29,799 research outputs found
Complexified Path Integrals and the Phases of Quantum Field Theory
The path integral by which quantum field theories are defined is a particular
solution of a set of functional differential equations arising from the
Schwinger action principle. In fact these equations have a multitude of
additional solutions which are described by integrals over a complexified path.
We discuss properties of the additional solutions which, although generally
disregarded, may be physical with known examples including spontaneous symmetry
breaking and theta vacua. We show that a consideration of the full set of
solutions yields a description of phase transitions in quantum field theories
which complements the usual description in terms of the accumulation of
Lee-Yang zeroes. In particular we argue that non-analyticity due to the
accumulation of Lee-Yang zeros is related to Stokes phenomena and the collapse
of the solution set in various limits including but not restricted to, the
thermodynamic limit. A precise demonstration of this relation is given in terms
of a zero dimensional model. Finally, for zero dimensional polynomial actions,
we prove that Borel resummation of perturbative expansions, with several
choices of singularity avoiding contours in the complex Borel plane, yield
inequivalent solutions of the action principle equations.Comment: 15 pages, 9 figures (newer version has better images
Physically Realistic Solutions to the Ernst Equation on Hyperelliptic Riemann Surfaces
We show that the class of hyperelliptic solutions to the Ernst equation (the
stationary axisymmetric Einstein equations in vacuum) previously discovered by
Korotkin and Neugebauer and Meinel can be derived via Riemann-Hilbert
techniques. The present paper extends the discussion of the physical properties
of these solutions that was begun in a Physical Review Letter, and supplies
complete proofs. We identify a physically interesting subclass where the Ernst
potential is everywhere regular except at a closed surface which might be
identified with the surface of a body of revolution. The corresponding
spacetimes are asymptotically flat and equatorially symmetric. This suggests
that they could describe the exterior of an isolated body, for instance a
relativistic star or a galaxy. Within this class, one has the freedom to
specify a real function and a set of complex parameters which can possibly be
used to solve certain boundary value problems for the Ernst equation. The
solutions can have ergoregions, a Minkowskian limit and an ultrarelativistic
limit where the metric approaches the extreme Kerr solution. We give explicit
formulae for the potential on the axis and in the equatorial plane where the
expressions simplify. Special attention is paid to the simplest non-static
solutions (which are of genus two) to which the rigidly rotating dust disk
belongs.Comment: 32 pages, 2 figures, uses pstricks.sty, updated version (October 7,
1998), to appear in Phys. Rev.
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
We consider the calculation of the master integrals of the three-loop massive
banana graph. In the case of equal internal masses, the graph is reduced to
three master integrals which satisfy an irreducible system of three coupled
linear differential equations. The solution of the system requires finding a matrix of homogeneous solutions. We show how the maximal cut can be
used to determine all entries of this matrix in terms of products of elliptic
integrals of first and second kind of suitable arguments. All independent
solutions are found by performing the integration which defines the maximal cut
on different contours. Once the homogeneous solution is known, the
inhomogeneous solution can be obtained by use of Euler's variation of
constants.Comment: 39 pages, 3 figures; Fixed a typo in eq. (6.16
The unified transform method for linear initial-boundary value problems: a spectral interpretation
It is known that the unified transform method may be used to solve any
well-posed initial-boundary value problem for a linear constant-coefficient
evolution equation on the finite interval or the half-line. In contrast,
classical methods such as Fourier series and transform techniques may only be
used to solve certain problems. The solution representation obtained by such a
classical method is known to be an expansion in the eigenfunctions or
generalised eigenfunctions of the self-adjoint ordinary differential operator
associated with the spatial part of the initial-boundary value problem. In this
work, we emphasise that the unified transform method may be viewed as the
natural extension of Fourier transform techniques for non-self-adjoint
operators. Moreover, we investigate the spectral meaning of the transform pair
used in the new method; we discuss the recent definition of a new class of
spectral functionals and show how it permits the diagonalisation of certain
non-self-adjoint spatial differential operators.Comment: 3 figure
The Pearcey Process
The extended Airy kernel describes the space-time correlation functions for
the Airy process, which is the limiting process for a polynuclear growth model.
The Airy functions themselves are given by integrals in which the exponents
have a cubic singularity, arising from the coalescence of two saddle points in
an asymptotic analysis. Pearcey functions are given by integrals in which the
exponents have a quartic singularity, arising from the coalescence of three
saddle points. A corresponding Pearcey kernel appears in a random matrix model
and a Brownian motion model for a fixed time. This paper derives an extended
Pearcey kernel by scaling the Brownian motion model at several times, and a
system of partial differential equations whose solution determines associated
distribution functions. We expect there to be a limiting nonstationary process
consisting of infinitely many paths, which we call the Pearcey process, whose
space-time correlation functions are expressible in terms of this extended
kernel.Comment: LaTeX 24 pages. Version 3 has an improved exposition and corrects a
minor erro
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