30 research outputs found

### Complex structures and the Elie Cartan approach to the theory of spinors

Each isometric complex structure on a 2$\ell$-dimensional euclidean space $E$
corresponds to an identification of the Clifford algebra of $E$ with the
canonical anticommutation relation algebra for $\ell$ ( fermionic) degrees of
freedom. The simple spinors in the terminology of E.~Cartan or the pure spinors
in the one of C. Chevalley are the associated vacua. The corresponding states
are the Fock states (i.e. pure free states), therefore, none of the above
terminologies is very good.Comment: 10

### Spectral stochastic processes arising in quantum mechanical models with a non-L2 ground state

A functional integral representation is given for a large class of quantum
mechanical models with a non--L2 ground state. As a prototype the particle in a
periodic potential is discussed: a unique ground state is shown to exist as a
state on the Weyl algebra, and a functional measure (spectral stochastic
process) is constructed on trajectories taking values in the spectrum of the
maximal abelian subalgebra of the Weyl algebra isomorphic to the algebra of
almost periodic functions. The thermodynamical limit of the finite volume
functional integrals for such models is discussed, and the superselection
sectors associated to an observable subalgebra of the Weyl algebra are
described in terms of boundary conditions and/or topological terms in the
finite volume measures.Comment: 15 pages, Plain Te

### Bose-Einstein condensate and Spontaneous Breaking of Conformal Symmetry on Killing Horizons

Local scalar QFT (in Weyl algebraic approach) is constructed on degenerate
semi-Riemannian manifolds corresponding to Killing horizons in spacetime.
Covariance properties of the $C^*$-algebra of observables with respect to the
conformal group PSL(2,\bR) are studied.It is shown that, in addition to the
state studied by Guido, Longo, Roberts and Verch for bifurcated Killing
horizons, which is conformally invariant and KMS at Hawking temperature with
respect to the Killing flow and defines a conformal net of von Neumann
algebras, there is a further wide class of algebraic (coherent) states
representing spontaneous breaking of PSL(2,\bR) symmetry. This class is
labeled by functions in a suitable Hilbert space and their GNS representations
enjoy remarkable properties. The states are non equivalent extremal KMS states
at Hawking temperature with respect to the residual one-parameter subgroup of
PSL(2,\bR) associated with the Killing flow. The KMS property is valid for
the two local sub algebras of observables uniquely determined by covariance and
invariance under the residual symmetry unitarily represented. These algebras
rely on the physical region of the manifold corresponding to a Killing horizon
cleaned up by removing the unphysical points at infinity (necessary to describe
the whole PSL(2,\bR) action).Each of the found states can be interpreted as a
different thermodynamic phase, containing Bose-Einstein condensate,for the
considered quantum field. It is finally suggested that the found states could
describe different black holes.Comment: 36 pages, 1 figure. Formula of condensate energy density modified.
Accepted for pubblication in Journal of Mathematical Physic

### Transition probabilities between quasifree states

We obtain a general formula for the transition probabilities between any
state of the algebra of the canonical commutation relations (CCR-algebra) and a
squeezed quasifree state. Applications of this formula are made for the case of
multimode thermal squeezed states of quantum optics using a general canonical
decomposition of the correlation matrix valid for any quasifree state. In the
particular case of a one mode CCR-algebra we show that the transition
probability between two quasifree squeezed states is a decreasing function of
the geodesic distance between the points of the upper half plane representing
these states. In the special case of the purification map it is shown that the
transition probability between the state of the enlarged system and the product
state of real and fictitious subsystems can be a measure for the entanglement.Comment: 13 pages, REVTeX, no figure

### Full regularity for a C*-algebra of the Canonical Commutation Relations. (Erratum added)

The Weyl algebra,- the usual C*-algebra employed to model the canonical
commutation relations (CCRs), has a well-known defect in that it has a large
number of representations which are not regular and these cannot model physical
fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs
of a countably dimensional symplectic space (S,B) and such that its
representation set is exactly the full set of regular representations of the
CCRs. This construction uses Blackadar's version of infinite tensor products of
nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalised
group algebra, explained below) for the \sigma-representation theory of the
abelian group S where \sigma(.,.):=e^{iB(.,.)/2}.
As an easy application, it then follows that for every regular representation
of the Weyl algebra of (S,B) on a separable Hilbert space, there is a direct
integral decomposition of it into irreducible regular representations (a known
result).
An Erratum for this paper is added at the end.Comment: An erratum was added to the original pape

### The Physical Principles of Quantum Mechanics. A critical review

The standard presentation of the principles of quantum mechanics is
critically reviewed both from the experimental/operational point and with
respect to the request of mathematical consistency and logical economy. A
simpler and more physically motivated formulation is discussed. The existence
of non commuting observables, which characterizes quantum mechanics with
respect to classical mechanics, is related to operationally testable
complementarity relations, rather than to uncertainty relations. The drawbacks
of Dirac argument for canonical quantization are avoided by a more geometrical
approach.Comment: Bibliography and section 2.1 slightly improve

### Thermal States in Conformal QFT. II

We continue the analysis of the set of locally normal KMS states w.r.t. the
translation group for a local conformal net A of von Neumann algebras on the
real line. In the first part we have proved the uniqueness of KMS state on
every completely rational net. In this second part, we exhibit several
(non-rational) conformal nets which admit continuously many primary KMS states.
We give a complete classification of the KMS states on the U(1)-current net and
on the Virasoro net Vir_1 with the central charge c=1, whilst for the Virasoro
net Vir_c with c>1 we exhibit a (possibly incomplete) list of continuously many
primary KMS states. To this end, we provide a variation of the
Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework:
if there is an inclusion of split nets A in B and A is the fixed point of B
w.r.t. a compact gauge group, then any locally normal, primary KMS state on A
extends to a locally normal, primary state on B, KMS w.r.t. a perturbed
translation. Concerning the non-local case, we show that the free Fermi model
admits a unique KMS state.Comment: 36 pages, no figure. Dedicated to Rudolf Haag on the occasion of his
90th birthday. The final version is available under Open Access. This paper
contains corrections to the Araki-Haag-Kaster-Takesaki theorem (and to a
proof of the same theorem in the book by Bratteli-Robinson). v3: a reference
correcte

### Abelian duality on globally hyperbolic spacetimes

We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism in a manifestly covariant way and without the assumption of compact Cauchy surfaces. We construct semi-classical configuration spaces and corresponding presymplectic Abelian groups of observables, which are quantized by the CCR-functor to the category of C*-algebras. We demonstrate explicitly how duality is implemented as a natural isomorphism between quantum field theories. We apply this formalism to develop a fully covariant quantum theory of self-dual fields

### Dynamical locality of the free scalar field

Dynamical locality is a condition on a locally covariant physical theory,
asserting that kinematic and dynamical notions of local physics agree. This
condition was introduced in [arXiv:1106.4785], where it was shown to be closely
related to the question of what it means for a theory to describe the same
physics on different spacetimes. In this paper, we consider in detail the
example of the free minimally coupled Klein--Gordon field, both as a classical
and quantum theory (using both the Weyl algebra and a smeared field approach).
It is shown that the massive theory obeys dynamical locality, both classically
and in quantum field theory, in all spacetime dimensions $n\ge 2$ and allowing
for spacetimes with finitely many connected components. In contrast, the
massless theory is shown to violate dynamical locality in any spacetime
dimension, in both classical and quantum theory, owing to a rigid gauge
symmetry. Taking this into account (equivalently, working with the massless
current) dynamical locality is restored in all dimensions $n\ge 2$ on connected
spacetimes, and in all dimensions $n\ge 3$ if disconnected spacetimes are
permitted. The results on the quantized theories are obtained using general
results giving conditions under which dynamically local classical symplectic
theories have dynamically local quantizations.Comment: 34pp, LaTeX2e. Version to appear in Annales Henri Poincar