Title
Time In Quantum Mechanics And Quantum Field Theory
Abstract
W Pauli pointed out that the existence of a self-adjoint time operator is incompatible with the semi-bounded character of the Hamiltonian spectrum. As a result, there has been much argument about the time-energy uncertainty relation and other related issues. In this paper, we show a way to overcome Pauli's argument. In order to define a time operator, by treating time and space on an equal footing and extending the usual Hamiltonian Ĥ to the generalized Hamiltonian Ĥμ (with Ĥ0 = Ĥ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrödinger equation i∂μψ = Ĥ μψ and Heisenberg equation dF̂/dxμ = ∂μF̂ + i[Ĥμ, F̂] are obtained, from which we have: (1) t is to Ĥ0 as xj, is to Ĥj (j = 1, 2, 3); likewise, t is to i∂0 as xj is to i∂j; (2) the proposed time operator is canonically conjugate to i∂0 rather than to Ĥ0, therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual ΔxμΔpμ ≥ 1/2 and the Mandelstam-Tamm treatment ΔxμΔHμ ≥ 1/2, have been formulated.
Publication Date
5-9-2003
Publication Title
Journal of Physics A: Mathematical and General
Volume
36
Issue
18
Number of Pages
5135-5147
Document Type
Article
Personal Identifier
scopus
DOI Link
https://doi.org/10.1088/0305-4470/36/18/317
Copyright Status
Unknown
Socpus ID
0038736745 (Scopus)
Source API URL
https://api.elsevier.com/content/abstract/scopus_id/0038736745
STARS Citation
Wang, Z. Y.; Chen, B.; and Xiong, C. D., "Time In Quantum Mechanics And Quantum Field Theory" (2003). Scopus Export 2000s. 1765.
https://stars.library.ucf.edu/scopus2000/1765