Probability Thermodynamics and Probability Quantum Field

Abstract

In this paper, we introduce probability thermodynamics and probability quantum fields. By probability we mean that there is an unknown operator, physical or nonphysical, whose eigenvalues obey a certain statistical distribution. Eigenvalue spectra define spectral functions. Various thermodynamic quantities in thermodynamics and effective actions in quantum field theory are all spectral functions. In the scheme, eigenvalues obey a probability distribution, so a probability distribution determines a family of spectral functions in thermodynamics and in quantum field theory. This leads to probability thermodynamics and probability quantum fields determined by a probability distribution. There are two types of spectra: lower bounded spectra, corresponding to the probability distribution with nonnegative random variables, and the lower unbounded spectra, corresponding to probability distributions with negative random variables. For lower unbounded spectra, we use the generalized definition of spectral functions. In some cases, we encounter divergences. We remove the divergence by a renormalization procedure. Moreover, in virtue of spectral theory in physics, we generalize some concepts in probability theory. For example, the moment generating function in probability theory does not always exist. We redefine the moment generating function as the generalized heat kernel, which makes the concept definable when the definition in probability theory fails. As examples, we construct examples corresponding to some probability distributions. Thermodynamic quantities, vacuum amplitudes, one-loop effective actions, and vacuum energies for various probability distributions are presented.