6,378 research outputs found
Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants
We construct a generalization of pure lattice gauge theory (LGT) where the
role of the gauge group is played by a tensor category. The type of tensor
category admissible (spherical, ribbon, symmetric) depends on the dimension of
the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the
category is the (symmetric) category of representations of a compact Lie group.
In the weak coupling limit we recover discretized BF-theory in terms of a
coordinate free version of the spin foam formulation. We work on general
cellular decompositions of the underlying manifold.
In particular, we are able to formulate LGT as well as spin foam models of
BF-type with quantum gauge group (in dimension <=4) and with supersymmetric
gauge group (in any dimension).
Technically, we express the partition function as a sum over diagrams
denoting morphisms in the underlying category. On the LGT side this enables us
to introduce a generalized notion of gauge fixing corresponding to a
topological move between cellular decompositions of the underlying manifold. On
the BF-theory side this allows a rather geometric understanding of the state
sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we
recover.
The construction is extended to include Wilson loop and spin network type
observables as well as manifolds with boundaries. In the topological (weak
coupling) case this leads to TQFTs with or without embedded spin networks.Comment: 58 pages, LaTeX with AMS and XY-Pic macros; typos corrected and
references update
An Introduction to Chiral Symmetry on the Lattice
The chiral symmetry of QCD is of central
importance for the nonperturbative low-energy dynamics of light quarks and
gluons. Lattice field theory provides a theoretical framework in which these
dynamics can be studied from first principles. The implementation of chiral
symmetry on the lattice is a nontrivial issue. In particular, local lattice
fermion actions with the chiral symmetry of the continuum theory suffer from
the fermion doubling problem. The Ginsparg-Wilson relation implies L\"uscher's
lattice variant of chiral symmetry which agrees with the usual one in the
continuum limit. Local lattice fermion actions that obey the Ginsparg-Wilson
relation have an exact chiral symmetry, the correct axial anomaly, they obey a
lattice version of the Atiyah-Singer index theorem, and still they do not
suffer from the notorious doubling problem. The Ginsparg-Wilson relation is
satisfied exactly by Neuberger's overlap fermions which are a limit of Kaplan's
domain wall fermions, as well as by Hasenfratz and Niedermayer's classically
perfect lattice fermion actions. When chiral symmetry is nonlinearly realized
in effective field theories on the lattice, the doubling problem again does not
arise. This review provides an introduction to chiral symmetry on the lattice
with an emphasis on the basic theoretical framework.Comment: (41 pages, to be published in Prog. Part. Nucl. Phys. Vol. 53, issue
1 (2004)
Four Dimensional Quantum Yang-Mills Theory and Mass Gap
A quantization procedure for the Yang-Mills equations for the Minkowski space
is carried out in such a way that field maps satisfying
Wightman axioms of Constructive Quantum Field Theory can be obtained. Moreover,
by removing the ultra violet cut off, the spectrum of the corresponding QCD
Hamilton operator is proven to be positive and bounded away from zero, except
for the case of the vacuum state, which has vanishing energy level. The whole
construction is gauge invariant. The particles corresponding to all solution
fields are bosons. As expected from QED, if the coupling constant converges to
zero, then so does the mass gap. The results are proved first for the model
with the bare coupling constant, and then for a model with a running coupling
constant by means of renormalization.Comment: With respect to the preceding version of this paper, the gauge
invariance of the construction has been proved and the construction of the
probability measure making the Hamiltonian QCD selfadjoint has been rewritten
with more clarit
SU(N) chiral gauge theories on the lattice
We extend the construction of lattice chiral gauge theories based on
non-perturbative gauge fixing to the non-abelian case. A key ingredient is that
fermion doublers can be avoided at a novel type of critical point which is only
accessible through gauge fixing, as we have shown before in the abelian case.
The new ingredient allowing us to deal with the non-abelian case as well is the
use of equivariant gauge fixing, which handles Gribov copies correctly, and
avoids Neuberger's no-go theorem. We use this method in order to gauge fix the
non-abelian group (which we will take to be SU(N)) down to its maximal abelian
subgroup. Obtaining an undoubled, chiral fermion content requires us to
gauge-fix also the remaining abelian gauge symmetry. This modifies the
equivariant BRST identities, but their use in proving unitarity remains intact,
as we show in perturbation theory. On the lattice, equivariant BRST symmetry as
well as the abelian gauge invariance are broken, and a judiciously chosen
irrelevant term must be added to the lattice gauge-fixing action in order to
have access to the desired critical point in the phase diagram. We argue that
gauge invariance is restored in the continuum limit by adjusting a finite
number of counter terms. We emphasize that weak-coupling perturbation theory
applies at the critical point which defines the continuum limit of our lattice
chiral gauge theory.Comment: 39 pages, 3 figures, A number of clarifications adde
Non-perturbative \lambda\Phi^4 in D=1+1: an example of the constructive quantum field theory approach in a schematic way
During the '70, several relativistic quantum field theory models in
and also in have been constructed in a non-perturbative way. That was
done in the so-called {\it constructive quantum field theory} approach, whose
main results have been obtained by a clever use of Euclidean functional
methods. Although in the construction of a single model there are several
technical steps, some of them involving long proofs, the constructive quantum
field theory approach contains conceptual insights about relativistic quantum
field theory that deserved to be known and which are accessible without
entering in technical details. The purpose of this note is to illustrate such
insights by providing an oversimplified schematic exposition of the simple case
of (with ) in . Because of the absence of
ultraviolet divergences in its perturbative version, this simple example
-although does not capture all the difficulties in the constructive quantum
field theory approach- allows to stress those difficulties inherent to the
non-perturbative definition. We have made an effort in order to avoid several
of the long technical intermediate steps without missing the main ideas and
making contact with the usual language of the perturbative approach.Comment: 63 pages. Typos correcte
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