Time Asymmetric Quantum Mechanics

The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone--von Neumann theorem, the solutions of the dynamical equations, the Schr\"odinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution. Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)). If one requires that scattering resonances of width $\Gamma$ and exponentially decaying states of lifetime $\tau=\frac{\hbar}{\Gamma}$ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution $t_{0}\leq t<\infty$, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system. The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry. From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics

Irreversible Quantum Mechanics in the Neutral K-System

The neutral Kaon system is used to test the quantum theory of resonance scattering and decay phenomena. The two dimensional Lee-Oehme-Yang theory with complex Hamiltonian is obtained by truncating the complex basis vector expansion of the exact theory in Rigged Hilbert space. This can be done for K_1 and K_2 as well as for K_S and K_L, depending upon whether one chooses the (self-adjoint, semi-bounded) Hamiltonian as commuting or non-commuting with CP. As an unexpected curiosity one can show that the exact theory (without truncation) predicts long-time 2 pion decays of the neutral Kaon system even if the Hamiltonian conserves CP.Comment: 36 pages, 1 PostScript figure include

On the Mass and Width of the Z-boson and Other Relativistic Quasistable Particles

The ambiguity in the definition for the mass and width of relativistic resonances is discussed, in particular for the case of the Z-boson. This ambiguity can be removed by requiring that a resonance's width $\Gamma$ (defined by a Breit-Wigner lineshape) and lifetime $\tau$ (defined by the exponential law) always and exactly fulfill the relation $\Gamma = \hbar/\tau$. To justify this one needs relativistic Gamow vectors which in turn define the resonance's mass M_R as the real part of the square root $\rm{Re}\sqrt{s_R}$ of the S-matrix pole position s_R. For the Z-boson this means that $M_R \approx M_Z - 26{MeV}$ and $\Gamma_R \approx \Gamma_Z-1.2{MeV}$ where M_Z and $\Gamma_Z$ are the values reported in the particle data tables.Comment: 23 page

Relating the Lorentzian and exponential: Fermi's approximation,the Fourier transform and causality

The Fourier transform is often used to connect the Lorentzian energy distribution for resonance scattering to the exponential time dependence for decaying states. However, to apply the Fourier transform, one has to bend the rules of standard quantum mechanics; the Lorentzian energy distribution must be extended to the full real axis $-\infty<E<\infty$ instead of being bounded from below $0\leq E <\infty$ (``Fermi's approximation''). Then the Fourier transform of the extended Lorentzian becomes the exponential, but only for times $t\geq 0$, a time asymmetry which is in conflict with the unitary group time evolution of standard quantum mechanics. Extending the Fourier transform from distributions to generalized vectors, we are led to Gamow kets, which possess a Lorentzian energy distribution with $-\infty<E<\infty$ and have exponential time evolution for $t\geq t_0 =0$ only. This leads to probability predictions that do not violate causality.Comment: 23 pages, no figures, accepted by Phys. Rev.

Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution

Whether one starts form the analytic S-matrix definition or the requirement of gauge parameter independence in renormalization theory, a relativistic resonance is given by a pole at a complex value s of energy squared. The complex number s does not define the mass and width separately and this definition does not lead to interfering Breit-Wigner if two or more resonances are involved. To accomplish both we invoke the decaying particle aspect of a resonance and associate to each pole a space of relativistic Gamow kets which transform irreducibly under causal Poincare transformations. A Gamow state has an exponential time evolution and one can choose of the many possible width parameters, that parameter as the width of the relativistic resonance which equals the inverse lifetime. This uniquely defines the mass and width parameters for a relativistic resonance. Two or more poles in the same partial wave are given by the sum of Breit-Wigners in the scattering amplitude and by a superposition of Gamow vectors with each Gamow vector corresponding to one Breit-Wigner. In addition to the sum of Breit-Wigners the scattering amplitude contains a background amplitude representing direct production of the final state (contact terms).This contact amplitude is associated to a background vector which is a continuous superposition of Lippmann-Schwinger states. Omitting this continuum gives the Weisskopf-Wigner approximation.Comment: 22 pages, REVTe

Measurement of the W-Pair Production Cross Section and W-Decay Branching Fractions in e+e−e^{+}e^{-} Interactions at s\sqrt{s}= 189 GeV

The data collected by the L3 experiment at LEP at a centre-of-mass energy of $188.6~\rm{Ge\kern -0.1em V}$ are used to measure the W-pair production cross section and the W-boson decay branching fractions. These data correspond to an integrated luminosity of 176.8~pb$^{-1}$. The total cross section for W-pair production, combining all final states, is measured to be $\sigma_{\rm{WW}}= 16.24 \pm 0.37~(stat.) \pm 0.22~(syst.)$~pb. Including our data collected at lower centre-of-mass energies, the hadronic branching fraction of the W-boson is determined to be $B(\rm{W} \rightarrow \rm{qq})= \left[ 68.20 \pm 0.68~(stat.) \pm 0.33~(syst.)\right]~\%$. The results agree with the Standard Model predictions.The data collected by the L3 experiment at LEP at a centre-of-mass energy of 188.6 GeV are used to measure the W-pair production cross section and the W-boson decay branching fractions. These data correspond to an integrated luminosity of 176.8pb^-1. The total cross section for W-pair production, combining all final states, is measured to be sigma_WW = 16.24 +/- 0.37(stat.) +/- 0.22(syst.) pb. Including our data collected at lower centre-of-mass energies, the hadronic branching fraction of the W-boson is determined to be B(W ->qq) = [68.20 +/- 0.68 (stat.) +/- 0.33 (syst.) ] %. The results agree with the Standard Model predictions.The data collected by the L3 experiment at LEP at a centre-of-mass energy of 188.6 GeV are used to measure the W-pair production cross section and the W-boson decay branching fractions. These data correspond to an integrated luminosity of 176.8 pb −1 . The total cross section for W-pair production, combining all final states, is measured to be σ WW =16.24±0.37 (stat.)±0.22 (syst.) pb. Including our data collected at lower centre-of-mass energies, the hadronic branching fraction of the W-boson is determined to be B (W→qq)=[68.20±0.68 (stat.)±0.33 (syst.)]%. The results agree with the Standard Model predictions