Physical Laws of Nature vs Fundamental First Principles Tian Ma, Shouhong Wang Supported in part by NSF, ONR and Chinese NSF http://www.indiana.edu/ fluid I. Laws of Gravity, Dark Matter and Dark Energy II. Principle of Interaction Dynamics III. Principle of Representation Invariance (PRI) IV. Unified Field Theory V. Summary 1
Inspired by the Albert Einstein s vision, our general view of Nature is as follows: Nature speaks the language of Mathematics: the laws of Nature are represented by mathematical equations, are dictated by a few fundamental principles, and take the simplest and aesthetic forms. We intend to derive experimentally verifiable laws of Nature based only on a few fundamental first principles, guided by experimental and observation evidences. 2
I. Laws of Gravity, Dark Matter and Dark Energy General relativity (Albert Einstein, 1915): The principle of equivalence says that the space-time is a 4D Riemannian manifold (M, g µν ) with g µν being regarded as gravitational potentials. The principle of general relativity requires that the law of gravity be covariant under general coordinate transformations, and dictates the Einstein-Hilbert action: (1) L EH ({g µν }) = M ( R + 8πG c 4 S ) gdx. The Einstein field equations are the Euler-Lagrangian equations of L EH : (2) R µν 1 2 g µνr = 8πG c 4 T µν, µ T µν = 0 3
Dark Matter: Theoretically for galactic rotating stars, one should have v 2 theoretical r = M observed G r 2. In 1970s, Vera Rubin observed the discrepancy between the observed velocity and theoretical velocity v observed > v theoretical, which implies that M = (the mass corresponding to v observed ) M theoretical > 0, which is called dark matter. Dark Energy was introduced to explain the observations since the 1990s by indicating that the universe is expanding at an accelerating rate. The 2011 Nobel Prize in Physics was awarded to Saul Perlmutter, Brian P. Schmidt and Adam G. Riess for their observational discovery on the accelerating expansion of the universe. 4
New Gravitational Field Equations (Ma-Wang, 2012): The presence of dark matter and dark energy implies that the energy-momentum tensor of visible matter T µν may no longer be conserved: µ T µν 0. Consequently, the variation of L EH must be taken under energy-momentum conservation constraint, leading us to postulate a general principle, called principle of interaction dynamics (PID) (Ma-Wang, 2012): (3) d L EH (g µν + λx µν ) = 0 X = {X µν } with µ X µν = 0. dλ λ=0 Then we derive a new set of gravitational field equations (4) R µν 1 2 g µνr = 8πG c 4 T µν µ Φ ν, [ ] 8πG µ c 4 T µν + µ Φ ν = 0. 5
The new term µ Φ ν cannot be derived from any existing f(r) theories, and from any scalar field theories. In other words, the term µ Φ ν does not correspond to any Lagrangian action density, and is the direct consequence of PID. The field equations (4) establish a natural duality: spin-2 graviton field {g µν } spin-1 dual vector graviton field {Φ µ } 6
Central Gravitational Field Consider a central gravitational field generated by a ball B r0 with radius r 0 and mass M. It is known that the metric of the central field for r > r 0 can be written in the form (5) ds 2 = e u c 2 dt 2 + e v dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ), u = u(r), v = v(r). Then the field equations (4) take the form v + 1 r (ev 1) = r 2 u φ, (6) u 1 r (ev 1) = r(φ 1 2 v φ ), ( 1 u + 2 u + 1 ) (u v ) = 2 r r φ. 7
We have derived in (Ma & Wang, 2012) an approximate gravitational force formula: (7) F = [ mc2 1r ] 2 eu (ev 1) rφ. This can be further simplified as (8) F = mmg ( 1 r 2 k ) 0 r + k 1r for r > r 0, k 0 = 4 10 18 Km 1, k 1 = 10 57 Km 3, demonstrating the presence of both dark matter and dark energy. Here the first term represents the Newton gravitation, the attracting second term stands for dark matter and the repelling third term is the dark energy. We note that our modified new formula is derived from first principles. 8
(Hernandez, Ma & Wang, 2015): Let x(s) def = (x 1 (s), x 2 (s), x 3 (s)) = ( e s u (e s ), e v(es) 1, e s φ (e s ) ). Then the nonautonomous gravitational field equations (6) become an autonomous system: x 1 = x 2 + 2x 3 1 2 x2 1 1 2 x 1x 3 1 2 x 1x 2 1 4 x2 1x 3, (9) x 2 = x 2 1 2 x 1x 3 x 2 2 1 2 x 1x 2 x 3, x 3 = x 1 x 2 + x 3 1 2 x 2x 3 1 4 x 1x 2 3, (x 1, x 2, x 3 )(s 0 ) = (α 1, α 2, α 3 ) with r 0 = e s 0. The asymptotically flat space-time geometry is represented by x = 0, which is a fixed point of the system (9). There is a two-dimensional stable manifold E s near x = 0, which can be 9
parameterized by (10) x 3 = h(x 1, x 2 ) = 1 2 x 1 + 1 2 x 2 + 1 16 x2 1 1 16 x2 2 + O( x 3 ). The field equations (9) are reduced to a two-dimensional dynamical system: x 1 = x 1 1 8 x2 1 1 8 x2 2 3 4 x 1x 2 + O( x 3 ), (11) x 2 = x 2 + 1 4 x2 1 x 2 2 1 4 x 1x 2 + O( x 3 ), with x 3 being slaved by (10). (x 1, x 2 )(s 0 ) = (α 1, α 2 ), 10
Figure 1: Only the orbits on Ω 1 with x 1 > 0 will eventually cross the x 2 -axis, leading to the sign change of x 1, and to a repelling gravitational force. 11
Asymptotic Repulsion Theorem (Hernandez-Ma-Wang, 15) For a central gravitational field, the following assertions hold true: 1) The gravitational force F is given by and is asymptotic zero: F = mc2 2 eu u, (12) F 0 if r. 2) If the initial value α in (11) is near the Schwarzschild solution with 0 < α 1 < α 2 /2, then there exists a sufficiently large r 1 such that the gravitational force F is repulsive for r > r 1 : (13) F > 0 for r > r 1. 12
The above theorem is valid provided the initial value α is small. Also, all physically meaningful central fields satisfy the condition (note that a black hole is enclosed by a huge quantity of matter with radius r 0 2MG/c 2 ). In fact, the Schwarzschild initial values are as (14) x 1 (r 0 ) = x 2 (r 0 ) = δ 1 δ, δ = 2MG c 2 r 0. The δ-factors for most galaxies and clusters of galaxies are (15) galaxies δ = 10 7, cluster of galaxies δ = 10 5. The dark energy phenomenon is mainly evident between galaxies and between clusters of galaxies, and consequently the above theorem is valid for central gravitational fields generated by both galaxies and clusters of galaxies. The asymptotic repulsion of gravity (dark energy) plays the role to stabilize the large scale homogeneous structure of the Universe. 13
Law of gravity (Ma-Wang, 2012) 1. (Einstein s PE). The space-time is a 4D Riemannian manifold {M, g µν }, with the metric {g µν } being the gravitational potential; 2. The Einstein PGR dictates the Einstein-Hilbert action (1); 3. The gravitational field equations (4) are derived using PID, and determine gravitational potential {g µν } and its dual vector field Φ µ = µ φ; 4. Gravity can display both attractive and repulsive effect, caused by the duality between the attracting gravitational field {g µν } and the repulsive dual vector field {Φ µ }, together with their nonlinear interactions governed by the field equations (4). 5. It is the nonlinear interaction between {g µν } and the dual field Φ µ that leads to a unified theory of dark matter and dark energy phenomena. 14
II. Principle of Interaction Dynamics (PID) Note: The new gravitational field equations, δl EH (g µν ) = µ Ψ ν, violate the least action principle, and instead are derived by the following PID: Principle of Interaction Dynamics (Ma-Wang, 2012). 1. For the four fundamental interactions, there exists a Lagrangian action (16) L({g µν }, A, ψ) = M L({g µν }, A, ψ) gdx where g µν is the gravitational field, A is a set of vector fields representing the gauge fields, and ψ is the wave functions of particles. The action obeys the principle of general relativity, the Lorentz and gauge invariances, and the principle of representation invariance (PRI). 15
2. The interaction fields (g µν, A, ψ) satisfy the variation equations 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. Note: The action is gauge invariant, but the field equations break the gauge symmetry. PID offers a simpler way of introducing Higgs fields from the first principle. 16
III. Principle of Representation Invariance Herman Weyl (1919): scale invariance g µν e α(x) g µν connection. leading to conformal James Clerk Maxwell (1861), Herman Weyl, Vladimir Fock and Fritz London: U(1) gauge invariance for quantum electrodynamics. Yang-Mills (1954): SU(2) gauge theory for strong interactions between nucleons associated with isospins of protons, neutrons:... A change of gauge means a change of phase factor ψ ψ, ψ = exp(iα)ψ, a change that is devoid of any physical consequences. Since ψ may depend on x, y, z, and t, the relative phase factor of ψ may at two different space-time points is therefore completely arbitrary. In other words, the arbitrariness in choosing the phase factor is local in character. 17
Maxwell Equations for electromagnetic fields: (17) (18) curl E 1 H = c t, div H = 0, curl H 1 E = c t + 4π J, c div E = 4πρ, where E, H are the electric and magnetic fields, and J is the current density, and ρ is the charge density. By (17), there is a vector A such that H = curl A, and there is an A0 such that E = 1 c A t + A 0. Take A µ = (A 0, A). Then the Maxwell equations can be written as (19) µ F µν = 4π c J ν with F µν = µ A ν ν A µ, J µ = ( cρ, J). 18
Question: What is the dynamic motion law of a charged electron? Dirac equations: (20) [iγ µ µ m] ψ = 0 ψ = (ψ 1, ψ 2, ψ 3, ψ 4 ) T γ µ = (γ 0, γ 1, γ 2, γ 3 ), γ 0 = σ 1 = ( ) 0 1, σ 1 0 2 = ( ) I 0, γ k = 0 I ( ) 0 i, σ i 0 3 = ( ) 0 σk σ k 0 ( ) 1 0 0 1 Special solutions: 1 e mc2 t/ 0 0, 0 e mc2 t/ 0 1 0, 0 e+mc2 t/ 0 0 1, 0 e+mc2 t/ 0 0 0 1 19
Experimentally, one cannot distinguish the two states ψ = e iθ(xµ) ψ and ψ. Mathematically, the phenomenon says that the Dirac equations are covariant under the U(1) gauge transformation (i.e., D µ ψ = e iθ D µ ψ): (21) ( ψ, õ) = ( e iθ(xµ) ψ, A µ 1 ) e µθ. The U(1) gauge symmetry and the Lorentz symmetry dictate the action: (22) L(A µ, ψ) = 1 4 F µνf µν + ψ [i(γ µ µ + iea µ ) m] ψdm, M whose variation is the quantum electrodynamics (QED) field equations: (23) (24) µ ( µ A ν ν A µ ) e ψγ ν ψ = 0 [iγ µ ( µ + iea µ ) m] ψ = 0 20
SU(N) Gauge Theory An SU(N) gauge theory describes an interacting N particle system: N Dirac spinors, representing the N (fermionic) particles: Ψ = (ψ 1,, ψ N ) T K def = N 2 1 gauge fields, representing the interacting potentials between these N particles: G a µ = (G a 0, G a 1, G a 2, G a 3) for a = 1,, K. 21
The Dirac equations for the N fermions are: (25) [iγ µ D µ m] Ψ = 0, D µ = µ + igg a µτ a. Consider the SU(N) gauge transformation (26) Ψ = ΩΨ Ω = e iθ a τ a SU(N), where {τ 1,, τ K } is a basis of the set of traceless Hermitian matrices with λ j kl being the structure constants. The covariance of Dirac eqs (25) is equivalent to D µ Ψ = ΩDµ Ψ, which implies that (27) Ga τ a = G a µωτ a Ω 1 + i g ( µω)ω 1. 22
Hence the SU(N) gauge transformation takes the following form: ( Ψ, Ga µ τ a, m) = (ΩΨ, G aµωτ a Ω 1 + ig ) ( µω)ω 1, ΩmΩ 1. Principle of Gauge Invariance. The electromagnetic, the weak, and the strong interactions obey gauge invariance: the Dirac or Klein-Gordon dynamical equations involved in the three interactions are gauge covariant, and the actions of the interaction fields are gauge invariant. Physically, gauge invariance refers to the conservation of certain quantum property of the underlying interaction. In other words, such quantum property of the N particles cannot be distinguished for the interaction, and consequently, the energy contribution of these N particles associated with the interaction is invariant under the general SU(N) phase (gauge) transformations. 23
Geometry of SU(N) gauge theory spinor bundle: M p (C 4 ) N gauge fields as connections: D µ = µ + igg a µτ a field strength as curvature: i g [D µ, D ν ] = i g ( µ + igg a µτ a )( ν + igg a ντ a ) i g ( ν + igg a ντ a )( µ + igg a µτ a ) =( µ G a ν ν G a µ)τ a ig [ G a µτ a, G b ντ b ] =( µ G a ν ν G a µ + gλ a bcg b µg c ν)τ a def = F a µντ a def = F µν Gauge invariance and and Lorentz invariance dictate the Yang-Mills action: [ (28) L Y M (G a µ, Ψ) = 1 ] 4 G abfµνf a µνb + Ψ (iγ µ D µ m) Ψ dx. M 24
Consider the following representation transformation of SU(N): (29) τ a = x b aτ b, X = (x b a) is a nondegenerate complex matrix. Then it is easy to verify that θ a, A a µ, G ab = 1 2 Tr(τ aτ b ) and λc ab under the representation transformation (29). are SU(N)-tensors PRI (Ma-Wang, 2012): The SU(N) gauge theory must be invariant under the representation transformation (29): the Yang-Mills action of the gauge fields is invariant, and the corresponding gauge field equations are covariant. 25
SU(N) gauge field equations (Ma-Wang, 2012) G ab [ ν F b νµ gλ b cdg αβ F c αµg d β] g Ψγµ τ a Ψ = (iγ µ D µ m)ψ = 0 [ µ 1 ] 4 k2 gx µ + α b G b µ φ a, Spontaneous gauge symmetry breaking: The terms ( µ + α b G b µ)φ a on the right-hand side of the gauge field equations break the gauge symmetry, and are induced the the principle of interaction dynamics (PID), first postulated by Ma-Wang (2012). PID takes the variation of the Lagrangian action under energy-momentum conservation constraint. The term α b G b µ is the (total) interaction potential, S 0 is the SU(N) charge potential, and F = g G 0 is the force. 26
By PRI, any linear combination of gauge potentials from two different gauge groups are prohibited by PRI. For example, the term αa µ + βw 3 µ in the electroweak theory violates PRI: The physical quantities Wµ, a Wµν, a λ a bc and Gw ab SU(2) representation transformation. are SU(2)-tensors under the W 3 µ is simply one particular component of the SU(2) tensor (W 1 µ, W 2 µ, W 3 µ). αa µ + βw 3 µ will have no meaning if we perform a transformation. The combination αa µ +βw 3 µ in the classical electroweak theory and in the standard model is due to the particular way of coupling the Higgs fields and the gauge fields. Also, the same difficulty appears in the Einstein route of unification by embedding U(1) SU(2) SU(3) into a larger Lie group. 27
IV. 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), Ma-Wang (2012) principle of representation invariance (PRI), Ma-Wang (2012) 28
By PRI, the Lagrangian action coupling the four fundamental interactions is naturally given by L = M [L EH + L EM + L W + L S + L D ] gdx, L EH = R + 8πG c 4 S, L EM = 1 4 F µνf µν, L W = 1 4 Gw abw a µνw µνb, L S = 1 4 Gs kls k µνs µνl, L D = Ψ(iγ µ D µ m)ψ. 29
Unified field model (Ma-Wang, 2012): (30) (31) (32) (33) (34) 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 iγ µ D µ mc [ 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 = [ µ 14 k2wx µ + eαe c A µ + g wαb w c ] c Λj cd gαβ SαµS c β d g s qγ µ τ k q = [ µ 14 k2sx µ + eαe c A µ + g wαa w c ] Ψ = 0. W a µ + g sα s k c Sk µ W a µ + g sα s k c Sk µ ] φ E, W b µ + g sα s k c Sk µ ] φ w a, W µ a + g sαj s ] c Sj µ φ s k, ] Φ ν, 30
Conclusions and Predictions of the Unified Field Model 1. Duality: The unified field model induces a natural duality: (35) {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,, 8. 2. 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. 31
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. Weak and Strong interaction charges and potentials: The two SU(2) and SU(3) 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}. 32
5. Strong and weak interaction potentials: For the first time, we derive Layered strong interactions potentials (Ma-Wang, 2012 & 13): (36) Φ s = Ng s (ρ) [ 1 r A ] (1 + kr)e kr, g s (ρ) = ρ ( ρw ρ ) 3 g s, where ρ w is the radius of the elementary particle (i.e. the w weakton), ρ is the particle radius, k > 0 is a constant with k 1 being the strong interaction attraction radius of this particle, and A is the strong interaction constant, which depends on the type of particles. These potentials match very well with experimental data. Again, strong interaction demonstrates both attraction and repelling behaviors. 33
Φ r r 34
Quark confinement: With these potentials, the binding energy of quarks can be estimated as (37) E q ( ρn ρ q ) 7 E n 10 20 E n 10 18 GeV, where E n 10 2 GeV is the binding energy of nucleons. This clearly explains the quark confinement. 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 10 50 at the atomic level, F e 10 56 at the molecular level. This demonstrates the short-range nature of strong interaction. 35
Weak Interaction potential: W = Ng w ( ρw ρ ) 3 e r/r 0 [ 1 r B ] 2r (1 + )e r/r 0 ρ r 0 where r 0 = 10 16, N is the number of weak charges, ρ is the radius of the particle, ρ w is the radius of the weaktons, and B is a constant. 36
IV. Summary We have derived experimentally verifiable laws of Nature based only on a few fundamental first principles, guided by experimental and observation evidences. We have discovered three fundamental principles: the principle of interaction dynamics (PID), (MA-Wang, 2012), the principle of representation invariance (PRI) (Ma-Wang, 2012), and the principle of symmetry-breaking (PSB) for unification (Ma-Wang, 12-14). PID takes the variation of the action under the energy-momentum conservation constraints, and is required by the dark matter and dark energy phenomena for gravity), by the quark confinements (for strong interaction), and by the Higgs field (for the weak interaction). 37
PRI requires that the gauge theory be independent of the choices of the representation generators. These representation generators play the same role as coordinates, and in this sense, PRI is a coordinate-free invariance/covariance, reminiscent of the Einstein principle of general relativity. In other words, PRI is purely a logic requirement for the gauge theory. PSB offers an entirely different route of unification from the Einstein unification route which uses large symmetry group. The three sets of symmetries the general relativistic invariance, the Lorentz and gauge invariances, as well as the Galileo invariance are mutually independent and dictate in part the physical laws in different levels of Nature. For a system coupling different levels of physical laws, part of these symmetries must be broken. 38
These three new principles have profound physical consequences, and, in particular, provide a new route of unification for the four interactions: 1) the general relativity and the gauge symmetries dictate the Lagrangian; 2) the coupling of the four interactions is achieved through PID and PRI in the field equations, which obey the PGR and PRI, but break spontaneously the gauge symmetry; 3) the unified field model can be easily decoupled to study individual interaction, when the other interactions are negligible; and 4) the unified field model coupling the matter fields using PSB. The new theory gives rise to solutions and explanations to a number of longstanding mysteries in modern theoretical physics, including e.g. 1) dark matter and dark energy phenomena, 2) the structure of back holes, 3) the structure and origin of our Universe, 4) the unified field theory, 5) quark confinement, and 6) mechanism of supernovae explosion and active galactic nucleus jets. 39