27,958 research outputs found
Berry's phase for compact Lie groups
The Lie group adiabatic evolution determined by a Lie algebra parameter
dependent Hamiltonian is considered. It is demonstrated that in the case when
the parameter space of the Hamiltonian is a homogeneous K\"ahler manifold its
fundamental K\"ahler potentials completely determine Berry geometrical phase
factor. Explicit expressions for Berry vector potentials (Berry connections)
and Berry curvatures are obtained using the complex parametrization of the
Hamiltonian parameter space. A general approach is exemplified by the Lie
algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.Comment: 24 pages, no figure
Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
We study geometric consistency relations between angles of 3-dimensional (3D)
circular quadrilateral lattices -- lattices whose faces are planar
quadrilaterals inscribable into a circle. We show that these relations generate
canonical transformations of a remarkable "ultra-local" Poisson bracket algebra
defined on discrete 2D surfaces consisting of circular quadrilaterals.
Quantization of this structure allowed us to obtain new solutions of the
tetrahedron equation (the 3D analog of the Yang-Baxter equation) as well as
reproduce all those that were previously known. These solutions generate an
infinite number of non-trivial solutions of the Yang-Baxter equation and also
define integrable 3D models of statistical mechanics and quantum field theory.
The latter can be thought of as describing quantum fluctuations of lattice
geometry.Comment: Plenary talk at the XVI International Congress on Mathematical
Physics, 3-8 August 2009, Prague, Czech Republi
Statistical Equilibrium in Quantum Gravity: Gibbs states in Group Field Theory
Gibbs states are known to play a crucial role in the statistical description
of a system with a large number of degrees of freedom. They are expected to be
vital also in a quantum gravitational system with many underlying fundamental
discrete degrees of freedom. However, due to the absence of well-defined
concepts of time and energy in background independent settings, formulating
statistical equilibrium in such cases is an open issue. This is even more so in
a quantum gravity context that is not based on any of the usual spacetime
structures, but on non-spatiotemporal degrees of freedom. In this paper, after
having clarified general notions of statistical equilibrium, on which two
different construction procedures for Gibbs states can be based, we focus on
the group field theory formalism for quantum gravity, whose technical features
prove advantageous to the task. We use the operator formulation of group field
theory to define its statistical mechanical framework, based on which we
construct three concrete examples of Gibbs states. The first is a Gibbs state
with respect to a geometric volume operator, which is shown to support
condensation to a low-spin phase. This state is not based on a pre-defined
symmetry of the system and its construction is via Jaynes' entropy maximisation
principle. The second are Gibbs states encoding structural equilibrium with
respect to internal translations on the GFT base manifold, and defined via the
KMS condition. The third are Gibbs states encoding relational equilibrium with
respect to a clock Hamiltonian, obtained by deparametrization with respect to
coupled scalar matter fields.Comment: v2 31 pages; typos corrected; section 2 modified substantially for
clarity; minor modifications in the abstract and introduction; arguments and
results unchange
Geometric flows and (some of) their physical applications
The geometric evolution equations provide new ways to address a variety of
non-linear problems in Riemannian geometry, and, at the same time, they enjoy
numerous physical applications, most notably within the renormalization group
analysis of non-linear sigma models and in general relativity. They are divided
into classes of intrinsic and extrinsic curvature flows. Here, we review the
main aspects of intrinsic geometric flows driven by the Ricci curvature, in
various forms, and explain the intimate relation between Ricci and Calabi flows
on Kahler manifolds using the notion of super-evolution. The integration of
these flows on two-dimensional surfaces relies on the introduction of a novel
class of infinite dimensional algebras with infinite growth. It is also
explained in this context how Kac's K_2 simple Lie algebra can be used to
construct metrics on S^2 with prescribed scalar curvature equal to the sum of
any holomorphic function and its complex conjugate; applications of this
special problem to general relativity and to a model of interfaces in
statistical mechanics are also briefly discussed.Comment: 18 pages, contribution to AvH conference Advances in Physics and
Astrophysics of the 21st Century, 6-11 September 2005, Varna, Bulgari
Quantum geometry of 3-dimensional lattices
We study geometric consistency relations between angles on 3-dimensional (3D)
circular quadrilateral lattices -- lattices whose faces are planar
quadrilaterals inscribable into a circle. We show that these relations generate
canonical transformations of a remarkable ``ultra-local'' Poisson bracket
algebra defined on discrete 2D surfaces consisting of circular quadrilaterals.
Quantization of this structure leads to new solutions of the tetrahedron
equation (the 3D analog of the Yang-Baxter equation). These solutions generate
an infinite number of non-trivial solutions of the Yang-Baxter equation and
also define integrable 3D models of statistical mechanics and quantum field
theory. The latter can be thought of as describing quantum fluctuations of
lattice geometry. The classical geometry of the 3D circular lattices arises as
a stationary configuration giving the leading contribution to the partition
function in the quasi-classical limit.Comment: 27 pages, 10 figures. Minor corrections, references adde
The locally covariant Dirac field
We describe the free Dirac field in a four dimensional spacetime as a locally
covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch,
using a representation independent construction. The freedom in the geometric
constructions involved can be encoded in terms of the cohomology of the
category of spin spacetimes. If we restrict ourselves to the observable algebra
the cohomological obstructions vanish and the theory is unique. We establish
some basic properties of the theory and discuss the class of Hadamard states,
filling some technical gaps in the literature. Finally we show that the
relative Cauchy evolution yields commutators with the stress-energy-momentum
tensor, as in the scalar field case.Comment: 36 pages; v2 minor changes, typos corrected, updated references and
acknowledgement
Holography and SL(2,\bR) symmetry in 2D Rindler spacetime
It is shown that it is possible to define quantum field theory of a massless
scalar free field on the Killing horizon of a 2D-Rindler spacetime. Free
quantum field theory on the horizon enjoys diffeomorphism invariance and turns
out to be unitarily and algebraically equivalent to the analogous theory of a
scalar field propagating inside Rindler spacetime, nomatter the value of the
mass of the field in the bulk. More precisely, there exists a unitary
transformation that realizes the bulk-boundary correspondence under an
appropriate choice for Fock representation spaces. Secondly, the found
correspondence is a subcase of an analogous algebraic correspondence described
by injective *-homomorphisms of the abstract algebras of observables generated
by abstract quantum free-field operators. These field operators are smeared
with suitable test functions in the bulk and exact 1-forms on the horizon. In
this sense the correspondence is independent from the chosen vacua. It is
proven that, under that correspondence the ``hidden'' SL(2,\bR) quantum
symmetry found in a previous work gets a clear geometric meaning, it being
associated with a group of diffeomorphisms of the horizon itself.Comment: Title changed, further minor changes, references added, accepted for
publication in J. Math. Phy
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