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Foundations of Physics

Physical principles in quantum field theory and in covariant harmonic oscillator formalism

D. Han, Y. S. Kim, Marilyn E. Noz

Published Dec 1, 1981• 11 citations
Publisher
DOI
10.1007/bf00727106

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Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of two-by-two matrices commonly called $SL(2,c)$. Thus the coupled harmonic oscillators can play the role of combining quantum mechanics with special relativity. It is therefore possible to relate the current issues of physics to the Lorentz-covariant formulation of quantum mechanics.

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The requirement of general covariance of quantum field theory (QFT) naturally leads to quantization based on the manifestly covariant De Donder-Weyl formalism. To recover the standard noncovariant formalism without violating covariance, fields need to depend on time in a specific deterministic manner. This deterministic evolution of quantum fields is recognized as a covariant version of the Bohmian hidden-variable interpretation of QFT. The reconcilation of quantum theory with general theory of relativity is still an unsolved problem. It is very likely that the successful reconcilation requires a radical reformulation of the basic principles of relativity, or that of quantum theory, or both. One obvious difference between quantum theory and general relativity is that quantum theory, in contrast with general relativity, is an undeterministic theory. Most 1 attempts towards the reconcilation start from the assumption that quantum gravity, just as any quantum theory, should also be an undeterministic theory. However, in contrast with this mainstream quantum-undeterministic paradigm, ’t Hooft suggests

•Jan 5, 2006•5 citations
Entanglement entropy (EE) is a pivotal quantity in quantum field theory (QFT) that captures the nonlocal correlations between subsystems, offering deep insights into the quantum structure of spacetime, field dynamics, and phase transitions. This work derives EE in QFT using the covariance matrix formalism, establishing its foundations through the von Neumann entropy of reduced density matrices. We analytically compute EE for coupled harmonic oscillators as a prototypical model, extending the results to quantum fields via perturbative expansions. In the context of Maxwell QFT, we analyze entropy corrections induced by external perturbations, revealing their fractal dynamics through the emergence of Julia sets. The study further incorporates noncommutative geometry, where the deformation of spacetime modifies the covariance matrix, field strength tensors, and modular indices. Numerical simulations validate the derived scaling laws, entropy corrections, and noncommutative effects. These findings provide a mathematically rigorous framework to explore the interplay between EE, field theory, and geometric structures in high-energy physics.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of two-by-two matrices commonly called SL(2,c)SL(2,c)SL(2,c). Thus the coupled harmonic oscillators can play the role of combining quantum mechanics with special relativity. It is therefore possible to relate the current issues of physics to the Lorentz-covariant formulation of quantum mechanics.
The requirement of general covariance of quantum field theory (QFT) naturally leads to quantization based on the manifestly covariant De Donder-Weyl formalism. To recover the standard noncovariant formalism without violating covariance, fields need to depend on time in a specific deterministic manner. This deterministic evolution of quantum fields is recognized as a covariant version of the Bohmian hidden-variable interpretation of QFT. The reconcilation of quantum theory with general theory of relativity is still an unsolved problem. It is very likely that the successful reconcilation requires a radical reformulation of the basic principles of relativity, or that of quantum theory, or both. One obvious difference between quantum theory and general relativity is that quantum theory, in contrast with general relativity, is an undeterministic theory. Most 1 attempts towards the reconcilation start from the assumption that quantum gravity, just as any quantum theory, should also be an undeterministic theory. However, in contrast with this mainstream quantum-undeterministic paradigm, ’t Hooft suggests