Electron-Positron Field

The above rules are readily applied to the case of the interacting with an external field for which... 

As was shown in [13], to include the relativistic recoil corrections in calculations of the energy levels, we must add to the standard Hamiltonian of the electron-positron field interacting with the quantized electromagnetic field and with the Coulomb field of the nucleus Vc an additional term. In the Coulomb gauge, this term is given by... 

The mathematical problem associated with the Dirac Hamiltonian, i.e. the starting point of the relativistic theory of atoms, can be phrased in simple terms. The electron-positron field can have states of arbitrarily negative energy. As a general feature of the Dirac spectrum this instability occurs even in the case of extended nuclei and even in the absence of any nucleus (free Dirac spectrum), the energy is not bounded from below. This gives rise to the necessity of renormalization and well-established renormalization schemes have been around for many decades. Despite their successful applications in physics, we may ask instead whether there exist states that allow for positivity of the energy. 

A rigorous mathematical model for the relativistic electron-positron field in the Hartree-Fock approximation has been recently proposed (Bach et al. 1999). It describes electrons and positrons with the Coulomb interaction in second quantization in an external field using generalized Hartree-Fock states. It is based on the standard QED Hamiltonian neglecting the magnetic interaction A = 0 and is motivated by a physical treatment of this model (Chaix and Iracane 1989 Chaix et al. 1989). 

The Dirac equation (7) can be considered as an equation for the components of the classical electron-positron field >I a(a ), 4 c(a ) (a is the spinor index). The Lagrangian for this classical field can be constructed as ... 

From the invariance of the action S with respect to translations in 4-coordinate space follow the expressions for the energy and momentum densities of the electron-positron field ... 

For the quantization of the electron-positron field the expansion of the arbitrary solution of the Dirac equation (x) with the fidl set of stationary... 

Now the operators (a ), (x) may be called the operators of the quantized electron-positron field. These operators are defined in the Fock space and act on the state vector ). The creation and annihilation operators satisfy the anti-commutation relations ... 

The expression for the Hamiltonian of the free electron-positron field according... 

The last term in I5q(94) presents the infinite energy of the vacuum and should be omitted. To avoid the occurence of the infinite vacuum energy both for the electromagnetic and electron-positron fields the normal product (7V-product) of the operators is traditionally used. Within the fV-product the positions of the creation operators are aJways to the left of the positions of the annihilation operators for the same particle. The infinite vacuum energy never occurs provided that the 7V-products are always used. 

The correct frame of description of interacting relativistic electrons is quantum electrodynamics (QED) where the matter field is the four-component operator-valued electron-positron field acting in the Fock space and depending on space-time = (ct, r) (x = (ct, —r)). Electron-electron interaction takes place via a photon field which is described by an operatorvalued four-potential A x ). Additionally, the system is subject to a static external classical (Bose condensed, c-number) field F , given by the four-potential (distinguished by the missing hat)... 

For a time-independent scalar potential, the electron-positron field operator, (a ), is expanded in a complete basis of four-component solutions of the time-dependent Dirac equation [19],... 

M. Huber, H. Siedentop. Solutions of the Dirac-Fock equations and the energy of the electron-positron field. Arch. Rut. Mech. Anal, 184(1) (2007) 1-22. 

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