Path space measure for the \(3+1\)-dimensional Dirac equation in momentum space. (English) Zbl 1046.81073
Summary: Nonstandard analysis is used to construct a measure over paths for the path integral solution to the Dirac equation in \(3+1\) dimension. Paths are considered in momentum space, because the Green function in the configuration space contains a derivative of the \(\delta\) function which keeps us from assigning a measure over paths. The solution is obtained not only as the standard part of a nonstandard path sum with respect to a nonstandard measure, but also as a standard path integral with respect to a standard measure extracted from the nonstandard one. The result is an extension of Gaveau’s work [B. Gaveau, J. Funct. Anal. 58, 310–319 (1984; Zbl 0562.35055)].
MSC:
| 81S40 | Path integrals in quantum mechanics |
Citations:
Zbl 0562.35055References:
| [1] | DOI: 10.1103/RevModPhys.20.367 · Zbl 1371.81126 · doi:10.1103/RevModPhys.20.367 |
| [2] | DOI: 10.1090/S0002-9939-1959-0108732-6 · doi:10.1090/S0002-9939-1959-0108732-6 |
| [3] | DOI: 10.1007/BF02791062 · Zbl 0418.35032 · doi:10.1007/BF02791062 |
| [4] | DOI: 10.1002/sapm1960391126 · Zbl 0096.06901 · doi:10.1002/sapm1960391126 |
| [5] | Ichinose T., Proc. Jpn. Acad. 58 pp 290– (1982) · Zbl 0518.28011 · doi:10.3792/pjaa.58.290 |
| [6] | Zastawniak T., Bull. Pol. Acad. Sci., Chem. 36 pp 341– (1989) |
| [7] | DOI: 10.1063/1.529433 · Zbl 0733.58006 · doi:10.1063/1.529433 |
| [8] | Nakamura T., J. Math. Phys. 38 pp 4052– (1997) · Zbl 0892.46082 · doi:10.1063/1.532083 |
| [9] | Zastawniak T., J. Math. Phys. 30 pp 1354– (1989) · Zbl 0689.70010 · doi:10.1063/1.528316 |
| [10] | DOI: 10.1016/0022-1236(84)90045-4 · Zbl 0562.35055 · doi:10.1016/0022-1236(84)90045-4 |
| [11] | Watanabe H., J. Funct. Anal. 65 pp 204– (1986) · Zbl 0603.28013 · doi:10.1016/0022-1236(86)90009-1 |
| [12] | Loeb P. A., Trans. Am. Nucl. Soc. 211 pp 113– (1975) · doi:10.1090/S0002-9947-1975-0390154-8 |
| [13] | Živaljević R. T., Math. Scand. 56 pp 276– (1985) · Zbl 0607.28010 · doi:10.7146/math.scand.a-12101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
