Bosonization and Current Algebra of Spinning Strings

We write down a general geometric action principle for spinning strings in $d$-dimensional Minkowski space, which is formulated without the use of Grassmann coordinates. Instead, it is constructed in terms of the pull-back of a left invariant Maurer-Cartan form on the $d$-dimensional Poincar\'e group to the world sheet. The system contains some interesting special cases. Among them are the Nambu string (as well as, null and tachyionic strings) where the spin vanishes, and also the case of a string with a spin current - but no momentum current. We find the general form for the Virasoro generators, and show that they are first class constraints in the Hamiltonian formulation of the theory. The current algebra associated with the momentum and angular momentum densities are shown, in general, to contain rather complicated anomaly terms which obstruct quantization. As expected, the anomalies vanish when one specializes to the case of the Nambu string, and there one simply recovers the algebra associated with the Poincar\'e loop group. We speculate that there exist other cases where the anomalies vanish, and that these cases give the bosonization of the known pseudoclassical formulations of spinning strings.Comment: Latex file, 29 p

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-Pauli Transitions From Spacetime Noncommutativity

There are good reasons to suspect that spacetime at Planck scales is noncommutative. Typically this noncommutativity is controlled by fixed "vectors" or "tensors" with numerical entries. For the Moyal spacetime, it is the antisymmetric matrix $\theta_{\mu\nu}$. In approaches enforcing Poincar\'e invariance, these deform or twist the method of (anti-)symmetrization of identical particle state vectors. We argue that the earth's rotation and movements in the cosmos are "sudden" events to Pauli-forbidden processes. They induce (twisted) bosonic components in state vectors of identical spinorial particles in the presence of a twist. These components induce non-Pauli transitions. From known limits on such transitions, we infer that the energy scale for noncommutativity is $\gtrsim 10^{24}\textrm{TeV}$. This suggests a new energy scale beyond Planck scale.Comment: 11 pages, 1 table, Slightly revised for clarity

Topology Change for Fuzzy Physics: Fuzzy Spaces as Hopf Algebras

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.Comment: LaTeX, 13 pages, v3: minor additions, added references, v4: corrected typos, to appear in IJMP

Duality in Fuzzy Sigma Models

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.Comment: Latex, 9 pages; misprints correcte

Covariant Quantum Fields on Noncommutative Spacetimes

A spinless covariant field $\phi$ on Minkowski spacetime \M^{d+1} obeys the relation $U(a,\Lambda)\phi(x)U(a,\Lambda)^{-1}=\phi(\Lambda x+a)$ where $(a,\Lambda)$ is an element of the Poincar\'e group \Pg and $U:(a,\Lambda)\to U(a,\Lambda)$ is its unitary representation on quantum vector states. It expresses the fact that Poincar\'e transformations are being unitary implemented. It has a classical analogy where field covariance shows that Poincar\'e transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the *-operation are in conflict so that there are no covariant Voros fields compatible with *, a result we found earlier. The notion of Drinfel'd twist underlying much of the preceding discussion is extended to discrete abelian and nonabelian groups such as the mapping class groups of topological geons. For twists involving nonabelian groups the emergent spacetimes are nonassociative.Comment: 20 page

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 $S_N$. 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

Bringing Up a Quantum Baby

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac's `Quantum Mechanics', then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving vectors in the domain of the Hamiltonian: The ``probability densities'' (hermitean forms) \psi^\dagger \chi for \psi,\chi in this domain generate an algebra from which the classical configuration space with its topology (and with further refinements of the axiom, its C^K and C^infinity structures) can be reconstructed using Gel'fand - Naimark theory. Classical topology is an attribute of only certain quantum states for these axioms, the configuration space emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.Comment: 23 pages, 2 figures ( ref. updated, no other changes