20 research outputs found
Explicit positive representation for weights on
It is an old idea to replace averages of observables with respect to a
complex weight by expectation values with respect to a genuine probability
measure on complexified space. This is precisely what one would like to get
from complex Langevin simulations. Unfortunately, these fail in many cases of
physical interest. We will describe method of deriving positive representations
by matching of moments and show simple examples of successful constructions. It
will be seen that the problem is greatly underdetermined.Comment: 8 pages, 3 figures. Material presented in Lattice 2017 conference in
Granad
Simulations of gaussian systems in Minkowski time
Many research programs aiming to deal with the sign problem were proposed
since the advent of lattice field theory. Several of these try to achieve this
by exploiting properties of analytic functions. This is also the case for our
study. There auxiliary complex variables are introduced and desired weight is
obtained after integrating them out. In this note we clarify conceptual
difficulties with this procedure encountered in previous works. In the process
we observe an exciting connection with thimbles and discover an interesting
hidden symmetry present in the problem. Problem of negative eigenvalues of the
action will be revisited and considered from a different perspective. As a
byproduct we perform simulations of simple quantum systems directly in
Minkowski time.Comment: The 36th Annual International Symposium on Lattice Field Theory -
LATTICE2018; 22-28 July, 2018; Michigan State University, East Lansing,
Michigan, US
Analyticity of the free energy for quantum Airy structures
It is shown that the free energy associated to a finite dimensional Airy
structure is an analytic function at each finite order of the
expansion. Semiclassical series itself is in general divergent. Calculations
are facilitated by putting the topological recursion equations into a form
exhibiting more explicitly the semiclassical geometry. This formulation
involves certain differential operators on the characteristic variety, which
are found to satisfy a Lie algebra cocycle condition. It is proven that this
cocycle is a coboundary. Developed formalism is applied in specific examples.
In the case of a divergent series, a simple resummation is performed.
Analytic properties of the obtained partition functions are investigated.Comment: Minor mistakes were correcte
Bosonization based on Clifford algebras and its gauge theoretic interpretation
We study the properties of a bosonization procedure based on Clifford algebra
valued degrees of freedom, valid for spaces of any dimension. We present its
interpretation in terms of fermions in presence of gauge fields
satisfying a modified Gauss' law, resembling Chern-Simons-like theories. Our
bosonization prescription involves constraints, which are interpreted as a
flatness condition for the gauge field. Solution of the constraints is
presented for toroidal geometries of dimension two. Duality between our model
and -form gauge theory is derived, which elucidates the
relation between the approach taken here with another bosonization map proposed
recently.Comment: Revised version. 36 pages, 6 figure
Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension
We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is,
operators on the half-line of the form . We describe their closed realizations
in the sense of the Hilbert space , allowing for
complex values of the parameters . In physical situations,
is proportional to the electric charge and is related to the
angular momentum.
We focus on realizations of homogeneous of degree .
They can be organized in a single holomorphic family of closed operators
parametrized by a certain 2-dimensional complex manifold. We describe the
spectrum and the numerical range of these realizations. We give an explicit
formula for the integral kernel of their resolvent in terms of Whittaker
functions. We also describe their stationary scattering theory, providing
formulas for a natural pair of diagonalizing operators and for the scattering
operator.
It is well-known that arise after separation of
variables of the Dirac-Coulomb operator in dimension 3. We give a simple
argument why this is still true in any dimension.
Our work is mainly motivated by a large literature devoted to distinguished
self-adjoint realizations of Dirac-Coulomb Hamiltonians. We show that these
realizations arise naturally if the holomorphy is taken as the guiding
principle. Furthermore, they are infrared attractive fixed points of the
scaling action. Beside applications in relativistic quantum mechanics,
Dirac-Coulomb Hamiltonians are argued to provide a natural setting for the
study of Whittaker (or, equivalently, confluent hypergeometric) functions
Generalized integrals of Macdonald and Gegenbauer functions
We compute bilinear integrals involving Macdonald and Gegenbauer functions.
These integrals are convergent only for a limited range of parameters. However,
when one uses generalized integrals they can be computed essentially without
restricting the parameters. The generalized integral is a linear functional
extending the standard integral to a certain class of functions involving
finitely many homogeneous non-integrable terms at the edpoints of the interval.
For generic values of parameters, generalized bilinear integrals of Macdonald
and Gegenbauer functions can be obtained by analytic continuation from the
region in which the integrals are convergent. In the case of integer parameters
we obtain expressions with explicit additional terms related to an anomaly,
namely the failure of the generalized integral to be scaling invariant.Comment: 38 page
Dynamical generalization of Yetter's model based on a crossed module of discrete groups
We construct a dynamical lattice model based on a crossed module of possibly
non-abelian finite groups. Its degrees of freedom are defined on links and
plaquettes, while gauge transformations are based on vertices and links of the
underlying lattice. We specify the Hilbert space, define basic observables
(including the Hamiltonian) and initiate a~discussion on the model's phase
diagram. The constructed model generalizes, and in appropriate limits reduces
to, topological theories with symmetries described by groups and crossed
modules, lattice Yang-Mills theory and -form electrodynamics. We conclude by
reviewing classifying spaces of crossed modules, with an emphasis on the direct
relation between their geometry and properties of gauge theories under
consideration.Comment: 55 pages, 27 figure
Nilpotent symmetries as a mechanism for Grand Unification
In the classic Coleman-Mandula no-go theorem which prohibits the unification of internal and spacetime symmetries, the assumption of the existence of a positive definite invariant scalar product on the Lie algebra of the internal group is essential. If one instead allows the scalar product to be positive semi-definite, this opens new possibilities for unification of gauge and spacetime symmetries. It follows from theorems on the structure of Lie algebras, that in the case of unified symmetries, the degenerate directions of the positive semi-definite invariant scalar product have to correspond to local symmetries with nilpotent generators. In this paper we construct a workable minimal toy model making use of this mechanism: it admits unified local symmetries having a compact (U(1)) component, a Lorentz (SL(2, ℂ)) component, and a nilpotent component gluing these together. The construction is such that the full unified symmetry group acts locally and faithfully on the matter field sector, whereas the gauge fields which would correspond to the nilpotent generators can be transformed out from the theory, leaving gauge fields only with compact charges. It is shown that already the ordinary Dirac equation admits an extremely simple prototype example for the above gauge field elimination mechanism: it has a local symmetry with corresponding eliminable gauge field, related to the dilatation group. The outlined symmetry unification mechanism can be used to by-pass the Coleman-Mandula and related no-go theorems in a way that is fundamentally different from supersymmetry. In particular, the mechanism avoids invocation of super-coordinates or extra dimensions for the underlying spacetime manifold
Beyond complex Langevin equations : a progress report
After a short review of one of proposals to avoid complex stochastic processes in Complex Langevin studies, the recent progress in the former is reported.
In particular, the new developments allow now to construct positive and normalizable representations for gaussian quantum mechanical, as well as field theoretical, path integrals directly in the Minkowski time. A relation to the idea of thimbles is also discussed