Lorentz Group Formulas∗

David N. Williams

Randall Laboratory of PhysicsUniversity of Michigan

Ann Arbor, MI 48109 –1040

Contents

1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Lorentz metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Pauli matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Transformation laws . . . . . . . . . . . . . . . . . . . . . . . 72.3 Spinor labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Homogeneous transformations . . . . . . . . . . . . . . . . . . 142.9 Planar transformations . . . . . . . . . . . . . . . . . . . . . . 15

3 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . 163.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Restriction to SU(2) . . . . . . . . . . . . . . . . . . . . . . . 183.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Extension to SL(2, C) . . . . . . . . . . . . . . . . . . . . . . . 22

4 Poincare group . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1 Poincare generators . . . . . . . . . . . . . . . . . . . . . . . . 23

∗February 13, 2018: This document is an expanded version of one of our compositionbooks of handwritten notes, dating from probably 1967.

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4.2 Casimir operators . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Finite-dimensional SL(2, C) generators . . . . . . . . . . . . . 254.4 Orbital generators . . . . . . . . . . . . . . . . . . . . . . . . . 274.5 Formal position operators . . . . . . . . . . . . . . . . . . . . 29

5 Nonzero mass particle . . . . . . . . . . . . . . . . . . . . . . . 335.1 Wigner representation . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . 335.1.2 Transformation law . . . . . . . . . . . . . . . . . . . . . 335.1.3 Infinitesimal generators . . . . . . . . . . . . . . . . . . 345.1.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1.5 Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.1.6 Canonical convention . . . . . . . . . . . . . . . . . . . 365.1.7 Helicity convention . . . . . . . . . . . . . . . . . . . . . 37

5.2 Spinor representation . . . . . . . . . . . . . . . . . . . . . . . 386 Zero mass particle . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.1 Wigner representation . . . . . . . . . . . . . . . . . . . . . . 396.2 Zwanziger representation . . . . . . . . . . . . . . . . . . . . . 39References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

These notes collect various formulas that we have found useful over the yearsfor the homogeneous and inhomogeneous Lorentz groups and their unitaryand spinorial representations. They are presented with a few comments,but without many details of the proofs.

2

1 Conventions

Our matrix notation is MT for transpose, M∗ for hermitean conjugate, andM for complex conjugate.

When M is a second rank tensor or spinor, or a transformation on tensoror spinor indices, whose components we want to regard as matrix elements,we surround it with parentheses or brackets and write the indices as alllower, with no dots. For example:

(M)µν = Mµν (M)αβ = M αβ

[Dj(U)

]αβ

= Dj(U)αβ

Our Lorentz signature is (+−−−).Unless otherwise stated, all homogeneous or inhomogeneous Lorentz

transformations, along with any of their linear representations, are regardedas active. That is, they move vectors rather than changing their coordinates.

1.1 Lorentz metric

Four-vector indices are indicated by lowercase Greek letters and three-vectorindices by lowercase Roman letters. Occasionaly we use x, y, z labels to dis-ambiguate three-vector indices. Momentum and position are traditionallycontravariant four vectors, with upper indices, and the corresponding gra-dients are covariant, with lower indices. We use notations like:

pµ = (p0,p) ∂µ = (∂0,∇) (1.1)

to indicate that the spatial components of pµ are the components of thethree-vector p, and the spatial components of ∂µ are those of the three-vector ∇.

The summation convention applies to repeated four-vector indices whenone is upper and the other lower, and to repeated three-vector indices irre-spective of upper or lower.

The homogeneuous Lorentz group is the set of real, 4×4 matrices Λsatisfying:

ΛTGΛ = G G =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

= GT = G−1 (1.2a)

det Λ = ±1 (1.2b)

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The proper Lorentz group L+ is the subset of unimodular matrices; the or-thochronous Lorentz group L↑ is the subset of matrices that preserve the signof the time component of time-like vectors; and the restricted Lorentz groupL↑+, which is also the connected subgroup, is the subset of orthochronous,unimodular matrices.

The Lorentz metric tensors gµν and gµν are both defined to be equalto the matrix elements (G)µν . They can be used to raise or lower anylower or upper four-vector index, by contracting on either index of g, arule that is consistent when applied to g itself. The matrix elements of thehomogeneous Lorentz transformation Λ are written as Λµ

ν , correspondingto contravariant transformations of Minkowski space. As usual for a co-variant coordinate transformation, lower indices transform according to thecontragredient ΛT−1, whose matrix elements are naturally written as Λµ

ν .Equation (1.2a) says that the latter is related to the former by lowering andraising with the Lorentz metric tensor. It can be handy in some manipula-tions to define Λµν or Λµν by lowering or raising the upper or lower index ofΛµ

ν , or by performing the opposite operations on Λµν . It is easy to check

from the definition of G that these operations are consistent.The fundamental isotropic tensors of the Lorentz group are the metric

tensor gµν and the pseudotensor alternating symbol εµνλρ, odd under spaceinversion:

g00 = −g11 = −g22 = −g33 = 1 gµν = gµν (1.3a)

ε0123 = −1 ε123 = 1 εµνλρ = −εµνλρ (1.3b)

Λµλ g

λν = gµλ Λλ

ν

(1.4a)Λµ

λ Λνρ g

λρ = gµν

Λµδ ε

δνλρ = εµκστ Λκ

ν Λσλ Λτ

ρ det Λ

Λµδ Λν

κ εδκλρ = εµνστ Λσ

λ Λτρ det Λ

(1.4b)Λµ

δ Λνκ Λλ

σ εδκσ

ρ = εµνλτ Λτρ det Λ

Λµδ Λν

κ Λλσ Λρ

τ εδκστ = εµνλρ det Λ

As a matter of fact, (1.4b) holds when Λ is any complex 4×4 matrix.

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In the immediately following formulas, parenthesized combinations oftensor indices represent determinants of components of g, and A tensorsare antisymmetrizing projections. For example, (µνδκ) = gµδ g

νκ − gµκ gνδ.

−εµνλρ εδκστ = 0!(µνλρδκστ

)= 0!·4!Aµνλρδκστ (1.5a)

−εµνλτ εδκστ = 1!(µνλδκσ

)= 1!·3!Aµνλδκσ (1.5b)

−εµνλρ εδκλρ = 2! (µνδκ) = 2!·2!Aµνκδ (1.5c)

−εµκλρ εδκλρ = 3! gµδ (1.5d)

−εµκλρ εµκλρ = 4! (1.5e)

1.2 Duals

Irreducible representations of the restricted Lorentz group for integer spinhave self and antiself duality properties that make it useful to include a fac-tor i in the definition of the dual. For antisymmetric, second-rank tensors:

Tµν = TAµν ≡ 1

2(Tµν − Tνµ) (1.6a)

TDµν ≡ i

2εµνλρ T λρ (TD)D = T (1.6b)

T SDµν ≡ 1

2

(T + TD

)µν

(T SD)D = T SD (1.6c)

TASDµν ≡ 1

2

(T − TD

)µν

(TASD)D = −TASD (1.6d)

(T SD)SD = T SD (TASD)ASD = TASD (1.6e)

(T SD)ASD = 0 (TASD)SD = 0 (1.6f)

Projection operators for the self and antiself dual parts of any second ranktensor are defined by:

P±(µν)(λρ) ≡14

(gµλ gνρ − gµρ gνλ)± i4εµνλρ (1.7a)

= P±(λρ)(µν) = P∓(λρ)(µν) (1.7b)

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P±(µν)(λρ)P±(λρ)

(στ) = P±(µν)(στ) (1.7c)

P±(µν)(λρ)P∓(λρ)

(στ) = 0 (1.7d)

Here self or antiself dual pairs of indices are surrounded by parentheses.The projection operators may of course be applied to any pair of indices ofa tensor of higher rank.

1.3 Units

Before putting ~ = c = 1, we have:

pµ = (E/c,p) = ~k = ~(ω/c,k) (1.8)

E =√m2c4 + p·pc2 ω/c =

√µ2 + k·k µ = mc/~ (1.9)

energy = MeV

momentum = MeV/c dim k = L−1 = dim MeV/~cmass = MeV/c2

From now on we put ~ = c = 1, as in:

pµ = (ω,p) ω =√m2 + p·p (1.10)

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2 Pauli matrices

2.1 Matrices

σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

)(2.1a)

[σi, σj

]= 2i εijk, σk

{σi, σj

}= 2 δij (2.1b)

σ = (σ1, σ2, σ3) = σ∗ σµ = (I,σ) σµ = (I,−σ) = σµ (2.1c)

12

Tr σi σj = δij12

Tr σµσν = gµν (2.1d)

Let zµ = (z0, z) be an arbitrary complex four vector. Then the mappingsbetween C4 and the complex 2×2 matrices defined by (2.2a) below areholomorphic, 1-1, onto, inverses of each other:

M ≡ z ·σ = z0I + z ·σ zµ = 12

TrMσµ (2.2a)

M2 = [(z0)2 + z ·z]I + 2z0 z ·σ (2.2b)

TrM = 2z0 TrM2 = 2[(z0)2 + z ·z] (2.2c)

detM = z ·z = 12

[(TrM)2 − TrM2

](2.2d)

2.2 Transformation laws

Let Λ be the image of A ∈ SL(2, C), Λ = Λ(±A), under the standard,two-to-one homomorphism of SL(2, C) onto L↑+. Then:

AσµA∗ = σν Λν

µ A∗−1 σµA−1 = σν Λνµ (2.3a)

Λµν = 1

2Tr(σµAσνA

∗) A = A∗ =⇒ Λ = ΛT (2.3b)

2.3 Spinor labels

When considered as a generic matrix, the elements of a 2×2 matrix Mare written as (M)αβ. When considered as a second rank spinor, espe-cially involving Pauli matrices or their products, the matrix elements ofM are written as lower, upper, lower dotted, or upper dotted indices, as

7

appropriate, corresponding respectively to the four faithful representationsof SL(2, C) defined by A, AT−1, A, and A∗−1. The matrix elements are

written according to their action towards the right: Aαβ, AT−1 α

β, Aαβ, and

A∗−1 αβ. All indices have the values ±1/2. Fortunately, one of the points ofthe spinor notation is that one rarely has to write down the matrix elementsof A variants. The contragredients AT−1 and A∗−1 can often be avoided byletting A and A act to the left, as in (2.8a) and (2.8b).

The spinor metric symbol ε raises and lowers spinor indices and relatescontragradient spinor transformation matrices, although not in exactly thesame way as g does for tensors. It implements unitary equivalence for lowerand upper indices of the same type, because of (2.4b). Dotted indices arenot equivalent to undotted indices.

ε ≡ iσ2 =

(0 1−1 0

)= −ε−1 (2.4a)

εMT ε−1 = M−1 detM (2.4b)

ε σµ ε−1 = σµ

T ε σµ ε−1 = σµ

T (2.4c)

σµαβ = (σµ)αβ σµαβ = (σµ)αβ = (σµ)αβ = σµαβ (2.4d)

εαβ = εαβ = εαβ = εαβ = (−1)12−α δα

−β = (ε)αβ = −(ε−1)αβ (2.4e)

The raising and lowering operations are written:

ηα = εαβ ηβ ηα = ε−1αβ ηβ = −εαβ ηβ = ηβ εβα (2.5a)

=⇒ ζα ηα = − ζα ηα (2.5b)

Thus a spinor index is raised by contracting on the right index of εαβ or εαβ,and lowered by contracting on the left index of εαβ or εαβ. The raising andlowering operations can be applied to σ, σ, and ε itself; and complex con-jugation converts between their undotted and dotted indices. The resultsturn out to be consistent and natural:

εαβ = εαγ εβδ εγδ εαβ = εγδ εγα εδβ (2.6a)

εαβ = εαγ εβδ εγδ εαβ = εγδ εγα εδβ (2.6b)

εαβ = εαβ εαβ = εαβ (2.6c)

8

σµαβ = (ε σµ ε

−1)αβ = (σTµ )αβ = σµ

βα (2.7a)

= (σµ)αβ = σµαβ (2.7b)

σµ αβ = (ε−1 σµ ε)αβ = (σTµ )αβ = σµβα (2.7c)

= (σµ)αβ = σµαβ (2.7d)

The Pauli matrices and the spinor metric are isotropic spinors underSL(2, C) because of (2.3a) and (2.4b):

Λµν Aα

γ Aβδ σνγδ = σµαβ Λµ

ν σν γδ Aγ

αAδβ = σµαβ (2.8a)

Aαγ Aβ

δ εγδ = εαβ εγδ AγαAδ

β = εαβ(2.8b)

Aαγ Aβ

δ εγδ = εαβ εγδ AγαAδ

β = εαβ

Note how we avoid having to write indices for AT−1 and A∗−1 by actingto the left with A and A on upper indices. It is sometimes handy to dothe analogous thing with Λ, as in (1.4b); but while allowing all positionsfor indices is a convenience for Λ, it is a notational burden for A and A.In fact it is not true that AT−1 is obtained from A by formally raising andlowering. Instead:

εαα′Aα′

β′εβ′β = (εA ε)αβ = −

(AT−1)

αβ= −AT−1α

β (2.9)

2.4 Orthogonality

The covariant Pauli matrices can be regarded as proportional to symbolsfor unitary transformation between vector and spinor forms of the irre-ducible (1

2, 12) representation of SL(2, C). The Lorentz-invariant trace iden-

tity (2.1d) has several equivalent spinor forms, which can be regarded asorthogonality relations for transformation from a vector to an equivalentrank-two spinor of type (1

2, 12) and back again.

There is an awkward point about the inverse orthgonality relation, fromrank-two spinor to vector to rank-two spinor. Namely, the Kronecker sym-bol with one index down and one up changes sign when formally raised andlowered, because:

εαβ = −εαβ = (I)αβ (2.10)

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To put it another way, there is no spinor δ such that δαβ and δαβ are both

equal to the Kronecker delta.In order to keep the convenient Kronecker notation, we define the spinor

Kronecker delta as an isotropic convenience symbol with only one configu-ration of indices, never to be raised or lowered :

δαβ = δα

β ≡ (I)αβ (2.11)

The dotted and undotted versions are in fact isotropic SL(2, C) spinors, butwe agree never to write down the other index configurations.

Some equivalent spinor forms of the trace orthogonality relation (2.1d)are given below in (2.12), and some for its inverse in (2.13a) and (2.13b):

12σµαβ σν

βα = 12σµ

βα σναβ = 12σµαβ σν

αβ = 12σµ

βα σνβα = gµν (2.12)

12σµα1β1

σµβ2α2 = 12σµβ1α1

σµα2β2 = δα1

α2 δβ1β2 (2.13a)

12σµα1β1

σµα2β2= 1

2σµβ1α1

σµβ2α2= εα1α2 εβ1β2 (2.13b)

It is straightforward albeit tedious to compute the l.h.s. of (2.13b) by ex-plicitly writing out the dot products of the four-vectors that result fromfixing the values of the spinor index pairs in σµα1β1

and σµα2β2, but the

calculation becomes trivial after a Clebsch-Gordan analysis. The l.h.s. isan isotropic spinor with two lower undotted and two lower dotted indices.Being invariant, the only nonzero irreducible component has to have zeroangular momentum content for each of the undotted and dotted pairs, andmust be proportional to εα1α2 εβ1β2 . The proportionality constant is fixed by

choosing, for example, (α1, α2, β1, β2) = (1/2, −1/2, 1/2, −1/2) and computing asingle dot product. The rest of the inverses can be derived by raising andlowering.

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2.5 Products

Let aµ = (a0,a), etc.

a·σ b·σ + b·σ a·σ = a·σ b·σ + b·σ a·σ = 2 a·b (2.14a)

a·σ b·σ = a·b+(b0 a− a0 b− ia× b

)·σ (2.14b)

b·σ a·σ = a·b+(a0 b− b0 a− i b× a

)·σ (2.14c)

a·σ b·σ = a·b+(a0 b− b0 a− ia× b

)·σ (2.14d)

b·σ a·σ = a·b+(b0 a− a0 b− i b× a

)·σ (2.14e)

σµ σν = gµν + i2εµνλρ σλ σρ (2.14f)

σµ σν = gµν − i2εµνλρ σλ σρ (2.14g)

12

(σµ σν − σν σµ) = (σµ σν)A = i

2εµνλρ σλσρ = (σµ σν)

SD (2.14h)

12

(σµ σν − σν σµ) = (σµ σν)A = − i

2εµνλρ σλσρ = (σµ σν)

ASD (2.14i)

[a, b, c]µ ≡ εµνλρ aνbλcρ (2.15a)

a·σ b·σ c·σ = a·σ b·c− b·σ a·c+ c·σ a·b+ i [a, b, c]·σ (2.15b)

a·σ b·σ c·σ = a·σ b·c− b·σ a·c+ c·σ a·b− i [a, b, c]·σ (2.15c)

σµ σν σλ = gµν σλ − gµλ σν + gνλ σµ − i εµνλρ σρ (2.15d)

σµ σν σλ = gµν σλ − gµλ σν + gνλ σµ + i εµνλρ σρ (2.15e)

i2εµνστ σσ στ σλ = σµ gνλ − σν gµλ − i εµνλρ σρ (2.15f)

i2εµνστ σσ στ σλ = −σµ gνλ + σν gµλ − i εµνλρ σρ (2.15g)

11

12

Tr (σµ σν σλ σρ) = gµν gλρ − gµλ gνρ + gµρ gνλ − i εµνλρ (2.16a)

12

Tr (σµ σν σλ σρ) = gµν gλρ − gµλ gνρ + gµρ gνλ + i εµνλρ (2.16b)

12

Tr[(σµσν)

A σλσρ

]= −

(gµλ gνρ − gµρ gνλ + i εµνλρ

)(2.16c)

= 12

Tr[σµσν (σλσρ)

A]

(2.16d)

= 12

Tr[(σµσν)

SD (σλσρ)SD]

(2.16e)

= −4P+(µν)(λρ) (2.16f)

12

Tr[(σµσν)

A σλσρ

]= −

(gµλ gνρ − gµρ gνλ − i εµνλρ

)(2.16g)

= 12

Tr[σµσν (σλσρ)

A]

(2.16h)

= 12

Tr[(σµσν)

ASD (σλσρ)ASD]

(2.16i)

= −4P−(µν)(λρ) (2.16j)

2.6 Boosts

It is generally easier to compute the effects of Λ ∈ L↑+, as well as the matrixelements of Λ, from the SL(2, C) versions.

Boost corresponding to (m, 0, 0, 0)→ p = (ω,p):

A(p) ≡√p·σm

=m+ p·σ√2m(m+ ω)

= A(p)∗ (2.17a)

p·σ = A(p) mσ0 A(p)∗ (2.17b)

A(p)−1 =

√p·σm

=m+ p·σ√2m(m+ ω)

(2.17c)

The matrix square roots in Eqs. (2.17a) and (2.17c) are positive definite.That can be seen by rotating p into the three-direction with a similaritytransformation in SU(2), which diagonalizes A(p) and A−1(p) into linear

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combinations of σ0 and σ3:

U A(p)U∗ =

m+ ω + |p|√

2m(m+ ω)0

0m+ ω − |p|√

2m(m+ ω)

(2.18a)

U A(p)−1 U∗ =

m+ ω − |p|√

2m(m+ ω)0

0m+ ω + |p|√

2m(m+ ω)

(2.18b)

Boost in terms of rapidity λ and direction n = p/|p|:

A(p) = exp(12λn·σ

)(2.19a)

= I cosh 12λ+ n·σ sinh 1

2λ (2.19b)

β =|p|ω

γ =1√

1− β2(2.19c)

coshλ = γ sinhλ = βγ (2.19d)

Equation (2.19b) follows from (2.19a) by substituting (n·σ)2 = I into thepower series expansion for the exponential, which converges everywhere. Itis straightforward to show that every hermitean, positive definite matrix inSL(2, C) has the form (2.19b).

Let x·σ = m+ p·σ, and Λ = Λ(p) = Λ[A(p)]. Then:

Λµν =

xµ xνm(m+ ω)

− gµν Λ−1 µν =xµ xν

m(m+ ω)− gµν (2.20)

Let uµ = pµ/m = (γ, γv). Then:

Λ00 = γ Λ0

i = Λi0 = γvi Λi

j = δij +γ2

1 + γvi vj (2.21a)

Λ =

u0 u

u ε+uu

1 + u0

(2.21b)

=

(coshλ n sinhλ

n sinhλ ε+ nn (coshλ− 1)

)n ≡ v

|v|(2.21c)

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2.7 Rotations

The covering group of the rotation subgroup SO(3) of L↑+ is the SU(2)subgroup of SL(2, C).

Rotations leave the x0 component of xµ as well as the magnitude of xinvariant:

U x0σ0 U∗ = x0σ0 =⇒ U∗ = U−1 (2.22a)

U x·σ U∗ = y ·σ =⇒ x·x = y ·y (2.22b)

Rotation by θ about the e = (e1, e2, e3) axis, e·e = 1:

U(θe) = exp(− i

2θ e·σ

)= U(−θe)−1 (2.23a)

= I cos 12θ − i e·σ sin 1

2θ (2.23b)

The proof that (2.23b) follows from (2.23a) imitates the power series argu-ment for boosts. A straightforward calculation shows that all elements ofSU(2) have the form (2.23b). Note that unitarity implies that:

U(θe)T = U(−θe) (2.24)

As an application of (2.23b), note that the spinor metric (2.4a) is arotation by −π about the y axis:e2 ≡ (0, 1, 0):

ε = U(−πe2) (2.25)

The three-dimensional rotation in L↑+ corresponding to θ and e is:

R(θe)00 = 1 R(θe)0i = R(θe)i0 = 0 (2.26a)

R(θe)ij = 12

Tr[σi U(θe)σj U(θe)∗

](2.26b)

= δij cos θ + eiej (1− cos θ)− εijk ek sin θ (2.26c)

2.8 Homogeneous transformations

The fact that A(p) and U(θe) separately parameterize the hermitean pos-itive definite and unitary matrices in SL(2, C) means, by polar decomposi-tion, that every element A has the form:

A = A(p)U(θe) (2.27a)

Corresponding to that, every element Λ in L↑+ has the form:

Λ = Λ(p)R(θe) (2.27b)

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2.9 Planar transformations

Let y1 ·y1 = y2 ·y2 ≡ y ·y:

M ≡ y ·y + y2 ·σ y1 ·σ (2.28a)

detM = 2 y ·y(y ·y + y2 ·y1

)(2.28b)

Let detM 6= 0, sgn y10 = sgn y2

0 when y ·y > 0, and q ·y1 = q ·y2 = 0:

A ≡ y ·y + y2 ·σ y1 ·σ√2 y ·y

(y ·y + y2 ·y1

) ∈ SL(2, C) (2.29a)

Ay1 ·σ A∗ = y2 ·σ A q ·σ A∗ = q ·σ (2.29b)

Λ(A)µν = gµν −1

y ·y + y2 ·y1

[y1µ y1ν + y1

µ y2ν + y2µ y2ν

− y ·y + 2 y2 ·y1y ·y

y2µ y1ν

](2.30a)

Λ(A) y1 = y2 Λ(A) q = q (2.30b)

15

3 Clebsch-Gordan coefficients

The usual Clebsch-Gordan (CG) coefficients for the addition of angular mo-menta also appear in the reduction of tensor products of finite-dimensionalrepresentations of SL(2, C) and L↑+ into irreducible components. Indeedthey serve that function for the proper, homogeneous complex Lorentzgroup, for which the covering group is SL(2, C)⊗SL(2, C). This sectionreviews our notational conventions for spinors of higher spin and discussesthe invariance of the CG coefficients under SL(2, C). In particular, CGcoefficients are isotropic spinors, just as gµν and εµνλρ are isotropic tensors.

3.1 Definition

There are various notations for the coefficients, but one dominant conven-tion for their values, that of Wigner [1, 2] and Condon and Shortley [3],which we call the standard phase convention. The standard convention hastwo parts, the first, for angular momentum in general:

(i) The choice of real, positive coefficients in the ladder recursion formulasto define the relative phases of normalized eigenstates of Jz in anirreducible subspace.

(ii) The choice of a real, positive amplitude for the transition betweentotal angular momentum eigenstates with maximum magnetic quan-tum number and product eigenstates of two angular momenta withmaximum magnetic quantum number in the first factor.

Rose [4] follows the standard convention,1 and his notation C(j1j2j;m1m2m)for the coefficient is common. The Particle Data Group adopts the standardconvention, and has an online link for CG coefficients, spherical harmonicsand d-functions.

Expressed in terms of normalized angular momentum eigenvectors withconventional phases, the CG coefficient is the transition amplitude:2

C(j1j2j;m1m2m) = 〈j1m1j2m2 |j1j2jm〉 (3.1a)

1Other authors who follow it include Blatt and Weisskopf [5], Edmonds [6], Fano andRacah [7], Messiah [8, 9], Merzbacher [10], Cohen-Tannoudji et al. [11, 12].

2 We do not follow the common labeling 〈j1j2m1m2 |j1j2jm〉 for the bra in the am-plitude. The notation in (3.1a) emphasizes the tensor product nature of the bra.

16

The normalized bra is a simultaneous eigenvector of the commuting obsev-ables,

J1 ·J1, J1z, J2 ·J2, J2z,

with eigenvalues,

j1(j1+1), m1, j2(j2+1), m2;

and the normalized ket is a simultaneous eigenvector of the commutingobservables,

J1 ·J1, J2 ·J2, J ·J , J z,

with eigenvalues,

j1(j1+1), 2(j2+1), j(j+1), m,

where J is the total angular momentum,

J = J1+J2.

The bras and kets are orthogonal unless

m = m1 +m2 , (3.1b)

because both are eigenvectors of Jz = J1z+J2z. It follows from the angularmomentum algebra that the j’s are nonnegative integers or half-integersthat satisfy the triangle condition,

|j1−j2| ≤ j ≤ j1+j2 . (3.1c)

The standard phase convention guarantees that the transition amplitudes,and hence the CG coefficients, are real; so it is also true that

C(j1j2j;m1m2m) = 〈j1j2jm|j1m1j2,m2〉 (3.1d)

Spinor notation for the CG coefficients replaces the magnetic quantumnumbersm1, m2, andm by appropriately lower or upper, undotted or dottedspinor indices for the corresponding spins, taking the same values as themagnetic quantum numbers [13], e.g., j, j−1, · · · − j. What is appropriateis determined by the transformation law of angular momentum eigenstates

17

under the representation of SU(2) defined by the angular momentum Liealgebra.

For spin-1/2, the infinitesimal generators of the self-representation ofSU(2), J = σ/2, obey the standard convention, so

(U)mm′ =⟨12m∣∣ exp(−iθe·σ/2)

∣∣12m′⟩

(3.2a)

In spite of the fact that we have written the spinor indices of the Paulimatrices as σαβ, the appropriate index type for U is:

Uαβ = (U)αβ (3.2b)

This clash of conventions is unavoidable. First of all, the assignment ofindex types for the SL(2, C) spinor calculus is, as far we know, historicallynear universal. Furthermore, having one undotted and one dotted indexfor the Pauli matrix three-vector is consonant with the complex conjugaterelationship between

⟨12m∣∣ and

∣∣12m⟩. But if we think of U as a special

A ∈ SL(2, C), it makes no sense to write Uαβ, because dotted and undottedspinor types are inequivalent under SL(2, C).

Although there is a clash, there is no paradox. Under SU(2), the for-merly inequivalent lower dotted and upper undotted indices are not onlyunitary equivalent; the matrix representations U and UT−1 are identical asunitary matrices. If we restrict to SU(2), it makes perfect sense to writethe spinor indices of the Pauli matrices as σµα

β, including the σ0 = I com-ponent.

3.2 Restriction to SU(2)

The basic properties of CG coefficients are determined by the angular mo-mentum algebra. That includes the construction of the unitary, irreduciblerepresentations of SU(2), the reduction of tensor products of those repre-sentations by CG coefficients, and the SU(2) invariance of the coefficients.We describe the facts here, and refer to any of several texts for the proofs.3

3Cf. Rose [4], Edmonds [6], Merzbacher [10], which all use the standard phase con-vention. Merzbacher is especially clear.

18

The standard unitary irreducible representations of SU(2) in the activeview4 are given by:[

Dj(U)]mm′ ≡ 〈jm| exp(−iθe·J) |jm′〉 =

[Dj(U)∗−1

]mm′ (3.3a)

Dj(U) = Dj(U) (3.3b)

Dj(U)T = Dj(UT) (3.3c)

Dj(U)∗ = Dj(U∗) (3.3d)

Dj(U)−1 = Dj(U−1) (3.3e)

Dj(U1)Dj(U2) = Dj(U1U2) (3.3f)

Dj(U)αβ ≡

[Dj(U)

]αβ

(3.3g)

The unitarity expressed in (3.3a) follows from the hermiticity of J . Per-sistence of complex conjugation in (3.3b) follows by conjugating the tran-sition amplitude in (3.3a) to change the sign of θ and transpose the mag-netic quantum numbers, then applying (2.24). Persistence of transposition(3.3c) follows from that of complex conjugation, plus unitarity. Persistenceof hermitian conjugation (3.3d) follows from that of complex conjugationand transposition, as does persistence of inversion (3.3e). The group com-position law (3.3f) is less obvious; it follows as a special case from (??) inSection ??, which expresses irreducible representations of SL(2, C) in termsof a tensor product of spin-1/2’s.

The raising and lowering symbol [ j ] is the natural generalization ofε. The evaluation in (3.4a) follows from (2.25) and explicit calculation ofDj[U(−πe2)]:5

[ j ]αβ = [ j ]αβ ≡[Dj(ε)

]αβ

= (−1)j−α δα−β (3.4a)

[ j ]αβ = [ j ]αβ ≡ [Dj(ε)]αβ =[Dj(ε)

]αβ

=[Dj(ε)

]αβ

= [ j ]αβ (3.4b)

4The minus sign in the exponential in (3.3a) corresponds to the active view; the pas-sive view would have a plus sign. Wigner [1] and [2], Rose [4, pp. 16,17,48], Merzbacher[10, p. 413], and Cohen-Tannoudji et al. [12] take the active view. Edmonds [6], Fanoand Racah [7] take the passive view.

5Cf. Edmonds [6, p. 59].

19

Thus the matrix [ j ] is real and unitary, with the action:

[ j ]Dj(U) [ j ]−1 = Dj(εUε−1

)= Dj

(UT−1) = Dj(U)T−1 (3.5a)

As with spin-1/2, we raise by contracting on the right index of [ j ]αβ or [ j ]αβ

and lower by contracting on the left index of [ j ]αβ or [ j ]αβ.The transformation law for angular momentum eigenstates is immediate

from (3.3a):

exp(−iθe·J) |jm〉 =∑m′

|jm′〉Dj(U)m′m (3.6a)

〈jm| exp(−iθe·J) =∑m′

Dj(U)mm′〈jm′| (3.6b)

This gives two transformation laws for CG coefficients:

〈jm1jm2| exp(−iθe·J) |j1j2jm〉 =∑m′

1m′2m

′

Dj1(U)m1

m′1 Dj2(U)m2

m′2 〈j1m′1j2m′2 |j1j2jm〉 (3.7a)

=∑m′

〈j1m1j2m2 |j1j2jm′〉Dj(U)m′m

〈j1j2jm| exp(−iθe·J) |jm1jm2〉 =∑m′

Dj(U)mm′〈j1j2jm′ |j1m1j2m2〉 (3.7b)

=∑

m′1m

′2m

′

〈j1j2jm|j1m′1j2m′2〉Dj1(U)m′1

m1 Dj2(U)m′2

m2

Among the two numerically equal ways of writing the CG coefficient,〈j1m1j2m2 |j1j2jm〉 transforms as an isotropic spinor with two lower indicesand one upper index,6 while 〈j1j2jm|j1m1j2m2〉 is isotropic with one lowerand two upper indices. As isotropic spinors they are not identical; raisingand lowering on one produces the other multiplied by a phase factor.

We define “the” CG spinor as the one that reduces two lower indices,corresponding to a tensor product of representations of SU(2), to a single

6This property is discussed by Wigner [2, pp. 292–296] and Edmonds [6, p. 46].

20

lower index, corresponding to an irreducible component of the equivalentdirect sum:

[ jj1j2 ]αα1α2 ≡ C(j1j2j;α1α2α) = 〈j1j2jα|j1α1j2α2〉

(3.8a)= [ jj1j2 ]αα1α2

[ jj1j2 ]αα1α2 ≡ [ jj1j2 ]αα1α2 (3.8b)

Here are some standard properties of CG coefficients in spinor notation.They all have obvious dotted versions, because everything is real.

Reduction to zero angular momentum:

[ 0jj ]0αβ = (−1)2j[ 0jj ]0

βα =[ j ]αβ√2j+1

(3.9)

Index inversion:

[ jj1j2 ]αα1α2 = (−1)2j [ jj1j2 ]αα1α2 (−1)2j = (−1)2j1+2j2 (3.10)

Symmetry:

[ jj2j1 ]αα2α1 = (−1)j−j1−j2 [ jj1j2 ]α

α1α2 (3.11)

Orthogonality:

δjj′ δαα′

= (−1)2j [ jj1j2 ]αα1α2 [ j′j1j2 ]α

′α1α2 (3.12a)

δα1

α′1 δα2

α′2 =

∑j

(−1)2j [ jj1j2 ]αα1α2 [ jj1j2 ]αα′

1α′2

(3.12b)

Equation (3.9), including the phase change for exchange of indices, followsby comparing (3.4a) with a calculation of the CG coefficient.7 The CG indexinversion formula (3.10) is the spinor form of magnetic quantum numberreflection symmetry:8

C(j1j2j;−m1,−m2,−m) = C(j1j2j;m1m2m) (3.13)

It follows from (3.4a).We defer the spinor version of the SU(2) invariance laws (3.7a) and

(3.7b) to Section 3.4, as a special case of SL(2, C) invariance.7Cf. [12, p. 1041]. Proportionality follows from the CG positive sign convention and

ladder recursion, and the normalization follows by taking traces on the l.h.s. of the CGorthogonality condition (3.12b).

8Cf. [12, pp. 1039,1041].

21

3.3 Orthogonality

3.4 Extension to SL(2, C)

22

4 Poincare group

The inhomogeneous Lorentz group, or Poincare group, has ten parameters,four for spacetime translations, and six for homogeneous Lorentz transfor-mations. The group elements may be written as (a,Λ), a ∈ R4, Λ ∈ L(4,R),aµ = (a0,a). The covering group of the connected part of the Poincaregroup P↑+ is the inhomogeneous SL(2, C) group, or iSL(2,C), with elements(a,A), A ∈ SL(2, C). Group laws:

(a,Λ) (a′,Λ′) = (a+ Λ a′,Λ Λ′) (4.1a)

(a,A) (a′, A)′ =(a+ Λ(A) a′, AA′

)(4.1b)

4.1 Poincare generators

For iSL(2,C), (or P↑+):[Mµν , Mλρ

]= −i

(gµλMνρ − gµρMνλ + gνρMµλ − gνλMµρ

)(4.2a)[

Pµ, Pν]

= 0 (4.2b)[Mµν , Pλ

]= −i

(gµλ Pν − gνλ Pµ

)(4.2c)

Scalar, vector, and second rank tensor operators relative to Mµν :[Mµν , S

]= 0 (4.3a)[

Mµν , Vλ]

= −i(gµλ Vν − gνλ Vµ

)(4.3b)[

Mµν , Tλρ]

= −i(gµλ Tνρ − gµρ Tνλ + gνρ Tµλ − gνλ Tµρ

)(4.3c)

Here S is a generic notation for any scalar. Later it will be used for spin.

23

Three-vector forms:

J =(M23,M31,M12

)Ji = 1

2εijkMjk (4.4a)

K =(M01,M02,M03

)Ki = M0i (4.4b)[

Ji, Jj]

= i εijk Jk (4.4c)[Ji, Kj

]= i εijkKk (4.4d)[

Ki, Kj

]= −i εijk Jk (4.4e)[

Ji, Pj]

= i εijk P k[Ji, P

0]

= 0 (4.4f)[Ki, P

j]

= i δij P0

[Ki, P

0]

= i P i (4.4g)

12MµνM

µν = J ·J −K ·K (4.4h)

18εµνλρMµνMλρ = J ·K = K ·J (4.4i)

For any four-vector operator V µ:[Ji, Vj

]= i εijk Vk

[Ji, V0

]= 0 (4.5a)[

Ki, Vj]

= −i δij V0[Ki, V0

]= −i Vi (4.5b)

The sign of the rotation generator J is unambiguous. The sign of theboost generator K is fixed by the physical interpretation of its orbital partas the Energieschwerpunkt operator, Eq. (4.30b). The exponential param-eterization of the active representation of the element (a,A) of iSL(2,C)corresponding to P µ,Mµν is

T (a)H(A) = exp(iP ·a

)exp(iλn·K

)exp(−iθ e·J

)(4.6)

4.2 Casimir operators

The Casimir invariants of the Poincare group are the mass and the magni-tude of spin. The mass operator

M2 = P ·P (4.7)

24

is a scalar function of P µ, and hence commutes will all generators of thePoincare group. We only consider M2 > 0 in these notes.

The spin is described in Section 5.1 via the Pauli-Lubanski vector:

wµ ≡ 12εµνλρ P νMλρ (4.8a)[

Pµ, wν]

= 0 (4.8b)[Mµν , wλ

]= −i

(gµλwν − gνλwµ

)(4.8c)[

wµ, wν]

= i εµνλρ P λwρ (4.8d)

= i(Pµ P

λMνλ − Pν P λMµλ +M2Mµν

)(4.8e)

4.3 Finite-dimensional SL(2, C) generators

M ≡ 12

(J + iK

)N ≡ 1

2

(J − iK

)(4.9a)[

Mi, Mj

]= i εijkMk

[Ni, Nj

]= i εijkNk

[Mi, Nj

]= 0 (4.9b)

The finite-dimensional, irreducible representations are labeled by (m.n),where m and n half integers:

M ·M = m(m+ 1) N ·N = n(n+ 1) (4.10a)

(s, 0) : K = −iJ M = J N = 0 (4.10b)

(0, s) : K = iJ M = 0 N = J (4.10c)

For any second rank, antisymetric tensor:

TDµν ≡ i

2εµνλρ T λρ Tµν = TDD

µν (4.11a)

T SDµν = 1

2

(Tµν + i

2εµνλρ T λρ

)(4.11b)

TASDµν = 1

2

(Tµν − i

2εµνλρ T λρ

)(4.11c)

Tµν = T SDµν + TASD

µν (4.11d)

TDµν = T SD

µν − TASDµν (4.11e)

25

The representations (s, 0) and (0, s) are respectively selfdual and anti-selfdual:

(s, 0) : Mµν = MSDµν MASD

µν = 0 (4.12a)

(0, s) : Mµν = MASDµν MSD

µν = 0 (4.12b)(12, 0)

: Mµν = i4

(σµ σν − σν σµ

)J = 1

2σ K = − i

2σ (4.12c)(

0, 12

): Mµν = i

4

(σµ σν − σν σµ

)J = 1

2σ K = i

2σ (4.12d)

The contravariant and covariant four-vector representations are equiva-lent to (1

2, 12). Generators for the contravariant vector representation:[

MVµν

]λρ = i (gλµ gρν − gλν gρµ) (4.13a)[

JVi

]jk = −i εijk

[JVi

]00 =

[JVi

]0j =

[JVi

]j0 = 0 (4.13b)[

KVi

]0j =

[KVi

]j0 = −i δij

[KVi

]00 =

[KVi

]jk = 0 (4.13c)

To calculate (4.13a), use the fact that near the identity the matrices Λ(A),A, and A∗ can be written as exponentials,9 which can be approximated by

Λλρ ≈ gλρ + i aµν

[MV

µν

]λρ aµν = −aνµ = aµν (4.14a)

A ≈ I − aµν 14

(σµ σν − σν σµ

)(4.14b)

A∗ ≈ I − aµν 14

(σν σµ − σµ σν

)(4.14c)

Then use Eqs. (2.3b) and (2.16b).The exponential forms of the vector representations for rotations and

boosts are:

R(θe) = exp(−i θ e·JV

)e = (e1, e2, e3) e·e = 1 (4.15a)

= I − i e·JV sin θ −(e·JV

)2(1− cos θ) (4.15b)

9It is well-known that every Λ ∈ L↑+ can be written as the exponential of an elementof its Lie algebra, while not every A ∈ SL(2, C) can. However, at least one of ±A is anexponential. A good exposition of this nontrivial matter can be found in [].

26

L(λn) = exp(in·KV

)n = (n1, n2, n3) n·n = 1 (4.16a)

= I + in·KV sinhλ−(n·KV

)2(coshλ− 1) (4.16b)

These formulas can be computed by power series manipulation with thehelp of(

e·JV)3

= −e·JV (4.17a)(n·KV

)3= n·KV (4.17b)

They can be connected to Eqs. (2.26a), (2.26c), and (2.21c) with the helpof [(

e·JV)n]0

0 = 1[(e·JV

)n]0i =

[(e·JV

)n]i0 = 0 (4.18a)(

e·JV)ij = −i εijk ek (4.18b)[(

e·JV)2]i

j = δij − eiej (4.18c)

4.4 Orbital generators

Let F be a linear space of sufficiently differentiable complex functions f(p),p ∈ R4. The following action defines a representation of L↑+ by linearoperators U(Λ) on F :

U(Λ)f(p) = f(Λ−1p) (4.19)

We use the same symbol U in later sections to indicate a unitary representa-tion; but here we are interested in infinitesimal generators for L↑+ on F onlyas differential operators, leaving aside any Hilbert space structure. Definethe orbital generators Lµν by the approximate action near the identity:

U(Λ)f(p) ≈ f(p) + i aµν Lµνf(p) ≈ f(p− i aµνMVµνp) (4.20)

where MVµν is the generator for the four-vector representation in Eq. (4.13a).

That equation together with Taylor expansion of the r.h.s. gives the result:

Lµν = i

(pµ∂

∂pν− pν

∂

∂pµ

)(4.21)

27

It is easy to check that these linear operators on F , together with P µ definedas multiplication by pµ, obey the Poincare Lie algebra.

Now consider the restriction of F to the mass shell. Let F be a linearspace of sufficiently differentiable functions f(p), p ∈ R3. Then a direct

calculation shows that the following linear operators on F also obey thePoincare Lie algebra:

L0j = iω∂

∂pj(4.22a)

Lij = i

(pi∂

∂pj− pj

∂

∂pi

)= −i

(pi∂

∂pj− pj ∂

∂pi

)(4.22b)

P µ = pµ = (ω,p) ω =√p·p+m2 m ≥ 0 (4.22c)

Definition. Consider the mass shell restriction map from F onto F , f(p) =f(p). A linear operator O on F is said to be restrictable to the mass shell

if there exists a linear operator O on F such that:

Of = Of (4.23)

Note that O is unique, and that the mapping O → O is an operator algebrahomomorphism.

Lemma 1. Lµν defined by Eqs. (4.22a) and (4.22b) is the mass shell re-striction of Lµν .

Proof.

∂f

∂pj(p) =

∂f

∂p0(ω,p)

∂ω

∂pj+∂f

∂pj(ω,p)

=∂f

∂p0(ω,p)

pj

ω+∂f

∂pj(ω,p)

= −pjω

∂f

∂p0(ω,p) +

∂f

∂pj(ω,p)

The proof for µν = 0j or j0 then follows directly from the definition of L0j,

Eq. (4.22a). For µν = ij, it follows from the definition of Lij, Eq. (4.22b),especially its antisymmetry.

This lemma justifies an informal notation10 where Lµν is intended to be10Sometimes seen, not recommended.

28

applied on the mass shell, but is written as in Eq. (4.21). Because of thealgebra homomorphism, the lemma makes it unnecessary to do a calculationto show that Lµν and P µ obey the Poincare algebra.

The following formula holds whether Ki is taken to be L0i, with P 0 =√P ·P +M2 and M fixed, or is taken to be L0i, with P 0 independent of

P : [Ki, f(P 0)

]= i P i f ′(P 0) (4.24)

4.5 Formal position operators

The “formal” in “formal position operator” refers to the neglect of anyHilbert space structure, hermiticity in particular, for operators on F or F .The basic requirements are that position operators be vectors under L↑+ orO+(3) and satisfy canonical commutation relations (CCR). Satisfying CCRamounts to four- or three-dimensional translation covariance.

Note that on F the operators

xµ = −i ∂∂pµ

(4.25)

satisfy the L↑+ vector law (4.3b), and CCR:[xµ, pν

]= −i gµν

[xµ, xν

]=[pµ, pν

]= 0 (4.26)

We call such operators Lorentz-covariant position operators. We call thecorresponding three-vector operators on F or F rotatation-covariant posi-tion operators.

Lemma 2. All Lorentz-covariant position operators on F , respectively,rotation-covariant position operators on F or F , have the form:

xµ = −i ∂∂pµ

+ pµ h(p·p) (4.27a)

xi = i∂

∂pi+ pi h(p·p) (4.27b)

It is a well-known fact that the CCR cannot be satisfied by physicalobservables for time and total energy in quantum mechanics, because that

29

forces the spectra of both to be unbounded above and below, which violatesthe stability of total energy. But as we said, formal position operatorsdisgregard Hilbert space structure.

Note that Lµν on F , as defined in Eq. (4.21), has the same expressionfor all Lorentz-covariant position operators, and is independent of h:

Lµν = −pµ xν + pν xµ = xµ pν − xν pµ (4.28)

The same applies to Lij on F and Lij on F in terms of rotation-covariantposition operators on the respective spaces.

Lemma 3. There is no Lorentz-covariant position operator on F .

Proof. Suppose that x is a rotation-covariant position operator on F . Thenfrom Eq. (4.3b), any x0 that extends xi to a fourvector must obey:[

L0i, xj]

= −i(g0j xi − gijx0

)= i δi

j x0 (4.29)

But after substituting Eq. (4.22a) for L0i and Eq. (4.27b) for xi into thecommutator, it is straightforward to check that there is no h for which it isproportional to δi

j.

For Lorentz-covariant position operators, the following lemma is a simplecorollary of the preceding lemma. However, we give an independent proof.

Lemma 4. Neither the time nor the spatial components of xµ on F can berestricted to the mass shell.

Proof. The point is that neither ∂/∂p0 nor ∂/∂pi is restrictable. Let f1 =

p0f(p) and f2 = ωf(p). Then f1 − f2 = 0. But consider

∂f1∂p0

= f + p0∂f

∂p0∂f2∂p0

= ω∂f

∂p0

∂f1∂pi

= p0∂f

∂pi∂f2∂pi

=pi

ωf + ω

∂f

∂pi

whence

∂f1∂pµ− ∂f2∂pµ

=pµωf

for which neither temporal nor spatial components vanish.

30

The upshot of the lemmas is that rotation-covariant position operatorscan be defined on F by Eq. (4.27b), but can be neither part of a four-vectornor the mass shell restriction of a rotation-covariant position operator onF . The next lemma states the role of a special choice of rotation-covariantposition.

Lemma 5. There is a unique rotation-covariant position operator on F forwhich L0i is the Energieschwerpunkt operator

L0i = 12

(xi ω + ω xi

)namely

xi = i∂

∂pi− i pi

2ω2

Proof. It is straightforward to check that putting the above xi into theEnergieschwerpunkt operator gives back Eq. (4.22a). Next, assume that an

xi exists for which L0i has the above form. Then

i ω∂

∂pi= 1

2

[xi, ω

]+ ωxi = i

pi

2ω+ ω xi

gives the result for xi.

Of course this same position operator can be used to define Lij. Forconvenience we collect the formulas:

xi = i∂

∂pi− i pi

2ω2(4.30a)

L0i = 12

(xi ω + ω xi

)(4.30b)

Lij = xi pj − xj pi (4.30c)

If we replace F by L2 (d3p/2ω), a straightforward calculation shows that xi

in Eq. (4.30a) is the only hermitean, rotation-covariant position operatorof the form in Eq. (4.27b). A similar calculation starting from Eqs. (4.22a)

and (4.22b) shows that Lµν is also hermitean. Instead we can just note thatthe hermiticity of xi makes the right-hand sides of Eqs. (4.30b) and (4.30c)manifestly hermitian.

31

The operator in Eq. (4.30a) is the Newton-Wigner position operator,which we denote by xinw when we want to be explicit.

Finally, let us mention that similarity transformation by a nonzero oper-ator function of P preserves the Poincare commutator algebra, as does anysimilarity transformation, but changes the action of L0i and xi on F . Wecan use that to transform the generators and Newton-Wigner position fromoperators hermitean on L2 (d3p/2ω) into operators hermitean on L2 (d3p):

ω−12 Lij ω

12 = −i

(pi∂

∂pj− pj ∂

∂pi)

= Lij (4.31a)

ω−12 L0i ω

12 = i ω

∂

∂pi+ i

pi

2ω= L0i + i

pi

2ω(4.31b)

ω−12 P µ ω

12 = pµ (4.31c)

ω−12 xinw ω

12 = i

∂

∂pi(4.31d)

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5 Nonzero mass particle

This section describes two equivalent, unitary, irreducible representations ofthe covering group iSL(2,C) of the Poincare group, for discrete mass m > 0and spin s, the Wigner and spinor representations.

5.1 Wigner representation

5.1.1 Hilbert space

The Hilbert space for the Wigner representation is

ψ(p)λ = 〈p, λ|ψ〉 ∈ L2(d3p/2ω)⊗ C2s+1 pµ = (ω,p) (5.1a)

〈ψ|ψ〉 =

∫d3p

2ωΣλ |ψ(p)λ|2 (5.1b)

5.1.2 Transformation law

In the following, note the distinction between U(A), which is a unitaryoperator on L2(d3p/2ω)⊗ C2s+1, and U(p), which is a 2×2 unitary matrixin SU(2). For A ∈ SL(2, C), Ds(A) is the (2s+1)-dimensional, irreduciblerepresentation (s,0), corresponding to a lower, undotted spinor index. Whenrestricted to the SU(2) subgroup, it coincides with the standard unitary,irreducible representation.

U(a,A) = exp(iP ·a) U(A) (5.2a)

〈p, λ| exp(iP ·a)ψ〉 = eip·a 〈p, λ|ψ〉 (5.2b)

〈p, λ|U(A)ψ〉 = Ds[B(p)−1AB(Λ−1p)

]λλ′⟨Λ−1p, λ′

∣∣ψ⟩ (5.2c)

B(p) =

√p·σm

U(p) U(p)σ3 U(p)∗ = e(p)·σ (5.2d)

〈U(A)ψ|U(A)ψ〉 = 〈ψ|ψ〉 U(A)∗ = U(A)−1 (5.2e)

The unitarity of U(A) expressed by Eq. (5.2e) follows from the unitarityof the argument of Ds in the transformation law (5.2c). That in turn fol-lows from the definition (5.2d) of B(p), and a threefold application of thetransformation law (2.3a) for covariant Pauli matrices:[

B(p)−1AB(Λ−1p)]mσ0

[B(Λ−1p)∗A∗B(p)−1∗

]= mσ0 (5.3)

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The combination B(p)−1AB(Λ−1p) ∈ SU(2) corresponds to a member ofthe Wigner little group of the rest frame value of the vector p.

5.1.3 Infinitesimal generators

We rename the infinitesimal generators (4.12a) and (4.12c) of the selfdual,(s,0) representation:

σµν(s, 0) = i2εµνλρ σλρ(s, 0) (5.4a)

σµν(1/2, 0) = i4

(σµ σν − σµ σµ) (5.4b)

Ds(A) σµν(s, 0) Ds(A−1

)= σλρ(s, 0) Λλ

µ Λρν (5.4c)

We use the names L = (L1, L2, L3) and K = (K1, K2, K3) for the on-shellorbital generators of rotations and boosts:

Li = 12εijk Ljk Ki = L0i (5.5)

Recall the definitions (4.22a) and (4.22b) for Lµν .To calculate the action of the generators Mµν of U(A), let the group

parameters aµν = −aνµ = aµν be sufficiently small to write:

A = exp[i aµνσµν(1/2, 0)]

Then

Ds(A) ≈ I + i aµνσµν(s, 0) (5.6a)

f(Λ−1p) ≈ f(p) + i aµνLµν f(p) (5.6b)

U(A) ≈ I + i aµνMµν (5.6c)

Ds[B(Λ−1p)

]λλ′⟨Λ−1p, λ′

∣∣ψ⟩≈ Ds

[B(p)

]λλ′ 〈p, λ′ |ψ〉

+ i aµν{[

Lµν , Ds[B(p)

]λλ′]〈p, λ′ |ψ〉+ Ds

[B(p)

]Lµν 〈p, λ′ |ψ〉

} (5.7)

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Hence

Mµν = Lµν + Ds[B(p)−1

] [Lµν , Ds

[B(p)

] ]+ Ds

[B(p)−1

]σµν(s, 0) Ds

[B(p)

](5.8a)

Sµν ≡ Ds[B(p)−1

] [Lµν , Ds

[B(p)

] ]+ Ds

[B(p)−1

]σµν(s, 0) Ds

[B(p)

](5.8b)

= Ds[B(p)−1

] [Lµν + σµν(s, 0), Ds

[B(p)

] ]+ σµν(s, 0) (5.8c)

Mµν = Lµν + Sµν (5.8d)

= Ds[B(p)−1

] [Lµν + σµν(s, 0)

]Ds[B(p)

](5.8e)

Here Eq. (5.8e) follows directly from the (5.8b) form of Sµν . Note thatin (5.8a–5.8e), B(p) and Ds

[B(p)

]may be replaced by the corresponding

operator functions of P , namely, B(P ) and Ds[B(P )

].

5.1.4 Spin

Let Λ(p) = Λ[B(p)]. The Pauli-Lubanski vector (4.8a) obeys

wµ = 12εµνλρ pνMλρ = 1

2εµνλρ pνSλρ (5.9a)

= 12

Ds[B(p)−1

]εµνλρ pνσλρ(s, 0) Ds

[B(p)

](5.9b)

= 12εµνλρ pνσλ′ρ′(s, 0) Λ(p)−1λ

′λ Λ(p)−1ρ

′ρ (5.9c)

= εν0ij 12mσij(s, 0) Λ(p)−1νµ (5.9d)

Spin operator:

S = (S1, S2, S3) (5.10a)

Si ≡1

mwµ Λ(p)µi = 1

2εijk σjk(s, 0) (5.10b)

w·w = −m2 S ·S (5.10c)

35

The definition is justified by the action

〈p, λ|Si ψ〉 = Siλλ′ 〈p, λ′ |ψ〉 (5.11a)

〈p, λ|S3 ψ〉 = λ 〈p, λ|ψ〉 (5.11b)⟨p, λ∣∣ (S1 ∓ iS2

)ψ⟩

=√

(s∓λ)(s±λ+1) 〈p, λ±1|ψ〉 (5.11c)

Recall that[P µ, wν

]= 0, and note that

[P µ, Si

]= 0.

5.1.5 Helicity

The helicity operator is defined as the projection of the total angular mo-mentum operator J = L + S onto the direction of the three-momentum,P·J/|P |. From definition (4.8a), the time component of the Pauli-Lubanskivector is related to the helicity:

w0 = −12εijk P jMjk = −P ·J (5.12)

Since P · L = 0, Eq. (5.8e) gives an expression that relates the helicityoperator to the spin operator (5.10b):

P ·J = Ds[B(p)−1

]p·S Ds

[B(p)

]= Ds

[U(p)∗

]p·S Ds

[U(p)

](5.13)

The last equality follows because, from Eq. (4.10b), the exponential formof the (s, 0) representation of the boost in B(p) is

Ds[A(p)

]= exp(λp·S/|p|) (5.14)

which commutes with p·S.

5.1.6 Canonical convention

There are two common conventions for the rotation U(p) in the definitionof B(p). This section covers the canonical convention:

U(p) = I B(p) = A(p) =√p·σ/m (5.15)

In this case the rotation part of Sµν simplifies:

Si = 12εijk σjk(s, 0) = Si (5.16)

36

Proof. The most direct argument11 is to restrict the transformation law(5.2c) to SU(2), and to use the rotation covariance of SL(2, C) boosts:

Ds(√

p·σ/m U√R−1p·σ/m

)= Ds

(√p·σ/m U

√R−1p·σ/m U∗U

)= Ds

(√p·σ/m

√p·σ/m U

)= Ds(U) (5.17)

Hence

〈p, λ|U(U)ψ〉 = Ds(U)λλ′⟨R−1p, λ′

∣∣ψ⟩ (5.18)

from which the spin part of the rotation generator is clearly the rotationgenerator of the (s, 0) representation.

We sketch a more involved approach. Think of the commutator inEq. (5.8c) as the action of the total generator on the rotation-covariant,second rank spinor function of p:

Ds(√

p·σ/m)λ

ρ ∈ F ⊗ C2s+1 ⊗ C2s+1

The above spinor function is invariant if R−1 is applied to its momentumargument at the same time that Ds(U)⊗Ds(U)−1 T is applied to its spinorindices. In this language, the total rotation generator must be zero, whichmeans the commutator in (5.8c) must vanish for the canonical conventionwhen restricted to rotations. This argument can be extended to covariantspinor functions, relative to either SU(2) or SL(2, C), of any rank with anynumber of vector arguments.

5.1.7 Helicity convention

The helicity convention for the rotation in B(p) obeys:

U(p)σ3 U(p)∗ = p·σ/|p| (5.19)

11Which I learned from Greg Weeks, private commnunication, 2012. I am astonishednot to have known this fact before.

37

This of course determines U(p) only up to a rotation about the 3-axis.From Eq. (5.13) it follows that in the helicity convention the helicity

operator is the third component of the spin operator:

P ·J = S3 (5.20)

5.2 Spinor representation

The spinor representation factors the transformations to and from the par-ticle rest frame out of the spin little group to produce a simple, (2s+1)-component spinor action, and adjusts the Hilbert-space metric to maintainthe unitarity of the Poincare group representation. As far as we know,this elegant idea originated independently with Henry Stapp [14] and HansJoos [15].12

12It was a key idea for Stapp’s M-function formulation of analytic S-matrix theory,in particular for his proof of the CPT theorem, and the connection between spin andstatistics. Joos presented it as a logical development in his definitive treatment of therepresentation theory of the Poincare group. Neither author was aware of the other’swork at the time of writing. Both works quickly became widely recognized.

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6 Zero mass particle

6.1 Wigner representation

6.2 Zwanziger representation

39

References

[1] Eugen Wigner, Gruppentheorie und ihre Anwendung auf die Quanten-mechanik der Atomspektren, (Friedrich Wieweg und Sohn, Akt.-Ges.,Braunschweig, 1931).

[2] Eugene P. Wigner, Group Theory, (Academic Press, New York, 1959).

[3] E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra,corrected edition (1st edition 1935), (Cambridge University Press, NewYork, 1951).

[4] Morris Edgar Rose, Elementary Theory of Angular Momentum, (JohnWiley & Sons, Inc., New York, 1957).

[5] John M. Blatt and Victor F. Weisskopf, Theoretical Nuclear Physics,(John Wiley & Sons, Inc., New York, 1952).

[6] A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd edi-tion (1st edition 1957), (Princeton University Press, Princeton, 1960).

[7] U. Fano and G. Racah, Irreducible Tensorial Sets, (Academic Press,New York, 1959).

[8] Albert Messiah, Mecanique quantique, tome 2, (Dunod, 1959).

[9] , Quantum Mechanics, vol. II, (North-Holland Publishing Com-pany, 1962).

[10] Eugen Merzbacher, Quantum Mechanics, 3rd edition (1st edition 1961,2nd edition 1970), (John Wiley & Sons. Inc., New York, 1998).

[11] Claude Cohen–Tannoudji, Bernard Diu, and Franck Laloe, Mecaniquequantique, tomes I, II, 2e edition (1ere edition 1973), (Hermann, Paris,1977).

[12] , Quantum Mechanics, vols. I, II, (John Wiley & Sons, NewYork, 1977).

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[13] David N. Williams, Construction of Invariant Scalar Amplitudes with-out Kinematical Singularities for Arbitrary-Spin Nonzero-Mass Two-Body Scattering Processes, (Ph.D. Dissertation, University of Califor-nia, Berkeley, 1963), Lawrence Radiation Laboratory Preprint UCRL11113, OSTI 4090663.

[14] Henry. P. Stapp, “Derivation of the CPT Theorem and the Connectionbetween Spin and Statistics from Postulates of the S-Matrix Theory”,Phys. Rev. 125, 2139–2162 (1962).

[15] Hans Joos, “Zur Darstellungstheorie der ihomogenen Lorentzgruppeals Grundlage quantenmechanischer Kinematik”, Fortschr. Phys. 10,65-146 (1962).

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