Applications Of The Heat Kernel Method To Quantum Field Theory And Statistic Physics

The heat kernel method is a very important method widely used in mathematicsand physics. The purpose of this method is to extract from a spectral sum function theinformation of a differential operator or the background information where theoperator is defined. During these decades, heat kernel methods have greatly promotedmathematical and physical developments. This is because heat kernel is easier tocalculate and its analytic properties are better comparing with all the other spectralfunctions. In this thesis, we first present an introduction of heat kernel methods indetail, and then we study several problems exist in quantum field theory and that instatistical physics with heat kernel methods.For applications in quantum field theory, we first provide the relations betweenheat kernels and scattering phase shifts. Then with these relations, we establish directconnections between heat kernel methods and scattering spectral methods (anotherimportant method in quantum field theory) on calculating one-loop quantumcorrections: We show that these two methods are not only formally equivalent, theirrenormalization treatments can also be included in the standard Feynman diagramrenormalization framework. Besides, with the relations between heat kernels andphase shifts, we for the first time obtain a large-time approximation of the heat kernelusing the low-energy approximation of phase shifts. We also obtain a high-energyexpansion of the phase shift, and express the coefficients of the expansion with heatkernel coefficients.For applications in statistical physics, we first provide the formula of the pressuredistribution on the boundary exerted by ideal quantum gases. Then we calculate thepressure on the surface of a rectangular box. We find that the pressure vanishes at theedges of a box, peaks at the middle of the surface, and the pressure magnitude fordifferent statistics satisfespFermi pclassical pBoseon every boundary point.Moreover, we discuss the relation between the surface pressure tensor distribution andthe generalized forces in thermodynamics, and find that the generalized forcecorresponding to a certain deformation can be expressed as a combination of thesurface pressure. Finally based on the grand potential with boundary modifications,we suggest that a confned ideal gas can exert forces whose effect is to reduce the totalsurface area of the boundary of an incompressible object. The force contributes to the adhesion of two thin flms in contact with each other. The force can also lead to therecoiling of a column, and it can compress an open container immersed in the gas.