Time Asymmetric Quantum Theory Research Articles

Boundary conditions, semigroups, quantum jumps, and the quantum arrow of time

Experiments on quantum systems are usually divided into preparation of states and the registration of observables. Using the traditional mathematical methods (the Hilbert space and Schwartz space of distribution theory), it is not possible to distinguish mathematically between observables and states. The Hilbert space as well as Schwartz space boundary conditions for the dynamical equations lead by mathematic theorems (Stone-von Neumann) to unitary group with —∞ < t < ∞. But in the experimental set-up, one clearly distinguishes between the preparation of a state and the registration of an observable in that state. Furthermore, a state must be prepared first before an observable can be measured in this state (causality). This suggests time asymmetric boundary conditions for the dynamical equations of quantum theory. Such boundary conditions have been provided by Hardy space in the Lax-Phillips theory for electromagnetic and acoustic scattering phenomena. The Paley-Wiener theorem for Hardy space then leads to semi-group and time asymmetry in quantum physics. It introduces a finite “beginning of time” t0 for a time asymmetric quantum theory, which have been observed as an ensemble of finite times t(i)0, the onset times of dark periods in the quantum jump experiments on a single ion.

Asymmetric Time Evolution and Reduced Dynamics

When using a time asymmetric quantum theory, one must identify the time evolution parameter with a duration in time rather than with a time coordinate value. This identification restricts the options for the quantum mechanical environment of open quantum systems. The restriction may be important for interpretational questions concerning irreversibility or entanglement, but there is no measurable difference between a reduced dynamics within a time symmetric theory or within a time asymmetric theory.

Resonance Phenomena and time asymmetric quantum mechanics

This article is a review of time asymmetric quantum theory and its consequences applied to the resonance and decay phenomena. We first give some phenomenological results about resonances and decaying states to support the popular idea that resonances characterized by a width \Gamma and decaying states characterized by a lifetime \tau are different appearances of the same physical entity. Based on Weisskopf-Wigner (WW) methods, one obtains approximately \frac{\hbar}{\tau}\approx \Gamma. However, using standard axioms of quantum physics it is not possible to establish a rigorous theory to which the various WW methods can be considered as approximations. In standard quantum theory, the set of states and the set of observables are mathematically identified and described by the same Hilbert space H. Modifying this Hilbert space axiom to a Hardy space axiom one distinguishes the prepared (in) states and detected (out) observables. This leads to semi-group time evolution and to beginnings of time for individual microsystems. As a consequence of this time asymmetric theory one derives \frac{\hbar}{\tau} = \Gamma as an exact relation, and this unifies resonances and decaying states. Finally, we show that this unification can also be extended to the relativistic regime.

Reply to ‘Comment on ‘On the inconsistency of the Bohm–Gadella theory with quantum mechanics’’

In this reply, we show that when we apply standard distribution theory to the Lippmann–Schwinger equation, the resulting spaces of test functions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. Also, we point out several differences between the ‘standard method’ of constructing rigged Hilbert spaces in quantum mechanics and the method used in time asymmetric quantum theory.

On the inconsistency of the Bohm–Gadella theory with quantum mechanics

The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation-registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann-Schwinger equation nor they should appear. It is also shown that the Bohm-Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm-Gadella theory must therefore be abandoned.

Time asymmetric quantum theory – III. Decaying states and the causal Poincaré semigroup

AbstractThese Gamow kets span an irreducible representation space for Poincaré transformations which, similar to the Wigner representations for stable particles, are characterized by spin (angular momentum of the partial wave amplitude) and complex mass (position of the resonance pole). The Poincaré transformations of the Gamow kets, as well as of the Lippmann‐Schwinger plane wave scattering states, form only a semigroup of Poincaré transformations into the forward light cone. Their transformation properties are derived. From these one obtains an unambiguous definition of resonance mass and width for relativistic resonances. The physical interpretation of these transformations for the Born probabilities and the problem of causality in relativistic quantum physics is discussed.

Time Asymmetric Quantum Theory and the Z-boson Mass and Width

A unified theory of resonance and decay phenomena is presented for the non-relativistic and the relativistic cases. It incorporates causality and time asymmetry. The relativistic quasistable particles are described by semigroup representations of the causal Poincaré transformations characterized by spin j and complex square mass sR = (MR — iΓR/2)2, where ΓR is the width of a relativistic Breit-Wigner scattering amplitude. The lifetime of the exponentially decaying state described by this semigroup representation [j, sR] is derived as τ = ␣/ΓR. This provides a criterion for the appropriate definition of the mass and width of a relativistic resonance.

Time asymmetric quantum theory and the ambiguity of the Z-boson mass and width

Relativistic Gamow vectors emerge naturally in a time asymmetric quantum theory as the covariant kets associated to the resonance pole \(s=s_R\) in the second sheet of the analytically continued S-matrix. They are eigenkets of the self-adjoint mass operator with complex eigenvalue \(\sqrt{s_R}\) and have exponential time evolution with lifetime \(\tau = - \hbar/2\mathrm{Im}\sqrt{s_R}\). If one requires that the resonance width \(\Gamma\) (defined by the Breit-Wigner lineshape) and the resonance lifetime \(\tau\) always and exactly fulfill the relation \(\Gamma=\hbar/\tau\), then one is lead to the following parameterization of \(s_R\) in terms of resonance mass \(M_R\) and width \(\Gamma_R\): \(s_R = (M_R - i\Gamma/2)^2\). Applying this result to the \(Z\)-boson implies that \(M_R \approx M_Z - 26\mbox{MeV}\) and $\Gamma_R \approx \Gamma_Z-1.2\mbox{MeV}$ are the mass and width of the {\it Z}-boson and not the particle data values \((M_Z,\Gamma_Z)\) or any other parameterization of the Z-boson lineshape. Furthermore, the transformation properties of these Gamow kets show that they furnish an irreducible representation of the causal Poincare semigroup, defined as a semi-direct product of the homogeneous Lorentz group with the semigroup of space-time translations into the forward light cone. Much like Wigner's unitary irreducible representations of the Poincare group which describe stable particles, these irreducible semigroup representations can be characterized by the spin-mass values \((j,s_R=(M_R-i\Gamma/2)^2)\).