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The Gamow vectors and the Schwinger effect

We introduce a `proper time' formalism to study the instability of the vacuum in a uniform external electric field due to particle production. This formalism allows us to reduce a quantum field theoretical problem to a quantum-mechanical one in a higher dimension. The instability results from the inverted oscillator structure which appears in the Hamiltonian. We show that the `proper time' unitary evolution splits into two semigroups. The semigroup associated with decaying Gamov vectors is related to the Feynman boundary conditions for the Green functions and the semigroup associated with growing Gamov vectors is related to the Dyson boundary conditions.


Introduction
The history of the unification of relativity and quantum mechanics began with the formulation of one-particle wave equations for irreducible representations of the Lorentz group (Klein-Gordon, Dirac, etc.). However this attempt was shown to be unsuccessful, due mainly to the appearance of negative energy solutions which was originally considered as a strong anomaly. Quantum field theory (QFT) arose as the solution of such difficulties shedding light on many new results as, for example, particle creation produced by classical fields. In a field theoretical language this effect is originated from the instability of the vacuum of the matter field, which privileges pair creation with respect to pair annihilation, due to the non-symmetrical choice of vacuum, compatible with the idea of the Dirac sea (all negative energy levels are filled). Such a process, which posed a trouble to the one-particle interpretation of the theory, can nevertheless be pictured as the possibility of tunneling from negative to positive energy states under the influence of the potential barrier produced by the external field [1,2]. This picture is extremely helpful for clarifying physical ideas, so we begin explaining it in Sec. 2.
The dissipative behavior of tunneling processes in nonrelativistic quantum mechanics has been extensively studied in the past years, and with this instability can be associated Gamow vectors, which serves as a representation of the exponential decay law [3]. The aim of this work is to study the intrinsic instability of the particle production mechanism. However it is neither clear how to introduce Gamow vectors in the standard first quantized theory nor at the level of second quantization. So we consider an alternative point of view given by a consistent first quantized formulation of relativistic quantum mechanics (RQM) based on a proper time method. 1 The key idea of this formulation is to replace the Dirac solution to the negative energy problem by the Stueckelberg interpretation [11,12,13], i.e. to consider (negative energy) particles travelling backward in time as antiparticles, introducing a fifth invariant parameter for labelling the evolution of the system. This formalism has the advantage that one can deal with a 'one-particle' configuration space of higher dimension instead of the infinite-dimensional problem of QFT. We discuss the main outlines of this formalism for charged scalar particles in Sec. 3, after that we consider the problem of these particles in an external constant and homogeneous electric field (Sec. 4) and then we introduce rigged Hilbert spaces associated to the choice of the boundary conditions for the propagators (Feynman and Dyson) and the corresponding proper time evolution of Gamow vectors generated by a semigroup of unitary operators (Sec. 5), closely related to a subjacent upsidedown oscillator structure.

The physical picture
Since the works of Heisenberg, Euler, and Kockel [14], we know that the vacuum is not an inert object. In fact it behaves as a dielectric in the presence of an external electric field, and such effect introduces the non-linear corrections to the Maxwell equations, to be discussed in Sec. 4. Vacuum polarization corresponds to a virtual pair creation and annihilation process, however actual pair creation can occur in the presence of external fields. Sauter was the first in estimate the probability of creating a pair in presence of a constant homogeneous electric field [1]. His reasoning was made at the level of the semiclassical limit of a first quantized theory. However they are so clear that the simple translation of his physical picture to the intrincate theoretical framework of quantum field theory, shed light on the understanding of the vacuum instability problem. But before discussing the Sauter derivation, let us remember the dictionary which allows us to translate the concepts of first quantization to QFT.
Often, textbooks repeat that we can do relativistic quantum mechanics of the Klein-Gordon field consistently and we can go straightforward to quantum field theory. Certainly, this is not true. In this position wee see a theoretical prejudice against negative energies and indefinite metric spaces. The first obstacle was climbed up by Stueckelberg and Feynman's [11,12,13] and a comprehensive review of the second can be found in the work of Feshbach and Villars [15]. Now let us concentrate in the Stueckelberg and Feynman ideas using the classical picture derived from the Klein-Gordon equation (h = c = 1) being D µ = ∂ µ + ieA µ the gauge covariant derivative and ψ(x) the complex scalar field. Using the WKB approximation in (1) we obtain the Hamilton-Jacobi equation in the classical limit where S is identified with the classical action and its gradient with the canonical momentum The Eq. (3) is the mass-shell condition for spinless particles and it is equivalent to the proper time velocity constraint (η µν = diag{+1, −1, −1, −1}) due to the proportionality between the velocity and the kinematical momentum The essence of the Stueckelberg-Feynman interpretation for antiparticles rests on a fundamental symmetry which appears in the equations of motion of charged particles when we reexpress them in function of the proper time. Then we can see from Eqs. (3) and (6) that the proper time reversal of the motion (which inverts the four-velocity, the action, and the canonical momenta) is equivalent to a charge conjugation in these equations. Then classical motions in which particles go backward in coordinate time ( dx 0 ds < 0), negative kinetic energy states according to Eq. (6), can be reinterpreted as charge conjugated particles (antiparticles) going forward in time. It is important to remark that our argument can be extrapolated to the semiclassical level. In fact, we see from the WKB wave function that the operation which conjugates the charge in the Klein-Gordon equation (the complex conjugation of the wave function) is equivalent to the operation which reverses the motion in the proper time (the inversion of the action). However a pure quantum explanation of this fundamental symmetry requires the PFT we develop in Sec. 3, in which the concept of proper time is introduced at the quantum level. By the moment we only use the fact that the sign of the kinetic energy is the gauge-invariant quantity which classifies particle and antiparticle states. Standard QFT is nothing else than the second quantization (many-particle theory) after changing the level of the natural vacuum (all positive kinetic energy states unfilled) by the notion of vacuum which rest on Dirac's sea idea for fermions (all negative kinetic energy states filled). Therefore a pair creation in the field theoretical language (a positive-kinetic-energy state plus a hole in the sea) corresponds to a transition from a negative-kinetic-energy state to a positive one in the first quantized theory. Note that the charge is also negative in a negative-kinetic-energy state, in such a way that it leaves a positive charged hole in the field theoretical language. It turns positive when we reverse the arrow of time according to the Stueckelberg-Feynman interpretation.
With this ideas in mind, we follow Sauter semiclassical argument. Suppose that we have an homogeneous constant electric field E = Ee 3 . Then the classical expression for a relativistic charged particle of total energy p 0 is where we have assumed that the kinetic energy is initially negative and the charge is −e (e > 0). Suppose that we have such an incoming particle (the outgoing antiparticle in field theoretical language) in the positive direction of the coordinate x 3 (p =pe 3 ). We see that it decelerates in the presence of the electric field, in such a way that the kinetic term in (7) decreases as the potential energy increases. There is a turning point a = p0−m eE at the coordinate x 3 for this classical particle. But, quantum-mechanically, the momentum p can become imaginary (p = ik) allowing the tunneling of this particle through the barrier. Inside the barrier we have the dispersion relation for the semiclassical wave associated with the particle. On the other side of the barrier, we assume that the dispersion relation is for the outgoing wave. It corresponds to a classical particle of positive kinetic energy, which begins its movement at the point b = p0−m eE .
We can estimate the probability of crossing the barrier using the Gamow formula derived from the WKB approximation where A is an undefined proportionality constant. Evaluating the integral we finally have In Sec. 4 we will see that Schwinger exact calculation can be fitted by Sauter expression choosing A = e 2 E 2 8π 3 . But the merit of this heuristic derivation lies on the physical picture it draws. The vacuum of field theory (the Dirac sea) in presence of an electric external field is like an infinite unstable nucleus, which is disintegrated by nucleons emission (our first quantized negative energy states going through the barrier). Another interesting picture can be traced at classical level solving proper time equations of motion [16] (Lorentz force law) For the case of constant electric field discussed above, adequately choosing the initial conditions, we obtain the hyperbolic trajectories Usually, in textbooks on electrodynamics, we find only one of the branches of the hyperbola, depending on the sign of the charge [17]. In this picture both particles and antiparticles evolve in the ordinary direction of coordinate time. 2 However for a given charge the two solutions follows from the equations of motion in proper time (that is analogous of what happens with the relativistic wave equations which admit both positive and negative energy solutions). Moreover according to the Stueckelberg and Feynman interpretation one branch corresponds to the particle and the other to the antiparticle. Now, let us compare the natural length-scale which appears in the classical problem (the minimal separation between the two branches of the hyperbola, 2m eE ) with the quantum scale of the Compton wavelength ( 1 m ) given by the uncertainty principle. We see that when these scales have the same order, particle and antiparticle trajectories overlap increasing the probability of creating a pair. In this case the strenght of the electric field must be of the order E ∼ e 2m 2 according to Sauter estimation for the tunneling process.
Closing this section note that our initial vacuum is unstable under the pair creation process and not under pair annihilation. It is due to we have filled all negative energy states. In Sec. 3 we will see that it is related with the analytical structure of the Feynman propagator.

The proper time formalism for the charge scalar field
In Sec. 2 we have seen that proper time can be introduced in the standard formulation only at the semiclassical level. Such lack of an invariant evolution parameter is the responsible of the main difficulty in reconciling the antagonistic formalisms of quantum mechanics and the theory of relativity [18]. In fact, while relativity deals with the space-time coordinates on an equal footing, quantum mechanics privileges an external absolute parameter to label the evolution of the state of the system. Therefore, in the second case, 'time' should have the properties of a c-number, unlike in the first case where, since the spatial coordinates are raised to the status of operators, Lorentz transformations should impose this character on the temporal coordinate as well. Thus, this dual role of "time" generates a trouble in RQM. This is basically the reason why a one-particle theory in the usual formulation of RQM finds several conceptual difficulties.
In the forties a PTF was developed for a possible unification between special relativity and quantum mechanics, within the framework of a consistent oneparticle theory. 3 However the price payed for PFT was giving up the concept of definite mass state. In fact, suppose that we want to develop a relativistic classical theory, in an explicit covariant way, putting the spatial coordinates and time in the same footing. We immediately notice that the Poisson bracket is incompatible with the classical mass constraint So the classical mass constraint (13) must be removed from the PTF, although nevertheless the standard RQM is recovered on-shell. Notice also that the Poincaré algebra must be enlarged to contain relation (12). Among the different extensions of the Poincaré group, whose algebra includes the canonical commutation relation corresponding to (12), the five-dimensional Galilei group [22] x is the alternative most frequently used in the literature. 4 We consider this formulation in this work. Actually the parameter τ , which gives the name to the formalism, is a priori independent of the classical proper time, but it can be related to it in the classical limit on mass shell. In the case of Galilean version of PTF, this parameter is nothing else that the parameter of the generalized Galilean boost, and it plays the role of a Newtonian time [23,24]. The temporal coordinate has a different status, and it is promoted to the rank of operator accordingly with the classical Poisson bracket (12). This dissociation of roles, solves the conflict exposed above. Of course, the notion of simultaneity and causality are the corresponding ones to Newtonian time, but standard relativistic notions, in coordinate time, are reobtained on shell [e.g. we will see that the retarded (causal) propagator of the PFT, on-shell is reduced to the Feynman one].
In this way the formalism closely copies the general outlines of the nonrelativistic quantum mechanics furnishing the theory with a well known structure. Let us outline it for the case of the spineless relativistic point particle.
Let {|x µ } (µ = 0, 1, 2, 3) be the basis of localized states of the position operator x µ for the charge of the system. This basis spans a linear space endowed with a scalar product, satisfying the normalization and the completeness conditions, In this coordinate representation the state of the system is represented by the wave function, belonging to a four-dimensional Hilbert space, defined on the space-time manifold. The position operator x µ and its canonical conjugate variable, the momentum p µ , which satisfy are given by (In Eq. (15) −i was chosen to preserve the sign in the ordinary relations for the spatial part.) In the Schroedinger picture, represents the probability density for the system to be at the space-time point x µ at "instant" τ . The wave function evolves with a Schroedinger equation, where H = η µν π µ π ν 2M is a super-Hamiltonian and M a super-mass parameter. The time reversal operator in the Wigner sense coincides with the charge conjugation operator This fact naturally introduces the Stueckelberg-Feynman interpretation in the formalism. Notice that from the Heisenberg equation of motion we have so, as we have mentioned above, at the classical level on the mass-shell the parameter τ is proportional to the classical proper time From now on, we re-scale the time calling s = 1 2M τ, using the Schwinger notation, which is more familiar in the literature. s must not be confused with the classical proper time. Schroedinger equation now reads In this case the super-Hamiltonian is re-scaled to H = η µν π µ π ν = π 2 .
The super-Hamiltonian as well as position and momentum operators are Hermitian in the inner product (14). A stationary solution of Eq. (16) is where ψ m 2 (x µ ) is a solution of a generalized Klein-Gordon equation, which is reinterpreted as an eigenvalue equation (the mass eigenvalue m 2 is real, but not restricted a priori to be positive).
We can see that the theory developed is formally identical to ordinary quantum mechanics. Then, all we have learned from this theory can be rewritten in the PTF. For example, let us consider the resolvent of the free Hamiltonian operator We see that R(z) is analytic in all the complex plane except for a cut along the positive real axis. R(z) is the extension to the complex plane of the inhomogeneous Green function of the Klein-Gordon equation The limiting values of R(z) approaching the positive real axis define two analytic functions in the lower (upper) half-plane. These functions are those defined in QFT by adding a negative (positive) small imaginary part to m 2 in order to give sense to the formal expression , (ǫ > 0).

In our notation, plus and minus correspond to the Feynman and Dyson propagators respectively
These functions can be analytically continued across the cut to the second Riemann-sheet on the upper (lower) complex half-plane, defining two analytic functions for all z ∈ C.
As it is well known, Feynman (Dyson) propagator can be obtained in the first quantized theory as the inhomogeneous Klein-Gordon Green function which propagates positive (negative) and negative (positive) energy states forward (backward) and backward (forward) in time, respectively. In the language of QFT Feynman propagator can be obtained as the mean value in the vacuum state of the time ordered product of field operators Eq. (19) in coordinate representation reads 4 The Heisenberg-Euler effective action and the particle creation The effective action S eff due to the interaction of the vacuum current with the external field such that leads to the Heisenberg-Euler corrections of the Maxwell equations. As it was shown by Schwinger [7] it can be obtained in the proper time formalism computing is the effective Lagrangian and C(x) is an additive constant determined in such a way that L eff (x) = 0 in the absence of external fields. We see from Eq. (21) that we have again reduced a field theoretical problem (the calculation of the effective action) to a quantum-mechanical one in a higher dimension. In fact, now our problem consists on evaluating the matrix element x| e iHs |x . Moreover we can reinterpret it as the persistence amplitude of the off-shell particle to remain at the point x µ of the space-time, and L eff (x) as the on-shell correlate of this amplitude per unit 'proper time'. The matrix element x| e iHs |x can be evaluated by path integrals or in the Heisenberg picture of the canonical formulation by means of an ingenious procedure developed by Schwinger departing from the integration of the Heisenberg equations of motion. Here we solve the eingenvalue problem in the Schroedinger picture. The reason is that through this procedure it is clearer that the proper time evolution splits into two semigroups.
Let us consider again the problem of a homogeneous and constant electric field. Using the conventions of Sec. 3 the super-Hamiltonian reads We see that it can be split in the Hamiltonian of an upsidedown harmonic oscillator plus the Hamiltonian of two free particles in the coordinates of the plane perpendicular to the electric field The eigenvalue problem for the free particle part has the standard plane-wave solution x 1 x 2 |p 1 p 2 . The inverted harmonic oscillator has pure imaginary frequency ω 2 = ±ieE. In Sec. 5 we see in detail that the two signs correspond to the spliting of the proper time evolution of this unstable system towards the future and towards the past respectively. The positive imaginary solution of the eingenvalue problem, corresponds to the generalized eingenstate representing a decaying Gamow vector for s > 0. 5 Using mass 'eingenstates' |p 1 p 2 , p 0 n of H we can easily evaluate the matrix element (see Appendix) . which coincides with the expression obtained by Schwinger [7] for the Heisenberg-Euler correction in the spinless case. Now let us discuss the particle creation process associated with the imaginary part of We can analytically continue the integrand to the lower half-plane, then the integral along the positive real axis becomes where Γ 0 is a path with the same end points. The integrand has poles in z ±n = ±i nπ eE . 6 Using the residues theorem we can rewrite L eff (z) as where the path Γ −1 has the same end points as Γ 0 in such a way that the closed counterclockwise contour Γ −1 ∪ (−Γ 0 ) encircles the first pole in the negative imaginary axis, z −1 = −i π eE . The probability of creating one-pair 7 per spacetime volume is given by Thus we finally have according to Sauter's estimation.
5 Gamow vectors associated with particle creation by an external field In Sec. 4 we have seen that an inverted harmonic oscillator structure appears in the Hamiltonian. We claim that the instability of the vacuum in the field theoretical language has its correlate in this simple unstable system at the level of the off-shell theory. We have noticed that this inverted oscillator has a pure imaginary frequency ω = ±ieE (antioscillator). This fact leads to a 'complex eigenvalue' problem which requires some technical points we are going to discuss in this section. We can solve the eingenvalue problem for this system from the solutions of the harmonic oscillator, obtaining two sets of 'complex eigenvalues' (which do 6 Note also that the resolvelt has a cut on the positive real axis and poles zn = ±ieE(n + 1 2 ) in the positive imaginary one. Since the effective Lagrangian can be written as L eff (z) =−i x| ln R(z) |x + C(x), the poles of the resolvent are related with the poles of the integrand. 7 The contribution corresponding to create two-pairs can be obtained taking a path which encircles the pole z −2 , and so on.
where K is the subset of Schwarz functions (S) of compact support and Z is the subset of Schwarz integer functions of exponential order, restricted to the real axis. {|v } and {|u } are two representations constructed with the generalized eigenvectors of the operators v = 1 √ 2 which are the analogues of the creation and annihilation operators for the harmonic oscillator. Therefore we can construct a pair of rigged Hilbert spaces: where L 2 (R) is a Hilbert space, Φ ± are dense subsets of L 2 (R) with their own complete nuclear topology, and Φ × ± are the dual spaces of Φ ± . The evolution operator U = e iHs is continuous on Φ ± and such that U † Φ ± ⊂ Φ ± , for s > < 0 only. Thus we can obtain the evolution in s of the pair of Gamow vectors as |p 0 n(s) = e −eE(n+1/2)s |p 0 n , p 0 n(s) = e eE(n+1/2)s | p 0 n , which are functionals in Φ × ± , respectively. We see that the pair of Gamow vectors represent a decaying state towards the future and a growing state from the past, respectively. These Gamow vectors are related via the s-time reversal Wigner operator K, 8 since it can be proved [25] that K : Φ ± −→ Φ ∓ , and therefore We have seen that the unitary temporal evolution splits into two semigroups. Let us interpret the physical meaning of such splitting. Coming back to the offshell propagator let us evaluate it for the Hamiltonian (22): = θ(s) dp 1 dp 2 dp 0 ∞ n=0 x| e iHs |p 1 p 2 p 0 n p 1 p 2 p 0 n| y , where we have used the completeness relation (34) since their generalized eigenvectors are well defined only for s > 0. It shows that the retarded condition in time s (Feynman prescription) is satisfied only for Gamow vectors decaying towards the future. Similarly the Dyson prescription is related to growing Gamow vectors. We can conclude that the choice of Feynman or Dyson prescriptions is a priori a conventional matter when we consider the whole Universe (or a closed isolated system). That is, for example, Feynman boundary condition for which particles propagate towards the future and antiparticles towards the past in coordinate time, is correlated with the Gamow vectors decaying towards the future in proper time. Then we have a unavoidable proper time asymmetry, coming from a proper time symmetrical theory which splits the dynamical evolution into two semigroups. But once we have made the Feynman choice, 'proper time' asymmetry is a substantial thing providing a privileged direction of time, the one in which Gamow vectors decay. Dyson choice only leads to a specular world, in which we can be living just now, if we conventionally interchange the role of past and future.

Appendix
The matrix element x| e iHs |x can be factorized as the matrix element for two free particles and a upside down harmonic oscillator .