arXiv subjects
Publications and source records attributed to A. P. Balachandran.
We elaborate on the role of quantum statistics in twisted Poincare invariant theories. It is shown that, in order to have twisted Poincare group as the symmetry of a quantum theory, statistics must be twisted. It is also confirmed that the removal of UV-IR mixing (in the absence of gauge fields) in such theories is a natural consequence.
We give formulations of noncommutative two dimensional gravities in terms of noncommutative gauge theories. We survey their classical solutions and show that solutions of the corresponding commutative theories continue to be solutions in the noncommutative theories as well. We argue that the existence of ``twisted'' diffeomorphisms, recently introduced in hep-th/0504183, is crucial for this conclusion.
The N=1 SUSY on S^2 and its fuzzy finite-dimensional matrix version are known. The latter regulates quantum field theories, and seems suitable for numerical work and capable of higher dimensional generalizations. In this paper, we study their instanton sectors. They are SUSY generalizations of U(1) bundles on S^2 and their fuzzy versions, and can be characterized by $k\in\mathbb{Z}$, the SUSY Chern numbers. In the no-instanton sector (k=0), N=2 SUSY can be chirally realized, the 3 new N=2 generators anticommuting with the ``Dirac'' operator defining the free action. If $k\neq 0$, the Dirac operator has zero modes which form an N=1 supermultiplet and an atypical representation of N=2 SUSY. They break the chiral SUSY generators by the Fujikawa mechanism. We have not found this mechanism for SUSY breakdown in the literature. All these phenomena occur also on the supersphere SUSY, the graded commutative limit of the fuzzy model. We plan to discuss that as well in a later work.
The Groenewold-Moyal plane is the algebra A_θ(R^(d+1)) of functions on R^(d+1) with the star-product as the multiplication law, and the commutator [x_μ,x_ν] =i θ_{μν} between the coordinate functions. Chaichian et al. and Aschieri et al. have proved that the Poincare group acts as automorphisms on A_θ(R^(d+1))$ if the coproduct is deformed. (See also the prior work of Majid, Oeckl and Grosse et al). In fact, the diffeomorphism group with a deformed coproduct also does so according to the results of Aschieri et al. In this paper we show that for this new action, the Bose and Fermi commutation relations are deformed as well. Their potential applications to the quantum Hall effect are pointed out. Very striking consequences of these deformations are the occurrence of Pauli-forbidden energy levels and transitions. Such new effects are discussed in simple cases.
Poincaré-invariant quantum field theories can be formulated on non-commutative planes if the coproduct on the Poincaré 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 $S$-Matrix. This result is in agreement with the previous results of Oeckl \cite{Oeckl:2000mar}.
We show that topological superfluid strings/vortices with flux tubes exist in the color-flavor locked (CFL) phase of color superconductors. Using a Ginzburg-Landau free energy we find the configurations of these strings. These strings can form during the transition from the normal phase to the CFL phase at the core of very dense stars. We discuss an interesting scenario for a network of strings and its evolution at the core of dense stars.
Waves on ``commutative'' spacetimes like R^d are elements of the commutative algebra C^0(R^d) of functions on R^d. When C^0(R^d) is deformed to a noncommutative algebra {\cal A}_θ(R^d) with deformation parameter θ({\cal A}_0 (R^d) = C^0(R^d)), waves being its elements, are no longer complex-valued functions on R^d. Rules for their interpretation, such as measurement of their intensity, and energy, thus need to be stated. We address this task here. We then apply the rules to interference and diffraction for d \leq 4 and with time-space noncommutativity. Novel phenomena are encountered. Thus when the time of observation T is so brief that T \leq 2 θw, where w is the frequency of incident waves, no interference can be observed. For larger times, the interference pattern is deformed and depends on \frac{θw}{T}. It approaches the commutative pattern only when \frac{θw}{T} goes to 0. As an application, we discuss interference of star light due to cosmic strings.
We explore the consequences of time-space noncommutativity in the quantum mechanics of atoms and molecules, focusing on the Moyal plane with just time-space noncommutativity ($[\hat{x}_μ,\hat{x}_ν]=iθ_{μν}$, $θ_{0i}\neqq 0$, $θ_{ij}=0$). Space rotations and parity are not automorphisms of this algebra and are not symmetries of quantum physics. Still, when there are spectral degeneracies of a time-independent Hamiltonian on a commutative space-time which are due to symmetries, they persist when $θ_{0i}\neqq 0$; they do not depend at all on $θ_{0i}$. They give no clue about rotation and parity violation when $θ_{0i}\neqq 0$. The persistence of degeneracies for $θ_{0i}\neqq 0$ can be understood in terms of invariance under deformed noncommutative ``rotations'' and ``parity''. They are not spatial rotations and reflection. We explain such deformed symmetries. We emphasize the significance of time-dependent perturbations (for example, due to time-dependent electromagnetic fields) to observe noncommutativity. The formalism for treating transition processes is illustrated by the example of nonrelativistic hydrogen atom interacting with quantized electromagnetic field. In the tree approximation, the $2s\to 1s +γ$ transition for hydrogen is zero in the commutative case. As an example, we show that it is zero in the same approximation for $θ_{0i}\ne 0$. The importance of the deformed rotational symmetry is commented upon further using the decay $Z^0 \to 2γ$ as an example.
In this work quantum physics in noncommutative spacetime is developed. It is based on the work of Doplicher et al. which allows for time-space noncommutativity. The Moyal plane is treated in detail. In the context of noncommutative quantum mechanics, some important points are explored, such as the formal construction of the theory, symmetries, causality, simultaneity and observables. The dynamics generated by a noncommutative Schrodinger equation is studied. We prove in particular the following: suppose the Hamiltonian of a quantum mechanical particle on spacetime has no explicit time dependence, and the spatial coordinates commute in its noncommutative form (the only noncommutativity being between time and a space coordinate). Then the commutative and noncommutative versions of the Hamiltonian have identical spectra.
In previous work, we developed quantum physics on the Moyal plane with time-space noncommutativity, basing ourselves on the work of Doplicher et al.. Here we extend it to certain noncommutative versions of the cylinder, $\mathbb{R}^{3}$ and $\mathbb{R}\times S^{3}$. In all these models, only discrete time translations are possible, a result known before in the first two cases. One striking consequence of quantised time translations is that even though a time independent Hamiltonian is an observable, in scattering processes, it is conserved only modulo $\frac{2π}θ$, where $θ$ is the noncommutative parameter. (In contrast, on a one-dimensional periodic lattice of lattice spacing $a$ and length $L=Na$, only momentum mod $\frac{2π}{L}$ is observable (and can be conserved).) Suggestions for further study of this effect are made. Scattering theory is formulated and an approach to quantum field theory is outlined.
Nonlinear `sigma' models in two dimensions have BPS solitons which are solutions of self- and anti-self-duality constraints. In this paper, we find their analogues for fuzzy sigma models on fuzzy spheres which were treated in detail by us in earlier work. We show that fuzzy BPS solitons are quantized versions of `Bott projectors', and construct them explicitly. Their supersymmetric versions follow from the work of S. Kurkcuoglu.
Fuzzy spaces are obtained by quantizing adjoint orbits of compact semi-simple Lie groups. Fuzzy spheres emerge from quantizing S^2 and are associated with the group SU(2) in this manner. They are useful for regularizing quantum field theories and modeling spacetimes by non-commutative manifolds. We show that fuzzy spaces are Hopf algebras and in fact have more structure than the latter. They are thus candidates for quantum symmetries. Using their generalized Hopf algebraic structures, we can also model processes where one fuzzy space splits into several fuzzy spaces. For example we can discuss the quantum transition where the fuzzy sphere for angular momentum J splits into fuzzy spheres for angular momenta K and L.
This paper discusses the formulation of the non-commutative Chern-Simons (CS) theory where the spatial slice, an infinite strip, is a manifold with boundaries. As standard star products are not correct for such manifolds, the standard non-commutative CS theory is not also appropriate here. Instead we formulate a new finite-dimensional matrix CS model as an approximation to the CS theory on the strip. A work which has points of contact with ours is due to Lizzi, Vitale and Zampini where the authors obtain a description for the fuzzy disc. The gauge fields in our approach are operators supported on a subspace of finite dimension N+ηof the Hilbert space of eigenstates of a simple harmonic oscillator with N, η\in Z^+ and N \neq 0. This oscillator is associated with the underlying Moyal plane. The resultant matrix CS theory has a fuzzy edge. It becomes the required sharp edge when N and ηgoes to infinity in a suitable sense. The non-commutative CS theory on the strip is defined by this limiting procedure. After performing the canonical constraint analysis of the matrix theory, we find that there are edge observables in the theory generating a Lie algebra with properties similar to that of a non-abelian Kac-Moody algebra. Our study shows that there are (η+1)^2 abelian charges (observables) given by the matrix elements (\cal A_i)_{N-1 N-1} and (\cal A_i)_{nm} (where n or m \geq N) of the gauge fields, that obey certain standard canonical commutation relations. In addition, the theory contains three unique non-abelian charges, localized near the N^th level. We show that all non-abelian edge observables except these three can be constructed from the abelian charges above. Using the results of this analysis we discuss the large N and ηlimit.
The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of the Ginsparg-Wilson algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. The theory reproduces the integrated U(1)_A anomaly and index theory correctly.
The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact connected Lie groups and G is simple. An elementary discussion of the differential geometric and bundle theoretic aspects of G/H, including its projective modules and complex, Kaehler and Riemannian structures, is presented for this purpose. An attractive feature of our approach is that it transparently shows obstructions to spin- and spin_c-structures. When a manifold is spin_c and not spin, U(1) gauge fields have to be introduced in a particular way to define spinors. Likewise, for manifolds like SU(3)/SO(3), which are not even spin_c, we show that SU(2) and higher rank gauge fields have to be introduced to define spinors. This result has potential consequences for string theories if such manifolds occur as D-branes. The spectra and eigenstates of the Dirac operator on spheres S^n=SO(n+1)/SO(n), invariant under SO(n+1), are explicitly found. Aspects of our work overlap with the earlier research of Cahen et al..
In spacetime dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are Nambu-Goldstone modes described by fields with values in G/H. In two-dimensional spacetimes as well, models where fields take values in G/H are of considerable interest even though in that case there is no spontaneous breaking of continuous symmetries. We consider such models when the world sheet is a two-sphere and describe their fuzzy analogues for G=SU(N+1), H=S(U(N-1)xU(1)) ~ U(N) and G/H=CP^N. More generally our methods give fuzzy versions of continuum models on S^2 when the target spaces are Grassmannians and flag manifolds described by (N+1)x(N+1) projectors of rank =< (N+1)/2. These fuzzy models are finite-dimensional matrix models which nevertheless retain all the essential continuum topological features like solitonic sectors. They seem well-suited for numerical work.
A study of the relation between topology change, energy and Lie algebra representations for fuzzy geometry in connection to $M$-theory is presented. We encounter two different types of topology change, related to the different features of the Lie algebra representations appearing in the matrix models of $M$-theory. From these studies, we propose a new method of obtaining non-commutative solutions for the non-Abelian $D$-brane action found by Myers. This mechanism excludes one of the two topology changing processes previously found in other non-commutative solutions of many matrix-based models in $M$-theory i.e. in M(atrix) theory, Matrix string theory and non-Abelian $D$-brane physics.
Many years ago Friedman and Sorkin [1] established the existence of certain topological solitonic excitations in quantum gravity called (topological) geons. Geons can have quantum numbers like charge and can be tensorial or spinorial having integer or half-odd integer spin. A striking result is that geons can violate the canonical spin-statistics connection [2,3]. Such violation induces novel physical effects at low energies. The latter will be small since the geon mass is expected to be of the order of Plank mass. Nevertheless, these effects are very striking and include CPT and causality violations and distortion of the cosmic microwave spectrum. Interesting relations of geon dynamics to supersymmetry are also discussed.