ua nt - p h / 05 05 14 1 v 2 1 2 Ju l 2 00 5 A time representation

The paper contains a proposal for an energy and time representation. We construct modes that correspond to fuzzy distributions around discrete values of energy or time. The modes form an orthogonal and complete set in the space of square integrable functions. Energy and time are self adjoint in the space spanned by the modes. The widths of the modes are analyzed as well as their energy-time uncertainty relations. The lower uncertainty attainable for the modes is shown. We also show times of arrival for massless particles. The Pauli theorem revisited Two arbitrary states of an elementary system can be transformed into each other by symmetry operations. This opens the door to express what can be observed of the system, i.e. the system properties, in terms of the generators of these symmetries. In the case of the Poincare group they are the momenta P̂ μ and the angular momenta and boosts M̂ . For the system to be elementary they are constrained by the mass shell condition P̂ 2 = m and by the spin condition Ŵ 2 = ms(s + 1). (The Pauli-Lubanski vector is defined as Ŵμ = ǫμναβP̂ M̂ , with h̄ = 1 in this paper unless otherwise specified). Notice now the conjunction of both, the four vector character of the momenta on one side, with the necessity of introducing conjugate operators ∗[email protected], [email protected]