2,531 research outputs found
An Algebraic Characterization of Vacuum States in Minkowski Space. III. Reflection Maps
Employing the algebraic framework of local quantum physics, vacuum states in
Minkowski space are distinguished by a property of geometric modular action.
This property allows one to construct from any locally generated net of
observables and corresponding state a continuous unitary representation of the
proper Poincare group which acts covariantly on the net and leaves the state
invariant. The present results and methods substantially improve upon previous
work. In particular, the continuity properties of the representation are shown
to be a consequence of the net structure, and surmised cohomological problems
in the construction of the representation are resolved by demonstrating that,
for the Poincare group, continuous reflection maps are restrictions of
continuous homomorphisms.Comment: 20 pages; change of title, reference added; version as to appear in
Commun. Math. Phy
Geometric modular action and spontaneous symmetry breaking
We study spontaneous symmetry breaking for field algebras on Minkowski space
in the presence of a condition of geometric modular action (CGMA) proposed
earlier as a selection criterion for vacuum states on general space-times. We
show that any internal symmetry group must commute with the representation of
the Poincare group (whose existence is assured by the CGMA) and each
translation-invariant vector is also Poincare invariant. The subspace of these
vectors can be centrally decomposed into pure invariant states and the CGMA
holds in the resulting sectors. As positivity of the energy is not assumed,
similar results may be expected to hold for other space--times.Comment: Dedicated to the memory of Siegfried Schlieder. 17 pages, no figures.
Revised version: simplified arguments and improved results; as to appear in
Annales H. Poincar
Further Representations of the Canonical Commutation Relations
We construct a new class of representations of the canonical commutation
relations, which generalizes previously known classes. We perturb the
infinitesimal generator of the initial Fock representation (i.e. the free
quantum field) by a function of the field which is square-integrable with
respect to the associated Gaussian measure. We characterize which such
perturbations lead to representations of the canonical commutation relations.
We provide conditions entailing the irreducibility of such representations,
show explicitly that our class of representations subsumes previously studied
classes, and give necessary and sufficient conditions for our representations
to be unitarily equivalent, resp. quasi-equivalent, with Fock, coherent or
quasifree representations
Quantum Probability Theory
The mathematics of classical probability theory was subsumed into classical
measure theory by Kolmogorov in 1933. Quantum theory as nonclassical
probability theory was incorporated into the beginnings of noncommutative
measure theory by von Neumann in the early thirties, as well. To precisely this
end, von Neumann initiated the study of what are now called von Neumann
algebras and, with Murray, made a first classification of such algebras into
three types. The nonrelativistic quantum theory of systems with finitely many
degrees of freedom deals exclusively with type I algebras. However, for the
description of further quantum systems, the other types of von Neumann algebras
are indispensable. The paper reviews quantum probability theory in terms of
general von Neumann algebras, stressing the similarity of the conceptual
structure of classical and noncommutative probability theories and emphasizing
the correspondence between the classical and quantum concepts, though also
indicating the nonclassical nature of quantum probabilistic predictions. In
addition, differences between the probability theories in the type I, II and
III settings are explained. A brief description is given of quantum systems for
which probability theory based on type I algebras is known to be insufficient.
These illustrate the physical significance of the previously mentioned
differences.Comment: 28 pages, LaTeX, typos removed and some minor modifications for
clarity and accuracy made. This is the version to appear in Studies in the
History and Philosophy of Modern Physic
Local Primitive Causality and the Common Cause Principle in Quantum Field Theory
If \{A(V)\} is a net of local von Neumann algebras satisfying standard axioms
of algebraic relativistic quantum field theory and V_1 and V_2 are spacelike
separated spacetime regions, then the system (A(V_1),A(V_2),\phi) is said to
satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of
projections A \in A(V_1), B \in A(V_2) correlated in the normal state \phi
there exists a projection C belonging to a von Neumann algebra associated with
a spacetime region V contained in the union of the backward light cones of V_1
and V_2 and disjoint from both V_1 and V_2, a projection having the properties
of a Reichenbachian common cause of the correlation between A and B. It is
shown that if the net has the local primitive causality property then every
local system (A(V_1),A(V_2),\phi) with a locally normal and locally faithful
state \phi and open bounded V_1 and V_2 satisfies the Weak Reichenbach's Common
Cause Principle.Comment: 14 pages, Late
Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools
We provide a brief overview of the basic tools and concepts of quantum field
theoretical scattering theory. This article is commissioned by the Encyclopedia
of Mathematical Physics, edited by J.-P. Francoise, G. Naber and T.S. Tsun, to
be published by the Elsevier publishing house.Comment: 14 pages, no figure
Covariant and quasi-covariant quantum dynamics in Robertson-Walker space-times
We propose a canonical description of the dynamics of quantum systems on a
class of Robertson-Walker space-times. We show that the worldline of an
observer in such space-times determines a unique orbit in the local conformal
group SO(4,1) of the space-time and that this orbit determines a unique
transport on the space-time. For a quantum system on the space-time modeled by
a net of local algebras, the associated dynamics is expressed via a suitable
family of ``propagators''. In the best of situations, this dynamics is
covariant, but more typically the dynamics will be ``quasi-covariant'' in a
sense we make precise. We then show by using our technique of ``transplanting''
states and nets of local algebras from de Sitter space to Robertson-Walker
space that there exist quantum systems on Robertson-Walker spaces with
quasi-covariant dynamics. The transplanted state is locally passive, in an
appropriate sense, with respect to this dynamics.Comment: 21 pages, 1 figur
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