Renormalization Proof for Spontaneously Broken Yang-Mills Theory with Flow Equations

Spontaneously broken Yang-Mills-Higgs Theory: A renormalization proof by flow equations

In this article*, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Sec.2), “it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms”; where in agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energy-momentum can only take rational values. In essence, the main scheme of this new mathematical axiomatic approach to fundamental laws of nature is as follows. First based on the assumption of rationality of D-momentum, by linearization (along with a parameterization procedure) of the Lorentz invariant energy-momentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford, and symmetric algebras) is derived. Then by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is derived uniquely as well. Each class of the derived general covariant field equations also includes a definite form of torsion field appeared as generator of the corresponding field’ invariant mass. In addition, it is shown that the (1+3)-dimensional cases of two classes of derived field equations represent a new general covariant massive formalism of bispinor fields of spin-2, and spin-1 particles, respectively. In fact, these uniquely determined bispinor fields represent a unique set of new generalized massive forms of the laws governing the fundamental forces of nature, including the Einstein (gravitational), Maxwell (electromagnetic) and Yang-Mills (nuclear) field equations. Moreover, it is also shown that the (1+2)-dimensional cases of two classes of these field equations represent a new general covariant massive formalism of bispinor fields of spin-3/2 and spin-1/2 particles, corresponding (asymptotically) to the Dirac and Rarita–Schwinger equations. As a particular consequence, it is shown that a certain massive formalism of general relativity – with a definite form of torsion field appeared originally as the generator of gravitational field’s invariant mass – is obtained only by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of a certain set of special relativistic algebraic matrix equations. In addition, in agreement with recent astronomical data, a new particular form of massive particle is identified (as the gauge boson of U(1) symmetry group) with invariant mass: m_γ ≈ 4.90571×10^-50 kg, which is generated by a definite coupled torsion field of the background space-time geometry. Moreover, based on the definite mathematical formalism of this axiomatic approach, along with the C, P and T symmetries (represented basically by the corresponding quantum operators) of the fundamentally derived field equations, it has been concluded that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times (where this conclusion, in particular, is based on the T-symmetry of these equations). It is shown that CPT is the only combined form of C, P, and T symmetries of interacting fields. In addition, on the basis of these discrete symmetries of derived field equations, it has been also shown that only left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents. Furthermore, it has been shown that the metric of background space-time is diagonalized for the uniquely derived fermion field equations (defined and expressed solely in (1+2)-dimensional space-time), where this property generates a certain set of additional symmetries corresponding uniquely to the "SU(2)_L⊗U(2)_R" symmetry group for spin-1/2 fermion fields (representing “1+3” generations of four fermions, including a group of eight leptons and a group of eight quarks), and also the "SU(2)_L⊗U(2)_R" and SU(3) gauge symmetry groups for spin-1 boson fields coupled to the spin-1/2 fermionic source currents. Hence, along with the known elementary particles, eight new elementary particles, including four new charge-less right-handed spin-1/2 fermions (including two leptons and two quarks), a spin-3/2 fermion, and also three new spin-1 (massive) bosons are predicted uniquely by this new mathematical axiomatic approach. As a particular result, based on the definite formulation of the derived Maxwell (and Yang-Mills) field equations, it has been also concluded that magnetic monopoles could not exist in nature.* *Note: "See also new published version of this article at: https://inspirehep.net/record/1387680/ ”. Comments: 90 Pages. A summary of a submitted and accepted research project (On "Foundations of Physics"), Ramin Zahedi, Japan, 2012 - 2015. This article has been invited and presented at the following international conferences: - The 2016 SIAM International Conference on Mathematical Aspects of Materials Science, Philadelphia, USA, 2016. (https://www.siam.org/meetings/ms16) - The 17th International Conference on Quantum Foundations: Quantum and Beyond, International Centre for Mathematical Modeling in Physics, Engineering and Cognitive Sciences (ICMM), Linnaeus University, Sweden, 2016. (https://lnu.se/en/qb/) -The 4th International Conference on New Frontiers in Physics, CERN Organized and Supported Conference (Europe), ICNFP-2015. (https://indico.cern.ch/e/icnfp2015/) - The 2016 International Conference on Algebraic Geometry and Mathematical Physics, University of Tromsø, Norway, 2016. (https://site.uit.no/) - The XXXVII Max Born International Symposium, International Conference on Non-commutative Geometry, Quantum Symmetries and Quantum Gravity (II), Wroclaw University, Poland, 2016. (http://ift.uni.wroc.pl/~mborn37/) - The 2016 GRavitational-wave Astronomy International Conference in Paris, Institute d‘Astrophysique de Paris (IAP), The University of Paris VI - Sorbonne University, CNRS, LERU, EUA, Paris, France, 'Supported by the European Union's 7th Framework Program: FP7/PEOPLE-2011-CIG,‘ 2016. (http://www.iap.fr/vie_scientifique/ateliers/GravitationalWave/2016/scripts/abstract.aspx) - The 22nd Internnational Australian Institute of Physics Congress (AIP), University of Queensland, Brisbane, Australia, 2016. (http://appc-aip2016.org.au) - The 21st International Conference on General Relativity and Gravitation, Columbia University, New York, USA, 2016. (http://www.gr21.org). External URLs (including preprints and earlier published versions of this research article): https://cds.cern.ch/record/1980381/ , https://inspirehep.net/record/1387680/, https://indico.cern.ch/event/344173/session/22/contribution/422/attachments/1140145/1646101/R.a.Zahedi--OnDiscretePhysics-Jan.2015-signed.pdf, http://eprints.lib.hokudai.ac.jp/dspace/handle/2115/59279/, https://core.ac.uk/display/41879427, http://www.gr21.org/GR21FullAbstractBooklet.pdf#section*.177, https://dx.doi.org/10.17605/OSF.IO/ZCHTP, https://dumas.ccsd.cnrs.fr/TDS-MACS/hal-01476703 , https://indico.cern.ch/event/344173/contribution/1740565/attachments/1140145/1726912/R.A.Zahedi--Forces.of.Nature.Laws-Jan.2015-signed.pdf , http://mts-ncomms.nature.com/ncomms_files/2017/07/10/00136200/00/136200_0_art_file_2445656_bsw9c4.pdf .

Critical behavior of polar modes in lead phosphate near the ferroelastic phase transition

Like moths attracted to a bright light, philosophers are drawn to glitz. So in discussing the notions of ‘gauge’, ‘gauge freedom’, and ‘gauge theories’, they have tended to focus on examples such as Yang–Mills theories and on the mathematical apparatus of fibre bundles. But while Yang–Mills theories are crucial to modern elementary particle physics, they are only a special case of a much broader class of gauge theories. And while the fibre bundle apparatus turned out, in retrospect, to be the right formalism to illuminate the structure of Yang–Mills theories, the strength of this apparatus is also its weakness: the fibre bundle formalism is very flexible and general, and, as such, fibre bundles can be seen lurking under, over, and around every bush. What is needed is an explanation of what the relevant bundle structure is and how it arises, especially for theories that are not initially formulated in fibre bundle language. Here I will describe an approach that grows out of the conviction that, at least for theories that can be written in Lagrangian/Hamiltonian form, gauge freedom arises precisely when there are Lagrangian/Hamiltonian constraints of an appropriate character. This conviction is shared, if only tacitly, by that segment of the physics community that works on constrained Hamiltonian systems.

We are interested in a gauge invariant coupling between four-dimensional Yang–Mills field and a three-brane that can fluctuate into higher dimensions. For this we interpret the Yang–Mills theory as a higher dimensional bulk gravity theory with dynamics that is governed by the Einstein action, and with a metric tensor constructed from the gauge field in a manner that displays the original gauge symmetry as an isometry. The brane moves in this higher dimensional spacetime under the influence of its bulk gravity, with dynamics determined by the Nambu action. This introduces the desired interaction between the brane and the gauge field in a way that preserves the original gauge invariance as an isometry of the induced metric. After a prudent change of variables the result can be interpreted as a gauge invariant and massive vector field that propagates in the original spacetime . The presence of the brane becomes entirely invisible, expect for the mass.

We give an overview of the use of combinatorics in renormalization of gauge theories, using the Connes–Kreimer Hopf algebra. We prove some physical results in perturbative quantum gauge theories without relying on the formal manipulations involving path integrals. Instead, we take a perturbative series of Feynman graphs as a starting point. After a careful setup and counting of Feynman graphs, we study the structure of the renormalization Hopf algebra of gauge theories on the level of Green’s functions. This involves Slavnov– Taylor identities, described by Hopf ideals, and Dyson–Schwinger equations, described by Hochschild cocycles [7]. As a new result, we prove the Kreimer’s gauge theory theorem formulated in [14].

In the previous paper,1 we derived the Abelian projected effective gauge theory as a low energy effective theory of the SU (N) Yang–Mills theory by adopting the maximal Abelian gauge. At that time, we have demonstrated the multiplicative renormalizability of the propagators for the diagonal gluon and the dual Abelian antisymmetric tensor field. In this paper, we show the multiplicative renormalizability of the Green's functions also for the off-diagonal gluon. Moreover, we complement the previous results by calculating the anomalous dimension and the renormalization group functions which are undetermined in the previous paper.

A basic problem of classical field theory, which has attracted growing attention over the past decade, is to find and classify all nonlinear deformations of linear abelian gauge theories. The physical interest in studying deformations is to address uniqueness of known nonlinear interactions of gauge fields and to look systematically for theoretical possibilities for new interactions. Mathematically, the study of deformations aims to understand the rigidity of the nonlinear structure of gauge field theories and to uncover new types of nonlinear geometrical structures. The first part of this paper summarizes and significantly elaborates a field-theoretic deformation method developed in earlier work. Some key contributions presented here are, firstly, that the determining equations for deformation terms are shown to have an elegant formulation using Lie derivatives in the jet space associated with the gauge field variables. Secondly, the obstructions (integrability conditions) that must be satisfied by lowest-order deformations terms for existence of a deformation to higher orders are explicitly identified. Most importantly, a universal geometrical structure common to a large class of nonlinear gauge theory examples is uncovered. This structure is derived geometrically from the deformed gauge symmetry and is characterized by a covariant derivative operator plus a nonlinear field strength, related through the curvature of the covariant derivative. The scope of these results encompasses Yang–Mills theory, Freedman–Townsend theory, and Einstein gravity theory, in addition to their many interesting types of novel generalizations that have been found in the past several years. The second part of the paper presents a new geometrical type of Yang–Mills generalization in three dimensions motivated from considering torsion in the context of nonlinear sigma models with Lie group targets (chiral theories). The generalization is derived by a deformation analysis of linear abelian Yang–Mills Chern–Simons gauge theory. Torsion is introduced geometrically through a duality with chiral models obtained from the chiral field form of self-dual (2+2) dimensional Yang–Mills theory under reduction to (2+1) dimensions. Field-theoretic and geometric features of the resulting nonlinear gauge theories with torsion are discussed.

We derive an explicit formula for factorizing an $n$-point closed string amplitude into open string amplitudes. Our results are phrased in terms of a momentum kernel which in the limit of infinite string tension reduces to the corresponding field theory kernel. The same momentum kernel encodes the monodromy relations which lead to the minimal basis of color-ordered amplitudes in Yang-Mills theory. There are interesting consequences of the momentum kernel pertaining to soft limits of amplitudes. We also comment on surprising links between gravity and certain combinations of kinematic and color factors in gauge theory.

The gauge structure of the four dimensional effective theory (4DET) arising from a pure Yang-Mills theory in five dimensions compactified on the orbifold $S^1/Z_2$ is reexamined on the basis of the BRST symmetry. The two scenarios that can arise are analyzed: if the gauge parameters propagate in the bulk, the excited Kaluza-Klein (KK) modes are gauge fields, but they are matter vector fields if these parameters are confined in the 3-brane. In the former case, it is shown that the 4DET is gauge invariant only if the compactification is carried out by using curvatures instead of gauge fields as fundamental objects. It is shown that the 4DET is governed by two types of gauge transformations, one determined by the KK zero modes of the gauge parameters and the other by the excited KK modes. The Dirac's method and the proper solution of the master equation are employed to show that the theory is subject to first class constraints. A gauge-fixing procedure to quantize the KK modes, that is covariant under the first type of gauge transformations, is introduced. The ghost sector induced by these gauge-fixing functions is derived on the basis of the BRST formalism. The effective quantum Lagrangian that links the interactions between light physics and heavy physics is presented. Concerning the radiative corrections of the excited KK modes on the light Green functions, the predictive character of this Lagrangian is stressed. In the case of the gauge parameters confined to the 3-brane, the known result in the literature is reproduced with some minor variants, although it is emphasized that the excited KK modes are not gauge fields, but matter fields that transform under the adjoint representation of $SU_4(N)$. The Dirac's method is employed to show that this theory is subject to both first and second class constraints.

We write a gravity theory with Yang–Mills-type action using the biconformal gauging of the conformal group. We show that the resulting biconformal Yang–Mills gravity theories describe 4-dim, scale-invariant general relativity in the case of slowly changing fields. In addition, we systematically extend arbitrary 4-dim Yang–Mills theories to biconformal space, providing a new arena for studying flat-space Yang–Mills theories. By applying the biconformal extension to a 4-dim pure Yang–Mills theory with conformal symmetry, we establish a 1-1, onto mapping between a set of gravitational gauge theories and 4-dim, flat-space gauge theories.

In this article a self-contained exposition of proving perturbative renormalizability of a quantum field theory based on an adaption of Wilson's differential renormalization group equation to perturbation theory is given. The topics treated include the spontaneously broken SU(2) Yang–Mills theory. Although mainly a coherent but selective review, the article contains also some simplifications and extensions with respect to the literature.

In this chapter we are concerned with other electrodynamic theories which show new features, not present in four-dimensional spinor QED. The new feature in scalar QED is the derivative coupling between the scalar field and the vector potential. It has important consequences for gauge invariance, in particular, the interaction gets modified by gauge invariance. Such derivative couplings also occur in Yang-Mills theories, and the same implications of gauge invariance appear here. This is briefly discussed in the Epilogue. The axial anomalies, that we describe next, provide an important example where gauge invariance partially breaks down. These anomalies also occur in the standard electro-weak theory and must be compensated there. This implies the lepton-quark duality, which means that to each lepton family (e, μ, τ) there must exist a corresponding quark family. Finally, if QED in two space plus one time dimensions is considered, the vacuum polarization shows unusual behaviour. This has the surprising consequence that the photon becomes massive. These examples illustrate how rich QED really is.

Part 1 Anomalies and fractional charge: non-canonical behaviour in canonical theories quantum mechanical symmetry breaking delta function potentials in two- and three-dimensional quantum mechanics update on anomalous theories effects of Dirac's negative energy sea on quantum numbers Fermion fractionization in physics the chiral anomaly. Part 2 Gauge theories and gravity: Yang-Mills vacuum as a bloch wave bifurcation and stability in Yang-Mills theory with sources Gauge theories in three dimensions (= at high temperature) planar gravity time travel? Gauge theories for gravity on a line. Part 3 Symmetry behavior: introducing scale symmetry hidden symmetry of the point magnetic monopole invariance, symmetry and periodicity in Gauge theories symmetry restoration at finite temperature mean field theory for non-equilibrium quantum fields. Part 4 Approaches to quantum theory following Dirac: canonical light-cone commutators and their applications invariant quantization, scale symmetry and Euclidean field theory (constrained) quantization without tears analysis on infinite-dimensional manifolds - Schrodinger representation for quantized fields. Part 5 Solitons, instantons and semi-classical quantum field theory: non-perturbative and topological methods in quantum field theory self-dual Chern-Simons solitions.

A gauge invariant exact renormalisation group. (I)