A method is given to compute an approximation to the noise kernel, defined as the symmetrized connected 2-point function of the stress tensor, for the conformally invariant scalar field in any spacetime conformal to an ultra-static spacetime for the case in which the field is in a thermal state at an arbitrary temperature. The most useful applications of the method are flat space where the approximation is exact and Schwarzschild spacetime where the approximation is better than it is in most other spacetimes. The two points are assumed to be separated in a timelike or spacelike direction. The method involves the use of a Gaussian approximation which is of the same type as that used by Page [1] to compute an approximate form of the stress tensor for this field in Schwarzschild spacetime. All components of the noise kernel have been computed exactly for hot flat space and one component is explicitly displayed. Several components have also been computed for Schwarzschild spacetime and again one component is explicitly displayed.
We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar -product combines the usual phase space * star and the noncommutative star-product. Having dealt with subtleties of ordering present in this problem we show that the Weyl correspondence of the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter θ-dependent, momentumdependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if θ = 0 there is no classical limit. Classical limit exists only if θ → 0 at least as fast as → 0, but this limit does not yield Newtonian mechanics, unless the limit of θ/ vanishes as θ → 0. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of θ and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context. AimIn this program of investigation we ask the question whether there is any structural similarity or conceptual connection between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. We want to see if our understanding of the quantum-classical correspondence acquired in the last decade can aid us in any way to understand the physical attributes and meanings of a noncommutative space from the vantage point of the ordinary commutative space. We find that the case of quantum to classical transition in the context of noncommutative geometry is quite different from that in the ordinary (commutative) space. Specifically, if θ = 0 there is no classical limit. Classical limit exists only if θ → 0 at least as fast as → 0, but this limit does not yield Newtonian mechanics, unless the limit of θ/ vanishes as θ → 0. We make explicit this relationship by deriving a superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM) and identifying the difference between the classical and the commutative limits. A superstar -product combines the usual phase space * star and the noncommutative -product [1].In this paper we focus on the nature of the commutative and classical limits of noncommutative quantum physics. We point out some subtleties which arise due to the ordering problem. When these issues are properly addressed we show that the classical correspondent to the NC Hamiltonian is indeed one with a θ-dependent, momentum-dependent shift in the coordinates. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of θ and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context. I. CRITERIA FOR CLASSICALITYWe open this di...
We present an analytic method based on the Hadamard-WKB expansion to calculate the self-force for a particle with scalar charge that undergoes radial infall in a Schwarzschild spacetime after being held at rest until a time t = 0. Our result is valid in the case of short duration from the start. It is possible to use the Hadamard-WKB expansion in this case because the value of the integral of the retarded Green's function over the particle's entire past trajectory can be expressed in terms of two integrals over the time period that the particle has been falling. This analytic result is expected to be useful as a check for numerical prescriptions including those involving mode sum regularization and for any other analytical approximations to self-force calculations.
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