The question about the existence of so-called ``hidden'' variables in quantum mechanics and the perception of the completeness of quantum mechanics are two sides of the same coin. Quantum analytical mechanics constitutes a completion of standard quantum mechanics based on the concept of stochastic trajectories in the configuration space of a quantum system. For particle systems, configuration space is made up out of their coordinates and, if relevant, their orientation. Quantum analytical mechanics derives equations of motion for these variables which allow a description of the measurement process as a dynamical physical process. After all, it is exactly these variables experiments are designed to interact with. The theory is not a replacement of Hilbert space quantum mechanics but a mathematical completion enriching our toolset for the description of quantum phenomena.
The question about the existence of so-called ``hidden'' variables in quantum mechanics and the perception of the completeness of quantum mechanics are two sides of the same coin. Quantum analytical mechanics constitutes a completion of standard quantum mechanics based on the concept of stochastic trajectories in the configuration space of a quantum system. For particle systems, configuration space is made up out of their coordinates and, if relevant, their orientation. Quantum analytical mechanics derives equations of motion for these variables which allow a description of the measurement process as a dynamical physical process. After all, it is exactly these variables experiments are designed to interact with. The theory is not a replacement of Hilbert space quantum mechanics but a mathematical completion enriching our toolset for the description of quantum phenomena.
We show that in the opposition to a rather common opinion quantum mechanics is not complete. It is possible to introduce so called hidden variable — in our model classical fields — and combine the statistical predictions of quantum mechanics with deterministic dynamics of those hidden variables. Quantum mechanics can be considered as an approximative description of physical processes based on neglecting by quantities of the magnitude o(α), where α is the dispersion of fluctuations of the Gaussian background field. In this paper we present the detailed presentation of theory of infinite‐dimensional phase space and derive main equations of quantum mechanics (e.g., Schrödinger’s equation, Heisenberg’s equation and von Neumann equation) from the Hamilton equation on the infinite‐dimensional symplectic space. We emphasize (to escape misunderstanding) that our paper is not about quantization of systems with infinite number of degrees of freedom, but about representation of quantum systems as classical systems with infinite number of degrees of freedom. We also investigate the purely mathematical problem of preserving of the dispersion of Gaussian fluctuations by Hamiltonian flows.
Quantum physics confronts contemporary metaphysicians with a fascinating and frustrating puzzle. It appears to be telling us something novel and surprising about the nature of the physical world, but it is hard to pin down exactly what it tells us. Hidden variable theories constitute a second major class of solutions to the measurement problem. They differ from the standard theory and spontaneous collapse theories in eschewing collapse altogether; according to hidden variable theories, the wavefunction always evolves according to the Schrodinger equation. The simplest and most radical response to the measurement problem is to assert that standard quantum mechanics without wavefunction collapse constitutes a complete physical theory. The many worlds theory is more flexible in this regard; according to the standard contemporary version, no presupposition is made concerning precisely which physical properties form the basis of measurement results. There are several approaches to quantum mechanics, any or none of which might yield an acceptable physical theory.
I explore the possibility that a quantum system S may be described completely by the combination of its standard quantum state ∣ψ⟩ and a (hidden) quantum state ∣ϕ⟩ (that lives in the same Hilbert space), such that the outcome of any standard projective measurement on the system S is determined once the two quantum states are specified. I construct an algorithm that retrieves the standard quantum-mechanical probabilities, which depend only on ∣ψ⟩, by assuming that the (hidden) quantum state ∣ϕ⟩ is drawn at random from some fixed probability distribution Pr(.) and by averaging over Pr(.). Contextuality and Bell nonlocality turn out to emerge automatically from this algorithm as soon as the dimension of the Hilbert space of S is larger than 2. If ∣ϕ⟩ is not completely random, subtle testable deviations from standard quantum mechanics may arise in sequential measurements on single systems.
A new hidden variable theory is proposed, according to which particles follows definite trajectories, as in Bohmian Mechanics or Nelson's stochastic mechanics; in the new theory, however, the trajectories are classical, i.e. Newtonian. This result is obtained by developing the following concepts: (i) the essential elements of a hidden variable theory are a set of trajectories and a measure defined on it; the Newtonian HCT will be defined by giving these two elements. (ii) The universal wave function has a tree structure, whose branches are generated by the measurement processes and are spatially disjoined. (iii) The branches have a classical structure, i.e. classical paths go along them; this property derives from the fact that the paths close to the classical ones give the main contribution to the Feynman propagator. (iv) Classical trajectories can give rise to quantum phenomena, like for instance the interference phenomena of the two-slit experiment, by violating the so called Independence Assumption, which is always implicitely made in the conceptual analysis of these phenomena.
Forty years after the advent of quantum mechanics the problem of hidden variables, that is, the possibility of imbedding quantum theory into a classical theory, remains a controversial and obscure subject. Whereas to most physicists the possibility of a classical reinterpretation of quantum mechanics remains remote and perhaps irrelevant to current problems, a minority have kept the issue alive throughout this period. (See Freistadt [5] for a review of the problem and a comprehensive bibliography up to 1957.) As far as results are concerned there are on the one hand purported proofs of the non-existence of hidden variables, most notably von Neumann’s proof, and on the other, various attempts to introduce hidden variables such as de Broglie [4] and Bohm [1] and [2]. One of the difficulties in evaluating these contradictory results is that no exact mathematical criterion is given to enable one to judge the degree of success of these proposals. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The Question of the Completeness of Quantum Mechanics When a classical system is in a pure state, that is, when its preparation represents maximal information, the value of every physical quantity is uniquely predicted. In other words, classical pure states are dispersion-free, and classical mechanics is deterministic. Only in case the state is nonpure, thus representing nonmaximal information about the system, does probability enter into the theory. Things are different with quantum systems. Even in case the state of the system is pure, it is impossible to predict a definite value of every physical quantity: only the probability distributions of the physical quantities are given. Pure quantum states have dispersion (though, of course, they can be dispersion-free for particular physical quantities), and probability enters into quantum theory at a much more fundamental level. The basic idea of so-called hidden-variable theories is to reject the fundamental role of probability in quantum theory and to argue that probability arises because quantum states, even the pure ones, do not represent the ultimate information about the system. Accordingly, it is conjectured that there are certain hidden variables, not yet subject to experimental detection, which would complete the information carried by the quantum states, thus giving rise to so precise a knowledge of the system that probabilities would disappear and precise values of all physical quantities would be uniquely determined.