194 research outputs found
Spin Dirac Operators on the Fuzzy 2-Sphere
The spin 1/2 Dirac operator and its chirality operator on the fuzzy 2-sphere
can be constructed using the Ginsparg-Wilson(GW) algebra
[arxiv:hep-th/0511114]. This construction actually exists for any spin on
, and have continuum analogues as well on the commutative sphere
or on . This is a remarkable fact and has no known analogue in
higher dimensional Minkowski spaces. We study such operators on and the
commutative and formulate criteria for the existence of the limit from
the former to the latter. This singles out certain fuzzy versions of these
operators as the preferred Dirac operators. We then study the spin 1 Dirac
operator of this preferred type and its chirality on the fuzzy 2-sphere and
formulate its instanton sectors and their index theory. The method to
generalize this analysis to any spin is also studied in detail.Comment: 16 pages, 1 tabl
Quantum Geons and Noncommutative Spacetimes
Physical considerations strongly indicate that spacetime at Planck scales is
noncommutative. A popular model for such a spacetime is the Moyal plane. The
Poincar\`e group algebra acts on it with a Drinfel'd-twisted coproduct. But the
latter is not appropriate for more complicated spacetimes such as those
containing the Friedman-Sorkin (topological) geons. They have rich
diffeomorphism groups and in particular mapping class groups, so that the
statistics groups for N identical geons is strikingly different from the
permutation group . We generalise the Drinfel'd twist to (essentially)
generic groups including to finite and discrete ones and use it to modify the
commutative spacetime algebras of geons as well to noncommutative algebras. The
latter support twisted actions of diffeos of geon spacetimes and associated
twisted statistics. The notion of covariant fields for geons is formulated and
their twisted versions are constructed from their untwisted versions.
Non-associative spacetime algebras arise naturally in our analysis. Physical
consequences, such as the violation of Pauli principle, seem to be the outcomes
of such nonassociativity.
The richness of the statistics groups of identical geons comes from the
nontrivial fundamental groups of their spatial slices. As discussed long ago,
extended objects like rings and D-branes also have similar rich fundamental
groups. This work is recalled and its relevance to the present quantum geon
context is pointed out.Comment: 41 page
Dirac operator on the q-deformed Fuzzy sphere and Its spectrum
The q-deformed fuzzy sphere is the algebra of
dim. matrices, covariant with respect to the adjoint action
of \uq and in the limit , it reduces to the fuzzy sphere
. We construct the Dirac operator on the q-deformed fuzzy
sphere- using the spinor modules of \uq. We explicitly obtain
the zero modes and also calculate the spectrum for this Dirac operator. Using
this Dirac operator, we construct the \uq invariant action for the spinor
fields on which are regularised and have only finite modes. We
analyse the spectrum for both being root of unity and real, showing
interesting features like its novel degeneracy. We also study various limits of
the parameter space (q, N) and recover the known spectrum in both fuzzy and
commutative sphere.Comment: 19 pages, 6 figures, more references adde
Twisted Poincare Invariance, Noncommutative Gauge Theories and UV-IR Mixing
In the absence of gauge fields, quantum field theories on the
Groenewold-Moyal (GM) plane are invariant under a twisted action of the
Poincare group if they are formulated following [1, 2, 3, 4, 5, 6]. In that
formulation, such theories also have no UV-IR mixing [7]. Here we investigate
UV-IR mixing in gauge theories with matter following the approach of [3, 4]. We
prove that there is UV-IR mixing in the one-loop diagram of the S-matrix
involving a coupling between gauge and matter fields on the GM plane, the gauge
field being nonabelian. There is no UV-IR mixing if it is abelian.Comment: 11 pages, 3 figure
Non-Linear Sigma Model on the Fuzzy Supersphere
In this note we develop fuzzy versions of the supersymmetric non-linear sigma
model on the supersphere S^(2,2). In hep-th/0212133 Bott projectors have been
used to obtain the fuzzy CP^1 model. Our approach utilizes the use of
supersymmetric extensions of these projectors. Here we obtain these (super)
-projectors and quantize them in a fashion similar to the one given in
hep-th/0212133. We discuss the interpretation of the resulting model as a
finite dimensional matrix model.Comment: 11 pages, LaTeX, corrected typo
UV-IR Mixing in Non-Commutative Plane
Poincar\'e-invariant quantum field theories can be formulated on
non-commutative planes if the coproduct on the Poincar\'e group is suitably
deformed \cite{Dimitrijevic:2004rf, Chaichian:2004za}.(See also especially
Oeckl \cite{Oeckl:1999jun},\cite{Oeckl:2000mar} and Grosse et
al.\cite{Grosse:2001mar}) As shown in \cite{Balachandran:2005eb}, this
important result of these authors implies modification of free field
commutation and anti-commutation relations and striking phenomenological
consequences such as violations of Pauli principle
\cite{Balachandran:2005eb,Bal3}. In this paper we prove that with these
modifications, UV-IR mixing disappears to all orders in perturbation theory
from the -Matrix. This result is in agreement with the previous results of
Oeckl \cite{Oeckl:2000mar}.Comment: Minor Changes in text and abstract, important references adde
Discrete Time Evolution and Energy Nonconservation in Noncommutative Physics
Time-space noncommutativity leads to quantisation of time and energy
nonconservation when time is conjugate to a compact spatial direction like a
circle. In this context energy is conserved only modulo some fixed unit. Such a
possibility arises for example in theories with a compact extra dimension with
which time does not commute. The above results suggest striking
phenomenological consequences in extra dimensional theories and elsewhere. In
this paper we develop scattering theory for discrete time translations. It
enables the calculation of transition probabilities for energy nonconserving
processes and has a central role both in formal theory and phenomenology.
We can also consider space-space noncommutativity where one of the spatial
directions is a circle. That leads to the quantisation of the remaining spatial
direction and conservation of momentum in that direction only modulo some fixed
unit, as a simple adaptation of the results in this paper shows.Comment: 17 pages, LaTex; minor correction
Quantum Aspects of the Noncommutative Sine-Gordon Model
In this paper, we first use semi-classical methods to study quantum field
theoretical aspects of the integrable noncommutative sine-Gordon model proposed
in [hep-th/0406065]. In particular, we examine the fluctuations at quadratic
order around the static kink solution using the background field method. We
derive equations of motion for the fluctuations and argue that at O(theta^2)
the spectrum of fluctuations remains essentially the same as that of the
corresponding commutative theory. We compute the one-loop two-point functions
of the sine-Gordon field and the additional scalar field present in the model
and exhibit logarithmic divergences, only some of which lead to UV/IR mixing.
We briefly discuss the one-loop renormalization in Euclidean signature and
comment on the obstacles in determining the noncommutativity corrections to the
quantum mass of the kink.Comment: 1+14 pages, 8 eps figures, Added references, Version to appear in
JHE
Quantum Fields on the Groenewold-Moyal Plane: C, P, T and CPT
We show that despite the inherent non-locality of quantum field theories on
the Groenewold-Moyal (GM) plane, one can find a class of , ,
and invariant theories. In particular, these are theories
without gauge fields or with just gauge fields and no matter fields. We also
show that in the presence of gauge fields, one can have a field theory where
the Hamiltonian is and invariant while the -matrix
violates and .
In non-abelian gauge theories with matter fields such as the electro-weak and
sectors of the standard model of particle physics, , ,
and the product of any pair of them are broken while
remains intact for the case . (Here , : coordinate functions,
constant.) When ,
it contributes to breaking also and . It is known that the
-matrix in a non-abelian theory depends on only through
. The -matrix is frame dependent. It breaks (the identity
component of the) Lorentz group. All the noncommutative effects vanish if the
scattering takes place in the center-of-mass frame, or any frame where
, but not otherwise. and are good symmetries of the theory in this special case.Comment: 18 pages, 1 figure, revised, 2 references adde
The Spin-Statistics Connection in Quantum Gravity
It is well-known that is spite of sharing some properties with conventional
particles, topological geons in general violate the spin-statistics theorem. On
the other hand, it is generally believed that in quantum gravity theories
allowing for topology change, using pair creation and annihilation of geons,
one should be able to recover this theorem. In this paper, we take an
alternative route, and use an algebraic formalism developed in previous work.
We give a description of topological geons where an algebra of "observables" is
identified and quantized. Different irreducible representations of this algebra
correspond to different kinds of geons, and are labeled by a non-abelian
"charge" and "magnetic flux". We then find that the usual spin-statistics
theorem is indeed violated, but a new spin-statistics relation arises, when we
assume that the fluxes are superselected. This assumption can be proved if all
observables are local, as is generally the case in physical theories. Finally,
we also show how our approach fits into conventional formulations of quantum
gravity.Comment: LaTeX file, 31 pages, 5 figure
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