Unified Field Equations Coupling Force Forces. Tian Ma, Shouhong Wang Supported in part by NSF and ONR

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1 Unified Field Equations Coupling Force Forces Tian Ma, Shouhong Wang Supported in part by NSF and ONR fluid 1

2 Outline I. Motivations II. PID III. Unified Field Equations Coupling Four Forces IV. Duality V. Electroweak Theory VI. Unified Theory of Dark Energy and Dark Matter 2

3 References: 1. Tian Ma & Shouhong Wang, Gravitational Field Equations and Theory of Dark Matter and Dark Energy, June, 2012, 2. Tian Ma & Shouhong Wang, Unified Field Equations Coupling Four Forces and Principle of Interaction Dynamics, September, 2012, arc-bin/mpa?yn=

4 I. Motivations Mystery of dark matter and dark energy Grand challenges for coupling all four interactions 4

5 II. Principle of Interaction Dynamics (PID) Due to the presence of dark energy and dark matter, the energy-momentum tensor T ij of normal matter is no longer conserved: D i (T ij ) 0. By an orthogonal decomposition theorem, there is a scalar function ϕ : M R such that (1) c4 T ij = T ij 8πG D id j ϕ, D i Tij = 0 ( L EH = R + 8πG ) gdx, S M D i [R ij 1 2 g ijr Namely ] c 4 = 0 = R ij 1 2 g ijr + 8πG c 4 T ij = 0 R ij 1 2 g ijr = 8πG c 4 T ij + D i D j ϕ, D i ( D i D j ϕ + 8πG ) c 4 T ij = 0 5

6 Equivalently: The new gravitational field equations (1) can be derived with constraint least action principle: lim λ 0 1 λ [L EH(g ij + λx ij ) L EH (g ij )] = (δl EH (g ij ), X) = 0 D i X ij = 0 This constraint least action leads us to postulate: Principle of Interaction Dynamics (PID): For all physical interactions there are Lagrangian actions (2) L(g, A, ψ) = M L(g ij, A, ψ) gdx, where A is a set of vector fields representing the gauge potentials, and ψ are the wave functions of particles. The states (g, A, ψ) are the extremum points of (2) with the (D + A)-free constraint. 6

7 III. Unified Field Equations Coupling Four Forces Lagrangian action functional L is the sum of the Einstein-Hilbert L EH, the GWS L GW S without Higgs field terms, and the standard QCD L QCD. (3) L = [L EH + L GW S + L QCD ] gdx L EH = R + 8πG c 4 S, L GW S = 1 4 W a µνw aµν 1 4 F µνf µν + 6 { Lk (iγ µ Dµ m k )L k + ψ } k R (iγ µ Dµ m l k)ψk R, k=1 L QCD = 1 4 F b µνf bµν + 6 k=1 q k (iγ µ Dµ m q k )q k 7

8 Thanks to the div A -free constraint, the Euler-Lagrangian of the action functional is balanced by gradient fields, resulting the unified field equations (4) coupling all four forces. The gradient fields break spontaneously the gauge-symmetries. This gives rise to a complete new mechanism for spontaneously breaking gauge-symmetries and for energy and mass generation, which provides similar outcomes as the Higgs mechanism. 8

9 Unified Field Equations Coupling Four Forces: (4) R µν 1 2 g µνr + 8πG c 4 T µν = (D µ + α 0 A µ + α W a Wµ a + α Bj Bµ)Φ j ν, 6 [ ] 1 µ ( µ A ν ν A µ ) + ie 2 L k γ µ L k + ψ k R γ µ ψk R k=1 = (D ν + β 0 A ν + β W a W a ν + β Bj B j ν)φ 0, µ W a µν g b=1 f bca g µα W a µνw c α + i g k=1 = (D ν + κ a 0A ν + κ a W bw b ν + κ a BjB j ν)φ a W, µ F b µν g k=1 g klb g µα B b µνb l α + i g k=1 = (D ν + θ b 0A ν + θ b W lw l ν + θ b BlB l ν)φ b B, L k γ ν σ a L k q k γ ν λ b q k (iγ µ Dµ m k )L k = 0 for k = 1, 2, 3, (iγ µ Dµ m l k)ψ R k = 0 for k = 1, 2, 3, 9 (iγ µ Dµ m q k )q k = 0 for k = 1,, 8.

10 T µν = δs + c4 [ δg ij 16πG gαβ WαµW a βν a + F αµ F βν + FαµF b βν b ] c 4 16πG g µν(l GW S +L QCD ). 10

11 IV. Duality The unified field equations (4) coupling all four forces induce a natural duality: (5) {g µν } Φ µ, A µ φ 0, Wµ a φ a W for a = 1, 2, 3, Bµ b φ b B for b = 1,, 8. Equivalently the graviton g, the photon γ, vector bosons W ± and Z, and the gluons g b, represented by the left-hand side of the above duality corresponds to the bosonic fields: vector boson Φ µ (massless with spin 1), and scalar bosons φ 0 (massless with spin 0), φ a W, φb B with spin 0. 11

12 Zero-point energy: The field φ 0, adjoint to the electromagnetic potential A µ, is a massless field with spin s = 0, and it needs to be confirmed experimentally. We think it is this particle field φ 0 zero-point energy. that causes the vacuum fluctuation or Eightfold Way mesons: We conjecture that the combination of the eight scalar bosons φ b B corresponding to gluons g b (1 b 8) leads to the Eightfold Way mesons: π +, π, π 0, K +, K, K 0, K0, η 0. 12

13 V. Electroweak Theory Higgs Bosons: The combination of the three Higgs bosonic fields on the right hand side of the duality induces three Higgs boson particles given by φ ± = 1 2 (φ 1 W ± iφ 2 W ), φ 3 W with φ ± having ± electric charges, and with φ 3 W being neutral. Note that the classical Weinberg-Salam theory induces one neutral Higgs boson particle. All three Higgs bosons deduced here possess masses, generated by the new mechanism. 13

14 In fact, consider only the electromagnetic and weak interactions and ignore the effect of other interactions, we derive a totally different electroweak theory. Again this electroweak theory produces the three vector bosons W ± and Z, as well as the three Higgs bosons. The spontaneous gauge-symmetry breaking is achieved by the constraint action without Higgs terms in the action functional. 14

15 VI Unified Theory of Dark Energy and Dark Matter New gravitational vector bosonic field: The new vector particle field Φ µ is massless with spin s = 1. This particle field corresponds to the scalar potential field ϕ in (1), caused by the non-uniform distribution of matter in the universe. The interaction between this particle field Φ µ and the graviton leads to a unified theory of dark matter and dark energy and explains the acceleration of expanding universe. 15

16 Consider gravity field alone: R ij 1 2 g ijr = 8πG c 4 T ij D i D j ϕ, D i R = 8πG c 4 T + Φ, M ( 8πG c 4 ) T ij + D i D j ϕ Φ gdx = 0 Consider a spherically symmetric central matter field with mass M and radius r 0. With the new field equations, we derive that the force exerted on an object with mass m is given by (6) F = mmg [ 1r 2 c2 2GM ( 2 + 2GM ) 1 dϕ c 2 r dr + = 0 ] c2 2GM Φ r, for r > r 0. The first term is the classical Newton gravitation, the second term is the coupling interaction between matter and the scalar potential ϕ, and the third term is the interaction generated by the scalar potential energy density Φ (R = Φ for r > r 0 ). 16

17 The sum ε = ε 1 + ε 2 of the scalar potential energy density ε 1 = c4 8πG Φ and the coupling interaction energy between the matter field and the scalar potential field ϕ ( ε 2 = c4 2 8πG r + 2MG ) dϕ c 2 r 2 dr, unifies dark matter and dark energy: ε > 0 ε < 0 represents dark energy, represents dark matter. 17

18 The interaction force formula (6) can be further simplified to derive the following approximate formula for r 0 < r < r km: (7) F = mmg [ 1 r 2 k ] 0 r + k 1r, where k 0 = km 1 and k 1 = km 3, which are estimated using rotation curves of galactic motion. Again, in (7), the first term represents the Newton gravitation, the attracting second term stands for dark matter and the repelling third term is the dark energy. 18