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

Socpus ID

0038736745 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/0038736745

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