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Yang-Mills Gravity and Yang-Mills Gravity and Accelerated Cosmic Expansion* Accelerated Cosmic Expansion*

((Based on a Model with Generalized Gauge Based on a Model with Generalized Gauge SymmetrySymmetry) )

Jong-Ping HsuJong-Ping Hsu

Physics Department, Univ. of Massachusetts Dartmouth,Physics Department, Univ. of Massachusetts Dartmouth,

North Dartmouth, Massachusetts 02747, USANorth Dartmouth, Massachusetts 02747, USA

*Collaborators: (1) Leonardo Hsu (Space-time symmetry and *Collaborators: (1) Leonardo Hsu (Space-time symmetry and quantum Yang-Mills gravity, World quantum Yang-Mills gravity, World

Scientific), Scientific),

(2) Kazuo Cottrell (‘A unified model with a generalized (2) Kazuo Cottrell (‘A unified model with a generalized gauge symmetry and its cosmological implications’.) gauge symmetry and its cosmological implications’.)

• (I) A BIG PICTURE OF SPACE-TIME: (I) A BIG PICTURE OF SPACE-TIME: • There exists a fundamental (flat) space-time There exists a fundamental (flat) space-time

symmetry framework that can encompass all symmetry framework that can encompass all interactions in physics, including gravity, and is valid interactions in physics, including gravity, and is valid for both inertial and non-inertial frames.for both inertial and non-inertial frames.

• (II) A UNIFIED PICTURE OF ALL FORCES: (II) A UNIFIED PICTURE OF ALL FORCES: • There exist fundamental gauge symmetries, which There exist fundamental gauge symmetries, which

dictate all basic interactions in nature. dictate all basic interactions in nature.

A. Gravity---Yang-Mills gravity---space-time A. Gravity---Yang-Mills gravity---space-time translational gauge symmetry Ttranslational gauge symmetry T44 (external, exact) (external, exact)

B. Electroweak---SUB. Electroweak---SU22 x U x U11 (spontaneous sym breaking) (spontaneous sym breaking)

• Strong force (QCD)---(SUStrong force (QCD)---(SU33))color color (exact) (exact)

• Cosmic baryonic (& leptonic) forces---UCosmic baryonic (& leptonic) forces---U1b 1b (exact)(exact)

• Such a unified model follows the Such a unified model follows the ideas of Glashow, Salam, Ward, and ideas of Glashow, Salam, Ward, and Weinberg.Weinberg.

• It can be formulated for both inertial It can be formulated for both inertial and non-inertial frames. and non-inertial frames.

• Symmetry appears to be the deepest Symmetry appears to be the deepest

foundation for our understanding of foundation for our understanding of the physical universe. the physical universe.

• Two basic frameworks in physics: Two basic frameworks in physics:

••1. Flat space-time: 1. Flat space-time:

• All field theories for electroweak and strong All field theories for electroweak and strong interactions interactions

2. Curved space-time: 2. Curved space-time:

• Einstein’s gravity Einstein’s gravity

• However, Einstein’s symmetry principle of However, Einstein’s symmetry principle of general coordinate invariance is a general coordinate invariance is a profound idea with highly non-trivial profound idea with highly non-trivial difficulties.difficulties.

• Why?Why?

• Gravity???Gravity???

• F. Dyson: (A founder of QED, together with F. Dyson: (A founder of QED, together with

Tomonaga, Schwinger, & Feynman) Tomonaga, Schwinger, & Feynman)

• Dyson stressed that Dyson stressed that

• ““The most glaring incompatibility of concepts The most glaring incompatibility of concepts in in

contemporary physics is that between contemporary physics is that between Einstein’s Einstein’s

principle of general coordinate invariance principle of general coordinate invariance and and all the modern schemes for quantum-all the modern schemes for quantum-mechanical description of naturemechanical description of nature.” .”

• (‘Missed Opportunity’, J. W. Gibbs Lecture at Amer. (‘Missed Opportunity’, J. W. Gibbs Lecture at Amer. Math. Soc. 1972) Math. Soc. 1972)

• This incompatibility is a MOTIVATION for our This incompatibility is a MOTIVATION for our research….. research…..

Gravity???Gravity???

E. P. Wigner, Symmetries and Reflections E. P. Wigner, Symmetries and Reflections

•(MIT Press, 1967) pp. 52-53 (MIT Press, 1967) pp. 52-53

• “• “The basic premise of this theory [general relativity] The basic premise of this theory [general relativity] is that coordinates are only auxiliary quantities which is that coordinates are only auxiliary quantities which can be given arbitrary values for every event. Hence, can be given arbitrary values for every event. Hence, the measurement of position, that is, of the space the measurement of position, that is, of the space coordinates, is certainly not a significant measurement coordinates, is certainly not a significant measurement if the postulates of the general theory are adopted…. if the postulates of the general theory are adopted…. Most of us have struggled with the problem of how, Most of us have struggled with the problem of how, under these premises, the general theory of relativity under these premises, the general theory of relativity can make meaningful statements and predictions at can make meaningful statements and predictions at all………all……….” .”

•Noether’s theorem II: No conservation of energy in Noether’s theorem II: No conservation of energy in GRGR

•Hilbert made a similar remark (around 1915)Hilbert made a similar remark (around 1915)

••Why should gauge symmetry in flat space-time Why should gauge symmetry in flat space-time be so successful for modeling all known be so successful for modeling all known interactions except gravity ? interactions except gravity ?

•Yang-Mills GravityYang-Mills Gravity (with a generalized gauge symmetry) (with a generalized gauge symmetry)

Yang-Mills Gravity enables us to have Yang-Mills Gravity enables us to have

A UNIFIED PICTURE OF ALL FORCES based on gauge symmetry: A UNIFIED PICTURE OF ALL FORCES based on gauge symmetry:

(1)(1)Yang-Mills gravityYang-Mills gravity---space-time translational gauge symmetry ---space-time translational gauge symmetry TT44

(2) Electroweak forces--- SU(2) Electroweak forces--- SU22 x U x U11 symmetry symmetry

(3) Strong force (QCD)--(SU(3) Strong force (QCD)--(SU33))color symm color symm

(4) (4) Baryonic forceBaryonic force---U---U1b 1b (accelerated cosmic expansion).(accelerated cosmic expansion).

(5) Leptonic force---U(5) Leptonic force---U1l 1l (accelerated cosmic expansion).(accelerated cosmic expansion).

To illustrate generalized gauge symmetry, I shall discuss (1) and To illustrate generalized gauge symmetry, I shall discuss (1) and (4) in this talk.(4) in this talk.

Gauge Symmetry in Flat SpacetimeGauge Symmetry in Flat Spacetime(a generalization of Yang-Mills’ internal gauge symmetry to (a generalization of Yang-Mills’ internal gauge symmetry to include external gauge symmetry)include external gauge symmetry)

• • Generalized Yang-Mills idea of gauge symmetry in Generalized Yang-Mills idea of gauge symmetry in

Flat 4-dim space-time----- a profound ideaFlat 4-dim space-time----- a profound idea

Local space-time translation gauge symmetry TLocal space-time translation gauge symmetry T44

•xxμμ→ x→ xμμ+Λ+Λμμ(x)(x), , ηημνμν=(1,-1,-1,-1), (c=ћ=1)=(1,-1,-1,-1), (c=ћ=1)

•ΛΛμμ(x): infinitesimal arbitrary function of space-time(x): infinitesimal arbitrary function of space-time

•4-dim displacement operator p4-dim displacement operator pνν=i ∂=i ∂νν=i∂/∂x=i∂/∂xνν

•T(4) gauge symmetry dictates the tensor fields φT(4) gauge symmetry dictates the tensor fields φμνμν. .

•Gauge covariant derivative ΔGauge covariant derivative Δμμ(x) :(x) :

•∂∂μμ→ ∂→ ∂μμ-- igφigφμνμνppνν = J = Jμνμν ∂ ∂νν= Δ= Δμμ..

•JJμνμν = η = ημνμν + gφ + gφμνμν, , φφμνμν = φ = φνμνμ..

• A Basic Observation:A Basic Observation:

• Dual interpretations ofDual interpretations of

****** xxμμ→ x’→ x’μμ=x=xμμ+Λ+Λμμ(x)(x)(ia) a local shift (translation) in (ia) a local shift (translation) in flatflat space-time, space-time,

(ib) an arbitrary infinitesimal coordinate (ib) an arbitrary infinitesimal coordinate transformation in transformation in flatflat space-time. space-time.

(ii) an arbitrary infinitesimal transformations of (ii) an arbitrary infinitesimal transformations of coordinates in coordinates in curvedcurved space-time. (GR) space-time. (GR)

******This is the key conceptual departure from that of GR.This is the key conceptual departure from that of GR.

(Early discussions of gravity based on flat space-time or (Early discussions of gravity based on flat space-time or translational gauge symmetry: A. A. Logunov, M.A. translational gauge symmetry: A. A. Logunov, M.A. Mestvirishvili, A.A. Vlasov, Y.M. Cho, N. Wu and others.)Mestvirishvili, A.A. Vlasov, Y.M. Cho, N. Wu and others.)

• Interpretation (Interpretation (ia,ibia,ib) ) Yang-Mills gravity in flat Yang-Mills gravity in flat

space-time (for both inertial and non-inertial frames).space-time (for both inertial and non-inertial frames).

• TT44 Gauge symmetry postulates the Gauge symmetry postulates the replacement in the Lagrangian:replacement in the Lagrangian:

• ∂∂μμ→ ∂→ ∂μμ+gφ+gφμνμν∂∂νν=J=Jμνμν ∂ ∂νν = Δ = Δμμ,, (c=ћ=1) (c=ћ=1)

• (i) g is not dimensionless, (i) g is not dimensionless, (dimension of (dimension of g=length)g=length)

• (ii) φ(ii) φμνμν is not a vector field is not a vector field• They differ from those in usual Yang-Mills gauge symmetryThey differ from those in usual Yang-Mills gauge symmetry..

• [Δ[Δμμ, Δ, Δνν]=C]=Cμναμνα ∂ ∂αα

TT44 Gauge curvature: C Gauge curvature: Cμναμνα

• CCμναμνα = J = Jμαμα(∂(∂ααJJνανα) - J) - Jνβνβ(∂(∂ββJJνανα), ),

• JJμνμν = η = ημνμν + gφ + gφμνμν, ,

• Lagrangian and Field EquationsLagrangian and Field Equations

• L= L= -- (1/2g (1/2g22)(C)(CμαβμαβCCμβαμβα -- C CμαμαααCCμβμβ

ββ) + L) + Lψψ ,,

• where Cwhere CμαβμαβCCμβαμβα = C = CμαβμαβCCμαβμαβ/2 ./2 .

• HHμνμν= = -- g g22TTμνμν

• HHμνμν = = -- ∂ ∂λλ{{JJλλααCCαμναμν -- J Jλλ

ααCCαβαβββηημνμν + C + Cμβμβ

ββJJνλνλ}} -- C Cμαβμαβ ∂ ∂ννJJαβαβ+ C+ Cμβμβ

ββ∂∂νν J Jαααα-C-Cλβλβ

ββ∂∂ννJJμμλλ

• TTμνμν = (1/2)[ = (1/2)[ψψiγiγμμ∂∂ννψ - (i∂ψ - (i∂ννψψ)γ)γμμψ]ψ]

• Interesting results:Interesting results:• In the limit of geometric-optics (i.e., classical limit), In the limit of geometric-optics (i.e., classical limit),

the wave eqs. of massive fermions and bosons the wave eqs. of massive fermions and bosons reduces to the same Hamilton-Jacobi type equationreduces to the same Hamilton-Jacobi type equation

•GGμνμν∂∂μμSS ∂∂ννSS = = mm22,, GGμνμν=η=ηαβαβJJαμαμJJβνβν ,,• where Gwhere Gμνμν appears to be an effective “Riemannian appears to be an effective “Riemannian

metric tensor” for (and only for) a classical object. metric tensor” for (and only for) a classical object.

• But for quantum fields and particles, the physical But for quantum fields and particles, the physical space-time is flat.space-time is flat.

• Maxwell’s eqs. Maxwell’s eqs. (classical limit) (classical limit) eikonal equation eikonal equation with a slightly different metric tensor Gwith a slightly different metric tensor GLL

μνμν

• • Effective curved space-time for the motion of Effective curved space-time for the motion of classical classical objects in Yang-Mills gravityobjects in Yang-Mills gravity

Experimental Results:Experimental Results:

•Perihelion shift-----`same’ as the usual resultPerihelion shift-----`same’ as the usual result

(within experimental accuracy)(within experimental accuracy)

•Red shift----`same’Red shift----`same’

•Gravitational quadrupole radiation-----`same’Gravitational quadrupole radiation-----`same’

•Bending of light ---- ‘Bending of light ---- ‘differentdifferent’’

• • Bending of Light Δφ=1.53” (only for light Bending of Light Δφ=1.53” (only for light rays with optical frequency)rays with optical frequency)

• • 12% smaller than the usual value 1.75”12% smaller than the usual value 1.75”• • Experimental accuracy: 10-20% Experimental accuracy: 10-20% (optical (optical frequency)frequency)

• Conclusions: A UNIFIED PICTURE OF ALL FORCESConclusions: A UNIFIED PICTURE OF ALL FORCES

• • A total unified model, including Yang-Mills gravity, A total unified model, including Yang-Mills gravity, based onbased on

• TT44 x (SU x (SU33))colorcolor x (SU x (SU22xUxU11) [xU) [xU1b1bxUxU1e1e]]

in flat space-time, with the total gauge covariant derivativein flat space-time, with the total gauge covariant derivative

δδμμ=∂=∂μμ+gφ+gφμνμν∂∂νν+ig G+ig Gμμaaλλaa/2 + if W/2 + if Wμμbb t t b b + if’ U+ if’ Uμμ++......

• • Where a=1,2,3……8 (λWhere a=1,2,3……8 (λaa =SU =SU33 generators) ; generators) ;

b=1,2,3 (tb=1,2,3 (tbb =SU =SU22 generators). generators).

• • One new conceptual result of Yang-Mills One new conceptual result of Yang-Mills gravity is that the apparent curvature of gravity is that the apparent curvature of space-time appears to be simply a space-time appears to be simply a manifestation of the flat space-time manifestation of the flat space-time translational gauge symmetry for the motion translational gauge symmetry for the motion of quantum particles in the classical limitof quantum particles in the classical limit..

•Accelerated cosmic expansion Accelerated cosmic expansion • based on a ‘generalized’ Ubased on a ‘generalized’ U11 gauge gauge

symmetry associated with conservation symmetry associated with conservation of baryon number (or charge):of baryon number (or charge):

• B’B’λλ(x) = B(x) = Bλ λ (x) + (x) + ΛΛλλ(x)(x), ,

• U’(x)=Ω(x)U(x), U’(x)=Ω(x)U(x), Ω(x) Ω(x) = = exp(-if )exp(-if )• Ū’(x)=Ū(x)ΩŪ’(x)=Ū(x)Ω-1-1(x), (x), • U(x)=fermion field, U(x)=fermion field,

• Ω(x)=path-dependent phase factorΩ(x)=path-dependent phase factor• In special case,in which In special case,in which ΛΛμμ(x)(x)= = ∂∂μμΛ(x)Λ(x) , the previous , the previous

generalized Ugeneralized U11 transformation simplify to the usual U transformation simplify to the usual U11 gauge transformation: Ω(x)= usual phase factorgauge transformation: Ω(x)= usual phase factor

• As usual, the generalized UAs usual, the generalized U11 gauge covariant gauge covariant

derivative is defined asderivative is defined as

∂∂μμ→ ∂→ ∂μμ-- iifBfBμμ= Δ= Δbbμμ

The UThe U11 gauge curvature is given by gauge curvature is given by

[[ΔΔbbμμ , , ΔΔbbνν]= if B]= if Bμνμν(x),(x),

wherewhere

BBμνμν(x)=(x)=∂∂ννBBμμ - - ∂∂μμBBνν,,

However, BHowever, Bμνμν(x) is not gauge invariant:(x) is not gauge invariant:

B’B’μνμν(x)=B(x)=Bμνμν(x)+ (x)+ ∂∂μμΛΛν ν (x)(x) - - ∂∂ννΛΛμμ(x)(x) ≠≠ B Bμνμν(x)(x)

• Only the divergence of the gauge curvature is Only the divergence of the gauge curvature is

gauge invariant:gauge invariant:

∂∂μμB’B’μνμν(x)=(x)=∂∂μμBBμνμν(x), (x),

• Provided the vector gauge function Provided the vector gauge function ΛΛμμ(x)(x) satisfy the constraintsatisfy the constraint

• ∂∂μμ∂∂μμΛΛν ν (x)(x) - - ∂∂νν∂∂μμΛΛμμ(x)(x) = 0 = 0

The generalized UThe generalized U1b1b gauge invariant Lagrangian: gauge invariant Lagrangian:

L= L= -- ( (LLbb22/2)/2) ∂∂μμFFμβμβ∂∂ννFFνβνβ + + ψψ[[iγiγμμ((∂∂μμ+ifB+ifBμμ)-m])-m]ψψ..

The baryonic gauge field equation is the fourth-order The baryonic gauge field equation is the fourth-order eq.eq.

∂∂22∂∂μμBBμνμν(x)- (f/L(x)- (f/Lbb22)) ψψγγμμBBμμψψ=0.=0.

• The static equation for BThe static equation for B00(r) is(r) is

LLbb22ΔΔΔΔBB00 = = (f/L(f/Lbb

22)) ψγ ψγ00ψψ..

For a spheric static solution of a point source, For a spheric static solution of a point source, we findwe find

BB00(r)=f/(8ΠL(r)=f/(8ΠLbb22) r .---------linear in r !) r .---------linear in r !

This linear potential will lead to a constant force This linear potential will lead to a constant force between baryons in the universe. This baryonic between baryons in the universe. This baryonic force will dominate the motion in extremely force will dominate the motion in extremely large distance, no matter how small the large distance, no matter how small the baryonic coupling constant f is. Such a baryonic baryonic coupling constant f is. Such a baryonic force resembles the U1 electromagnetic force force resembles the U1 electromagnetic force and it is repulsive between two baryons (protons and it is repulsive between two baryons (protons and neutrons).and neutrons).

• Experimental test Experimental test of accelerated cosmic of accelerated cosmic expanison due to baryonic force.expanison due to baryonic force.

• Consider a supernova with mass mConsider a supernova with mass mss located in a located in a sphere of roughly 100 billion galaxies (as reveal by sphere of roughly 100 billion galaxies (as reveal by Hubble). We idealize baryonic galaxies as points Hubble). We idealize baryonic galaxies as points uiformly distributed in a big sphere with a radius Ruiformly distributed in a big sphere with a radius Roo and a constant baryon density. We can calculate and a constant baryon density. We can calculate the total force of the sphere that acts on a the total force of the sphere that acts on a supernova at a distance r < Rsupernova at a distance r < Roo. We obtain*. We obtain*

• dd22rr/dt/dt22=(9f=(9f22M)/(8LM)/(8Lbb22mmpp

22))[[1-1-rr22/{5R/{5Roo22}}]]((rr/R/Roo), [Gauge]), [Gauge]

• For comparison, in the conventional model with a For comparison, in the conventional model with a cosmological constant in Einstein equation, one hascosmological constant in Einstein equation, one has

• dd22rr/dt/dt22= C = C r,r, C = const. [General Rela.]C = const. [General Rela.]**JP Hsu and L. Hsu,”A model of cosmic acceleration of a supernova and JP Hsu and L. Hsu,”A model of cosmic acceleration of a supernova and

exp.”exp.”

• Conclusions:Conclusions:

• Yang-Mills gravity suggests that the Yang-Mills gravity suggests that the apparent curvature of space-time appears to apparent curvature of space-time appears to be simply a manifestation of the flat space-be simply a manifestation of the flat space-time translational gauge symmetry for the time translational gauge symmetry for the motion of quantum particles in the classical motion of quantum particles in the classical limit.limit.

• We can have a field-theoretic understanding We can have a field-theoretic understanding of the accelerated cosmic expansion based of the accelerated cosmic expansion based on a generalized gauge symmetry (involving on a generalized gauge symmetry (involving baryon number conservation, vector gauge baryon number conservation, vector gauge functions and path-dependent phases.) functions and path-dependent phases.)