Zusammenfassung und Ausblick

Fliegner, Susanne

doi:10.1007/978-3-476-03354-3_4

Cited by 1 publication

(7 citation statements)

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“…The quantum mechanical path integral has proved useful in computing Green's functions at one-loop order [10][11][12][13] and beyond [14][15][16]; this suggests using this approach to examine a n (x 0 , ∆) for ∆ = 0. Although our method is not identical to that of [7], the two approaches are similar and both results agree when ∆ = 0. The representation of M xy = x| exp 1 2 1 √ g (∂ µ − iA µ )g µν √ g(∂ ν − iA ν ) t |y in terms of a quantum mechanical path integral is not uniquely specified [17][18][19], as discussed in [14].…”

Section: Introductionmentioning

confidence: 67%

“…The path integral in (7) can be evaluated by systematic functional differentiation of the standard result [21,24,12]…”

Section: Expanding M Xymentioning

confidence: 99%

“…From (8), it is easily seen that no term in (7) will involve factors of ∆ 2 . From this observation, combined with simple power counting arguments, it is straightforward to tabulate the possible contributions to the various coefficients a n (x 0 , ∆) (see table I).…”

Section: Expanding M Xymentioning

confidence: 99%

“…(When a temperature dependent QMPI is considered as in [12], then the temperature provides a second dimensionful parameter which must be considered). The coefficients of the various contributions can be easily determined by appropriately choosing L and N in (7) and then systematically applying (8). For example, if L = n, N = 0, then it is apparent that the contribution to a n (x 0 , ∆) proportional to V n (x 0 ) is 1 n!…”

Section: Expanding M Xymentioning

confidence: 99%

“…This expansion when ∆ = 0 is extremely useful when examining certain properties of the generating functional at one-loop order; in particular, the divergence structure of a theory at one-loop order can be discerned. Among the approaches used to evaluate a n (x 0 , 0) are the perturbative solution to the heat equation [1,2], the use of pseudo-differential operators [3], working in momentum space [4], systematically rearranging a Schwinger expansion of (1a) in powers of A and V into an expression of the form (1b) [5,6] and representing (1a) as a quantum mechanical path integral (QMPI) hence expanding it in powers of t [7,8]. The only places of which the authors are aware where a n (x 0 , ∆) is considered for ∆ = 0 are in [2] and [6].…”

Section: Introductionmentioning

confidence: 99%

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Off-diagonal elements of the DeWitt expansion from the quantum-mechanical path integral

Dilkes

McKeon

1996

Phys. Rev. D

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The DeWitt expansion of the matrix element M xy = x| exp −[ 1 2 (p − A) 2 + V ]t |y , (p = −i∂) in powers of t can be made in a number of ways. For x = y (the case of interest when doing one-loop calculations) numerous approaches have been employed to determine this expansion to very high order; when x = y (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 M xy by a quantum mechanical path integral. We also generalize our method to the case of curved space, allowing us to determine the DeWitt expansion ofM xy = x| exp 1t|y 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 M xy as a quantum mechanical path integral can be assessed. Furthermore, the exact dependence ofM xy 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.

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