Heat Kernel and Loop Currents by the Generating Function Method

The generating function method is applied to the trace of the heat kernel and the one-loop effective action derived from the covariant perturbation theory. The basis of curvature invariants of second order for the heat kernel (Green function) is built and simple rules for form factor manipulations are proposed. The results are checked by deriving the Schwinger-DeWitt series of the heat kernel and divergences of one-loop currents.Comment: 12 pages, LaTeX, to appear in: Heat Kernel Techniques and Quantum Gravity, Discourses in Mathematics and Its Applications, No. 4, ed. by S. A. Fulling, (Texas A&M University, College Station, Texas, 1995

The method of the kernel of the evolution equation in the theory of gravity

The method of covariant perturbation theory allowed for the computation of the kernel of the evolution equation on a spin Riemannian manifold. The proposed axiomatic definition of the covariant effective action introduces the universal scale parameter, with the length square dimensionality, into a dimensionless mathematical theory. It is shown that this geometrical result has a physical meaning of the action of field theory, including gravity. Two lowest tensor order terms of this functional are independent of a spin group and local. They reproduce the action of general relativity with the cosmological constant. The current value of the universal scale can be determined with the measured Hubble constant. This scale parameter considered as a physical variable can let us build cosmological theories axiomatically.Comment: 10 pages, author's translatio

A geometrical theory of thermal phenomena based on the kernel of the evolution equation

The kernel of the evolution equation is used to build a mathematical theory of thermal phenomena of gaseous and condensed matter. The group velocity of sound and the molar density are proposed to be its two thermal variables that replace the over-complete set of temperature, pressure and volume. The defining constants of the New SI (2019) of physical units are used to form a dimensionless variable of the new geometrical thermal theory. The dimensionless thermal sum generates, by the variation principle, equations on physical observables. Energy and entropy are neither defined nor needed in the present formalism. The molar specific heat is derived and its temperature dependence is verified with the experimental data for mono-atomic gases. Critique of the postulates and notions of traditional statistical thermodynamics is given.Comment: 31 page