Canonical Quantum Statistics of an Isolated Schwarzschild Black Hole with a Spectrum E_n = sigma sqrt{n} E_P

Many authors - beginning with Bekenstein - have suggested that the energy levels E_n of a quantized isolated Schwarzschild black hole have the form E_n = sigma sqrt{n} E_P, n=1,2,..., sigma =O(1), with degeneracies g^n. In the present paper properties of a system with such a spectrum, considered as a quantum canonical ensemble, are discussed: Its canonical partition function Z(g,beta=1/kT), defined as a series for g<1, obeys the 1-dimensional heat equation. It may be extended to values g>1 by means of an integral representation which reveals a cut of Z(g,beta) in the complex g-plane from g=1 to infinity. Approaching the cut from above yields a real and an imaginary part of Z. Very surprisingly, it is the (explicitly known) imaginary part which gives the expected thermodynamical properties of Schwarzschild black holes: Identifying the internal energy U with the rest energy Mc^2 requires beta to have the value (in natural units) beta = 2M(lng/sigma^2)[1+O(1/M^2)], (4pi sigma^2=lng gives Hawking's beta_H), and yields the entropy S=[lng/(4pi sigma^2)] A/4 + O(lnA), where A is the area of the horizon.Comment: 14 pages, LaTeX A brief note added which refers to previous work where the imaginary part of the partition function is related to metastable states of the syste

Action-angle variables for dihedral systems on the circle

A nonrelativistic particle on a circle and subject to a cos^{-2}(k phi) potential is related to the two-dimensional (dihedral) Coxeter system I_2(k), for k in N. For such `dihedral systems' we construct the action-angle variables and establish a local equivalence with a free particle on the circle. We perform the quantization of these systems in the action-angle variables and discuss the supersymmetric extension of this procedure. By allowing radial motion one obtains related two-dimensional systems, including A_2, BC_2 and G_2 three-particle rational Calogero models on R, which we also analyze.Comment: 8 pages; v2: references added, typos fixed, version for PL

Noncanonical quantum optics

Modification of the right-hand-side of canonical commutation relations (CCR) naturally occurs if one considers a harmonic oscillator with indefinite frequency. Quantization of electromagnetic field by means of such a non-CCR algebra naturally removes the infinite energy of vacuum but still results in a theory which is very similar to quantum electrodynamics. An analysis of perturbation theory shows that the non-canonical theory has an automatically built-in cut-off but requires charge/mass renormalization already at the nonrelativistic level. A simple rule allowing to compare perturbative predictions of canonical and non-canonical theories is given. The notion of a unique vacuum state is replaced by a set of different vacua. Multi-photon states are defined in the standard way but depend on the choice of vacuum. Making a simplified choice of the vacuum state we estimate corrections to atomic lifetimes, probabilities of multiphoton spontaneous and stimulated emission, and the Planck law. The results are practically identical to the standard ones. Two different candidates for a free-field Hamiltonian are compared.Comment: Completely rewritten version of quant-ph/0002003v2. There are overlaps between the papers, but sections on perturbative calculations show the same problem from different sides, therefore quant-ph/0002003v2 is not replace

On the two-body problem in general relativity

We consider the two-body problem in post-Newtonian approximations of general relativity. We report the recent results concerning the equations of motion, and the associated Lagrangian formulation, of compact binary systems, at the third post-Newtonian order (1/c^6 beyond the Newtonian acceleration). These equations are necessary when constructing the theoretical templates for searching and analyzing the gravitational-wave signals from inspiralling compact binaries in VIRGO-type experiments.Comment: 10 pages, to appear in a special issue of Comptes Rendus de l'Academie des Sciences, Paris, on the subject "Missions Spatiales en Physique Fondamentale

Maximal independent families and a topological consequence

AbstractFor κ⩾ω and X a set, a family A⊆P(X) is said to be κ-independent on X if |⋂A∈FAf(A)|⩾κ for each F∈[A]<ω and f∈{−1,+1}F; here A+1=A and A−1=X⧹A.Theorem 3.6For κ⩾ω, some A⊆P(κ) with |A|=2κ is simultaneously maximal κ-independent and maximal ω-independent on κ. The family A may be chosen so that every two elements of κ are separated by 2κ-many elements of A.Corollary 5.4For κ⩾ω there is a dense subset D of {0,1}2κ such that each nonempty open U⊆D satisfies |U|=d(U)=κ and no subset of D is resolvable. The set D may be chosen so that every two of its elements differ in 2κ-many coordinates.Remarks(a) Theorem 3.6 answers affirmatively a question of Eckertson [Topology Appl. 79 (1997) 1–11]. Two proofs are given here. (b) Parts of Corollary 5.4 have been obtained by other methods by Feng [Topology Appl. 105 (2000) 31–36] and (for κ=ω) by Alas et al. [Topology Appl. 107 (2000) 259–273]

The Hitting Times with Taboo for a Random Walk on an Integer Lattice

For a symmetric, homogeneous and irreducible random walk on d-dimensional integer lattice Z^d, having zero mean and a finite variance of jumps, we study the passage times (with possible infinite values) determined by the starting point x, the hitting state y and the taboo state z. We find the probability that these passages times are finite and analyze the tails of their cumulative distribution functions. In particular, it turns out that for the random walk on Z^d, except for a simple (nearest neighbor) random walk on Z, the order of the tail decrease is specified by dimension d only. In contrast, for a simple random walk on Z, the asymptotic properties of hitting times with taboo essentially depend on the mutual location of the points x, y and z. These problems originated in our recent study of branching random walk on Z^d with a single source of branching

Diverse homogeneous sets

A set H [subset of or equal to] [omega] is said to be diverse with respect to a partition [Pi] of [omega] if at least two pieces of [Pi] have an infinite intersection with H. A family of partitions of [omega] has the Ramsey property if, whenever [[omega]]2 is two-colored, some monochromatic set is diverse with respect to at least one partition in the family. We show that no countable collection of even infinite partitions of [omega] has the Ramsey property, but there always exists a collection of 1 finite partitions of [omega] with the Ramsey property.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30188/1/0000573.pd

Canonical form of Euler-Lagrange equations and gauge symmetries

The structure of the Euler-Lagrange equations for a general Lagrangian theory is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas gauge coordinates and their derivatives enter in the right hand sides of the equations as arbitrary functions of time. The reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. Moreover, the reduction procedure allows one to reveal the gauge identities between the Euler-Lagrange equations. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time, it is proven that for local theories all the gauge generators are local in time operators.Comment: 27 pages, LaTex fil

Third post-Newtonian dynamics of compact binaries: Noetherian conserved quantities and equivalence between the harmonic-coordinate and ADM-Hamiltonian formalisms

A Lagrangian from which derive the third post-Newtonian (3PN) equations of motion of compact binaries (neglecting the radiation reaction damping) is obtained. The 3PN equations of motion were computed previously by Blanchet and Faye in harmonic coordinates. The Lagrangian depends on the harmonic-coordinate positions, velocities and accelerations of the two bodies. At the 3PN order, the appearance of one undetermined physical parameter \lambda reflects an incompleteness of the point-mass regularization used when deriving the equations of motion. In addition the Lagrangian involves two unphysical (gauge-dependent) constants r'_1 and r'_2 parametrizing some logarithmic terms. The expressions of the ten Noetherian conserved quantities, associated with the invariance of the Lagrangian under the Poincar\'e group, are computed. By performing an infinitesimal ``contact'' transformation of the motion, we prove that the 3PN harmonic-coordinate Lagrangian is physically equivalent to the 3PN Arnowitt-Deser-Misner Hamiltonian obtained recently by Damour, Jaranowski and Sch\"afer.Comment: 30 pages, to appear in Classical and Quantum Gravit

Around the Hossz\'u-Gluskin theorem for nn-ary groups

We survey results related to the important Hossz\'u-Gluskin Theorem on $n$-ary groups adding also several new results and comments. The aim of this paper is to write all such results in uniform and compressive forms. Therefore some proofs of new results are only sketched or omitted if their completing seems to be not too difficult for readers. In particular, we show as the Hossz\'u-Gluskin Theorem can be used for evaluation how many different $n$-ary groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the mentioned theorem can be also used for investigation of $\mathcal{Q}$-independent subsets of semiabelian $n$-ary groups for some special families $\mathcal{Q}$ of mappings