The Einstein–Podolsky–Rosen Effect: Paradox or Gate?

The Einstein–Podolsky–Rosen (EPR) paradox represents one of the most controversial aspects of quantum mechanics (QM). In this paper, we suggest that it can be solved by taking into account the fact that physical quantum phenomena can be extended backward in time (i.e. we take into account two arrows of time instead of one). We derive such a strong statement as a consequence of symmetries and conservation laws implying field equations which are invariant under time reversal. Our approach, violating Einstein's locality postulate, confirms QM predictions and explains the failure of Bell's inequalities.     

Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom

This paper gives a thorough investigation on formulating and solving quantum problems by extended analytical mechanics that extends canonical variables to complex domain. With this complex extension, we show that quantum mechanics becomes a part of analytical mechanics and hence can be treated integrally with classical mechanics. Complex canonical variables are governed by Hamilton equations of motion, which can be derived naturally from Schrödinger equation. Using complex canonical variables, a formal proof of the quantization axiom p → pˆ = −i, which is the kernel in constructing quantum-mechanical systems, becomes a one-line corollary of Hamilton mechanics. The derivation of quantum operators from Hamilton mechanics is coordinate independent and thus allows us to derive quantum operators directly under any coordinate system without transforming back to Cartesian coordinates. Besides deriving quantum operators, we also show that the various prominent quantum effects, such as quantization, tunneling, atomic shell structure, Aharonov–Bohm effect, and spin, all have the root in Hamilton mechanics and can be described entirely by Hamilton equations of motion.     

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra А on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ . We show that every a ∊ А defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita–Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra А which can be used to define a state dependent dynamics; i.e., the pair (А, ϕ), where ϕ is a state on А, is a “dynamic object.” Only if certain additional conditions are satisfied, the Connes–Nikodym–Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (А, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.     

Transformation theory ofq-quantum mechanics

Although there is no empirical motivation for replacing the commutators of dynamically conjugate operators in quantum mechanics byq-commutators, it appears possible to construct a consistent mathematical formulism based on this idea. To examine such a possibility further, we have studied the relation of this proposal to the Schwinger action principle, since the entire quantum mechanical formulism may be inferred from this principle. In particular, we have discussed the quantum transformation theory within this framework.To Julian Schwinger, 1918–1994, one of the creators of quantum field theory, and a giant of twentieth-century physics     

A quantum electromechanical device: the electromechanical single-electron pillar

A new nanoelectromechanical device is introduced, useful for quantum electromechanics. The focus will be on single-electron transistors with a mechanical degree of freedom. The technical approach as well as the experimental realization of a new vertical mechanical single-electron tunneling device are discussed. This transistor is fabricated in a semiconductor material, forming a nanopillar between source and drain contacts. This concept can readily be transferred to large scale fabrication, being of importance for building integrated sensors and amplifier stages for quantum electromechanical circuits. Operation of the device at room temperature in the frequency range of 350–400 MHz is presented. A straightforward theoretical model of device operation is given.     

Distinguishable equivalent particles with symmetrized wave functions

The quantum formalism ofdistinguishable, yetequivalent particles (with symmetric or antisymmetric wave functions) is here worked out. The result is an entirely explicit formulation of the way in which classical mechanics emerges from quantum mechanics for such particles. Distinguishability is achieved at the cost of dynamical precision; the two are, in fact, complementary.Research sponsored in part by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. 557–67.     

Hidden variable models for quantum theory cannot have any local part

It was shown by Bell that no local hidden variable model is compatible with quantum mechanics. If, instead, one permits the hidden variables to be entirely nonlocal, then any quantum mechanical predictions can be recovered. In this Letter, we consider general hidden variable models which can have both local and nonlocal parts. We show the existence of (experimentally verifiable) quantum correlations that are incompatible with any hidden variable model having a nontrivial local part, such as the model proposed by Leggett.     

How Does an Electron Tell the Time?

We examine a model of a digital clock to clarify the origin of the spacetime approach in special relativity. Specifically, we consider a two photon clock and assemble a statistical mechanics of such clocks to see how Minkowski space relates to local finite frequency clock behaviour. The result suggests that finite frequency clocks measure spacetime area and it is this feature that provides a simple mechanism behind Minkowski space on large scales. The same feature appears to implicate quantum mechanics on small scales.     

Contextuality within quantum mechanics manifested in subensemble mean values

For spin-1/2 particles, using a suitable Mach–Zehnder-type setup with a spin-flipper, we argue that it is a direct consequence of the quantum mechanical treatment that an experimentally verifiable subensemble mean of the measured values of an arbitrarily chosen spin variable exhibits dependence on the choice of a comeasurable ‘path’ observable. This, in turn, enables inferring path–spin contextuality at the level of individual measured values of spin that are predetermined using a relevant hidden-variable model applied to our setup.     

The formalism of equilibrium quantum statistical mechanics revisited

It is shown that the traditional formalism of equilibrium quantum statistical mechanics may fully be incorporated into a general macro-observable approach to quantum statistical mechanics recently proposed by the same author. ^(1,2) In particular, the partition functions which in the traditional approach are assumed to connect nonnormalized density operators with thermodynamic functions are reinterpreted as functions connecting so-called quantum mechanical effect operators with state parameters. It is argued that these functions although only part of a much richer internal structure of the macro-observable are sufficient to cope with all problems one usually encounters in equilibrium quantum statistical mechanics. p]Denn eigentlich unternehmen wir umsonst, das Wesen eines Dinges auszudrücken. Wirkungen werden wir gewahr, und eine vollständige Geschichte dieser Wirkungen umfaßte wohl allenfalls das Wesen jenes Dinges. Johann W. v. Goethe Farbenlehre     

Calculating the C Operator in PT-symmetric Quantum Mechanics

It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition it is cumbersome to calculate C in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method can be used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory.     

Failure of Standard Quantum Mechanics for the Description of Compound Quantum Entities

We reformulate the separated quantum entities theorem, i.e., the theorem that proves that two separated quantum entities cannot be described by means of standard quantum mechanics, within the fully elaborated operational Geneva–Brussels approach to quantum axiomatics, where the basic mathematical structure is that of a State Property System. We give arguments that show that the core of this result indicates a failure of standard quantum mechanics, and not just some peculiar shortcoming due to the axiomatic approach to quantum mechanics itself.     

Violation of Heisenberg's uncertainty relations on individual particles within subset ofγ photons ine + e − → 2γ pair creation

It is shown in this work that, if one accepts (a) the absolute energymomentum conservation in EPR-type sources (such as thee ⁺ e ^– 2 source analyzed here) and (b) the validity of the quantum mechanical formalism, then the standard Copenhagen interpretation of quantum measurement itself implies experimental violations of the Heisenberg uncertainty relations for certain observable subsets of photons emitted by such EPR sources. It thus follows from the quantum formalism itself that the uncertainty relations are not valid for all observable individual particles—in direct violation with an essential assumption of Bohr's and Heisenberg's philosophical outlook.     

Nonstandard extension of quantum logic and Dirac's bra-ket formalism of quantum mechanics

An extension of the quantum logical approach to the axiomatization of quantum mechanics usingnonstandard analysis methods is proposed. The physical meaning of a quantum logic as a lattice of propositions is conserved by its nonstandard extension. But not only the usual Hubert space formalism of quantum mechanics can be derived from the nonstandard extended quantum logic. Also the Dirac bra-ket quantum mechanics can be derived as a consequence of such an extended quantum logic.     

Elucidation of the probabilistic structure of quantum mechanics and definition of a compatible joint probability

A new integrated view on the probabilistic organization of quantum mechanics is constructed. It is then proved that for superposition state vectors the theoretical quantum mechanical distribution for the momentum observable is devoid of operational definition, and hence cannot be the source of conditions of compatibility to be imposed upon a researched joint probability concept. A compatible joint probability concept is outlined.Publisher's note: The full paper will appear inFoundations of Physics, Volume 13, Number 4, April 1983.     

Quantum limitations on the sensitivity of gravitational wave detectors with free masses

The problem of recording a classical disturbance by tracking the coordinate of a free particle is examined within the scope of nonrelativistic quantum mechanics. The absence of the fundamental limitation on the sensitivity — the standard quantum limit — is proven. An arbitrarily small disturbance can be recorded with preparation of the system in a quantum state having a negative quantum correlation coefficient between the observable coordinate and momentum. It is shown that it belongs to the collective coherent states — the condensed states. Arguments are presented for the absence of fundamental quantum limits on the magnitude of the recordable disturbance in the measurement of an arbitrary observable with a continuous spectrum.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 14–19, September, 1988.The author thanks A. A. Grib for useful discussions and interest in the work.     

A probabilistic approach to quantum mechanics based on ‘tomograms’

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.     

Remarks on quantum symmetry

We discuss the concept of Quantum Symmetry in quantum field theory, and in particular the role of the gauge principle. We present a scheme how quantum symmetries can be realized in a Hilbert space, and sketch its construction from the theory of superselection sectors of the gauge invariant (observable) quantities. The approach is independent of (Drinfeld's) quantum groups.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.     

A new approach for the justification of ensembles in quantum statistical mechanics — I

In this and the following paper, a new approach for the justification of ensembles in statistical mechanics is given. The essential physical idea is that a measurement is an average of values arising from disjoint regions in three-space. This idea is given a mathematical basis in terms of a class of operators called local operators, and the first paper is devoted primarily to the development of the properties of local operators. In particular, a complete characterization of the bounded local operators on ₂ spaces of finite measure is given. Two results of importance for statistical mechanics are also derived. First, it is shown that the observables of quantum mechanics are local operators. Second, it is shown that the expectation value of an observable for a pure state can be written formally as an ensemble average. In the following paper, these results are used to develop a new approach for the justification of statistical ensembles.This work was supported in part by research grants from the National Science Foundation and the U.S. Public Health Service. The material of this paper is contained in a doctoral dissertation submitted by the author to the University of Oregon (1969).     

Generalized gauge independence and the physical limitations on the von Neumann measurement postulate

An analysis is presented of the significance and consequent limitations on the applicability of the von Neumann measurement postulate in quantum mechanics. Directly observable quantities, such as the expectation value of the velocity operator, are distinguished from mathematical constructs, such as the expectation value of the canonical momentum, which are not directly observable. A simple criterion to distinguish between the two types of operators is derived. The non-observability of the electromagnetic four-potentials is shown to imply the non-measurability of the canonical momentum. The concept of a mechanical gauge is introduced and discussed. Classically the Lagrangian is nonunique within a total time derivative. This may be interpreted as the freedom of choosing a mechanical (M) gauge function. In quantum mechanics it is often implicitly assumed that the M-gauge vanishes. However, the requirement that directly observable quantities be independent of the arbitrary mechanical gauge is shown to lead to results analogous to those derived from the requirement of electromagnetic gauge independence of observables. The significance of the above to the observability of transition amplitudes between field-free energy eigenstates in the presence (and absence) of electromagnetic fields is discussed. E- and M-gauge independent transition amplitudes between field-free energy eigenstates in the absence of electromagnetic fields are defined. It is shown that, in general, such measurable amplitudes cannot be defined in the presence of externally applied time-dependent fields. Transition amplitudes in the presence of time-independent fields are discussed. The path dependence of previous derivations of E-gauge independent Hamiltonians and/or transition amplitudes in the presence of electromagnetic fields are related to the inherent M-gauge dependence of these quantities in the presence of such fields.