Relativistic invariance of the vacuum
Authors
Abstract
Relativistic invariance of the vacuum is (or follows from) one of the Wightman axioms, which is commonly believed to be true. Without these axioms, here we present a direct and general proof of continuous relativistic invariance of all realtime vacuum correlations of fields, not only scattering (forward in time), based on closed time path formalism. The only assumptions are basic principles of relativistic quantum field theories: the relativistic invariance of the Lagrangian, of the form including known interactions (electromagnetic, weak and strong), and standard rules of quantization. The proof is in principle perturbative leaving a possibility of spontaneous violation of invariance. Time symmetry is, however, manifestly violated.
1 Introduction
Relativistic invariance of the vacuum is a basic property of all quantum field theories based on relativistically invariant dynamics, directly following from one of Wightman axioms [1]. The (third) Wightman postulate assumes the existence of a single ground state with zero energymomentum eigenvalue [2]. The postulate is generally accepted by high energy community [3] although the problem is more sophisticated than it looks and the validity of the axiom itself has been questioned [4, 5]. However, to postulate zero energy is in conflict with renormalization which may add indefinite contribution. Moreover, this postulate should be redundant since the vacuum is already determined by the Lagrangian and the rules of quantization. Indeed, without the Wightman axiom, the invariance has been already proved for special cases of relativistic quantum field theories: single fields (electromagnetic, current, etc.), free theories, Feynman (forward) timeordered (in–out) correlations of fields, scattering processes, second order and some other classes of correlations [6–11], but never in general (for arbitrarily timeordered correlations). A large part of high energy community wrongly claim it is already proved in general [12]. For instance, a naive proof by reconstruction of vacuum wavefunction out of Feynman correlations fails because such a reconstruction is incomplete and not unique. A general proof requires to show Lorentz invariance of all arbitrarily ordered correlations at real spacetime points. The invariance could be certainly proved only if the underlying dynamics is invariant and conserved charges are zero. It has been suggested that Nature may violate the invariance directly (by noninvariant dynamics) or spontaneously (by a Higgslike mechanism) [13]. We do not discuss these two last possibilities here.
In this paper, we confirm the common intuition by a direct proof, not relying on Wightman axioms. Although formally general, it is better to understand it as perturbative. We show that zerotemperature vacuum correlations of fields at real spacetime points are invariant under continuous transformations of a reference frame. We shall use the framework of the closed time path formalism (CTP) [14–16] where correlations are defined on the complex path (going downwards with respect to imaginary part). It is defined in a particular reference frame, hence the invariance is not manifest. We only assume basic, accepted principles of relativistic quantum field theories: (i) relativistically invariant Lagrangian, including known interactions (electromagnetic, weak and strong, specified in detail later in the paper) and (ii) rules of quantization (standard construction of field expectation values, consistent with CTP). However, the discrete transformations like time reversal and parity have to be treated separately. If the underlying dynamics is invariant under charge conjugation or parity reflection then the proof remains valid for these transformations [1]. Unfortunately, even if the dynamics is symmetric with respect to time reversal, then CTP is not (except special cases, e.g. at spacelike points or in scattering problems). In the same way, CTP is in general not invariant under joint charge–parity–time reversal in contrast to dynamics. It closely related to violation of time symmetry in quantum noninvasive measurements [17].
The paper is organized as follows. We first recall the construction of the complextime path correlations, defining relevant correlations of fields. Next, we show that the correlations are independent of the particular shape of the path. Then we prove the Lorentz invariance of the zerotemperature nocharge vacuum for continuous transformations. Then we discuss possible pitfalls in other attempts of the proof based e.g. on momentum representation, which is dedicated to all who claim that the invariance is trivial or easy extension of existing proofs [12]. Finally, we will see that time symmetry is absent in CTP. Throughout the paper, we use the convention ħ=c=1 and β=1/k _{ B } T (inverse temperature).
2 Closed time path formalism and definitions
3 Contour shape independence
4 Lorentz invariance
It is natural to expect that Green functions are invariant under relativistic Lorentz transformations at zero temperature if the Lagrangian is Lorentzinvariant. However, starting from CTP makes it not obvious because the contour prefers some time direction. Intuitively, we expect that this should not bring about any problem but the warning light comes already from the noninvariance due to finite temperatures, which enters only as a jump in the contour. Below, we show that the Green functions at real times in zerotemperature vacuum are indeed invariant under continuous Lorentz transformations in Lorentzinvariant field theories but not necessarily under time reversal.
4.1 Relativistic notation in CTP
The rest of conventions, including Lorentz generators J, spinors and Lagrangian density \(\mathcal{L}\) (becomes Lagrangian L when spaceintegrated) are consistent with textbooks [7, 8, 10, 40], which we recall in Appendix C for completeness.
The great advantage of Lagrangian and pathintegral formalism is that it seems to be perfectly Lorentzinvariant at first sight. However, the possible caveats are hidden in the shape of CTP, which is bent into imaginary time but not spatial coordinates. The common sense intuition tells us that the Lorentz invariance must be broken by finite temperatures, which essentially dictate the contour jump. Moreover, even at zero temperature, the contour retains preferred time direction so Lorentz invariance is not manifest and selfevident.
4.2 Formal proof of invariance

Lorentz transformations are analytically continued to complex spacetime.

We ignored possible problems for x ^{0}→±i∞.

Convergence and renormalization is not discussed.
To give more insight to the proof, dispel possible objections and make it robust, we will now reformulate it in terms of perturbative diagrams showing that each diagram separately is Lorentzinvariant.
It is enough to prove that JG=0 for all Lorentz generators J and all Green functions G—defined by (3) at real times, because every finite continuous transformation can be written as Λ=expwJ and (d/dw)ΛG=JΛG=JG _{ w } where ΛG=G _{ w } is always some Green function at real times.
4.3 Free theories
Nonzero temperature β<∞ will break Lorentz invariance, which means that there is a fourvector related to temperature. There is no unique choice. One can define β ^{ μ } [41, 42] with β ^{0}=β and β ^{ k }=0, k=1,2,3 in a particular reference frame. However, good alternatives are k _{ B } T ^{ μ }=β ^{ μ }/β⋅β or \(u^{\mu}=\beta^{\mu}/\sqrt{\beta\cdot\beta}\), see also a general discussion [43]. However, one has to remember that the zero temperature limit means infinitely timelike β ^{ μ }. The zero fourvector β limit, when Lorentz invariance might be obvious, corresponds unfortunately to infinite temperature so this is not what we are looking for.
4.4 Interaction
This is our main result. The proof has been obtained in the time domain, without stretching the real time to infinities. Note the analogy to contour shape independence. The heart of the proof is the fact that analytic continuation of time does not hurt the generating function. It is quite intuitive but never before shown explicitly.
5 Incomplete proofs
It is tempting to ask: Why not to prove Lorentz invariance using simply energymomentum representation [44–48]? This approach seems attractive and practical but it has to be supplemented by arguments following from spacetime CTP representation. The main problems arise for offshell Green functions (not describing scattering processes) and field mass/strength renormalization. They cannot be resolved without returning to spacetime representation.
Having presented above the complications in energymomentum space, we conclude that the proof is fully correct in the spacetime domain, while in the momentum/Fourier space it would require assuming at best several additional technical rules and may not be general (for very complicated diagrams). Still, most of practical calculations are easier performed in the latter case, with spacetime arguments used only when the momentum rules are ambiguous.
Another attempt of proof requires reconstruction of vacuum wavefunction out of in–out Green functions. This is possible in simple quantum models, including free theories, finite and harmonic systems but not in fully interacting relativistic theory because renormalizing fields, especially for fermions, make the information in in–out functions incomplete and insufficient to uniquely reconstruct vacuum wavefunction. On the other hand, dimensional regularization is useless when proving Lorentz invariance because Lorentz transformations are defined only for integer dimensions. Lastly, an often heard argument is analyticity of Green functions [12], but how to define it on CTP? The condition of nonincreasing imaginary part makes it impossible to define analytic conditions (Cauchy–Riemann) because for free real times the middle one is pinched—the imaginary part of the derivative is not allowed. One cannot allow the contour to go upwards in imaginary direction because the energy spectrum is bounded from below and open form above, making integrals nonrenormalizably divergent. On the other hand analyticity with respect to real s is questionable because of turning points t _{0}(s _{0}) and crossings (t _{ n }≃t _{ m }). Such a proof will be restricted to a particular branch of CTP, e.g. upper/lower real or vertical imaginary part, but not general. One cannot also use GellMann and Low theorem [54], which could simply extend the proof from free theories to interacting ones by adiabatic switching, because it works itself in a preferred frame. Any attempt to prove Lorentz invariance in general will ultimately fail or arrive at the proof here presented.
6 Absence of timereversal symmetry
The momentum representation hides the fact that the Lorentz invariance is not necessarily valid for time reversal, which is only true for second order correlations or if the factorization (36) holds for higher order correlations. Note that the reversal p ^{0}→−p ^{0} break the time symmetry because of θ(±p ^{0}) factors in G _{±∓}. Trying to reverse branches +↔− does not help either, because then G _{±±} gets conjugated, so there is no way to restore original Green function. If we both reverse p ^{0} and make conjugation then we will return to the original Green function but the time will be also twice reversed which is again not what we wanted to have. The time (generally charge–parity–time) symmetry is valid only for scattering processes or spacelike separated points. This lack of symmetry is fundamental and especially surprising in context of noninvasive measurements which are naturally defined on CTP [17].
7 Conclusions
We have shown that the closed time path formalism is consistent with relativistic invariance at zero temperature. The presented picture pathintegral Lagrangian fits well into covariant relativistic framework with the analytic continuation of time as the only nontrivial extra element. It is remarkable that complex time glues statistical mechanics, time evolution and relativistic symmetry without any problems. The fact that CTP is invariant only under continuous transformations, not time reversal, shows that there the properties of CTP still differ from scattering processes. An open issue is whether the Lorentz symmetry can be shown nonperturbatively (what about spontaneous breaking?) and for quantum gravity.
Acknowledgements
I thank W. Belzig, A. Neumaier, S. Mrówczyński and P. Chankowski for fruitful discussions and motivation and for hospitality of University of Konstanz, where part of this work has been completed.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Appendix A: Hamiltonian formulation of CTP
The traditional starting point of every quantummechanical problem is the Hermitian Hamiltonian operator \(\hat{H}\) in a given Hilbert space. The physical quantities are described by Hermitian operators \(\hat {X}_{k}\) corresponding to their classical counterparts X _{ k } (numbers). Depending on the measurement scheme, there exists a correspondence between operators and measurement outcomes, \(\hat{X}_{k}\to X_{k}\). Evolution and state is described by Hamiltonian \(\hat{H}(t)\to\hat {H}+\hat{H}_{1}(t)\), where \(\hat{H}_{1}(t)\) is the time dependent part, due to changing external forces, which are assumed to be absent for Re t<0. We shall assume that t lies on CTP as in Fig. 1. Every conserved quantity can be incorporated into the definition of the Hamiltonian, \(\hat {H}\to\hat{H}\mu_{Q}\hat{Q}\). For the main purpose of this paper, \(\hat{H}_{1}\) and \(\hat{Q}\) can be omitted. However, the whole CTP formalism works perfectly also for nonzero \(\hat{H}_{1}\), which is necessary in problems involving external (changing) forces and charges [55].
Performing a Taylor expansion of exponentials in (A.1) one can explicitly check that the result does not depend on a particular shape of CTP [38, 39], as long as (1) is satisfied and t _{0} is greater than all interesting times. In such a check it is important to note that \(\hat{H}\) is timeindependent along the wiggly part of the contour, so the ordering does not matter there. On the other hand (A.1) is useful also for deriving the thermodynamic functions and transport coefficients, e.g. the Kubo formula [32–34].
Usually one separates free harmonic Hamiltonian \(\hat{H}_{0}\) writing \(\hat{H}(t)=\hat{H}_{0}+\hat{H}_{I}(t)\). The remaining calculations are usually performed perturbatively, with the help of the free twopoint Green’s functions and Wick’s theorem [56, 57], which allows a decoupling of the free manypoint Green functions into products of twopoint functions, see below.
A.1 Free Green functions
A.2 Wick theorem
The Wick theorem states essentially that every manypoint Green function for a quadratic Hamiltonian splits into products of twopoint functions. This fact is analogous to the property of Fourier transforms of Gaussian functions, which are again Gaussian. Wick theorem is useful for interactions—one proceeds perturbatively, expanding (A.1) in interaction strength.
A.3 Hamiltonian–Lagrangian equivalence for bosonic fields
A.4 Hamiltonian–Lagrangian equivalence for fermionic fields
Appendix B: Technical aspects of path integrals
B.1 Wick theorem for path integrals
B.2 Conservation laws
B.3 Hubbard–Stratonovich transformation and renormalization
Appendix C: Lorentz transformations
Apart from continuous transformations there exist two special discrete transformations, time reversal x ^{0}→−x ^{0} and parity (mirror) inversion x ^{ k }→−x ^{ k }, k=1,2,3. One can also make charge conjugation e→−e (the interaction strength). The proof of Lorentz invariance of zero temperature vacuum will be valid only for continuous transformations. If Lagrangian is invariant with respect to parity inversion or charge conjugation then we can include it, too (which is e.g. not the case for weak interactions).

a product/sum of Lorentz scalars is a Lorentz scalar,

\(\bar{\psi}\phi\) and A are Lorentz scalars,

\(\bar{\psi}\gamma^{\nu}\phi\) and A ^{ ν } and ∂ ^{ ν } A are Lorentz vectors,

\(\bar{\psi}\gamma^{\nu}\gamma^{\mu}\phi\) and ∂ ^{ μ } A ^{ ν } are Lorentz tensors, etc.,

A product of Lorentz vectors (tensors) is a Lorentz scalar if the Greek indices always come in raised/lowered pairs \({}^{\mu}_{\mu}\) and Einstein summation is performed.
When considering weak interactions, one should include also pseudoscalars \(\bar{\psi_{k}}\gamma^{5}\psi_{l}\), \(A_{\mu}\bar{\psi}_{k}\gamma^{5}\gamma^{\mu}\psi_{l}\) where γ ^{5}=iγ ^{0} γ ^{1} γ ^{2} γ ^{3}, which transforms as scalar under continuous transformations but changes sign under parity inversion.