Relativistic invariance of the vacuum

Open Access
Regular Article - Theoretical Physics


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 real-time 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 energy-momentum 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) time-ordered (in–out) correlations of fields, scattering processes, second order and some other classes of correlations [6, 7, 8, 9, 10, 11], but never in general (for arbitrarily time-ordered 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 non-invariant dynamics) or spontaneously (by a Higgs-like 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 zero-temperature 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, 15, 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 space-like 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 complex-time 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 zero-temperature no-charge 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/kBT (inverse temperature).

2 Closed time path formalism and definitions

The complex closed time path C is a specially chosen curve of time t in the complex plane as a function of a real parameter s. The curve is continuous with nonincreasing \(\operatorname{Im}t\) (important for convergence of integrals, involving energies bounded from below and open from above) except a jump of iβ, which becomes imaginary infinity at zero temperature, see Fig. 1. The physical part of the contour is located on the real axis (both upper and lower). In particular, the curve in Fig. 1a is parametrized by real s∈[si,sf]→t(s) with
$$\begin{aligned} \begin{aligned} &\operatorname{Im}(\mathrm{d}t/\mathrm{d}s)\leq0,\qquad t(s_f)=t(s_i)-\mathrm{i}\beta, \\ & \operatorname{Im} t\to0_+\quad\mbox{and}\quad \mathrm{Re}\,\mathrm{d}t/\mathrm{d}s>0 \quad\mbox{for }s\in[s_+,s_0], \\ & \operatorname{Im} t\to0_-\quad\mbox{and}\quad \mathrm{Re}\,\mathrm{d}t/\mathrm{d}s<0 \quad\mbox{for }s\in[s_0,s_-], \\ & t(s_\pm)=0_\pm, \qquad t(s_0)=t_0, \quad s_i<s_+<s_0<s_-<s_f. \end{aligned} \end{aligned}$$
The generalized contour in Fig. 1b contains multiple real parts.
Fig. 1

The closed time path in the complex-time plane. All the path cannot go upwards in the imaginary direction. The finite temperature implies the jump iβ, which extends to infinity at zero temperature. Apart from the above conditions, the shape of the path is arbitrary. We stress that the path is ordered by the real parameter s. All the important times have been marked. The shift between real parts is infinitesimal only for better visibility but in fact can be zero. (a) The simple in–out–in (shortly in–in) contour with a single pair of real parts (between 0 and t0). (b) The generalized contour for arbitrarily ordered correlations

To facilitate an easier notation throughout the paper we define Heaviside and Dirac functions for complex-time arguments t,
$$\begin{aligned} \begin{aligned} &\theta\bigl(t-t'\bigr)=\theta \bigl(s-s'\bigr), \\ & \delta\bigl(t-t'\bigr)=\delta\bigl(s-s'\bigr)/( \mathrm{d}t/\mathrm{d}s), \\ &\partial_t=(\mathrm{d}t/\mathrm{d}s)^{-1}\partial_s, \qquad \mathrm{d}t=(\mathrm{d}t/\mathrm{d}s)\,\mathrm{d}s \end{aligned} \end{aligned}$$
where, as usual, θ(ss′)=1 for s>s′ and 0 otherwise while ∫d(ss′)=1 and δ(ss′)=0 for ss′. Therefore θ defines a special order of times along the path, and it may be converse to the usual order. This is important since only s is the real, ordering parameter, while t is complex and thus has no natural ordering. Note that in all contour time-integrals we essentially integrate over s.
Let us take the generating functional for quantum correlations (often referred to as Green functions) with respect to the action, making use of the Lagrangian approach, namely [11, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]
$$\begin{aligned} G \equiv& \biggl\langle \prod_k A_k(t_k)\prod_l \phi_l(t_l) \biggr\rangle \\ =& \prod_k\frac{\delta}{\delta \mathrm{i}\chi_k(t_k)}\prod _l\frac {\delta}{\delta \mathrm{i}\eta_l(t_l)}\mathrm{e}^{S[\chi,\eta]} \bigg|_{\chi =0,\eta=0} \end{aligned}$$
$$ \mathrm{e}^{S[\chi,\eta]}=\frac{\int DAD\phi\,\mathrm{e}^{\int_C \mathrm{i}\,\mathrm{d}t(L+\sum_k(\chi_k(t)A_k(t)+\eta_k(t)\phi_k(t))}}{ \int DAD\phi\,\mathrm{e}^{\int_C \mathrm{i}\,\mathrm{d}t\,L}} $$
with auxiliary commuting (c-) field χ (χχ′=χχ) and anticommuting Grassmann field η (ηη′=−ηη). Here tk can be located everywhere on the path (upper real, lower real, left complex part). In the above, A and ϕ are generic bosonic and fermionic (Grassmann) fields, respectively, and the Lagrangian is a time-local function of fields and their time derivatives, namely \(L=L[A,\dot{A},\phi,\dot{\phi}]\). The integral ∫Cdt is over the just defined time path, and should be read as an integral over real s, ∫ds(dt/ds). The boundary conditions for fields are A(ti)=A(tf) and ϕ(ti)=−ϕ(tf), for bosonic and fermionic field, respectively (Kubo–Martin–Schwinger condition [32, 33, 34]). Standard (in–out) correlations are obtained for the auxiliary fields nonzero only on the upper real part of the contour. General correlations involve both parts of the contour (in–out–in or in–in) or even more (in–in–in, etc.). The path-integral formulation of CTP is equivalent to Hamiltonian formulation, as we show in Appendix A. The standard issues such as Wick theorem, conservation laws and renormalization are resolved in CTP analogously to in–out correlations, see Appendix B.

3 Contour shape independence

The fact that we can freely wiggle the contour is reasonably believed for CTP [24, 35, 36, 37], but a direct proof is very instructive [38, 39]. Let us prove that (4) is independent of a particular shape of CTP, because the proof of Lorentz invariance will be analogous. It follows from the invariance of DA and under infinitesimal transformation δt(s) parametrized by real s (the variation may change endpoints but the jump must be kept constant)
$$\begin{aligned} \begin{aligned} &A(t)\to \mathrm{e}^{\delta t\partial_t}A(t)=A(t)+\delta t \partial_t A(t)+\cdots, \\ &\phi(t)\to \mathrm{e}^{\delta t\partial_t} \phi(t)=\phi(t)+\delta t\partial_t \phi(t)+\cdots \end{aligned} \end{aligned}$$
which gives L(t+δt)=exp(δt∂t)L(t). This means that the effect of the contour changes can be absorbed into a shift under time integral and a such it disappears (analogously to an integral over a total derivative). We stress that this proof is valid only if do not change the shape of the real part of the contour, except shifting t0 if it lies beyond all relevant times. It is essential that the fields are smooth.
An alternative, more explicit proof relies on perturbative approach. One starts with Gaussian, diagonalized Lagrangian
$$\begin{aligned} L_0 =& \sum_k \bigl[ \psi^\ast_k(\mathrm{i}\partial_{t}-E_k) \psi_k + A_k^\ast(\mathrm{i} \partial_{t}-E_k)A_k \\ &{}+\dot{B}_k^2/2-E_k^2B_k^2/2 \bigr]. \end{aligned}$$
In the above, we have denoted complex ψ instead of real ϕ and distinguished the linear time derivative bosonic field A from the quadratic B. The two-point Green function for B reads
$$ \bigl\langle B_k(t)B_l\bigl(t'\bigr) \bigr\rangle _0=\frac{\delta_{kl}}{2E_k}\sum_\pm \mp\frac{\mathrm{e}^{\pm \mathrm{i}|t'-t|E_k}}{1-\mathrm{e}^{\pm\beta E_k}}, $$
where |t′−t|=(tt′)θ(tt′)+(t′−t)θ(t′−t). For A we get (A.7) and for ψ (A.8).
If we add interaction term, L=L0+LI, and LI depends on fields appearing in L0, the generating function (4)—by virtue of the Wick theorem (B.1)—will be a sum of integrals represented by Feynman diagrams, with lines (propagators) corresponding to free Green functions (A.7), (A.8) and (7) and vertices corresponding to LI. Suppose that t(s) is varied by infinitesimal δt. Let us consider the variation of Green functions. Note that
$$ \delta \bigl\langle X(t)X^\ast\bigl(t'\bigr) \bigr\rangle _0= \bigl(\delta t\partial _t+\delta t' \partial_{t'} \bigr) \bigl\langle X(t)X^\ast\bigl(t'\bigr) \bigr\rangle _0 $$
where X=A,B,ψ and subscript 0 denotes averaging (3) with respect to L0. The free Green function depends only on endpoint times t, t′, not the shape of CTP between them.
Now, each physical diagram corresponds to average of the type
$$ \int_C \mathrm{d}t_1\cdots \mathrm{d}t_N \biggl\langle \prod_{n\leq N} L_I(t_n) \prod_{k>N} X_k(t_k) \biggr\rangle _0, $$
where X=ψ,A,B and tk lie in the real (right hand) part of the contour for k>N. So we get
$$\begin{aligned} &{\delta\int_C \mathrm{d}t_1\cdots \mathrm{d}t_N \biggl\langle \prod_n L_I(t_n)\prod_k X_k(t_k) \biggr\rangle _0} \\ &{\quad= \int \mathrm{d}s_1\cdots \mathrm{d}s_N\sum _{m\leq N}\frac{\partial }{\partial s_m} \biggl\langle \delta t_n \prod_{n\leq N} L_I(t_n)\prod _{k>N} X_k(t_k) \biggr\rangle _0,} \end{aligned}$$
where we used Wick-decoupling into products of free Green functions, Leibniz rule δ(AB)=BδA+AδB, δ dt=dδt=ds∂sδt, (8), inverse Leibniz rule applied to s and back Wick-coupling. The final result of the shape variation integral is zero, which completes the proof.

4 Lorentz invariance

It is natural to expect that Green functions are invariant under relativistic Lorentz transformations at zero temperature if the Lagrangian is Lorentz-invariant. 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 non-invariance due to finite temperatures, which enters only as a jump in the contour. Below, we show that the Green functions at real times in zero-temperature vacuum are indeed invariant under continuous Lorentz transformations in Lorentz-invariant field theories but not necessarily under time reversal.

4.1 Relativistic notation in CTP

Let us briefly recall relativistic notation. The components of position four-vector are denoted xμ, μ=0,1,2,3, with time x0=t and three spatial coordinates xk, k=1,2,3. Fields, functions of position will be denoted as f(x) where x=(x0,x1,x2,x3). The derivatives are denoted by μ=/∂xμ. From now on, Greek indices are reserved for four-vectors and higher Lorentz-like tensors and structures. It is extremely important for the purpose of this paper that the time x0(s) is here a complex function of the real parameter s. In particular the time derivative 0 is translated into the real derivative s by
$$ \partial_0=\bigl(\mathrm{d}x^0/\mathrm{d}s\bigr)^{-1} \partial/\partial s. $$
Similarly, we introduce Dirac four-delta, including complex time,
$$ \delta^4\bigl(x-x'\bigr)=\frac{\delta(s-s')}{\mathrm{d}x^0/\mathrm{d}s} \delta \bigl(x^1-x^{\prime1}\bigr)\delta\bigl(x^2-x^{\prime2} \bigr)\delta\bigl(x^3-x^{\prime3}\bigr) $$
and four-integral ∫Cd4x with d4x=dx0 dx1 dx2 dx3 and dx0=(dt/ds) ds. For a compact notation of relative four position including the complex-time component of the four-vector on CTP we will denote |xx′|μ with |xx′|1,2,3=(xx′)1,2,3 but
$$ \bigl|x-x'\bigr|^0=\theta\bigl(x^0-x^{\prime0} \bigr) \bigl(x^0-x^{\prime0}\bigr)+\theta\bigl(x^{\prime 0}-x^{0} \bigr) \bigl(x^{\prime0}-x^{0}\bigr), $$
where θ(x0x′0)=θ(ss′) applies to complex x0 and x′0 along CTP.

The rest of conventions, including Lorentz generators J, spinors and Lagrangian density \(\mathcal{L}\) (becomes Lagrangian L when space-integrated) are consistent with textbooks [7, 8, 10, 40], which we recall in Appendix C for completeness.

The essential postulate of QFT is that the Lagrangian density is a Lorentz-invariant. It means that the generating functional (4) can be written now as
$$\begin{aligned} \mathrm{e}^{S[\chi,\xi,\eta]} \propto&\int DADBD\bar{\psi }D\psi \\ &{}\times \exp\int_C \mathrm{i}\,\mathrm{d}^4x \biggl(\mathcal{L}+ \sum_k(\chi_kA_k+\xi _k\cdot B_k \\ &{} +\bar{\eta}_k\psi_k+ \bar{\psi}_k\eta_k) \biggr) \end{aligned}$$
where all fields are functions of spacetime, e.g. ηk(x) with complex time x0=t, \(\mathcal{L}\) transforms as a scalar field and normalization imposes constraint S[0,0,0]=0. Here Ak denotes Lorentz scalars and \(B^{\mu}_{k}\)—four-vectors with the dot ⋅ denoting Minkowski scalar product. In our convention, assuming that \(\mathcal{L}\) is a scalar we also assumed that average charge densities are zero. The first Minkowski coordinate (time) has the jump \(x^{0}_{i}-x^{0}_{f}=i\beta\).
We stress that the charge conservation, Ward–Takahashi identity, holds in CTP if the Lagrangian does not mix different classes of fermions [10]
$$ \bigl\langle \partial_\mu j^\mu_N\cdots\bigr\rangle =0, \quad j_N^\mu=\sum_{n\in N} \bar{\psi}_n\gamma^\mu\psi_n. $$

The great advantage of Lagrangian and path-integral formalism is that it seems to be perfectly Lorentz-invariant 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 self-evident.

4.2 Formal proof of invariance

We shall now present a brief formal proof of Lorentz invariance of vacuum generating function (4), with the auxiliary field nonzero only for real times (right hand part of the contour) at zero temperature kBT=0 or β=+∞. Note that a zero temperature makes the jump on CTP between ti and tf infinite. Due to the freedom of choice of the shape, we assume the contour in Fig. 1 with ti→i∞ and tf→−i∞, see Fig. 2. Applying a boost Λ to the fields AΛA, BΛB, ψΛψ we have the complex-time invariance of Lagrangian
$$\begin{aligned} &{\int_C \mathrm{d}^4x\mathcal{L}[A,B,\psi]\to\int \mathrm{d}^4x\mathcal{L}(\varLambda A,\varLambda B,\varLambda\psi)} \\ &{\quad= \int_C \mathrm{d}^4x\varLambda_0 \mathcal{L}[A,B,\psi]=\int_C \mathrm{d}^4x\mathcal{L}[A,B,\psi]} \end{aligned}$$
where we have used the fact that both the measure d4x and the integration boundary is invariant under Lorentz transformations and Λ0f(x)=f(Λ−1x). Moreover D(ΛA)=DA, D(ΛB)=DB and D(Λψ)=. All the exponent in the generating functional (14) is a Lorentz scalar so the above reasoning remains valid. We write
$$\begin{aligned} &{\int DADBD\bar{\psi}D\psi\; \exp\int_C \mathrm{i}\, \mathrm{d}^4x \biggl(\mathcal{L}[A,B,\psi]} \\ &{\qquad{}+\sum_k(\varLambda\chi_kA_k+ \varLambda\xi_k\cdot B_k+\varLambda \bar{\eta}_k \psi_k+\bar{\psi}_k\varLambda\eta_k) \biggr)} \\ &{\quad =\int D(\varLambda A)D(\varLambda B)D(\varLambda\bar{\psi})D(\varLambda\psi )\exp\int _C \mathrm{i}\,\mathrm{d}^4x} \\ &{\qquad{}\times \biggl(\mathcal{L}[\varLambda A,\varLambda B,\varLambda\psi]} \\ &{ \qquad{}+ \sum_k(\varLambda\chi_kA_k+ \varLambda\xi_k\cdot B_k+\varLambda \bar{\eta}_k \varLambda\psi_k +\varLambda\bar{\psi}_k\varLambda \eta_k) \biggr)} \\ &{\quad = \int DADBD\bar{\psi}D\psi\; \exp\int_C \mathrm{i}\, \mathrm{d}^4x} \\ &{\qquad{}\times \biggl(\mathcal{L}[ A,B,\psi]} \\ &{\qquad{} +\sum_k( \chi_kA_k+\xi_k\cdot B_k+\bar {\eta}_k\psi_k +\bar{\psi}_k\eta_k) \biggr),} \end{aligned}$$
which proves the invariance. However, we had to assume that Λ acts on functions of complex arguments, not just real domain, as time is complex. To be correct we should work only in the real domain (s,x1,x2,x3) and define Λ by Taylor expansion of generator exponential
$$ \varLambda=\mathrm{e}^{\mathrm{i}\omega_{\mu\nu}J^{\mu\nu}}=\sum_{n=0}^\infty \frac{1}{n!}\bigl(\mathrm{i}\omega_{\mu\nu}J^{\mu\nu} \bigr)^n. $$
This is equivalent to real Lorentz transformation for real times, which is the case we need (Green functions at real times). The above proof, although formally correct, is hardly acceptable because one uses ill-defined mathematical operations. The vague issues are:
  • Lorentz transformations are analytically continued to complex spacetime.

  • We ignored possible problems for x0→±i∞.

  • Convergence and renormalization is not discussed.

Fig. 2

Zero-temperature CTP. The boundary times ti and tf are sent to imaginary infinity, i∞ and −i∞, respectively. Similarly to Fig. 1 we can take the simple contour (a) and the generalized one (b)

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 Lorentz-invariant.

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=JGw where ΛG=Gw is always some Green function at real times.

4.3 Free theories

We shall first prove Lorentz invariance of vacuum zero-temperature Green functions in free theories, where they can be found exactly. The Lagrangian density for free scalar theory reads
$$ 2\mathcal{L}_0=\partial_\mu A\partial^\mu A-m^2A^2, $$
where m is the mass of the field. The Green function (propagator) satisfies Klein–Gordon equation
$$ \bigl(\partial_\mu\partial^\mu+m^2\bigr)\bigl\langle A(x)A\bigl(x'\bigr)\bigr\rangle _0=-\mathrm{i}\delta^4\bigl(x-x'\bigr), $$
where acts on x (we shall write ′ if acting on x′). The solution reads
$$\begin{aligned} G_0\bigl(x,x'\bigr) =&\bigl\langle A(x)A \bigl(x'\bigr)\bigr\rangle _0 \\ =&\int \frac{\mathrm{d}^3p}{2p^0(2\pi)^3}\sum_\pm\mp\frac{\mathrm{e}^{\pm \mathrm{i}p^\mu|x-x'|_\mu}}{ 1-\mathrm{e}^{\pm\beta p^0}} \end{aligned}$$
where pμ is the four momentum with \(p^{0}=\sqrt{{\mathbf{p}}^{2}+m^{2}}\). It is important that (21) is valid for time components, x0 and x′0 along the CTP contour as defined in Fig. 1 and (13). For the Dirac field,
$$ \mathcal{L}_0=\bar{X}\bigl(\mathrm{i}\gamma^\mu \partial_\mu-m\bigr)X $$
we have
$$ \bigl(-\mathrm{i}\gamma^\mu\partial_\mu+m\bigr)_{bc} \bigl\langle X_c(x)\bar {X}_a\bigl(x' \bigr)\bigr\rangle _0=-\mathrm{i}\delta_{ab}\delta^4 \bigl(x-x'\bigr), $$
with the summation over c and mab=ab. Now, we get two cases. If X=A (bosonic field, unphysical, usually appearing in renormalization) then A(ti)=A(tf) but if X=ψ (fermionic field) then ψ(ti)=−ψ(tf). The general solution reads
$$ \bigl\langle X_a(x)\bar{X}_b\bigl(x' \bigr)\bigr\rangle _0=\bigl(\mathrm{i}\gamma^\mu \partial_\mu +m\bigr)_{ab}G^X_0 \bigl(x,x'\bigr), $$
$$\begin{aligned} \begin{aligned} &G^A_0\bigl(x,x'\bigr)=G_0 \bigl(x,x'\bigr),\\ & G^\psi_0 \bigl(x,x'\bigr)= \int\frac{\mathrm{d}^3p}{2p^0(2\pi)^3}\sum_\pm\mp \frac{\mathrm{e}^{\pm \mathrm{i}p^\mu|x-x'|_\mu}}{ 1+\mathrm{e}^{\pm\beta p^0}}. \end{aligned} \end{aligned}$$
For vector fields Bμ, the internal Lorentz structure is simply added by metric tensor 〈Bμ(x)Bν(x′)〉0=agμνG0(x,x′). Possible problems occur for expressions of the type \(\partial_{\mu}\partial'_{\nu}G_{0}\) because the derivative produces unphysical contact terms ∼δ(x0x′0) [40]. Fortunately, they appear always in pairs with a renormalization field, which cancels the singularity.

Nonzero temperature β<∞ will break Lorentz invariance, which means that there is a four-vector related to temperature. There is no unique choice. One can define βμ [41, 42] with β0=β and βk=0, k=1,2,3 in a particular reference frame. However, good alternatives are kBTμ=βμ/ββ 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 time-likeβμ. The zero four-vector β limit, when Lorentz invariance might be obvious, corresponds unfortunately to infinite temperature so this is not what we are looking for.

In the zero-temperature limit β→+∞, (21) and (25) take the form
$$ G^\psi_0\bigl(x,x'\bigr)=G_0 \bigl(x,x'\bigr)=\int\frac{\mathrm{d}^3p}{2p^0(2\pi)^3}\mathrm{e}^{- \mathrm{i}p^\mu|x-x'|_\mu}. $$
Applying Lorentz generator
$$ J_0=J_{0x}+J_{0x'}, \qquad J^{\mu\nu}_{0x}= \mathrm{i}\bigl(x^\mu\partial^\nu-x^\nu \partial^\mu\bigr) $$
acting on x and x′ by J0x and J0x, respectively, one can check explicitly (a lengthy textbook exercise) that JG=0 (so ΛG=G) when applied to (26), so the free theory is indeed Lorentz-invariant (there are no additional problems with transformation of internal vector or spinor structure). Remember that the time derivative in J is always taken along the contour, namely with complex-time derivative 0 (11).

4.4 Interaction

The rest of the proof is similar to the shape-independence proof. Again, it is important that all time components below, \(x^{0}_{k}\), run along the CTP contour as defined in Fig. 1 (in particular Fig. 2 at zero temperature) and (13). The Green functions
$$ \biggl\langle \prod_{k} X_k(x_k) \biggr\rangle $$
are, by virtue of the Wick theorem, represented by sums of diagrams with external vertices k (not integrated), internal vertices (integrated) corresponding to the interaction part of the Lagrangian density \(\mathcal{L}_{I}\) and connecting lines corresponding to free Green functions (21) and (25). We want to show that J vanishes when applied to the whole diagram. Suppose that the diagram has fixed number N of internal vertices 1..N and K external ones N+1..N+K. then we have to show that
$$ \int \mathrm{d}^4x_1\cdots \mathrm{d}^4x_N \sum_{l>N} J^l \biggl\langle \prod _{n\leq N}\mathcal{L}_I(x_n) \prod_{k>N} X_k(x_k) \biggr\rangle _0=0, $$
where Xk represent fields whose expectation value define the Green function, Jm acts on all fields meeting at the vertex m. The average is taken with respect to free Lagrangian density \(\mathcal{L}_{0}\). We decouple (29) into free Green functions using Wick theorem and use Leibniz rule. Since Lorentz transformations do not affect free Green functions, we have
$$ \sum_{m=1}^{N+K} J^m \biggl\langle \prod_{n\leq N}\mathcal{L}_I(x_n) \prod_{k>N} X_k(x_k) \biggr\rangle _0=0. $$
Since \(\mathcal{L}_{I}\) is a Lorentz scalar we can change \(\sum_{n}J^{n}\to \sum_{n} J^{n}_{0}\). Finally, we perform integration
$$ \int \mathrm{d}^4x\, J^{\mu\nu}_0\cdots $$
over each vertex nN (x=xn), which yields only boundary values due to ∫dtxk0=∫dsxks, see Eq. (C.7). However, free propagators vanish exponentially at large distances and, at zero temperature, with x0→±i∞. Hence (31) vanishes along with the left hand side of (29), which completes the proof.

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 energy-momentum representation [44, 45, 46, 47, 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 off-shell Green functions (not describing scattering processes) and field mass/strength renormalization. They cannot be resolved without returning to spacetime representation.

The energy-momentum representation can be used only on the real part of the contour stretched to the whole real axis (t0→+∞ and t(s±)→−∞), taking separately its upper and lower branch, as Fourier transform is only defined in real axis. The Green function gets additional indices to keep track of the branch, \(G_{uv}(x,x')=G(x_{u},x'_{v})\) where u,v=± and \(x^{0}_{\pm}=x^{0}\pm \mathrm{i}\epsilon\), ϵ→0+. We have
$$ G\bigl(x,x'\bigr)=\int\frac{\mathrm{d}^4p}{(2\pi)^4}G(p)\mathrm{e}^{-\mathrm{i}p^\mu(x_\mu -x'_\mu)}, $$
where, in the space (+,−),
$$\begin{aligned} G^{\pm}(p) =&\left ( \begin{array}{@{}c@{\quad}c@{}} \mathrm{i}(p^2-m^2+\mathrm{i}\varepsilon)^{-1}&2\pi\delta_-(p^2-m^2)\\ 2\pi\delta_+(p^2-m^2)&-\mathrm{i}(p^2-m^2-\mathrm{i}\varepsilon)^{-1} \end{array} \right ) \\ &{}\pm2\pi\delta\bigl(p^2-m^2\bigr)n^\pm \bigl(\bigl|p^0\bigr|\bigr)\left ( \begin{array}{@{}c@{\quad}c@{}} 1&1\\ 1&1 \end{array} \right ), \end{aligned}$$
where δ±(p2m2)=θp0)δ(p2m2), the upper index +/− denotes bosonic/fermionic field with n±(q)=(eβq∓1)−1. We denoted ε→0+ defined for momenta to distinguish it from ϵ defined in spacetime.
In the zero-temperature limit n→0 and the free Green functions are manifestly Lorentz-invariant. The problems arise for interactions. Now, the interacting Green function takes on renormalization corrections,
$$\begin{aligned} G_{0++}(p) =&\frac{\mathrm{i}}{p^2-m^2+\mathrm{i}\varepsilon}\to G_{++}(p) \\ =& \frac {\mathrm{i}Z(p)}{p^2-M^2+\mathrm{i}\varepsilon}, \end{aligned}$$
where Z(p) is regular at p2=M2. The renormalization is obtained by summation of Dyson series of self-energy contributions. However, such step is ill-defined for p2=M2 because ε dominates in this regime. This can be cured by a slight tilt of the real-time contour as in Fig. 3 [10]. In fact, the new contour contains the essential feature of Fig. 2—the endpoints in ±i∞. However, for G±∓ the trick with the contour does not help. We are forced to simply accept
$$ \delta_\pm\bigl(p^2-m^2\bigr)\to Z \delta_\pm\bigl(p^2-M^2\bigr) $$
because this part is only defined for p2=M2. One can circumvent the problem by ad hoc defining extrapolation of G±± to G±∓ and the p2=M2 case. Another argument is the fluctuation dissipation theorem, which imposes relations between elements of G [49, 50], or mass derivative [36]. However, this is only the case of elementary particles like electron and photon. Composed bound states like nucleons, atoms, and molecules get their renormalization corrections, too, but they cannot be simply reduced to a single dressed propagator line. In the time domain, bound states do not lead to any problems because the poles/deltas in propagators appear only in the infinite time limit and we do not have to sum infinite series of diagrams.
Fig. 3

The tilted contour which circumvents the problem of pole renormalization p2m2M2

Further problems may arise from pinch singularities [51], which may occur for products of propagators [(p2m2+iε)(p2m2−iε)]−1. Fortunately, a careful analysis makes them always cancel [46, 47]. Having the invariance of in–out correlations, it would be easy to prove Lorentz invariance if all the Green functions factorized at zero temperature [41, 48], namely
$$ \Bigl\langle \prod X_k(p_k) \Bigr\rangle = \biggl\langle \prod_{k+} X_{k}(p_k) \biggr\rangle \biggl\langle \prod_{k-} X_k(p_k) \biggr\rangle $$
where k± denotes fields defined on the upper(+)/lower(−) branch by the Fourier transform \(X_{\pm}(p)= \int \mathrm{d}^{4}x_{\pm} \mathrm{e}^{\mathrm{i}p^{\mu}x_{\mu\pm}}X_{\pm}(x_{\pm})\). An attempt to prove the factorization requires generalization of CTP to the contours like that in Fig. 4 with varying σ [41, 44, 45, 48]. If σ=β/2 then the factorization occurs indeed for β→∞. However, not every Green function is σ-independent [52, 53]. From translational invariance the nonvanishing Green functions satisfy ∑p=0. The new vertical part of the contour gives additional factors \(\mathrm{e}^{\pm\sigma p^{0}}\) to each propagator from lower to upper branch (+) or vice versa (−). These factors cancel only if
$$ \sum_{k+}p^0_k=\sum _{k-}p^0_k=0, $$
which means that the frequencies/energies of upper and lower branch separately add up to zero as seen in Fig. 5. This situation is common to scattering processes but not in general. It holds e.g. for in–out correlations when all times are on the upper real part due to time shift invariance but not necessarily for in–in correlations when times are located on both real parts (upper and lower).
Fig. 4

The separated upper and lower real contour branches 0≤σβ

Fig. 5

The particle view of factorization mechanism. The Green function is σ-independent only if \(p^{0}_{1+}+p^{0}_{2+}+p^{0}_{3+}+p^{0}_{4+}+p^{0}_{5+}=0\) and \(p^{0}_{6-}+p^{0}_{7-}+p^{0}_{8-}=0\)

Having presented above the complications in energy-momentum 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 t0(s0) and crossings (tntm). 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 Gell-Mann 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 time-reversal 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 p0→−p0 break the time symmetry because of θp0) 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 p0 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 space-like 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 path-integral 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.



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.

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© The Author(s) 2013

Authors and Affiliations

  1. 1.Faculty of PhysicsUniversity of WarsawWarsawPoland

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