Quantum field theory overcomes Einstein's objections against quantum mechanics

It is shown that if zero-point oscillations are assumed to exist, quantum field theory will lack the determinism of classical physics. Scientific Research Institute for Nuclear Physics, Moscow University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 71–74, May, 1996.     

Quantum structures: An attempt to explain the origin of their appearance in nature

We explain quantum structure as due to two effects: (a) a real change of state of the entity under the influence of the measurement and (b) a lack of knowledge about a deeper deterministic reality of the measurement process. We present a quantum machine, with which we can illustrate in a simple way how the quantum structure arises as a consequence of the two mentioned effects. We introduce a parameter that measures the size of the lack of knowledge of the measurement process, and by varying this parameter, we describe a continuous evolution from a quantum structure (maximal lack of knowledge) to a classical structure (zero lack of knowledge). We show that for intermediate values of we find a new type of structure that is neither quantum nor classical. We apply the model to situations of lack of knowledge about the measurement process appearing in other aspects of reality. Specifically, we investigate the quantumlike structures that appear in the situation of psychological decision processes, where the subject is influenced during the testing and forms some opinions during the testing process. Our conclusion is that in the light of this explanation, the quantum probabilities are epistemic and not ontological, which means that quantum mechanics is compatible with a determinism of the whole.     

Randomness in Classical Mechanics and?Quantum Mechanics

The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in the classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations: Δq>0 and Δp>0, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero).     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Anomalous isotropic luminescence from anthracene sandwich dimers

A classical statistical probability amplitude is introduced whose square modulus is the distribution function. This enables the analogy between classical statistical mechanics and quantum mechanics to be completed. The analogy is developed until quantum statistical derivations can be used in classical statistical mechanics. Two master equations are found: the classical equivalent of the Pauli Master Equation, and a generally valid master equation. Well-known classical equations are deduced from these in a special representation. Interference terms are found and discussed.     

Probability and logical structure of statistical theories

A characterization of statistical theories is given which incorporates both classical and quantum mechanics. It is shown that each statistical theory induces an associated logic and joint probability structure, and simple conditions are given for the structure to be of a classical or quantum type. This provides an alternative for the quantum logic approach to axiomatic quantum mechanics. The Bell inequalities may be derived for those statistical theories that have a classical structure and satisfy a locality condition weaker than factorizability. The relation of these inequalities to the issue of hidden variable theories for quantum mechanics is discussed and clarified.     

Relativity and indeterminism

It is well known that Albert Einstein adhered to a deterministic world view throughout his career. Nevertheless, his developments of the special and general theories of relativity prove to be incompatible with that world view. Two different forms of determinism—classical Laplacian determinism and the determinism of isolated systems—are considered. Through careful considerations of what concretely is involved in predicting future states of the entire universe, or of isolated systems, it is shown that the demands of the theories of relativity make these deterministic positions untenable.     

Unification theory of classical statistical uncertainty relation and quantum uncertainty relation and its applications

General classical statistical uncertainty relation is deduced and generalized to quantum uncertainty relation. We give a general unification theory of the classical statistical and quantum uncertainty relations, and prove that the classical limit of quantum mechanics is just classical statistical mechanics. It is shown that the classical limit of the general quantum uncertainty relation is the general classical uncertainty relation. Also, some specific applications show that the obtained theory is self-consistent and coincides with those from physical experiments.     

Bohmian Mechanics,the Quantum-Classical Correspondence and the Classical Limit: The Case of the Square Billiard

Square billiards are quantum systems complying with the dynamical quantum-classical correspondence. Hence an initially localized wavefunction launched along a classical periodic orbit evolves along that orbit, the spreading of the quantum amplitude being controlled by the spread of the corresponding classical statistical distribution. We investigate wavepacket dynamics and compute the corresponding de Broglie-Bohm trajectories in the quantum square billiard. We also determine the trajectories and statistical distribution dynamics for the equivalent classical billiard. Individual Bohmian trajectories follow the streamlines of the probability flow and are generically non-classical. This can also hold even for short times, when the wavepacket is still localized along a classical trajectory. This generic feature of Bohmian trajectories is expected to hold in the classical limit. We further argue that in this context decoherence cannot constitute a viable solution in order to recover classicality.     

Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory

No Heading We show that the Dirac-von Neumann formalism for quantum mechanics can be obtained as an approximation of classical statistical field theory. This approximation is based on the Taylor expansion (up to terms of the second order) of classical physical variables – maps f : Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures ρ on Ω having zero mean value and dispersion σ²(ρ) ≈ h. This viewpoint to the conventional quantum formalism gives the possibility to create generalized quantum formalisms based on expansions of classical physical variables in the Taylor series up to terms of nth order and considering statistical states ρ having dispersion σ²(ρ) = hⁿ (for n = 2 we obtain the conventional quantum formalism).     

经典不确定关系到量子不确定关系的推广及对世界稳定性的应用

一般地引入了经典统计力学的联合概率密度分布,并把其密度函数一般地推广为一般的复函数和它的复共厄的乘积,进而自然地把经典统计力学平均值推广到量子力学的平均值,首次依据联合概率密度分布把经典统计力学的不确定关系严格地推广到量子力学的不确定关系,给出了经典统计力学与量子力学中不确定关系的对应关系,发现了人们对经典统计力学与量子力学不确定关系推导时所各遗漏的一个重要的条件,对不确定关系与此相关的所有的相关文章和书都需修正.最后一般地阐明了世界的稳定性与不确定关系的联系,同时一般地给出了不确定关系失效的条件.     

Renormalization and Quantum Scaling of Frenkel–Kontorova Models

We generalise the classical Transition by Breaking of Analyticity for the class of Frenkel–Kontorova models studied by Aubry and others to non-zero Planck’s constant and temperature. This analysis is based on the study of a renormalization operator for the case of irrational mean spacing using Feynman’s functional integral approach. We show how existing classical results extend to the quantum regime. In particular we extend MacKay’s renormalization approach for the classical statistical mechanics to deduce scaling of low frequency effects and quantum effects. Our approach extends the phenomenon of hierarchical melting studied by Vallet, Schilling and Aubry to the quantum regime     

Nonlinear Schrödinger equations from prequantum classical statistical field theory

We derive some important features of the standard quantum mechanics from a certain classical-like model—prequantum classical statistical field theory, PCSFT. In this approach correspondence between classical and quantum quantities is established through asymptotic expansions. PCSFT induces not only linear Schrödinger's equation, but also its nonlinear generalizations. This coupling with “nonlinear wave mechanics” is used to evaluate the small parameter of PCSFT.     

Stochastic phase spaces and master Liouville spaces in statistical mechanics

The concept of probability space is generalized to that of stochastic probability space. This enables the introduction of representations of quantum mechanics on stochastic phase spaces. The resulting formulation of quantum statistical mechanics in terms of -distribution functions bears a remarkable resemblance to its classical counterpart. Furthermore, both classical and quantum statistical mechanics can be formulated in one and the same master Liouville space overL ²(). A joint derivation of a classical and quantum Boltzman equation provides an illustration of the practical uses of these formalisms.Supported in part by an NRC grant.     

Quantum probability and unified approach to quantization and dynamics

A simplified derivation of the Gudder-Hemion quantum probability formula is proposed. Defining configurations as the classical (q, p) deterministic states and generalized action as the (quantum) generating function of a canonical transformation, we obtain the usual quantization rules (for arbitrary polynomial quantities) and derive the Schrödinger wave equation on the same grounds. This approach suggests a statistical interpretation of the wave function in terms of the classical canonical transformations.     

On probability theory and probabilistic physics—Axiomatics and methodology

A new formulation involving fulfillment of all the Kolmogorov axioms is suggested for acomplete probability theory. This proves to be not a purely mathematical discipline. Probability theory deals with abstract objects—images of various classes of concrete objects—whereas experimental statistics deals with concrete objects alone. Both have to be taken into account. Quantum physics and classical statistical physics prove to be different aspects ofone probabilistic physics. The connection of quantum mechanics with classical statistical mechanics is examined and the origin of the Schrödinger equation is elucidated. Attention is given to the true meaning of the wave-corpuscle duality, and the incompleteness of nonrelativistic quantum mechanics is explained.