17,037 research outputs found
Weak Coherent State Path Integrals
Weak coherent states share many properties of the usual coherent states, but
do not admit a resolution of unity expressed in terms of a local integral. They
arise e.g. in the case that a group acts on an inadmissible fiducial vector.
Motivated by the recent Affine Quantum Gravity Program, the present article
studies the path integral representation of the affine weak coherent state
matrix elements of the unitary time-evolution operator. Since weak coherent
states do not admit a resolution of unity, it is clear that the standard way of
constructing a path integral, by time slicing, is predestined to fail. Instead
a well-defined path integral with Wiener measure, based on a continuous-time
regularization, is used to approach this problem. The dynamics is rigorously
established for linear Hamiltonians, and the difficulties presented by more
general Hamiltonians are addressed.Comment: 21 pages, no figures, accepted by J. Math. Phy
Line defects and 5d instanton partition functions
We consider certain line defect operators in five-dimensional SUSY gauge
theories, whose interaction with the self-dual instantons is described by 1d
ADHM-like gauged quantum mechanics constructed by Tong and Wong. The partition
function in the presence of these operators is known to be a generating
function of BPS Wilson loops in skew symmetric tensor representations of the
gauge group. We calculate the partition function and explicitly prove that it
is a finite polynomial of the defect mass parameter , which is an essential
property of the defect operator and the Wilson loop generating function. The
relation between the line defect partition function and the qq-character
defined by N. Nekrasov is briefly discussed.Comment: 17 pages, 1 figure; typos fixed, references corrected; version to be
published in JHE
Turning the Quantum Group Invariant XXZ Spin-Chain Hermitian: A Conjecture on the Invariant Product
This is a continuation of a previous joint work with Robert Weston on the
quantum group invariant XXZ spin-chain (math-ph/0703085). The previous results
on quasi-Hermiticity of this integrable model are briefly reviewed and then
connected with a new construction of an inner product with respect to which the
Hamiltonian and the representation of the Temperley-Lieb algebra become
Hermitian. The approach is purely algebraic, one starts with the definition of
a positive functional over the Temperley-Lieb algebra whose values can be
computed graphically. Employing the Gel'fand-Naimark-Segal (GNS) construction
for C*-algebras a self-adjoint representation of the Temperley-Lieb algebra is
constructed when the deformation parameter q lies in a special section of the
unit circle. The main conjecture of the paper is the unitary equivalence of
this GNS representation with the representation obtained in the previous paper
employing the ideas of PT-symmetry and quasi-Hermiticity. An explicit example
is presented.Comment: 12 page
Spontaneous breakdown of Lorentz symmetry in scalar QED with higher order derivatives
Scalar QED is studied with higher order derivatives for the scalar field
kinetic energy. A local potential is generated for the gauge field due to the
covariant derivatives and the vacuum with non-vanishing expectation value for
the scalar field and the vector potential is constructed in the leading order
saddle point expansion. This vacuum breaks the global gauge and Lorentz
symmetry spontaneously. The unitarity of time evolution is assured in the
physical, positive norm subspace and the linearized equations of motion are
calculated. Goldstone theorem always keeps the radiation field massless. A
particular model is constructed where the the full set of standard Maxwell
equations is recovered on the tree level thereby relegating the effects of
broken Lorentz symmetry to the level of radiative corrections.Comment: 14 pages, to appear in Phys. Rev.
Random words, quantum statistics, central limits, random matrices
Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved
[math.CO/9906120] that the expected shape \lambda of the semi-standard tableau
produced by a random word in k letters is asymptotically the spectrum of a
random traceless k by k GUE matrix. In this article we give two arguments for
this fact. In the first argument, we realize the random matrix itself as a
quantum random variable on the space of random words, if this space is viewed
as a quantum state space. In the second argument, we show that the distribution
of \lambda is asymptotically given by the usual local limit theorem, but the
resulting Gaussian is disguised by an extra polynomial weight and by reflecting
walls. Both arguments more generally apply to an arbitrary finite-dimensional
representation V of an arbitrary simple Lie algebra g. In the original
question, V is the defining representation of g = su(k).Comment: 11 pages. Minor changes suggested by the refere
A rigorous path integral for quantum spin using flat-space Wiener regularization
Adapting ideas of Daubechies and Klauder [J. Math. Phys. {\bf 26} (1985)
2239] we derive a rigorous continuum path-integral formula for the semigroup
generated by a spin Hamiltonian. More precisely, we use spin-coherent vectors
parametrized by complex numbers to relate the coherent representation of this
semigroup to a suitable Schr\"odinger semigroup on the Hilbert space
of Lebesgue square-integrable functions on the Euclidean plane . The
path-integral formula emerges from the standard Feynman-Kac-It\^o formula for
the Schr\"odinger semigroup in the ultra-diffusive limit of the underlying
Brownian bridge on . In a similar vein, a path-integral formula can be
constructed for the coherent representation of the unitary time evolution
generated by the spin Hamiltonian.Comment: revised versio
Topological BF Theories in 3 and 4 Dimensions
In this paper we discuss topological BF theories in 3 and 4 dimensions.
Observables are associated to ordinary knots and links (in 3 dimensions) and to
2-knots (in 4 dimensions). The vacuum expectation values of such observables
give a wide range of invariants. Here we consider mainly the 3-dimensional
case, where these invariants include Alexander polynomials, HOMFLY polynomials
and Kontsevich integrals.Comment: 25 pages, latex, no figures. Transmission problems have been solve
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