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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 14, 2020• 0 citations

Abstract

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.

DOI: 10.2299/jsp.24.123

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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.

The basic particles of quanta are quarks and leptons. Quarks are combined with each other and form neutrons and protons. Neutrons and protons are combined with each other and form quanta called atomic nuclei, which are considered to vibrate. This vibration is considered to be expressed by a wave equation. In this session, we show that the one-dimensional crystal described in Session 5 satisfies a quantized wave equation. Moreover, the one-dimensional crystal is equivalent to an LC ladder circuit, which consists of a coil and a capacitor, as also described in Session 5. Therefore, the applicability of circuit theory to atomic nuclei is shown by assuming that atomic nuclei satisfy the quantized wave equation as LC ladder circuits.

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.

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.

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.