Gamow Vectors and Representations of the Poincaré Semigroup

The foundations of time asymmetric quantum theory are reviewed and are applied to the construction of relativistic Gamow vectors. Relativistic Gamow vectors are obtained from the resonance pole of the S-matrix and furnish an irreducible representation of the Poincaré semigroup. They have all the properties needed to represent relativistic quasistable particles and can be used to fix the definition of mass and width of relativistic resonances like the Z-boson. Most remarkably, they have only a semigroup time evolution into the forward light cone—expressing time asymmetry on the microphysical level. 1 Time Asymmetric Quantum Mechanics In classical physics one has time symmetric dynamical equations with time asymmetric boundary conditions [1, 2]. These time asymmetric boundary conditions come in pairs: given one time asymmetric boundary condition, its time reversed boundary condition can also be formulated mathematically. For example in classical electrodynamics one has retarded and advanced solutions of the time symmetric dynamical (Maxwell) equations or in general relativity one has time asymmetric big bang and Plenary talk at the International Symposium on Quantum Theory and Symmetries, July 1999, Goslar, Germany. Written from transparencies and notes with N. L. Harshman and M. J. Mithaiwala. big crunch solutions of Einstein’s time symmetric equation. Except for a few prominent cases of pedagogical importance (e.g. stationary states or cyclic evolutions), the physics of our world is predominantly time-asymmetric. Somehow nature chooses one of the pair of time asymmetric boundary conditions. The standard quantum mechanics in Hilbert space [3] does not allow time asymmetric boundary conditions for the Schrödinger or von Neumann equation [4]. However this is a consequence of themathematical properties of the Hilbert space and need not imply that quantum physics is strictly time symmetric. It would be incredible if classical electrodynamics had a radiation arrow of time and quantum electrodynamics did not also have an arrow of time. In quantum physics Peierls and Siegert considered many years ago time asymmetric solutions with purely outgoing boundary conditions [5]. The choice of appropriate dense subspaces Φ+ and Φ− of the (complete) Hilbert space H allows the formulation of time asymmetric boundary conditions: Φ+ ⊂ H for the out-states {ψ−} of scattering theory which are actually observables as defined by the registration apparatus (detector), and Φ− ⊂ H for the in-states {φ+} which are prepared states as defined by the preparation apparatus (accelerator). Time asymmetric quantum theory distinguishes meticulously between states {φ+} and observables {ψ−}. Two different dense subspaces of the Hilbert space H are chosen, Φ− = {φ+} and Φ+ = {ψ−}. The standard Hilbert space quantum theory uses H for both, {ψ−} = {φ+} = H and as a result is time symmetric with a reversible unitary group time evolution. In the theory of scattering and decay, a pair of time asymmetric boundary conditions can be heuristically implemented by choosing inand out-plane wave “states” |E+〉 and |E−〉 which are solutions of the Lippmann-Schwinger equation [6] |E〉 = |E〉+ 1 E −H ± i0 |E〉 = Ω |E〉, (1) where (H − V )|E〉 = E|E〉. The energy distribution of the incident beam is given by |〈+E|φ+〉|2 = |〈E|φin〉|2 and the energy resolution of the detector (which for perfect efficiency is the energy distribution of the detected “out-states”) is measured as |〈−E|ψ−〉|2 = |〈E|ψout〉|2. The sets {|E±〉} are the basis system that is used for the Dirac basis vector expansion of the in-states φ ∈ Φ− and the out-states (observables) ψ ∈ Φ+: ψ = ∑