Mathematical Formulation of Our Theory

Satoru Yamamoto

BongJu Kim

In mathematical aspect, we introduce quantum algorithm and the mathematical structure of quantum computer. Quantum algorithm is expressed by linear algebra on a finite dimensional complex inner product space. The mathematical formulations of quantum mechanics had been established in around 1930, by von Neumann. The formulation uses functional analysis, linear algebra and probability theory. The knowledge of the mathematical formulation of QM is enough quantum mechanical knowledge for approaching to quantum algorithm and it might be efficient way for mathematicians that starting with mathematical formulations of QM. We explain the mathematical formulations of quantum mechanics briefly, quantum bits, quantum gates, quantum discrete Fourier transformation, Deutsch's algorithm and Shor's algorithm.

Martin Pieper

Borros Arneth

Modern theoretical physics is formulated through several highly successful mathematical frameworks, including Green-function theory, quantum mechanics, quantum field theory, gauge theory and statistical mechanics. Although these theories employ different mathematical objects and were developed independently, they exhibit remarkable mathematical compatibility. Green functions naturally appear in quantum field theory, Hilbert-space methods underlie quantum mechanics, gauge theories are formulated geometrically, and statistical mechanics connects microscopic dynamics with macroscopic thermodynamics. The mathematical origin of this compatibility remains an open conceptual question. This paper investigates whether these apparently different frameworks can be understood as representations of a common underlying mathematical structure. We introduce the Architecture Principle, according to which the primitive object is not a Green function, wave function or Hilbert space, but a Green-Function Architecture describing propagation, interaction, closed transport and structural consistency. Green functions, quantum-mechanical states, gauge transport and statistical ensembles are interpreted as different mathematical representations of the same underlying architecture. The proposed framework does not modify the established equations of physics. Instead, it provides a common structural language that relates several major branches of theoretical physics while preserving their existing mathematical formalisms. The paper develops the basic concepts of the Architecture Principle, introduces an architectural invariant, formulates the Representation Principle and discusses representative applications. Finally, the principal mathematical problems required for a complete representation theory are identified.

K. Kong Wan

A physical theory often admits different mathematical descriptions. Classical mechanics has three familiar formulations: the traditional Newton’s formulation, the Lagrange’s and the Hamilton’s formulations. It is easy to see that these are physically equivalent. Quantum mechanics also admits many different formulations. This chapter discusses how different mathematical descriptions of a probability theory based on a Hilbert space can be physically equivalent. The prescription does not single out a unique Hilbert space nor a unique set of unit vectors and operators for the description of a given quantum system. This non-uniqueness does not matter since operators and unit vectors themselves are not physically measurable quantities. Two different mathematical descriptions of a quantum system are physically equivalent if the unit vectors and selfadjoint operators for the representation of states and observables in the two descriptions are related by a simultaneous unitary transformation.

Valter Moretti