Phase-space structure of the Dirac vacuum

Abstract

We study the phase-space correlation function for the Dirac vacuum in the presence of simple field configurations. Our formalism rests on the Wigner transform of the Dirac-Heisenberg correlation function of the Dirac field coupled to the electromagnetic field. A self-consistent set of equations obeyed by the 16 components of the phase-space correlation function and by the electric and magnetic field is derived. Our approach is manifestly gauge invariant. A closed system of integro-differential equations is obtained neglecting the quantum fluctuations of the electromagnetic field as should be appropriate for strong fields. These equations are an extension of the Vlasov equations used in the description of plasma. Our first applications address the production of particles in strong external fields. We set a framework for the inclusion of the back reaction of produced particles and for the description of the local changes of the vacuum state. I. I N T R O D U C T I O NThe structure of the QCD vacuum has been considered an important element for the understanding of stronginteraction physics [I]. T h e electroweak vacuum I-Iiggs field is believed to be the origin of all elementary-particle masses [2]. T h e QED vacuuln in strong fields has been explored both theoretically and experimentally in heavyion collisions and there is a significant discrepancy between the observed particle spectra and the theoretical predictions [3]. All this calls for a renewed effort to develop a systematic framework to describe the structure of the vacuum in a manner which would be to a large degree independent of perturbative quantum field theory. Despite its historical name, the vacuum state is not empty; it is populated with myriads of virtual particles which endow it with a rich structure. The standard, perturbative formulation of quantum field theory deeinphasizes this fact by treating the vacuum as just the simplest possible reference state: the one with the lowest possible energy and with all the relevant symmetry properties. All the remaining state vectors that are introduced to model real physical situations are defined relative to this unique reference state by building excitations on top of the ground state. This pragmatic approach enables one to calculate effectively the elements of the S matrix with the help of appropriate Feynman diagrams without ever paying attention to the problem of the internal structure of the vacuum and to its time evolution.In our opinion, in order to obtain an effective description of the local vacuum structure and its time evolution, we should use a single time parameter. This will enable us to treat the &ED vacuum as a genuine dynamical system with its fully prescribed dynamics, its conservation laws, and a whole new category of physical phenomena that take place in the phase space of virtual particles. There is an obvious drawback that the correlation functions will not be manifestly covariant, although the theory will remain relativistically in-variant. On the other hand, we will gain by introducin...

Relativistic Wigner functions

“…Second, the requirement of the gauge invariance of the Wigner function is fulfilled by the insertion of the line integral in the exponent that compensates for the changes of the phases of the Dirac field operators. Thus, the final expression for the 16-component Wigner function introduced in [3] is:…”

Relativistic kinetic equations for electromagnetic, scalar, and pseudoscalar interactions

“…The main advantage of the equal-time formulation lies in the fact that the initial value of the equal-time Wigner function can be directly obtained from the corresponding initial field operators, since there is only one time scale in the equal-time formulation. Therefore some quantum problems such as pair production [5,6] in strong electric fields have so far only be solved in the equal-time formulation [3,7]. The advantages of the covariant formulation, on the other hand, are explicit Lorentz invariance and the feature that the kinetic equations naturally split into a transport equation of Vlasov-Boltzmann type and a generalized mass-shell condition [1,2] which makes explicit the off-shell effects generated by the collisions in the system.…”

“…One of the recent progresses in transport theory [1,2] is the establishment of transport equations for spinor [3] and scalar [4] equal-time Wigner functions with abelian gauge interaction. The main advantage of the equal-time formulation lies in the fact that the initial value of the equal-time Wigner function can be directly obtained from the corresponding initial field operators, since there is only one time scale in the equal-time formulation.…”

“…[8], we write the equal-time Wigner function W 3 (x, p) as the energy average of the covariant Wigner function W 4 (x, p) and decompose it in a similar way [3] into its spinor components:…”

Dirac open-quantum-system dynamics: Formulations and simulations

“…An alternative quantum field theoretic formulation of the Wigner function for Dirac fermions has also been put forth [57,[61][62][63][64][65][66].…”