The Weyl correspondence and path integrals

Journal Article · Wed Jan 01 04:00:00 EST 1975 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.522468· OSTI ID:4184007

Mizrahi, Maurice M.

The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrodinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h times an ''effective action'' taken along it. The latter is the time integral of pdotq-h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h$sup 2$, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)P-eA (Q)$sup 2$+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl's rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.

• Research Organization: Center for Relativity, University of Texas, Austin, Texas 78712
• Sponsoring Organization: USDOE
• NSA Number: NSA-33-004550
• OSTI ID: 4184007
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 11 Vol. 16; ISSN JMAPAQ; ISSN 0022-2488
• Publisher: American Institute of Physics (AIP)
• Country of Publication: United States
• Language: English

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