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arXiv:0709.4325 · Journal of Physics A Mathematical and Theoretical

Crypto-harmonic oscillator in higher dimensions: classical and quantum aspects

Subir Ghosh, Bibhas Ranjan Majhi

Published Jan 29, 2008• 4 citations

Abstract

We study complexified harmonic oscillator models in two and three dimensions. Our work is a generalization of the work of Smilga (2007 Preprint 0706.4064 ( J. Phys. A: Math. Theor. at press)) who initiated the study of these Crypto-gauge invariant models that can be related to PT -symmetric models. We show that rotational symmetry in higher spatial dimensions naturally introduces more constraints (in contrast to Smilga (2007 Preprint 0706.4064 ( J. Phys. A: Math. Theor. at press)) where one deals with a single constraint) with a much richer constraint structure. Some common as well as distinct features in the study of the same Crypto-oscillator in different dimensions are revealed. We also quantize the two dimensional Crypto-oscillator.

DOI: 10.1088/1751-8113/41/6/065306

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We study complexified harmonic oscillator models in two and three dimensions. Our work is a generalization of the work of Smilga (2007 Preprint 0706.4064 ( J. Phys. A: Math. Theor. at press)) who initiated the study of these Crypto-gauge invariant models that can be related to PT -symmetric models. We show that rotational symmetry in higher spatial dimensions naturally introduces more constraints (in contrast to Smilga (2007 Preprint 0706.4064 ( J. Phys. A: Math. Theor. at press)) where one deals with a single constraint) with a much richer constraint structure. Some common as well as distinct features in the study of the same Crypto-oscillator in different dimensions are revealed. We also quantize the two dimensional Crypto-oscillator.

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This paper investigates the application of non-linear harmonic oscillators within the framework of quantum field theory, seeking to provide a deeper understanding of fundamental forces and their dynamics. The core focus is to develop a novel method, termed the "quantum-geometric" algorithm, that leverages geometric properties to represent and model quantum fields. Traditional approaches often struggle with the inherent complexities of quantum mechanics, and this research proposes a shift towards a more intuitive and geometric representation, potentially offering new insights into the nature of quantum phenomena. We present a preliminary algorithm and discuss its implications for both theoretical modeling and the potential for future advancements in quantum field theory.