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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, 2018• 2 citations
Publisher

Abstract

In Session 10, we showed that the telegrapher's equations can be extended using the Riccati differential equation and that a new circuit element can be obtained. In Session 11, we showed that the Dirac equation, which is used in quantum theory, can be expressed as the extended telegrapher's equations and that circuit theory can be applied to the Dirac equation. In this session, we demonstrate that a new circuit element can be obtained by applying reactive power to the Dirac equation. In this case, energy given by Einstein's well-known formula E = mc2 should be considered for the discussion on an ordinary world that does not require the theory of relativity.

DOI
10.2299/jsp.22.17

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In Session 10, we showed that the telegrapher's equations can be extended using the Riccati differential equation and that a new circuit element can be obtained. In Session 11, we showed that the Dirac equation, which is used in quantum theory, can be expressed as the extended telegrapher's equations and that circuit theory can be applied to the Dirac equation. In this session, we demonstrate that a new circuit element can be obtained by applying reactive power to the Dirac equation. In this case, energy given by Einstein's well-known formula E = mc2 should be considered for the discussion on an ordinary world that does not require the theory of relativity.
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.
In quantum theory, a wave function expresses the state of matter waves and is not considered to physically exist in general. Equations that are satisfied by a wave function are called wave equations. In classical physics, wave equations are well known and are derived from the Maxwell equations. However, the Maxwell equations do not define important physical properties, such as the lossless property of circuits. The lossless property is represented by the cascade matrix defined in analog circuit theory. The definition of the lossless property in wave equations became possible owing to the telegrapher's equations introduced by Heaviside. The cascade matrix can be determined from uniform transmission lines that can be obtained from the telegrapher's equations and those extended using the Riccati differential equation.
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.
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.
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.