1,047 research outputs found
Remarks on the GNS Representation and the Geometry of Quantum States
It is shown how to introduce a geometric description of the algebraic
approach to the non-relativistic quantum mechanics. It turns out that the GNS
representation provides not only symplectic but also Hermitian realization of a
`quantum Poisson algebra'. We discuss alternative Hamiltonian structures
emerging out of different GNS representations which provide a natural setting
for quantum bi-Hamiltonian systems.Comment: 20 page
From the Jordan product to Riemannian geometries on classical and quantum states
The Jordan product on the self-adjoint part of a finite-dimensional
-algebra is shown to give rise to Riemannian metric
tensors on suitable manifolds of states on , and the covariant
derivative, the geodesics, the Riemann tensor, and the sectional curvature of
all these metric tensors are explicitly computed. In particular, it is proved
that the Fisher--Rao metric tensor is recovered in the Abelian case, that the
Fubini--Study metric tensor is recovered when we consider pure states on the
algebra of linear operators on a finite-dimensional
Hilbert space , and that the Bures--Helstrom metric tensors is
recovered when we consider faithful states on .
Moreover, an alternative derivation of these Riemannian metric tensors in terms
of the GNS construction associated to a state is presented. In the case of pure
and faithful states on , this alternative geometrical
description clarifies the analogy between the Fubini--Study and the
Bures--Helstrom metric tensor.Comment: 32 pages. Minor improvements. References added. Comments are welcome
Thermal equilibrium states for quantum fields on non-commutative spacetimes
Fully Poincar\'e covariant quantum field theories on non-commutative Moyal
Minkowski spacetime so far have been considered in their vacuum
representations, i.e. at zero temperature. Here we report on work in progress
regarding their thermal representations, corresponding to physical states at
non-zero temperature, which turn out to be markedly different from both,
thermal representations of quantum field theory on commutative Minkowski
spacetime, and such representations of non-covariant quantum field theory on
Moyal Minkowski space with a fixed deformation matrix.Comment: 20 pages. Contribution to the proceedings of the conference 'Quantum
Mathematical Physics', Regensburg, 29.09.-02.10.201
Inequivalent coherent state representations in group field theory
In this paper we propose an algebraic formulation of group field theory and
consider non-Fock representations based on coherent states. We show that we can
construct representations with infinite number of degrees of freedom on compact
base manifolds. We also show that these representations break translation
symmetry. Since such representations can be regarded as quantum gravitational
systems with an infinite number of fundamental pre-geometric building blocks,
they may be more suitable for the description of effective geometrical phases
of the theory
The universal C*-algebra of the electromagnetic field
A universal C*-algebra of the electromagnetic field is constructed. It is
represented in any quantum field theory which incorporates electromagnetism and
expresses basic features of this field such as Maxwell's equations, Poincar\'e
covariance and Einstein causality. Moreover, topological properties of the
field resulting from Maxwell's equations are encoded in the algebra, leading to
commutation relations with values in its center. The representation theory of
the algebra is discussed with focus on vacuum representations, fixing the
dynamics of the field.Comment: 17 pages, 1 figure; v2: minor corrections, version as to appear in
Lett. Math. Phys. Dedicated to the memory of D. Kastler and J.E. Roberts; v3
improvement of layou
Background independent quantizations: the scalar field II
We are concerned with the issue of quantization of a scalar field in a
diffeomorphism invariant manner. We apply the method used in Loop Quantum
Gravity. It relies on the specific choice of scalar field variables referred to
as the polymer variables. The quantization, in our formulation, amounts to
introducing the `quantum' polymer *-star algebra and looking for positive
linear functionals, called states. Assumed in our paper homeomorphism
invariance allows to derive the complete class of the states. They are
determined by the homeomorphism invariant states defined on the CW-complex
*-algebra. The corresponding GNS representations of the polymer *-algebra and
their self-adjoint extensions are derived, the equivalence classes are found
and invariant subspaces characterized. In the preceding letter (the part I) we
outlined those results. Here, we present the technical details.Comment: 51 pages, LaTeX, no figures, revised versio
Algebraic Quantum Theory on Manifolds: A Haag-Kastler Setting for Quantum Geometry
Motivated by the invariance of current representations of quantum gravity
under diffeomorphisms much more general than isometries, the Haag-Kastler
setting is extended to manifolds without metric background structure. First,
the causal structure on a differentiable manifold M of arbitrary dimension
(d+1>2) can be defined in purely topological terms, via cones (C-causality).
Then, the general structure of a net of C*-algebras on a manifold M and its
causal properties required for an algebraic quantum field theory can be
described as an extension of the Haag-Kastler axiomatic framework.
An important application is given with quantum geometry on a spatial slice
within the causally exterior region of a topological horizon H, resulting in a
net of Weyl algebras for states with an infinite number of intersection points
of edges and transversal (d-1)-faces within any neighbourhood of the spatial
boundary S^2.Comment: 15 pages, Latex; v2: several corrections, in particular in def. 1 and
in sec.
A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II : Supmech and Quantum Systems
Supmech, which is noncommutative Hamiltonian mechanics \linebreak (NHM)
(developed in paper I) with two extra ingredients : positive observable valued
measures (PObVMs) [which serve to connect state-induced expectation values and
classical probabilities] and the `CC condition' [which stipulates that the sets
of observables and pure states be mutually separating] is proposed as a
universal mechanics potentially covering all physical phenomena. It facilitates
development of an autonomous formalism for quantum mechanics. Quantum systems,
defined algebraically as supmech Hamiltonian systems with non-supercommutative
system algebras, are shown to inevitably have Hilbert space based realizations
(so as to accommodate rigged Hilbert space based Dirac bra-ket formalism),
generally admitting commutative superselection rules. Traditional features of
quantum mechanics of finite particle systems appear naturally. A treatment of
localizability much simpler and more general than the traditional one is given.
Treating massive particles as localizable elementary quantum systems, the
Schrdinger wave functions with traditional Born interpretation appear
as natural objects for the description of their pure states and the
Schrdinger equation for them is obtained without ever using a
classical Hamiltonian or Lagrangian. A provisional set of axioms for the
supmech program is given.Comment: 55 pages; some modifications in text; improved treatment of
topological aspects and of Noether invariants; results unchange
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