Two Fundamental Principles of Nature s Interactions

Transcription

1 Two Fundamental Principles of Nature s Interactions Tian Ma, Shouhong Wang Supported in part by NSF, ONR and Chinese NSF fluid I. Gravity and Principle of Interaction Dynamics PID) II. Principle of Representation Invariance PRI) III. Unified Field Theory 1

2 I. Gravity and Principle of Interaction Dynamics PID) General relativity Einstein, 1915): Postulated two basic principles: 1) the principle of general relativity, and 2) the principle of equivalence, which lead to the space-time manifold is a 4D Riemannian manifold M, g µν ) tensor fields transform under the transformation group: ϕ µ p) x ν )p M. Einstein field equations: R µν 1 2 g µνr = 8πG c 4 T µν, µ T µν = 0. are the Euler-Lagrangian equations of the Einstein-Hilbert functional L EH g µν ) = M R + 8πG ) gdx c 4 gµν S µν 2

3 Principle of Interaction Dynamics PID) Ma-Wang, 2012): Least action with energy-momentum conservation constraints. With PID, we derive the following gravitational field equations: L EH g µν ) = M R + 8πG ) gdx c 4 S δleh = R µν 1 2 g µνr + 8πG c 4 T µν δl EH g µν ), X) = 0 µ X µν = 0 = R µν 1 2 g µνr = 8πG c 4 T µν µ Φ ν ) 8πG D µ c 4 T µν + µ Φ ν = 0 Note: The new term µ Φ ν cannot be derived 1) from any existing fr) theories, and 2) from any scalar field theories. 3

4 The new vector particle field Φ µ is a spin-1 massless bosonic particle: Φ ν = e c A µ µ Φ ν + 8πG c 4 µ T µν The nonlinear 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. A spherically symmetric central matter field with mass M and radius r 0 exerts a force on an object with mass m. [ 1 F = m g 00 = mmg r 2 + k ] 0 r k 1r where k 0 = km 1, k 1 = km 3. Gravity can display both attractive and repulsive behaviors, and the new gravitational field equations give rise to a new unified theory for dark energy and dark matter. 4

5 Principle of Interaction Dynamics PID, MW-2012). 1. For all interactions in Nature, there exist a Lagrangian functional L{g µν }, A, ψ) = L{g µν }, A, ψ) gdx M 3. The interaction fields g µν, A, ψ) are extrema of L with div A -free constraints: δ δg µν L{g ij}, A, ψ) = µ + α b A b µ)φ ν, δ δa a µ L{g ij }, A, ψ) = µ + β a) b A b µ)φ a, δ δψ L{g ij}, A, ψ) = 0. PID offers a simpler way of introducing Higgs fields from first principles! 5

6 II. Principle of Representation Invariance PRI) Principle of gauge invariance, Maxwell 1861), Weyl 1919), Klein 1938), Yang-Mills 1954): Certain physical properties of fermionic particles Ψ are not distinguishable under the SUN) gauge transformations: gauge fields: A a µ a = 1,, N 2 1), connection: D µ = µ + iga a µτ a Dirac fields: Ψ = ψ 1,, ψ N) t curvature: F µν = F a µντ a = i g [D µ, D µ ] = µ A a ν ν A a µ + gλ a bca b µa c ν)τ a gauge transformation: Ux) = e iθa x)τ a SUN), Ψx) = Ux)Ψx), Ã a µτ a = i g µu)ψ + UA a µτ a U 1 where θ a = θ a x) 1 a N 2 1) are real parameters, and the traceless Hermitian matrices τ a are generators of SUN) with [τ a, τ b ] = τ a τ b τ b τ a = iλ c ab τ c. 6

7 Principle of Representation Invariance PRI) Ma-Wang, 2012): Physical laws for an SUN) gauge theory are independent of representations of SUN): Transformation of the generators τ a = {τ 1,, τ K }: τ a = x b aτ b, X = x b a). θ a, A a µ, and λ c ab are SUN)-tensors under this transformation. G ab = 1 4N λc ad λd cb = 1 2 Trτ aτ b ) is a symmetric positive definite 2nd-order covariant SUN)-tensor, which can be regarded as a Riemannian metric on SUN). The representation invariant action and gauge field equations are L = 1 4 G abg µα g νβ FµνF a αβ b + Ψ [ iγ µ µ + iga a µτ a ) m ] Ψ, M { [ Gab ν Fνµ b gλ b cdg αβ FαµA c d ] β g Ψγµ τ a Ψ = µ + α b A b µ)φ a, iγ µ D µ m)ψ = 0 Dirac eqs for fermions 7

8 III. Unified Field Theory The unified field model is derived based on the following principles: principles of general relativity and Lorentz invariance Einstein 1905, 1915) principle of gauge invariance, postulated by J. C. Maxwell 1861), H. Weyl 1919), O. Klein 1938), C. N. Yang & R. L. Mills 1954). spontaneous symmetry breaking by Y. Nambu 1960, Y. Nambu & G. Jona- Lasinio 1961 principle of interaction dynamics PID), M.-W. 2012) principle of representation invariance PRI), M.-W. 2012) Lagrangian action functional is the natural combination of the Einstein-Hilbert functional, the standard U1)-QED, SU2)-weak and SU3)-strong interaction actions. 8

9 Unified field model Ma-Wang, 2012): 1) 2) 3) 4) 5) R µν 1 2 g µνr + 8πG c 4 T µν = ν F νµ e ψγ µ ψ = G w ab G s kj [ ν W b νµ g w = [ ν S j νµ g s = [ µ eαe c A µ g wαa w c [ µ eαe c A µ g wαa w c ] c λb cdg αβ WαµW c β d g w Lγµ σ a L mh c ) 2 xµ eαe [ µ iγ µ Dµ m)ψ = 0. c Λj cd gαβ SαµS c β d [ µ + 1 mπ c 4 ] W a µ g sα s k c Sk µ c A µ g wαb w c g s qγ µ τ k q ) 2 xµ eαe c A µ g wαa w c W a µ g sα s k c Sk µ ] φ E, ] Φ ν, W µ b g sαk s ] c Sk µ φ w a, W µ a g sαj s ] c Sj µ φ s k, 9

10 Conclusions and Predictions of the Unified Field Model 1. Duality: The unified field model induces a natural duality: 6) {g µν } massless graviton) Φ µ, A µ photon) φ E, Wµ a massive bosons W ± & Z) φ w a for a = 1, 2, 3, Sµ k massless gluons) φ s k for k = 1,, Decoupling and Unification: An important characteristics is that the unified model can be easily decoupled. Namely, Both PID and PRI can be applied directly to individual interactions. For gravity alone, we have derived modified Einstein equations, leading to a unified theory for dark matter and dark energy. 10

11 3. Spontaneous symmetry breaking and mass generation mechanism: We obtained a much simpler mechanism for mass generation and energy creation, completely different from the classical Higgs mechanism. This new mechanism offers new insights on the origin of mass. 4. The two SU2) and SU3) constant vectors {αa w } and {αk s }, containing 11 parameters, represent the portions distributed to the gauge potentials by the weak charge g w and strong charge g s. We define, e.g., the total potential S µ : S µ = α s ks k µ ={S 0, S 1, S 2, S 3 } ={strong charge potential, strong rotational potential}. 11

12 5. Strong interaction Quantum Chromodynamics based on an SU3) gauge theory Yukawa, Yang- Mills, Gell-Mann, O. Greenberg,...) Layered strong interactions potentials Ma-Wang, 2012 & 13): based on two new principles: principle of interaction dynamics PID), and principle of representation invariance PRI): S 0 = g s ρ) [ 1 r A s ρ 1 + r ) ] e r/r ρw, g s ρ) = g s R ρ ) 3, where A s is a constant depending on the particle type, and R is the attracting radius of strong interactions given by { cm for w and quarks, R = cm for hadrons. 12

13 Φ r r These potentials match very well with experimental data. Again, strong interaction demonstrates both attraction and repelling behaviors. 13

14 Quark confinement: With these potentials, the binding energy of quarks can be estimated as 7) E q ρn ρ 0 ) 6 E n E n GeV, where E n 10 2 GeV is the binding energy of nucleons. This clearly explains the quark confinement. 14

15 Short-range nature of strong interaction: With the strong interaction potentials, at the atom/molecule scales, the ratio between strong force and electromagnetic attraction force is F a F e = 9g 2 s ρ0 ρ a ) 6 / e 2 { at the atomic level, at the molecular level. This demonstrates the short-range nature of strong interaction. 15

16 6. Weak interactions An SU2) gauge theory Ma-Wang, 2012 & 2013: Layered gauge weak interaction potentials: W 0 = g w ρ)e r/r 0 [ 1 r A w ρ 1 + 2r ) ] e r/r ρw 0, g w ρ) = g w r 0 ρ ) 3 where ρ w and ρ are the radii of the constituent weakton and the particle, A w is a constant depending on the types of particles, and r cm is the radius of weak interaction. The potentials show that the weak interaction is also short-ranged The weak interaction/force demonstrates both attraction and repelling behaviors. 16

17 7. Stability of matters The attraction and repelling nature of all four forces leads to the stability of the matter in the universe. 17