Non-hermitian radial momentum operator and path integrals in polar coordinates

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dc.creatorFujikawa, Kazuo
dc.date2008-05-26
dc.date2008-07-15
dc.date.accessioned2026-07-07T11:51:06Z
dc.date.available2026-07-07T11:51:06Z
dc.descriptionA salient feature of the Schrödinger equation is that the classical radial momentum term $p_{r}^{2}$ in polar coordinates is replaced by the operator $\hat{P}^{\dagger}_{r} \hat{P}_{r}$, where the operator $\hat{P}_{r}$ is not hermitian in general. This fact has important implications for the path integral and semi-classical approximations. When one defines a formal hermitian radial momentum operator $\hat{p}_{r}=(1/2)((\frac{\hat{\vec{x}}}{r}) \hat{\vec{p}}+\hat{\vec{p}}(\frac{\hat{\vec{x}}}{r}))$, the relation $\hat{P}^{\dagger}_{r} \hat{P}_{r}=\hat{p}_{r}^{2}+\hbar^{2}(d-1)(d-3)/(4r^{2})$ holds in $d$-dimensional space and this extra potential appears in the path integral formulated in polar coordinates. The extra potential, which influences the classical solutions in the semi-classical treatment such as in the analysis of solitons and collective modes, vanishes for $d=3$ and attractive for $d=2$ and repulsive for all other cases $d\geq 4$. This extra term induced by the non-hermitian operator is a purely quantum effect, and it is somewhat analogous to the quantum anomaly in chiral gauge theory.
dc.descriptionA comment on the difference of the hermitian radial momentum operator in the present context of path integrals and in the conventional operator Schroedinger problem is added. To be published in Prog. Theor. Phys.. 17 pages
dc.identifierhttps://arxiv.org/abs/0805.3879
dc.identifierhttp://arxiv.org/abs/0805.3879
dc.identifierProg.Theor.Phys.120:181-195,2008
dc.identifierdoi:10.1143/PTP.120.181
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/203781
dc.subjectHigh Energy Physics - Theory
dc.subjectQuantum Physics
dc.titleNon-hermitian radial momentum operator and path integrals in polar coordinates
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