56 research outputs found
The Yang-Mills equations on the universal cosmos
AbstractGlobal existence and regularity of solutions for the Yang-Mills equations on the universal cosmos M̃, which has the form R1 × S3 for each of an 8-parameter continuum of factorizations of M̃ as time × space, are treated by general methods. The Cauchy problem in the temporal gauge is globally soluble in its abstract evolutionary form with arbitrary data for the field ⊕ potential in L2,r(S3) ⊕ L2,r + 1(S3), where r is an integer >1 and L2,r denotes the class of sections whose first r derivatives are square-integrable; if r = 1, the problem is soluble locally in time. When r is 3 or more the solution is identifiable with a classical one; if infinite, the solution is in C∞(M̃). These results extend earlier work and approaches [1–5]. Solutions of the equations on Minkowski space-time M0 extend canonically (modulo gauge transformations) to solutions on M̃ provided their Cauchy data are moderately smooth and small near spatial infinity. Precise asymptotic structures for solutions on M0 follow, and in turn imply various decay estimates. Thus the energy in regions uniformly bounded in direction away from the light cone is O(¦x0¦−5), where x0 is the Minkowski time coordinate; analysis solely in M0 [8,9] earlier yielded the estimate O(¦x0¦−2) applicable to the region within the light cone. Similarly it follows that the action integral for a solution of the Yang-Mills equations in M0 is finite, in fact absolutely convergent
Singular operators on boson fields as forms on spaces of entire functions on Hilbert space
AbstractInvariant scales of entire analytic functions on Hilbert space are introduced and applied. Singular operators represented by sesquilinear forms on spaces of regular vectors are given explicit integral representations via kernels that are entire functions on the direct sum of the Hilbert space with its dual. The Weyl (or, exponentiated boson field) operators act smoothly and irreducibly on corresponding spaces of entire functions. Arbitrary symplectic operators on a single-particle Hilbert space are shown to be implementable on the corresponding boson field by appropriate generalized operators
Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration
Optimal hypercontractivity bounds for the fermion oscillator semigroup are
obtained. These are the fermion analogs of the optimal hypercontractivity
bounds for the boson oscillator semigroup obtained by Nelson. In the process,
several results of independent interest in the theory of non-commutative
integration are established. {}.Comment: 18 p., princeton/ecel/7-12-9
Testing axioms for Quantum Mechanics on Probabilistic toy-theories
In Ref. [1] one of the authors proposed postulates for axiomatizing Quantum
Mechanics as a "fair operational framework", namely regarding the theory as a
set of rules that allow the experimenter to predict future events on the basis
of suitable tests, having local control and low experimental complexity. In
addition to causality, the following postulates have been considered: PFAITH
(existence of a pure preparationally faithful state), and FAITHE (existence of
a faithful effect). These postulates have exhibited an unexpected theoretical
power, excluding all known nonquantum probabilistic theories. Later in Ref. [2]
in addition to causality and PFAITH, postulate LDISCR (local discriminability)
and PURIFY (purifiability of all states) have been considered, narrowing the
probabilistic theory to something very close to Quantum Mechanics. In the
present paper we test the above postulates on some nonquantum probabilistic
models. The first model, "the two-box world" is an extension of the
Popescu-Rohrlich model, which achieves the greatest violation of the CHSH
inequality compatible with the no-signaling principle. The second model "the
two-clock world" is actually a full class of models, all having a disk as
convex set of states for the local system. One of them corresponds to the "the
two-rebit world", namely qubits with real Hilbert space. The third model--"the
spin-factor"--is a sort of n-dimensional generalization of the clock. Finally
the last model is "the classical probabilistic theory". We see how each model
violates some of the proposed postulates, when and how teleportation can be
achieved, and we analyze other interesting connections between these postulate
violations, along with deep relations between the local and the non-local
structures of the probabilistic theory.Comment: Submitted to QIP Special Issue on Foundations of Quantum Informatio
Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field
The long-time asymptotics is analyzed for all finite energy solutions to a
model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the
nonlinearity concentrated at a single point: each finite energy solution
converges as time goes to plus or minus infinity to the set of all ``nonlinear
eigenfunctions'' of the form \psi(x)e\sp{-i\omega t}. The global attraction
is caused by the nonlinear energy transfer from lower harmonics to the
continuous spectrum and subsequent dispersive radiation.
We justify this mechanism by the following novel strategy based on inflation
of spectrum by the nonlinearity. We show that any omega-limit trajectory has
the time-spectrum in the spectral gap [-m,m] and satisfies the original
equation. This equation implies the key spectral inclusion for spectrum of the
nonlinear term. Then the application of the Titchmarsh Convolution Theorem
reduces the spectrum of each omega-limit trajectory to a single harmonic in
[-m,m].
The research is inspired by Bohr's postulate on quantum transitions and
Schroedinger's identification of the quantum stationary states to the nonlinear
eigenfunctions of the coupled U(1)-invariant Maxwell-Schroedinger and
Maxwell-Dirac equations.Comment: 29 pages, 1 figur
Modular Structure and Duality in Conformal Quantum Field Theory
Making use of a recent result of Borchers, an algebraic version of the
Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e.
the Tomita-Takesaki modular group associated with the von Neumann algebra of a
wedge region and the vacuum vector concides with the evolution given by the
rescaled pure Lorentz transformations preserving the wedge. A similar geometric
description is valid for the algebras associated with double cones. Moreover
essential duality holds on the Minkowski space , and Haag duality for double
cones holds provided the net of local algebras is extended to a pre-cosheaf on
the superworld , i.e. the universal covering of the Dirac-Weyl
compactification of . As a consequence a PCT symmetry exists for any
algebraic conformal field theory in even space-time dimension. Analogous
results hold for a Poincar\'e covariant theory provided the modular groups
corresponding to wedge algebras have the expected geometrical meaning and the
split property is satisfied. In particular the Poincar\'e representation is
unique in this case.Comment: 23 pages, plain TeX, TVM26-12-199
Unitary Representations of Unitary Groups
In this paper we review and streamline some results of Kirillov, Olshanski
and Pickrell on unitary representations of the unitary group \U(\cH) of a
real, complex or quaternionic separable Hilbert space and the subgroup
\U_\infty(\cH), consisting of those unitary operators for which g - \1
is compact. The Kirillov--Olshanski theorem on the continuous unitary
representations of the identity component \U_\infty(\cH)_0 asserts that they
are direct sums of irreducible ones which can be realized in finite tensor
products of a suitable complex Hilbert space. This is proved and generalized to
inseparable spaces. These results are carried over to the full unitary group by
Pickrell's Theorem, asserting that the separable unitary representations of
\U(\cH), for a separable Hilbert space \cH, are uniquely determined by
their restriction to \U_\infty(\cH)_0. For the classical infinite rank
symmetric pairs of non-unitary type, such as (\GL(\cH),\U(\cH)), we
also show that all separable unitary representations are trivial.Comment: 42 page
Cosmological spacetimes balanced by a scale covariant scalar field
A scale invariant, Weyl geometric, Lagrangian approach to cosmology is
explored, with a a scalar field phi of (scale) weight -1 as a crucial
ingredient besides classical matter \cite{Tann:Diss,Drechsler:Higgs}. For a
particularly simple class of Weyl geometric models (called {\em Einstein-Weyl
universes}) the Klein-Gordon equation for phi is explicitly solvable. In this
case the energy-stress tensor of the scalar field consists of a vacuum-like
term Lambda g_{mu nu} with variable coefficient Lambda, depending on matter
density and spacetime geometry, and of a dark matter like term. Under certain
assumptions on parameter constellations, the energy-stress tensor of the
phi-field keeps Einstein-Weyl universes in locally stable equilibrium. A short
glance at observational data, in particular supernovae Ia (Riess ea 2007),
shows interesting empirical properties of these models.Comment: 28 pages, 1 figure, accepted by Foundations of Physic
Multiplicativity of completely bounded p-norms implies a new additivity result
We prove additivity of the minimal conditional entropy associated with a
quantum channel Phi, represented by a completely positive (CP),
trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is
restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We
show that this follows from multiplicativity of the completely bounded norm of
Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten
p-norm on matrices; we also give an independent proof based on entropy
inequalities. Several related multiplicativity results are discussed and
proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map
and the corresponding completely bounded norm are achieved for positive
semi-definite matrices. Physical interpretations are considered, and a new
proof of strong subadditivity is presented.Comment: Final version for Commun. Math. Physics. Section 5.2 of previous
version deleted in view of the results in quant-ph/0601071 Other changes
mino
Contractions, deformations and curvature
The role of curvature in relation with Lie algebra contractions of the
pseudo-ortogonal algebras so(p,q) is fully described by considering some
associated symmetrical homogeneous spaces of constant curvature within a
Cayley-Klein framework. We show that a given Lie algebra contraction can be
interpreted geometrically as the zero-curvature limit of some underlying
homogeneous space with constant curvature. In particular, we study in detail
the contraction process for the three classical Riemannian spaces (spherical,
Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three
relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a
different perspective, we make use of quantum deformations of Lie algebras in
order to construct a family of spaces of non-constant curvature that can be
interpreted as deformations of the above nine spaces. In this framework, the
quantum deformation parameter is identified as the parameter that controls the
curvature of such "quantum" spaces.Comment: 17 pages. Based on the talk given in the Oberwolfach workshop:
Deformations and Contractions in Mathematics and Physics (Germany, january
2006) organized by M. de Montigny, A. Fialowski, S. Novikov and M.
Schlichenmaie
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