Static Electron-Positron Pair Creation in
Strong Fields for a Nonlinear Dirac model
We consider the Hartree-Fock approximation of Quantum Electrodynamics, with the exchange term neglected. We prove that the probability of static electron-positron pair creation for the Dirac vacuum polarized by an external field of strength behaves as for large enough. Our method involves two steps. First we estimate the vacuum expectation of general quasi-free states in terms of their total number of particles, which can be of general interest. Then we study the asymptotics of the Hartree-Fock energy when which gives the expected bounds.
- 1 Estimate on the probability to create pairs
- 2 On the distribution of quasi-free states in the -particle spaces
- 3 Asymptotics of the Polarized Vacuum Energy in strong external fields
- 4 Proof of Theorem 1
- A Product States
In 1930, Dirac  suggested the idea of identifying the vacuum with a sea of virtual electrons with negative kinetic energy. His theory implies that when a sufficiently strong source of energy is provided to the vacuum, some virtual electrons are excited into real electrons, leaving “holes” in the Dirac sea. These holes can be interpreted as positrons, the anti-particles of the electrons, which were experimentally observed in 1933 by Anderson . The extraction of an electron from the Dirac sea is usually called electron-positron pair creation. Sauter , and Heisenberg-Euler  considered the possibility that an external electromagnetic field could excite the Dirac sea to create those pairs. Schwinger  then computed the probability of dynamical pair creation by a constant, uniform, external electric field in the framework of Quantum Electrodynamics (QED). The specific phenomenon of pair production triggered by an external, non-quantized field is thus labeled the Schwinger effect. It is remarkable that this effect is different from the absorption of photons by the vacuum, which is another possible source for pair creation. The Schwinger effect is based on the fact that the vacuum acts as a polarizable medium which can decay into electron-positron pairs when excited by a sufficiently strong electric field. Although the modern formulation of QED no longer describes the vacuum as a sea of virtual particles, Dirac’s theory is still valid in the mean-field approximation .
Experimentally, pair creation in electric fields has not been observed yet because it is only non-negligible in a very strong field. However, recent progress in laser physics have permitted to create very strong fields, making the observation of the Schwinger effect possible in the near future [9, 30, 5].
One has to distinguish between dynamical and static pair creation. Dynamical pair creation consists in studying the time evolution of the vacuum state when an external field is progressively turned on, so that a pair consisting of a scattering electron and a corresponding hole in the Dirac sea is created. The external field is then progressively switched off, and one has to check if the pair still exists when the field is completely turned off. Static pair creation, on the other hand, consists in the study of the absolute ground state (the polarized vacuum) of the Hamiltonian in an external field. Therefore, it is a time-independent process. In this context, the vacuum with an additional particle is energetically more favorable than the vacuum without particle. Static pair creation is easier to study than dynamical pair creation, but it is also a bit less relevant from the physical point of view.
When the interactions between particles are neglected (the so-called linear case), static pair creation was mathematically studied by Klaus and Scharf . They proved that the probability of pair creation becomes 1 when the strength of the positive external field sufficiently increases such that an eigenvalue of the Hamiltonian of the system crosses zero. In the linear case, dynamical pair creation is a very involved phenomenon, whose properties were mathematically understood very recently. Nenciu [22, 23] proved that there is a discontinuity in the probability to create pairs as the strength of a specific external field increases, in the adiabatic limit. Later on, Pickl and Dürr [25, 24] proved that the probability of pair creation tends to 1 in the adiabatic limit, for general over-critical external fields, by carefully studying the resonances created by the eigenvalues diving into the essential spectrum of the Hamiltonian of the system.
This article is devoted to the mathematical study of static pair creation in a nonlinear model describing the polarized vacuum, taking into account the interactions between particles. This model was first proposed by Chaix and Iracane  in 1989 and it has recently been given a solid mathematical ground in a series of papers by Gravejat, Hainzl, Lewin, Séré, and Solovej [13, 14, 17, 15, 10]. As in those papers, the main difficulty of our work is the nonlinearity of the model. The more involved study of dynamical pair creation for the same model will be the subject of future work.
In the considered model, the polarized vacuum in a potential generated by a density of charge is described by an operator on (a density matrix). This operator is a solution to the nonlinear equation
where is the (free) Dirac operator and is any self-adjoint operator such that with . Discarding the operator , we see that is the ground state in the grand canonical ensemble of a dressed Dirac operator with density of charge perturbed by the density of the vacuum. The polarized vacuum therefore interacts with itself.
We shall consider the operator in the limit . Of our particular interest is the probability that pairs are generated, which is a nonlinear function of (see Section 1.1 below). The usual picture [20, 27, 12] is that if the first eigenvalue of is negative, as showed in Figure 1, then the vacuum becomes charged and the probability of creating at least one pair is 1. In the linear case, Hainzl  showed that the charge of the vacuum in the external density is exactly the number of eigenvalues (counted with multiplicity) of the operator crossing 0 when we increase from to . However, because of the nonlinearity of the model we study, detecting for which values of the first eigenvalue will cross 0 is very difficult. However, the probability of pair creation can be very close to 1 without any crossing, as we will explain in Section 1.1.
More precisely, we prove that the probability of static pair creation behaves as (see Theorem 1), where is the charge of a nucleus put in the vacuum, and is a constant depending on different parameters of the model such as the cut-off or the shape of the nucleus. The proof relies on the large- asymptotics of the polarized vacuum energy, which is obtained by using an appropriate trial state. This implies that the average number of particles of the polarized vacuum is of order . We then use general estimates showing that the probability to create pairs for a quasi-free quantum state is bigger than , where is the average number of particles of the quantum state and is a universal constant. Since for the polarized vacuum , the result follows.
The paper is organized as follows. In Section 1 we introduce the Bogoliubov-Dirac-Fock model and we state our main result. In Section 2, we prove the general estimates on quasi-free states on Fock space, which are of independent interest. In the end of Section 2 we come back to our particular setting. In Section 3, we study the large- asymptotics of the polarized vacuum energy. Finally, in Section 4 we prove Theorem 1 using the tools developed in Section 2 and 3. In Appendix A, we recall some properties of product states, which are used in Section 2.
Acknowledgments. I sincerely thank Mathieu Lewin for his precious guidance and constant help. I also acknowledge support from the ERC MNIQS-258023 and from the ANR “NoNAP” (ANR-10-BLAN 0101) of the French ministry of research.
1. Estimate on the probability to create pairs
1.1. Probability to create a pair
The first quantity to define is the probability to create a pair. Let be (separable) Hilbert spaces, representing the one particle (resp. anti-particle) space. The natural space to describe a system with an arbitrary number of particles/anti-particles is the Fock space
with the usual notation for any Hilbert space and with the convention . We also define the vacuum state where . Recall that a state over can be defined 111For convenience, we will later define a state as a linear form on the tensor product of CAR algebras , which in our context does not change anything since all the states we consider are normal. as a positive linear functional with , where is the set of all bounded linear operators on . Notice that any normalized defines a state (called pure state) by the formula , where is the usual inner product on . Following [24, Corollary 4.1], [31, Eq. (10.154)], and [23, Section 2], we define the probability for a state to create a particle/anti-particle pair by
where is the orthogonal projection on . For a pure state , we have .Therefore, if and only if (the vacuum has probability zero to create pairs), while if and only if . In the latter case, notice that does not litterally contain pairs, in the sense that its number of particles may not be equal to its number of anti-particles. This definition merely measures the probability that a state contains real particles/anti-particles.
Typically, represents the free (or bare) vacuum and we want to measure the probability of a perturbation of , representing the polarized (or dressed) vacuum in the presence of an external electric field, to have pairs. Assuming that is a pure quasi-free state, we have the well-known formula (see e.g. [31, Theorem 10.6], [4, Theorem 2.2], or [15, Theorem 5])
where (resp. ) is the free particle (resp. anti-particle) creation operator, (resp. ) is an orthonormal set for (resp. ), and .
From the formula (1.2), we see that as soon as or . Moreover, in this case real particles in the states and real anti-particles in the states have been created. In the linear case, Klaus and Scharf  proved that if the external field is strong enough. However, there can be a high probability to create pairs even when . Indeed, since in this case we have
one sees that is close to 1 when the are large enough. One simple condition is that is large enough, by the inequality
Note that this is indeed (half) the average total number of particle of the state (number of particle + number of anti-particle),
where is the usual number operator on (see formula (2.15)). Hence is close to 1 when is large enough. While the non-vanishing case can be interpreted as the creation of real particles, this second explanation for an increasing can be interpreted as a “virtual pair creation”. In this article, we study an analog of “virtual pair creation” for more general states than those given by formula (1.2).
1.2. Static pair creation in the reduced BDF approximation
In this article, we will rather use the reduced Bogoliubov-Dirac-Fock approximation, a non-linear model enabling to describe an interacting vacuum in which is a function of itself. It was introduced by Hainzl, Lewin, Séré and Solovej in a series of articles [13, 14, 17, 15] after the pioneering work of Chaix, Iracane, and Lions [6, 7]. We will now briefly recall the model and the results needed for our study.
In units where , the reduced Bogoliubov-Dirac-Fock (rBDF) energy functional is the (formal) difference between the energy of the state and that of the free vacuum , with the exchange term dropped. It depends only on the variable ,
Here, is the coupling constant and is the external charge density belonging to the Coulomb space
endowed with the inner product (the hat denotes the Fourier transform222Our convention is .). We also use the notation for any . In order to define the domain of the rBDF energy functional, let us fix a cut-off and define the one-particle Hilbert space
The operator is the usual Dirac operator on , where are the Dirac matrices acting on ,
and are the Pauli matrices,
The operator stabilizes , and its restriction to defines a bounded operator on , which we still denote by . For convenience we introduce . We denote by the Schatten class of all bounded operators on the Hilbert space such that . For any operator on and for any , we let and we define
It is a Banach space endowed with the norm
For any , we define its generalized trace by
and its density by for all , where denotes the matrix kernel of . This density is well defined since implies that is smooth. Furthermore, it is proved in [15, Lemma 1] that for any . We conclude that the rBDF energy functional is well-defined on the convex set
Notice that the kinetic part of the rBDF energy is well defined since
The variational set is the convex hull of , where is the density matrix of a pure Hartree-Fock state, which is a Hilbert-Schmidt perturbation of the free vacuum . The rigorous derivation of the rBDF energy functional and the motivation for this functional setting can be found in [17, 13].
For any and , the rBDF energy functional admits global minimizers on . Minimizers are not necessarily unique, but they always share the same density . Any minimizer satisfies the self-consistent equation
where is a self-adjoint operator such that and . Hence, uniqueness holds if and only if . Notice that since the density is unique, the operator is itself unique. Any minimizer of on is interpreted as a generalized one-particle density matrix of a BDF state (see Section 2.6) representing the polarized vacuum in the potential . When there is a unique minimizer , it is a difference of two projectors and hence it is the generalized one-particle density matrix of a pure state. We emphasize that there is no charge constraint in this minimization problem: In fact, the polarized vacuum could (and should) be charged when is very large. In this case, one may think that an electron-positron pair is created, with the positron sent to infinity.
We want to estimate in terms of , and confirm the picture that the stronger the field, the more pairs are created. As a consequence, we will fix a non-zero density (interpreted as the shape of the external charge density) and study for large. Our main result is the following.
The assumption allows us to have an explicit estimate. If we remove this assumption, we can still prove the weaker result that as . In the sequel, by rescaling if needed, we will assume that
If , we expect an asymptotics lower than , but we are unable to prove it.
We will see that as .
Theorem 1 says that in a very strong field, , the probability to create at least one electron-positron pair is very close to 1. It is reasonable to think that for some sufficiently large , the first eigenvalue of crosses 0 in which case . However, determining the behaviour of the eigenvalue of as increases is difficult because of the nonlinearity of the model and because we are in a regime far from being perturbative. For all these reasons, the estimate (1.6) on is the best we have so far. For very large , one expects that many electron-positron pairs will be generated. We conjecture that we have indeed as for all , where is the orthogonal projector on the -particle space in Fock space (see Section 2). This would mean that for large , the probability to create at least -pairs is very close to 1. Our method of proof only gives this result for . However, if we assume that is a pure quasi-free state for all large enough (which is the case if ), then the conjecture follows from Proposition 2.6 below.
1.3. Strategy of the proof
The proof is separated into two parts. The first one consists in estimating the energy of the polarized vacuum,
from above by . We will also give a lower bound to show that the power is optimal, although we only need the upper bound for the proof of Theorem 1. From this estimate, we then infer that the average number of particles (counted relatively to that of the free vacuum, see Section 2.6) in the polarized vacuum satisfies
The precise statements of these results and their proofs can be found in Section 3.
In a second part, we prove an estimate on the vacuum expectation for a quasi-free state , in terms of its average number of particle . These estimates are of independent interest and therefore we also provide several other estimates for the distribution of quasi-free states in the -particle spaces. These results are contained in Section 2. Finally, in Section 4, we combine the two parts and prove Theorem 1.
2. On the distribution of quasi-free states in the -particle spaces
In this section, we consider general quasi-free states. Only in Section 2.6 we come back to our particular situation of pair creation. We start by introducing the notation used throughout this section.
Let be a complex, separable Hilbert space whose inner product is linear in the second argument. We also need an anti-linear operator such that , where is another complex Hilbert space333Recall that the adjoint of an anti-linear operator is defined as for all and . Typically, one chooses and the complex conjugation, or and . Here we keep abstract because this will be useful for the construction of BDF states in Section 2.6.. Let be the associated Fock space with . We still denote by the vacuum vector. For , we denote by the orthogonal projection on . We recall from Section 1.1 that is the space of all linear bounded operators on . Let be the CAR unital -subalgebra of generated by the usual creation (resp. annihilation operators) (resp. ), for . We denote by the particle number operator on ,
for any orthonormal basis in . Then for all . A state on is a non-negative linear functional which is normalized: . A state is called normal if there exists a non-negative operator on (sometimes called the density matrix of ) such that and for all . Of particular interest are the pure states which are normal states with for with . We define the average particle number of as
The one-particle density matrix (1-pdm) of is the operator defined by
for all . It is a self-adjoint operator on , satisfying . In the same fashion, we define its pairing matrix which is a linear operator on by
for all . It satisfies . Moreover, if we define the operator on by block
then , see [4, Lemma 2.1]. This last relation implies that
in the sense of quadratic forms on . Notice also that . A state is called quasi-free if for any operators which are either a or a for any , then for any and
where is the set of permutations of which verify and for all , and is the parity of the permutation . The relation (2.3) is called the Wick formula. From this definition, we see that a quasi-free state is completely determined by its density matrices . We recall [4, Theorem 2.3]
For any such that with additionally , there exists a unique quasi-free state on with finite number of particle such that is its 1-particle density matrix and its pairing matrix. Furthermore, is normal: there exists with and such that for all .
Now, we need some terminology, which is not universal in the literature. We call Hartree-Fock (HF) states the quasi-free states with and , because when such states are pure, they are usual Slater determinants. Quasi-free states with and , are called Hartree-Fock-Bogoliubov (HFB) states. Pure HFB states are particularly simple since they are Bogoliubov rotations of the vacuum . The aim of this section is to study the distribution of quasi-free states in the particle subspaces , in terms of . Our results are different for HF or HFB, pure or mixed states.
Quasi-free states are also called Gaussian states, in particular because they can be written as (limits of) Gibbs states of quadratic Hamiltonians (i.e. normal states with density matrices , where is a quadratic Hamiltonian). The Gaussian character of quasi-free states is however deeper. In this section we will show that the distribution of a quasi-free state over the different , that is , also has some Gaussian characteristics. More precisely, we will provide estimates of the form
This estimate means that a quasi-free state which has a large average number of particles necessarily has an exponentially small vacuum expectation
Let us explain the picture in a commutative setting. Let for be a Gaussian function such that . Then is the average position of , as desbribed in Figure 2. Now if goes to , the whole function moves to infinity and in particular becomes smaller and smaller: . In other words, as the average position of goes to infinity, goes to zero (and this is true for any with fixed). We will prove a similar fact for quasi-free states. Indeed, for any quasi-free state we have , which is the analog of . We also know that is the average number of particle of ; it is the analog of . We want to prove that when is large, then the main part of lives in the high- particle spaces, that is “follows” its average number of particles, as shown in Figure 2. The analog of in this case is and we thus want to prove that goes to zero as goes to , for any quasi-free state . A natural extension of this result would be that also goes to zero for any fixed .
We will provide explicit estimates depending on the properties of the quasi-free state (pure, mixed, HF or HFB). In the most general case of mixed HFB states, we only derive a bound on the vacuum expectation. The following table tells us where each case is treated.
In spite of their usefulness, we have not found the following estimates in the literature. One main reason is probably that does not belong to the CAR algebra, hence only makes sense for normal states.
A useful tool for the proofs of the following results is the notion of product state. A product state is a state on when each is a state on . While this notion is intuitive, we recall how it is precisely defined in Appendix A.
2.3. Hartree-Fock case
Proposition 2.3 (HF case).
Let be a quasi-free state with and . Then for any we have
We also have the following estimate
for all while if , where .
This estimate implies that for any fixed , as , which is the expected behaviour.
A more theoretical corollary of (2.4) is that for any fixed vanishing at infinity, goes to zero as goes to . This uses the fact that the algebra of these s is generated by the . In the same fashion, one can also prove that as , for any fixed compact operator .
Since is trace-class it can be diagonalized in an orthonormal basis , . For all , let be the unique quasi-free state on having as 1-pdm and as its pairing matrix. Then by Proposition A.2 in Appendix A, one has . Moreover, since
where is the isometry between and defined in Appendix A, we have
To prove (2.5), we notice that for all , , and we identify the coefficients of in . This yields
where we used that for all . Notice from the first equality that if . ∎
2.4. Pure Hartree-Fock-Bogoliubov case
Proposition 2.6 (Pure HFB case).
Let a quasi-free pure state with . Then for any we have
We also have the following estimate for all with and
while if or .
It is well-known [4, Theorem 2.6] that is pure if and only if , which is equivalent to and . The operator is trace-class and is anti-hermitian and Hilbert-Schmidt, hence both and are diagonalizable. Since they commute, they are simultaneously diagonalizable. Remember that any anti-hermitian can be diagonalized in blocks corresponding to its kernel and blocks. Hence there exists a decomposition , with such that
For all , and stabilize ;
If then ;
If then with and .
In particular, where is the quasi-free state on with 1-pdm and pairing matrix . Let us now prove that for all
where is the number operator on . First we consider the case , and let be a normalized vector. Then and so that . Therefore
Since , we have if hence or . In both cases we have
Now suppose and let be an orthonormal basis of such that in this basis and have the form given above. Then and , so that
We deduce that