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Quantization and Quantum-Like Phenomena: A Number Amplitude Approach

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Abstract

Historically, quantization has meant turning the dynamical variables of classical mechanics that are represented by numbers into their corresponding operators. Thus the relationships between classical variables determine the relationships between the corresponding quantum mechanical operators. Here, we take a radically different approach to this conventional quantization procedure. Our approach does not rely on any relations based on classical Hamiltonian or Lagrangian mechanics nor on any canonical quantization relations, nor even on any preconceptions of particle trajectories in space and time. Instead we examine the symmetry properties of certain Hermitian operators with respect to phase changes. This introduces harmonic operators that can be identified with a variety of cyclic systems, from clocks to quantum fields. These operators are shown to have the characteristics of creation and annihilation operators that constitute the primitive fields of quantum field theory. Such an approach not only allows us to recover the Hamiltonian equations of classical mechanics and the Schrödinger wave equation from the fundamental quantization relations, but also, by freeing the quantum formalism from any physical connotation, makes it more directly applicable to non-physical, so-called quantum-like systems. Over the past decade or so, there has been a rapid growth of interest in such applications. These include, the use of the Schrödinger equation in finance, second quantization and the number operator in social interactions, population dynamics and financial trading, and quantum probability models in cognitive processes and decision-making. In this paper we try to look beyond physical analogies to provide a foundational underpinning of such applications.

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Notes

  1. Notice also, that since N is an unbounded operator then so are Z and Z +. Thus, properly speaking, we should have \(\left [ Z,Z^{+}\right ]^{++}=I\), where ++ denotes closure.

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Acknowledgments

The auhors are sincerely grateful to Fabio Bagarello for his help in clarifying a number of the mathematical technicalities that arise in the paper. Thanks are also due to an unknown referee for his/her valuable comments on the structure of the paper.

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Correspondence to T. R. Robinson.

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Robinson, T.R., Haven, E. Quantization and Quantum-Like Phenomena: A Number Amplitude Approach. Int J Theor Phys 54, 4576–4590 (2015). https://doi.org/10.1007/s10773-015-2726-8

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