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 Schrodinger 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 Schrodinger 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.
Highlights
The importance of the theoretical framework of classical mechanics in early attempts to develop quantum theory in the mid to late 1920s is well documented [1], [2]
Canonical quantization became the focus of intense research by Jordan, Born, Dirac and others in the years 1926 and 1927 [6], which laid the essential theoretical foundations to much of the quantum mechanics and, what is considered to be, the even more fundamental quantum ...eld theory that we know today
We have obtained the properties, Eqs. (29), (45) and (46), of the number amplitude operator, Z and its adjoint Z+, that are interpreted as creation and annihilation operators in the second quantization formalism, and that serve as the primitive quantum ...elds of the theory
Summary
The importance of the theoretical framework of classical mechanics in early attempts to develop quantum theory in the mid to late 1920s is well documented [1], [2]. Even in later stages of the search for a systematic quantum formalism, classical mechanics, the elements involving Hamilton’s equations and canonical variables, continued to in‡uence the often intuitive developments Heisenberg, in his famous paper of 1925 [3] had recognized the importance of replacing abelian variables of classical mechanics by non-abelian ones in quantum mechanics. Wigner [9], from a di¤erent point of view from Feynman’s, asked to what extent canonical quantization was essential, while retaining the Hamiltonian function of canonical variables from the classical theory Wigner answered his own question, to some extent, by showing that there were alternatives to the usual canonical commutation relations, that were still consistent with Hamilton’s equations.
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