To apply the abstract quantum formalism to a particular physical system, one must specify the precise form of the relevant measurement and symmetry transformation operators. These operators are determined by a set of rules, the correspondence rules of quantum theory. The physical origin of these rules is obscure, and their physical interpretation and their degree of generality is presently unclear. In this paper, we show that all of the commonly-used correspondence rules can be systematically derived from a new physical principle, the Average-Value Correspondence Principle. This principle shows that the correspondence rules result from the systematic translation of relations between measurement results known to hold in a classical model of a system, providing these rules with a clear physical interpretation, and clearly demarcating their domain of applicability.
To apply the abstract quantum formalism to a particular physical system, one must specify the precise form of the relevant measurement and symmetry transformation operators. These operators are determined by a set of rules, the correspondence rules of quantum theory. The physical origin of these rules is obscure, and their physical interpretation and their degree of generality is presently unclear. In this paper, we show that all of the commonly-used correspondence rules can be systematically derived from a new physical principle, the Average-Value Correspondence Principle. This principle shows that the correspondence rules result from the systematic translation of relations between measurement results known to hold in a classical model of a system, providing these rules with a clear physical interpretation, and clearly demarcating their domain of applicability.
In a companion paper (hereafter referred to as Paper I), we have presented an attempt to derive the finite-dimensional abstract quantum formalism from a set of physically comprehensible assumptions. In this paper, we formulate a correspondence principle, the Average-Value Correspondence Principle, that allows relations between measurement outcomes which are known to hold in a classical model of a system to be systematically taken over into the quantum model of the system. Using this principle, we derive the explicit form of the temporal evolution operator (thereby completing the derivation of the abstract quantum formalism begun in Paper I), and derive many of the formal rules (such as operator rules, commutation relations, and Dirac's Poisson bracket rule) that are needed to apply the abstract quantum formalism to model particular physical systems.
In previous work [1], we have presented an attempt to derive the finite‐dimensional abstract quantum formalism from a set of physically comprehensible assumptions. In this paper, we continue the derivation of the quantum formalism by formulating a correspondence principle, the Average‐Value Correspondence Principle, that allows relations between measurement outcomes which are known to hold in a classical model of a system to be systematically taken over into the quantum model of the system, and by using this principle to derive many of the correspondence rules (such as operator rules, commutation relations, and Dirac's Poisson bracket rule) that are needed to apply the abstract quantum formalism to model particular physical systems.
In a companion paper (hereafter referred to as Paper I), we have presented an attempt to derive the finite-dimensional abstract quantum formalism within the framework of information geometry. In this paper, we formulate a correspondence principle, the Average-Value Correspondence Principle, that allows relations between measurement results which are known to hold in a classical model of a system to be systematically taken over into the quantum model of the system. Using this principle, we derive the explicit form of the temporal evolution operator (thereby completing the derivation of the abstract quantum formalism begun in Paper I), and derive many of the correspondence rules (such as operator rules, commutation relations, and Dirac's Poisson bracket rule) that are needed to apply the abstract quantum formalism to model particular physical systems.
Abstract Quantum mechanics is one of the basic theories of modern physics. Here, the famous Schrödinger equation and the differential operators representing mechanical quantities in quantum mechanics are derived, just based on the principle that the translation invariance (symmetry) of a system in Hamiltonian mechanics should be preserved in quantum mechanics. Moreover, according to the form of the differential operators, the commutation relation in quantum mechanics between the generalized coordinate and the generalized momentum can be directly obtained. We believe that the results in this paper are very useful for understanding the physical origin of quantum mechanics.
In this paper and a companion paper, we attempt to systematically investigate the possibility that the concept of information may enable a derivation of the quantum formalism from a set of physically comprehensible postulates. To do so, we formulate an abstract experimental set-up and a set of assumptions based on generalizations of experimental facts that can be reasonably taken to be representative of quantum phenomena, and on theoretical ideas and principles, and show that it is possible to deduce the quantum formalism. In particular, we show that it is possible to derive the abstract quantum formalism for finite-dimensional quantum systems and the formal relations, such as the canonical commutation relationships and Dirac's Poisson Bracket rule, that are needed to apply the abstract formalism to particular systems of interest. The concept of information, via an information-theoretic invariance principle, plays a key role in the derivation, and gives rise to some of the central structural features of the quantum formalism.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].