A non-perturbative study of 4d U(1) non-commutative gauge theory -- the fate of one-loop instability

Recent perturbative studies show that in 4d non-commutative spaces, the trivial (classically stable) vacuum of gauge theories becomes unstable at the quantum level, unless one introduces sufficiently many fermionic degrees of freedom. This is due to a negative IR-singular term in the one-loop effective potential, which appears as a result of the UV/IR mixing. We study such a system non-perturbatively in the case of pure U(1) gauge theory in four dimensions, where two directions are non-commutative. Monte Carlo simulations are performed after mapping the regularized theory onto a U(N) lattice gauge theory in d=2. At intermediate coupling strength, we find a phase in which open Wilson lines acquire non-zero vacuum expectation values, which implies the spontaneous breakdown of translational invariance. In this phase, various physical quantities obey clear scaling behaviors in the continuum limit with a fixed non-commutativity parameter $\theta$, which provides evidence for a possible continuum theory. The extent of the dynamically generated space in the non-commutative directions becomes finite in the above limit, and its dependence on $\theta$ is evaluated explicitly. We also study the dispersion relation. In the weak coupling symmetric phase, it involves a negative IR-singular term, which is responsible for the observed phase transition. In the broken phase, it reveals the existence of the Nambu-Goldstone mode associated with the spontaneous symmetry breaking.Comment: 29 pages, 23 figures, references adde

The Fuzzy Disc

We introduce a finite dimensional matrix model approximation to the algebra of functions on a disc based on noncommutative geometry. The algebra is a subalgebra of the one characterizing the noncommutative plane with a * product and depends on two parameters N and theta. It is composed of functions which decay exponentially outside a disc. In the limit in which the size of the matrices goes to infinity and the noncommutativity parameter goes to zero the disc becomes sharper. We introduce a Laplacian defined on the whole algebra and calculate its eigenvalues. We also calculate the two--points correlation function for a free massless theory (Green's function). In both cases the agreement with the exact result on the disc is very good already for relatively small matrices. This opens up the possibility for the study of field theories on the disc with nonperturbative methods. The model contains edge states, a fact studied in a similar matrix model independently introduced by Balachandran, Gupta and Kurkcuoglu.Comment: 17 pages, 8 figures, references added and correcte

Probing the fuzzy sphere regularisation in simulations of the 3d \lambda \phi^4 model

We regularise the 3d \lambda \phi^4 model by discretising the Euclidean time and representing the spatial part on a fuzzy sphere. The latter involves a truncated expansion of the field in spherical harmonics. This yields a numerically tractable formulation, which constitutes an unconventional alternative to the lattice. In contrast to the 2d version, the radius R plays an independent r\^{o}le. We explore the phase diagram in terms of R and the cutoff, as well as the parameters m^2 and \lambda. Thus we identify the phases of disorder, uniform order and non-uniform order. We compare the result to the phase diagrams of the 3d model on a non-commutative torus, and of the 2d model on a fuzzy sphere. Our data at strong coupling reproduce accurately the behaviour of a matrix chain, which corresponds to the c=1-model in string theory. This observation enables a conjecture about the thermodynamic limit.Comment: 31 pages, 15 figure

Classical Gravity on Fuzzy Space-Time

A review is made of recent efforts to find relations between the commutation relations which define a noncommutative geometry and the gravitational field which remains as a shadow in the commutative limit.Comment: Lecture given at the 30th International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow, Germany, August 27-31, 1996; 11 Pages LaTe

Spectral geometry with a cut-off: topological and metric aspects

Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutative geometry, and focus on Connes distance associated to a spectral triple (A, H, D). A high momentum (short distance) cut-off is implemented by the action of a projection P on the Dirac operator D and/or on the algebra A. This action induces two new distances. We individuate conditions making them equivalent to the original distance. We also study the Gromov-Hausdorff limit of the set of truncated states, first for compact quantum metric spaces in the sense of Rieffel, then for arbitrary spectral triples. To this aim, we introduce a notion of "state with finite moment of order 1" for noncommutative algebras. We then focus on the commutative case, and show that the cut-off induces a minimal length between points, which is infinite if P has finite rank. When P is a spectral projection of $D$, we work out an approximation of points by non-pure states that are at finite distance from each other. On the circle, such approximations are given by Fejer probability distributions. Finally we apply the results to Moyal plane and the fuzzy sphere, obtained as Berezin quantization of the plane and the sphere respectively.Comment: Reference added. Minor corrections. Published version. 38 pages, 2 figures. Journal of Geometry and Physics 201

The One-Plaquette Model Limit of NC Gauge Theory in 2D

It is found that noncommutative U(1) gauge field on the fuzzy sphere S^2_N is equivalent in the quantum theory to a commutative 2-dimensional U(N) gauge field on a lattice with two plaquettes in the axial gauge A_1=0. This quantum equivalence holds in the fuzzy sphere-weak coupling phase in the limit of infinite mass of the scalar normal component of the gauge field. The doubling of plaquettes is a natural consequence of the model and it is reminiscent of the usual doubling of points in Connes standard model. In the continuum large N limit the plaquette variable W approaches the identity 1_{2N} and as a consequence the model reduces to a simple matrix model which can be easily solved. We compute the one-plaquette critical point and show that it agrees with the observed value \bar{\alpha}_*=3.35. We compute the quantum effective potential and the specific heat for U(1) gauge field on the fuzzy sphere S^2_{N} in the 1/N expansion using this one-plaquette model. In particular the specific heat per one degree of freedom was found to be equal to 1 in the fuzzy sphere-weak coupling phase of the gauge field which agrees with the observed value 1 seen in Monte Carlo simulation. This value of 1 comes precisely because we have 2 plaquettes approximating the NC U(1) gauge field on the fuzzy sphere.Comment: 46 pages,4 figures,typos corrected,references adde

Fredkin Gates for Finite-valued Reversible and Conservative Logics

The basic principles and results of Conservative Logic introduced by Fredkin and Toffoli on the basis of a seminal paper of Landauer are extended to d-valued logics, with a special attention to three-valued logics. Different approaches to d-valued logics are examined in order to determine some possible universal sets of logic primitives. In particular, we consider the typical connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As a result, some possible three-valued and d-valued universal gates are described which realize a functionally complete set of fundamental connectives.Comment: 57 pages, 10 figures, 16 tables, 2 diagram