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.
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.
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 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.
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 this paper, we show how the framework of information geometry, the natural geometry of discrete probability distributions, can be used to derive the quantum formalism. The derivation rests upon a few elementary features of quantum phenomena, such as complementarity and global gauge invariance. When appropriately formulated within an information-geometric framework, and combined with a novel information-theoretic principle, these features lead to the abstract quantum formalism for finite-dimensional quantum systems, and the result of Wigner's theorem. By means of a correspondence principle, several correspondence rules of quantum theory, such as the canonical commutation relationships, are also systematically derived. The derivation suggests that information geometry is directly or indirectly responsible for many of the central structural features of the quantum formalism, such as the importance of square roots of probability and the occurrence of sinusoidal functions of phases in a pure quantum state. Global gauge invariance is shown to play a crucial role in the emergence of the formalism in its complex form.
Abstract The reconstruction of quantum physics has been connected with the interpretation of the quantum formalism, and has continued to be so with the recent deeper consideration of the relation of information to quantum states and processes. This recent form of reconstruction has mainly involved conceiving quantum theory on the basis of informational principles, providing new perspectives on physical correlations and entanglement that can be used to encode information. By contrast to the traditional, interpretational approach to the foundations of quantum mechanics, which attempts directly to establish the meaning of the elements of the theory and often touches on metaphysical issues, the newer, more purely reconstructive approach sometimes defers this task, focusing instead on the mathematical derivation of the theoretical apparatus from simple principles or axioms. In its most pure form, this sort of theory reconstruction is fundamentally the mathematical derivation of the elements of theory from explicitly presented, often operational principles involving a minimum of extra‐mathematical content. Here, a representative series of specifically information‐based treatments—from partial reconstructions that make connections with information to rigorous axiomatizations, including those involving the theories of generalized probability and abstract systems—is reviewed.