Hausdorff dimensions and gauge theories (original) (raw)
Scale-dependent Hausdorff dimensions in 2d gravity
Physics Letters B, 2014
By appropriate scaling of coupling constants a one-parameter family of ensembles of two-dimensional geometries is obtained, which interpolates between the ensembles of (generalized) causal dynamical triangulations and ordinary dynamical triangulations. We study the fractal properties of the associated continuum geometries and identify both global and local Hausdorff dimensions.
International Journal of Modern Physics A, 1986
In order to make it operationally accessible, it is proposed that the notion of the dimension of space-time be based on measure-theoretic concepts, thus admitting the possibility of noninteger dimensions. It is found then, that the Hausdorff covering procedure is operationally unrealizable because of the inherent finite space-time resolution of any real experiment. We therefore propose to define an operational dimension which, due to the quantum nature of the coverings, is smaller than the idealized Hausdorff dimension. As a consequence of the dimension of space-time less than four, relativistic quantum field theory becomes finite. Also, the radiative corrections of perturbation theory are sensitive on the actual value of the dimension 4–ε. Present experimental results and standard theoretical predictions for the electromagnetic moment of the electron seem to suggest a nonvanishing value for ε.
Geometry of non-Hausdorff spaces and its significance for physics
Journal of Mathematical Physics, 2011
Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be far-reaching and illuminating. Examples of situations in which the Hausdorff relation is of the total type, i.e., when it identifies all points of the considered space, are the space of Penrose tilings and space-times of some cosmological models with strong curvature singularities. With every Hausdorff relation a groupoid can be associated, and a convolutive algebra defined on it allows one to analyze the space that otherwise would remain intractable. The regular representation of this algebra in a bundle of Hilbert spaces leads to a von Neumann algebra of random operators. In this way, a probabilistic description (in a generalized sense) naturally takes over when the concept of point looses its meaning. In this situation counterparts of the position and m...
Arithmetical Investigations, 2008
In Chap. 7 we study the representation GL d+1 (Zp) → U (H) where H = L 2 (P d (Zp)). The commutant of GL d+1 (Zp) is generated by the functions on Ω d := B d,1 \GL d+1 /B d,1. Note that the measure on the space Ω d is induced from the Haar measure on GL d+1 (Zp) and is given by the β-measure with the appropriate parameters. Ω d parameterizes the relative position of two lines in P d (Zp) which is given by the "angle" between them, a real number in the real case and an integer in the p-adic cases. Remember that for real prime η GL d+1 (Zη) = O d+1 if η is real, U d+1 if η is complex, B d,1 (Zη) = O d × O1 if η is real, U d × U1 if η is complex. Here O d is the orthogonal group and U d is the unitary group of size d × d. Then, in the real case, we get a finite angle sin θ between two lines. In the p-adic case, we get an integer N. Hence we have
The notion of a random gauge and its interpretation
Quantum Studies: Mathematics and Foundations
We introduce the concept of a random gauge. We propose two distinct types of random gauge that can be defined based on the concept of phase noise in scattering theory. In the context of quantum physics, we discuss a variety of possible realizations of this concept that can be connected to various Aharonov-other effects and make some connections with relativity as well.
Intrinsic and extrinsic geometry of random surfaces
Physics Letters B, 1992
We prove that the extrinsic Hausdorff dimension is always greater than or equal to the intrinsic Hausdorff dimension in models of triangulated random surfaces with action which is quadratic in the separation of vertices. We furthermore derive a few naive scaling relations which relate the intrinsic Hausdorff dimension to other critical exponents. These relations suggest that the intrinsic Hausdorff dimension is infinite if the susceptibility does not diverge at the critical point.
Stochastic quantization of gauge theories
Annals of Physics, 1982
The equivalence between a scalar quantum field theory in D dimensions and its classical counterpart in D + 2 dimensions which is coupled to an external random source with Gaussian correlations was observed by previous authors. This stochastic quantization is extended to gauge theories. The proof exploits the supersymmetry formalism suggested by Parisi and Sourlas.
Gauge Couplings and Group Dimensions in the Standard Model
Eprint Arxiv Hep Th 0107144, 2001
The gauge field term in the Standard Model Lagrangian is slightly rewritten, suggesting that the three gauge couplings have absorbed factors which depend on the dimensions of the corresponding gauge groups. The ratios of the physical couplings may turn out to be dominated by these factors, with deviations due to quantum corrections.
Self-consistent nonperturbative anomalous dimensions
Journal of Physics A: Mathematical and General, 2003
A self-consistent treatment of two and three point functions in models with trilinear interactions forces them to have opposite anomalous dimensions. We indicate how the anomalous dimension can be extracted non-perturbatively by solving and suitably truncating the topologies of the full Dyson-Schwinger set of equations. The first step requires a sensible ansatz for the full vertex part, which conforms to first order perturbation theory at least. We model this vertex to obtain typical transcendental relations between anomalous dimension and coupling constant g which coincide with known results to order g 4 .