Fermion field

While Svozil certainly does a credible job in questioning how a local fermion field theory on a tessellated space might still be made able to circumvent the species doubling problem and thereby be converted (at least formally) into a CA-like theory, one could also argue that Svozil does not take his argument as far as it could be... 

Fermion Fields Consider now a fermion field -ip and make the same sort of substitution as above dfjtp —> ipn+p — ipn - The fermionic action becomes... 

It is easy to see that this expression has two minima within the Brillouin zone. One minimum is at fc = 0 and gives the correct continuum limit. The other, however, is at k = 7t/a and carries an infinite momentum as the lattice spacing a 0. In other words, discretizing the fermion field leads to the unphysical problem of species doubling. (In fact, since there is a doubling for each space-time dimension, this scheme actually results in 2 = 16 times the expected number of fermions.)... 

If if>ln(x), to(x) are the fermion field operators discussed at length above, then... 

Thus the current operator indeed transforms like a vector. This must be the case in order that the equation Qdu(x) = ju(x) transform properly, assuming the transformation property (11-267) for Au(x). We now inquire briefly into tike question of the uniqueness of the U(ia) operator, in particular into the question of the phase associated with the fermion field operator. Note that the phase of the photon field operator is uniquely determined (Eq. (11-267)) by the fact that An is a hermitian field which commutes with the total charge operator Q. The negaton-positon field operator on the other hand does not commute with the total charge operator, in fact... 

Similar results can be derived for fermionic fields (H. Queiroz et.al.,). In such a situation, we can start with the propagator given by... 

It is worth emphasizing that for the fermionic field, we have antiperi-odic KMS conditions, which when considered in terms of space com-pactification, coincide withthe physical bag-model conditions (A.P.C. Mal-bouisson et.al., 2004 A. Chodos et.ah, 1974 C.A. Lutken et.al., 1988). Such a result will be used in Section 5. Now we will be concerned with the algebraic elements of the thermal theories. 

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. 

We have shown that generalizations of the TFD Bogoliubov transformation allow a calculation, in a very direct way, of the Casimir effect at finite temperature for cartesian confining geometries. This approach is applied to both bosonic and fermionic fields, making very clear the... 

It is interesting to note that we have calculated the casimir pressure at finite temperature for parallel plates, a square wave-guide and a cubic box. For a fermion field in a cubic box with an edge of 1.0 fm, which is of the order of the nuclear dimensions, the critical temperature is 100 MeV. Such a result will have implications for confinement of quarks in nucleons. However such an analysis will require a realistic calculation, a spherical geometry, with full account of color and flavor degrees of freedom of quarks and gluons. 

For the fermions it is convenient to define the dressed fermion fields... 

To construct the low energy effective theory of the fermionic system, we rewrite the fermion fields as... 

Besides the nuclear attraction, fextix) could also include additional external fields, if present. j/(x) denotes the fermion field operator of the interacting, inhomogeneous system characterized by if, p x ) is the corresponding fermion four current operator. 

In quantum field theory, the gauge field is determined by its Lagrangian density, and the fermion field, by the Dirac Lagrangian density ... 

In order to describe the interaction between the gauge and fermion fields, the following equation is used ... 

In the case of quantum field theory the section determines the Hilbert space of states under a certain gauge. This choice of gauge then determines the unitary representation of the Hilbert space. We may then replace the section with the fermion field /, which acts on the Fock space of states. It is then apparent that a gauge transformation A t > A t + 84 is associated with a unitary transform of the fermion field v / > v / I 8 /. The unitary transformation of the fermion... 

This demonstrates the association between the unitary transformation of the fermion field and the gauge theory. 

As this concerns the nature of non-Abelian electrodynamics, we will pursue the matter of a GUT that incorporates non-Abelian electrodynamics. This GUT will be an 50(10) theory as outlined above. We have that an extended electro-weak theory that encompasses non-Abelian electrodynamics is spin(4) = 51/(2) x 517(2). This in turn can be embedded into a larger 50(10) algebra with spin(6) = 517(4). 50(10) may be decomposed into 517(2) x 517(2) x 517(4). This permits the embedding of the extended electro weak theory with 517(4), which may contain the nuclear interactions as 517(4) 51/(3) x 1/(1). In the following paragraphs we will discuss the nature of this gauge theory and illustrate some basic results and predictions on how nature should appear. We will also discuss the nature of fermion fields in an 517(2) x 51/(2) x 51/(4) theory. 

Note that the integration over the fermionic fields in (33) should be implemented exactly while the assumed integration over A t) will be taken with the saddle point accuracy. Now taking integrals in (28) for the probability we obtain... 

Ultimately, the theory must be consistent with quantum electrodynamics, which reduces in the absence of radiative terms to a time-dependent equation, in which i/ is interpreted as a fermion field operator Y state function field equation of motion is equivalent to simultaneous equations... 

In quantized theory, this is an operator in the fermion field algebra. Assuming mo = 0, the mean value (0 Af 0) vanishes in the reference vacuum state because all momenta and currents cancel out. In a single-electron state a) = al 0), a self-energy (more precisely, self-mass) is defined by Smc2 = a Mc2 a) = a / d3x y0(—eji)i/ a). Only helicity-breaking virtual transitions can contribute to this electromagnetic self-mass. 

The 5(7(2) gauge field W), has three components, corresponding to the isospin vector of matrices r, with no relationship to the coordinate space ct, x. By implication, the fermion field i// is a set of spinors, one for each value of the isospin index. The covariant derivative... 

Considering only the SU(2) field interaction, the Euler-Lagrange equation for the fermion field is... 

As in the case of the electromagnetic self-mass, the implied dynamical mass increment is infinite unless perturbation-theory sums are truncated by a renormalization cutoff procedure. In analogy to electrodynamics, each fermion field acquires an incremental dynamical mass through interaction with the gauge field. This implies in electroweak theory that neutrinos must acquire such a dynamical mass from their interaction with the SUIT) gauge field. For a renormalized Dirac fermion in an externally determined SUIT) gauge field, the Lagrangian density is... 

Transformation properties of W/x can be derived by considering an infinitesimal Sl/(2) gauge transformation, IJ = I — igx(x)-T, forxC ) -> 0. The corresponding infinitesimal transformation of the fermion field is... 

The Lagrangian density for a massless fermion field interacting with the S(J(2) gauge field is... 

The fermion field equation ihy D f = 0 is implied by independent variation of ft. The Euler-Lagrange equations for the gauge field follow from... 

Energy-momentum conservation is expressed by dvT = 0 for a closed system. If Tfi were a symmetric tensor (when converted to 7 /x"), this would be assured because i f Tfi = 0 by construction. Since the gauge field part of the tensor deduced from Noether s theorem is not symmetric, this requires special consideration, as discussed below. A symmetric energy-momentum tensor is required for any eventual unification of quantum field theory and general relativity [422], The fermion field energy and momentum are... 

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

Here eo is the magnitude of the electronic charge. The equation of motion for the massless fermion field is... 

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