1. review research Yang-Mills Gravity and Accelerated Cosmic Expansion* ( Based on a Model with Generalized Gauge Symmetry ) Jong-Ping Hsu Physics Department, Univ. of Massachusetts Dartmouth, North Dartmouth, Massachusetts 02747, USA *Collaborators: (1) Leonardo Hsu (Space-time symmetry and quantum Yang-Mills gravity, World Scientific), (2) Kazuo Cottrell (‘A unified model with a generalized gauge symmetry and its cosmological implications’.) (2) Kazuo Cottrell (‘A unified model with a generalized gauge symmetry and its cosmological implications’.)

2. (I) A BIG PICTURE OF SPACE-TIME: (I) A BIG PICTURE OF SPACE-TIME: There exists a fundamental (flat) space-time symmetry framework that can encompass all interactions in physics, including gravity, and is valid for both inertial and non-inertial frames. There exists a fundamental (flat) space-time symmetry framework that can encompass all interactions in physics, including gravity, and is valid 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. A. Gravity---Yang-Mills gravity---space-time translational gauge symmetry T 4 (external, exact) B. Electroweak---SU 2 x U 1 (spontaneous sym breaking) C. Strong force (QCD)---(SU 3 ) color (exact) D. Cosmic baryonic (& leptonic) forces---U 1b (exact)

3. Such a unified model follows the ideas of Glashow, Salam, Ward, and Weinberg. Such a unified model follows the ideas of Glashow, Salam, Ward, and Weinberg. It can be formulated for both inertial and non-inertial frames. It can be formulated for both inertial and non-inertial frames. Symmetry appears to be the deepest Symmetry appears to be the deepest foundation for our understanding of the physical universe.

4. 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 interactions All field theories for electroweak and strong interactions 2. Curved space-time: Einstein’s gravity Einstein’s gravity However, Einstein’s symmetry principle of general coordinate invariance is a profound idea with highly non-trivial difficulties. However, Einstein’s symmetry principle of general coordinate invariance is a profound idea with highly non-trivial difficulties. Why? Why?

5. Gravity??? Gravity??? F. Dyson: (A founder of QED, together with F. Dyson: (A founder of QED, together with Tomonaga, Schwinger, & Feynman) Dyson stressed that Dyson stressed that “The most glaring incompatibility of concepts in “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 quantum-mechanical description of nature.” (‘Missed Opportunity’, J. W. Gibbs Lecture at Amer. Math. Soc. 1972) (‘Missed Opportunity’, J. W. Gibbs Lecture at Amer. Math. Soc. 1972) This incompatibility is a MOTIVATION for our research….. This incompatibility is a MOTIVATION for our research…..

6. Gravity??? 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] is that coordinates are only auxiliary quantities which can be given arbitrary values for every event. Hence, the measurement of position, that is, of the space coordinates, is certainly not a significant measurement if the postulates of the general theory are adopted…. Most of us have struggled with the problem of how, under these premises, the general theory of relativity can make meaningful statements and predictions at all……….” “The basic premise of this theory [general relativity] is that coordinates are only auxiliary quantities which can be given arbitrary values for every event. Hence, the measurement of position, that is, of the space coordinates, is certainly not a significant measurement if the postulates of the general theory are adopted…. Most of us have struggled with the problem of how, under these premises, the general theory of relativity can make meaningful statements and predictions at all……….” Noether’s theorem II: No conservation of energy in GR Noether’s theorem II: No conservation of energy in GR Hilbert made a similar remark (around 1915) Hilbert made a similar remark (around 1915)

7. Why should gauge symmetry in flat space-time be so successful for modeling all known interactions except gravity ? Why should gauge symmetry in flat space-time be so successful for modeling all known interactions except gravity ? Yang-Mills Gravity (with a generalized gauge symmetry) Yang-Mills Gravity (with a generalized gauge symmetry) 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)Yang-Mills gravity---space-time translational gauge symmetry T 4 (2) Electroweak forces--- SU 2 x U 1 symmetry (3) Strong force (QCD)--(SU 3 ) color symm (4) Baryonic force---U 1b (accelerated cosmic expansion). (5) Leptonic force---U 1l (accelerated cosmic expansion). To illustrate generalized gauge symmetry, I shall discuss (1) and (4) in this talk.

8. Gauge Symmetry in Flat Spacetime (a generalization of Yang-Mills’ internal gauge symmetry to 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 idea Local space-time translation gauge symmetry T 4 x μ → x μ +Λ μ (x), η μν =(1,-1,-1,-1), (c=ћ=1) x μ → x μ +Λ μ (x), η μν =(1,-1,-1,-1), (c=ћ=1) Λ μ (x): infinitesimal arbitrary function of space-time Λ μ (x): infinitesimal arbitrary function of space-time 4-dim displacement operator p ν =i ∂ ν =i∂/∂x ν 4-dim displacement operator p ν =i ∂ ν =i∂/∂x ν T(4) gauge symmetry dictates the tensor fields φ μν. T(4) gauge symmetry dictates the tensor fields φ μν. Gauge covariant derivative Δ μ (x) : Gauge covariant derivative Δ μ (x) : ∂ μ → ∂ μ - igφ μν p ν = J μν ∂ ν = Δ μ. ∂ μ → ∂ μ - igφ μν p ν = J μν ∂ ν = Δ μ. J μν = η μν + gφ μν, φ μν = φ νμ. J μν = η μν + gφ μν, φ μν = φ νμ.

9. A Basic Observation: A Basic Observation: Dual interpretations of Dual interpretations of ***x μ → x’ μ =x μ +Λ μ (x) (ia) a local shift (translation) in flat space-time, (ib) an arbitrary infinitesimal coordinate transformation in flat space-time. (ii) an arbitrary infinitesimal transformations of coordinates in curved space-time. (GR) *** This is the key conceptual departure from that of GR. (Early discussions of gravity based on flat space-time or translational gauge symmetry: A. A. Logunov, M.A. Mestvirishvili, A.A. Vlasov, Y.M. Cho, N. Wu and others.)

10. Interpretation (ia,ib)  Yang-Mills gravity in flat space-time (for both inertial and non-inertial frames). Interpretation (ia,ib)  Yang-Mills gravity in flat space-time (for both inertial and non-inertial frames). T 4 Gauge symmetry postulates the replacement in the Lagrangian: T 4 Gauge symmetry postulates the replacement in the Lagrangian: ∂ μ → ∂ μ +gφ μν ∂ ν =J μν ∂ ν = Δ μ, (c=ћ=1) ∂ μ → ∂ μ +gφ μν ∂ ν =J μν ∂ ν = Δ μ, (c=ћ=1) (i) g is not dimensionless, (dimension of g=length) (i) g is not dimensionless, (dimension of g=length) (ii) φ μν is not a vector field (ii) φ μν is not a vector field They differ from those in usual Yang-Mills gauge symmetry. They differ from those in usual Yang-Mills gauge symmetry. [Δ μ, Δ ν ]=C μνα ∂ α [Δ μ, Δ ν ]=C μνα ∂ α T 4 Gauge curvature: C μνα C μνα = J μα (∂ α J να ) - J νβ (∂ β J να ), C μνα = J μα (∂ α J να ) - J νβ (∂ β J να ), J μν = η μν + gφ μν, J μν = η μν + gφ μν,

11. Lagrangian and Field Equations Lagrangian and Field Equations L= - (1/2g 2 )(C μαβ C μβα - C μα α C μβ β ) + L ψ, L= - (1/2g 2 )(C μαβ C μβα - C μα α C μβ β ) + L ψ, where C μαβ C μβα = C μαβ C μαβ /2. where C μαβ C μβα = C μαβ C μαβ /2. H μν = - g 2 T μν H μν = - g 2 T μν H μν = - ∂ λ { J λ α C αμν - J λ α C αβ β η μν + C μβ β J νλ } H μν = - ∂ λ { J λ α C αμν - J λ α C αβ β η μν + C μβ β J νλ } - C μαβ ∂ ν J αβ + C μβ β ∂ ν J α α -C λβ β ∂ ν J μ λ - C μαβ ∂ ν J αβ + C μβ β ∂ ν J α α -C λβ β ∂ ν J μ λ T μν = (1/2)[ψiγ μ ∂ ν ψ - (i∂ ν ψ)γ μ ψ] T μν = (1/2)[ψiγ μ ∂ ν ψ - (i∂ ν ψ)γ μ ψ]

12. Interesting results: Interesting results: In the limit of geometric-optics (i.e., classical limit), the wave eqs. of massive fermions and bosons reduces to the same Hamilton-Jacobi type equation In the limit of geometric-optics (i.e., classical limit), the wave eqs. of massive fermions and bosons reduces to the same Hamilton-Jacobi type equation G μν ∂ μ S ∂ ν S = m 2, G μν =η αβ J αμ J βν, G μν ∂ μ S ∂ ν S = m 2, G μν =η αβ J αμ J βν, where G μν appears to be an effective “Riemannian metric tensor” for (and only for) a classical object. where G μν appears to be an effective “Riemannian metric tensor” for (and only for) a classical object. But for quantum fields and particles, the physical space-time is flat. But for quantum fields and particles, the physical space-time is flat. Maxwell’s eqs.  (classical limit)  eikonal equation with a slightly different metric tensor G L μν Maxwell’s eqs.  (classical limit)  eikonal equation with a slightly different metric tensor G L μν Effective curved space-time for the motion of classical objects in Yang-Mills gravity Effective curved space-time for the motion of classical objects in Yang-Mills gravity

13. Experimental Results: Perihelion shift-----`same’ as the usual result Perihelion shift-----`same’ as the usual result (within experimental accuracy) Red shift----`same’ Red shift----`same’ Gravitational quadrupole radiation-----`same’ Gravitational quadrupole radiation-----`same’ Bending of light ---- ‘different’ Bending of light ---- ‘different’ Bending of Light Δφ=1.53” (only for light rays with optical frequency) Bending of Light Δφ=1.53” (only for light rays with optical frequency) 12% smaller than the usual value 1.75” 12% smaller than the usual value 1.75” Experimental accuracy: 10-20% (optical frequency) Experimental accuracy: 10-20% (optical frequency)

14. Conclusions: A UNIFIED PICTURE OF ALL FORCES Conclusions: A UNIFIED PICTURE OF ALL FORCES A total unified model, including Yang-Mills gravity, based on A total unified model, including Yang-Mills gravity, based on T 4 x (SU 3 ) color x (SU 2 xU 1 ) [xU 1b xU 1e ] T 4 x (SU 3 ) color x (SU 2 xU 1 ) [xU 1b xU 1e ] in flat space-time, with the total gauge covariant derivative δ μ =∂ μ +gφ μν ∂ ν +ig G μa λ a /2 + if W μb t b + if’ U μ +... Where a=1,2,3……8 (λ a =SU 3 generators) ; Where a=1,2,3……8 (λ a =SU 3 generators) ; b=1,2,3 (t b =SU 2 generators). b=1,2,3 (t b =SU 2 generators). One new conceptual result of Yang-Mills gravity is that the apparent curvature of space-time appears to be simply a manifestation of the flat space-time translational gauge symmetry for the motion of quantum particles in the classical limit. One new conceptual result of Yang-Mills gravity is that the apparent curvature of space-time appears to be simply a manifestation of the flat space-time translational gauge symmetry for the motion of quantum particles in the classical limit.

15. Accelerated cosmic expansion Accelerated cosmic expansion based on a ‘generalized’ U 1 gauge symmetry associated with conservation of baryon number (or charge): based on a ‘generalized’ U 1 gauge symmetry associated with conservation of baryon number (or charge): B’ λ (x) = B λ (x) + Λ λ (x), B’ λ (x) = B λ (x) + Λ λ (x), U’(x)=Ω(x)U(x), Ω(x) = exp(-if ) U’(x)=Ω(x)U(x), Ω(x) = exp(-if ) Ū’(x)=Ū(x)Ω -1 (x), Ū’(x)=Ū(x)Ω -1 (x), U(x)=fermion field, U(x)=fermion field, Ω(x)=path-dependent phase factor Ω(x)=path-dependent phase factor In special case,in which Λ μ (x)= ∂ μ Λ(x), the previous generalized U 1 transformation simplify to the usual U 1 gauge transformation: Ω(x)= usual phase factor In special case,in which Λ μ (x)= ∂ μ Λ(x), the previous generalized U 1 transformation simplify to the usual U 1 gauge transformation: Ω(x)= usual phase factor

16. As usual, the generalized U 1 gauge covariant derivative is defined as As usual, the generalized U 1 gauge covariant derivative is defined as ∂ μ → ∂ μ - ifB μ = Δ bμ The U 1 gauge curvature is given by [Δ bμ, Δ bν ]= if B μν (x), where B μν (x)=∂ ν B μ - ∂ μ B ν, However, B μν (x) is not gauge invariant: B’ μν (x)=B μν (x)+ ∂ μ Λ ν (x) - ∂ ν Λ μ (x) ≠ B μν (x)

17. Only the divergence of the gauge curvature is gauge invariant: Only the divergence of the gauge curvature is gauge invariant: ∂ μ B’ μν (x)=∂ μ B μν (x), Provided the vector gauge function Λ μ (x) satisfy the constraint Provided the vector gauge function Λ μ (x) satisfy the constraint ∂ μ ∂ μ Λ ν (x) - ∂ ν ∂ μ Λ μ (x) = 0 ∂ μ ∂ μ Λ ν (x) - ∂ ν ∂ μ Λ μ (x) = 0 The generalized U 1b gauge invariant Lagrangian: L= - (L b 2 /2) ∂ μ F μβ ∂ ν F νβ + ψ [iγ μ (∂ μ +ifB μ )-m]ψ. The baryonic gauge field equation is the fourth-order eq. ∂ 2 ∂ μ B μν (x)- (f/L b 2 ) ψγ μ B μ ψ=0.

18. The static equation for B 0 (r) is The static equation for B 0 (r) is L b 2 ΔΔB 0 = (f/L b 2 ) ψγ 0 ψ. For a spheric static solution of a point source, we find B 0 (r)=f/(8ΠL b 2 ) r.---------linear in r ! This linear potential will lead to a constant force between baryons in the universe. This baryonic force will dominate the motion in extremely large distance, no matter how small the baryonic coupling constant f is. Such a baryonic force resembles the U1 electromagnetic force and it is repulsive between two baryons (protons and neutrons).

19. Experimental test of accelerated cosmic expanison due to baryonic force. Experimental test of accelerated cosmic expanison due to baryonic force. Consider a supernova with mass m s located in a sphere of roughly 100 billion galaxies (as reveal by Hubble). We idealize baryonic galaxies as points uiformly distributed in a big sphere with a radius R o and a constant baryon density. We can calculate the total force of the sphere that acts on a supernova at a distance r < R o. We obtain* Consider a supernova with mass m s located in a sphere of roughly 100 billion galaxies (as reveal by Hubble). We idealize baryonic galaxies as points uiformly distributed in a big sphere with a radius R o and a constant baryon density. We can calculate the total force of the sphere that acts on a supernova at a distance r < R o. We obtain* d 2 r/dt 2 =(9f 2 M)/(8L b 2 m p 2 ) [ 1-r 2 /{5R o 2 } ] (r/R o ), [Gauge] d 2 r/dt 2 =(9f 2 M)/(8L b 2 m p 2 ) [ 1-r 2 /{5R o 2 } ] (r/R o ), [Gauge] For comparison, in the conventional model with a cosmological constant in Einstein equation, one has For comparison, in the conventional model with a cosmological constant in Einstein equation, one has d 2 r/dt 2 = C r,C = const. [General Rela.] d 2 r/dt 2 = C r,C = const. [General Rela.] * JP Hsu and L. Hsu,”A model of cosmic acceleration of a supernova and exp.”

20. Conclusions: Conclusions: Yang-Mills gravity suggests that the apparent curvature of space-time appears to be simply a manifestation of the flat space-time translational gauge symmetry for the motion of quantum particles in the classical limit. Yang-Mills gravity suggests that the apparent curvature of space-time appears to be simply a manifestation of the flat space-time translational gauge symmetry for the motion of quantum particles in the classical limit. We can have a field-theoretic understanding of the accelerated cosmic expansion based on a generalized gauge symmetry (involving baryon number conservation, vector gauge functions and path-dependent phases.) We can have a field-theoretic understanding of the accelerated cosmic expansion based on a generalized gauge symmetry (involving baryon number conservation, vector gauge functions and path-dependent phases.)