1,720,994 research outputs found
From Peierls brackets to a generalized Moyal bracket for type-I gauge theories
No full textIn the space-of-histories approach to gauge fields and their quantization, the Maxwell,
Yang–Mills and gravitational field are well known to share the property of being type-I
theories, i.e. Lie brackets of the vector fields which leave the action functional invariant
are linear combinations of such vector fields, with coefficients of linear combination given
by structure constants. The corresponding gauge-field operator in the functional integral
for the in-out amplitude is an invertible second-order differential operator. For such an
operator, we consider advanced and retarded Green functions giving rise to a Peierls
bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on
the space of all group-invariant functionals in two cases only: either the gauge-fixing is
arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a
linear functional of gauge fields, which are generic points of the space of histories. In
both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting
the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal
bracket is defined that reduces to i hbar times the Peierls bracket to lowest order in hbar
Multimomentum maps in general relativity
No full textThe properties of multimomentum maps on null hypersurfaces, and their relation with the constraint analysis of General Relativity, are described. Unlike the case of spacelike hypersurfaces, some constraints which are second class in the Hamiltonian formalism turn out to contribute to the multimomentum map
A new family of gauges in linearized general relativity
No full textFor vacuum Maxwell theory in four dimensions, a supplementary condition exists (due
to Eastwood and Singer) which is invariant under conformal rescalings of the metric, in agreement
with the conformal symmetry of the Maxwell equations. Thus, starting from the de Donder gauge,
which is not conformally invariant but is the gravitational counterpart of the Lorenz gauge, one
can consider, led by formal analogy, a new family of gauges in general relativity, which involve
fifth-order covariant derivatives of metric perturbations. The admissibility of such gauges in the
classical theory is first proven in the cases of linearized theory about flat Euclidean space or flat
Minkowski spacetime. In the former, the general solution of the equation for the fulfillment of the
gauge condition after infinitesimal diffeomorphisms involves a 3-harmonic 1-form and an inverse
Fourier transform. In the latter, one needs instead the kernel of powers of the wave operator, and
a contour integral. The analysis is also used to put restrictions on the dimensionless parameter
occurring in the DeWitt supermetric, while the proof of admissibility is generalized to a suitable
class of curved Riemannian backgrounds. Eventually, a non-local construction of the tensor field
is obtained which makes it possible to achieve conformal invariance of the above gauges
Conformally invariant gauge conditions in electromagnetism and general relativity
No full textThe construction of conformally invariant gauge conditions for Maxwell and Einstein theories on a manifold M
is found to involve two basic ingredients. First, covariant derivatives of a linear gauge (e.g. Lorenz or de Donder),
completely contracted with the tensor field representing the metric on the vector bundle of the theory. Second,
the addition of a compensating term, obtained by covariant differentiation of a suitable tensor field built from
the geometric data of the problem. The existence theorem for such a gauge in gravitational theory is here proved
when the manifold M is endowed with a m-dimensional positive-definite metric g. An application to a generally
covariant integral formulation of the Einstein equations is also outlined
A CFT description of the BTZ black hole: Topology versus Geometry (or Thermodynamics versus Statistical Mechanics)
No full textIn this paper we review the properties of the black hole entropy in the light of a general conformal field theory treatment. We find that the properties of horizons of the BTZ black holes in ADS_{3}, can be described in terms of an effective unitary CFT_{2} with central charge c=1 realized in terms of the Fubini-Veneziano vertex operators.
It is found a relationship between the topological properties of the black hole solution and the infinite algebra extension of the conformal group in 2D, SU(2,2), i.e. the Virasoro Algebra, and its subgroup SL(2,Z) which generates the modular symmetry. Such a symmetry induces a duality for the black hole solution with angular momentum J\neq 0. On the light of such a global symmetry we reanalyze the Cardy formula for CFT_{2} and its possible generalization to D>2 proposed by E. Verlinde
On the ADM equations for general relativity
No full textThe Arnowitt-Deser-Misner (ADM)
evolution equations for the induced metric and the
extrinsic-curvature tensor of the spacelike surfaces which foliate
the space-time manifold in canonical general relativity
are a first-order system of quasi-linear
partial differential equations, supplemented by the constraint
equations. Such equations are here mapped into another
first-order system. In particular, an evolution equation for the
trace of the extrinsic-curvature tensor K
is obtained whose solution is related to
a discrete spectral resolution of a three-dimensional elliptic
operator P of Laplace type. Interestingly, all
nonlinearities of the original equations give rise to the potential
term in P. An example of this construction is given in the case of
a closed Friedmann-Lemaitre-Robertson-Walker universe.
Eventually, the ADM equations are re-expressed as a coupled
first-order system for the induced metric and the trace-free
part of K. Such a system is written in a form which clarifies how
a set of first-order differential operators and their inverses, jointly
with spectral resolutions of operators of Laplace type, contribute
to solving, at least in principle, the original ADM system
The role of elliptic operators in the initial-value problem for general relativity
No full textThe Arnowitt–Deser–Misner (ADM) equations are deeply in-tertwined with discrete spectral resolutions of an elliptic operator of Laplace type associated with the spacelike hypersurfaces which foliate the space-time manifold, and the non-linearities of the four-dimensional hyperbolic theory are mapped into the potential term occurring in this operator. The ADM equations are here re-expressed as a coupled first-order system for the induced metric and the trace-free part of the extrinsic-curvature tensor, and their formulation in terms of integral equations is studied
Linear form of canonical gravity
No full textRecent work in the literature has shown that general relativity can be
formulated in terms of a jet bundle which, in local coordinates, has five entries: local
coordinates on Lorentzian space-time, tetrads, connection one-forms, multivelocities
corresponding to the tetrads and multivelocities corresponding to the connection
one-forms. The derivatives of the Lagrangian with respect to the latter class of
multivelocities give rise to a set of multimomenta which naturally occur in the
constraint equations. Interestingly, all the constraint equations of general relativity
are linear in terms of this class of multimomenta. This construction has been then
extended to complex general relativity, where Lorentzian space-time is replaced by
a four-complex-dimensional complex-Riemannian manifold. One then finds a
holomorphic theory where the familiar constraint equations are replaced by a set of
equations linear in the holomorphic multimomenta, providing such multimomenta
vanish on a family of two-complex-dimensional surfaces. In quantum gravity, the
problem arises to quantize a real or a holomorphic theory on the extended space
where the multimomenta can be defined
Nonlocality and ellipticity in a gauge invariant quantization
No full textThe quantum theory of a free particle in two dimensions with nonlocal boundary conditions
on a circle is known to lead to surface and bulk states. Such a scheme is here
generalized to the quantized Maxwell field, subject to mixed boundary conditions. If
the Robin sector is modified by the addition of a pseudo-differential boundary operator,
gauge-invariant boundary conditions are obtained at the price of dealing with gauge-
field and ghost operators which become pseudo-differential. A good elliptic theory is
then obtained if the kernel occurring in the boundary operator obeys certain summability
conditions, and it leads to a peculiar form of the asymptotic expansion of the
symbol. The cases of ghost operator of negative and positive order are studied within
this framework
Multimomentum maps on null hypersurfaces
No full textThis paper studies the application of multimomentum maps to the
constraint analysis of general relativity on null hypersurfaces. It is shown that, unlike
the case of spacelike hypersurfaces, some constraints which are second class in the
Hamiltonian formalism turn out to contribute to the multimomentum map. To
recover the whole set of secondary constraints found in the Hamiltonian formalism, it
is necessary to combine the multimomentum map with those particular Euler-
Lagrange equations which are not of evolutionary type. The analysis is performed on
the outgoing null cone only
- …
