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Register a new userManifold Topology, Observables and Gauge Group
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G.Morchio
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The relation between manifold topology, observables and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated to positions and displacements on the manifold. The guiding, physically motivated, principles are i) locality, i.e. the generating role of the algebras localized in small, topological trivial, regions, ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, iii) the exclusion of additional local degrees of freedom with respect to the Schroedinger representation. The locally normal representations of the resulting observable algebra are classified by unitary representations of the fundamental group of the manifold, which actually generate an observable, topological, subalgebra. The result is confronted with the standard approach based on the introduction of the universal covering ${tilde{cal M}}$ of $cal{M}$ and on the decomposition of $L^2({tilde{cal M}})$ according to the spectrum of the fundamental group, which plays the role of a gauge group. It is shown that in this way one obtains all the representations of the observables iff the fundamental group is amenable. The implications on the observability of the Permutation Group in Particle Statistics are discussed.
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In a spacetime divided into two regions $U_1$ and $U_2$ by a hypersurface $Sigma$, a perturbation of the field in $U_1$ is coupled to perturbations in $U_2$ by means of the holographic imprint that it leaves on $Sigma$. The linearized gluing field equation constrains perturbations on the two sides of a dividing hypersurface, and this linear operator may have a nontrivial null space. A nontrivial perturbation of the field leaving a holographic imprint on a dividing hypersurface which does not affect perturbations on the other side should be considered physically irrelevant. This consideration, together with a locality requirement, leads to the notion of gauge equivalence in Lagrangian field theory over confined spacetime domains. Physical observables in a spacetime domain $U$ can be calculated integrating (possibly non local) gauge invariant conserved currents on hypersurfaces such that $partial Sigma subset partial U$. The set of observables of this type is sufficient to distinguish gauge inequivalent solutions. The integral of a conserved current on a hypersurface is sensitive only to its homology class $[Sigma]$, and if $U$ is homeomorphic to a four ball the homology class is determined by its boundary $S = partial Sigma$. We will see that a result of Anderson and Torre implies that for a class of theories including vacuum General Relativity all local observables are holographic in the sense that they can be written as integrals of over the two dimensional surface $S$. However, non holographic observables are needed to distinguish between gauge inequivalent solutions.
Quantum Darwinism posits that information becomes objective whenever multiple observers indirectly probe a quantum system by each measuring a fraction of the environment. It was recently shown that objectivity of observables emerges generically from the mathematical structure of quantum mechanics, whenever the system of interest has finite dimensions and the number of environment fragments is large [F. G. S. L. Brand~ao, M. Piani, and P. Horodecki, Nature Commun. 6, 7908 (2015)]. Despite the importance of this result, it necessarily excludes many practical systems of interest that are infinite-dimensional, including harmonic oscillators. Extending the study of Quantum Darwinism to infinite dimensions is a nontrivial task: we tackle it here by using a modified diamond norm, suitable to quantify the distinguishability of channels in infinite dimensions. We prove two theorems that bound the emergence of objectivity, first for finite energy systems, and then for systems that can only be prepared in states with an exponential energy cut-off. We show that the latter class of states includes any bounded-energy subset of single-mode Gaussian states.
We consider a construction of observables by using methods of supersymmetric field theories. In particular, we give an extension of AKSZ-type observables using the Batalin-Vilkovisky structure of AKSZ theories to a formal global version with methods of formal geometry. We will consider the case where the AKSZ theory is split which will give an explicit construction for formal vector fields on base and fiber within the formal global action. Moreover, we consider the example of formal global generalized Wilson surface observables whose expectation values are invariants of higher-dimensional knots by using $BF$ field theory. These constructions give rise to interesting global gauge conditions such as the differential Quantum Master Equation and further extensions.
We analize the impact of two-loop renormalization group equations on the $SU(3)_ctimes SU(2)_wtimes U(1)_Y$ gauge couplings unification in various supersymmetric theories. In general the presence of superfields in higher representation than the doublet spoil the gauge couplings unification at one-loop. The situation is more interesting when the renormalization group equations are calculated at two-loop. In this case we show that the unification of the gauge couplings can be achieved for models with triplet superfield(s). In the analysis of the models with triplet superfield(s) we show that the dimensionless couplings do not have a Landau pole in their evolution at high energies but they run to a nontrivial ultraviolet fixed point.
In the instanton ensemble the fermionic zero modes collectivize and break chiral symmetry. Recent studies of resulting zero mode zone confirm its very small width and overall importance for lattice simulations. Confinement however has been related with completely different topological objects, the magnetic monopoles. Instanton constituents -- instanton dyons, discovered at nonzero holonomy by Pierre van Baal and others -- are able to explain both confinement and chiral symmetry breaking. The talk summarizes recent works deriving the instanton-dyon mutual interactions, and statistical studies of their ensemble. At high density the screening is robust enough to do it analytically, in the mean-field-type approach: we call this limit Dense Dyonic Plasma (DDP). Above $T_c$ the classical interaction between the dyons induce strong correlations and should be studied by direct numerical simulations. Those works are now in progress.
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