Unambiguous quantization from the maximum classical correspondence that is self-consistent: the slightly stronger canonical commutation rule Dirac missed

Dirac's identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac's canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E. H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical "shadow world" which can manifest negative densities in phase space.Comment: 12 pages, Final publication in Foundations of Physics; available online at http://www.springerlink.com/content/k827666834140322

Perturbation theory of the space-time non-commutative real scalar field theories

The perturbative framework of the space-time non-commutative real scalar field theory is formulated, based on the unitary S-matrix. Unitarity of the S-matrix is explicitly checked order by order using the Heisenberg picture of Lagrangian formalism of the second quantized operators, with the emphasis of the so-called minimal realization of the time-ordering step function and of the importance of the $\star$-time ordering. The Feynman rule is established and is presented using $\phi^4$ scalar field theory. It is shown that the divergence structure of space-time non-commutative theory is the same as the one of space-space non-commutative theory, while there is no UV-IR mixing problem in this space-time non-commutative theory.Comment: Latex 26 pages, notations modified, add reference

Dirac versus Reduced Quantization of the Poincar\'{e} Symmetry in Scalar Electrodynamics

The generators of the Poincar\'{e} symmetry of scalar electrodynamics are quantized in the functional Schr\"{o}dinger representation. We show that the factor ordering which corresponds to (minimal) Dirac quantization preserves the Poincar\'{e} algebra, but (minimal) reduced quantization does not. In the latter, there is a van Hove anomaly in the boost-boost commutator, which we evaluate explicitly to lowest order in a heat kernel expansion using zeta function regularization. We illuminate the crucial role played by the gauge orbit volume element in the analysis. Our results demonstrate that preservation of extra symmetries at the quantum level is sometimes a useful criterion to select between inequivalent, but nevertheless self-consistent, quantization schemes.Comment: 24 page

Symplectic evolution of Wigner functions in markovian open systems

The Wigner function is known to evolve classically under the exclusive action of a quadratic hamiltonian. If the system does interact with the environment through Lindblad operators that are linear functions of position and momentum, we show that the general evolution is the convolution of the classically evolving Wigner function with a phase space gaussian that broadens in time. We analyze the three generic cases of elliptic, hyperbolic and parabolic Hamiltonians. The Wigner function always becomes positive in a definite time, which is shortest in the hyperbolic case. We also derive an exact formula for the evolving linear entropy as the average of a narrowing gaussian taken over a probability distribution that depends only on the initial state. This leads to a long time asymptotic formula for the growth of linear entropy.Comment: this new version treats the dissipative cas

Hamiltonian dynamics and spectral theory for spin-oscillators

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.Comment: 32 page

Diffeomorphisms as Symplectomorphisms in History Phase Space: Bosonic String Model

The structure of the history phase space $\cal G$ of a covariant field system and its history group (in the sense of Isham and Linden) is analyzed on an example of a bosonic string. The history space $\cal G$ includes the time map $\sf T$ from the spacetime manifold (the two-sheet) $\cal Y$ to a one-dimensional time manifold $\cal T$ as one of its configuration variables. A canonical history action is posited on $\cal G$ such that its restriction to the configuration history space yields the familiar Polyakov action. The standard Dirac-ADM action is shown to be identical with the canonical history action, the only difference being that the underlying action is expressed in two different coordinate charts on $\cal G$. The canonical history action encompasses all individual Dirac-ADM actions corresponding to different choices $\sf T$ of foliating $\cal Y$. The history Poisson brackets of spacetime fields on $\cal G$ induce the ordinary Poisson brackets of spatial fields in the instantaneous phase space ${\cal G}_{0}$ of the Dirac-ADM formalism. The canonical history action is manifestly invariant both under spacetime diffeomorphisms Diff$\cal Y$ and temporal diffeomorphisms Diff$\cal T$. Both of these diffeomorphisms are explicitly represented by symplectomorphisms on the history phase space $\cal G$. The resulting classical history phase space formalism is offered as a starting point for projection operator quantization and consistent histories interpretation of the bosonic string model.Comment: 45 pages, no figure

The Quantum Mechanical Arrows of Time

The familiar textbook quantum mechanics of laboratory measurements incorporates a quantum mechanical arrow of time --- the direction in time in which state vector reduction operates. This arrow is usually assumed to coincide with the direction of the thermodynamic arrow of the quasiclassical realm of everyday experience. But in the more general context of cosmology we seek an explanation of all observed arrows, and the relations between them, in terms of the conditions that specify our particular universe. This paper investigates quantum mechanical and thermodynamic arrows in a time-neutral formulation of quantum mechanics for a number of model cosmologies in fixed background spacetimes. We find that a general universe may not have well defined arrows of either kind. When arrows are emergent they need not point in the same direction over the whole of spacetime. Rather they may be local, pointing in different directions in different spacetime regions. Local arrows can therefore be consistent with global time symmetry.Comment: 9 pages, 4 figures, revtex4, typos correcte

Cosmological perturbations and short distance physics from Noncommutative Geometry

We investigate the possible effects on the evolution of perturbations in the inflationary epoch due to short distance physics. We introduce a suitable non local action for the inflaton field, suggested by Noncommutative Geometry, and obtained by adopting a generalized star product on a Friedmann-Robertson-Walker background. In particular, we study how the presence of a length scale where spacetime becomes noncommutative affects the gaussianity and isotropy properties of fluctuations, and the corresponding effects on the Cosmic Microwave Background spectrum.Comment: Published version, 16 page

Sensitivity to measurement perturbation of single atom dynamics in cavity QED

We consider continuous observation of the nonlinear dynamics of single atom trapped in an optical cavity by a standing wave with intensity modulation. The motion of the atom changes the phase of the field which is then monitored by homodyne detection of the output field. We show that the conditional Hilbert space dynamics of this system, subject to measurement induced perturbations, depends strongly on whether the corresponding classical dynamics is regular or chaotic. If the classical dynamics is chaotic the distribution of conditional Hilbert space vectors corresponding to different observation records tends to be orthogonal. This is a characteristic feature of hypersensitivity to perturbation for quantum chaotic systems.Comment: 11 pages, 6 figure

On plane wave and vortex-like solutions of noncommutative Maxwell-Chern-Simons theory

We investigate the spectrum of the gauge theory with Chern-Simons term on the noncommutative plane, a modification of the description of the Quantum Hall fluid recently proposed by Susskind. We find a series of the noncommutative massive ``plane wave'' solutions with polarization dependent on the magnitude of the wave-vector. The mass of each branch is fixed by the quantization condition imposed on the coefficient of the noncommutative Chern-Simons term. For the radially symmetric ansatz a vortex-like solution is found and investigated. We derive a nonlinear difference equation describing these solutions and we find their asymptotic form. These excitations should be relevant in describing the Quantum Hall transitions between plateaus and the end transition to the Hall Insulator.Comment: 17 pages, LaTeX (JHEP), 1 figure, added references, version accepted to JHE