1. Yang-Mills Gravity (vs. Einstein Gravity) Yang-Mills Gravity (vs. Einstein Gravity) Jong-Ping Hsu Jong-Ping Hsu Physics Department Univ. of Massachusetts Dartmouth

3. Einstein’s original idea for the future of physics: Geometrization of physics Geometrization of physics Riemann Geometry Riemann Geometry Weyl Geometry Weyl Geometry Finsler Geometry, …… Finsler Geometry, …… ******************************* ******************************* Fiber Bundle (Yang-Mills approach) Fiber Bundle (Yang-Mills approach)

4. F. Dyson: (A founder of QED, together with Tomonaga, Schwinger, & Feynman) He stressed that “The most glaring incompatibility of concepts in contemporary physics is that between the principle of general coordinate invariance and a quantum-mechanical description of all of nature.’ (‘Missed Opportunity’ talk at Amer. Math. Soc. ~1970) (‘Missed Opportunity’ talk at Amer. Math. Soc. ~1970) Quantum gravity appears to be the last challenge to the powerful gauge symmetry of the Yang-Mills theory.

5. Yang-Mills Gravity,T(4)x….. Motivations : (1) It is very hard for conventional field theory to describe gravity. Why? (ds 2 =……) (2) Is there a hidden Yang-Mills gauge symmetry in the Hilbert-Einstein Lagrangian? (which gauge group?) Ning Wu, Y.M. Cho, …. ‘yes’. Really? Explicit calculations….. (3) Using Yang-Mills’ Approach based on Flat space-time to investigate and improve the relationship between Quantum Mechanics and Gravity. Yang-Mills Gravity,T(4)x….. Motivations : (1) It is very hard for conventional field theory to describe gravity. Why? (ds 2 =……) (2) Is there a hidden Yang-Mills gauge symmetry in the Hilbert-Einstein Lagrangian? (which gauge group?) Ning Wu, Y.M. Cho, …. ‘yes’. Really? Explicit calculations….. (3) Using Yang-Mills’ Approach based on Flat space-time to investigate and improve the relationship between Quantum Mechanics and Gravity.

6. Gauge Symmetry Symmetry appears to be the deepest foundation for our understanding of the physical world. Two established conservation laws: Two established conservation laws: (1) T(4) symmetry: Energy-Momentum Conservation In curved space-time :Hayashi & Nakano, Prog. Theor. Phys(1967) Utiyama & Fukuyama, P.T.P. (1971); Y.M. Cho, P.R. (1976), *** In Flat space-time : Ning Wu, Commun. Theor. Phys. 2002-2004 In Flat space-time : Ning Wu, Commun. Theor. Phys. 2002-2004 JPH, Int. J. Mod. Phys. A, 2006. *** JPH, Int. J. Mod. Phys. A, 2006. *** (2) U(1) symmetry: Baryon number conservation T.D.Lee & C. N. Yang, P.R. (1955), [“Cosmic Lee-Yang force”] Tosa, Marshak and S. Okubo, P. R. D (1983). {{ {It is straightforward to include (electron-) lepton number conservation: T(4)xU(1)xU(1)}}} *** SUGGESTION: Gauge symmetry is powerful for cancellations of ultraviolet divergences of interacting fields ONLY in the framework of flat spacetime, but not in curved spacetime. (Why?......) (Why?......)

7. Why accelerated expansion? Cosmic repulsive constant force between baryons (A simple picture in flat spacetime approximation) L=-(L s 2 /4)  B   B  + L’( u- and d-quarks) L=-(L s 2 /4)  B   B  + L’( u- and d-quarks) B  =   B -  B  B  =   B -  B       B   g’J  = 0  g’=g/(3L s 2 ) linear (static) potential: B o  r linear (static) potential: B o  r Eötvös-type experiments: g’<10 -62 /cm 2 Eötvös-type experiments: g’<10 -62 /cm 2 d 2 r/dt 2 =g grav +g B, g B =constant d 2 r/dt 2 =g grav +g B, g B =constant (General relativity with a cosmological constant: d 2 r/dt 2 =g grav +      r ) Peebles et al, Rev. Mod. Phys. 2003 (General relativity with a cosmological constant: d 2 r/dt 2 =g grav +      r ) Peebles et al, Rev. Mod. Phys. 2003 Experimental test? r-dependent acceleration? Experimental test? r-dependent acceleration?

8. Fourier transform of the Feynman propagator with vanishing Fourier transform of the Feynman propagator with vanishing time-component k 0. * The only force with a field-theoretic interpretation. Why linear potential*?

9. Gauge Symmetry in Flat Spacetime Yang-Mills Approach in Flat 4-dim space-time Yang-Mills Approach in Flat 4-dim space-time Translation Gauge Symmetry T(4) x   x  +   (x)   (x): infinitesimal arbitrary function. x   x  +   (x)   (x): infinitesimal arbitrary function. 4-dim spacetime displacement operator p = i ∂ 4-dim spacetime displacement operator p = i ∂ T(4) gauge symmetry dictates  . (c= ћ =1) Gauge covariant derivative ∆  : ∂   ∂  -ig   p = J  ∂ = ∆ . J  =   + g  ,   =    J  =   + g  ,   =   

10. A Basic Observation: Dual interpretation of x   x’  =x  +   (x) (i) a shift of spacetime (ii)an arbitrary infinitesimal transformation of coordinates.  Yang-Mills gravity for both inertial and non-inertial frames

11. A physical quantity Q  1……  n  1……  m (x) Infinitesimal T(4) gauge transformation: Q  1 …  n  1 …  m (x)  {Q  1 …  n 1 … m (x)  ∂ Q  1 …  n 1 … m (x)}  (∂x’  1 / ∂x 1 )… (∂x’  m / ∂x m )  (∂x’  1 / ∂x 1 )… (∂x’  m / ∂x m )  (∂x  1 / ∂x’  1 )… (∂x  n / ∂x’  n )

12. For examples: Q(x)  Q(x) -  ∂ Q(x) A  (x)  A  (x)-  ∂ A  (x)+A ∂   A  (x)  A  (x) -  ∂ A  (x) - A ∂   A  (x)  A  (x) -  ∂ A  (x) - A ∂   A  (x)  A  (x) -  ∂ A  (x) - A  ∂  - A ∂   - A  ∂  - A ∂   ……… etc. Formally similar to the Lie variations in Riemann Geo.

13. General frames of Reference ( inertial and non-inertial frames with zero curvature tensor ) metric tensor P  (x)        =(+  ) ( in the limit of zero acceleration ) Example: constant linear acceleration  o P  (x)=(W o 2, -1,-1,-1), W o 2 =  2 (  o -2 +  o x) General frames of Reference ( inertial and non-inertial frames with zero curvature tensor ) metric tensor P  (x)        =(+  ) ( in the limit of zero acceleration ) Example: constant linear acceleration  o P  (x)=(W o 2, -1,-1,-1), W o 2 =  2 (  o -2 +  o x)

14. Inertial F I  Accelerated frame F F I (w I,x I,y I,z I )  F(w,x,y,z) w I =  [x+1/(  o  o 2 )] -  o /(  o  o ), w I =  [x+1/(  o  o 2 )] -  o /(  o  o ), x I =  [x+1/(  o  o 2 )] - 1/(  o  o ), y I =y, z I =z. x I =  [x+1/(  o  o 2 )] - 1/(  o  o ), y I =y, z I =z. { dw I =  [W o dw+  dx], dx I =  [dx+  W o dw],}  =  o w+  o,  o =1/(1-  o 2 ) 1/2,  =  o w+  o,  o =1/(1-  o 2 ) 1/2,  =1/(1-  2 ) 1/2,  o = constant linear acceleration  =1/(1-  2 ) 1/2,  o = constant linear acceleration Principle of limiting 4-dimensional symmetry: In the limit  o  0, all accelerated transformations must reduce to Lorentz transformations Principle of limiting 4-dimensional symmetry: In the limit  o  0, all accelerated transformations must reduce to Lorentz transformations

15. T(4) Gauge symmetry requires the replacement: ∂   ∂  +g   ∂  J  ∂  ∆ Gauge covariant derivative: ∆ [∆ , ∆ ]=C  ∂  Gauge curvature: C , C  = J  (∂  J  ) - J  (∂  J  ), J  =   + g  ,   =   T(4) Gauge symmetry requires the replacement: ∂   ∂  +g   ∂  J  ∂  ∆ Gauge covariant derivative: ∆ [∆ , ∆ ]=C  ∂  Gauge curvature: C , C  = J  (∂  J  ) - J  (∂  J  ), J  =   + g  ,   =  

16. (1/2)   (    )(   )    (1/2)G  (    )(    ) G  =   +2g   +g 2       Two interpretations: (A) The spacetime really becomes curved. ( Following Einstein….). (B) As if the space-time becomes curved. (Yang-Mills) “Effective metric tensor” G  is due to the presence of the tensor field in flat spacetime. The real physical spacetime is still flat for Yang-Mills Theory. For simplicity of discussion, let us consider Only inertial frames (P  (x) =  , D  =   ):

17. Lagrangian and Field Equation L=  (1/2 g 2 )(C  C   C   C    )+L  L=  (1/2 g 2 )(C  C   C   C    )+L   H  =  g 2 T    H  =   [J  C   J  C     + C   J ]   C   J  + C     J   - C    J    C   J  + C     J   - C    J  T  = (1/2)[  i          ]

18. Linearized tensor field equation, T(4)                                     g  T    T  ).  The same as those in GR.  Higher order terms are different from those in GR.

19.        Simplified field equation:        g  T     T  )   g S   usual retarded potential:   x,t) =  ( g/4  )  d 3 x ’ S  ( x ’,t-| x - x ’ |)/| x - x ’ |,  Newtonian approximation gives g =(8  G) 1/2 g    g     Gm/r, etc. Gravitational Radiation Gauge condition:        Simplified field equation:        g  T     T  )   g S   usual retarded potential:   x,t) =  ( g/4  )  d 3 x ’ S  ( x ’,t-| x - x ’ |)/| x - x ’ |,  Newtonian approximation gives g =(8  G) 1/2 g    g     Gm/r, etc.

20.      g  T  + t   )      g  T  + t   ) To a second order approximation, the energy-momentum of the gravitational field t  is To a second order approximation, the energy-momentum of the gravitational field t  is t  = (      (      t  = (      (                                                                                                              (    

21. Gravitational Radiation In the wave zone at a distance much larger than the dimension of the source, the solution of the field can be approximated by a plane wave: In the wave zone at a distance much larger than the dimension of the source, the solution of the field can be approximated by a plane wave:   x  e  exp(-ik.xe   exp(ik.x)  Take the average of t  over a region space and time much larger than the wavelengths of the radiated waves: = 2k  k e  e*  + k  k e e*  

22. The total power P o emitted by a rotating body ----rotating around one of the principal axes of the ellipsoid of inertia At twice the rotating frequency  i.e. At twice the rotating frequency  i.e.   =2  P o (2  )=32G  6 I 2 e 2 /5. P o (2  )=32G  6 I 2 e 2 /5. I=moment of inertia, I=moment of inertia, e= equatorial ellipticity e= equatorial ellipticity The same as that obtained in GR. The same as that obtained in GR.

23. Remarks difference between Yang-Mills gravity and GR in the 2nd order approximation The advance of the perihelion for one revolution of the planet The advance of the perihelion for one revolution of the planet  =[6Gm/P][1  3(E o 2  m p 2 )/(4m p 2 )],  =[6Gm/P][1  3(E o 2  m p 2 )/(4m p 2 )], Where P=M 2 /(m p 2 Gm), M=angular momentum Where P=M 2 /(m p 2 Gm), M=angular momentum The difference can be tested if the velocity of the Mercury is about 0.1c The difference can be tested if the velocity of the Mercury is about 0.1c Bending of Light Bending of Light   =[4Gm  o /M][1  18G 2 m 2  o 2 /M 2 ] The correction term is too small to test. The correction term is too small to test.

24. Conclusions Yang-Mills gravity [based on T(4) x U(1) in flat spacetime] is viable, and can provide a field- theoretic explanation of the accelerated expansion of the universe. Yang-Mills gravity [based on T(4) x U(1) in flat spacetime] is viable, and can provide a field- theoretic explanation of the accelerated expansion of the universe. Quadrupole radiations cannot be distinguished from that of GR by known experiments Quadrupole radiations cannot be distinguished from that of GR by known experiments It suggests that classical gravity with an effective metric tensor shows up only in the limit of geometric optics (i.e., classical limit) of field theory. It suggests that classical gravity with an effective metric tensor shows up only in the limit of geometric optics (i.e., classical limit) of field theory. The energy-momentum tensor and its conservation in Yang-Mills gravity are The energy-momentum tensor and its conservation in Yang-Mills gravity arewell-defined.

25. Conjectures It should be possible to quantize Yang-Mills gravity in flat spacetime, and the maximum interaction vertex for gravitons is the 4-vertex (in Feynman rules) [ The generators of T(4) group do not have the usual constant matrix representation.] It should be possible to quantize Yang-Mills gravity in flat spacetime, and the maximum interaction vertex for gravitons is the 4-vertex (in Feynman rules) [ The generators of T(4) group do not have the usual constant matrix representation.] The divergence in higher order amplitudes in Yang-Mills gravity should be less than that in GR (with ∞ -vertex of gravitons) The divergence in higher order amplitudes in Yang-Mills gravity should be less than that in GR (with ∞ -vertex of gravitons) A field with a fourth-order differential equation can lead to a linear potential, which may have something to do with quark confinement A field with a fourth-order differential equation can lead to a linear potential, which may have something to do with quark confinement