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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 Jan 1, 2015• 0 citations
Publisher

Abstract

A lossless transmission line of length l is called a unit element. By showing numerical examples, we demonstrate that resonance and eigen-oscillation can be obtained using a unit element as a circuit element. Because resonance and eigen-oscillation are responses in the steady state, the transient response that occurs until the circuit reaches the steady state should be considered. Although the transient response has been conventionally considered a phenomenon of exponential attenuation that depends on a time constant, we numerically demonstrate that the transient response of a unit element is a phenomenon of discrete attenuation that can be expressed using a z-transform. In addition, resonance and eigen-oscillation involve reactive power, which is not related to energy, because the transient response of the circuit using a unit element is closely related to the phases expressed by the z-transform. We also numerically show that the reactive power is confined in the circuit.

DOI
10.2299/jsp.19.209

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A lossless transmission line of length l is called a unit element. By showing numerical examples, we demonstrate that resonance and eigen-oscillation can be obtained using a unit element as a circuit element. Because resonance and eigen-oscillation are responses in the steady state, the transient response that occurs until the circuit reaches the steady state should be considered. Although the transient response has been conventionally considered a phenomenon of exponential attenuation that depends on a time constant, we numerically demonstrate that the transient response of a unit element is a phenomenon of discrete attenuation that can be expressed using a z-transform. In addition, resonance and eigen-oscillation involve reactive power, which is not related to energy, because the transient response of the circuit using a unit element is closely related to the phases expressed by the z-transform. We also numerically show that the reactive power is confined in the circuit.
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.
Physical phenomena showing both particle and wave natures at the same time exhibit resonance. In addition, resonance is a physical phenomenon occurring in the steady state. Transient and steady-state responses can be discriminated when transmission lines are used. We demonstrate that (1) extended telegrapher's equations can be obtained using the Riccati differential equation, (2) lossless transmission lines called Pasteur, Tellegen, and bi-isotropic (BI) media can be characterized using the extended telegrapher's equations, and (3) the resonance conditions can be determined for each of these media.
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 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.