Observer dependence of bubble nucleation and Schwinger pair production

Observer dependence of bubble nucleation and Schwinger pair production

Jaume Garriga, Departament de Física Fonamental i
Institut de Ciències del Cosmos, Universitat de Barcelona,
Martí i Franquès 1, 08028 Barcelona, SpainInstitute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Sugumi Kanno, Departament de Física Fonamental i
Institut de Ciències del Cosmos, Universitat de Barcelona,
Martí i Franquès 1, 08028 Barcelona, SpainInstitute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Misao Sasaki, Departament de Física Fonamental i
Institut de Ciències del Cosmos, Universitat de Barcelona,
Martí i Franquès 1, 08028 Barcelona, SpainInstitute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Jiro Soda, Departament de Física Fonamental i
Institut de Ciències del Cosmos, Universitat de Barcelona,
Martí i Franquès 1, 08028 Barcelona, SpainInstitute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Alexander Vilenkin Departament de Física Fonamental i
Institut de Ciències del Cosmos, Universitat de Barcelona,
Martí i Franquès 1, 08028 Barcelona, SpainInstitute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract

Pair production in a constant electric field is closely analogous to bubble nucleation in a false vacuum. The classical trajectories of the pairs are Lorentz invariant, but it appears that this invariance should be broken by the nucleation process. Here, we use a model detector, consisting of other particles interacting with the pairs, to investigate how pair production is seen by different Lorentzian observers. We focus on the idealized situation where a constant external electric field is present for an infinitely long time, and we consider the in-vacuum state for a charged scalar field that describes the nucleating pairs. The in-vacuum is defined in terms of modes which are positive frequency in the remote past. Even though the construction uses a particular reference frame and a gauge where the vector potential is time dependent, we show explicitly that the resulting quantum state is Lorentz invariant. We then introduce a “detector” particle which interacts with the nucleated pairs, and show that all Lorentzian observers will see the particles and antiparticles nucleating preferentially at rest in the detector’s rest frame. Similar conclusions are expected to apply to bubble nucleation in a sufficiently long lived vacuum. We also comment on certain unphysical aspects of the Lorentz invariant in-vacuum, associated with the fact that it contains an infinite density of particles. This can be easily remedied by considering Lorentz breaking initial conditions.

Keywords:
Schwinger pair creation, bubble nucleation, Lorentz violation
\preprint

YITP-12-68, KUNS-2410

1 Introduction

Vacuum decay through bubble nucleation was first analyzed in the ground-breaking paper by Voloshin, Kobzarev and Okun (VKO) VKO (). Their analysis also uncovered an unexpected problem. In the semiclassical picture, the bubble is formed momentarily at rest; then it expands, rapidly approaching the speed of light. Lorentz invariance of the vacuum requires that the probability of nucleating a bubble should be the same in all inertial frames. This suggests that the rate of bubble formation per unit spacetime volume should include an integral over the rest frame of nucleation, that is, over the Lorentz group. This integral is of course divergent, so one obtains a meaningless infinite answer 111For a recent discussion of related issues, see Dvali (); GSV (); GV12 ()..

An elegant resolution of the paradox was given by Coleman Coleman (), who developed an instanton method for calculating the bubble nucleation rate. The instanton in this case is invariant, and its Lorentzian continuation is invariant with respect to Lorentz boosts. From this Coleman concluded that “an expanding bubble looks the same to all Lorentz observers, and to integrate over the Lorentz group is to erroneously count the same final state many times”.

There is little doubt that Coleman’s calculation of the nucleation rate is correct. However, some intriguing questions about the bubble nucleation process remain unanswered. The Lorentzian continuation of the instanton solution describes a bubble which contracts from infinite size, bounces at a minimum radius, and then re-expands. This solution is Lorentz invariant, but it appears that the contracting part of it is unphysical and needs to be cut off. However, the cutoff would necessarily break the Lorentz symmetry, and we would be back to the VKO problem. The answer that is usually given is that the semiclassical approximation breaks down near the nucleation point (that is, near the bounce at the minimal radius). So there is no well defined cutoff surface. The bubble emerges from a fuzzy quantum region, and only its late-time asymptotic behavior has classical meaning.

This may be true, but we can still ask how different observers would see the nucleation process. An observer can fill space with detectors measuring the scalar field of the bubble. This would yield a complete spacetime history of the nucleation process. In particular, the rest frame of bubble nucleation may be determined in this way.

One can anticipate several possible scenarios that can emerge from such an analysis. (A) It could be that the frame of nucleation simply coincides with the rest frame of the detectors. In other words, each observer will see bubbles forming at rest in her own rest frame. (B) It is conceivable that the contracting part of the bubble history is not completely cut off and can be at least partially observed. In fact, in the case of scenario (A), an observer moving relative to the detectors will see the detection begin on the contracting part of the bubble trajectory. (C) Yet another possibility is that the frame of nucleation is influenced by how the decaying false vacuum was set up. If the false vacuum has zero energy, its spacetime is flat, and the space will be filled by nucleating bubbles in a finite amount of time. This implies that the false vacuum could not have existed forever; it must have been created in some manner in the past. We could imagine, for example, that the false vacuum was set up on some spacelike surface, e.g., on a hyperplane . The preferred frame of reference specified by this choice could in principle determine the frame of bubble nucleation.

If the false vacuum has positive energy, it is described by de Sitter space, and the bubble nucleation process can continue forever. However, the geometry of the full de Sitter space is similar to that of the worldsheet of a bubble: it describes a universe contracting from infinite size, bouncing at a minimum radius and re-expanding. Bubbles would fill the universe during the contracting phase, so one needs to introduce a cutoff. This would break the de Sitter symmetry and define a preferred frame222The conclusion that the false vacuum state cannot be extended to the infinite past has been shown to hold in the most general case, without assuming homogeneity or isotropy Borde (); BGV ().. One might think that the memory of the preferred frame would gradually fade away and would have no effect on bubbles nucleating at late times. However, it was shown in GGV () that a peculiar “persistence of memory” effect does not fade away and persists at arbitrarily late times. Thus the possibility (C) that the frame of nucleation is influenced by the initial conditions remains a viable option.

In the present paper we shall attempt to address these issues using pair production in a constant electric field as a model for bubble nucleation. We shall consider charged spin-0 particles, described by a complex scalar field , in a constant electric field in a -dimensional Minkowski space. Such an electric field is Lorentz invariant, and pair production by the Schwinger process Schwinger:1951nm () is closely analogous to bubble nucleation. It is conceivable that despite the invariance of the background, the Lorentz symmetry could be spontaneously broken by the quantum state of the -field. Indeed, particles and antiparticles of the pairs are accelerated by the electric field in opposite directions, and since the number of the pairs is constantly growing, one can expect a growing expectation value of the electric current . Any non-zero value of would break the Lorentz invariance. However, the situation is complicated by the fact that the current is formally infinite and needs to be renormalized.

Hence, the first part of our project is to investigate the invariance properties of the in-vacuum state, defined by requiring that the mode functions of the -field are positive-frequency at . This quantum state has been extensively discussed in the literature (see, e.g., Grib (); Brout:1995rd (); Massar (); Gabriel (); Tanji:2008ku () and references therein), but to our knowledge its invariance properties have not yet been studied. We show by a direct calculation that the in-vacuum state is Lorentz invariant.

Nonetheless, as we shall see, divergent terms in the expectation values of physical observables, like the current or the energy-momentum tensor , do not seem to admit any Lorentz invariant regulators. The unusual properties of the in-vacuum state are due to the fact that at any finite time it contains an infinite number of out-particles. In a more realistic model, the electric field would be cancelled by the back-reaction of the created pairs, and only a finite density of particles would be produced. Alternatively, in the context where the electric field is external, we can still consider states where the number density of pairs is finite at any given moment of time. For instance, we could choose initial conditions such that the number density of pairs vanishes on some initial surface 333 The number density of pairs is not a sharply defined quantity. Nonetheless, it can be defined by using the instantaneous Hamiltonian diagonalization. Such states (with vanishing particle number at ) have been discussed in Refs. Grib () and Tanji:2008ku (). The initial number of particles could also be rigorously defined by considering an electric field which vanishes at and then turned on at some time in the past. An abrupt turn-on at an instant of time was considered in Refs. Tanji:2008ku (); Gavrilov:1996pz (); Gavrilov:2012jk (), and an adiabatic turn-on with was discussed in Ref. Grib (). We shall come back to this issue in Section 4.

Despite its somewhat unphysical properties, we find that the in-vacuum is a useful laboratory for studying the bubble nucleation process, at least in the limit where the initial conditions are removed sufficiently far in the past. In this paper we shall consider tree-level interactions between the pairs and the detector. The kinematics of these interactions is such that, out of the infinite bath of created particles, only those whose momentum relative to the detector is in a certain range will have a chance to interact with it. In this sense, most particles in the bath are invisible, and their infinite density is irrelevant.

To be specific, we model the detector by introducing two additional scalar fields: a charged scalar and a neutral scalar , with the interaction Lagrangian

 Lint=−g(ϕ†ψχ+ψ†ϕχ), (1)

where is a coupling constant. This model has been studied in great detail by Massar and Parentani Massar () and by Gabriel et al Gabriel (), who used it to investigate the Unruh effect for an accelerated detector.

We start with a charged -particle in the initial state. It interacts with -antiparticles of the pairs via444Here and below we use asterisk to denote antiparticles. and thus has a finite lifetime . We shall calculate the momentum distribution of -particles in the final state and use it to deduce the momentum distribution of the created -pairs. To achieve this goal, we will have to consider the interaction (1) with a time-dependent coupling, , so that the detector is turned on for a finite period of time . We shall also briefly discuss the case when the role of the detector is played by the neutral -particle. All our results point in the direction of option (A) – that the frame of pair (and bubble) nucleation is determined by the frame of the detector.

The paper is organized as follows. In Section 2, we introduce the in-vacuum state and review the Schwinger pair production using the method of Bogoliubov coefficients. In Section 3, we show that the in-vacuum is Lorentz invariant. In Section 4, we calculate the two-point function in this vacuum and discuss the expectation value of the current, pointing out the pathologies associated with the infinite density of particles in the in-vacuum. We argue that these can be remedied by considering Lorentz breaking initial conditions. In Section 5, we set up our detector model. The final distribution of -particles and the observed momentum distribution of the pairs are calculated in Sections 6 and 7. Finally, our results are summarized and discussed in Section 8. Some technical details of the calculations are presented in the Appendices.

2 Schwinger pair production

We consider a constant electric field in ()-dimensions. Spin-zero charged particles that are being pair produced by the Schwinger effect are described by a complex scalar field ; the corresponding action is

 S = ∫d2x[−ημν(∂μ+ieAμ)ϕ∗(∂ν−ieAν)ϕ−m2ϕ∗ϕ−14FμνFμν]. (2)

Here, and are the mass and the electric charge of the particles,

 Fμν=∂μAν−∂νAμ=−ϵμνE, (3)

is the unit anti-symmetric tensor with , and is the value of the electric field. It is clear from this expression that a constant electric field in ()-dimensions is Lorentz invariant. More specifically, an observer with a 2-velocity sees the electric field given by

 Eμ=Fμνuν=Enμ,nμ≡uνϵνμ, (4)

where is a unit spacelike vector normal to .

For this electric field, we may choose a Lorentz-covariant gauge in which the gauge field is given by

 Aμ=12Eϵμνxν. (5)

However, since the calculation in this gauge seems technically more involved, we choose a non-covariant gauge where

 Aμ=12E(ϵμνxν−∂μ(x0x1)). (6)

This gives the components of the gauge field as

 Aμ=(At,Ax)=(0,−Et), (7)

where .

2.1 The in-vacuum and the Bogoliubov coefficients

The variation of the action with respect to gives the field equation,

 [−∂2t+(∂x−ieAx)2−m2]ϕ=0. (8)

We expand the field in terms of the creation and anihilation operators,

 ϕ(t,x)=∫dk(2π)1/2(akϕk(t)+b†−kϕ∗k(t))eikx, (9)

where the mode functions satisfy the equation,

 [d2dt2+m2+(k+eEt)2]ϕk=0. (10)

The canonical commutation relations lead to

 [ak,a†k′]=δ(k−k′),[bk,b†k′]=δ(k−k′), (11)

and the normalization condition,

 i(ϕ∗k(t)∂tϕk(t)−ϕk(t)∂tϕ∗k(t))=1. (12)

Linearly independent solutions of Eq.(10) can be expressed in terms of the parabolic cylinder functions,

 ϕ±k(z)∝Dν∗[±(1−i)z], (13)

where

 z≡√eE(t+keE),ν=−1+iλ2 ,λ≡m2eE. (14)

The general solution is given by a linear superposition of . We choose

 ϕk(z)=1(2eE)1/4eiπ4ν∗Dν∗[−(1−i)z], (15)

where the coefficient is chosen in order to satisfy the normalization condition (12). For future reference, we give a useful integral representation of the parabolic cylinder functions,

 DΛ(Z)=e−Z24Γ(−Λ)∫∞0dwe−Zw−w22w−Λ−1,forR(Λ)<0. (16)

From the asymptotic expansion formula,

 Dp(z)∼e−z24zp,for|z|≫1,|z|≫|p|,|arg z|<34π, (17)

we find

 ϕk≈1(2eE)1/4(√2|z|)ν∗ei2z2,forz≪− |ν|. (18)

Hence at , indicating that these mode functions are positive frequency in the “in” region at . The corresponding vacuum state , defined by is the in-vacuum, which has no particles in the asymptotic “in” region.

Because of the non-trivial background, the positive frequency mode function at does not remain positive frequency at finite , and in particular it is given by a linear combination of the positive and negative frequency functions at . The positive frequency functions at are related by a Bogoliubov transformation,

 ϕk=αkϕoutk+βkϕout∗k, (19)

where

 ϕoutk(z)=1(2eE)1/4e−iπ4νDν[(1+i)z], (20)

is the positive frequency mode function at and . We can check that the asymptotic expansion of indeed gives

 ϕoutk≈1(2eE)1/4(√2z)νe−i2z2,forz≫|ν|. (21)

Using a linear relation,

 Dν∗[−(1−i)z]=eiπν∗Dν∗[(1−i)z]+√2πΓ(−ν∗)e−iπ2νDν[(1+i)z], (22)

we can read off the Bogoliubov coefficients,

 αk=√2πΓ(−ν∗)eiπ4(ν∗−ν),βk=eiπν∗. (23)

This gives

 |βk|2=e−πλ=e−πm2eE. (24)

2.2 Momentum distribution and Lorentz invariance

The Bogoliubov coefficients are simply related to the momentum distribution of particles in the “out” state at ,

 dndk=12π|βk|2. (25)

This can in turn be related to the rate of the Schwinger process, ie, the rate of pair creation of charged particles. The mixing between positive and negative frequency modes in the in-vacuum mode functions (15) occurs in the interval

 |z|≲|ν|, (26)

centered at (or ). Hence, the number density of particles created between time and is given by

 n=12π∫−eEt0−eEtdk |βk|2. (27)

 dndt=(eE2π)e−πm2eE. (28)

This is the Schwinger formula for the rate of pair creation.

An important feature of the momentum distribution (25) is that it is independent of . The total density of created particles, obtained by integration over , is infinite. This is not surprising, since the pair creation process was going on at a constant rate for an infinite time. A more realistic calculation would include back-reaction of the pairs on the electric field, so the field would gradually decrease and only a finite density of pairs would be produced. But according to Eq. (28), in a weak electric field with pair creation is a very slow process, and the field can remain nearly constant for a very long time. Thus, one can expect that our idealized treatment should apply in some limiting sense. We shall see, however, that taking the limit and interpreting the result is not always straightforward.

It should be noted that in Eq. (25) is the canonical momentum, which is related to the physical momentum by

 kphys=k+eEt. (29)

The only subtlety is in determining the range of this distribution. Pair creation in a given mode occurs in a spacetime region where the negative frequency contribution to the mode function becomes significant. As we discussed, this occurs at . Hence, Eq. (25) is valid in the range

 −eEt≲k<∞. (30)

The uncertainty in the lower limit of this range can be estimated from (26), which (assuming ) leads to the uncertainty

 Δkmin∼λ1/2m,forλ≫1. (31)

Note that at future infinity, the distribution of out particles is given by (25) in the full range . The final distribution is then Lorentz invariant555This conclusions follows from the Lorentz invariance of the phase space element Landau ().. On the other hand, at any finite time, the invariance is broken by the lower cut-off in Eq. (30).

Eq. (29) leads to , and since are independent of , we have

 dndkphys=12π|βk|2. (32)

From (29) and (30), the range of physical momentum for the created particles is

 0≲kphys<∞. (33)

Hence, we expect to see particles which predominantly have positive momentum

 dndkphys≈12π|βk|2θ(kphys), (34)

and similarly

 dn∗dkphys≈12π|βk|2θ(−kphys) (35)

for antiparticles. This restriction on is intuitively plausible. Pairs are produced at rest; then particles and antiparticles are accelerated by the field in opposite directions. In the limit of infinite time, one can expect to find particles having arbitrarily large momentum, but one would not find particles with (or antiparticles with )666Note, however, that the uncertainty in the lower limit for is of order . For , this seems to leave some room for highly relativistic particles with both signs of the physical momentum..

A potential problem with this picture is that the cutoff at clearly breaks Lorentz invariance. This is not necessarily impossible. On one hand, even though the background is Lorentz invariant, the invariance could be broken by the in-vacuum state which we are using for the -field. This would correspond to option (C) in the Introduction, where the frame of nucleation might be influenced by how the decaying vacuum is set up. Finally, we must consider the possibility that the in-vacuum is Lorentz invariant, but all observers see the distributions (34) and (35) for the momenta of particles and anti-particles relative to the detector’s rest frame. This would correspond to option (A) in the Introduction.

Our next task is therefore to investigate the Lorentz invariance of the adiabatic in-vacuum.

3 Lorentz invariance of the in-vacuum

To examine the behavior of the in-vacuum under Lorentz transformations, we compare two in-vacua defined with respect to different rest frames which are related by a Lorentz boost. To do so, we consider a gauge transformation that relates the components of the gauge field in the two frames, such that has the same form given by Eq. (7) in each frame. A positive frequency mode function of at in the gauge (7) reads

 ϕk(z)eikx;z=√eE(t+keE), (36)

with from Eq. (15). We consider another Lorentz frame given by

 ¯t=γ(t−vx),¯x=γ(x−vt);γ=(1−v2)−1/2. (37)

In the barred frame, if we adopt the gauge , the positive frequency modes at are

 0¯ϕ¯k(¯z)ei¯k¯x;¯z=√eE(¯t+¯keE), (38)

where the prefix ‘0’ means the gauge

 ¯A¯0=0. (39)

Note that the functions are the same as with the replacement ,

 0¯ϕ¯k(¯z)=ϕ¯k(¯z). (40)

To compare the positive frequency function in the barred frame with the one in the original unbarred frame, we have to express the barred frame positive frequency fuction in terms of the original coordinates . To do so, we perform a gauge trasformation from to gauge.

The electromagnetic vector potential trasforms from the barred frame to the original frame as

 0Aμ=∂¯xν∂xμ¯A¯ν, (41)

where is the components of the potential in the gauge (39) in the coordinates . This gives

 0A0=∂¯x∂t¯A¯1=γvE¯t=γ2vE(t−vx), (42) 0A1=∂¯x∂x¯A¯1=−γE¯t=−γ2E(t−vx). (43)

We perform the gauge transformation from this gauge to the gauge (7),

 Aμ=0Aμ−∂μΩ, (44)

so that is given by (7). Namely we set

 A0=0A0−∂0Ω=0,A1=0A1−∂1Ω=−Et. (45)

This can be easily solved. We find

 Ω=Eγ2v2(t2−2vtx+x2). (46)

Under this gauge transformation, since the covariant derivative of ,

 (∂μ−ieAμ)ϕ, (47)

is gauge-invariant, the mode function trasforms as

 0¯ϕ¯k→¯ϕ¯k=0¯ϕ¯ke−ieΩ. (48)

Thus the positive frequency mode function in the barred frame is expressed in terms of the original coordinates as

 ¯ϕ¯kei¯k¯x;¯ϕ¯k(t,x)=ϕ¯k(¯z)e−ieΩ(t,x), (49)

where

 ¯z=√eEγ(t−vx)+¯k√eE,¯x=γ(x−vt). (50)

A simple way to examine the Lorentz invariance of the in-vacuum is to evaluate the Klein-Gordon inner product

 K≡∫t=constdx(ϕk∂t¯ϕ¯k−¯ϕ¯k∂tϕk)e−ikx+i¯k¯x (51)

This product characterizes the mixing of positive with negative frequency modes. If is non-vanishing, it means that the vacua defined by and by are intrinsically different, hence the in-vacuum is not Lorentz invariant. The inner product (51) is computed in Appendix A using the exact mode functions, and the result is

 K=0. (52)

This shows that the in-vacuum (hence also the out-vacuum) is Lorentz invariant.

We shall see in the next Section that this state has certain unphysical features. These would actually make it pathological when we consider interactions beyond the tree level.

4 Two-point function, Current, and Lorentz breaking quantum states

A quantity of particular interest for our problem is the expectation value of the electric current . The physical picture of particles and antiparticles produced at a constant rate and driven in opposite directions by the electric field, suggests that there should be a steadily growing electric current,

 ∂⟨J1⟩∂t=C, (53)

where , while the charge density remains equal to zero,

 ⟨J0⟩=0. (54)

On the other hand, the expectation value of any vector in a Lorentz invariant quantum state should be zero. Since we found that the in-vacuum is Lorentz invariant, this indicates that we should have . Once again, there is some tension between Lorentz invariance and the physical picture of pair production. Let us investigate this question in some detail.

Formally, the current can be expressed as

 Jμ=ie2(ϕ†Dμϕ−ϕ(Dμϕ)†)+h.c. (55)

A naive calculation of the expectation value of in the in-vacuum gives an infinite answer. One way to regulate this sort of infinity, which respects Lorentz and gauge invariance, is to use the point splitting method. The idea is to take the two field operators in the products in Eq. (55) at different spacetime points, and , and then take the limit . The current expectation value can then be expressed in terms of the 2-point function,

 G(1)=G++G−, (56)

where

 G+(xμ,yμ)≡in⟨0|ϕ†(xμ) e−ie∫yxAνdxν ϕ(yμ)|0⟩in, (57)
 G−(xμ,yμ)≡in⟨0|ϕ(yμ) e−ie∫yxAνdxν ϕ†(xμ)|0⟩in, (58)

and the Wilson line has been inserted to make the 2-point function gauge invariant. The expectation value of the current is given by

 ⟨Jμ⟩=ie2limxν→yν(∂∂yμ−∂∂xμ)G(1)(yν−xν)=ielimΔxν→0∂G(1)(Δxν)∂Δxμ, (59)

where .

The 2-point functions are calculated in Appendix B. The result is

 G+(Δxμ)=|α|24πΓ(−ν∗)Wiλ/2,0(ieEΔs2/2)√ieEΔs2/2,forΔt−Δx>0 (60)

and

 G+(Δxμ)=|α|24πΓ(−ν)W−iλ/2,0(−ieEΔs2/2)√−ieEΔs2/2,forΔt−Δx<0. (61)

Here, are Whittaker functions, , and is the Bogoliubov coefficient given by Eq. (23). We have omitted the index of momentum , since is independent of . The function is simply related to ,

 G−(Δt,Δx)=G+(−Δt,Δx). (62)

For this is given by (157), and for it is given by (158).

The first thing we note is that the 2-point functions and are Lorentz invariant. This is a further manifestation of the Lorentz invariance of the in-vacuum. The function is also invariant with respect to time reversal,

 G(1)(Δt,Δx)=G(1)(−Δt,Δx), (63)

but not with respect to spatial reflection,

 G(1)(Δt,−Δx)=G(1)(Δt,Δx)∗. (64)

The latter is not surprising, since the electric field defines a preferred direction on the -axis. is invariant under a simultaneous transformation , . Note also that the functions are not invariant under time reversal; hence the in-vacuum state is not time reversal invariant, as one might expect.

For the calculation of the current (59), we are interested in the 2-point functions in the limit of . This is given by

 G+(Δxμ)≈−|α|24π[ln(ieEΔs22)+ψ(12−iλ2)+2γ] (65)

for and by a complex conjugate expression for . Here, is Euler’s constant. The leading term in the limit of large mass, , is

 G±(Δxμ)≈−|α|24πln(m2Δs24). (66)

The singularity structure of the 2-point functions , and therefore of is rather unusual. Most importantly, the leading divergent term at is different from that for a free field: the coefficient multiplying the logarithm in (65) has an extra factor of , compared to the case of . In terms of the standard terminology, the 2-point function is not of the Hadamard form. Another non-Hadamard feature is the -function discontinuity arising from the different forms of at .

The non-standard singularity structure of the 2-point functions makes the regularization of the expectation values of the current and of the energy-momentum tensor problematic. The point splitting method with Hadamard subtraction cannot be used, and even if one does not insist on the Hadamard form of the 2-point functions, the coincidence limits of and of its derivatives are ill defined, because of the discontinuity on the light cone. The Pauli-Villars regularization also appears to fail, since the coefficient multiplying the leading divergence in (65) is mass-dependent. This seems to indicate that Lorentz invariant regularization of the expectation values of observables is impossible in our in-vacuum.

As mentioned in the introduction, this is to be expected, since the Lorentz invariant in-state contains an infinite number of created pairs. In other words, the divergent current is caused by “actual” particles, corresponding to the non-vanishing occupation numbers of all momentum states with (rather than by zero point fluctuations, whose effect can be subtracted by using Lorentz invariant counterterms).

In Appendix C we calculate the expectation value of using a direct momentum space regularization,

 |k|≪kmax, (67)

and then taking the limit . Physically, the current is due to the created pairs, and therefore the regulator (67) amounts to ignoring the contribution from pairs with . Noting that the mixing between positive and negative frequency modes occurs near the time when , we can think of (67) as a proxy for the situation where the electric field is only turned on for a finite interval of time777The expectation value of the current with an electric field turned on for a finite period of time has been studied by a number of authors (e.g., Refs.Novikov:1980ni (); Krotov:2010ma (); Gavrilov:2007hq (); Gavrilov:2012jk ()). In this case, Lorentz invariance is explicitely broken and the total number of produced pairs is finite. The rate of current growth during the period when the field is nearly constant agrees with our result (53), (68)., . The regulator (67) is not manifestly Lorentz invariant and, not surprisingly, the result which is obtained by using it is not Lorentz invariant either; it agrees with Eqs. (53),(54) with

 C=e2Eπe−πm2eE. (68)

This is what one would expect on physical grounds: the rate of current growth is proportional to the rate of Schwinger pair production.

In view of the unphysical properties of the Lorentz invariant in-vacuum, we are led to the conclusion that (in the presence of a constant electric field), no physical state can be Lorentz invariant.

To illustrate this point more explicitly, we may consider the closely related example of massless spinor QED in dimensions Peskin (). Aside from the conserved electric current , this model also has an axial current, which in dimensions is related to the electric current by

 ~Jμ=e¯ψγμγ5ψ=ϵμνJν. (69)

Here, . As is well known the axial current would be classically conserved, but any gauge invariant (and Lorentz invariant) regulator will lead to a nonzero divergence given by the anomaly equation Adler (); Jackiw ():

 ∂μ~Jμ=e2Eπ. (70)

This equation is of course valid even if the electric field is not constant, but here we are primarily interested in the case where it is, and hence the background is Lorentz invariant. Combining Eqs. (69) and (70), the renormalized electric current satisfies

 ϵμν∂μJν=e2Eπ. (71)

This is consistent with our Eqs. (53),(54),(68) in the massless limit . Note that even though Eq. (71) for the current has a covariant form, it does not admit Lorentz-invariant solutions. In this sense, Lorentz invariance is spontaneously broken: quantum states satisfying Eq. (71) cannot be Lorentz invariant.

5 A model detector

We now turn to the question of how the pair production process is seen by different observers. To this end, we introduce a model detector which involves two additional scalar fields: a charged scalar and a neutral scalar , with the interaction

 Sint=−g∫d2x[ϕ†ψχ+ψ†ϕχ]=−∫dtHint, (72)

where is a coupling constant. We denote the masses of , and particles by , and , respectively. The charge of -particles is the same as that of -particles, as required by gauge invariance of the action.

Suppose first that we have a charged -particle in the initial state. It will scatter on a -antiparticle of the pairs, producing a in the final state. Such an interaction represents a “detection” of , with playing the role of a domain wall in our model. Alternatively, we can have a neutral particle in the initial state, in which case the detection process yields a -particle in the final state.

It will be useful to consider the kinematics of the scattering process. Starting with

 ψϕ∗→χ, (73)

let , and be the momenta of , and , respectively. Momentum conservation requires that

 q−k=p. (74)

We remind the reader that is the canonical momentum of , which is related to the physical momentum by (and opposite sign for -antiparticle). Similarly, for -particle, . For the neutral -particle, . Hence, Eq. (74) implies

 qphys−kphys=pphys, (75)

which simply states the momentum conservation at the moment of collision.

In the absence of an electric field, we also have energy conservation,

 √q2phys+m2ψ+√k2phys+m2ϕ=√p2+m2χ. (76)

Squaring this equation and combining with (74), we obtain, after some algebra,

 2m2ψ kphys=M2 qphys±√q2phys+m2ψ√M4−4m2ϕm2ψ, (77)

where

 M2≡m2χ−m2ψ−m2ϕ. (78)

For a given momentum of the probe , there are two values of for which scattering is possible: one corresponding to the -particle coming from the left and the other from the right. Note also that the scattering is kinematically allowed only if

 mχ>mψ+mϕ. (79)

Similar relations can be written for the neutral probe scattering . The corresponding relation for the masses is .

In the presence of an electric field, the energy conservation (76) does not strictly apply. One way to think about this is that the interaction process is not momentary and involves some time uncertainty . Charged particles are accelerated by the electric field during this time interval, so their energy is changed. Alternatively, energy non-conservation can be understood in terms of the Unruh effect Massar (); Gabriel (). The charged detector particle moves with an acceleration and is exposed to an effective temperature . Processes with energy fluctuation can therefore occur with a Boltzmann suppressed probability . For example, a -particle can decay, , even when , so the decay would be kinematically forbidden in the absence of an electric field.

Apart from the -pair production, -particle pairs will also be produced by the electric field. In addition, triplets of and of will be spontaneously produced from the vacuum. Various interaction processes in our model are illustrated in Fig. 1. In order to avoid false detections, we have to choose the parameters of the model so that processes shown in Figs. 1(a) and 1(c) are suppressed compared to the true detection process shown in Fig. 1(b). This can be achieved by requiring that

 mψ≫mϕ;(mχ−mψ)mψ≫m2ϕ. (80)

The first and the second of these conditions ensure the suppression of processes 1(a) and 1(c), respectively.

We shall calculate the amplitudes for the detection processes like (73) to lowest order of perturbation theory in the coupling , but exactly with respect to the background electric field . We first expand all the fields in terms of the in-vacuum creation and annihilation operators as we did in (9),

 ϕ = ∫dk(2π)1/2(akϕk+b†−kϕ∗k)eikx,ϕ†=∫dk(2π)1/2(b−kϕk+a†kϕ∗k)e−ikx, (81) ψ = ∫dq(2π)1/2(dqψq+f†−qψ∗q)eiqx,ψ†=∫dq(2π)1/2(f−qψq+d†qψ∗q)e−iqx, (82) χ = ∫dp(2π)1/2(cpχp+c†−pχ∗p)eipx. (83)

Here, , and are positive frequencty mode functions of the respective fields at . For and fields they are given by Eq. (15) with masses and , respectively. For the field,

 χp=1√2ωpeipx−iωpt, (84)

where . Since is neutral, is also positive frequency at . The canonical commutation relations are

 [ak,a†k′]=δ(k−k′),[bk,b†k′]=δ(k−k′), [dq,d†q′]=δ(q−q′), [fq,f†q′]=δ(q−q′), [cp,c†p′]=δ(p−p′). (85)

In the interaction picture, the initial state at evolves to a final state at , which is given by

 |f⟩ = exp[−i∫∞−∞dt′Hint(t′)] |i⟩ (86) ≃ |i⟩−i∫∞−∞dt′Hint(t′)|i⟩, (87)

where in the second line we kept only terms up to linear order in .

With a -particle detector, the initial state is

 |q⟩=d†q |0⟩, (88)

where stands for the in-vacuum state, and we can examine the detection process by studying the distribution of -particles in the final state,

 dNχdp=⟨f| c†pcp |f⟩⟨q|q⟩ (89)

To the lowest order in , the out-expectation value in (89) can be expressed as

 ⟨f| c†pcp |f⟩ = ∫∞−∞dt′∫∞−∞dt′′∫dk1dp1∫dk′1dp′1 (90) × ⟨q|Hint(t′′)|p′1,k′1⟩ ⟨p′1,k′1| c†pcp |p1,k1⟩ ⟨p1,k1|Hint(t′)|q⟩, (91)

where we have inserted two sets of intermediate states defined in the in-vacuum,

 |p′,k′⟩=c†p′a†k′|0⟩. (92)

Strictly speaking, we also had to include intermediate states of the form

 d†qf†q′c†p′a†k′|0⟩ (93)

and

 d†qd†q′c†p′b†k′|0⟩, (94)

which would also give non-vanishing matrix elements of . Inclusion of these states would account for the and triplet creation processes of Fig. 1(c). We disregard these processes, assuming that the conditions (80) are satisfied.

Using Eqs. (85), we have

 ⟨p′1,k′1| c†pcp |p1,k1⟩=δ(p−p′1)δ(p1−p)δ(k1−k′1) , (95)

and Eq. (91) simplifies to

 ⟨f| c†pcp |f⟩ = ∫dk |M(p,q,k)|2, (96)

where

 M(p,q,k)=∫∞−∞dt′⟨p,k|Hint(t′)|q⟩ . (97)

Substituting the mode expansions (83) into Eq. (72) for , we obtain

 M(p,q,k)=g√2π δ(q−p−k)∫∞−∞dt′ ϕ∗kψqχ∗p . (98)

Here, the delta-function enforces the momentum conservation, . Substituting this in Eq. (96), we find

 ⟨f| c†pcp |f⟩=g22π δ(0)∣∣∣ ∫∞−∞dt ϕ∗q−p ψq χ∗p ∣∣∣2. (99)

Finally, substituting into (89) and taking into account the normalization

 ⟨q|q⟩=δ(0) , (100)

we obtain the momentum distribution of -particles in the out-state,

 dNχdp=12π∣∣∣ g∫∞−∞dt ϕ∗q−p ψq χ∗p ∣∣∣2≡12π∣∣Aχ(p;q)∣∣2 . (101)

In the alternative setup, with a neutral -particle in the initial state, the final distribution of -particles is given by

 dNψdq=⟨f| doutq†d