antineutrino current, it turns out that the analog of (5.164) is

<p'l/f |P> = -MP))V(1 + iys)MP')- (5165)

The projection operator P+ in (5.165) again follows from (5.159), but the reason for reversing the momentum in the spinors in (5.165) must await the discussion of charge conjugation in Chapter 6.

Finally, note that even for q2 = (p' — p)2 / 0, we assume that (5.164) and (5.165) do not develop anomalous magnetic-moment contributions or form factors, because neutrinos interact only weakly with matter (Chapter 13). Such weak interaction experiments detect two types of neutrinos, asso-ciated with electrons and muons, respectively.

Spin-§ Particles. It is also possible, and again useful, to construct a Dirac bispinor wave function for a spin-| free particle. In Section 2.E we built up a spin-f two-component spinor by the Clebsch-Gordan combination (2.62) of a spin-1 polarization vector and a spin4 spinor, with the spin-! combination (1 x \ = | -(- \) removed by the condition (2.63), <T;<p, = 0. These rest-frame statements can be expressed in bispinor language by using the Dirac-Pauli representation and then boosting up to a general momentum frame with p2 = m2 (recall —dij^g^ — P^pv/m2). This leads to the helicity sum

<X)(p) = Z <ii n» m^\py""\p), (5.166) JL'X"

Wave Equations for Other Fermi Particles 95

where such spin-| covariant bispinors satisfy a free-particle Dirac equation (Rarita and Schwinger 1941, Auvil and Brehm 1966)

(P - m)M„(p) = 0, (5.167)

with (2.63) replaced by (see Problem 5.11)

y"H„(p) = 0. (5.168)

Given (5.167) and (5.168), u^ automatically obeys the weaker conditions

(p2 - m2)uJp) = 0, /?X(P) = 0- (5.169) A polarization or spin sum can then be formed from these spin-| bispinors

in analogy with the spin-1 polarization sum (4.46) and the Dirac projection operator (5.69). In the rest frame this projection operator is

^;i=i d > r = * u - ^ (5.170)

because (2.63) requires CT,^ = ^ijOj = 0, a condition obviously satisfied by (5.170), since aiai = 7>. The normalization of (5.170) is chosen so that ^ijSPjk = SPik- Then the boosted version of (5.170) is (see Problem 5.11)

^ + m ) + > ( r , + ^ _ „ , ) ( v , + £;)

O + my

(5.171a)

(5.171b)

(5.171c)

As long as the particles are on mass shell, this scheme can be extended to arbitrarily high spin [see e.g. Fronsdal (1958), Scadron (1968)].

There are, however, alternative formulations of spin-| wave functions. Moreover, for p2 # m2 or for wave equations involving interactions, ambi-guities arise and such theories are no longer unique. In the context of lagran-gian field theory, interactions involving massive particles with spin-| and higher (and sometimes even spin 1) are "nonrenormalizable" (see Chapter 15). Luckily, nature has been kind enough to see to it that most of the fundamental particles detected so far have low spin.

General references on the Dirac equation are: Dirac (1958), Hamilton (1959), Schweber (1961), Bjorken and Drell (1964), Muirhead (1965), Sak-urai (1967), Pilkuhn (1967), Bethe and Jackiw (1968), Schiff (1968), Baym (1969), Berestetskn et al. (1971), Jauch and Rohrlich (1976).

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5. Relativistic Wave Equations and their Derivation · 5.2 The Klein–Gordon Equation 119 and the...

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Transcript of 5. Relativistic Wave Equations and their Derivation · 5.2 The Klein–Gordon Equation 119 and the...