On the role of coherent states in quantum foundations

Coherent states, and the Hilbert space representations they generate, provide ideal tools to discuss classical/quantum relationships. In this paper we analyze three separate classical/quantum problems using coherent states, and show that useful connections arise among them. The topics discussed are: (1) a truly natural formulation of phase space path integrals; (2) how this analysis implies that the usual classical formalism is ``simply a subset'' of the quantum formalism, and thus demonstrates a universal coexistence of both the classical and quantum formalisms; and (3) how these two insights lead to a complete analytic solution of a formerly insoluble family of nonlinear quantum field theory models.Comment: ICQOQI'2010, Kiev, Ukraine, May-June 2010, Conference Proceedings (9 pages

The Utility of Coherent States and other Mathematical Methods in the Foundations of Affine Quantum Gravity

Affine quantum gravity involves (i) affine commutation relations to ensure metric positivity, (ii) a regularized projection operator procedure to accomodate first- and second-class quantum constraints, and (iii) a hard-core interpretation of nonlinear interactions to understand and potentially overcome nonrenormalizability. In this program, some of the less traditional mathematical methods employed are (i) coherent state representations, (ii) reproducing kernel Hilbert spaces, and (iii) functional integral representations involving a continuous-time regularization. Of special importance is the profoundly different integration measure used for the Lagrange multiplier (shift and lapse) functions. These various concepts are first introduced on elementary systems to help motivate their application to affine quantum gravity.Comment: 15 pages, Presented at the X-International Conference on Symmetry Methods in Physic

Generalized Affine Coherent States: A Natural Framework for Quantization of Metric-like Variables

Affine variables, which have the virtue of preserving the positive-definite character of matrix-like objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherent-state overlap function in terms of suitable path-integral formulations.Comment: 17 pages, LaTeX, no figure

Coherent State Approach to Time Reparameterization Invariant Systems

For many years coherent states have been a useful tool for understanding fundamental questions in quantum mechanics. Recently, there has been work on developing a consistent way of including constraints into the phase space path integral that naturally arises in coherent state quantization. This new approach has many advantages over other approaches, including the lack of any Gribov problems, the independence of gauge fixing, and the ability to handle second-class constraints without any ambiguous determinants. In this paper, I use this new approach to study some examples of time reparameterization invariant systems, which are of special interest in the field of quantum gravity

Noncanonical Quantization of Gravity. I. Foundations of Affine Quantum Gravity

The nature of the classical canonical phase-space variables for gravity suggests that the associated quantum field operators should obey affine commutation relations rather than canonical commutation relations. Prior to the introduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constraints are introduced following the projection operator method which involves no gauge fixing, no complicated moduli space, nor any auxiliary fields. The result, which is only qualitatively sketched in the present paper, involves another reproducing kernel with which inner products are defined for the physical Hilbert space and which is obtained through a reduction of the original reproducing kernel. Several of the steps involved in this general analysis are illustrated by means of analogous steps applied to one-dimensional quantum mechanical models. These toy models help in motivating and understanding the analysis in the case of gravity.Comment: minor changes, LaTeX, 37 pages, no figure

Coherent states for the hydrogen atom

We construct wave packets for the hydrogen atom labelled by the classical action-angle variables with the following properties. i) The time evolution is exactly given by classical evolution of the angle variables. (The angle variable corresponding to the position on the orbit is now non-compact and we do not get exactly the same state after one period. However the gross features do not change. In particular the wave packet remains peaked around the labels.) ii) Resolution of identity using this overcomplete set involves exactly the classical phase space measure. iii) Semi-classical limit is related to Bohr-Sommerfield quantization. iv) They are almost minimum uncertainty wave packets in position and momentum.Comment: 9 pages, 2 figures, minor change in language and journal reference adde

Gazeau−-Klauder squeezed states associated with solvable quantum systems

A formalism for the construction of some classes of Gazeau$-$Klauder squeezed states, corresponding to arbitrary solvable quantum systems with a known discrete spectrum, are introduced. As some physical applications, the proposed structure is applied to a few known quantum systems and then statistical properties as well as squeezing of the obtained squeezed states are studied. Finally, numerical results are presented.Comment: 18 pages, 12 figure

Lattice simulations of real-time quantum fields

We investigate lattice simulations of scalar and nonabelian gauge fields in Minkowski space-time. For SU(2) gauge-theory expectation values of link variables in 3+1 dimensions are constructed by a stochastic process in an additional (5th) ``Langevin-time''. A sufficiently small Langevin step size and the use of a tilted real-time contour leads to converging results in general. All fixed point solutions are shown to fulfil the infinite hierarchy of Dyson-Schwinger identities, however, they are not unique without further constraints. For the nonabelian gauge theory the thermal equilibrium fixed point is only approached at intermediate Langevin-times. It becomes more stable if the complex time path is deformed towards Euclidean space-time. We analyze this behavior further using the real-time evolution of a quantum anharmonic oscillator, which is alternatively solved by diagonalizing its Hamiltonian. Without further optimization stochastic quantization can give accurate descriptions if the real-time extend of the lattice is small on the scale of the inverse temperature.Comment: 36 pages, 15 figures, Late

Photon-added coherent states as nonlinear coherent states

The states $|\alpha,m>$, defined as ${a^{\dagger}}^{m}|\alpha>$ up to a normalization constant and $m$ is a nonnegative integer, are shown to be the eigenstates of $f(\hat{n},m)\hat{a}$ where $f(\hat{n},m)$ is a nonlinear function of the number operator $\hat{n}$. The explicit form of $f(\hat{n},m)$ is constructed. The eigenstates of this operator for negative values of $m$ are introduced. The properties of these states are discussed and compared with those of the state $|\alpha,m >$.Comment: Rev Tex file with two figures as postscript files attached. Email: [email protected]