Individuation in Quantum Mechanics

It has been claimed that the Principle of the Identity of Indiscernibles (PII) is incompatible with quantum mechanics, considered as a complete theory. Van Fraassen has argued specifically that a conflict between the two arises due to the requirements of Bose-Einstein statistics when imposed on two-particle quantum states. It is shown here that this apparent contradiction of the PII with quantum mechanics can be removed by the introduction of a natural criterion of individuality.     

Elementary Quantum Mechanics in a Space-Time Lattice

Studies of quantum fields and gravity suggest the existence of a minimal length, such as Planck length. It is natural to ask how the existence of a minimal length may modify the results in elementary quantum mechanics (QM) problems familiar to us. In this paper we address a simple problem from elementary non-relativistic quantum mechanics, called “particle in a box”, where the usual continuum (1+1)-space-time is supplanted by a space-time lattice. Our lattice consists of a grid of λ ₀×τ ₀ rectangles, where λ ₀, the lattice parameter, is a fundamental length (say Planck length) and, we take τ ₀ to be equal to λ ₀/c. The corresponding Schroedinger equation becomes a difference equation, the solution of which yields the q-eigenfunctions and q-eigenvalues of the energy operator as a function of λ ₀. The q-eigenfunctions form an orthonormal set and both q-eigenfunctions and q-eigenvalues reduce to continuum solutions as λ ₀→0. The corrections to eigenvalues because of the assumed lattice is shown to be O(l₀²)O(\lambda_{0}^{2}). We then compute the uncertainties in position and momentum, Δx, Δp for the box problem and study the consequent modification of Heisenberg uncertainty relation due to the assumption of space-time lattice, in contrast to modifications suggested by other investigations.     

Classical Mechanics and Quantum Mechanics

The Newton equation of motion is derived from quantum mechanics.     

Quantum Space-Time: A Review

The review presents systematically the results of studies which develop an idea of quantum properties of space-time in the microworld or near exotic objects (black holes, magnetic monopoles and others). On the basis of this idea motion equations of nonrelativistic and relativistic particles are studied. It is shown that introducing concept of quantum space-time at small distances (or near superdense matter) leads to an additional force giving rise to appearance of spiral-like behaviour of a particle along its classical trajectory. Given method is generalized to nonrelativistic quantum mechanics and to motion of a particle in gravitational force. In the latter case, there appears to be an antigravitational effect in the motion of a particle leading to different value of free-fall time (at least for gravitational force of exotic objects) for particles with different masses. Gravitational consequences of quantum space-time and tensor structures of physical quantities are inveatigated in detail. From experimental data on testing relativity and anisotropy of inertia estimation L ≦ 10⁻²² cm on the value of the fundamental length is obtained.     

Zitterbewegung in Quantum Mechanics

The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with structural features inherent in the Dirac equation. This paper develops a self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung and electromagnetic interactions. The model admits periodic solutions with quantized energy, and the correct magnetic moment is generated by charge circulation. It attributes to the electron an electric dipole moment rotating with ultrahigh frequency, and the possibility of observing this directly as a resonance in electron channeling is analyzed in detail. Correspondence with the Dirac equation is discussed. A modification of the Dirac equation is suggested to incorporate the rotating dipole moment.     

No-Signaling in Quantum Mechanics

We review some concepts and reasonings regarding the notion of no-signaling and its relation to quantum mechanics in bipartite Bell-type scenarios. We recapitulate the no-signaling property of joint conditional probability distributions in geometrical and information theoretic terms. We summarize the reasons why quantum mechanics does not enable instantaneous communication. We make some comments on quantum field theoretic aspects.     

Recurrence in Quantum Mechanics

We first compare the mathematical structure of quantum and classical mechanics when both are formulated in a C^*-algebraic framework. By using finite von Neumann algebras, a quantum mechanical analogue of Liouville's theorem is then proposed. We proceed to study Poincaré recurrence in C^*-algebras by mimicking the measure theoretic setting. The results are interpreted as recurrence in quantum mechanics, similar to Poincaré recurrence in classical mechanics.     

Consciousness and Quantum Mechanics

For a quantum mechanical measurement to be complete, John von Neumann and others assumed that a conscious observer must be present to affect a reduction or collapse of the state function. Also, William James believed that the influence of consciousness on physical bodies is required by the demands of biological evolution. The author shows how both of these ideas might be correct if there exists a neurological mechanism that responds to the presence of an inside observer of a kind defined in a previous paper. An experiment is proposed to test the hypothetical mechanism.     

Time-Symmetric Quantum Mechanics

A time-symmetric formulation of nonrelativistic quantum mechanics is developed by applying two consecutive boundary conditions onto solutions of a time- symmetrized wave equation. From known probabilities in ordinary quantum mechanics, a time-symmetric parameter P₀ is then derived that properly weights the likelihood of any complete sequence of measurement outcomes on a quantum system. The results appear to match standard quantum mechanics, but do so without requiring a time-asymmetric collapse of the wavefunction upon measurement, thereby realigning quantum mechanics with an important fundamental symmetry.     

The Kantowski-Sachs Space-Time in Loop Quantum Gravity

We extend the ideas introduced in the previous work to a more general space-time. In particular we consider the Kantowski-Sachs space time with space section with topology . In this way we want to study a general space time that we think to be the space time inside the horizon of a black hole. In this case the phase space is four dimensional and we simply apply the quantization procedure suggested by loop quantum gravity and based on an alternative to the Schroedinger representation introduced by H. Halvorson. Through this quantization procedure we show that the inverse of the volume density and the Schwarzschild curvature invariant are upper bounded and so the space time is singularity free. Also in this case we can extend dynamically the space time beyond the classical singularity. PACS number: 04.60.Pp, 04.70.Dy     

Possibility and Time in Quantum Mechanics

In the discourse of quantum mechanics it is usual to say that non-commuting observables cannot have definite values at the same time, or that they cannot be simultaneously measured. But, what does the term ‘cannot’ mean in this context? Does it stand for impossible? Should Heisenberg’s principle be read in terms of uncertainty or of indeterminacy? On the other hand, whereas the debates about the nature of time in classical and relativistic mechanics have been many and varied, the question about the nature of time in quantum mechanics has not received the same attention, especially when compared to the large amount of literature on interpretive issues. The purpose of this paper is to show that, under a realist interpretation of quantum mechanics, these two matters, possibility and time, are strongly related. The final aim is to argue that, when possibility and actuality are conceived as irreducible modes of being, they are correlated to two different notions of time that can be distinguished in the quantum realm: parameter-time and event-time.     

Quantum Mechanics is About Quantum Information

I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles. The notion of quantum information is to be understood as a new physical primitive---just as, following Einsteins special theory of relativity, a field is no longer regarded as the physical manifestation of vibrations in a mechanical medium, but recognized as a new physical entity in its own right.     

Hamiltonian Formalism in Quantum Mechanics

Abstract

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures.

To Vladimir Igorevich Arnol’d with admiration, on occasion of his 60^th birthday.