Spontaneous Lorentz Violation: The Case of Infrared QED

It is by now clear that infrared sector of QED has an intriguingly complex structure. Based on earlier pioneering works on this subject, two of us recently proposed a simple modification of QED by constructing a generalization of the $U(1)$ charge group of QED to the"Sky"group incorporating the known spontaneous Lorentz violation due to infrared photons, but still compatible in particular with locality. There it was shown that the"Sky"group is generated by the algebra of angle dependent charges and a study of its superselection sectors has revealed a manifest description of spontaneous breaking of Lorentz symmetry. We further elaborate this approach here and investigate in some detail the properties of charged particles dressed by the infrared photons. We find that Lorentz violation due to soft photons may be manifestly codified in an angle dependent fermion mass modifying therefore the fermion dispersion relations. The fact that the masses of the charged particles are not Lorentz invariant affects their spin content too.Time dilation formulae for decays should also get corrections. We speculate that these effects could be measured possibly in muon decay experiments.


Introduction
In quantum field theory (QFT), observables are local and generate the algebra of local observables A . In contrast, time evolution and global symmetries are not local. In Lagrangian field theories, their generators involve integrals of fields over all of space. They are thus not elements of A , but instead generate automorphisms of A . They are elements of the automorphism group AutA of A .
The group of automorphisms of A generated by conjugation using unitary elements of A is the group InnA of inner automorphisms. It is a normal subgroup of AutA . The group AutA /InnA is the outer automorphism group OutA .
The automorphisms generated by some global symmetries may be equivalent to inner ones. If that is not the case, then they define non-trivial elements of OutA .
In QFT, we choose an irreducible representation ρ or a superselection sector of A . It can happen that a global symmetry transformation cannot be implemented by a unitary or antiunitary operator in the representation space H of A . In that case, the symmetry is said to be spontaneously broken. Since elements of A act by definition on H , InnA cannot be spontaneously broken. So that can happen only if S / ∈ InnA . Spontaneous breaking by a Higgs field can be understood in this framework. Thus no local operation can change the asymptotic expression φ ∞ of the Higgs field φ. Hence φ ∞ is a label for the representation ρ. If a symmetry changes φ ∞ , it changes the representation.
Hence it is spontaneously broken.
The mechanism of spontaneous breaking can be illustrated even with the 3 × 3 matrix algebra M 3 (C). In its irreducible representation ρ, which is three-dimensional, its unitary subgroup U 3 is represented irreducibly by a representation we can name as 3. Complex conjugation is an automorphism of M 3 (C). It changes 3 to the inequivalent representation3 and hence ρ to the inequivalent representationρ. This automorphism is thus spontaneously broken in the representation ρ.
The Poincaré group is an automorphism group of the observables of QED. It transforms elements of A nontrivially. Instead, the electric charge Q, at least classically, is the total electric flux at infinity and hence commutes with all elements of A . Its different values q go into the labels for the different irreducible representations of A . They are similar to the Casimir operators of Lie algebras. Since Q is Poincaré invariant, Poincaré symmetry is compatible with charge superselection. The latter does not cause spontaneous Poincaré violation.
In QED, infrared photons accumulate at infinity and create a non-zero electromagnetic field f µν there as nicely shown by Buchholz [4] and Fröhlich et al. [1,2]. Since local operations cannot affect f µν , this field also labels superselection sectors. But f µν is not Lorentz invariant. Therefore Lorentz symmetry is spontaneously broken in QED.
The group U (1) of QED is based on electric charge which classically is a measure of the net flux of electric field E (E i = f 0i ) on the sphere S 2 ∞ at infinity. In previous work, we extended U (1) to the "Sky" group G sky which is sensitive to all the partial waves of E on S 2 ∞ . It is superselected because f 0i is. But elements of G sky /U (1) are not Lorentz invariant and cause spontaneous Lorentz violation.
In quantum theory, the infrared cloud on S 2 ∞ is incorporated in the state vectors, the charged states being dressed by a coherent state of photons [1,2,12,13]. We have constructed the dressing operator in [5] using a closed form ω. Previous work [1,2] determine the coherent state of the photon and hence ω. Thus we can write the dressed charged particle states. A simple twist of the electron (or charged particle) mass is sensitive to the coherent state. The twisted mass is not Lorentz invariant. It affects the dispersion relation, the spin content of the particle and time delays in decays. These can be measured. Naturally it also leads to Lorentz non-invariant scattering amplitudes in charged sectors which too can be measured.
In section 2, we recall the vertex operator dressing the charged vector states with a cloud of infrared photons. It is gauge, but not in general Lorentz invariant. A new result we describe is its incorporation in the Lagrangian. We can set it equal to 1 by adding the term proportional to F ∧ ω, F : Electromagnetic field , ω : closed two-form , to the action. It is remarkably close to the θ-term θ 8π 2 Tr F ∧ F, F : curvature two-form , (1.2) in QCD which induces P and T violation. Just like the θ-term, (1.1) is a surface term and does not affect equations of motion. By QED, we always mean QED in (3 + 1)-dimensional spacetime, or QED 4 . But we can also consider QED 3 in (2 + 1)-dimensional spacetime where analogous considerations regarding infrared photons apply [6]. In that case, the dressing operator involves a closed one-form ω, so that on R 2,1 , ω = dα, with α a scalar function. The term in the action which absorbs the vertex operator from state vectors is proportional to which is close to the standard Chern-Simons term. In order to see this, we can interpret A in A ∧ F as dα. Section 3 reviews the proposed mass twist introduced in [5]. We establish that the twist is compatible with locality. Still it is affected by the infrared cloud at S 2 ∞ . Thus inertia is affected by the "vacuum" as in Mach's principle. In a similar manner, the Higgs field at infinity induces vector meson masses. Section 4 briefly indicates the changes in our considerations for QED 3 . In particular, in this case the ω in the vertex operator is a closed one-form, ω = dα on R 2,1 . Section 5 summarizes the part of Fröhlich et al. [2] helpful in the construction of ω. In section 6, we examine the modified dispersion relations of charged particles which violate Lorentz invariance. It seems possible to measure this violation. There are also similar violations in scattering amplitudes which are also in principal measurable.
For related recent work with emphasis on BMS group, see [15].
2 The Vertex Operator for QED 4 Coherent States

Preliminaries
The gauge transformations generated by the Gauss law will be denoted as the group G ∞ 0 , the subscript 0 denoting that it is connected to identity. Its elements g approach the identity when its spatial argument x approaches infinity, Its Lie algebra is spanned by where E i is the electric field, J 0 is the charge density and where the subscript 0 indicates compact support and superscript ∞ indicates infinite differentiability, both of which are standard notations. The Gauss law is imposed in the theory by requiring that G(Λ) vanishes for any choice of the test function Λ ∈ C ∞ 0 so that G ∞ 0 → {1} on quantum states.
Another manner to state this requirement is the following. If C is the space of connections and G ∞ 0 acts on C by gauge transformations, the principle bundle for gauge theories without matter is C /G ∞ 0 , that is, the quotient of C by the action of G ∞ 0 . The generators Q 0 (χ) of the charge group also come from the Gauss law, but now χ is required to approach a constant at infinity, The charge group they generate will be denoted by For QED with spatial slice R 3 , we can choose χ( x) = 1 for all x conveniently since G(Λ) → 0 in quantum theory. That gives the standard expression for charge. The Sky group G sky has generators where the function χ ∞ on S 2 ∞ need not be constant. As a result, Q(χ) need not even be rotationally invariant modulo a G(Λ). Hence G sky breaks Lorentz invariance.
The Sky group acts trivially on the charge zero sector which has no infrared cloud. But in charged sectors, state vectors are twisted by a coherent state vector of the infrared cloud, and hence, G sky can act by a non-trivial representation breaking Lorentz invariance.
We note that G sky has one-dimensional irreducible representations. That is because it is abelian. In fact, since the action of Q(χ) on quantum state vectors depends only on χ(n), we can write, All the Q(χ ∞ lm Y lm ) commute among themselves. Hence each of them generates a onedimensional abelian group.
Suppose we are given a standard representation of G sky , Then the vertex operator where ω is a closed two-form, dω = 0, (2.14) maps |· to the infrared-dressed states |· ω as we show below: |· is a Gauss law compatible vector (assuming that is the case with |· ). But Q(χ) does not commute with V (ω). Let ω ∞ be the asymptotic expression for ω, while let χ ∞ be the asymptotic χ as before, Then with we obtain with the central element These c's generate the centre of the algebra of U 's and V 's. This algebra resembles the Weyl algebra. If where |· denotes any vector transforming just by U (1) of charge of Sky, then can transform non-trivially under Sky, If ω ∞ is not zero, then the sector H ω of the Hilbert space spanned by {|· ω } breaks Lorentz symmetry.

Incorporation of Vertex Operator in Lagrangian
Let E j denote the electric field. It is conjugate to the potential A j : where * ω j = jkl ω kl . Note that in (2.27), the ket |· and bra ·| may represent different vectors. Thus the Hamiltonian of F µν , The field conjugate to A i is thus shifted from E i to E i for the zero-twist state vectors . We can accomplish this shift by adding to the QED Lagrangian density L a term ∆L ω = 1 2 µνλρ F µν ω λρ , that is, writing That is because the coefficient of ∂ 0 A i in the added term ∆L ω gives the contribution * ω i to the conjugate momentum of A i . We observe also that ∆L ω involves ω 0i which are new. The covariant-looking ∆L ω , if it is not to affect equations of motion, has to be a total divergence. This means that or ω as a two-form in 4-dimensions must be closed : The term ijk0 F ij ω k0 in ∆L ω may also be obtained by a Lorentz transformation of F 0i and ω jk from 0ijk F 0i ω jk . The presence of ω 0i in ∆L ω means that it can in general dress the electron with an infrared cloud containing both electric and magnetic fields.
It is natural to identify the asymptotic part of ω µν with Buchholz's f µν [4]. As observed in the introduction, ∆L ω resembles the QCD θ-term. Note also that ω µν depends on the charged particle. Hence it transforms under CPT in a manner required to maintain CPT invariance of (2.31).

The Twisted Mass for QED 4
The infrared cloud leading to Lorentz violation brightens the sphere S 2 ∞ at spatial infinity. We need a model for observing its effects. In setting up this model, we can be guided by the spontaneous symmetry breakdown due to the Higgs field.

On the Higgs Field
Spontaneous symmetry breakdown is caused by the Higgs field at spatial infinity. One may imagine then that it cannot be observed by local physics. But that is not the case. It makes itself felt in two ways, under two circumstances: • In a non-gauge theory, it creates Goldstone bosons. They are described by quantizing the local fluctuations around the constant Higgs field configuration. They affect the spectrum of the Hamiltonian by creating massless particles and eliminating the spectral gap of the Hamiltonian between vacuum and a particle state.
• In a gauge theory, appropriate gauge fields consume the Higgs fluctuations and become massive. Thus the opposite happens regarding the Hamiltonian spectrum: a gapless spectrum with massless vector fields gets gapped. Mach suggested that inertia is affected by the ambient background [10]. That is what happens here to the inertial mass of the vector fields. It is important to observe that the acquired mass of vector fields depend on their quantum numbers. For instance, the W ± and Z masses are different.
• The passage from the massless to the massive phase of the gauge field is a gauge transformation. It is affected by the gauge group element obtained from the polar decomposition of the Higgs field. It preserves locality so that the massive vector theory is a local theory. We regard this as also an important fact.

Twisting the Mass
In general, Letχ denote those test functions for which the l = 0 term is absent, If the charged particle state is not dressed by the infra-photons, that is, on states |· ω=0 , Q(χ)|· 0 = χ ∞ 00 q|· 0 . does not change the physics in the sector with label ω = 0 where the mass term is unaffected. Furthermore m(χ)ψψ is local, since mψψ is local and the twist e iQ(χ) commutes with local observables by gauge invariance. This property is independent of ω labeling state vectors. The Weyl relation is still valid. If Therefore with ω ∞ = da, Therefore the effect of the ω-twist is to replace m with m cos In view of equation (3.10), we may interpretχ ∞ as follows. Let an experiment measure the mass of the charged particle over all directions of S 2 ∞ with a sensitivity in the direction x given byχ ∞ . Then the resultant value for the mass is given by (3.10).

Calculation of ω
The vertex operator creates a coherent state of infrared photons which depends on the charged particle state |· 0 on which it operates. This coherent state has been calculated in a form convenient for us by Fröhlich et al. [2]. We will use their results without proof. Other important works relevant to us on this coherent state are by Kibble [11], Eriksson [12] and Gervais and Zwanziger [13].
The results of Fröhlich et al. can be described as follows. In the Coulomb gauge, the free electromagnetic field has the expression where The commutation relations are Let | p, e |0 denote the tensor product of a single non-interacting charged particle vector of charge e and momentum p and the photon vacuum, where we have suppressed spin labels. Then the dressed electron state vector surrounded by its infrared cloud is where as indicated, ω depends on p.
we can set V (ω) = 1 and instead replace A i by σ p (A i ) where, as in [2] σ p (a i )(k) = a i (k) + i ( * ω p (k) i ) . (4.8) The map σ p : a i → σ p (a i ) (4.9) and its adjoint define an automorphism of the algebra of creation and annihilation operators. For N charged particles of momenta p a and charges e a (a : 1 , , · · · , N ), the dressed state vector is where the Fourier transform of * ω pa is * ω pa (k) = e a p a · k p a − p a ·kk . Lorentz invariance can be spoilt by the presence of the second term in (4.8). For the case of a single charged particle, for Lorentz invariance, if U (1) is the unitary operator for Lorentz transformation Λ on the quantum Hilbert space, we require that However, this is not the case. Instead, we have that [1,2] The unit vectork denotes the direction in which we observe the "sky" of the source. Thus Lorentz symmetry is broken. It is remarkable that the Lorentz invariance breaking second term in (4.15) has no dependence on charged particle momentum. As a consequence, in the N charged particle sector, this term depends only on the total charge Q andk. Thus, and * Ω Q (k) is given by (4.15) with Q for e. Writing P a = a p a , (4.18) as the total charged particle momentum, we can capture the features of Lorentz invariance breaking in the N charged particle sector by replacing V ( ω pa ) by V (ω P = ω p ). As regards Lorentz invariance, we loose no information since does not induce Lorentz invariance violation.
In scattering theory, however, with widely separated particles in the in or out state vector, the above replacement may not be appropriate as it is nonlocal. Instead, it seems best to use V ( a ω pa ) which dresses each charged particle with its on infrared cloud.
If Λ is a rotation, * Ω Q (k) vanishes. This suggests that the rotational symmetry is preserved. However, we note that composition of two boosts can produce a rotation, suggesting trouble with the latter too. This point requires further clarification.

Consequences of Twisted Electron Mass
The consistency of twisting the charged spinor mass to capture Lorentz breaking raises issues about its effect on locality and perturbation theory. We tentatively suggest the following answers.
As we observed, Q(χ) commutes with local observables. Hence locality seems unaffected.
In perturbation theory, we encounter the charged particle propagator in the internal lines. This is a vacuum expectation value and the twisted mass becomes the untwisted one on vacuum. So it appears that no internal lines are affected. Hence the Lorentz breaking term affects only external lines and thereby scattering amplitudes. This conclusion is supported by the Lagrangian (2.31). The Lorentz breaking term is a surface term at infinity.
The twisted mass term m cos Q(χ)ψψ is a local operator. Its correlators restricted to local space-time regions should not be affected by the twist. We can try to observe the effect of mass twist in dispersion relation and in scattering where the electron is the dressed one. Then the mass term m of the electron with momentum p in the scattering amplitude is changed to Now, as noted previously, we have * ω p (x) i = d 3 k * ω p (k) i e i k· x , We can take the large r limit of the expression (5.6) following Gervais and Zwanziger [13].
where we use k 0 = ω and and > 0 gives the high frequency cut-off. Hence * ω p (x) i = lim This gives (5.3).The pole at p ·k = 0 of * ω p (k) i can be treated as a principle value, while the pole due to k ·x + i −2 term in (5.11) is well-defined by the → 0 + prescription.

On Lorentz-Violating Mass: Dispersion Relation and Particle Spin are Changed
We can write (5.3) as follows We would like to remark that the photon cloud surrounding the charged particle has the electric field * ω p (x) i as alluded to after (3.10). The functionχ ∞ (x) takes moments of this field: they are determined by the experimental arrangement measuring say the mass. Let R ∈ SO(3). It acts on the functionχ as (Rχ)(x) =χ(R −1 x). Then by using the rotational invariance of the two measures, we get m(Rp , Rχ) = m(p ,χ) . (5.13) Thus m is a rotationally invariant function of its arguments. It is useful to write out the integral in the argument of the cosine in (5.12) as I := I 1 − I 2 (5.14) where By rotational invariance I is a scalar function ofχ ∞ andp. Since I 2 is independent ofp, it vanishes. We show this explicitly in the Appendix. From eachχ lm , we can form a scalar by coupling it to a polynomial of degree l inp. The integral in (5.15) is a linear combination of these scalars. Since the integral in (5.15) is even inp, only even l = 0 need be considered in this linear combination.(Recall that by our assumption the l = 0 component ofχ is 0.) The simplestχ to consider is thus l = 2. For l = 2, we can writeχ where χ ij is symmetric and traceless. Therefore the integral I 1 takes the form where C depends only on ε. We show in the appendix that as ε goes to 0, Thus finally m(p ,χ) = m cosh(e(2π) 3 χ ijpipj ) (5.20) , we can if desired plot the twisted mass say forp = (0, 0, 1). Forp = (0, 0, 1) and the simple choice . Hence the integral I is changed to I = Cχ ij (Λp) i (Λp) j .

Mass Twist changes Spin
Thep dependence of mass persists even when the charged particle is brought to rest, the specificp dependence in the mass being determined by the process by which the particle is brought to rest. The origin in thep-space is getting "blown up" [14]. Now in the Poincarè representation theory for a massive particle, mass is assumed to be a scalar and spin is introduced by attaching an irreducible representation (IRR) of SU (2) to the vector state in the rest frame. The twisted mass, however, is not a rotational scalar. By (5.18), it depends on χ ijpipj and all its powers for the choice made above for χ. Thus standard spin such as 1/2 of the muon acquires all its orbital excitations, which depend on χ ∞ (x). Its (2n) th power is suppressed by the coefficient α n , with α being the fine structure constant. This phenomenon will affect decay selection rules (and of courses scattering). Further analysis of this observation is called for.
For a sensitive experiment on the isotropy of space, see [16].

Non-Abelian Superselection and Higgs Symmetry breaking
Non-Abelian super-selection rules play a role even in the familiar phenomenon where a complex Higgs field breaks a U (1) symmetry spontaneously. We conclude this paper with this observation.
We consider U (1) gauge symmetry broken by a complex scalar field ϕ(x). Let f R be test functions supported in r ≥ R. Then commutes with all observables supported in r < R. So S R for R → ∞, denoted by S ∞ , is superselected. We note that is superselected too and (6.1) and (6.2) do not commute. We have that We assume that S ∞ = 0. Both S ∞ and Q(ξ) commute with all local observables. Therefore, our previous arguments [5] lead to the conclusion that one of them must be spontaneously broken.
We explain the above argument briefly in our context. If we diagonalize S ∞ , that defines a superselection sector. However, Q(ξ) then changes it by (6.2). Hence it is spontaneously broken.
If we diagonalize S ∞ , as we do in superconductivity, we get a domain D 1 for the Hamiltonian H which makes it self-adjoint. The operator e iQ(ξ) changes this domain: it is spontaneously broken.
We can also opt to diagonalize e iQ(ξ) . Then U (1) is preserved, but at the expense that H is no longer defined: the integral of energy density diverges classically. We can still define e −itH , finding first a domain in which S ∞ is diagonal and then extending this unitary operator to the vectors with e iQ(ξ) diagonal. (A unitary operator can act on all vectors of the Hilbert space.) We call such a domain D 2 .
If P E is the projector for energies less than E for H on D 1 , it is bounded and hence defined in D 2 . A legitimate question is, whether if we reconstruct H from D 2 , then we will get the same or similar low energy spectrum of H as from D 1 . If that is the case, then there may be a new approach to spontaneous symmetry breaking. Now, (A.10) Observe that (K 1 ) ijk must be of the form (K 1 ) ijk = f 1 (ε)k ikjkk + f 2 (ε) δ ijkk + δ jkki + δ kikj . (A.11) so that K 1 = f 1 (p ·k) 3 + 3f 2 (p ·k) . (A.12) Contracting (K 1 ) ijk withk ikjkk , we get (A.13) Contracting (K 1 ) ijk with δ ijkk + δ kikj + δ jkki , we get (A.14) The sign of the O(ε 0 ) term is fixed by using the Cauchy principal value formula Therefore, ·k. (A.20) Replacing K = K 1 + K 2 into the expression for I 1 and integrating gives We finally get, in the limit ε → 0,