Constitution of objects in classical mechanics and in quantum mechanics

The constitution of objects is discussed in classical mechanics and in quantum mechanics. The requirement of objectivity and the Galilei invariance of classical and quantum mechanics leads to the postulate of covariance which must be fulfilled by observable quantities. Objects are then considered as carriers of these covariant observables and turn out to be representations of the Galilei group. Individual systems can be defined in classical mechanics by their trajectories in phase space. However, in quantum mechanics the characterization of individuals can only be achieved approximately by means of unsharp observables.     

Zwitters: Particles between quantum and classical

We describe both quantum particles and classical particles in terms of a classical statistical ensemble, with a probability distribution in phase space. By use of a wave function in phase space both can be treated in the same quantum formalism. Quantum particles are characterized by a specific choice of observables and time evolution of the probability density. Then interference and tunneling are found within classical statistics. Zwitters are (effective) one-particle states for which the time evolution interpolates between quantum and classical particles. Experimental bounds on a small parameter can test quantum mechanics.     

Physical uniformities on the state space of nonrelativisitic quantum mechanics

Uniformities describing the distinguishability of states and of observables are discussed in the context of general statistical theories and are shown to be related to distinguished subspaces of continuous observables and states, respectively. The usual formalism of quantum mechanics contains no such physical uniformity for states. Using recently developed tools of quantum harmonic analysis, a natural one-to-one correspondence between continuous subspaces of nonrelativistic quantum and classical mechanics is established, thus exhibiting a close interrelation between physical uniformities for quantum states and compactifications of phase space. General properties of the completions of the quantum state space with respect to these uniformities are discussed.     

On classical representations of finite-dimensional quantum mechanics

In the case of a finite-dimensional Hilbert space, it is shown that quantum mechanics can be embedded into discrete classical probability theory. In particular, states can be represented as stochastic vectors and observables as random variables such that all probabilities and expectation values are given in classical terms.     

On the representation of quantum mechanics on phase space

It is shown that Hilbert-space quantum mechanics can be represented on phase space in the sense that the density operators can be identified with phase-space densities and the observables can be described by functions on phase space. In particular, we consider phase-space representations of quantum mechanics which are related to certain joint position-momentum observables.     

Why Do the Quantum Observables Form a Jordan Operator Algebra?

The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.     

Classical signal model for quantum channels

Recently it was shown that the main distinguishing features of quantum mechanics (QM) can be reproduced by a model based on classical random fields, the so-called prequantum classical statistical field theory (PCSFT). This model provides a possibility to represent averages of quantum observables, including correlations of observables on subsystems of a composite system (e.g., entangled systems), as averages with respect to fluctuations of classical (Gaussian) random fields. We consider some consequences of the PCSFT for quantum information theory. They are based on our previous observation that classical Gaussian channels (important in classical signal theory) can be represented as quantum channels. Now we show that quantum channels can be represented as classical linear transforms of classical Gaussian signals.     

Amplitude phase-space model for quantum mechanics

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrödinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.     

Quantization postulate in quantum mechanics

The procedure of transition from classical observables to quantum (operator) observables in quantum mechanics is discussed. By an example it is shown that, even in simple cases, the method of self-adjoint extensions of formal differential expressions for defining physical observables as operators is not equivalent to the procedure of forming operator functions corresponding to these observables. This inequivalence is not a formal one but has physical consequences connected with the compatibility of observables.     

Canonical transformations in quantum mechanics

This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular classes of quantum canonical transformations.     

Mathematical foundations of quantum field theory: Fermions,gauge fields,and supersymmetry part I: Lattice field theories

In this paper we explore the mathematical foundations of quantum field theory. From the mathematical point of view, quantum field theory involves several revolutions in structure just as severe as, if not more than, the revolutionary change involved in the move from classical to quantum mechanics. Ordinary quantum mechanics is based upon real-valued observables which are not all compatible. We will see that the proper mathematical understanding of Fermi fields involves a new concept of probability theory, the graded probability space. This new concept also yields new points of view concerning ergodic theorems in statistical mechanics.     

On the Representation of Quantum Mechanics on a Classical Sample Space

Under the common viewpoint of statistical maps,the concept of observables in quantum mechanics and inclassical probability theory are discussed and compared.It is shown that, by means of injective statistical maps, quantum mechanics can to a certain extentbe reformulated in classical terms. Some characteristicexamples are considered.     

The Weyl representation in classical and quantum mechanics

The position representation of the evolution operator in quantum mechanics is analogous to the generating function formalism of classical mechanics. Similarly, the Weyl representation is connected to new generating functions described by chords and centres in phase space. Both classical and quantal theories relie on the group of translations and reflections through a point in phase space. The composition of small time evolutions leads to new versions of the classical variational principle and to path integrals in quantum mechanics. The strong resemblance between the two theories allows a clear derivation of the semiclassical limit in which observables evolve classically in the Weyl representation. The restriction of the motion to the energy shell in classical mechanics is the basis for a full review of the semiclassical Wigner function and the theory of scars of periodic orbits. By embedding the theory of scars in a fully uniform approximation, it is shown that the region in which the scar contribution is oscillatory is separated from a decaying region by a caustic that touches the shell along the periodic orbit and widens quadratically within the energy shell.     

Principles of maximally classical and maximally realistic quantum mechanics

Recently Auberson, Mahoux, Roy and Singh have proved a long standing conjecture of Roy and Singh: In 2N-dimensional phase space, a maximally realistic quantum mechanics can have quantum probabilities of no more than N+1 complete commuting cets (CCS) of observables coexisting as marginals of one positive phase space density. Here I formulate a stationary principle which gives a nonperturbative definition of a maximally classical as well as maximally realistic phase space density. I show that the maximally classical trajectories are in fact exactly classical in the simple examples of coherent states and bound states of an oscillator and Gaussian free particle states. In contrast, it is known that the de Broglie-Bohm realistic theory gives highly nonclassical trajectories.     

Cognition versus Constitution of Objects: From Kant to Modern Physics

Classical mechanics in phase space as well as quantum mechanics in Hilbert space lead to states and observables but not to objects that may be considered as carriers of observable quantities. However, in both cases objects can be constituted as new entities by means of invariance properties of the theories in question. We show, that this way of reasoning has a long history in physics and philosophy and that it can be traced back to the transcendental arguments in Kant’s critique of pure reason.     

The pure state space of quantum mechanics as Hermitian symmetric space

The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kähler manifold. The classical principles of quantum mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non-linear geometrical context.     

Proposed experiment to test the bounds of quantum correlations

The combination of quantum correlations appearing in the Clauser-Horne-Shimony-Holt inequality can give values between the classical bound, 2, and Tsirelson's bound, 2 x square root of 2. However, for a given set of local observables, there are values in this range which no quantum state can attain. We provide the analytical expression for the corresponding bound for a parametrization of the local observables introduced by Filipp and Svozil, and describe how to experimentally trace it using a source of singlet states. Such an experiment will be useful to identify the origin of the experimental errors in Bell's inequality-type experiments and could be modified to detect hypothetical correlations beyond those predicted by quantum mechanics.     

Universal local symmetries and nonsuperposition in classical mechanics

In the Hilbert space formulation of classical mechanics, pioneered by Koopman and von Neumann, there are potentially more observables than in the standard approach to classical mechanics. In this Letter, we show that actually many of those extra observables are not invariant under a set of universal local symmetries which appear once the Koopman and von Neumann formulation is extended to include the evolution of differential forms. Because of their noninvariance, those extra observables have to be removed. This removal makes the superposition of states in the Koopman and von Neumann formulation, and as a consequence also in classical mechanics, impossible.