On the Classical Limit in Bohm’s Theory

The standard means of seeking the classical limit in Bohmian mechanics is through the imposition of vanishing quantum force and quantum potential for pure states. We argue that this approach fails, and that the Bohmian classical limit can be realized only by combining narrow wave packets, mixed states, and environmental decoherence.     

Properties of Segal’s Entropy for Quantum Systems

Properties of Segal’s entropy for semifinite and finite von Neumann algebras are investigated. In particular, its invariance with respect to a trace-preserving normal *-homomorphism is studied, as well as norm-continuity in the trace norm on the set of bounded in the operator norm density matrices.     

Thermodynamic theory of the Tolman’s length

There still exists controversy on the sign and magnitude of the Tolman’s length and the Tolman’s gap.Further experimental,computational and theoretical investigations on them are needed to solve this problem.In 2006,Blokhuis and Kuipers obtained a rigorous relationship between the Tolman’s length and other thermodynamic quantities for the single-component liquid-vapour system.In the present paper,we derive two general relationships between the Tolman’s length and other thermodynamic quantities for the single-component liquid-vapour system.The relationship derived by Blokhuis and Kuipers and an earlier result turn out to be two special cases of our results.     

On Lieb's Conjecture for the Wehrl Entropy¶of Bloch Coherent States

Lieb@s conjecture for the Wehrl entropy of Bloch coherent states is proved for spin 1 and spin 3/2. Using a geometric representation we solve the entropy integrals for states of arbitrary spin and evaluate them explicitly in the cases of spin 1, 3/2, and 2. We also give a group theoretic proof for all spin of a related inequality. Received: 2 March 1999 / Accepted: 7 May 1999     

On the thermodynamics of the solidification of the Earth’s core

It has been shown that in the model of solidification in the inner solid Earth’s core, a phase transition at the interface between the solid core and outer liquid core is stepwise in the presence of a relatively small static shear modulus and shear acoustic vibrations in the inner core. The entropy and volume jumps, temperature hysteresis, and other thermodynamic derivatives certainly related to each other have been calculated taking into account the possible critical behavior of the system and using geophysical data.     

Gravitational Lensing in the Strong Field Limit for Kar’s Metric

In this paper we calculate the strong field limit deflection angle for a light ray passing near a scalar charged spherically symmetric object, described by a metric which comes from the low-energy limit of heterotic string theory. Then, we compare the expansion parameters of our results with those obtained in the Einstein’s canonical frame, obtained by a conformal transformation, and we show that, at least at first order, the results do not agree.     

Response to Nagle’s Criticism of My Proposed Definition of the Entropy

In a recent paper, Nagle criticized the new definition of entropy that I had proposed in an earlier work. In the examples for which Nagle claims my definition fails, he took a formula that I had derived for one set of experiments and used it to represent my definition for other experiments. However, the formulas obtained from my definition depend on the specific experimental observables. If my definition is correctly applied to Nagles experiments, no contradictions remain.     

Classical Projections of Quantum Mechanics and the Limit ħ → 0

The convergence of the dynamics of classical projection to the dynamics of the classical limit is investigated for 0. A mistake from a previous paper is pointed out, and the correct version of the result is given. A new, similar result is presented if the function generating the Hamiltonian of both the classical projection and the classical limit is a polynomial.     

On Zurek’s Derivation of the Born Rule

Recently, W. H. Zurek presented a novel derivation of the Born rule based on a mechanism termed environment-assisted invariance, or envariance [W. H. Zurek, Phys. Rev. Lett. 90(2), 120404 (2003)]. We review this approach and identify fundamental assumptions that have implicitly entered into it, emphasizing issues that any such derivation is likely to face.     

On the CHSH Form of Bell’s Inequalities

A common mistake present in the derivation of the usually known as the CHSH form of Bell’s inequalities is pointed out. References and comments to the correct approach are given. This error does not alter the final result and only affects the logical consistency of the derivation, but since it seems to be a widespread misconception regarding the roll and interpretation of the of use of hidden variables in Bell’s theorem it is considered to be of general interest.     

On the Road Map of Vogel’s Plane

We define “population” of Vogel’s plane as points for which universal character of adjoint representation is regular in the finite plane of its argument. It is shown that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. We find all their solutions: classical series of simple Lie algebras (including an “odd symplectic” one), \({D_{2,1,\lambda}}\) superalgebra, the line of sl(2) algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely \({knm=4k+4n+2m+12,}\) contains all simple Lie algebras, except so\({(2N+1).}\) Among isolated solutions are, besides exceptional simple Lie algebras, so called \({\mathfrak{e}_{7\frac{1}{2}}}\) algebra and also two other similar unidentified objects with positive dimensions. In addition, there are 47 isolated solutions in “unphysical semiplane” with negative dimensions. Isolated solutions mainly belong to the few lines in Vogel plane, including some rows of Freudenthal magic square. Universal dimension formulae have an integer values on all these solutions at least for first three symmetric powers of adjoint representation.     

Cornelius Lanczos’s Derivation of the Usual Action Integral of Classical Electrodynamics

The usual action integral of classical electrodynamics is derived starting from Lanczos’s electrodynamics – a pure field theory in which charged particles are identified with singularities of the homogeneous Maxwell’s equations interpreted as a generalization of the Cauchy–Riemann regularity conditions from complex to biquaternion functions of four complex variables. It is shown that contrary to the usual theory based on the inhomogeneous Maxwell’s equations, in which charged particles are identified with the sources, there is no divergence in the self-interaction so that the mass is finite, and that the only approximations made in the derivation are the usual conditions required for the internal consistency of classical electrodynamics. Moreover, it is found that the radius of the boundary surface enclosing a singularity interpreted as an electron is on the same order as that of the hypothetical “bag” confining the quarks in a hadron, so that Lanczos’s electrodynamics is engaging the reconsideration of many fundamental concepts related to the nature of elementary particles.     

On the Geodesic Nature of Wegner’s Flow

Wegner’s method of flow equations offers a useful tool for diagonalizing a given Hamiltonian and is widely used in various branches of quantum physics. Here, generalizing this method, a condition is derived, under which the corresponding flow of a quantum state becomes geodesic in a submanifold of the projective Hilbert space, independently of specific initial conditions. This implies the geometric optimality of the present method as an algorithm of generating stationary states. The result is illustrated by analyzing some physical examples.     

Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons

We present a new proof of the convergence of the N ?particle Schrödinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in \({\hbar}\) , up to an exponentially small remainder. For \({\hbar = 0}\) , the classical dynamics in the mean-field limit is given by the Vlasov equation.     

Niels Bohr’s Generalization of Classical Mechanics

We clarify Bohrs interpretation of quantum mechanics by demonstrating the central role played by his thesis that quantum theory is a rational generalization of classical mechanics. This thesis is essential for an adequate understanding of his insistence on the indispensability of classical concepts, his account of how the quantum formalism gets its meaning, and his belief that hidden variable interpretations are impossible.     

Fluctuations in the Thermodynamic Limit of Focussing Cubic Schrödinger

We consider the statistical mechanics of a complex field Z whose dynamics are governed by the focussing cubic Schrödinger equation. Here the Hamiltonian $$H = \int {_\Omega } \left[ {\frac{1}{2}\left| {\nabla Z} \right|^2 - \frac{1}{4}\left| Z \right|^4 } \right]dx$$ is unbounded from below, preventing the natural Gibbs measure from being normalizable. This difficulty may be circumvented⁽⁵⁾ by taking Ω the circle of perimeter L and fixing the mean-square (which is conserved by the dynamics): $\int {_0^L } \left| Z \right|^2 dx = LD$ for positive “density” D. The resulting (probability) measure on paths is absolutely continuous to the two-dimensional Wiener measure and is known to be invariant under the flow.^{(2, 7)} One way to extend this picture to the whole-line flow is to take the thermodynamic limit (L↑∞). Unfortunately, the unboundedness of H causes vast local concentration of the field as L increases and leads to collapse at L=∞.⁽¹¹⁾ Here we attempt to capture fluctuations away from this collapse by performing a joint continuum and infinite-volume limit for an appropriate lattice ensemble. The result is that, for high density, the scaled paths go over into a White Noise.     

Thermodynamic Limit for Spin Glasses. Beyond the Annealed Bound

Using a correlation inequality of Contucci and Lebowitz for spin glasses, we demonstrate existence of the thermodynamic limit for finite-dimensional spin glasses, without the assumption of the annealed bound. Using this method we can weaken the hypotheses for this result beyond what exists in the literature.     

Quantum Computing’s Classical Problem,Classical Computing’s Quantum Problem

Tasked with the challenge to build better and better computers, quantum computing and classical computing face the same conundrum: the success of classical computing systems. Small quantum computing systems have been demonstrated, and intermediate-scale systems are on the horizon, capable of calculating numeric results or simulating physical systems far beyond what humans can do by hand. However, to be commercially viable, they must surpass what our wildly successful, highly advanced classical computers can already do. At the same time, those classical computers continue to advance, but those advances are now constrained by thermodynamics, and will soon be limited by the discrete nature of atomic matter and ultimately quantum effects. Technological advances benefit both quantum and classical machinery, altering the competitive landscape. Can we build quantum computing systems that out-compute classical systems capable of some \(10^{30}\) logic gates per month? This article will discuss the interplay in these competing and cooperating technological trends.