Renormalizable Quantum Gauge Theory of Gravity

The quantum gravity is formulated based on the principle of local gauge invariance. The model discussed in this paper has local gravitational gauge symmetry, and gravitational field is represented by gauge field. In the leading-order approximation, it gives out classical Newton's theory of gravity. In the first-order approximation and for vacuum, it gives out Einstein's general theory of relativity. This quantum gauge theory of gravity is a renormalizable quantum theory.     

Renormalizable Quantum Gauge Theory of Gravity

The quantum gravity is formulated based on the principle of local gauge invariance. The model discussedin this paper has local gravitational gauge symmetry, and gravitational field is represented by gauge field. In the leading-order approximation, it gives out classical Newton‘s theory of gravity. In the first-order approximation and for vacuum,it gives out Einstein‘s general theory of relativity. This quantum gauge theory of gravity is a renormalizable quantumtheory.     

Unified Theory of Fundamental Interactions

Based on local gauge invariance, four different kinds of fundamental interactions in nature are unified in a theory which has SU(3)C( )SU SU(2)L( )U(1)( )s Gravitational Gauge Group gauge symmetry. In this approach,gravitational field, like electromagnetic field, intermediate gauge field, and gluon field, is represented by gauge potential.Four kinds of fundamental interactions are formulated in the similar manner, and therefore can be unified in a direct or semi-direct product group. The model discussed in this paper is a renormalizable quantum model and can be regarded as an extension of the standard model to gravitational interactions, so it can be used to study quantum effects of gravitational interactions.     

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.     

The spinor representation,U(1)Γ embedding,and symmetry-breaking patterns in SO(14)

An SO(14) gauge theory with a spinor representation is presented as an extension of the flavor-unifying SU(7) model. The global Γ symmetry in the SU(7) theory becomes a local gauge symmetry as a part of the SO(14). The quarks are classified by 3 groups and the leptons by 2 groups according to the Γ quantum number. Three patterns of spontaneous symmetry breaking are considered, and only one of them is shown to be a viable choice.     

Are There Dynamical Laws?

The nature of a physical law is examined, and it is suggested that there may not be any fundamental dynamical laws. This explains the intrinsic indeterminism of quantum theory. The probabilities for transition from a given initial state to a final state then depends on the quantum geometry that is determined by symmetries, which may exist as relations between states in the absence of dynamical laws. This enables the experimentally well-confirmed quantum probabilities to be derived from the geometry of Hilbert space and gives rise to effective probabilistic laws. An arrow of time which is consistent with the one given by the second law of thermodynamics, regarded as an effective law, is obtained. Symmetries are used as the basis for a new proposed paradigm of physics. This naturally gives rise to the gravitational and gauge fields from the symmetry group of the standard model and a general procedure for obtaining interactions from any symmetry group.     

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.     

Review of AdS/CFT Integrability, Chapter IV.4: Integrability in QCD and {\fancyscript{N} <4 } SYM

There is a growing amount of evidence that QCD (and four-dimensional gauge theories in general) possesses a hidden symmetry which does not exhibit itself as a symmetry of classical Lagrangians but is only revealed on the quantum level. In this review we consider the scale dependence of local gauge invariant operators and high-energy (Regge) behavior of scattering amplitudes to explain that the effective QCD dynamics in both cases is described by completely integrable systems that prove to be related to the celebrated Heisenberg spin chain and its generalizations.     

Gravitational Gauge Interactions of Scalar Field

Quantum gauge theory of gravity is formulated based on gauge principle. Because the Lagrangian has strict local gravitational gauge symmetry, gravitational gauge theory is a perturbatively renormalizable quantum theory. Gravitational gauge interactions of scalar field are studied in this paper. In quantum gauge theory of gravity, scalar field minimal couples to gravitational field through gravitational gauge covariant derivative. Comparing the Lagrangian for scalar field in quantum gauge theory of gravity with the corresponding Lagrangian in quantum fields in curved space-time, the definition for metric in curved space-time in geometry picture of gravity can be obtained, which is expressed by gravitational gauge field. In classical level, the Lagrangian and Hamiltonian approaches are also discussed.     

Gravitational Gauge Interactions of Scalar Field

Quantum gauge theory of gravity is formulated based on gauge principle. Because the Lagrangian hasstrict local gravitational gauge symmetry, gravitational gauge theory is a perturbatively renormalizable quantum theory.Gravitational gauge interactions of scalar field are studied in this paper. In quantum gauge theory of gravity, scalar fieldminimal couples to gravitational field through gravitational gauge covariant derivative. Comparing the Lagrangian forscalar field in quantum gauge theory of gravity with the corresponding Lagrangian in quantum fields in curved space-time, the definition for metric in curved space-time in geometry picture of gravity can be obtained, which is expressedby gravitational gauge field. In classical level, the Lagrangian and Hamiltonian approaches are also discussed.     

Gauge anomaly cancellation in chiral gauge theories

We consider chiral fermions interacting minimally with abelian and non-abelian gauge fields. Using a path integral approach and exploring the consequences of a mechanism of symmetry restoration, we show that the gauge anomaly has null expectation value in the vacuum for both cases (abelian and non-abelian). We argue that the same mechanism has no possibility to cancel the chiral anomaly, what eliminates competition between chiral and gauge symmetry at full quantum level. We also show that the insertion of the gauge anomaly in arbitrary gauge invariant correlators gives a null result, which points towards anomaly cancellation in the subspace of physical state vectors.     

Discrete accidental symmetry for a particle in a constant magnetic field on a torus

A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the 1/r and r² potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge-Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the θ-vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters θ_x and θ_y explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.     

Covariant gauges and point sources in general relativity

The tree graph contributions to the vacuum expectation value of the quantized gravitational field produced by a point mass source are found to diverge. These divergences can be removed only by giving the source a finite extension, and it is first necessary to analyze the corresponding classical situation before making a comparison with the quantum theory. In this paper, the model for such an extended particle is taken to be a spherical shell of initially static pressure-free dust. Without solving the Einstein equations explicity, a coordinate-independent mass renormalization formula can be derived, valid at the moment of time symmetry, which relates the total mass of the system to the bare mass of the source and its invariant radius. The equations are then solved for various choices of coordinate systems, allowing the invariant radius of the shell to be expressed in terms of its coordinate dependent extension. The results are in agreement with those obtained previously by Arnowitt, Deser, and Misner. The work of these authors is generalized to include coordinate frames for which the metric is discontinuous across the shell. Aside from any intrinsic interest, such a generalization is necessary since the most convenient coordinate system for the quantum calculations, namely the covariant de Donder (harmonic) gauge, falls into this category. By expanding the total mass of the source in terms of its bare mass and harmonic coordinate extension, the classical Schwarzschild solution may be cast into a form which facilitates a direct comparison with the quantum theory in the de Donder gauge.     

First integrals of motion in a gauge covariant framework, Killing-Maxwell system and quantum anomalies

Hidden symmetries in a covariant Hamiltonian framework are investigated. The special role of the Stackel-Killing and Killing-Yano tensors is pointed out. The covariant phase-space is extended to include external gauge fields and scalar potentials. We investigate the possibility for a higher-order symmetry to survive when the electromagnetic interactions are taken into account. Aconcrete realization of this possibility is given by the Killing-Maxwell system. The classical conserved quantities do not generally transfer to the quantized systems producing quantum gravitational anomalies. As a rule the conformal extension of the Killing vectors and tensors does not produce symmetry operators for the Klein-Gordon operator.     

Spontaneous symmetry breaking in local gauge quantum field theory; the Higgs mechanism

Spontaneous symmetry breakings in indefinite metric quantum field theories are analyzed and a generalization of the Goldstone theorem is proved. The case of local gauge quantum field theories is discussed in detail and a characterization is given of the occurrence of the Higgs mechanism versus the Goldstone mechanism. The Higgs phenomenon is explained on general grounds without the introduction of the so-called Higgs fields. The basic property is the relation between the local internal symmetry group and the local group of gauge transformations of the second kind. Spontaneous symmetry breaking ofc-number gauge transformations of the second kind is shown to always occur if there are charged local fields. The implications about the absence of mass gap in the Wightman functions and the occurrence of massless particles associated with the unbroken generators in the Higgs phenomenon are discussed.     

Hamiltonian interpretation of anomalies

A family of quantum systems parametrized by the points of a compact space can realize its classical symmetries via a new kind of nontrivial ray representation. We show that this phenomenon in fact occurs for the quantum mechanics of fermions in the presence of background gauge fields, and is responsible for both the nonabelian anomaly and Witten's SU(2) anomaly. This provides a hamiltonian interpretation of anomalies: in the affected theories Gauss' law cannot be implemented. The analysis clearly shows why there are no further obstructions corresponding to higher spheres in configuration space, in agreement with a recent result of Atiyah and Singer.     

The anomaly in the Slavnov identity for N = 1 supersymmetric Yang-Mills theories

We solve the consistency conditions of the BRS symmetry in a general N = 1 sypersymmetric Yang-Mills theory with semi-simple gauge group. As a result we find uniquely the supersymmetric extension of the chiral anomaly. Its coefficient is calculated in one loop and does not, in general, vanish. This corrects our earlier statement on the absence of this anomaly.