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 to the generalized Hamiltonian _μ (with ₀ = ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrödinger equation i∂_μψ = _μψ and Heisenberg equation d/dx^μ = ∂_μ + i[_μ, ] are obtained, from which we have: (1) t is to ₀ as x_j is to _j (j = 1, 2, 3); likewise, t is to i∂₀ as x_j is to i∂_j; (2) the proposed time operator is canonically conjugate to i∂₀ rather than to ₀, therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual Δx_μΔp_μ ≥ 1/2 and the Mandelstam–Tamm treatment Δx_μΔH_μ ≥ 1/2, have been formulated.

PACS numbers: 03.65.Xp, 03.70.+k, 03.65.Ta

Print publication: Issue 18 (9 May 2003)
Received 18 November 2002, in final form 18 March 2003
Published 23 April 2003