arXiv subjects
Publications and source records attributed to Detlev Buchholz.
This article contrasts the concepts of a proper condensate and the Onsager-Penrose criterion for Bose-Einstein condensation in the form of a debate between two proponents of the respective concepts. The prologue briefly introduces the two criteria. The epilogue contains remarks on properties of the underlying resolvent algebra that are physically relevant in this context.
The limit states of canonical ensembles of a large number of interacting bosons at a given temperature, which are confined by harmonic forces, are studied in the framework of the resolvent algebra. It is shown that the limits satisfy the KMS condition or are ground states, regardless of the type of interaction. In case of attractive forces, where the ensembles collapse, observables that become meaningless in the limit disappear from the limit representations. For repulsive forces, this can also happen if condensates with an infinite number of particles in the same state (proper condensates) appear in the limit. The resulting structures and their interpretation are illustrated by a simple model. The study of vanishing harmonic forces (thermodynamic limit) involves changes of the dynamics. It is conveniently based on derivations acting on the algebra. They are given by the commutator of the Hamiltonians with the elements of the algebra. To ensure that the images remain in the algebra, the interaction must be regularized. This is accomplished in a manner that has only a minor impact on the dynamics and may be of broader interest. With this input a relation between the strength of the confining harmonic forces and the number of particles in the ensembles is derived from the condition that the limit states are to be stationary (invariant) under the adjoint action of the unconfined, spatially homogeneous limit dynamics. This relation encompasses the conditions that are frequently used in studies of Bose-Einstein condensates.
The creation of electrically charged states and the resulting electromagnetic fields are considered in space-time regions in which such experiments can actually be carried out, namely in future-directed light cones. Under the simplifying assumption of external charges, charged states are formed from neutral pairs of opposite charges, with one charge being shifted to light-like infinity. It thereby escapes observation. Despite the fact that this charge moves asymptotically at the speed of light, the resulting electromagnetic field has a well-defined energy operator that is bounded from below. Moreover, due to the spatiotemporal restrictions, the transverse electromagnetic field (the radiation) has no infrared singularities in the light cone. They are quenched and the observed radiation can be described by states in the Fock space of photons. The longitudinal field between the charges (giving rise to Gauss's law) disappears for inertial observers in an instant. This is consistent with the fact that there is no evidence for the existence of longitudinal photons. The results show that the restrictions of operations and observations to light cones, which are dictated by the arrow of time, amount to a Lorentz-invariant infrared cutoff.
Some advantages of the algebraic approach to many body physics, based on resolvent algebras, are illustrated by the simple example of non-interacting bosons which are confined in compact regions with soft boundaries. It is shown that the dynamics of these systems converges to the spatially homogeneous dynamics for increasing regions and particle numbers and a variety of boundary forces. The corresponding correlation functions of thermal equilibrium states also converge in this limit. Depending on the filling of the regions with particles, the limits can either be spatially homogeneous, including the Bose-Einstein condensates, or they become inhomogeneous with varying, but finite local particle densities. In case of this spontaneous breakdown of the spatial symmetry, the presence of condensates can be established by exhibiting temporal correlations over large temporal distances (memory effects).
Let Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents (aQ + bP - i r)^{-1} for real constants a,b,r=/=0. This implies that the basic resolvents form a total set (norm dense span) in the C*-algebra R generated by the resolvents, termed resolvent algebra. So the basic resolvents share this property with the unitary Weyl operators, which span the Weyl algebra. These results obtain for finite systems of particles in any number of dimensions. The resolvent algebra of infinite systems (quantum fields), being the inductive limit of its finitely generated subalgebras, is likewise spanned by its basic resolvents.
Algebraic quantum field theory is a general mathematical framework for relativistic quantum physics, based on the theory of operator algebras. It comprises all observable and operational aspects of a theory. In its framework the entire state space of a theory is covered, starting from the vacuum over arbitrary configurations of particles to thermal equilibrium and non-equilibrium states. It provides a solid foundation for structural analysis, the physical interpretation of the theory and the development of new constructive schemes. This survey is commissioned by the Encyclopedia of Mathematical Physics, edited by M. Bojowald and R.J. Szabo. It is to be published by the Elsevier publishing house.
Based on the hypothesis that the (non-reversible) arrow of time is intrinsic in any system, no matter how small, the consequences are discussed. Within the framework of local quantum physics it is shown how such a semi-group action of time can consistently be extended to that of the group of spacetime translations in Minkowski space. In presence of massless excitations, however, there arise ambiguities in the theoretical extensions of the time translations to the past. The corresponding loss of quantum information on states upon time is determined. Finally, it is explained how the description of operations in classical terms combined with constraints imposed by the arrow of time leads to a quantum theoretical framework. These results suggest that the arrow of time is fundamental in nature and not merely a consequence of statistical effects on which the Second Law is based.
Within the framework of the universal algebra of the electromagnetic field, the impact of globally neutral configurations of external charges on the field is analyzed. External charges are not affected by the field, but they induce localized automorphisms of the universal algebra. Gauss's law implies that these automorphisms cannot be implemented by unitary operators involving only the electromagnetic field, they are outer automorphisms. The missing degrees of freedom can be incorporated in an enlargement of the universal algebra, which can concretely be represented by exponential functions of gauge fields and an abelian algebra describing the external charges. In this manner, gauge fields manifest themselves in the framework of gauge invariant observables. The action of the automorphisms on the vacuum state gives rise to representations of the electromagnetic field with vanishing global charge, which are locally disjoint from the vacuum representation. This feature disappears in the enlarged universal algebra of the electromagnetic field. The energy content of the states is well defined in both cases and bounded from below. The passage from these globally neutral states to charged states and the determination of their energy content are also being discussed.
All causal Lie products of solutions of the Klein-Gordon equation and the wave equation in Minkowski space are determined. The results shed light on the origin of the algebraic structures underlying quantum field theory.
Within the framework of the algebra of canonical commutation relations in Euclidean space, a long range order between particles in bounded regions is established in states with a sufficiently large particle number. It occurs whenever homogeneous proper (infinite) condensates form locally in the states in the limit of infinite densities. The condensates are described by eigenstates of the momentum operator, covering also those cases, where they are streaming with a constant velocity. The arguments given are model independent and lead to a new criterion for the occurrence of condensates. It makes use of a novel approach to the identification of condensates, based on a characterization of regular and singular wave functions.
In this article a novel characterization of Bose-Einstein condensates is proposed. Instead of relying on occupation numbers of a few dominant modes, which become macroscopic in the limit of infinite particle numbers, it focuses on the regular excitations whose numbers stay bounded in this limit. In this manner, subspaces of global, respectively local regular wave functions are identified. Their orthogonal complements determine the wave functions of particles forming proper (infinite) condensates in the limit. In contrast to the concept of macroscopic occupation numbers, which does not sharply fix the wave functions of condensates in the limit states, the notion of proper condensates is unambiguously defined. It is outlined, how this concept can be used in the analysis of condensates in models. The method is illustrated by the example of trapped non-interacting ground states and their multifarious thermodynamic limits, differing by the structure of condensates accompanying the Fock vacuum. The concept of proper condensates is also compared with the Onsager-Penrose criterion, based on the analysis of eigenvalues of one-particle density matrices. It is shown that the concept of regular wave functions is useful there as well for the identification of wave functions forming proper condensates.
A novel C*-algebraic framework is presented for relativistic quantum field theories, fixed by a Lagrangean. It combines the postulates of local quantum physics, encoded in the Haag-Kastler axioms, with insights gained in the perturbative approach to quantum field theory. Key ingredients are an appropriate version of Bogolubov's relative $S$-operators and a reformulation of the Schwinger-Dyson equations. These are used to define for any classical relativistic Lagrangean of a scalar field a non-trivial local net of C*-algebras, encoding the resulting interactions at the quantum level. The construction works in any number of space-time dimensions. It reduces the longstanding existence problem of interacting quantum field theories in physical spacetimeto the question of whether the C*-algebras so constructed admit suitable states, such as stable ground and equilibrium states. The method is illustrated on the example of a non-interacting field and it is shown how to pass from it within the algebra to interacting theories by relying on a rigorous local version of the interaction picture.
A universal C*-algebra of gauge invariant operators is presented, describing the electromagnetic field as well as operations creating pairs of static electric charges having opposite signs. Making use of Gauss' law, it is shown that the string-localized operators, which necessarily connect the charges, induce outer automorphisms of the algebra of the electromagnetic field. Thus they carry additional degrees of freedom which cannot be created by the field. It reveals the fact that gauge invariant operators encode information about the presence of non-observable gauge fields underlying the theory. Using the Gupta-Bleuler formalism, concrete implementations of the outer automorphisms by exponential functions of the gauge fields are presented. These fields also appear in unitary operators inducing the time translations in the resulting representations of the universal algebra.
The virtues of resolvent algebras, compared to other approaches for the treatment of canonical quantum systems, are exemplified by infinite systems of non-relativistic bosons. Within this framework, equilibrium states of trapped and untrapped bosons are defined on a fixed C*-algebra for all physically meaningful values of the temperature and chemical potential. Moreover, the algebra provides the tools for their analysis without having to rely on 'ad hoc' prescriptions for the test of pertinent features, such as the appearance of Bose-Einstein condensates. The method is illustrated in case of non-interacting systems in any number of spatial dimensions and sheds new light on the appearance of condensates. Yet the framework also covers interactions and thus provides a universal basis for the analysis of bosonic systems.
The framework of dynamical C*-algebras for scalar fields in Minkowski space, based on local scattering operators, is extended to theories with locally perturbed kinetic terms. These terms encode information about the underlying spacetime metric, so the causality relations between the scattering operators have to be adjusted accordingly. It is shown that the extended algebra describes scalar quantum fields, propagating in locally deformed Minkowski spaces. Concrete representations of the abstract scattering operators, inducing this motion, are known to exist on Fock space. The proof that these representers also satisfy the generalized causality relations requires, however, novel arguments of a cohomological nature. They imply that Fock space representations of the extended dynamical C*-algebra exist, involving linear as well as kinetic and pointlike quadratic perturbations of the field.
The essence of the path integral method in quantum physics can be expressed in terms of two relations between unitary propagators, describing perturbations of the underlying system. They inherit the causal structure of the theory and its invariance properties under variations of the action. These relations determine a dynamical algebra of bounded operators which encodes all properties of the corresponding quantum theory. This novel approach is applied to non-relativistic particles, where quantum mechanics emerges from it. The method works also in interacting quantum field theories and sheds new light on the foundations of quantum physics.
A recent idea, put forward by Mund, Rehren and Schroer, is discussed; it suggests that in gauge quantum field theory one can replace the point-localized gauge fields by string-localized vector potentials built from gauge invariant observables and a principle of string-independence. Based on a kinematical model, describing unmovable (static) fields carrying opposite charges, it is shown that these string-localized potentials cannot be used for the description of the gauge bridges between electrically charged fields. These bridges are needed in order to ensure the validity of Gauss' law. This observation does not preclude the existence of Poincaré invariant theories, describing the coupling of string-localized gauge invariant potentials to matter fields. But these potentials are not a full-fledged substitute for the gauge fields in ``usual'' quantum electrodynamics.
Based on the assumption that time evolves only in one direction and mechanical systems can be described by Lagrangeans, a dynamical C*-algebra is presented for non-relativistic particles at atomic scales. Without presupposing any quantization scheme, this algebra is inherently non-commutative and comprises a large set of dynamics. In contrast to other approaches, the generating elements of the algebra are not interpreted as observables, but as operations on the underlying system; they describe the impact of temporary perturbations caused by the surroundings. In accordance with the doctrine of Nils Bohr, the operations carry individual names of classical significance. Without stipulating from the outset their `quantization', their concrete implementation in the quantum world emerges from the inherent structure of the algebra. In particular, the Heisenberg commutation relations for position and velocity measurements are derived from it. Interacting systems can be described within the algebraic setting by a rigorous version of the interaction picture. It is shown that Hilbert space representations of the algebra lead to the conventional formalism of quantum mechanics, where operations on states are described by time-ordered exponentials of interaction potentials. It is also discussed how the familiar statistical interpretation of quantum mechanics can be recovered from operations.