203 research outputs found
Notions of Infinity in Quantum Physics
In this article we will review some notions of infiniteness that appear in
Hilbert space operators and operator algebras. These include proper
infiniteness, Murray von Neumann's classification into type I and type III
factors and the class of F{/o} lner C*-algebras that capture some aspects of
amenability. We will also mention how these notions reappear in the description
of certain mathematical aspects of quantum mechanics, quantum field theory and
the theory of superselection sectors. We also show that the algebra of the
canonical anti-commutation relations (CAR-algebra) is in the class of F{/o}
lner C*-algebras.Comment: 11 page
Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
In a recent paper Bender and Mannheim showed that the unequal-frequency
fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in
which the energy eigenvalues are real and bounded below, the Hilbert-space
inner product is positive definite, and time evolution is unitary. Central to
that analysis was the recognition that the Hamiltonian of the
model is PT symmetric. This Hamiltonian was mapped to a conventional
Dirac-Hermitian Hamiltonian via a similarity transformation whose form was
found exactly. The present paper explores the equal-frequency limit of the same
model. It is shown that in this limit the similarity transform that was used
for the unequal-frequency case becomes singular and that becomes a
Jordan-block operator, which is nondiagonalizable and has fewer energy
eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart.
Thus, the equal-frequency PT theory emerges as a distinct realization of
quantum mechanics. The quantum mechanics associated with this Jordan-block
Hamiltonian can be treated exactly. It is shown that the Hilbert space is
complete with a set of nonstationary solutions to the Schr\"odinger equation
replacing the missing stationary ones. These nonstationary states are needed to
establish that the Jordan-block Hamiltonian of the equal-frequency
Pais-Uhlenbeck model generates unitary time evolution.Comment: 39 pages, 0 figure
Twisted duality of the CAR-Algebra
We give a complete proof of the twisted duality property M(q)'= Z M(q^\perp)
Z* of the (self-dual) CAR-Algebra in any Fock representation. The proof is
based on the natural Halmos decomposition of the (reference) Hilbert space when
two suitable closed subspaces have been distinguished. We use modular theory
and techniques developed by Kato concerning pairs of projections in some
essential steps of the proof.
As a byproduct of the proof we obtain an explicit and simple formula for the
graph of the modular operator. This formula can be also applied to fermionic
free nets, hence giving a formula of the modular operator for any double cone.Comment: 32 pages, Latex2e, to appear in Journal of Mathematical Physic
Breaking axi-symmetry in stenotic flow lowers the critical transition Reynolds number
Flow through a sinuous stenosis with varying degrees of non-axisymmetric shape variations and at Reynolds number ranging from 250 to 750 is investigated using direct numerical simulation (DNS) and global linear stability analysis. At low Reynolds numbers (Re < 390), the flow is always steady and symmetric for an axisymmetric geometry. Two steady state solutions are obtained when the Reynolds number is increased: a symmetric steady state and an eccentric, non-axisymmetric steady state. Either one can be obtained in the DNS depending on the initial condition. A linear global stability analysis around the symmetric and non-axisymmetric steady state reveals that both flows are linearly stable for the same Reynolds number, showing that the first bifurcation from symmetry to antisymmetry is subcritical. When the Reynolds number is increased further, the symmetric state becomes linearly unstable to an eigenmode, which drives the flow towards the nonaxisymmetric state. The symmetric state remains steady up to Re = 713, while the non-axisymmetric state displays regimes of periodic oscillations for Re ≥ 417 and intermittency for Re & 525. Further, an offset of the stenosis throat is introduced through the eccentricity parameter E. When eccentricity is increased from zero to only 0.3% of the pipe diameter, the bifurcation Reynolds number decreases by more than 50%, showing that it is highly sensitive to non-axisymmetric shape variations. Based on the resulting bifurcation map and its dependency on E, we resolve the discrepancies between previous experimental and computational studies. We also present excellent agreement between our numerical results and previous experimental resultsThis is the author accepted manuscript. The final version is available from AIP via http://dx.doi.org/10.1063/1.493453
Relativity and the low energy nd Ay puzzle
We solve the Faddeev equation in an exactly Poincare invariant formulation of
the three-nucleon problem. The dynamical input is a relativistic
nucleon-nucleon interaction that is exactly on-shell equivalent to the high
precision CDBonn NN interaction. S-matrix cluster properties dictate how the
two-body dynamics is embedded in the three-nucleon mass operator. We find that
for neutron laboratory energies above 20 MeV relativistic effects on Ay are
negligible. For energies below 20 MeV dynamical effects lower the nucleon
analyzing power maximum slightly by 2% and Wigner rotations lower it further up
to 10 % increasing thus disagreement between data and theory. This indicates
that three-nucleon forces must provide an even larger increase of the Ay
maximum than expected up to now.Comment: 29 pages, 2 ps figure
Ground state representations of loop algebras
Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point in
S^1 and identifying the real line with the punctured circle, we consider the
subalgebra Sg of Lg of rapidly decreasing elements on R. We classify the
translation-invariant 2-cocycles on Sg. We show that the ground state
representation of Sg is unique for each cocycle. These ground states correspond
precisely to the vacuum representations of Lg.Comment: 22 pages, no figur
Entanglement, Haag-duality and type properties of infinite quantum spin chains
We consider an infinite spin chain as a bipartite system consisting of the
left and right half-chain and analyze entanglement properties of pure states
with respect to this splitting. In this context we show that the amount of
entanglement contained in a given state is deeply related to the von Neumann
type of the observable algebras associated to the half-chains. Only the type I
case belongs to the usual entanglement theory which deals with density
operators on tensor product Hilbert spaces, and only in this situation
separable normal states exist. In all other cases the corresponding state is
infinitely entangled in the sense that one copy of the system in such a state
is sufficient to distill an infinite amount of maximally entangled qubit pairs.
We apply this results to the critical XY model and show that its unique ground
state provides a particular example for this type of entanglement.Comment: LaTeX2e, 34 pages, 1 figure (pstricks
First Order Relativistic Three-Body Scattering
Relativistic Faddeev equations for three-body scattering at arbitrary
energies are formulated in momentum space and in first order in the two-body
transition-operator directly solved in terms of momentum vectors without
employing a partial wave decomposition. Relativistic invariance is incorporated
within the framework of Poincare invariant quantum mechanics, and presented in
some detail.
Based on a Malfliet-Tjon type interaction, observables for elastic and
break-up scattering are calculated up to projectile energies of 1 GeV. The
influence of kinematic and dynamic relativistic effects on those observables is
systematically studied. Approximations to the two-body interaction embedded in
the three-particle space are compared to the exact treatment.Comment: 26 pages, 13 figure
Nonlinear theory of mirror instability near threshold
An asymptotic model based on a reductive perturbative expansion of the drift
kinetic and the Maxwell equations is used to demonstrate that, near the
instability threshold, the nonlinear dynamics of mirror modes in a magnetized
plasma with anisotropic ion temperatures involves a subcritical
bifurcation,leading to the formation of small-scale structures with amplitudes
comparable with the ambient magnetic field
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