15,930 research outputs found

### Instantons in Quantum Mechanics and Resurgent Expansions

Certain quantum mechanical potentials give rise to a vanishing perturbation
series for at least one energy level (which as we here assume is the ground
state), but the true ground-state energy is positive. We show here that in a
typical case, the eigenvalue may be expressed in terms of a generalized
perturbative expansion (resurgent expansion). Modified Bohr-Sommerfeld
quantization conditions lead to generalized perturbative expansions which may
be expressed in terms of nonanalytic factors of the form exp(-a/g), where a > 0
is the instanton action, and power series in the coupling g, as well as
logarithmic factors. The ground-state energy, for the specific Hamiltonians, is
shown to be dominated by instanton effects, and we provide numerical evidence
for the validity of the related conjectures.Comment: 12 pages, LaTeX; further typographical errors correcte

### Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions

We consider specific quantum mechanical model problems for which perturbation
theory fails to explain physical properties like the eigenvalue spectrum even
qualitatively, even if the asymptotic perturbation series is augmented by
resummation prescriptions to "cure" the divergence in large orders of
perturbation theory. Generalizations of perturbation theory are necessary which
include instanton configurations, characterized by nonanalytic factors
exp(-a/g) where a is a constant and g is the coupling. In the case of
one-dimensional quantum mechanical potentials with two or more degenerate
minima, the energy levels may be represented as an infinite sum of terms each
of which involves a certain power of a nonanalytic factor and represents itself
an infinite divergent series. We attempt to provide a unified representation of
related derivations previously found scattered in the literature. For the
considered quantum mechanical problems, we discuss the derivation of the
instanton contributions from a semi-classical calculation of the corresponding
partition function in the path integral formalism. We also explain the relation
with the corresponding WKB expansion of the solutions of the Schroedinger
equation, or alternatively of the Fredholm determinant det(H-E) (and some
explicit calculations that verify this correspondence). We finally recall how
these conjectures naturally emerge from a leading-order summation of
multi-instanton contributions to the path integral representation of the
partition function. The same strategy could result in new conjectures for
problems where our present understanding is more limited.Comment: 66 pages, LaTeX; refs. to part II preprint update

### Proof of Razumov-Stroganov conjecture for some infinite families of link patterns

We prove the Razumov--Stroganov conjecture relating ground state of the O(1)
loop model and counting of Fully Packed Loops in the case of certain types of
link patterns. The main focus is on link patterns with three series of nested
arches, for which we use as key ingredient of the proof a generalization of the
Mac Mahon formula for the number of plane partitions which includes three
series of parameters

### Jucys-Murphy elements and Weingarten matrices

We provide a compact proof of the recent formula of Collins and Matsumoto for
the Weingarten matrix of the orthogonal group using Jucys-Murphy elements.Comment: v2: added a referenc

### Random vector and matrix and vector theories: a renormalization group approach

Random matrices in the large N expansion and the so-called double scaling
limit can be used as toy models for quantum gravity: 2D quantum gravity coupled
to conformal matter. This has generated a tremendous expansion of random matrix
theory, tackled with increasingly sophisticated mathematical methods and number
of matrix models have been solved exactly. However, the somewhat paradoxical
situation is that either models can be solved exactly or little can be said.
Since the solved models display critical points and universal properties, it is
tempting to use renormalization group ideas to determine universal properties,
without solving models explicitly. Initiated by Br\'ezin and Zinn-Justin, the
approach has led to encouraging results, first for matrix integrals and then
quantum mechanics with matrices, but has not yet become a universal tool as
initially hoped. In particular, general quantum field theories with matrix
fields require more detailed investigations. To better understand some of the
encountered difficulties, we first apply analogous ideas to the simpler O(N)
symmetric vector models, models that can be solved quite generally in the large
N limit. Unlike other attempts, our method is a close extension of Br\'ezin and
Zinn-Justin. Discussing vector and matrix models with similar approximation
scheme, we notice that in all cases (vector and matrix integrals, vector and
matrix path integrals in the local approximation), at leading order,
non-trivial fixed points satisfy the same universal algebraic equation, and
this is the main result of this work. However, its precise meaning and role
have still to be better understood

### Renormalization of gauge theories and master equation

The evolution of ideas which has led from the first proofs of the
renormalizability of non-abelian gauge theories, based on Slavnov--Taylor
identities, to the modern proof based on the BRS symmetry and the master
equation is recalled. This lecture has been delivered at the {\bf Symposium in
the Honour of Professor C. N. Yang}, Stony-Brook, May 21-22 1999.Comment: 9 pages, TeX, with private macros: zmacxxx.tex, lfont.te

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