1,842 research outputs found
The asymmetric ABAB matrix model
In this letter, it is pointed out that the two matrix model defined by the
action S=(1/2)(tr A^2+tr B^2)-(alpha_A/4) tr A^4-(alpha_B/4) tr B^4-(beta/2)
tr(AB)^2 can be solved in the large N limit using a generalization of the
solution of Kazakov and Zinn-Justin (who considered the symmetric case
alpha_A=alpha_B). This model could have useful applications to 3D Lorentzian
gravity.Comment: 7 pages, 1 figur
The transition temperature of the dilute interacting Bose gas for internal degrees of freedom
We calculate explicitly the variation of the Bose-Einstein
condensation temperature induced by weak repulsive two-body interactions
to leading order in the interaction strength. As shown earlier by general
arguments, is linear in the dimensionless product
to leading order, where is the density and the scattering length. This
result is non-perturbative, and a direct perturbative calculation of the
amplitude is impossible due to infrared divergences familiar from the study of
the superfluid helium lambda transition. Therefore we introduce here another
standard expansion scheme, generalizing the initial model which depends on one
complex field to one depending on real fields, and calculating the
temperature shift at leading order for large . The result is explicit and
finite. The reliability of the result depends on the relevance of the large
expansion to the situation N=2, which can in principle be checked by systematic
higher order calculations. The large result agrees remarkably well with
recent numerical simulations.Comment: 10 pages, Revtex, submitted to Europhysics Letter
Unified Treatment of Even and Odd Anharmonic Oscillators of Arbitrary Degree
We present a unified treatment, including higher-order corrections, of
anharmonic oscillators of arbitrary even and odd degree. Our approach is based
on a dispersion relation which takes advantage of the PT-symmetry of odd
potentials for imaginary coupling parameter, and of generalized quantization
conditions which take into account instanton contributions. We find a number of
explicit new results, including the general behaviour of large-order
perturbation theory for arbitrary levels of odd anharmonic oscillators, and
subleading corrections to the decay width of excited states for odd potentials,
which are numerically significant.Comment: 5 pages, RevTe
Constraining renormalon effects in lattice determination of heavy quark mass
The Borel summation technique of infrared renormalons is applied to the
lattice determination of heavy quark mass. With Borel summation a physical
heavy quark pole mass and binding energy of a heavy-light meson can be defined
in a rigorous and calculable manner. A notable feature of the Borel summation,
compared to the usual perturbative cancellation of IR renormalons, is an
automatic scale separation. The two approaches of handling renormalon
divergence are compared in the B-meson as well as in an (imaginary) heavy-light
meson with a mass much larger than the inverse of the lattice spacing.Comment: References and NNLO analysis added. Version to appear in Phys Rev
Symmetric path integrals for stochastic equations with multiplicative noise
A Langevin equation with multiplicative noise is an equation schematically of
the form dq/dt = - F(q) + e(q) xi, where e(q) xi is Gaussian white noise whose
amplitude e(q) depends on q itself. I show how to convert such equations into
path integrals. The definition of the path integral depends crucially on the
convention used for discretizing time, and I specifically derive the correct
path integral when the convention used is the natural, time-symmetric one that
time derivatives are (q_t - q_{t-\Delta t}) / \Delta t and coordinates are (q_t
+ q_{t-\Delta t}) / 2. [This is the convention that permits standard
manipulations of calculus on the action, like naive integration by parts.] It
has sometimes been assumed in the literature that a Stratanovich Langevin
equation can be quickly converted to a path integral by treating time as
continuous but using the rule \theta(t=0) = 1/2. I show that this prescription
fails when the amplitude e(q) is q-dependent.Comment: 8 page
Gerbes and Heisenberg's Uncertainty Principle
We prove that a gerbe with a connection can be defined on classical phase
space, taking the U(1)-valued phase of certain Feynman path integrals as Cech
2-cocycles. A quantisation condition on the corresponding 3-form field strength
is proved to be equivalent to Heisenberg's uncertainty principle.Comment: 12 pages, 1 figure available upon reques
Knot theory and matrix integrals
The large size limit of matrix integrals with quartic potential may be used
to count alternating links and tangles. The removal of redundancies amounts to
renormalizations of the potential. This extends into two directions: higher
genus and the counting of "virtual" links and tangles; and the counting of
"coloured" alternating links and tangles. We discuss the asymptotic behavior of
the number of tangles as the number of crossings goes to infinity.Comment: chapter of the book Random Matrix Theory, Eds Akemann, Baik and Di
Francesc
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