386 research outputs found
Spectral noncommutative geometry and quantization: a simple example
We explore the relation between noncommutative geometry, in the spectral
triple formulation, and quantum mechanics. To this aim, we consider a dynamical
theory of a noncommutative geometry defined by a spectral triple, and study its
quantization. In particular, we consider a simple model based on a finite
dimensional spectral triple (A, H, D), which mimics certain aspects of the
spectral formulation of general relativity. We find the physical phase space,
which is the space of the onshell Dirac operators compatible with A and H. We
define a natural symplectic structure over this phase space and construct the
corresponding quantum theory using a covariant canonical quantization approach.
We show that the Connes distance between certain two states over the algebra A
(two ``spacetime points''), which is an arbitrary positive number in the
classical noncommutative geometry, turns out to be discrete in the quantum
theory, and we compute its spectrum. The quantum states of the noncommutative
geometry form a Hilbert space K. D is promoted to an operator *D on the direct
product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization
of the family of the triples (A, H, D).Comment: 7 pages, no figure
The uses of Connes and Kreimer's algebraic formulation of renormalization theory
We show how, modulo the distinction between the antipode and the "twisted" or
"renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes
the proofs of equivalence of the (corrected) Dyson-Salam,
Bogoliubov-Parasiuk-Hepp and Zimmermann procedures for renormalizing Feynman
amplitudes. We discuss the outlook for a parallel simplification of
computations in quantum field theory, stemming from the same algebraic
approach.Comment: 15 pages, Latex. Minor changes, typos fixed, 2 references adde
Gravity coupled with matter and foundation of non-commutative geometry
We first exhibit in the commutative case the simple algebraic relations
between the algebra of functions on a manifold and its infinitesimal length
element . Its unitary representations correspond to Riemannian metrics and
Spin structure while is the Dirac propagator ds = \ts \!\!---\!\! \ts =
D^{-1} where is the Dirac operator. We extend these simple relations to
the non commutative case using Tomita's involution . We then write a
spectral action, the trace of a function of the length element in Planck units,
which when applied to the non commutative geometry of the Standard Model will
be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian
coupled to gravity. The internal fluctuations of the non commutative geometry
are trivial in the commutative case but yield the full bosonic sector of SM
with all correct quantum numbers in the slightly non commutative case. The
group of local gauge transformations appears spontaneously as a normal subgroup
of the diffeomorphism group.Comment: 30 pages, Plain Te
Fluctuation Operators and Spontaneous Symmetry Breaking
We develop an alternative approach to this field, which was to a large extent
developed by Verbeure et al. It is meant to complement their approach, which is
largely based on a non-commutative central limit theorem and coordinate space
estimates. In contrast to that we deal directly with the limits of -point
truncated correlation functions and show that they typically vanish for provided that the respective scaling exponents of the fluctuation
observables are appropriately chosen. This direct approach is greatly
simplified by the introduction of a smooth version of spatial averaging, which
has a much nicer scaling behavior and the systematic developement of Fourier
space and energy-momentum spectral methods. We both analyze the regime of
normal fluctuations, the various regimes of poor clustering and the case of
spontaneous symmetry breaking or Goldstone phenomenon.Comment: 30 pages, Latex, a more detailed discussion in section 7 as to
possible scaling behavior of l-point function
A Class of Bicovariant Differential Calculi on Hopf Algebras
We introduce a large class of bicovariant differential calculi on any quantum
group , associated to -invariant elements. For example, the deformed
trace element on recovers Woronowicz' calculus. More
generally, we obtain a sequence of differential calculi on each quantum group
, based on the theory of the corresponding braided groups . Here
is any regular solution of the QYBE.Comment: 16 page
Polynomial rings of the chiral models
Via explicit diagonalization of the chiral fusion matrices, we
discuss the possibility of representing the fusion ring of the chiral SU(N)
models, at level K=2, by a polynomial ring in a single variable when is odd
and by a polynomial ring in two variables when is even.Comment: 10 pages, LaTex (ioplppt.sty
Carnot-Caratheodory metric and gauge fluctuation in Noncommutative Geometry
Gauge fields have a natural metric interpretation in terms of horizontal
distance. The latest, also called Carnot-Caratheodory or subriemannian
distance, is by definition the length of the shortest horizontal path between
points, that is to say the shortest path whose tangent vector is everywhere
horizontal with respect to the gauge connection. In noncommutative geometry all
the metric information is encoded within the Dirac operator D. In the classical
case, i.e. commutative, Connes's distance formula allows to extract from D the
geodesic distance on a riemannian spin manifold. In the case of a gauge theory
with a gauge field A, the geometry of the associated U(n)-vector bundle is
described by the covariant Dirac operator D+A. What is the distance encoded
within this operator ? It was expected that the noncommutative geometry
distance d defined by a covariant Dirac operator was intimately linked to the
Carnot-Caratheodory distance dh defined by A. In this paper we precise this
link, showing that the equality of d and dh strongly depends on the holonomy of
the connection. Quite interestingly we exhibit an elementary example, based on
a 2 torus, in which the noncommutative distance has a very simple expression
and simultaneously avoids the main drawbacks of the riemannian metric (no
discontinuity of the derivative of the distance function at the cut-locus) and
of the subriemannian one (memory of the structure of the fiber).Comment: published version with additional figures to make the proof more
readable. Typos corrected in this ultimate versio
General Relativity in terms of Dirac Eigenvalues
The eigenvalues of the Dirac operator on a curved spacetime are
diffeomorphism-invariant functions of the geometry. They form an infinite set
of ``observables'' for general relativity. Recent work of Chamseddine and
Connes suggests that they can be taken as variables for an invariant
description of the gravitational field's dynamics. We compute the Poisson
brackets of these eigenvalues and find them in terms of the energy-momentum of
the eigenspinors and the propagator of the linearized Einstein equations. We
show that the eigenspinors' energy-momentum is the Jacobian matrix of the
change of coordinates from metric to eigenvalues. We also consider a minor
modification of the spectral action, which eliminates the disturbing huge
cosmological term and derive its equations of motion. These are satisfied if
the energy momentum of the trans Planckian eigenspinors scale linearly with the
eigenvalue; we argue that this requirement approximates the Einstein equations.Comment: 6 pages, RevTe
Magnetic Field Measurement with Ground State Alignment
Observational studies of magnetic fields are crucial. We introduce a process
"ground state alignment" as a new way to determine the magnetic field direction
in diffuse medium. The alignment is due to anisotropic radiation impinging on
the atom/ion. The consequence of the process is the polarization of spectral
lines resulting from scattering and absorption from aligned atomic/ionic
species with fine or hyperfine structure. The magnetic field induces precession
and realign the atom/ion and therefore the polarization of the emitted or
absorbed radiation reflects the direction of the magnetic field. The atoms get
aligned at their low levels and, as the life-time of the atoms/ions we deal
with is long, the alignment induced by anisotropic radiation is susceptible to
extremely weak magnetic fields (G). In fact,
the effects of atomic/ionic alignment were studied in the laboratory decades
ago, mostly in relation to the maser research. Recently, the atomic effect has
been already detected in observations from circumstellar medium and this is a
harbinger of future extensive magnetic field studies. A unique feature of the
atomic realignment is that they can reveal the 3D orientation of magnetic
field. In this article, we shall review the basic physical processes involved
in atomic realignment. We shall also discuss its applications to
interplanetary, circumstellar and interstellar magnetic fields. In addition,
our research reveals that the polarization of the radiation arising from the
transitions between fine and hyperfine states of the ground level can provide a
unique diagnostics of magnetic fields in the Epoch of Reionization.Comment: 30 pages, 12 figures, chapter in Lecture Notes in Physics "Magnetic
Fields in Diffuse Media". arXiv admin note: substantial text overlap with
arXiv:1203.557
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