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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 24, 2019• 0 citations

Abstract

Circuit theory includes steady-state responses that are obtained using the Laplace transform; however, they are not included in classical physics. By applying the Laplace transform to the Dirac equation as a basic equation of quantum theory, we can examine the steady-state response in quantum theory and determine the spin 1/2 as a characteristic phenomenon in this theory.

DOI: 10.2299/jsp.23.35

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Circuit theory includes steady-state responses that are obtained using the Laplace transform; however, they are not included in classical physics. By applying the Laplace transform to the Dirac equation as a basic equation of quantum theory, we can examine the steady-state response in quantum theory and determine the spin 1/2 as a characteristic phenomenon in this 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.

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 the steady-state of lossless circuits, resonance can be obtained using the Laplace transform. We focus on commensurate transmission line circuits constructed using eight unit elements as an example and determine the resonance, transient response, and nonstationary response of the circuits using a z-transform. In old quantum theory, light quanta are assumed to satisfy the heat conduction equation. When considering the response of light quanta as a physical phenomenon from the viewpoint of circuit theory, we can interpret that the heat conduction equation describes the propagation of waves on a lossy distributed RC transmission line, in which light quanta repeatedly collide with each other. We demonstrate that light quanta are different from photons, which are interpreted as satisfying the Maxwell's equations and propagating on a lossless transmission line.

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 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 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.