69 research outputs found
Quasi-particles, thermodynamic consistency and the gap equation
The thermodynamic properties of superconducting electrons are usually studied
by means of the quasi-particles distribution; but in this approach, the ground
state energy and the dependence of the chemical potential on the electron
density cannot be determined. In order to solve these problems, the
thermodynamic potentials are derived by means of the Bogoliubov-Valatin
formalism. The thermodynamic potentials can be obtained by computing the free
energy of a gas of quasi-particles, whose energy spectrum is conditional on the
gap function. However, the nontrivial dependence of the gap on the temperature
jeopardises the validity of the standard thermodynamic relations. In this
article it is shown how the thermodynamic consistency (i.e. the validity of the
Maxwell relations) is recovered, and the correction terms to the
quasi-particles potentials are computed. It is shown that the
Bogoliubov-Valatin transformation avoids the problem of the thermodynamic
consistency of the quasi-particle approach; in facts, the correct
identification of the variables, which are associated with the quasi-particles,
leads to a precise calculation of the quasi-particles vacuum energy and of the
dependence of the chemical potential on the electron density. The stationarity
condition for the grand potential coincides with the gap equation, which
guarantees the thermodynamic consistency. The expressions of various
thermodynamic potentials, as functions of the (T,V,N) variables, are produced
in the low temperature limit; as a final check, a rederivation of the
condensation energy is presented.Comment: 16 page
Functional integration and abelian link invariants
The functional integral computation of the various topological invariants,
which are associated with the Chern-Simons field theory, is considered. The
standard perturbative setting in quantum field theory is rewieved and new
developments in the path-integral approach, based on the Deligne-Beilinson
cohomology, are described in the case of the abelian U(1) Chern-Simons field
theory formulated in S^1 x S^2.Comment: 20 pages, 4 figures, Contribution to the Proceedings of the workshop
"Chern-Simons Gauge theory: 20 years after", Bonn, August 200
Critical angular velocity for vortex lines formation
For helium II inside a rotating cylinder, it is proposed that the formation
of vortex lines of the frictionless superfluid component of the liquid is
caused by the presence of the rotating quasi-particles gas. By minimising the
free energy of the system, the critical value Omega_0 of the angular velocity
for the formation of the first vortex line is determined. This value
nontrivially depends on the temperature, and numerical estimations of its
temperature behaviour are produced. It is shown that the latent heat for a
vortex formation and the associated discontinuous change in the angular
momentum of the quasi-particles gas determine the slope of Omega_0 (T) via some
kind of Clapeyron equation.Comment: 16 page
Schwinger-Dyson functional in Chern-Simons theory
In perturbative SU(N) Chern-Simons gauge theory, it is shown that the
Schwinger-Dyson equations assume a quite simplified form. The generating
functional of the correlation functions of the curvature is considered; it is
demonstrated that the renormalized Schwinger-Dyson functional is related with
the generating functional of the correlation functions of the gauge connections
by some kind of duality transformation.Comment: 11 page
Representations for creation and annihilation operators
A new representation -which is similar to the Bargmann representation- of the
creation and annihilation operators is introduced, in which the operators act
like "multiplication with" and like "derivation with respect to" a single real
variable. The Hilbert space structure of the corresponding states space is
produced and the relations with the Schroedinger representation are derived.
Possible connections of this new representation with the asymptotic wave
functions of the gauge-fixed quantum Chern-Simons field theory and (2+1)
gravity are pointed out. It is shown that the representation of the fields
operator algebra of the Chern-Simons theory in the Landau gauge is not a
*-representation; the consequences on the evolution of the states in the
semiclassical approximation are discussed.Comment: 15 page
Path-integral invariants in abelian Chern-Simons theory
We consider the Chern-Simons gauge theory defined in a general closed
oriented 3-manifold ; the functional integration is used to compute the
normalized partition function and the expectation values of the link
holonomies. The nonperturbative path-integral is defined in the space of the
gauge orbits of the connections which belong to the various inequivalent
principal bundles over ; the different sectors of the configuration space
are labelled by the elements of the first homology group of and are
characterized by appropriate background connections. The gauge orbits of flat
connections, whose classification is also based on the homology group, control
the extent of the nonperturbative contributions to the mean values. The
functional integration is achieved in any 3-manifold , and the corresponding
path-integral invariants turn out to be strictly related with the abelian
Reshetikhin-Turaev surgery invariants
Deligne-Beilinson cohomology and abelian links invariants
For the abelian Chern-Simons field theory, we consider the quantum functional
integration over the Deligne-Beilinson cohomology classes and we derive the
main properties of the observables in a generic closed orientable 3-manifold.
We present an explicit path-integral non-perturbative computation of the
Chern-Simons links invariants in the case of the torsion-free 3-manifolds
, and
Topological gauge fixing
We implement the metric-independent Fock-Schwinger gauge in the abelian
quantum Chern-Simons field theory defined in . The expressions
of the various components of the propagator are determined. Although the gauge
field propagator differs from the Gauss linking density, we prove that its
integral along two oriented knots is equal to the linking number
Abelian link invariants and homology
We consider the link invariants defined by the quantum Chern-Simons field
theory with compact gauge group U(1) in a closed oriented 3-manifold M. The
relation of the abelian link invariants with the homology group of the
complement of the links is discussed. We prove that, when M is a homology
sphere or when a link -in a generic manifold M- is homologically trivial, the
associated observables coincide with the observables of the sphere S^3. Finally
we show that the U(1) Reshetikhin-Turaev surgery invariant of the manifold M is
not a function of the homology group only, nor a function of the homotopy type
of M alone.Comment: 18 pages, 3 figures; to be published in Journal of Mathematical
Physic
Three-manifold invariant from functional integration
We give a precise definition and produce a path-integral computation of the
normalized partition function of the abelian U(1) Chern-Simons field theory
defined in a general closed oriented 3-manifold. We use the Deligne-Beilinson
formalism, we sum over the inequivalent U(1) principal bundles over the
manifold and, for each bundle, we integrate over the gauge orbits of the
associated connection 1- forms. The result of the functional integration is
compared with the abelian U(1) Reshetikhin-Turaev surgery invariant
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