Gauge Symmetry and Transverse Symmetry Transformations in Gauge Theories

The transverse symmetry transformations associated with the normal symmetry transformations are proposed to build the transverse constraints on the basic vertices in gauge theories. I show that, while the BRST symmetry in non-Abelian gauge theory QCD (Quantum Chromodynamics) leads to the Slavnov-Taylor identity for the quark-gluon vertex which constrains the longitudinal part of thevertex, the transverse symmetry transformation associated with the BRST symmetry enables to derive the transverse Slavnov-Taylor identity for the quark-gluon vertex, which constrains the transverse part of the quark-gluon vertex from the gauge symmetry of QCD.     

Gauge Symmetry and Transverse Symmetry Transformations in Gauge Theories

The transverse symmetry transformations associated with the normal symmetry transformations are proposed to build the transverse constraints on the basic vertices in gauge theories. I show that, while the BRST symmetry in non-Abelian gauge theory QCD （Quantum Chromodynamics） leads to the Slavnov-Taylor identity for the quark-gluon vertex which constrains the longitudinal part of the vertex, the transverse symmetry transformation associated with the BRST symmetry enables to derive the transverse Slavnov-Taylor identity for the quark-gluon vertex, which constrains the transverse part of the quark-gluon vertex from the gauge symmetry of QCD.     

Gauge and Space-Time Symmetry Unification

Unification ideas suggest an integral treatment of fermion and boson spin andgauge-group degrees of freedom. Hence, a generalized quantum field equation,based on Dirac's, is proposed and investigated which contains gauge and flavorsymmetries, determines vector gauge field and fermion solution representations,and fixes their mode of interaction. The simplest extension of the theory with a6-dimensional Clifford algebra has an SU(2) _L × U(1) symmetry, which isassociated with the isospin and the hypercharge, their vector carriers, two-flavorcharged and chargeless leptons, and scalar particles. A mass term producesbreaking of the symmetry to an electromagnetic U(1), and a Weinberg's angle_W with sin ²(_W) = 0.25. A more realistic 8D extension gives coupling constantsof the respective groups g = 1/2 .707 and g = 1/6 .408, with thesame _W.     

Ultrametric Broken Replica Symmetry RaMOSt

We propose an ultrametric breaking of replica symmetry for diluted spin glasses in the framework of Random Multi-Overlap Structures (RaMOSt). Our approach permits to bound the free energy through a trial function that depends on a set of numbers over which one has to take the infimum.Such trial function is a first (ultrametric and factorized) example of a bound in the intersection of the probability spaces of the iterative and the RaMOSt theories, and it shows that a “direct dilution” of the Parisi Ansatz is not always exact.     

Reparametrization Invariance as Gauge Symmetry

Reparametrization invariance treated as a gaugesymmetry shows some specific peculiarities. We studythese peculiarities both from a general point of viewand by concrete examples. We consider the canonical treatment of reparametrization-invariantsystems in which one fixes the gauge on the classicallevel by means of time-dependent gauge conditions. Insuch an approach one can interpret different gauges as different reference frames. We discuss therelation between different gauges and the problem ofgauge invariance in this case. Finally, we establish ageneral structure of reparametrizations and itsconnection with the zero-Hamiltonian phenomenon.     

Quantum Gauge General Relativity

Based on gauge principle, a new model on quantum gravity is proposed in the frame work of quantum gauge theory of gravity. The model has local gravitational gauge symmetry, and the field equation of the gravitational gauge field is just the famous Einstein‘s field equation. Because of this reason, this model is called quantum gauge general relativity, which is the consistent unification of quantum theory and general relativity. The model proposed in this paper is a perturbatively renormalizable quantum gravity, which is one of the most important advantage of the quantum gauge general relativity proposed in this paper. Another important advantage of the quantum gauge general relativity is that it can explain both classical tests of gravity and quantum effects of gravitational interactions, such as gravitational phase effects found in COW experiments and gravitational shielding effects found in Podkletnov experiments.     

Quantum Gauge General Relativity

Based on gauge principle, a new model on quantum gravity is proposed in the frame work of quantum gauge theory of gravity. The model has local gravitational gauge symmetry, and the field equation of the gravitational gauge field is just the famous Einstein‘s field equation. Because of this reason, this model is called quantum gauge general relativity, which is the consistent unification of quantum theory and general relativity. The model proposed in this paper is a perturbatively renormalizable quantum gravity, which is one of the most important advantage of the quantum gauge general relativity proposed in this paper. Another important advantage of the quantum gauge general relativity is that it can explain both classical tests of gravity and quantum effects of gravitational interactions, such as gravitational phase effects found in COW experiments and gravitational shielding effects found in Podkletnov experiments.     

Fuzzy Sets in Macroscopic Quantum Systems

We express quantum properties by quantum fuzzyset functions, and these by generalized transitionprobabilities. The property of beingexcited is fuzzy for one two-level atoms, butshown to be crisp for infinitely many ones.     

Macroscopic Approach to Quantum Optics

This paper investigates how an approach alternative to the canonical quantization schemes may be used to describe Quantum Optics phenomena. By utilizing the approach pioneered by Keldysh, we derive equations for the time dependent correlation functions of the quantized optical fields. These contain the coupling to matter in linear and nonlinar response functions which replace the material parameters of phenomenological macroscopic theories. We present these results as alternatives to existing theoretical methods in Quantum Optics. The paper presents the general formulation of the theory, derives the equations in some specific cases relating to non‐linear optics, and solves some illustrative special cases. We regain known results but also some additional terms deriving from the quantum fluctuations of the material media.     

Enhanced Gauge Symmetry and Braid Group Actions

 Enhanced gauge symmetry appears in Type II string theory (as well as F- and M-theory) compactified on Calabi–Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry algebra, which itself acts on the deformation space of the variety in a compatible way. Extensions of this result are given for nontrivial B-fields on K3 surfaces, explaining physical restrictions on the B-field, as well as for elliptic fibrations. The present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local A _2n -configuration in a threefold having global ℤ/2 monodromy. Received: 28 October 2002 / Accepted: 9 December 2002 Published online: 28 May 2003 Communicated by R.H. Dijkgraaf     

Spin-2 Quantum Gauge Theories and Perturbative Gauge Invariance

In the framework of causal perturbation theory we analyze the gauge structure of a massless self-interacting quantum tensor field. We look at this theory from a pure field theoretical point of view without assuming any geometrical aspect from general relativity. To first order in the perturbation expansion of the S-matrix we derive necessary and sufficient conditions for such a theory to be gauge invariant, by which we mean that the gauge variation of the self-coupling with respect to the gauge charge operator Q is a divergence in the sense of vector analysis. The most general trilinear self-coupling of the graviton field turns out to be the one derived from the Einstein–Hilbert action plus divergences and coboundaries.     

Symmetry and Topology in Quantum Logic

A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic, gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic. PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta.     

The Corolla Polynomial for Spontaneously Broken Gauge Theories

In Kreimer and Yeats (Electr. J. Comb. 41–41, 2013), Kreimer et al. (Annals Phys. 336, 180–222, 2013) and Sars (2015) the Corolla Polynomial \( \mathcal C ({\Gamma }) \in \mathbb C [a_{h_{1}}, \ldots , a_{h_{\left \vert {\Gamma }^{[1/2]} \right \vert }}]\) was introduced as a graph polynomial in half-edge variables \(\{a_{h}\}_{h \in {\Gamma }^{[1/2]}}\) over a 3-regular scalar quantum field theory (QFT) Feynman graph Γ. It allows for a covariant quantization of pure Yang-Mills theory without the need for introducing ghost fields, clarifies the relation between quantum gauge theory and scalar QFT with cubic interaction and translates back the problem of renormalizing quantum gauge theory to the problem of renormalizing scalar QFT with cubic interaction (which is super renormalizable in 4 dimensions of spacetime). Furthermore, it is, as we believe, useful for computer calculations. In Prinz (2015) on which this paper is based the formulation of Kreimer and Yeats (Electr. J. Comb. 41–41, 2013), Kreimer et al. (Annals Phys. 336, 180–222, 2013) and Sars (2015) gets slightly altered in a fashion specialized in the case of the Feynman gauge. It is then formulated as a graph polynomial \(\mathcal C ({\Gamma } ) \in \mathbb C [a_{h_{1 \pm }}, \ldots , a_{h_{\left \vert {\Gamma }^{[1/2]} \right \vert } \vphantom {h}_{\pm }}, b_{h_{1}}, \ldots , b_{h_{\left \vert {\Gamma }^{[1/2]} \right \vert }}] \) in three different types of half-edge variables \( \{a_{h_{+}} , a_{h_{-}} , b_{h}\}_{h \in {\Gamma }^{[1/2]}} \). This formulation is also suitable for the generalization to the case of spontaneously broken gauge theories (in particular all bosons from the Standard Model), as was first worked out in Prinz (2015) and gets reviewed here.     

Quantum Group Symmetry in Hofstadter Problem

We show that there is a quantum Slq（2） group symmetry in Hofstadter problem on square lattice. The cyclic representation of the quantum group is discussed and its application for computing the degeneracy density of the model is shown.     

Macroscopic Quantum Coherence in Antiferromagnetic Molecular Magnets

The macroscopic quantum coherence in a biaxial antiferromagnetic molecular magnet in the presence of magnetic field acting parallel to its hard anisotropy axis is studied within the two-sublattice model.On the basis of instanton technique in the spin-coherent-state path-integral representation,both the rigorous Wentzel-Kramers-Brillouin exponent and pre-exponential factor for the ground-state tunnel splitting are obtained.We find that the quantum fluctuations around the classical paths can not only induce a new quantum phase previously reported by Chiolero and Loss (Phys.Rev.Lett.80(1998)169),but also have great influence on the intensity of the ground-state tunnel splitting.Those features clearly have no analogue in the ferromagnetic molecular magnets.We suggest that they may be the universal behaviors in all antiferromagnetic molecular magnets.The analytical results are complemented by exact diagonalization calculation.     

Quantum Group Symmetry in Hofstadter Problem

We show that there is a quantum Sl_q(2) group symmetry in Hofstadter problem on square lattice. The cyclic representation of the quantum group is discussed and its application for computing the degeneracy density of the model is shown.     

Symmetry and Evolution in Quantum Gravity

We propose an operator constraint equation for the wavefunction of the Universe that admits genuine evolution. While the corresponding classical theory is equivalent to the canonical decomposition of General Relativity, the quantum theory contains an evolution equation distinct from standard Wheeler–DeWitt cosmology. Furthermore, the local symmetry principle—and corresponding observables—of the theory have a direct interpretation in terms of a conventional gauge theory, where the gauge symmetry group is that of spatial conformal diffeomorphisms (that preserve the spatial volume of the Universe). The global evolution is in terms of an arbitrary parameter that serves only as an unobservable label for successive states of the Universe. Our proposal follows unambiguously from a suggestion of York whereby the independently specifiable initial data in the action principle of General Relativity is given by a conformal geometry and the spatial average of the York time on the spacelike hypersurfaces that bound the variation. Remarkably, such a variational principle uniquely selects the form of the constraints of the theory so that we can establish a precise notion of both symmetry and evolution in quantum gravity.     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

Short-Range Repulsion and Broken Chiral Symmetry in Low-Energy Scattering

This paper is an essentially modified version of the chapter “Short-range repulsion” from the English edition [1] of the book “Dispersion theories of strong interactions at low-energies” (in the Russian edition [1] this chapter is absent). Unlike the English version, we have employed the concept of chiral symmetry and ist consequences for the low-energy characteristics of strong interactions as boundary conditions on the solution of dispersion equations. The introduction of the short-range repulsion “potentials” into the low-energy equations for lower partial waves makes it possible to eliminate the main difficulties of the purely elastic low-energy (Pele) approximation. There is then a possibility in principle of obtaining solutions with small s-wave scattering lengths and broad resonances. The use of threshold conditions resulting from chiral symmetry allows us (under certain additional conditions) to express the main resonance scattering parameters in terms of the pion decay characteristics. Formulas are presented, by means of one of which the ϱ meson mass m_ϱ is expressed in terms of the pion mass μ and the decay constant ƒ_π from the PCAC condition (Eq. (5.26) and the other (Eq. (5.27)) expresses the ϱ meson width Γ via μƒ_π and m_π and is a generalization of the well-known KSFR relation taking into account unitarity corrections. Similar results have been obtained for the Δ₃₃ resonance in pion-nucleon scattering. Thus, using the broken chiral symmetry approximation and unitarity dispersion equations for low-energy ππ and πN scattering we have obtained masses, life-time and coupling constants for p-wave resonances by specifying only the pion and nucleon masses, their life-times and the Fermi coupling constant. Reported at the Conference on the High Energy Physics in the Institute for Theoretical Physics, Kiev, the Ukrainian SSR, October 1969.