Wu-Sheng Dai

We reveal a duality in the traveling-wave generalized Korteweg-de Vries (GKdV) equation. Based on this duality, we propose a method for constructing families of exactly solvable equations. Equations related by the duality transformation form a duality family, enabling the derivation of solutions for all members from any single solved equation.

The Padé approximant performs the pole analysis in statistical mechanics and particle physics. The Padé approximant can recover the singularity information lost in the polynomial approximation given by perturbation methods but also introduces redundant singularities. This paper presents a method to identify redundant singularities by analyzing the approximation itself rather than relying on the original function or underlying physical principles.

In this paper, we present a Mach’s principle-based model of the origin of mass. According to Mach’s principle, the mass of matter arises from the coupling with the rest of the Universe rather than being an inherent property of matter itself. This model, in a sense, provides a model realization of Mach’s principle. In this model, the mass of matter is determined by spacetime. We illustrate the model by solving exact solutions in various spacetimes, such as static spherical, Robertson-Walker, and static cylindrical spacetimes.

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 quantum field theory. This leads to probability thermodynamics and probability quantum fields determined by a probability distribution. In constructing spectral functions, we encounter a problem. The conventional definition of spectral functions applies only to lower bounded spectra. In our scheme, however, 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. To take the lower unbounded spectra into account, we generalize the definition of spectral functions by analytical continuation. 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 introduced in this paper, which makes the concept definable when the definition in probability theory fails. Thermodynamic quantities, vacuum amplitudes, one-loop effective actions, and vacuum energies for various probability distributions are presented.

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 quantum field theory. This leads to probability thermodynamics and probability quantum fields determined by a probability distribution. In constructing spectral functions, we encounter a problem. The conventional definition of spectral functions applies only to lower bounded spectra. In our scheme, however, 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. To take the lower unbounded spectra into account, we generalize the definition of spectral functions by analytical continuation. In some cases, we encounter divergences. We remove the divergence by a renormalization procedure. 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 introduced in this paper, which makes the concept definable when the definition in probability theory fails. 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.

We propose an approach for calculating one-loop effective actions and vacuum energies in quantum field theory. Spectral functions are functions defined by the eigenvalues of an operator. One-loop effective actions and vacuum energies in quantum field theory, as well as scattering phase shifts and scattering amplitudes in quantum mechanics, are all spectral functions. If a transformation between different spectral functions is identified, we can obtain a spectral function from another through the transformation. In this paper, we convert quantum mechanical methods for calculating scattering phase shifts and scattering amplitudes into quantum field theory methods for calculating one-loop effective actions and vacuum energies. As examples, the Born approximation and the WKB approximation in quantum mechanics are converted into quantum field theory methods. We also calculate the one-loop effective action and vacuum energy of scalar fields in the Schwarzschild spacetime and the Reissner-Nordstr\"{o}m spacetime as examples. Some integral representations of the Bessel function are given in appendices.

We build a model of the origin of physical constants, including masses and coupling constants. We consider the quantum correction of masses and coupling constants. Some exactly solved leading quantum corrections are given. In the model, the physical constant originates from a coupling between the matter field and a background field. We show that if such a background field as it should be in the real physical world is a quantum field, then the physical constant will have a space- and time-dependent quantum correction and will no longer be a constant. We build a scalar field model and a mechanics model. In these two models, we discuss the quantum correction of masses and coupling constants in the field framework and in the mechanics framework.