User:Jhsujhsu/Quantum and classical Yang-Mills gravity

Translation gauge symmetry in flat space-time
Yang-Mills gravity is a new formulation of gravity based on the Yang-Mills-Utiyama-Weyl symmetry framework for internal and external gauge groups. The gauge symmetry group for Yang-Mills gravity is the external translation group T4 in flat four-dimensional space-time. Yang-Mills gravity is consistent with experiments and bring gravity back to the arena of quantum mechanics in flat space-time with the Poincare group. One new result of Yang-Mills gravity is that the apparent curvature of space-time appears to be simply a manifestation of the flat space-time translation gauge symmetry for the motion of quantum particle in the classical limit. The flat space-time translation gauge symmetry naturally allows the unification of gravity with electroweak and strong forces. This was accomplished within the framework of Yang-Mills-Utiyama-Weyl framework, following the unification ideas of Glashow, Salam, Ward and Weinberg. Such a view suggests that all known fundamental interactions in nature appear to be manifestations of U1, SU2, SU3 and T4 gauge symmetries. The intimate relationship between interactions and gauge symmetries was advocated by Utiyama, Yang, Mills and others. Yang-Mills gravity with space-time translation gauge group is essential to avoid the difficulties in the framework of curved space-time as discussed by physicists and mathematicians. For example, F. Dyson stressed that "the most glaring incompatibility of concepts in contemporary physics is that between Einstein's principle of general coordinate invariance and all the modern schemes for a quantum-mechanical description of nature."

Combined local translations and general coordinate transformations
The Lagrangian of Yang-Mills gravity is assumed to be invariant under a local space-time translation: $$x^{\mu} \to x'^{\mu}=x^{\mu}+\Lambda^{\mu}(x)$$. It suffices to consider the infinitesimal and arbitrary vector function $$\Lambda^{\mu}(x)$$. This infinitesimal space-time translations have two conceptual interpretations: (i)  They are local translations in flat space-time, and (ii) they are general coordinate transformations. This dual role of local space-time translation implies that local translation gauge invariance is inseparable from general coordinate invariance in flat space-time. Note that Einstein followed the general coordinate invariance in curved space-time and hence, employed space-time curvature and a covariant derivative to formulate his theory of gravitation. In contrast, Yang-Mills gravity follows the local translation invariant in flat space-time and hence, uses $$T_4$$ gauge curvature and a gauge covariant derivative to formulate a theory of gravity.

Based on Yang-Mills gravity, one is able to bring gravitational interaction back to the arena of quantum mechanics and gauge field theories in flat space-time. In inertial frame of reference, the fundamental space-time symmetry group is the Poincare group, which contains two sub-groups: the global translation group and the Lorentz group. The Abelian sub-group of space-time translations leads to the conservation of energy-momentum. The external $$T_4$$ gauge symmetry in flat space-time enables one to construct a quantum theory of gravity that is consistent with experiments in inertial frames with well-defined space-time coordinates.

Classical Yang-Mills gravity
In usual electromagnetic and Yang-Mills-type gauge theories are associated with internal gauge groups, one assumes the replacement, $$\partial_\mu \to \partial_\mu + ifB_\mu(x), \ \ \ (c=\hbar=1),$$

in the Lagrangian. The gauge field $$B_\mu(x)=B_\mu^a(x) t^a$$ involves constant matrix representations of the generators $$t^a$$ of internal gauge groups.

However,  in Yang-Mills gravity, the generators of the external space-time translation group $$T_4$$ are the momentum operators $$p_\mu=i\partial /\partial x^\mu= i\partial_\mu $$. Hence, one assumes the replacement,

$$

\partial _\mu \to \partial_\mu  - ig\phi_{\mu}^{\nu} p_\nu \equiv \Delta_\mu \equiv J_{\mu}^{\nu}\partial_\nu. \ \ \ \ \ \

J_{\mu}^{\nu}=\delta_{\mu}^{\nu}+g\phi_{\mu}^{\nu},$$           $$  \ \ \ \  \eta_{\mu\nu}=(1,-1,-1,-1),$$

similar to that in the non-Abelian gauge theories. This replacement is the basic postulate for the new formulation of Yang-Mills gravity. Since the generators of the $$T_4$$ group are $$p_\mu=i\partial_\mu$$, one has a symmetric tensor gauge field (i.e., a spin-2 field) rather than the usual 4-vector field (i.e., a spin-1 field) in the $$T_4$$ gauge covariant derivative $$\Delta_\mu$$.

Two immediate consequences of the new replacements for the $$T_4$$ group are as follows:

(i) Because the term $$   g\phi_{\mu}^{\nu} \partial_\nu $$ in the  $$T_4$$  replacement has the same sign as its complex conjugate, this property implies that the gravitational interaction leads to only one kind of force for all particles and anti-particles. This property differs from the electromagnetic interaction due to the replacement $$\partial_\mu \to \partial_\mu + ieA_\mu(x)$$, where the difference in sign of the complex conjugate implies the existence of both attractive and repulsive forces.

(ii)  The $$T_4$$ gauge covariant derivative $$ \Delta_\mu \equiv J_{\mu}^{\nu}\partial_\nu$$ in the Lagrangian has a remarkable property. Namely, it dictates that, in the classical or short-wave limit, all particle wave equations in flat space-time, except the gravitational wave equations, reduce to the Hamilton-Jacobi type equation with the same `effective Riemann metric tensor' $$ G_{\mu\nu}(x)$$,

$$ G^{\mu\nu}(\partial_\mu S)(\partial_\nu S)- m^2 = 0, \ \ \ \ m^2 > 0,    \ \ \   G^{\mu\nu}(x) = \eta_{\alpha\beta}J^{\alpha\mu}(x)J^{\beta\nu}(x). $$

The Hamilton-Jacobi-type shows the effects of a universal gravitational force acting on all classical objects. Because it is formally the same as the corresponding equation in Einstein's gravity based on curved space-time. It implies that classical objects, such as planets, behave as if they were in a `curved space-time,' even though the underlying physical space-time for quantum particles is actually flat.

Experimental tests of Yang-Mills gravity
In the first order approximation, Yang-Mills gravity is consistent with the experimentally measured gravitational red shift. In the second order approximation, it correctly predicts the perihelion shift of Mercury and the gravitational quadrupole radiation of binary pulsars. However, in the equation for the light ray in the geometric-optics limit, one has a slightly different effective metric tensor. As a result, Yang-Mills gravity predicts an angle $$\Delta \phi \approx 1.53''$$  for the deflection of light ray with optical frequencies by the sun. It is about 12% smaller than the observed values 1.75 '', whose accuracy is no better than10-20%.

Quantum Yang-Mills gravity
The $$T_4$$ gauge curvatures, $$ C_{\mu\nu\alpha}$$, are  given by

$$ C_{\mu\nu\alpha}= J_{\mu\sigma}\partial^{\sigma} J_{\nu\alpha}-J_{\nu\sigma} \partial^{\sigma} J_{\mu\alpha}=\Delta_{\mu}J_{\nu\alpha}-\Delta_{\nu}J_{\mu\alpha}=-C_{\nu\mu\alpha}.$$

The gauge invariant action $$S_{\phi\psi}$$ for symmetric spin 2 field $$\phi_{\mu\nu}=\phi_{\nu\mu}$$ in flat space-time is assumed to be

$$

S_{\phi}=\int L_{\phi} d^{4}x, \ \ \  $$           $$ \ \ \ \ \ L_{\phi}= \frac{1}{2g^2}\left (C_{\mu\alpha\beta}C^{\mu\beta\alpha}- C_{\mu\alpha}^{ \ \ \  \alpha}C^{\mu\beta}_{ \ \ \  \beta} \right).$$$$

$$The gauge-fixing term  is necessary for the quantization of fields with gauge symmetry. One may choose the usual gauge condition of the form

$$

\partial^\mu J_{\mu\nu} - \frac{\zeta}{2} \partial_\nu J = Y_{\nu}, \ \ \ J=

J^{\lambda}_\lambda=\delta^{\mu}_{\mu}- g\phi, \ \ \ \phi=\phi^{\lambda}_\lambda,$$

where $$Y_{\nu}$$ is a suitable function of space-time.

Quantum Yang-Mills gravity in flat space-time is based on the gauge invariant action $$S_{\phi}$$, which allows one to derive the rules for Feynman diagrams involving gravitational tensor fields and the associated `ghost particles.'