Dirac Field

Abstract. Within the context of the Thermofield Dynamics, we introduce generalized Bogoliubov transformations which accounts simultaneously for spatial com-pactification and thermal effects. As a specific application of such a formalism, we consider the Casimir effect for Maxwell and Dirac fields at finite temperature. Particularly, we determine the temperature at which the Casimir pressure for a massless fermionic field in a cubic box changes its nature from attractive to repulsive. This critical temperature is approximately 100 MeV when the edge of the cube is of the order of the confining length ( 1 fm) for baryons. 

On the other hand, for the free-massless Dirac field, the doubled Green function is given by... 

We shall be concerned with the doubled operators describing the energy-momentum tensor of the free Maxwell and Dirac fields according to the tilde conjugation rules, we have, respectively ... 

Similarly, from Eq. (24), we find the Casimir energy and pressure for the Dirac field confined between parallel plates, with anti-periodic boundary conditions, as ... 

Considering such choices for the parameters ay and using the explicit form of Gq1(x — x +y) in the 4-dimensional space-time (corresponding to N = 3), we obtain the renormalized o-dependenl, energy-momentum tensor in the general case, for both Maxwell and Dirac fields ... 

Notice that the results obtained so far (Eqs. (19) and (23) for the Maxwell field and Eqs. (20) and (24) for the Dirac field) are particular cases of the above expressions, corresponding to a = (/3,0,0,0) and a = (0,0,0,i2L) respectively. Another important aspect is that T 11) ) is traceless in both cases, as it should be. Now, we will apply these general results to some specific examples. 

We initially consider the massless Dirac field confined between parallel planes. In this case, we take a = (/3,0,0, i2L) and = +1 in Eq. (29)... 

We teach our students that the infinite zero-point energy that arises when a free relativistic scalar (or Dirac-) field theory is canonically quantized can be subtracted (discarded) by a suitable redefinition of the energy-origin in other words, by normal ordering. However, the energy-origin can be re-defined only once, and only in homogenous space (i.e. without boundary conditions) and... 

In a previous work [33] we suggest an effective approach to study of conditional symmetry of the nonlinear Dirac equation based on its Lie symmetry. We have observed that all the Poincare-invariant ansatzes for the Dirac field i(x) can be represented in the unified form by introducing several arbitrary elements (functions) ( ), ( ),..., ( ). As a result, we get an ansatz for the field /(x) that reduces the nonlinear Dirac equation to system of ordinary differential equations, provided functions ,( ) satisfy some compatible over-determined system of nonlinear partial differential equations. After integrating it, we have obtained a number of new ansatzes that cannot in principle be obtained within the framework of the classical Lie approach. 

The total Lagrangian X = JS G + JS D + JS , then involves the interaction between fermions and the gauge field. The Dirac field will be generically considered to be the electron and the gauge theory will be considered to be the non-Abelian electromagnetic field. The theory then describes the interaction between electrons and photons. A gauge theory involves the conveyance of momentum form one particle (electron) to another by the virtual creation and destruction of a vector boson (photon) that couples to the two electrons. The process can be diagrammatically represented as... 

The mass-density i/r (x, t)y0trn/r(x, t) defines an invariant mass integral for the Dirac field,... 

For the Dirac field in an externally determined Maxwell field, the Lagrangian density including a renormalized mass term is... 

The formulation of a relativistic n-electron Hamiltonian is much more problematic, since one must combine the theory of the Dirac field with the full electromagnetic field, and quantize both fields. The construction of an n-electron Hamiltonian in this context is outside of the scope of this chapter. However, we can consider approximations to the exact electron interaction, which are valid to low orders in c and the derivation of which does not require the quantization of the electromagnetic field. 

In the limit to the Minkowski spacetime, the Dirac field equation of electron (Equation 12.26) is combined with the Schrodinger field equation of the ath nucleus as follows ... 

Non-relativistic Hartree-Fock fields were used for light atoms (T. G. Strand and R. A. Bonham, J. Chem. Phys., 1964, 40, 1686), and Thomas-Fermi-Dirac fields for the heavy atoms (R. A. Bonham and T. O. Strand, J. Chem. Phys., 1963,39, 2200). 

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