29 research outputs found
Ultraviolet Finite Quantum Field Theory on Quantum Spacetime
We discuss a formulation of quantum field theory on quantum space time where
the perturbation expansion of the S-matrix is term by term ultraviolet finite.
The characteristic feature of our approach is a quantum version of the Wick
product at coinciding points: the differences of coordinates q_j - q_k are not
set equal to zero, which would violate the commutation relation between their
components. We show that the optimal degree of approximate coincidence can be
defined by the evaluation of a conditional expectation which replaces each
function of q_j - q_k by its expectation value in optimally localized states,
while leaving the mean coordinates (q_1 + ... + q_n)/n invariant.
The resulting procedure is to a large extent unique, and is invariant under
translations and rotations, but violates Lorentz invariance. Indeed, optimal
localization refers to a specific Lorentz frame, where the electric and
magnetic parts of the commutator of the coordinates have to coincide *).
Employing an adiabatic switching, we show that the S-matrix is term by term
finite. The matrix elements of the transfer matrix are determined, at each
order in the perturbative expansion, by kernels with Gaussian decay in the
Planck scale. The adiabatic limit and the large scale limit of this theory will
be studied elsewhere.
--
*) S. Doplicher, K. Fredenhagen, and J.E.Roberts, Commun. Math. Phys. 172,
187 (1995) [arXiv:hep-th/0303037]Comment: LaTeX (using amsmath, amssymb), 23 pages, 1 figure. Dedicated to
Rudolf Haag on the occasion of his 80th birthday. See also: hep-th/0303037,
hep-th/0201222. Second version: minor changes in exposition, two references
added. To appear on Commun. Math. Phy
Non Local Theories: New Rules for Old Diagrams
We show that a general variant of the Wick theorems can be used to reduce the
time ordered products in the Gell-Mann & Low formula for a certain class on non
local quantum field theories, including the case where the interaction
Lagrangian is defined in terms of twisted products.
The only necessary modification is the replacement of the
Stueckelberg-Feynman propagator by the general propagator (the ``contractor''
of Denk and Schweda)
D(y-y';tau-tau')= - i
(Delta_+(y-y')theta(tau-tau')+Delta_+(y'-y)theta(tau'-tau)), where the
violations of locality and causality are represented by the dependence of
tau,tau' on other points, besides those involved in the contraction. This leads
naturally to a diagrammatic expansion of the Gell-Mann & Low formula, in terms
of the same diagrams as in the local case, the only necessary modification
concerning the Feynman rules. The ordinary local theory is easily recovered as
a special case, and there is a one-to-one correspondence between the local and
non local contributions corresponding to the same diagrams, which is preserved
while performing the large scale limit of the theory.Comment: LaTeX, 14 pages, 1 figure. Uses hyperref. Symmetry factors added;
minor changes in the expositio
Field Theory on Noncommutative Spacetimes: Quasiplanar Wick Products
We give a definition of admissible counterterms appropriate for massive
quantum field theories on the noncommutative Minkowski space, based on a
suitable notion of locality. We then define products of fields of arbitrary
order, the so-called quasiplanar Wick products, by subtracting only such
admissible counterterms. We derive the analogue of Wick's theorem and comment
on the consequences of using quasiplanar Wick products in the perturbative
expansion.Comment: 22 pages, 2 figures, v2: minor changes, v3: minor changes, reference
adde
Propagators and Matrix Basis on Noncommutative Minkowski Space
We describe an analytic continuation of the Euclidean Grosse-Wulkenhaar and
LSZ models which defines a one-parameter family of duality covariant
noncommutative field theories interpolating between Euclidean and Minkowski
space versions of these models, and provides an alternative regularization to
the usual Feynman prescription. This regularization allows for a matrix model
representation of the field theories in terms of a complex generalization of
the usual basis of Landau wavefunctions. The corresponding propagators are
calculated and identified with the Feynman propagators of the field theories.
The regulated quantum field theories are shown to be UV/IR-duality covariant.
We study the asymptotics of the regularized propagators in position and matrix
space representations, and confirm that they generically possess a comparably
good decay behaviour as in the Euclidean case.Comment: 45 pages; v2: clarifying comments added; v3: further clarifying
comments added; Final version published in Physical Review
On the unitarity problem in space/time noncommutative theories
It is shown that the violation of unitarity observed in space/time
noncommutative field theories is due to an improper definition of quantum field
theory on noncommutative spacetime.Comment: 7 pages; typos corrected, sign convention changed on p.
Translation Invariance, Commutation Relations and Ultraviolet/Infrared Mixing
We show that the Ultraviolet/Infrared mixing of noncommutative field theories
with the Gronewold-Moyal product, whereby some (but not all) ultraviolet
divergences become infrared, is a generic feature of translationally invariant
associative products. We find, with an explicit calculation that the phase
appearing in the nonplanar diagrams is the one given by the commutator of the
coordinates, the semiclassical Poisson structure of the non commutative
spacetime. We do this with an explicit calculation for represented generic
products.Comment: 24 pages, 1 figur
Quantum Field Theory on Quantum Spacetime
Condensed account of the Lectures delivered at the Meeting on {\it
Noncommutative Geometry in Field and String Theory}, Corfu, September 18 - 20,
2005.Comment: 10 page
Minimal length in quantum space and integrations of the line element in Noncommutative Geometry
We question the emergence of a minimal length in quantum spacetime, comparing
two notions that appeared at various points in the literature: on the one side,
the quantum length as the spectrum of an operator L in the Doplicher
Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical
noncommutative spacetime; on the other side, Connes' spectral distance in
noncommutative geometry. Although on the Euclidean space the two notions merge
into the one of geodesic distance, they yield distinct results in the
noncommutative framework. In particular on the Moyal plane, the quantum length
is bounded above from zero while the spectral distance can take any real
positive value, including infinity. We show how to solve this discrepancy by
doubling the spectral triple. This leads us to introduce a modified quantum
length d'_L, which coincides exactly with the spectral distance d_D on the set
of states of optimal localization. On the set of eigenstates of the quantum
harmonic oscillator - together with their translations - d'_L and d_D coincide
asymptotically, both in the high energy and large translation limits. At small
energy, we interpret the discrepancy between d'_L and d_D as two distinct ways
of integrating the line element on a quantum space. This leads us to propose an
equation for a geodesic on the Moyal plane.Comment: 29 pages, 2 figures. Minor corrections to match the published versio
DFR Perturbative Quantum Field theory on Quantum Space Time, and Wick Reduction
We discuss the perturbative approach a` la Dyson to a quantum field theory
with nonlocal self-interaction :phi*...*phi:, according to Doplicher,
Fredenhagen and Roberts (DFR). In particular, we show that the Wick reduction
of non locally time--ordered products of Wick monomials can be performed as
usual, and we discuss a very simple Dyson diagram.Comment: 15 pages, pdf has active hyperlinks. To appear in the proceedings of
the conference on "Rigorous quantum Field Theory", held at Saclay on July
19-21, 2004, on the occasion of Jacques Bros' 70th birthda
Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime
We derive new space-time uncertainty relations (STUR) at the fundamental
Planck length from quantum mechanics and general relativity (GR), both in
flat and curved backgrounds. Contrary to claims present in the literature, our
approach suggests that no minimal uncertainty appears for lengths, but instead
for minimal space and four-volumes. Moreover, we derive a maximal absolute
value for the energy density. Finally, some considerations on possible
commutators among quantum operators implying our STUR are done.Comment: Final version published in "Class. Quantum Grav.