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 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 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 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.
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 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.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
``Quantum'' stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of ``measurement'' and (possibly non-local) ``correlations'' to the discussion. Crucially, all this comes with a ``probabilistic calculus''. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism: (i) While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all! (ii) Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description. (iii) The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism. In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven. Surprisingly, in this seemingly purely qualitative setting even the quantitative Born rule arises, that is the rule which tells you how to calculate the probabilities. Indeed, each such category has as endomorphism Hom of the tensor unit an abelian semiring of `scalars', and a special subset of these scalars will play the role of weights: each scalar induces a natural transformation which propagates through physical processes, and when a `state' undergoes a `measurement', the composition of the corresponding morphisms gives rise to the weight. Hence the probabilistic weights live within the category of processes. J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955). S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130 A more reader friendly version entitled `Quantum information flow, concretely, abstractly' is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf