Photon Quantization with Lorentz Violation

Don Colladay

New College of Florida

Talk presented at Miami 2012

(work done in collaboration with Patrick McDonald)

Overview of Talk

• Review of Gupta-Blueler Method of Photon Quantization

• Application to SME Photon Sector

• Issues with momentum-space expansions

Review of Gupta-Bleuler Quantization

Basic idea of Gupta-Bleuler is to add a gauge-fixing term to theLagrangian that allows for all four components of the vectorpotential to be quantized in a covariant manner

• Has advantage of maintaining Lorentz covariance explicitlyin the quantization procedure, a great advantage for calcu-lations

• Procedure introduces negative-norm states that must be re-moved using some condition on the physical Hilbert-spacestates

– S. Gupta, 1950; K. Bleuler, 1950

Formulation in the conventional case starts with Lagrangian

L = −1

4F2 −

λ

2(∂ ·A)2

where λ is a ”gauge-fixing” term

Calculation of the conjugate momenta yield

πj = F j0

and

π0 = −λ∂ ·A

(Note that π0 = 0 when λ = 0 destroying covariance)

Quantization is imposed using equal-time commutators

[Aρ(t, ~x), πν(t, ~y)] = iη νρ δ

3(~x− ~y)

and

[Aρ(t, ~x), Aν(t, ~y)] = [πρ(t, ~x), πν(t, ~y)] = 0

in analogy with Poisson-Bracket approach

Choosing λ = 1 (Feynman gauge) decouples the commutators

[Aρ(t, ~x), Aν(t, ~y)] = iηρνδ3(~x− ~y)

and

[Aρ(t, ~x), Aν(t, ~y)] = [Aρ(t, ~x), Aν(t, ~y)] = 0

(other choices for λ are not as simple...)

Expansion of the field in terms of mode operators yields

Aµ(x) =∫

d3~p

(2π)32p0

∑α

(aα(~p)εµα(~p)e−ip·x + a†α(~p)ε∗µα (~p)eip·x

)

• p0 = |~p| are positive frequency solutions to the dispersion

relation (p2 = 0)

• α = 0,1,2,3 runs through all four polarization vectors

• εµα vectors can be freely chosen as orthonormal basis

If εµα are chosen so that εµ0 = {1,0,0,0}, commutation relations

imply that mode operators satisfy

[aα(~p), a†β(~p′)] = −ηαβ2p0(2π)3δ3(~p− ~p′)

Due to the presence of the metric, a†0(~p) operators are uncon-

ventional and produce negative-norm states when they act on

vacuum

To eliminate them, impose a requirement on the physical states

〈ψ|(∂ ·A)|ψ〉 = 0, for |ψ〉 ∈ Hphysor, equivalently,

(∂ ·A+)|ψ〉 = 0

where (+) represents the positive-frequency part of the field

In terms of mode operators, the condition is

(a0(~p)− a3(~p))|ψ〉 = 0

provided the 3 direction is taken to point along the momentum

The hamiltonian takes the form

H =∫

d3~p

(2π)32p0

3∑α=1

a†α(~p)aα(~p)− a†0(~p)a0(~p)

the scalar polarization states create negative-energy states, but

the physical condition on Hphys protects the physical states from

any negative energy problems

SME Photon Sector (minimal extension, CPT-even terms)

SME photon lagrangian is given by

L = −1

4F2 + (kF )µναβF

µνFαβ − λ(∂ ·A)2

Calculation of πµ yields

πj = F j0 + kj0αβFαβ, π0 = −λ∂µAµ

Setting λ = 1 and imposing the quantization condition

[Aρ(t, ~x), πν(t, ~y)] = iη νρ δ

3(~x− ~y)

is commensurate with the Gupta-Blueler approach

Conversion to commutation relations involving A gives

[Ai(t, ~x), Aj(t, ~y)] = −iRijδ3(~x− ~y)

where Rij is the inverse matrix of δij − 2(kF )oioj, and the other

relations are conventional

It is convenient to put this into more covariant notation by set-

ting R00 = −1 and R0i = 0

[Aµ(t, ~x), Aν(t, ~y)] = −iRµνδ3(~x− ~y)

Calculation of the Hamiltonian H = πµAµ − L yields (after ap-propriate partial integrations in H =

∫d3~xH)

H =1

2

((∂jA

j)2 − (A0)2 +A0∂0(∂ ·A) + ~E2 + ~B2)

−k0i0jF0iF0j +

1

4kijklF

ijF kl

The commutation relations produce the expected action of thehamiltonian on the fields as the generator of time translations

i[H,Aµ] = ∂0Aµ

Similarly, the three-momentum operator

P i =∫d3~x

(πj∂iAj − (∂ ·A)∂iA0

)satisfies

i[P i, Aµ] = ∂iAµ

indicating that the cannonical quantization works

Gauge condition can be implemented by restricting physical state

space Hphys so that

〈ψ|(∂ ·A)|ψ〉 = 0, for |ψ〉 ∈ HphysGauge-terms then drop out of the expectation value of the hamil-

tonian and momentum as in the conventional case

Useful classification of kF terms uses type of ”dual”

(kF )µναβ =1

4εµνρσεαβγδ(kF )ρσγδ

Can split into

kF = kSDF ⊕ kASDF

• Self-dual components → no birefringence and Tr(kF ) 6= 0

• Anti-self-dual components → birefringence and Tr(kF ) = 0

(Also Weyl-tensor part of kF )

Experimental bounds on these

• birefringent terms can be bounded using cosmological tests

at 10−32 level

• non-birefringent terms bounded using laboratory scale exper-

iments at 10−14 - 10−17 level

See Data Tables for Lorentz and CPT violation for details

– A. Kostelecky and N. Russell, arXiv:0801.0287

The self-dual kF components can be handled using a coordinate

redefinition (at lowest-order in the free photon theory) and are

therefore not particularly interesting from a theoretical point of

view

Much more interesting are the anti-self-dual terms that lead to

birefringence effects, these terms cause fundamental issues in

the usual Fourier expansion technique (next part of talk...)

Issues with momentum-space expansion

The conventional expansion of the field takes form

Aµ(x) =∫

d3~p

(2π)3

∑α

1

2p0

(aα(~p)εµα(~p)e−ip·x + a†α(~p)ε∗µα (~p)eip·x

)where the p0-factor has a dependence on α, due to the birefrin-

gence

Plugging this into the commutator yields terms of the form

∑α,β

(1

p0′

)εµα(~p)ενβ(~p′)[aα(~p), a†β(~p′)]e−i(p·x−p

′·y)

To eliminate time dependence and generate the delta function,

let the commutator take the usual form

[aα(~p), a†β(~p′)] = −ηαβ2p0(2π)3δ3(~p− ~p′)

implies condition on ε vectors∑α,β

ηαβεµα(~p)ενβ(~p) = −Rµν

Analogy: Vierbein formalism in general relativity looks like this...

Special case to see if this condition feasible:

Let k0103 be anti-self-dual (birefringent) and related symmetric

components be only nonzero terms.

Energies are determined by a polynomial of form

p4(f(p20)) = 0

where f is a second-degree polynomial in p0 yielding two solutions

Plot of energy surfaces are perturbed spheres

Other solution looks like

Can get overlapping energy surfaces near cusp points

If stay away from this area, polarization sum seems to work fine

For example, if let ~p point in the 2-direction it is possible to

choose the normalization of the polarization vectors such that∑α,β

ηαβεµα(~p)ενβ(~p) = −Rµν

holds in agreement with the commutation relations

However, there can be problems where the degeneracies occur,

for example, if let ~p point in the 3-direction, one of the physical

modes becomes degenerate with the gauge mode

In this case, there are only three polarization vectors, an insuffi-

cient number to satisfy the commutation rules

It may be the case that this type of singular behavior only occurs

on a ”set of measure zero” in momentum space, but we are still

working on proving this...

Conclusions

• Gupta-Blueler quantization of the SME photon sector ap-

pears to work well when coordinate-space functions are con-

sidered

• It is unclear how to expand the momentum-space functions

to impose the quantization condition consistently for all di-

rections of the momenta

• We conjecture that ”problem points” may be a set of mea-

sure zero so that the quantization may be imposed consis-

tently on almost all of momentum space