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A gauge-invariant UV-IR mixing and the corresponding phase transition for U(1) fields on the fuzzy sphere
Get PDFFrom a string theory point of view the most natural gauge action on the fuzzy sphere S_L^2 is the Alekseev-Recknagel-Schomerus action which is a particular combination of the Yang-Mills action and the Chern-Simons term. The differential calculus on the fuzzy sphere is 3−dimensional and thus the field content of this model consists of a 2-dimensional gauge field together with a scalar fluctuation normal to the sphere. For U(1) gauge theory we compute the quadratic effective action and shows explicitly that the tadpole diagrams and the vacuum polarization tensor contain a gauge-invariant UV-IR mixing in the continuum limit L→∞ where L is the matrix size of the fuzzy sphere. In other words the quantum U(1) effective action does not vanish in the commutative limit and a noncommutative anomaly survives. We compute the scalar effective potential and prove the gauge-fixing-independence of the limiting model L = ∞ and then show explicitly that the one-loop result predicts a first order phase transition which was observed recently in simulation. The one-loop result for the U(1) theory is exact in this limit. It is also argued that if we add a large mass term for the scalar mode the UV-IR mixing will be completely removed from the gauge sector. It is found in this case to be confined to the scalar sector only. This is in accordance with the large L analysis of the model. Finally we show that the phase transition becomes harder to reach starting from small couplings when we increase M
Noncommutative U(1) Gauge Theory As a Non-Linear Sigma Model
Get PDFNoncommutative U(1) gauge theory in 4−dimensions is shown to be equivalent in some scaling limit to an ordinary non-linear sigma model in 2−dimensions . The model in this regime is solvable and the corresponding exact beta function is found. We also show that classical U(n) gauge theory on R^(d−2) × R_θ^2 can be approximated by a sequence of ordinary (d − 2)−dimensional Georgi-Glashow models with gauge groups U(n(L + 1)) where L + 1 is the matrix size of the regularized noncommutative plane R_θ^2
Large deviations provide good approximation to queueing system with dynamic routing
Get PDFWe consider a system with two infinite-buffer FCFS servers (of speed one). The arrivals processes are three independent Poisson flows Ξ_i , of rates λ_i, i = 0, 1, 2, each with IID task service times. The tasks from Ξ_i are directed to server i, i = 1, 2 (dedicated traffic). The tasks from Ξ_0 are directed to the server that has the shorter workload in the buffer at the time of arrival (opportunistic traffic). We compare the analytical data for the large deviation (LD) probabilities for the virtual waiting time in flow Ξ_0 and empercial delay freqencies from simulations
A matrix phase for the ϕ⁴ scalar field on the fuzzy sphere
Get PDFThe critical properties of the real φ^4 scalar field theory are studied numerically on the fuzzy sphere. The fuzzy sphere is a matrix (non commutative) discretisation of the algebra of functions on the usual two dimensional sphere. It is also one of the simplest examples of a non commutative space to study field theory on. Aside from the usual disordered and uniform phases present in the commutative scalar field theory, we find and discuss in detail a new phase with spontaneously broken rotational invariance, called matrix phase because the geometry of the fuzzy sphere, as expressed by the kinetic term, becomes negligible there. This gives some further insight on the effect of UV—IR mixing, the unusual behaviour which arises naturally when taking the commutative limit of a non commutative field theory
Short distance behaviour of the effective string
Get PDFWe study the Polyakov loop correlator in the (2+1) dimensional Z 2 gauge model. An algorithm that we have presented recently, allows us to reach high precision results for a large range of distances and temperatures, giving us the opportunity to test predictions of the effective Nambu–Goto string model. Here we focus on the regime of low temperatures and small distances. In contrast to the high temperature, large distance regime, we find that our numerical results are not well described by the two loop-prediction of the Nambu–Goto model. In addition we compare our data with those for the SU(2) and SU(3) gauge models in (2+1) dimensions obtained by other authors. We generalize the result of Lüscher and Weisz for a boundary term in the interquark potential to the finite temperature case
Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance
Get PDFA covariant approach towards a theory of deformations is developed to examine both the first and second variation of the Helfrich-Canham Hamiltonian — quadratic in extrinsic curvature — which describes fluid vesicles at mesoscopic scales. Deformations are decomposed into tangential and normal components; At first order, tangential deformations may always be identified with a reparametrization; at second order, they differ. The relationship between tangential deformations and reparametrizations, as well as the coupling between tangential and normal deformations, is examined at this order for both the metric and the extrinsic curvature tensors. Expressions for the expansion to second order in deformations of geometrical invariants constructed with these tensors are obtained; in particular, the expansion of the Hamiltonian to this order about an equilibrium is considered. Our approach applies as well to any geometrical model for membranes
Quantum Black Holes: the Event Horizon as a Fuzzy Sphere
Get PDFModeling the event horizon of a black hole by a fuzzy sphere leads us to modify some suggestions in the literature concerning black hole mass spectra. We derive a formula for the mass spectrum of quantum black holes in terms of four integers which define the area, angular momentum, electric and magnetic charge of the black hole. Although the event horizon becomes a commutative sphere in the classical limit a vestige of the quantum theory still persists in that the event horizon stereographically projects onto the non-commutative plane. We also suggest how the classical bounds on extremal black holes might be modified in the quantum theory
Large deviations for the local particle densities
Get PDFWe analyze the relations between the large deviation principle of the “local” particle densities of the x− and k−spaces respectively. Here the k−space means the space of momentums (the Fourier transform counterpart of the x− space). This study gives new insights on the results of papers [2], where the authors have found the corresponding large deviation principle of the local particle density in the x− space. In particular, for a very large class of stable Hamiltonians we show that the “local” particle densities (x− and k−spaces) are equal to each other from the point of view of the large deviation principle. In other words, the “local” particle densities in the x− and k−spaces are in this case exponentially equivalent [1]. Applying this result to the specific case of the Perfect Bose Gas, we found an alternative proof to the one done in [2]
The Random Walk in Generalised Quantum Theory
Get PDFOne can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we “quantize” the classical random walk by finding, subject to a certain condition of “strong positivity”, the most general Markovian, translationally invariant “decoherence functional” with nearest neighbor transitions