Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics

We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous representations in a nested scale of Hilbert spaces. We also construct a couple of examples illustrative of the key features of group representations in rigged Hilbert spaces. Finally, we establish a simple criterion for the integrability of an operator Lie algebra in a rigged Hilbert space

Solutions of Quantum Gravity Coupled to the Scalar Field

We consider the Wheeler-De Witt equation for canonical quantum gravity coupled to massless scalar field. After regularizing and renormalizing this equation, we find a one-parameter class of its solutions.Comment: 8 pages, LaTe

Black Hole Solution of Quantum Gravity

We present a spherically symmetric and static exact solution of Quantum Einstein Equations. This solution is asymptotically (for large $r$) identical with the black hole solution on the anti--De Sitter background and, for some range of values of the mass possesses two horizons. We investigate thermodynamical properties of this solution.Comment: Plain Latex, 10 page

Irreversible Quantum Mechanics in the Neutral K-System

The neutral Kaon system is used to test the quantum theory of resonance scattering and decay phenomena. The two dimensional Lee-Oehme-Yang theory with complex Hamiltonian is obtained by truncating the complex basis vector expansion of the exact theory in Rigged Hilbert space. This can be done for K_1 and K_2 as well as for K_S and K_L, depending upon whether one chooses the (self-adjoint, semi-bounded) Hamiltonian as commuting or non-commuting with CP. As an unexpected curiosity one can show that the exact theory (without truncation) predicts long-time 2 pion decays of the neutral Kaon system even if the Hamiltonian conserves CP.Comment: 36 pages, 1 PostScript figure include

Topological Black Holes in Quantum Gravity

We derive the black hole solutions with horizons of non-trivial topology and investigate their properties in the framework of an approach to quantum gravity being an extension of Bohm's formulation of quantum mechanics. The solutions we found tend asymptotically (for large $r$) to topological black holes. We also analyze the thermodynamics of these space-times.Comment: 4pages, no figures, plain LaTe

Relativistic Partial Wave Analysis Using the Velocity Basis of the Poincare Group

The velocity basis of the Poincare group is used in the direct product space of two irreducible unitary representations of the Poincare group. The velocity basis with total angular momentum j will be used for the definition of relativistic Gamow vectors.Comment: 14 pages; revte

Semiclassical approximation with zero velocity trajectories

We present a new semiclassical method that yields an approximation to the quantum mechanical wavefunction at a fixed, predetermined position. In the approach, a hierarchy of ODEs are solved along a trajectory with zero velocity. The new approximation is local, both literally and from a quantum mechanical point of view, in the sense that neighboring trajectories do not communicate with each other. The approach is readily extended to imaginary time propagation and is particularly useful for the calculation of quantities where only local information is required. We present two applications: the calculation of tunneling probabilities and the calculation of low energy eigenvalues. In both applications we obtain excellent agrement with the exact quantum mechanics, with a single trajectory propagation.Comment: 16 pages, 7 figure

Gamow-Jordan Vectors and Non-Reducible Density Operators from Higher Order S-Matrix Poles

In analogy to Gamow vectors that are obtained from first order resonance poles of the S-matrix, one can also define higher order Gamow vectors which are derived from higher order poles of the S-matrix. An S-matrix pole of r-th order at z_R=E_R-i\Gamma/2 leads to r generalized eigenvectors of order k= 0, 1, ... , r-1, which are also Jordan vectors of degree (k+1) with generalized eigenvalue (E_R-i\Gamma/2). The Gamow-Jordan vectors are elements of a generalized complex eigenvector expansion, whose form suggests the definition of a state operator (density matrix) for the microphysical decaying state of this higher order pole. This microphysical state is a mixture of non-reducible components. In spite of the fact that the k-th order Gamow-Jordan vectors has the polynomial time-dependence which one always associates with higher order poles, the microphysical state obeys a purely exponential decay law.Comment: 39 pages, 3 PostScript figures; sub2.eps may stall some printers and should then be printed out separately; ghostview is o.

Resonances, Unstable Systems and Irreversibility: Matter Meets Mind

The fundamental time-reversal invariance of dynamical systems can be broken in various ways. One way is based on the presence of resonances and their interactions giving rise to unstable dynamical systems, leading to well-defined time arrows. Associated with these time arrows are semigroups bearing time orientations. Usually, when time symmetry is broken, two time-oriented semigroups result, one directed toward the future and one directed toward the past. If time-reversed states and evolutions are excluded due to resonances, then the status of these states and their associated backwards-in-time oriented semigroups is open to question. One possible role for these latter states and semigroups is as an abstract representation of mental systems as opposed to material systems. The beginnings of this interpretation will be sketched.Comment: 9 pages. Presented at the CFIF Workshop on TimeAsymmetric Quantum Theory: The Theory of Resonances, 23-26 July 2003, Instituto Superior Tecnico, Lisbon, Portugal; and at the Quantum Structures Association Meeting, 7-22 July 2004, University of Denver. Accepted for publication in the Internation Journal of Theoretical Physic