Interacting Bosons and Fermions in 3D Optical Lattice Potentials
Particle Physics of the early Universe€¦ · Lecture to the electrodynamics – a system of...
Transcript of Particle Physics of the early Universe€¦ · Lecture to the electrodynamics – a system of...
Particle Physics of the early Universe
Alexey BoyarskySpring semester 2015
Why do you need that?
Plan of today’s lecture
Let us apply the formalism that we developed in the previousLecture to the electrodynamics – a system of interacting fermionsplus dynamical electromagnetic field
We will consider three different cases:
– Electron is scattering in the external electromagnetic field– Electron is scattering on the (dynamical) proton– Electron is scattering on its own anti-particle (positron)
Alexey Boyarsky PPEU 1
Electron scattering in Coulomb field
In non-relativistic quantum mechanics if Hamiltonian has the formH = H0+ V then the probability of transition between an initial stateψi(x) and the final state ψf(x) of unperturbed Hamiltonian H0 isgiven by (Landau & Lifshitz, vol. 3, § 43):
dwif =2π
~|Vif |2δ(Ei − Ef)dνf (1)
where |Vif | is the matrix element between initial and final states; and dnf is thenumber of final states with the energy Ef (degeneracy of the energy level).
The interaction that perturbs the Hamiltonian is given by
Vint = eγ0γµAµ(x) (2)
recall that electric current jµ = eψ(x)γµψ(x)
0Following Bjoren & Drell, Sec. 7.1
Alexey Boyarsky PPEU 2
Electron scattering in Coulomb field
If we consider static point source with the Coulomb field
A0(x) =Ze
4π|x|, ~A = 0 (3)
and wave-functions1
ψi(x) = C1us(pi)eipi·x , ψf(x) = C2ur(pf)e
ipf ·x (4)
Find C1 that ψi in (4) has the correct normalization per unit flux in the relativisticcase. Normalization of u is given by (8)
Using (2–4) we write the matrix element
Vif = C1C2ur(pf)γ0us(pf)
∫d
3xe
ix·(pi−pf )A0(x) (5)
1Here us, ur are 4-component spinors – solution of the Dirac equations (γ · p − m)us = 0,ur(γ · p+m) = 0, s = ±, r = ± – polarizations of spin.
Alexey Boyarsky PPEU 3
Electron scattering in Coulomb field
Degeneracy of a final state with Ef is given by
dνf = 2× d3pf(2π)3
∫p0>0
dp0 δ(p2 −m2)
︸ ︷︷ ︸density of states
=d3pf
(2π)3Ef(6)
Eq. (6) and normalization of the final state ψf should agree witheach other in such a way that∫
dνf ψf(x)ψf(x′) = δ(3)(x− x′) (7)
Notice that this condition fixes the normalization of the spinor uf inEq. (4).
us(pi) and ur(pf) carry the information about spin-polarizations ofinitial and final states. We can sum over these states (i.e. the
Alexey Boyarsky PPEU 4
Electron scattering in Coulomb field
experiment does not measure the polarizations). This can be doneusing the identity
∑s
us(p)us(p) =
(/p+m
2m
)(8)
(take into account (/p−m)(/p+m) = p2 −m2)
As a result we get
dwif = 2π|Vif |2δ(Ei − Ef)d3pf
(2π)3Ef
= Z2(4πα)
2|C1|2|C2|2|ur(pf)γ0us(pi)|2
|pi − pf |4d3pf
(2π)3Efδ(Ei − Ef)
(9)
Alexey Boyarsky PPEU 5
Electron scattering in Coulomb field
as a result the sum over initial and final states in (9) becomes
∑r,s
|ur(pi)γ0us(pf)|2 =
∑r,s
urγ0ususγ
0ur = Tr
(γ
0/pi +m
2mγ
0/pf +m
2m
)
the spin sum rule leads toaveraging over initial polarizations
1
2
∑s
∑r
|urγ0us|2 ≡ F (pi, pf ,m) = (EiEf + pipf +m
2)/2m
2 (10)
summing over the final polarizations
Using () and representing d3pf = dΩp2fdpf we find the differential
cross-sectiondσ
dΩ=
Z2α2mEi
4|pi|4 sin4(θ2)F (pi, pf ,m) (11)
– which coincides with the familiar Rutherford scattering (up tothe function F (. . . ) that depends on the spins/polarizations of theparticles
Alexey Boyarsky PPEU 6
Electron scattering on proton
Consider next the situation when the electromagnetic field iscreated by other particle (“proton”)
While the formulas (4)–(6) remain true, the expression for Aµchanges
Notice that in general Aµ(x) created by the moving proton will betime-dependent so instead of Eq. (1) the formula
dwif =2π
~|Vif |2δ(Ei − Ef − q0)dnf (12)
where q0 is the frequency of the perturbation Aµ(t,x) ∝ e−iq0t
If proton is described by a spinor
Ψ = U(P )e−iP ·x, U(P )–4 component spinor (13)
Alexey Boyarsky PPEU 7
Electron scattering on proton
then its electric current is
Jµ(y) = Ψf(y)γ
µΨi(y) (14)
(the form of Ψi and Ψf is the same as Eq. (4) with m → Mp anddifferent momenta)
The electromagnetic field obeys the Klein-Gordon equation
Aµ = Jµ or Aµ(x) =1
Jµ(y) (15)
Very naively, the operator −1 (inverse to the Klein-Gordonoperator) can be easily constructed if one considers (15) in Fourierspace:2
p2Aµ(q) = Jµ(q) or Aµ(q) =
1
q2Jµ(q) (16)
2We denote by Aµ(p) and Jµ(p) Fourier transform
Alexey Boyarsky PPEU 8
Electron scattering on proton
where
Jµ(q) ≡∫d
4x e−iq·x
Ψf(x)γµΨi(x) (17)
=
√√√√ M2p
E(p)i E
(p)f
Ur(Pi)γµUs(Pf)
∫d
4x e−i(q−Pf−Pf )·x (18)
=
√√√√ M2p
E(p)i E
(p)f
Ur(Pi)γµUs(Pf)δ(4)
(q − Pf − Pf) (19)
Notice that in Eq. (5) we only need Aµ(pi − pf). The resultingexpression is then equivalent to (5) if one substitutes
γ0 Z
|q|2→ γ
µ 1
q2
√√√√ M2p
E(p)i E
(p)f
Ur(Pi)γµUs(Pf)δ(4)
(q − Pf − Pf) (20)
where q = pf − pi – transferred 4-momentum and q is its spatial component.4-spinors Us and Ur are the in- and out- 4-spinors of a proton.
Alexey Boyarsky PPEU 9
Electron scattering on proton
From Eq. (5) and (20) we find
Vif =
∫d3x
∫d3q (uγµu)e−i(q−pi+pf)·xAµ(q) (21)
where Aµ(q) is given by r.h.s. of (20):
The resulting Vif is proportional to
Vif ∝ δ3(pi − pf + P i − P f) (22)
That is the result looks like a scattering of electron in externalfield (9) where the external field Aµ is the field created by the proton(14).
As a result the probability dwif has the form very similar to Eq. (9)
Alexey Boyarsky PPEU 10
Electron scattering on proton
for Z = 1 and
dwif =(. . . )
∣∣ur(pf)γµus(pi)∣∣2∣∣Ur′(Pf)γµUs′(Pi)∣∣2|pi − pf |4
× δ(Ei + E(p)i − Ef − E
(p)f )δ
3(pi − pf + P i − P f)
(23)
Alexey Boyarsky PPEU 11
Interaction of light with the Dirac sea
Interaction of charged particles goes via exchange of virtual photonquanta
Alexey Boyarsky PPEU 12
Electron-positron scattering3
+Electron-positron scattering
Two differences betweenpositron and proton. One istrivial – mass (not important forour computations so far).
blue diagram is similar to whatwe have seen before (electron-proton scattering with proton →positron).
Another difference: electron plus positron can convert into aphoton
red diagram : electron+positron disappear in the intermediate stateand only a new particle (photon) stays
⇒need a quantum theory of photon2See Bjoren & Drell, Sec. 7.9
Alexey Boyarsky PPEU 13
Electromagnetic field as collection ofoscillators 4
Consider the solution of wave equation for vector potential ~A ( impose
A0 = 0 and div ~A = 0)
1
c2∂2A
∂t2−∆A = 0 (24)
Solution
A(x, t) =
∫d3k
(2π)3
[ak(t)eik·x + a∗k(t)e−ik·x
](25)
where the complex functions ak(t) have the following time dependence
ak(t) = ake−iωkt, ωk = |k| (26)
3See Landau & Lifshitz, Vol. 4, §2
Alexey Boyarsky PPEU 14
Generalized coordinate and momentum
ak(t) and a∗k(t) obey the following equations:
ak(t) = −iωkak(t) , a∗k(t) = iωka∗k(t) (27)
Notice that we re-wrote partial differential equation (25) secondorder in time into a set of ordinary differential equations for infiniteset of functions ak(t) and a∗k(t). Any solution of the freeMaxwell’s equation is parametrized by the (infinite) set of complexnumber ak,a
∗k
To make the meaning of Eqs. (30) clear, let us introduce theirimaginary and real parts.
Qk ≡ ak + a∗k
P k ≡ −iωk
(ak − a∗k
) dynamics=⇒
Qk = P k
P k = −ω2kQk
(28)
Alexey Boyarsky PPEU 15
Hamiltonian of electromagnetic field
Hamiltonian (total energy) of electromagnetic field is given by
H =1
2
∫d3x
[E2 +B2
]=
1
2
∫d3k
(2π)3
[E2
k +B2k
](29)
Using mode expansion (25) and definition (28) we can write withfrequencies ωk
H[Qk,P k
]=
1
2
∫d3k
(2π)3
[Pk
2 + ω2kQk
2]
(30)
Therefore dynamical equations (28) are nothing by the Hamiltonianequations
Qk =∂H∂Pk
, Pk = − ∂H∂Qk
(31)
with Hamiltonian (30)
Alexey Boyarsky PPEU 16
Hamiltonian of electromagnetic field
Eqs. (30)–(31) describe Hamiltonian dynamics of a sum ofindependent oscillators with frequencies ωk
Classical electromagnetic field can be consideredas an infinite sum of oscillators with frequencies ωk
Recall: for quantum mechanical oscillator, described by theHamiltonian
Hosc = − ~2
2m
d2
dx2+mω2
2x2 (32)
one can introduce creation and annihilation operators:
a† =1√
2~mω(mωx+ ~∂x) ; a =
1√2~mω
(mωx− ~∂x) (33)
Commutation [a, a†] = 1
Hamiltonian can be rewritten as Hosc = ~ω(a†a+ 12)
Alexey Boyarsky PPEU 17
Properties of creation/annihilationoperators
Commutation [a, a†] = 1
If one defines a vacuum |0〉, such that a |0〉 = 0 (Fock vacuum) thena state |n〉 ≡ (a†)n |0〉 is the eigenstate of the Hamiltonian (32) withEn = ~ω(n+ 1
2), n = 0, 1, . . .
Given |n〉, n > 0, a† |n〉 =√n+ 1 |n+ 1〉 and a |n〉 =
√n |n− 1〉
Time evolution of the operators a, a†:
i~∂a
∂t= [Hosc, a] (34)
and Hermitian conjugated for a†
Alexey Boyarsky PPEU 18
Birth of quantum field theory
Dirac (1927) proposes to treat radiation as a collection of quantumoscillators
Paul A.M. Dirac Quantum theory of emission and absorption of radiation
Proc.Roy.Soc.Lond. A114 (1927) 243
Take the classical solution (25)
A(x, t) =
∫d3k
(2π)3
[ak(t)eik·x + a∗k(t)e−ik·x
]
Introduce creation/annihilation operators ak, a†k
[ak, a†p] = ~δk,p [ak, ap] = 0 (35)
Alexey Boyarsky PPEU 19
Birth of quantum field theory
Replace Eq. (25) with a quantum operator
A(x, t) =
∫d3k
(2π)3
[ak(t)eik·x + a†k(t)e−ik·x
](36)
Operator a†k creates photon with momentum k and frequencyωk
Operator ak destroys photon with momentum k andfrequency ωk (if exists in the initial state)
State without photons↔ Fock vacuum:
ak |0〉 = 0 ∀k (37)
Alexey Boyarsky PPEU 20
Birth of quantum field theory
State with N photons with momenta k1,k2, . . . ,kN :
|k1,k2, . . . ,kN〉 = a†k1a†k2
. . . a†kN |0〉 (38)
Let us find the average of the operator (36) between a vacuum andthe 1-photon state |k〉 ≡ a†k |0〉 (recall that we are working in the gauge A0 = 0)
〈0| A(x)(a†k |0〉
)=ε(k)√2ωk
e−iωkt+ik·x (39)
where the polarization 3-vectors ε(k) are such that ε · k = 0 andε2 = 1'
&
$
%
Quantum electrodynamics (QED) – first quantum field theory hasbeen created. Free fields with interaction treated perturbatively infine-structure constant: α = e2
~c
Alexey Boyarsky PPEU 21
Second order perturbation theory
The Dirac Hamiltonian H0 is perturbed by V given byV = γ0γµAµ(x)
Let us start with the blue diagram
We could have proceeded as in the case of electron-protoninteraction – find the electromagnetic field, created by the positronand compute the scattering of electron in this field. Instead, we dothis computation in a different way, and then demonstrate that theresult is the same
A full system of intermediate states |n〉 contains both electron,positron and photon. That is– the initial state |i〉 = |e−(p1), e+(q1)〉 ⊗ |0〉– the final state |f〉 = |e−(p′1), e+(q′1)〉 ⊗ |0〉– and the intermediate state |n〉 = |e−(p′1), e+(q1)〉 ⊗ |k〉(momentum of electron in the intermediate state is equal to its final momentum!)
Alexey Boyarsky PPEU 22
Computation of the matrix element
u(p1) v(q1)
u(p′1) v(q′1)
Intermediate 3-particle state
– the initial state |i〉 = |e−(p1), e+(q1)〉 ⊗ |0〉.
The energy of initial state: Ei = E(q1) + E(p1)
– The intermediate state |n〉 = |e−(p′1), e+(q1)〉⊗|k〉.
The energy of intermediate state: En =E(q1) + E(p′1)± ωk
– The final state |f〉 = |e−(p′1), e+(q′1)〉 ⊗ |0〉.
The matrix element for the process can be described as follows:
Mif =
∫d3k
(2π)3〈i| V |n〉 1
Ei − En〈n| V |f〉 (40)
The matrix element Vin ≡(〈0| ⊗ 〈e+e−|
)V(|e+e−〉 ⊗ |k〉
)is given
by the following expression
Alexey Boyarsky PPEU 23
Computation of the blue diagram
Vin =u(p′1)γ
µu(p1)
εµ√2ωk
∫d
3x e
i(p1−p′1−k)·x
ei(E(p1)−ωk−E(p′1)
)t+ ωk → −ωk
=u(p′1)γ
µu(p1)
εµ√2ωk
δ(3)
(p1 − p′1 − k)e
i(E(p1)−ωk−E(p′1)
)t+ ωk → −ωk
(41)(the positron state is the same in |i〉 and |n〉 and therefore
⟨e+∣∣ ∣∣e+
⟩= 1)
Notice that the expression (41) contains explicit time dependence as E(p1) 6=ωk + E(p′1). This is the indication of the fact that 3-momentum and energycannot be conserved at the same time.
Similarly Vnf = δ(3)(q1−q′1 +k)v(q1)γµv(q′1)εµ√2ωk
ei(E(q1)+ωk−E(q′1))t
As a result Mif is given by (after the integral over dn ≡ d3k(2π)3):
Mif =
∫d3k
(2π)3
VinVnfEi − En
=
Alexey Boyarsky PPEU 24
Computation of the blue diagram
Mif = ei(Ei−Ef)tδ(3)(p′1 + q′1 − p1 − q1)(v(q1)γµv(q′1))
(u(p′1)γµu(p1)
)(E(p1)− E(p′1)
)2 − ω2k
∣∣∣∣k=p1−p′1
(42)
Show that the denominator of (42) is equal to 1k2 where kµ is 4-vector equal to
the difference (p1 − p′1)µ.
notice, that time dependence will cancel out from (42) when we takeinto account δ(Ei − Ef)
Blue diagram can be thought of as 2nd order perturbation theorywith the 3-particle intermediate state |n〉, containing photon.
Notice that the matrix element (42) coincides with the matrixelement (22) (taking into account (21))
Alexey Boyarsky PPEU 25
Computation of the red diagram
There is another intermediate state where there is only photon —red diagram
In this case we have different intermediate state: |n〉 = |0〉 ⊗ |k〉.The energy of intermediate state: En = ±ωk
Matrix element
Vin ≡⟨e+e−
∣∣ V |k〉 = v(q1)γµu(p1)εµ√2ωk
∫d3x ei(p1+q1−k)·xei(Ei−ωk)t
=v(q1)γµu(p1)εµ√2ωk
δ(3)(p1 + q1 − k)ei(Ei−ωk)t + ωk ↔ −ωk(43)
where the initial energy Ei = E(p1) + E(q1) .
Similarly, for the Vnf element we get
Vnf = u(p′1)γµv(q′1)εµ√2ωk
δ(3)(p′1 + q′1 − k)ei(ωk−Ef)t (44)
Alexey Boyarsky PPEU 26
Computation of the red diagram
As a result, we get:
Mif =
∫dn
VinVnf
Ei − En
= ei(Ei−Ef )t
δ(3)
(p′1 + q
′1 − p1 − q1)
(v(q1)γµu(p1))(u(p′1)γ
µv(q′1))
E2i − ω2
k
∣∣∣∣k=p1+q1
(45)
. . . here Ei = E(p1) + E(q1) – initial energy of electron+positron,k = p1 + q1 – their total momentum of the pair
. . . again one can show that E2i −ω2
k = k2 where k is a 4-momentumof the intermediate photon, k = p1 + q1
Alexey Boyarsky PPEU 27
Total matrix element
Amplitudes of two processes (blue and red on the Figure) shouldbe added together before | . . . |2 is taken. That is the probability ofthe process is proportional to |M1+M2|2 rather than |M1|2+|M2|2(interference terms are present)
|M|2 = (. . . )
∣∣∣∣∣∣∣∣∣∣i
(u(p′1)γµu(p1)
)(v(q1)γµv(q′1)
)(p1 − p′1)2︸ ︷︷ ︸blue diagram
− i(u(p′1)γµv(q′1)
)(v(q1)γµu(p1)
)(p1 + q1)2︸ ︷︷ ︸red diagram
∣∣∣∣∣∣∣∣∣∣
2
(46)
where . . . is a prefactor, depending on energies/masses of particles.
Perturbative series in V
Mif = M(1)if +M
(2)if + . . .
We saw that M (1)if is equal to zero (Eq. (1)). If non-zero, M (1)
if ∝ e
(charge in V )
Alexey Boyarsky PPEU 28
Total matrix element
We found M (2)if (2nd order perturbation theory in V )
This expression M (2)if is proportional to e2
This parameter e is known experimentally to be small. Theexpansion parameter of electron-photon interaction is known as thefine structure constant α ≡ e2
~c ≈ 1137
Indeed, consider next order in expansion. It includes moreintermediate states – higher order in e
Alexey Boyarsky PPEU 29
Next order
There are e+e− scattering processes that go through severalintermediate states:
M(3)if =
∫dn1
∫dn2
Vin1Vn1n2Vn2f
(Ei − En1)(En1 − En2)
M(4)if =
∫dn1
∫dn2
∫dn3
Vin1Vn1n2Vn2n3Vn3f
(Ei − En1)(En1 − En2)(En2 − En3)
Alexey Boyarsky PPEU 30
Virtual particles
In the above computations we have explicitly separate space andtime.
The intermediate states were “physical” (energy and momentumwere related via E2 = p2 +m2), but only 3-momentum conservationwas imposed in every vertex. The total (initial - final) energy wasconserved, but for intermediate states it was not
It is possible to construct explicitly Lorentz-invariant technique forcomputation of such matrix elements
This is called Feynman technique. Its rules are presented in
Alexey Boyarsky PPEU 31
Feynman rules
There are three types of objects in constructing Feynman graphs:
– external lines (real particles)– internal lines (virtual particles)– vertices (interaction points)
To each external fermion line one associates a spinor u, v , etc.according to the following rule:
To each external photon line one associates a polarization vector
Alexey Boyarsky PPEU 32
Feynman rules
Each virtual line adds a propagator:
– Virtual fermion:
SF (p) =i(/p+m)
p2 −m2 + iε
– Virtual photon:
Dµν(p) =−iηµνp2 + iε
Each vertex (two fermion lines plus one photon line) receives afactor −ieγµ
Alexey Boyarsky PPEU 33
Feynman rules
Energy-momentum conservation is imposed at every vertex
Alexey Boyarsky PPEU 34
Electron-positron scattering
Let us repeat the computation of electron-positron scattering
u(p1) v(q1)
u(p′1) v(q′1)
Dµν(p1 − p′1)
M =
(u(p′1)(−ieγµ)u(p1)
)(v(q1)(−ieγµ)v(q′1)
)(p1 − p′1)2
Similarly
M =
(u(p′1)(−ieγµ)v(q′1)
)(v(q1)(−ieγµ)u(p1)
)(p1 + q1)2
Alexey Boyarsky PPEU 35
Compton scattering
Write general form of matrix elements for Compton scattering
Derive differential cross-section in non-relativistic case (photonenergy ω me) and in ultra-relativistic case (ω me)
Peskin & Schroeder, Sec. 5.5
Alexey Boyarsky PPEU 36
Pair creation
Compute cross-section of γ + γ → e+ + e− (pair production)
Derive differential cross-section in ultra-relativistic case (ω me)
Peskin & Schroeder, Sec. 5.5
Alexey Boyarsky PPEU 37
Onther consequences of Dirac theory ofpositrons
Photons are bosons (particles of spin = 1). Electrons/positrons arefermions particles of spin = 1/2. Therefore, angular momentumconservation means that photon couples to electron + positron
Photons could produce electron-positron pairs. However, the
process γ → e+e− is not possible if all particle are “real” (i.e.
photon obeys E = cp, electron/positron E =√p2c2 +m2
ec4 – “on-shell
conditions”)
Instead, a pair of photons can produce electron-positron pair via
γ + γ → e+e− :γ
k1
k2
γ
e−
e+
p1
p2
where (k1 + k2)2 ≥ 4m2e
Similarly, electron-positron pair can annihilate into a pair of
Alexey Boyarsky PPEU 38
Onther consequences of Dirac theory ofpositrons
photons
Kinematically, the red electron is virtual (i.e. for it E 6=√p2c2 +m2c4
– check this)
γk1
k2
γ
γ
γ
k′2
k′1If energies of incoming photons aresmaller than twice the electron mass (i.e.(k1 + k2)2 < 4m2
e) photons produceonly virtual electron-positron pair whichcan then “annihilate” into another pair ofphotons – light-on-light scattering
Charge screening:
Alexey Boyarsky PPEU 39
Systems with many particles
Presence of the negative-energy levels means that you can createparticle-antiparticle pairs out of “nowhere”
Particles in the pair can be real, but they can be also virtual (i.e.E2 − p2 6= m2)
According to the Heisenberg uncertainty relation ∆E ∆t & 1, ifone measures the state of system two times, separated by a shortperiod ∆t 1/m, one will find a state with 1, 2, 3, ... additional pairs.
It means that we no longer work with definite number of particles:number of particles may change! (Contrary to non-relativisticquantum mechanics)
We need an approach that naturally takes into account states withdifferent number of particles (we will return to this point in thisLecture)
Alexey Boyarsky PPEU 40
Birth of quantum field theory
Quantum electrodynamics (QED) – first quantum field theory.Free fields with interaction treated perturbatively in fine-structureconstant: α = e2
~c
Divergencies? Many answers beyond tree-level (1st order)perturbation theory were infinite, because one had to sum upcontributions from infinite number of virtual particles with growingenergies Ep = ±
√m2 + p2
Alexey Boyarsky PPEU 41