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Journal of Signal Processing

Circuit Theory Based on New Concepts and Its Application to Quantum Theory

Nobuo Nagai, Takashi Yahagi

Published May 24, 2018• 0 citations
Publisher

Abstract

In quantum mechanics and quantum theory, there are interactive phenomena that are dependent on time in the theory of relativity as well as phenomena that are independent of time, such as those discussed with renormalization, gauge theory, and the reduction of wave packets. The steady-state response in circuit theory is independent of time derivative terms. In this session, we attempt to apply circuit theory to renormalization, gauge theory, and the reduction of wave packets on the basis of the steady-state response and losslessness.

DOI
10.2299/jsp.22.135

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In quantum mechanics and quantum theory, there are interactive phenomena that are dependent on time in the theory of relativity as well as phenomena that are independent of time, such as those discussed with renormalization, gauge theory, and the reduction of wave packets. The steady-state response in circuit theory is independent of time derivative terms. In this session, we attempt to apply circuit theory to renormalization, gauge theory, and the reduction of wave packets on the basis of the steady-state response and losslessness.
In this lecture series, we attempt to construct a theoretical system of quantum theory that emphasizes the wave nature (in other words, that is based on circuit theory) and assumes quanta to be circuit elements. Circuit equations given by voltage and current have steady-state and transient responses. The steady-state response is expressed by complex numbers. For lossless circuits, the active and reactive powers can be distinguished, and the active power is expressed by a conserved quantity. In physics, physical phenomena are considered to occur as a result of some motion, suggesting the importance of motion. In circuit theory, a transient response is generated when a steady state transits to another steady state. In the transient response, the active and reactive powers cannot be distinguished. Therefore, the transient response is considered to be a physical phenomenon without a conserved quantity and to require special techniques involving difficult problems, such as renormalization.
To apply circuit theory to quantum theory, we review the problems of circuit theory described in the past sessions. In concrete, the physical properties of circuits, such as losslessness, steady-state response, and reactive power, are reexamined. It is shown that the Laplace transform is more advantageous than Heaviside's operational calculus and that active power can be conserved when a lossless circuit is used. It is also demonstrated that a forward wave exp(-jβx) travels clockwise because exp(jωt) in the Laplace transform is required for separating the variables of partial differential equations related to waves.
The Schrödinger equation is derived from the concept that particles of matter such as electrons have a wave nature, and it is used to express phenomena specific to quantum mechanics, such as the advancing wave and the quantum tunneling effect. In this session, we apply circuit theory to the Schrödinger equation and define voltage and current using wave functions that satisfy the Schrödinger equation. By this method, we show that a lossless transmission line or circuit can be obtained. Thus, a circuit that satisfies the Schrödinger equation is lossless and is not directly related to heat, but the reactive power is related to the regulation of energy. Therefore, such a circuit can be discussed in a theoretical system that does not rely on probability theory.
In Session 8, we demonstrated that circuit theory can be applied to the Schrödinger equation, a basic equation in quantum mechanics. In this session, we attempt to apply circuit theory to the Dirac equation using the Riccati differential equation and demonstrate that physical phenomena such as spin 1/2 can be explained on the basis of circuit theory. To this end, we obtain the one-dimensional Dirac equation for the complex equivalent circuit for a physical phenomenon. We show that the Dirac equation can be regarded as a type of low-energy effective field theory in quantum theory and that circuit theory can be applied to the Dirac equation.
The strong and weak interactions in quantum theory represent the physical phenomena that bind quanta. We attempt to apply impedance matching that represents the connection of circuit elements in analog circuit theory to these interactions. The interactions in quantum theory are related to the phase in gauge theory. The phase can be given to the connected state of circuit elements using cascade matrices, which can determine the phase of circuits. We show that asymmetric LC ladder circuits are suitable for representing the interactions of quanta in terms of the phase as well as the quantized wave equations. We also show that the equivalent circuits for the connection of quarks and gluons can be represented as LC ladder circuits.