Dynamical Ambiguities in Singular Gravitational Field

We consider particle dynamics in singular gravitational field. In 2d spacetime the system splits into two independent gravitational systems without singularity. Dynamical integrals of each system define $sl(2,R)$ algebra, but the corresponding symmetry transformations are not defined globally. Quantization leads to ambiguity. By including singularity one can get the global $SO(2.1)$ symmetry. Quantization in this case leads to unique quantum theory.Comment: 7 pages, latex, no figures, submitted for publicatio

New Generalized Verma Modules and Multilinear Intertwining Differential Operators

The present paper contains two interrelated developments. First, are proposed new generalized Verma modules. They are called k-Verma modules, k\in N, and coincide with the usual Verma modules for k=1. As a vector space a k-Verma module is isomorphic to the symmetric tensor product of k copies of the universal enveloping algebra U(g^-), where g^- is the subalgebra of lowering generators in the standard triangular decomposition of a simple Lie algebra g = g^+ \oplus h \oplus g^- . The second development is the proposal of a procedure for the construction of multilinear intertwining differential operators for semisimple Lie groups G . This procedure uses k-Verma modules and coincides for k=1 with a procedure for the construction of linear intertwining differential operators. For all k central role is played by the singular vectors of the k-Verma modules. Explicit formulae for series of such singular vectors are given. Using these are given explicitly many new examples of multilinear intertwining differential operators. In particular, for G = SL(2,R) are given explicitly all bilinear intertwining differential operators. Using the latter, as an application are constructed (n/2)-differentials for all n\in 2N, the ordinary Schwarzian being the case n=4. As another application, in a Note Added we propose a new hierarchy of nonlinear equations, the lowest member being the KdV equation.Comment: 30 pages, plain TEX with harvmac; corrected misprint

Weight bases of Gelfand-Tsetlin type for representations of classical Lie algebras

This paper completes a series devoted to explicit constructions of finite-dimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers dealt with C and D types). A weight basis for each representation of the Lie algebra o(2n+1) is constructed. The basis vectors are parametrized by Gelfand--Tsetlin-type patterns. Explicit formulas for the matrix elements of generators of o(2n+1) in this basis are given. The construction is based on the representation theory of the Yangians. A similar approach is applied to the A type case where the well-known formulas due to Gelfand and Tsetlin are reproduced.Comment: 29 pages, Late

Physical Vacuum Properties and Internal Space Dimension

The paper addresses matrix spaces, whose properties and dynamics are determined by Dirac matrices in Riemannian spaces of different dimension and signature. Among all Dirac matrix systems there are such ones, which nontrivial scalar, vector or other tensors cannot be made up from. These Dirac matrix systems are associated with the vacuum state of the matrix space. The simplest vacuum system realization can be ensured using the orthonormal basis in the internal matrix space. This vacuum system realization is not however unique. The case of 7-dimensional Riemannian space of signature 7(-) is considered in detail. In this case two basically different vacuum system realizations are possible: (1) with using the orthonormal basis; (2) with using the oblique-angled basis, whose base vectors coincide with the simple roots of algebra E_{8}. Considerations are presented, from which it follows that the least-dimension space bearing on physics is the Riemannian 11-dimensional space of signature 1(-)& 10(+). The considerations consist in the condition of maximum vacuum energy density and vacuum fluctuation energy density.Comment: 19 pages, 1figure. Submitted to General Relativity and Gravitatio

Information-disturbance tradeoff in estimating a maximally entangled state

We derive the amount of information retrieved by a quantum measurement in estimating an unknown maximally entangled state, along with the pertaining disturbance on the state itself. The optimal tradeoff between information and disturbance is obtained, and a corresponding optimal measurement is provided.Comment: 4 pages. Accepted for publication on Physical Review Letter

On the fidelity of two pure states

The fidelity of two pure states (also known as transition probability) is a symmetric function of two operators, and well-founded operationally as an event probability in a certain preparation-test pair. Motivated by the idea that the fidelity is the continuous quantum extension of the combinatorial equality function, we enquire whether there exists a symmetric operational way of obtaining the fidelity. It is shown that this is impossible. Finally, we discuss the optimal universal approximation by a quantum operation.Comment: LaTeX2e, 8 pages, submitted to J. Phys. A: Math. and Ge

Quantum state estimation and large deviations

In this paper we propose a method to estimate the density matrix \rho of a d-level quantum system by measurements on the N-fold system. The scheme is based on covariant observables and representation theory of unitary groups and it extends previous results concerning the estimation of the spectrum of \rho. We show that it is consistent (i.e. the original input state \rho is recovered with certainty if N \to \infty), analyze its large deviation behavior, and calculate explicitly the corresponding rate function which describes the exponential decrease of error probabilities in the limit N \to \infty. Finally we discuss the question whether the proposed scheme provides the fastest possible decay of error probabilities.Comment: LaTex2e, 40 pages, 2 figures. Substantial changes in Section 4: one new subsection (4.1) and another (4.2 was 4.1 in the previous version) completely rewritten. Minor changes in Sect. 2 and 3. Typos corrected. References added. Accepted for publication in Rev. Math. Phy

The BV-algebra structure of W_3 cohomology

We summarize some recent results obtained in collaboration with J. McCarthy on the spectrum of physical states in $W_3$ gravity coupled to $c=2$ matter. We show that the space of physical states, defined as a semi-infinite (or BRST) cohomology of the $W_3$ algebra, carries the structure of a BV-algebra. This BV-algebra has a quotient which is isomorphic to the BV-algebra of polyvector fields on the base affine space of $SL(3,C)$. Details have appeared elsewhere. [Published in the proceedings of "Gursey Memorial Conference I: Strings and Symmetries," Istanbul, June 1994, eds. G. Aktas et al., Lect. Notes in Phys. 447, (Springer Verlag, Berlin, 1995)]Comment: 8 pages; uses macros tables.tex and amssym.def (version 2.1 or later

Fusion rules for admissible representations of affine algebras: the case of A2(1)A_2^{(1)}

We derive the fusion rules for a basic series of admissible representations of $\hat{sl}(3)$ at fractional level $3/p-3$. The formulae admit an interpretation in terms of the affine Weyl group introduced by Kac and Wakimoto. It replaces the ordinary affine Weyl group in the analogous formula for the fusion rules multiplicities of integrable representations. Elements of the representation theory of a hidden finite dimensional graded algebra behind the admissible representations are briefly discussed.Comment: containing two TEX files: main file using input files harvmac.tex, amssym.def, amssym.tex, 19p.; file with figures using XY-pic package, 6p. Correction in the definition of general shifted weight diagra

Covariant quantum measurements which maximize the likelihood

We derive the class of covariant measurements which are optimal according to the maximum likelihood criterion. The optimization problem is fully resolved in the case of pure input states, under the physically meaningful hypotheses of unimodularity of the covariance group and measurability of the stability subgroup. The general result is applied to the case of covariant state estimation for finite dimension, and to the Weyl-Heisenberg displacement estimation in infinite dimension. We also consider estimation with multiple copies, and compare collective measurements on identical copies with the scheme of independent measurements on each copy. A "continuous-variables" analogue of the measurement of direction of the angular momentum with two anti-parallel spins by Gisin and Popescu is given.Comment: 8 pages, RevTex style, submitted to Phys. Rev.