Photon Quantization with Lorentz Violation
Transcript of Photon Quantization with Lorentz Violation
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Photon Quantization with Lorentz Violation
Don Colladay
New College of Florida
Talk presented at Miami 2012
(work done in collaboration with Patrick McDonald)
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Overview of Talk
• Review of Gupta-Blueler Method of Photon Quantization
• Application to SME Photon Sector
• Issues with momentum-space expansions
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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
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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)
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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...)
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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
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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
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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
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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
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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)
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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
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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
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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 )
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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
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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...)
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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)
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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...
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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
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Plot of energy surfaces are perturbed spheres
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Other solution looks like
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Can get overlapping energy surfaces near cusp points
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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
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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...
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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