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arXiv:1312.7741

Time irreversibility in the quantum systems with infinite number of particles

Huo Yuping

Published Dec 24, 2013• 0 citations
arXivHTMLPDF

Abstract

The time irreversible evolution of quantum systems with infinite number of particles (QSINP) was studied within a newly constructed algebraic framework. The QSINP could be described by the quantum infinite lattice field. The *-Algebra R is the set of all dynamic variables of the QSINP, in which a special addition and multiplication operations were defined. The full pure state vector (FPSV) ρ, the pure state vector {ρ_f} and the equivalent relations within them are also well defined. The set of all pure state vectors, which are equivalent to the FPSV ρ, is a Hilbert Space {H_ρ}, if the addition, the inner product and the norm were defined in it. The set of all linear transformation {N_ρ} on {H_ρ} is isomorphic to R, and {{N_ρ},{H_ρ}} is the representation of R, associated with ρ. The *-Algebra R has infinitely many non-equivalent irreducible GNS constructions (representations), associated with different nonequivalent FPSVs respectively. It is proved that, the dynamical motions of the QSINP is totally within the GNS construction associated with the initial pure state vector; However, the time reversal transformation makes the initial FPSV and its corresponding dynamical evolution into another non-equivalent GNS construction. Therefore, within the GNS construction associated with the FPSV, the dynamics of the QSINP is time irreversible. Finally, due to the Liouville operator has real continuous spectrum within the GNS construction, the longtime asymptotic solution of Liouville equation in the GNS construction could be treated by the formal scattering theory. The Master equation with dissipative term was obtained, which is formally irrelevant of the initial state and the corresponding GNS construction. This master equation could be regarded as the evolution equation of the QSINP on R.

DOI
10.48550/arxiv.1312.7741

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The time irreversible evolution of quantum systems with infinite number of particles (QSINP) was studied within a newly constructed algebraic framework. The QSINP could be described by the quantum infinite lattice field. The *-Algebra R is the set of all dynamic variables of the QSINP, in which a special addition and multiplication operations were defined. The full pure state vector (FPSV) ρ, the pure state vector {ρ_f} and the equivalent relations within them are also well defined. The set of all pure state vectors, which are equivalent to the FPSV ρ, is a Hilbert Space {H_ρ}, if the addition, the inner product and the norm were defined in it. The set of all linear transformation {N_ρ} on {H_ρ} is isomorphic to R, and {{N_ρ},{H_ρ}} is the representation of R, associated with ρ. The *-Algebra R has infinitely many non-equivalent irreducible GNS constructions (representations), associated with different nonequivalent FPSVs respectively. It is proved that, the dynamical motions of the QSINP is totally within the GNS construction associated with the initial pure state vector; However, the time reversal transformation makes the initial FPSV and its corresponding dynamical evolution into another non-equivalent GNS construction. Therefore, within the GNS construction associated with the FPSV, the dynamics of the QSINP is time irreversible. Finally, due to the Liouville operator has real continuous spectrum within the GNS construction, the longtime asymptotic solution of Liouville equation in the GNS construction could be treated by the formal scattering theory. The Master equation with dissipative term was obtained, which is formally irrelevant of the initial state and the corresponding GNS construction. This master equation could be regarded as the evolution equation of the QSINP on R.
It has been shown by Flesia and Piron that the scattering theory of Lax and Phillips, developed primarily for the treatment of hyperbolic systems (e.g., classical scattering of electromagnetic waves on a finite target), can be applied to the quantum theory. In order to construct the correspondence, the continuously infinite family of (isomorphic) Hilbert spaces associated with each value of the time (are taken to be a direct integral representation (with a choice of measure) of a larger Hilbert space which includes the variable t in its measure space. We show that this theory, for which the evolution contracted to the subspace orthogonal to the incoming and outgoing subspaces is a contractive semigroup, provides a natural description of the irreversible motion of an unstable quantum system
It is difficult to incorporate the concept of interacting physical particles in the frame of conventional quantum mechanics. It comes from the fact that in this formalism the Hamiltonian plays a dual role as the generator of motion and as the observable associated with the energy. As a result it is easy to define either a representation in which particles interact but are not well defined (a representation in which the Hamiltonian is not diagonal) or a representation in which the units are well defined but do not interact. We expect that in the search for a representation in which we may speak of both well defined and still interacting particles, the distinction between reversible and irreversible processes should play a basic role. Certainly we do not want to eliminate irreversible processes such as scattering, production or decay which are basically related to the mechanisms through which we observe particles. To distinguish between reversible and irreversible processes we have to work in a representation such that a Liapounov function (the entropy) may be ascribed to the time evolution of the system. The introduction of such a representation through a specific class of non-unitary transforms is described. We then propose newmore » quantum rules directly applicable to dissipative systems with a Liapounov function. For systems with a discrete spectrum these new quantum rules reduce trivially to the usual quantum rules as formulated by Born, Jordan and Heisenberg. The theory is applied to a simple field theoretical model - the so-called Friedrichs model-for which all calculations can be performed exactly.« less
The state vector evolution in the interaction of initial measured pure state with collective quantum system or the field with a very large number of degrees of freedom N is analysed in a nonperturbative QED formalism. As the example the measurement of the electron final state scattered on nucleus or neutrino is considered.In the nonperturbative field theory the complete manifold of the system states is nonseparable i.e. is described by tensor product of infinitely many independent Hilbert spaces. The interaction of this system with the measured state can result in the final states which belong to different Hilbert spaces which corresponds to different values of some classical observables,i.e. spontaneous symmetry breaking occurs. Interference terms (IT) between such states in the measurement of any Hermitian observable are infinitely small and due to it the final pure states can't be distinguished from the mixed ones, characteristic for the state collapse. The evolution from initial to final system state is nonunitary and become formally irreversible in the limit of infinite time. The electromagnetic (e-m) bremmstrahlung produced in the electron scattering process contain the unrestricted number of soft photons which radiation flux become classic observable. Analoguous processes which occurs in the second kind phase transitions in ferromagnetic and phonon excitations in cristall lattice are considered briefly.
A particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one-to-one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.
This paper investigates the relationship between subsystems and time in a closed nonrelativistic system of interacting bosons and fermions. It is possible to write any state vector in such a system as an unentangled tensor product of subsystem vectors, and to do so in infinitely many ways. This requires the superposition of different numbers of particles, but the theory can describe in full the equivalence relation that leads to a particle-number superselection rule in conventionally defined subsystems. Time is defined as a functional of subsystem changes, thus eliminating the need for any reference to an external time variable. The dynamics of the unentangled subsystem decomposition is derived from a variational principle of dynamical stability, which requires the decomposition to change as little as possible in any given infinitesimal time interval, subject to the constraint that the state of the total system satisfy the Schroedinger equation. The resulting subsystem dynamics is deterministic. This determinism is regarded as a conceptual tool that observers can use to make inferences about the outside world, not as a law of nature. The experiences of each observer define some properties of that observer's subsystem during an infinitesimal interval of time (i.e., the present moment); everything else must be inferred from this information. The overall structure of the theory has some features in common with quantum Bayesianism, the Everett interpretation, and dynamical reduction models, but it differs significantly from all of these. The theory of information described here is largely qualitative, as the most important equations have not yet been solved. The quantitative level of agreement between theory and experiment thus remains an open question.
The microscopic theory of irreversible processes that we developed is summarized and illustrated, using as a simple example the Friedrichs model. Our approach combines the Poincaré's point of view (dynamical interpretation of irreversibility) with the Gibbs-Einstein ensemble point of view. It essentially consists in a nonunitary transformation theory based on the symmetry properties of the Liouville equation and dealing with continuous spectrum. The second law acquires a microscopic content in terms of a Liapounov function which is a quadratic functional of the density operator. In our new representation of dynamics, which is defined for a restricted set of observables and states, this functional takes a universal form. We obtain, in this way, a semi-group description, the generator of which contains a part directly related to the microscopic entropy production. The Friedrichs model gives us a simple field theoretical example for which the entropy production can be evaluated. The thermodynamical meaning of life-times is explicitly displayed. The transition from pure states to mixtures, as well as the occurrence of long tails in thermodynamic systems, are also briefly discussed.