In the first part of this work (http://www.arxiv.org/abs/quant-ph/0509044), it was shown that the Klein-Gordon-Maxwell electrodynamics in the unitary gauge allows natural elimination of the particle wave function and describes independent evolution of the electromagnetic field. Therefore, the electromagnetic field can be regarded as the guiding field in the Bohm interpretation of quantum mechanics. An extension of those results to the Dirac-Maxwell electrodynamics was less general, but represented at least an interesting toy model of quantum theory. Another model based on the Dirac-Maxwell electrodynamics is considered in this work. The model also typically allows elimination of the wave function and describes independent evolution of the electromagnetic field.
Schroedinger (Nature, v.169, p.538 (1952)) demonstrated that, contrary to the widespread belief, charged particles may be described by real fields. Therefore the sets of solutions with real-valued charged fields are considered in the present work for some versions of (non-second-quantized) quantum electrodynamics (for Dirac spinors "real-valued" is understood as "satisfying the Majorana condition"). In some of the versions any solution may be obtained from a solution from those sets by a gauge transform. The solutions from those sets have common features suggesting a natural interpretation alo…
The complex-valued quantum mechanics considers quantum motion on the complex plane instead of on the real axis, and studies the variations of a particle complex position, momentum and energy along a complex trajectory. On the basis of quantum Hamilton-Jacobi formalism in the complex space, we point out that having complex-valued motion is a universal property of quantum systems, because every quantum system is actually accompanied with an intrinsic complex Hamiltonian originating from the equation. It is revealed that the conventional real-valued quantum mechanics is a special case of the comp…
The interaction of a moving charged particle with its coherent electromagnetic field is analysed in the framework of non-relativistic quantum mechanics. It is shown that, when this interaction is taken into account, a spatially localized state may have a mean energy lower than the one corresponding to a delocalized state.
This paper addresses the single-valued requirement for quantum wave functions when they are analytically continued in the spatial coordinates. This is particularly relevant for de Broglie-Bohm, hydrodynamic, or stochastic models of quantum mechanics where the physical basis for single-valuedness has been questioned. It first constructs a large class of multi-valued wave functions based on knotted vortex filaments familiar in fluid mechanics, and then it argues that for free particles, these systems will likely radiate electromagnetic radiation if they are charged or have multipolar moments, an…
The conformability of angular observales (angular momentum and azimuthal angle) with the mathematical rules of quantum mechanics is a question which still rouses debates. It is valued negatively within the existing approaches which are restricted by two amendable presumptions. If the respective presumptions are removed one can obtain a general approach in which the mentioned question is valued positively.
Physicists describe nature using mathematics as the natural language, and for quantum mechanics, it prefers to use complex numbers. However, whether complex numbers are really necessary for the theory has been debated ever since its birth. Recently, it has been shown that a three-party correlation created in entanglement swapping scenarios comprising independent states and measurements cannot be reproduced using only real numbers. Previous experiments have conceptually supported the predication, yet not satisfying the independent state preparations and measurements simultaneously. Here, we imp…
We examine the longstanding problem of introducing a time observable in Quantum Mechanics; using the formalism of positive-operator-valued measures we show how to define such an observable in a natural way and we discuss some consequences.
Quantum mechanics of bending of a nonrelativistic monoenergetic charged particle beam by a dipole magnet is studied in the paraxial approximation. The transfer map for the position and momentum components of a particle of the beam between two transverse planes at different points on the curved optic axis of the system is derived starting with the nonrelativistic Schrödinger equation. It is found that the quantum transfer map contains the classical transfer map as the main part and there are tiny quantum correction terms. The negligibly small quantum corrections explain the remarkable success o…
We show that a large class of dissipative systems can be brought to a canonical form by introducing complex co-ordinates in phase space and a complex-valued hamiltonian. A naive canonical quantization of these systems lead to non-hermitean hamiltonian operators. The excited states are unstable and decay to the ground state . We also compute the tunneling amplitude across a potential barrier.
The induced fractional zero-point canonical angular momentum on charged particles by the Aharonov - Bohm (AB) vector potential is realized via modified combined traps. It explores new features for this type of quantum effects: In a limit of vanishing mechanical kinetic energy the AB vector potential alone cannot induce a fractional zero-point canonical angular momentum on charged particles at the quantum mechanical level in the AB magnetic field-free region; But for the case of the AB vector potential with another one of a "spectator" magnetic field the AB vector potential induces a fractional…
No physical measurement can be performed with infinite precision. This leaves a loophole in the standard no-go arguments against non-contextual hidden variables. All such arguments rely on choosing special sets of quantum-mechanical observables with measurement outcomes that cannot be simulated non-contextually. As a consequence, these arguments do not exclude the hypothesis that the class of physical measurements in fact corresponds to a dense subset of all theoretically possible measurements with outcomes and quantum probabilities that \emph{can} be recovered from a non-contextual hidden var…
We discuss the dynamics of a classical spinless quantum particle carrying electric charge and constrained to move on a non singular static surface in ordinary three dimensional space in the presence of arbitrary configurations of time independent electric currents. Starting from the canonical action in the embedding space we show that a charged particle with charge $q$ couples to a term linear in $qA^3M$, where $A^3$ is the transverse component of the electromagnetic vector potential and $M$ is the mean curvature in the surface. This term cancels exactly a curvature contribution to the orbital…
Fundamental solutions for the free Dirac electron and Einstein photon equations in position coordinates are constructed as matrix valued functionals on the space of bump functions. It is shown that these fundamental solutions are related by a unitary transform via the Cauchy distribution in imaginary time. We study the way the classical relativistic mechanics of the free particle comes from the quantum mechanics of the free Dirac electron.
We present a self-contained formulation of spin-free nonrelativistic quantum mechanics that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories, obtained by extremizing an action and satisfying energy conservation. The theory applies for arbitrary configuration spaces and system dimensionalities. Various beneficial ramifications - theoretical, computational, and interpretational - are discussed.
In this work, we give the wave equations of relativistic and non-relativistic quantum mechanics which are different from the Schrödinger and Klein-Gordon equation, and we also give the new relativistic wave equation of a charged particle in an electromagnetic field
Analysis of quantum mechanics with real-valued Schrödinger equation,single-event quantum-path dynamics, Mauprtuis path in parameter space, and branching paths beyond semiclassics
It is the purpose of the present contribution to demonstrate that the generalization of the concept of a quantum mechanical observable from the Hermitian operator of standard quantum mechanics to a positive operator-valued measure is not a peripheral issue, allegedly to be understood in terms of a trivial nonideality of practical measurement procedures, but that this generalization touches the very core of quantum mechanics, viz. complementarity and violation of the Bell inequalities.
Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular, th…
We consider the dynamics of a charged particle interacting with background electromagnetic field under the influence of linearized gravitational waves in the long wave-length and low-velocity limit. Following the prescription in \cite{speli}, the system is quantized and the Hamiltonian is then solved by using standard algebraic iterative methods. The solution is in conformity with the classical analysis and shows the possibility of tuning the frequency by changing the magnetic field to set up resonance.
It is shown that Bell's theorem fails for the Clifford algebra valued local realistic variables. This is made evident by exactly reproducing quantum mechanical expectation value for the EPR-Bohm type spin correlations observable by means of a local, deterministic, Clifford algebra valued variable, without necessitating either remote contextuality or backward causation. Since Clifford product of multivector variables is non-commutative in general, the spin correlations derived within our locally causal model violate the CHSH inequality just as strongly as their quantum mechanical counterparts.
The supersymmetric quantum mechanics of a two-dimensional non-relativistic particle subject to external magnetic and electric fields is studied in a superfield formulation and with the typical non-minimal coupling of (2+1) dimensions. Both the N=1 and N=2 cases are contemplated and the introduction of the electric interaction is suitably analysed.
Quantum mechanics over real numbers fully reproduces standard quantum theory
Centro Brasileiro de Pesquisas Fı́sicas（巴西物理研究中央中心）；National Institute of Science and Technology for Complex Systems（复杂系统科学与技术国家研究所）；Faculty of Mathematics, University of Bialystok（比亚韦斯托克大学数学学院）
The problem of quantum harmonic oscillator with "regular+random" square frequency, subjected to "regular+random external force, is considered in framework of representation of the wave function by complex-valued random process. Average transition probabilities are calculated. Stochastic density matrix method is developed, which is used for investigation of thermodinamical characteristics of the system, such as entropy and average energy.
Relations between Hamiltonian mechanics and quantum mechanics are studied. It is stressed that classical mechanics possesses all the specific features of quantum theory: operators, complex variables, probabilities (in case of ergodic systems). The Planck constant and the Fock space appear after putting a dynamical system in a thermal bath. For harmonic oscillator in a thermal bath, the probability amplitudes can be identified with the complex valued phase functions $f(q+ip)$ describing small deviations from the equilibrium state, when the time of relaxation is large. A chain of such oscillator…
According to our modal-Hamiltonian interpretation (MHI) of quantum mechanics, the Hamiltonian of the closed system defines the set of its definite-valued observables. This definition seems to be incompatible with the pointer basis selected by the environment-induced decoherence (EID) of the open system. In this paper we argue that decoherence can be understood from a closed system perspective which (i) shows that the incompatibility between MHI and EID is only apparent, and (ii) solves certain conceptual challenges that the EID program still has to face.
Motion of a charged particle in uniform magnetic field has been studied in detail, classically as well as quantum mechanically. However, classical dynamics of a charged particle in non-uniform magnetic field is solvable only for some specific cases. We present in this paper, a general integral equation for some specific class of non-uniform magnetic field and its solutions for some of them. We also examine the supersymmetry of Hamiltonians in exponentially decaying magnetic field with radial dependence and conclude that this kind of non-uniformity breaks supersymmetry.
The quantum Jarzynski equality is an important theorem of modern quantum thermodynamics. We show that the Jarzynski equality readily generalizes to relativistic quantum mechanics described by the Dirac equation. After establishing the conceptual framework we solve a pedagogical, yet experimentally relevant, system analytically. As a main result we obtain the exact quantum work distributions for charged particles traveling through a time-dependent vector potential evolving under Schrödinger as well as under Dirac dynamics, and for which the Jarzynski equality is verified. Special emphasis is pu…
We study theoretically the possibilities of coupling the quantum mechanical motion of a trapped charged particle (e.g. ion or electron) to quantum degrees of freedom of superconducting devices, nano-mechanical resonators and quartz bulk acoustic wave resonators. For each case, we estimate the coupling rate between the charged particle and its macroscopic counterpart and compare it to the decoherence rate, i.e. the rate at which quantum superposition decays. A hybrid system can only be considered quantum if the coupling rate significantly exceeds all decoherence rates. Our approach is to examin…
We construct an effective commutative Schrödinger equation in Moyal space-time in $(1+1)$-dimension where both $t$ and $x$ are operator-valued and satisfy $\left[ \hat{t}, \hat{x} \right] = i θ$. Beginning with a time-reparametrised form of an action we identify the actions of various space-time coordinates and their conjugate momenta on quantum states, represented by Hilbert-Schmidt operators. Since time is also regarded as a configuration space variable, we show how an `induced' inner product can be extracted, so that an appropriate quantum mechanical interpretation is obtained. We then disc…
The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill-behaved at turning points. In contrast, quantum trajectory methods--based on Bohmian mechanics (quantum hydrodynamics)--are characterized by infinite forces and erratic trajectories near nodes, but also well-behaved, single-valued representational functions. In this paper, we unify these two approaches into a single method that captures the best features of both, and in addition, satisfies the correspondence principle. Stationary eige…
Recently, it is shown that the extended phase space formulation of quantum mechanics is a suitable technique for studying the quantum dissipative system. Here, as an application of this formalism, we consider a dissipative system of charged particles interacting with an external time dependent electric field. Such a system has been investigating by Buch and Denman, and two solutions with completely different structure have been obtained for Schrödinger's equation in two different gauges. We demonstrate how both gauges lead to the same conductivity by generalizing the gauge transformations to t…
Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoherent states are diagonal with respect to a fixed orthonormal basis, i.e., they can be seen as arising from a von Neumann measurement. Here, we introduce and study a generalization to a resource theory of coherence defined with respect to the most general quantum measurement, i.e., to an arbitrary positive-operator-valued measure (POVM). We establish POVM-based coherence measures and POVM-incoherent operations which coincide…
We analyse a proposition which considers quantum theory as a mere tool for calculating probabilities for sequences of outcomes of observations made by an Observer, who him/herself remains outside the scope of the theory. Predictions are possible, provided a sequence includes at least two such observations. Complex valued probability amplitudes, each defined for an entire sequence of outcomes, are attributed to Observer's reasoning, and the problem of wave function's collapse is dismissed as a purely semantic one. Our examples include quantum "weak values", and a simplified version of the "dela…
Bohm's interpretation of Quantum Mechanics leads to the derivation of a Quantum Kinetic Equation (QKE): in the present work, propagation of waves in charged quantum gases is investigated starting from this QKE. Dispersion relations are derived for fully and weakly degenerate fermions and bosons (these latter above critical temperature), and the differences underlined. Use of a kinetic equation permits investigation of "Landau-type" damping: it is found that the presence of damping in fermion gases is dependent upon the degree of degeneracy, whereas it is always present in boson gases. In fully…
We describe here the coherent formulation of electromagnetism in the non-relativistic quantum-mechanical many-body theory of interacting charged particles. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because a manifold of misinterpretations emerged especially regarding the magnetic field and gauge invariance. The situation was determined by the historical development of quantum mechanics, starting from the Schrödinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body th…
In the Copenhagen interpretation the Heisenberg uncertainty relation is interpreted as the mathematical expression of the concept of complementarity, quantifying the mutual disturbance necessarily taking place in a simultaneous or joint measurement of incompatible observables. This interpretation has already been criticized by Ballentine a long time ago, and has recently been challenged in an experimental way. These criticisms can be substantiated by using the generalized formalism of positive operator-valued measures, from which a new inequality can be derived, precisely illustrating the Cope…
The Schrödinger equation is shown to be equivalent to a constrained Liouville equation under the assumption that phase space is extended to Grassmann algebra valued variables. For onedimensional systems, the underlying Hamiltonian dynamics has a N=2 supersymmetry. Potential applications to more realistic theories are briefly discussed.
Based on the Stueckelberg-Horwitz-Piron theory of covariant quantum mechanics on curved spacetime, we solved wave equation for a charged covariant harmonic oscillator in the background of charged static spherically symmetric black hole. Using Greens functions , we found asymptotic form for the wave function in the lowest mode (s-mode) and in higher moments. It has been proven that for s-wave, in a definite range of solid angles, the differential cross section depends effectively to the magnetic and electric charges of the black hole.
By using the multipolar gauge it is shown that the quantum mechanics of an electrically charged particle moving in a prescribed classical electromagnetic field (wave mechanics) may be expressed in a manner that is gauge invariant, except that the only gauge functions that are allowable in a gauge transformation are those that consist of the sum of a function that depends only on the space coordinates and a term that is the product of a constant and the time coordinate. The multipolar gauge specifies the specific set of potentials that are best suited for use in the Schroedinger equation.
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not…
The thesis is devoted to the phase space representation of relativistic quantum mechanics. For a class of observables with matrix-valued Weyl symbols proportional to the identity matrix, the Weyl-Wigner-Moyal formalism is proposed. The evolution equations are found to coincide with their counterparts in relativistic quantum mechanics with non-local Hamiltonian. The difference between the theories is connected with peculiarities of the constraints on the initial conditions. Effective increase in coherence between eigenstates of the Hamiltonian is found. Relativistic coherent states that take in…
In order to realize supersymmetric quantum mechanics methods on a four dimensional classical phase-space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star-product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jay…
Algebra of quantum mechanics via classical phonons. II: Klein-Gordon dynamics, the Heisenberg formalism, the Dirac canonical commutation rule and the Poincare algebra through the continuous Poisson bracket formalism
An induced fractional zero-point angular momentum of charged particles by the Bohm-Aharonov (B-A) vector potential is realized via a modified combined trap. It explores a "spectator" mechanism in this type of quantum effects: In the limit of the kinetic energy approaching one of its eigenvalues the B-A vector potential alone cannot induce a fractional zero-point angular momentum at quantum mechanical level in the B-A magnetic field-free region; But when there is a "spectator" magnetic field the B-A vector potential induces a fractional zero-point angular momentum. The "spectator" does not cont…
We explore the possibility of engineering quantum states of a charged mechanical oscillator by coupling it to a stream of atoms in superpositions of high-lying Rydberg states. Our scheme relies on the driving of a two-phonon resonance within the oscillator by coupling it to an atomic two-photon transition. This approach effectuates a controllable open system dynamics on the oscillator that permits the dissipative creation of squeezed and other non-classical states which are central to applications such as sensing and metrology or for studies of fundamental questions concerning the boundary bet…
Algebra of quantum mechanics via classical phonons. I: The Schrodinger equation as the Newtonian equation of motion and quantum observables as classical averages
We use the formulation of the quantum mechanics of first quantized Klein-Gordon fields given in the first of this series of papers to study relativistic coherent states. In particular, we offer an explicit construction of coherent states for both charged and neutral (real) free Klein-Gordon fields as well as for charged fields interacting with a constant magnetic field. Our construction is free from the problems associated with charge-superselection rule that complicated the previous studies. We compute various physical quantities associated with our coherent states and present a detailed inve…
In (quant-ph/0701141) Rajeev studied quantization of the damped simple harmonic oscillator and introduced a complex-valued Hamiltonian (which is normal). In this note we point out that the quantization is interpreted as a quantum mechanics with {\bf complex time}. We also present a problem on quantization of classical control systems.
A Hamiltonian approach is presented to study the two dimensional motion of damped electric charges in time dependent electromagnetic fields. The classical and the corresponding quantum mechanical problems are solved for particular cases using canonical transformations applied to Hamiltonians for a particle with variable mass. The Green's function is constructed and, from it, the motion of a Gaussian wave packet is studied in detail.
The generation, as well as the detection, of gravitational radiation by means of charged superfluids is considered. One example of such a "charged superfluid" consists of a pair of Planck-mass-scale, ultracold "Millikan oil drops," each with a single electron on its surface, in which the oil of the drop is replaced by superfluid helium. When levitated in a magnetic trap, and subjected to microwave-frequency electromagnetic radiation, a pair of such "Millikan oil drops" separated by a microwave wavelength can become an efficient quantum transducer between quadrupolar electromagnetic and gravita…
We recast Dirac's Lagrangian in quantum mechanics in the language of vector bundles and show that the action is an operator-valued connection one-form. Phases associated with change of frames of reference are seen to be total differentials in the transformation of the action. The relativistic case is discussed and we show that it gives the correct phase in the non-relativistic limit for uniform acceleration.
