Advances in Classical Field Theory

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We introduce a three independent functions field formalism for stationary and non-stationary barotropic flows. This is less than the four variables which appear in the standard equations of fluid dynamics which are the velocity field v and the density ρ.

Equations of motion and energy-momentum tensors are obtained for a dissipative medium sustaining electromagnetic polarizations using a field formalism. A previous work has been simplified by reducing the number of independent vector fields interacting with the sink modes. A relativistic formalism of the same is also suggested.

Usually the electromagnetic field is naively thought as given by six independent quantities E and B. However in chapter 2 the field was presented as dependent on a four-component vector potential. In this chapter a different approach is formulated, in which electromagnetic field is determined by two scalar complex functions. This approach will serve as a starting point for the following variational analysis given in the next chapter, where the electromagnetic field is shown to depend only on a single complex function.

Possible variational principles for excitation of an electromagnetic field in a wave guide are discussed. Our emphasis is not on the calculation of the modal shapes, which is common in previous art, but rather on the calculation of modal amplitude evolution, which are important in electron devices such as free electron lasers and gyrotrons. The variational formalism allows us to show that the electromagnetic field can be defined by a single complex function.

It will be shown that barotropic magnetohydrodynamics is equivalent to a four function field theory, reducing the number of equations and variables needed to describe the theory from seven (the magnetic field B the velocity field v and the density p )to four function field theory possesses a novel double infinite symmetry group. Non trivial topologies and the differences between stationary and non-stationary flows will be given detailed attention.

It is stated in many text books that the any metric appearing in general relativity should be locally Lorentzian i.e. of the type ημv = diag (1,−1,−1,−1) this is usually presented as an independent axiom of the theory, which can not be deduced from other assumptions. In this work we show that the above assertion is a consequence of a standard linear stability analysis of the Einstein equations and need not be assumed.

”In a spiral galaxy, the ratio of dark-to-light matter is about a factor of ten. That’s probably a good number for the ratio of our ignorance-to-knowledge. We’re out of kindergarten, but only in about third grade.” Vera Rubin

By a special representation of the number density operator and working with a modulus-phase formalism we obtain the dissipative current, needed to ensure particle number (or charge) conservation in the conducting system coupled to an environment. This generalizes and simplifies previous derivations aimed at the Lindblad type of master equations. In addition to a part depending linearly on the Hamiltonian current, we obtain also a pseudo-curl contribution to the dissipative current.

The study of non Abelian gauge theories is thought to be in the realm of quantum field theory, having to do mainly with the study of elementary particle phenomena. The non Abelian gauge theory widely known as the ”standard model” describes correctly all elementary particles and forces except gravity.

Our goal here is to show that non Abelian gauge theories appear also in physical settings in which the field need not be quantized (second quantization). Hence they can be considered as classical fields which are the subject of this book. In particular it will be shown that those fields appear in the study of molecular systems is the frame work of the Born - Oppenheimer theory.

Kramers-Kronig or dispersion relations that arise from the analytic behavior of electromagnetic and other classical waves are here adapted to quantum waves. In the framework of theories that view the wave packet (wp) collapse as a time-continuous process, we postulate analytic properties for the component-amplitudes in a wp as functions of a complex time. We then construct a model which embodies the removal of the non-selected components in the aftermath of the measurement, to be ultimately followed by a thermalequilibrium like superposition state of the system. Conjugate relations hold between component-moduli and phases and these show that a non-selected component (one that vanishes in the measurement) acquires within the duration of the collapse a fast oscillating phase factor. Thus, by virtue of mathematical properties, both phase-decoherence and amplitude-decay have to occur in a collapse process, indifferently to the physical mechanism.

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