27 research outputs found
Off-Diagonal Elements of the DeWitt Expansion from the Quantum Mechanical Path Integral
The DeWitt expansion of the matrix element M_{xy} = \left\langle x \right|
\exp -[\case{1}{2} (p-A)^2 + V]t \left| y \right\rangle, in
powers of can be made in a number of ways. For (the case of interest
when doing one-loop calculations) numerous approaches have been employed to
determine this expansion to very high order; when (relevant for
doing calculations beyond one-loop) there appear to be but two examples of
performing the DeWitt expansion. In this paper we compute the off-diagonal
elements of the DeWitt expansion coefficients using the Fock-Schwinger gauge.
Our technique is based on representing by a quantum mechanical path
integral. We also generalize our method to the case of curved space, allowing
us to determine the DeWitt expansion of \tilde M_{xy} = \langle x| \exp
\case{1}{2} [\case{1}{\sqrt {g}} (\partial_\mu - i
A_\mu)g^{\mu\nu}{\sqrt{g}}(\partial_\nu - i A_\nu) ] t| y \rangle by use of
normal coordinates. By comparison with results for the DeWitt expansion of this
matrix element obtained by the iterative solution of the diffusion equation,
the relative merit of different approaches to the representation of as a quantum mechanical path integral can be assessed. Furthermore, the
exact dependence of on some geometric scalars can be
determined. In two appendices, we discuss boundary effects in the
one-dimensional quantum mechanical path integral, and the curved space
generalization of the Fock-Schwinger gauge.Comment: 16pp, REVTeX. One additional appendix concerning end-point effects
for finite proper-time intervals; inclusion of these effects seem to make our
results consistent with those from explicit heat-kernel method
Method to compute the stress-energy tensor for the massless spin 1/2 field in a general static spherically symmetric spacetime
A method for computing the stress-energy tensor for the quantized, massless,
spin 1/2 field in a general static spherically symmetric spacetime is
presented. The field can be in a zero temperature state or a non-zero
temperature thermal state. An expression for the full renormalized
stress-energy tensor is derived. It consists of a sum of two tensors both of
which are conserved. One tensor is written in terms of the modes of the
quantized field and has zero trace. In most cases it must be computed
numerically. The other tensor does not explicitly depend on the modes and has a
trace equal to the trace anomaly. It can be used as an analytic approximation
for the stress-energy tensor and is equivalent to other approximations that
have been made for the stress-energy tensor of the massless spin 1/2 field in
static spherically symmetric spacetimes.Comment: 34 pages, no figure
Quantum Theory in Accelerated Frames of Reference
The observational basis of quantum theory in accelerated systems is studied.
The extension of Lorentz invariance to accelerated systems via the hypothesis
of locality is discussed and the limitations of this hypothesis are pointed
out. The nonlocal theory of accelerated observers is briefly described.
Moreover, the main observational aspects of Dirac's equation in noninertial
frames of reference are presented. The Galilean invariance of nonrelativistic
quantum mechanics and the mass superselection rule are examined in the light of
the invariance of physical laws under inhomogeneous Lorentz transformations.Comment: 25 pages, no figures, contribution to Springer Lecture Notes in
Physics (Proc. SR 2005, Potsdam, Germany, February 13 - 18, 2005
Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity
Starting with the Chern-Simons formulation of (2+1)-dimensional gravity we
show that the gravitational interactions deform the Poincare symmetry of flat
space-time to a quantum group symmetry. The relevant quantum group is the
quantum double of the universal cover of the (2+1)-dimensional Lorentz group,
or Lorentz double for short. We construct the Hilbert space of two gravitating
particles and use the universal R-matrix of the Lorentz double to derive a
general expression for the scattering cross section of gravitating particles
with spin. In appropriate limits our formula reproduces the semi-classical
scattering formulae found by 't Hooft, Deser, Jackiw and de Sousa Gerbert.Comment: 45 pages, amslate
Symplectic Dirac operators and Mpc -structures
Given a symplectic manifold (M, ω) admitting a metaplectic structure, and choosing a positive ω-compatible almost complex structure J and a linear connection preserving ω and J, Katharina and Lutz Habermann have constructed two Dirac operators D and D acting on sections of a bundle of symplectic spinors. They have shown that the commutator [DD] is an elliptic operator preserving an infinite number of finite dimensional subbundles. We extend the construction of symplectic Dirac operators to any symplectic manifold, through the use of Mpc structures. These exist on any symplectic manifold and equivalence classes are parametrized by elements in H2(MZ) . For any Mpc structure, choosing J and a linear connection as before, there are two natural Dirac operators, acting on the sections of a spinor bundle, whose commutator is elliptic. Using the Fock description of the spinor space allows the definition of a notion of degree and the construction of a dense family of finite dimensional subbundles; the operator stabilizes the sections of each of those