Yang-Mills Gravity Yang-Mills Gravity (vs. Einstein Gravity)(vs. Einstein Gravity)

Jong-Ping HsuJong-Ping Hsu

Physics DepartmentPhysics Department

Univ. of Massachusetts Univ. of Massachusetts DartmouthDartmouth

Einstein’s original idea for the future of Einstein’s original idea for the future of physics:physics:

• Geometrization of physicsGeometrization of physics

• Riemann GeometryRiemann Geometry

• Weyl GeometryWeyl Geometry

• Finsler Geometry, ……Finsler Geometry, ……

• **************************************************************

• Fiber Bundle (Yang-Mills approach)Fiber Bundle (Yang-Mills approach)

F. Dyson:F. Dyson: (A founder of QED, together with (A founder of QED, together with Tomonaga, Schwinger, & Feynman)Tomonaga, Schwinger, & Feynman)

He stressed thatHe stressed that““The most glaring incompatibility of The most glaring incompatibility of

concepts in contemporary physics is that concepts in contemporary physics is that between the principle of general between the principle of general coordinate invariance and a quantum-coordinate invariance and a quantum-mechanical description of all of nature.’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 Quantum gravity appears to be the last challenge to the powerful gauge symmetry of the Yang-Mills theory.powerful gauge symmetry of the Yang-Mills theory.

Yang-Mills Gravity ,T(4)x…..Yang-Mills Gravity ,T(4)x….. MotivationsMotivations: :

(1) It is very hard for conventional field (1) It is very hard for conventional field theory to describe gravity. Why? theory to describe gravity. Why? (ds(ds22=……)=……)

(2) Is there a hidden Yang-Mills gauge (2) Is there a hidden Yang-Mills gauge symmetry in the Hilbert-Einstein symmetry in the Hilbert-Einstein Lagrangian? Lagrangian? (which gauge group?)(which gauge group?)

Ning Wu, Y.M. Cho, …. ‘yes’Ning Wu, Y.M. Cho, …. ‘yes’. . Really? Explicit calculations…..Really? Explicit calculations…..

(3) Using Yang-Mills’ Approach based on Flat (3) Using Yang-Mills’ Approach based on Flat space-time to investigate and improve the space-time to investigate and improve the relationship between Quantum Mechanics relationship between Quantum Mechanics and Gravity.and Gravity.

Gauge SymmetryGauge SymmetrySymmetry appears to be the deepest foundation for Symmetry appears to be the deepest foundation for our understanding of the physical world.our understanding of the physical world.

• Two established conservation laws:Two established conservation laws:(1) T(4) symmetry: Energy-Momentum Conservation(1) T(4) symmetry: Energy-Momentum Conservation In curved space-timeIn curved space-time :Hayashi & Nakano, Prog. Theor. Phys(1967) :Hayashi & Nakano, Prog. Theor. Phys(1967) Utiyama & Fukuyama, P.T.P. (1971); Y.M. Cho, P.R. (1976),Utiyama & Fukuyama, P.T.P. (1971); Y.M. Cho, P.R. (1976),*** ***

In Flat space-timeIn Flat space-time: Ning Wu, Commun. Theor. Phys. 2002-2004: 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(2) U(1) symmetry: Baryon number conservation T.D.Lee & C. N. Yang, P.R. (1955), [“Cosmic Lee-Yang force”]T.D.Lee & C. N. Yang, P.R. (1955), [“Cosmic Lee-Yang force”]

Tosa, Marshak and S. Okubo, P. R. D (1983).Tosa, Marshak and S. Okubo, P. R. D (1983).{{{{{It is straightforward to include (electron-) lepton number conservation: {It is straightforward to include (electron-) lepton number conservation: T(4)xU(1)xU(1)}}}T(4)xU(1)xU(1)}}}

****** SUGGESTION: Gauge symmetry is powerful for cancellations of SUGGESTION: Gauge symmetry is powerful for cancellations of ultraviolet divergences of interacting fields ONLY in the framework ultraviolet divergences of interacting fields ONLY in the framework of flat spacetime, but not in curved spacetime.of flat spacetime, but not in curved spacetime.

(Why?......)(Why?......)

Why accelerated expansion?Why accelerated expansion?Cosmic repulsive constant force between Cosmic repulsive constant force between baryonsbaryons(A simple picture in flat spacetime approximation)(A simple picture in flat spacetime approximation)

• L=-(LL=-(Lss22/4)/4)BBBB+ L’(+ L’(u- and d-quarks)u- and d-quarks)

• BB==BB- - BB

BB g’J g’J = 0 = 0 g’=g/(3L g’=g/(3Lss22))

• linear (static) potential: Blinear (static) potential: Boo r r• Eötvös-type experiments: g’<10Eötvös-type experiments: g’<10-62-62/cm/cm22

• dd22rr/dt/dt22==gggravgrav++ggB B , , ggBB=constant=constant

• (General relativity with a cosmological constant: (General relativity with a cosmological constant:

dd22rr/dt/dt22==gggravgrav++rr )) Peebles et al, Rev. Mod. Phys. 2003Peebles et al, Rev. Mod. Phys. 2003

• Experimental test? r-dependent acceleration?Experimental test? r-dependent acceleration?

• Fourier transform of the Feynman Fourier transform of the Feynman propagator with vanishing propagator with vanishing

time-component ktime-component k00..

* The only force with a field-theoretic interpretation.* The only force with a field-theoretic interpretation.

1(k2)2

eikrd3k= - r

Why linear potential*?Why linear potential*?

Gauge Symmetry in Flat Gauge Symmetry in Flat SpacetimeSpacetime

• Yang-Mills Approach in Flat 4-dim space-timeYang-Mills Approach in Flat 4-dim space-time

Translation Gauge Symmetry T(4) Translation Gauge Symmetry T(4) xx x x++(x) (x) (x): infinitesimal arbitrary function.(x): infinitesimal arbitrary function.

4-dim spacetime displacement operator p4-dim spacetime displacement operator p = = ii ∂∂

T(4) gauge symmetry T(4) gauge symmetry dictates dictates .. (c=(c=ћћ=1)=1)

Gauge covariant derivative Gauge covariant derivative ∆∆ : :

∂∂ ∂ ∂-ig-igpp = J= J ∂∂

= ∆= ∆..

JJ = = + g+ g, , = =

A Basic Observation:A Basic Observation:Dual interpretation of Dual interpretation of xx x’ x’=x=x++(x) (x) (i) a shift of spacetime(i) a shift of spacetime(ii)an arbitrary infinitesimal (ii)an arbitrary infinitesimal transformation of transformation of coordinates.coordinates.

Yang-Mills gravity for Yang-Mills gravity for both inertial and non-inertial both inertial and non-inertial framesframes

A physical quantity A physical quantity Q Q1……1……nn

1……1……mm(x)(x)

Infinitesimal T(4) gauge transformation:Infinitesimal T(4) gauge transformation:

QQ11……nn

11……mm(x)(x){Q{Q11……nn

11……mm(x)(x)∂∂QQ11……nn

11……

mm

(x)}(x)}

(∂x’(∂x’11/ ∂x/ ∂x11)… (∂x’)… (∂x’mm/ ∂x/ ∂xmm))

(∂x(∂x11/ ∂x’/ ∂x’11)… (∂x)… (∂xnn/ ∂x’/ ∂x’nn))

For examples:For examples:

Q(x) Q(x) Q(x) - Q(x) - ∂∂Q(x) Q(x)

AA(x)(x)AA(x)- (x)- ∂∂AA(x)+A(x)+A∂∂

AA(x)(x)AA(x) - (x) - ∂∂AA(x) - A(x) - A∂∂

AA(x)(x)AA (x) - (x) - ∂∂AA(x) (x)

- A- A∂∂ - A- A∂∂

……… ……… etc.etc.

FormallyFormally similar to the Lie variations in Riemann Geo. similar to the Lie variations in Riemann Geo.

General frames of ReferenceGeneral frames of Reference((inertial and non-inertial frames with zero curvature inertial and non-inertial frames with zero curvature

tensortensor) )

metric tensor Pmetric tensor P(x)(x)

=(+=(+) ) ((in the limit of zero accelerationin the limit of zero acceleration))

Example: constant linear acceleration Example: constant linear acceleration oo

P P(x)=(W(x)=(Woo22, -1,-1,-1), W, -1,-1,-1), Woo

22==22

((oo-2-2

+ + oox) x)

Inertial FInertial FII Accelerated Accelerated frame Fframe FFFII(w(wII,x,xII,y,yII,z,zII))F(w,x,y,z)F(w,x,y,z)• wwII==[x+1/([x+1/(oooo

22)] - )] - oo/(/(oooo),),• xxII==[x+1/([x+1/(oooo

22)] - 1/()] - 1/(oooo), y), yII=y, z=y, zII=z.=z.

{ dw{ dwI I ==[W[Woodw+dw+dx] , dxdx] , dxI I ==[dx+[dx+WWoodw] ,}dw] ,}

= = oow+w+o,o, o o =1/(1- =1/(1- oo22))1/21/2, ,

=1/(1- =1/(1- 22))1/21/2, , oo= constant linear acceleration= constant linear acceleration

• Principle of limiting 4-dimensional symmetry: In the Principle of limiting 4-dimensional symmetry: In the limit limit oo 0, all accelerated transformations must 0, all accelerated transformations must reduce to Lorentz transformationsreduce to Lorentz transformations

T(4) Gauge symmetry requires T(4) Gauge symmetry requires the replacement:the replacement: ∂∂ ∂ ∂+g+g∂∂JJ ∂∂ ∆ ∆

Gauge covariant derivative: ∆Gauge covariant derivative: ∆

[∆[∆, ∆, ∆]=C]=C ∂∂

Gauge curvature: Gauge curvature: CC

, , CC = J= J(∂(∂JJ

) - J) - J(∂(∂JJ),),

JJ = = + g+ g, , = =

(1/2) (1/2) (()()() ) (1/2)G (1/2)G(()()())

GG==+2g+2g +g +g22

Two interpretations:Two interpretations:(A) The spacetime really becomes curved. ((A) The spacetime really becomes curved. (Following Following

Einstein….).Einstein….).

(B) As if the space-time becomes curved. (Yang-(B) As if the space-time becomes curved. (Yang-Mills)Mills)

““Effective metric tensor” GEffective metric tensor” G is due to the presence of the tensor is due to the presence of the tensor field in flat spacetime.field in flat spacetime.

The real physical spacetime is still flat for Yang-Mills Theory.The real physical spacetime is still flat for Yang-Mills Theory.

For simplicity of discussion, let us consider For simplicity of discussion, let us consider Only inertial frames (POnly inertial frames (P(x) = (x) = , , DD==):):

Lagrangian and Field Equation Lagrangian and Field Equation

L=L= (1/2 (1/2gg22)(C)(CCC C CCC

)+L)+L

HH= = gg22TT

HH= = [J[J

CC J JCC

+ C+ CJJ]]

CCJJ+ C+ CJJ

- C- CJJ

TT = (1/2)[ = (1/2)[ii ]]

Linearized tensor field equation, Linearized tensor field equation, T(4)T(4)

ggTT TT) ) ..

The same as those in GR.The same as those in GR.

Higher order terms are different from those in Higher order terms are different from those in

GR .GR .

Gravitational Radiation

Gauge condition:

Simplified field equation:Simplified field equation:

g gTT TT

) ) ggSS

usual retarded potential:usual retarded potential:

xx,t),t)= = ((g/4g/4))dd33x’Sx’S((xx’,t-|’,t-|xx--xx’|)/|’|)/|xx--xx’|,’|,Newtonian approximation givesNewtonian approximation givesgg=(8=(8G)G)1/21/2 gg g g Gm/r , Gm/r , etc.etc.

ggTT+ t+ t)) • To a second order approximation, the energy-momentum of the To a second order approximation, the energy-momentum of the

gravitational field gravitational field ttisis

• tt= (= (((

((

Gravitational RadiationGravitational Radiation• In the wave zone at a distance much larger than the dimension of the 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:source, the solution of the field can be approximated by a plane wave:

xexeexp(-ik.xeexp(-ik.xeexp(ik.x)exp(ik.x)

Take the average of tTake the average of t over a region space over a region space and time much larger than the wavelengths and time much larger than the wavelengths of the radiated waves:of the radiated waves:

<t<t>=>=2k2kkkeee*e*+ k+ kkkeee*e*

The total power PThe total power Poo emitted by a emitted by a rotating body ----rotating around one rotating body ----rotating around one of the principal axes of the ellipsoid of of the principal axes of the ellipsoid of inertiainertia• At twice the rotating frequency At twice the rotating frequency i.e.i.e.

=2 =2

•PPoo(2(2)=32G)=32G66II22ee22/5./5.

• I=moment of inertia, I=moment of inertia, • e= equatorial ellipticitye= equatorial ellipticity

•The same as that obtained in The same as that obtained in GR.GR.

RemarksRemarksdifference between Yang-Mills gravity and difference between Yang-Mills gravity and GRGRin the 2nd order approximationin 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 =[6Gm/P][1 3(E 3(Eoo2 2 m mpp

22)/(4m)/(4mpp22)],)],

• Where P=MWhere P=M22/(m/(mpp22Gm), M=angular momentumGm), M=angular momentum

• The difference can be tested if the velocity of the The difference can be tested if the velocity of the Mercury is about 0.1cMercury is about 0.1c

• Bending of LightBending of Light

=[4Gm=[4Gmoo/M][1 /M][1 18G 18G22mm22oo22/M/M2 2 ]]

• The correction term is too small to test.The correction term is too small to test.

ConclusionsConclusions• Yang-Mills gravity [Yang-Mills gravity [based on T(4) x U(1) in flat based on T(4) x U(1) in flat

spacetimespacetime] is viable, and can provide a field-] is viable, and can provide a field-theoretic explanation of the accelerated expansion theoretic explanation of the accelerated expansion of the universe.of the universe.

• Quadrupole radiations cannot be distinguished Quadrupole radiations cannot be distinguished from that of GR by known experimentsfrom that of GR by known experiments

• It suggests that classical gravity with an effective It suggests that classical gravity with an effective metric tensor shows up only in the limit of metric tensor shows up only in the limit of geometric optics (i.e., classical limit) of field theory.geometric optics (i.e., classical limit) of field theory.

• The energy-momentum tensor and its conservation The energy-momentum tensor and its conservation in Yang-Mills gravity are in Yang-Mills gravity are well-defined.well-defined.

ConjecturesConjectures• It should be possible to quantize Yang-Mills It should be possible to quantize Yang-Mills

gravity in flat spacetime, and the maximum gravity in flat spacetime, and the maximum interaction vertex for gravitons is the 4-vertex interaction vertex for gravitons is the 4-vertex (in Feynman rules) [(in Feynman rules) [The generators of T(4) group The generators of T(4) group do not have the usual constant matrix representationdo not have the usual constant matrix representation.].]

• The divergence in higher order amplitudes in The divergence in higher order amplitudes in Yang-Mills gravity should be less than that in Yang-Mills gravity should be less than that in GR (with ∞ -vertex of gravitons)GR (with ∞ -vertex of gravitons)

• A field with a fourth-order differential equation A field with a fourth-order differential equation can lead to a linear potential, which may have can lead to a linear potential, which may have something to do with quark confinementsomething to do with quark confinement