Physical Laws of Nature vs Fundamental First Principles

Tian Ma, Shouhong Wang

Supported in part by NSF, ONR and Chinese NSFhttp://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 Natureare represented by mathematical equations, are dictated by a fewfundamental principles, and take the simplest and aesthetic forms.

• We intend to derive experimentally verifiable laws of Nature based only on a fewfundamental 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 Riemannianmanifold (M, gµν) with gµν being regarded as gravitational potentials.

• The principle of general relativity requires that the law of gravity be covariantunder general coordinate transformations, and dictates the Einstein-Hilbertaction:

(1) LEH(gµν) =

∫M

(R+

8πG

c4S

)√−gdx.

• The Einstein field equations are the Euler-Lagrangian equations of LEH:

(2) Rµν −1

2gµνR = −8πG

c4Tµν, ∇µTµν = 0

3

Dark Matter: Theoretically for galactic rotating stars, one should have

v2theoretical

r=Mobserved G

r2.

In 1970s, Vera Rubin observed the discrepancy between the observed velocity andtheoretical velocity

vobserved > vtheoretical,

which implies that

4M = (the mass corresponding to vobserved)−Mtheoretical > 0,

which is called dark matter.

Dark Energy was introduced to explain the observations since the 1990s byindicating that the universe is expanding at an accelerating rate. The 2011 NobelPrize 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-momentumtensor of visible matter Tµν may no longer be conserved: ∇µTµν 6= 0.

Consequently, the variation of LEH must be taken under energy-momentumconservation constraint, leading us to postulate a general principle, called principleof interaction dynamics (PID) (Ma-Wang, 2012):

(3)d

dλ

∣∣∣λ=0

LEH(gµν + λXµν) = 0 ∀X = Xµν with ∇µXµν = 0.

Then we derive a new set of gravitational field equations

(4)

Rµν −1

2gµνR = −8πG

c4Tµν −∇µΦν,

∇µ[

8πG

c4Tµν +∇µΦν

]= 0.

5

• The new term ∇µΦν cannot be derived

– from any existing f(R) theories, and– from any scalar field theories.

In other words, the term ∇µΦν does not correspond to any Lagrangian actiondensity, and is the direct consequence of PID.

• The field equations (4) establish a natural duality:

spin-2 graviton field gµν ←→ spin-1 dual vector graviton field Φµ

6

Central Gravitational Field

Consider a central gravitational field generated by a ball Br0 with radius r0 andmass M . It is known that the metric of the central field for r > r0 can be writtenin the form

(5) ds2 = −euc2dt2 + evdr2 + r2(dθ2 + sin2 θdϕ2), u = u(r), v = v(r).

Then the field equations (4) take the form

(6)

v′ +1

r(ev − 1) = −r

2u′φ′,

u′ − 1

r(ev − 1) = r(φ′′ − 1

2v′φ′),

u′′ +

(1

2u′ +

1

r

)(u′ − v′) = −2

rφ′.

7

We have derived in (Ma & Wang, 2012) an approximate gravitational force formula:

(7) F =mc2

2eu[−1

r(ev − 1)− rφ′′

].

This can be further simplified as

(8)F = mMG

(− 1

r2− k0r

+ k1r

)for r > r0,

k0 = 4× 10−18Km−1, k1 = 10−57Km−3,

demonstrating the presence of both dark matter and dark energy. Here the firstterm represents the Newton gravitation, the attracting second term stands for darkmatter and the repelling third term is the dark energy. We note that our modifiednew formula is derived from first principles.

8

(Hernandez, Ma & Wang, 2015):

Let x(s)def= (x1(s), x2(s), x3(s)) =

(esu′(es), ev(e

s) − 1, esφ′(es)). Then the non-

autonomous gravitational field equations (6) become an autonomous system:

(9)

x′1 = −x2 + 2x3 −1

2x21 −

1

2x1x3 −

1

2x1x2 −

1

4x21x3,

x′2 = −x2 −1

2x1x3 − x22 −

1

2x1x2x3,

x′3 = x1 − x2 + x3 −1

2x2x3 −

1

4x1x

23,

(x1, x2, x3)(s0) = (α1, α2, α3) with r0 = es0.

• The asymptotically flat space-time geometry is represented by x = 0, which is afixed point of the system (9).

• There is a two-dimensional stable manifold Es near x = 0, which can be

9

parameterized by

(10) x3 = h(x1, x2) = −1

2x1 +

1

2x2 +

1

16x21 −

1

16x22 +O(|x|3).

The field equations (9) are reduced to a two-dimensional dynamical system:

(11)

x′1 = −x1 −1

8x21 −

1

8x22 −

3

4x1x2 +O(|x|3),

x′2 = −x2 +1

4x21 − x22 −

1

4x1x2 +O(|x|3),

(x1, x2)(s0) = (α1, α2),

with x3 being slaved by (10).

10

Figure 1: Only the orbits on Ω1 with x1 > 0 will eventually cross the x2-axis,leading to the sign change of x1, and to a repelling gravitational force.

11

Asymptotic Repulsion Theorem (Hernandez-Ma-Wang, 15) For a centralgravitational field, the following assertions hold true:

1) The gravitational force F is given by

F = −mc2

2euu′,

and is asymptotic zero:

(12) F → 0 if r →∞.

2) If the initial value α in (11) is near the Schwarzschild solution with 0 < α1 <α2/2, then there exists a sufficiently large r1 such that the gravitational forceF is repulsive for r > r1:

(13) F > 0 for r > r1.

12

• The above theorem is valid provided the initial value α is small. Also, allphysically meaningful central fields satisfy the condition (note that a black holeis enclosed by a huge quantity of matter with radius r0 2MG/c2).

In fact, the Schwarzschild initial values are as

(14) x1(r0) = x2(r0) =δ

1− δ, δ =

2MG

c2r0.

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 betweenclusters of galaxies, and consequently the above theorem is valid for centralgravitational fields generated by both galaxies and clusters of galaxies.

The asymptotic repulsion of gravity (dark energy) plays the role to stabilize thelarge 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, gµν, withthe metric gµν 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 determinegravitational potential gµν and its dual vector field Φµ = ∇µφ;

4. Gravity can display both attractive and repulsive effect, caused by the dualitybetween the attracting gravitational field gµν and the repulsive dual vectorfield Φµ, together with their nonlinear interactions governed by the fieldequations (4).

5. It is the nonlinear interaction between gµν and the dual field Φµ that leads toa unified theory of dark matter and dark energy phenomena.

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II. Principle of Interaction Dynamics (PID)

Note: The new gravitational field equations, δLEH(gµν) = −∇µΨν, violate theleast 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(gµν, A, ψ) =

∫M

L(gµν, A, ψ)√−gdx

where gµν is the gravitational field, A is a set of vector fields representing thegauge fields, and ψ is the wave functions of particles.

The action obeys the principle of general relativity, the Lorentz and gaugeinvariances, and the principle of representation invariance (PRI).

15

2. The interaction fields (gµν, A, ψ) satisfy the variation equations of L withdivA-free constraints:

δ

δgµνL(gij, A, ψ) = (∇µ + αbA

bµ)Φν,

δ

δAaµL(gij, A, ψ) = (∇µ + β

(a)b Abµ)φa,

δ

δψL(gij, A, ψ) = 0.

Note: The action is gauge invariant, but the field equations break the gaugesymmetry.

PID offers a simpler way of introducing Higgs fields from the first principle.

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III. Principle of Representation Invariance

• Herman Weyl (1919): scale invariance gµν → eα(x)gµν leading to conformalconnection.

• 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 nucleonsassociated 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 ψmay depend on x, y, z, and t, the relative phase factor of ψ may at twodifferent 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:

curl ~E = −1

c

∂ ~H

∂t, div ~H = 0,(17)

curl ~H =1

c

∂ ~E

∂t+

4π

c~J, div ~E = 4πρ,(18)

where ~E, ~H are the electric and magnetic fields, and ~J is the current density, andρ 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 +∇A0.

Take Aµ = (A0, ~A). Then the Maxwell equations can be written as

(19) ∂µFµν =4π

cJν with Fµν = ∂µAν − ∂νAµ, Jµ = (−cρ, ~J).

18

Question: What is the dynamic motion law of a charged electron?

Dirac equations:

(20) [iγµ∂µ −m]ψ = 0 ψ = (ψ1, ψ2, ψ3, ψ4)T

γµ = (γ0, γ1, γ2, γ3), γ0 =

(I 00 −I

), γk =

(0 σk−σk 0

)σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)Special solutions:

e−mc2t/~

1000

, e−mc2t/~

0100

, e+mc2t/~

0010

, e+mc2t/~

0001

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Experimentally, one cannot distinguish the two states ψ = eiθ(xµ)ψ and ψ.

Mathematically, the phenomenon says that the Dirac equations are covariant underthe U(1) gauge transformation (i.e., Dµψ = eiθDµψ):

(21) (ψ, Aµ) =

(eiθ(x

µ)ψ,Aµ −1

e∂µθ

).

The U(1) gauge symmetry and the Lorentz symmetry dictate the action:

(22) L(Aµ, ψ) =

∫M

−1

4FµνF

µν + ψ [i(γµ∂µ + ieAµ)−m]ψdM,

whose variation is the quantum electrodynamics (QED) field equations:

∂µ(∂µAν − ∂νAµ)− eψγνψ = 0(23)

[iγµ(∂µ + ieAµ)−m]ψ = 0(24)

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 = (ψ1, · · · , ψN)T

• K def= N2 − 1 gauge fields, representing the interacting potentials between these

N particles:Gaµ = (Ga0, G

a1, G

a2, G

a3) for a = 1, · · · ,K.

21

The Dirac equations for the N fermions are:

[iγµDµ −m] Ψ = 0, Dµ = ∂µ + igGaµτa.(25)

Consider the SU(N) gauge transformation

(26) Ψ = ΩΨ ∀Ω = eiθaτa ∈ SU(N),

where τ1, · · · , τK is a basis of the set of traceless Hermitian matrices with λjklbeing the structure constants.

The covariance of Dirac eqs (25) is equivalent to DµΨ = ΩDµΨ, which impliesthat

(27) Gaτa = GaµΩτaΩ−1 +

i

g(∂µΩ)Ω−1.

22

Hence the SU(N) gauge transformation takes the following form:

(Ψ, Gaµτa, m) =

(ΩΨ, GaµΩτaΩ

−1 +i

g(∂µΩ)Ω−1, ΩmΩ−1

).

Principle of Gauge Invariance. The electromagnetic, the weak, and thestrong interactions obey gauge invariance:

• the Dirac or Klein-Gordon dynamical equations involved in the threeinteractions are gauge covariant, and• the actions of the interaction fields are gauge invariant.

Physically, gauge invariance refers to the conservation of certain quantum propertyof the underlying interaction. In other words, such quantum property of the Nparticles cannot be distinguished for the interaction, and consequently, the energycontribution of these N particles associated with the interaction is invariant underthe general SU(N) phase (gauge) transformations.

23

Geometry of SU(N) gauge theory

• spinor bundle: M ⊗p (C4)N

• gauge fields as connections: Dµ = ∂µ + igGaµτa

• field strength as curvature:

i

g[Dµ, Dν] =

i

g(∂µ + igGaµτa)(∂ν + igGaντa)−

i

g(∂ν + igGaντa)(∂µ + igGaµτa)

=(∂µGaν − ∂νGaµ)τa − ig

[Gaµτa, G

bντb]

=(∂µGaν − ∂νGaµ + gλabcG

bµG

cν)τa

def= F aµντa

def= Fµν

• Gauge invariance and and Lorentz invariance dictate the Yang-Mills action:

(28) LYM(Gaµ,Ψ) =

∫M

[−1

4GabF aµνFµνb + Ψ (iγµDµ −m) Ψ

]dx.

24

Consider the following representation transformation of SU(N):

(29) τa = xbaτb, X = (xba) is a nondegenerate complex matrix.

Then it is easy to verify that θa, Aaµ, Gab = 12Tr(τaτ

†b ) and λcab are SU(N)-tensors

under the representation transformation (29).

PRI (Ma-Wang, 2012): The SU(N) gauge theory must be invariant underthe 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)

Gab[∂νF bνµ − gλbcdgαβF cαµGdβ

]− gΨγµτaΨ =

[∂µ −

1

4k2gxµ + αbG

bµ

]φa,

(iγµDµ −m)Ψ = 0

• Spontaneous gauge symmetry breaking: The terms (∂µ + αbGbµ)φa on the

right-hand side of the gauge field equations break the gauge symmetry, andare induced the the principle of interaction dynamics (PID), first postulated byMa-Wang (2012).

• PID takes the variation of the Lagrangian action under energy-momentumconservation constraint.

• The term αbGbµ is the (total) interaction potential, S0 is the SU(N) charge

potential, and F = −g∇G0 is the force.

26

By PRI, any linear combination of gauge potentials from two different gaugegroups are prohibited by PRI.

For example, the term αAµ + βW 3µ in the electroweak theory violates PRI:

• The physical quantities W aµ , W a

µν, λabc and Gwab are SU(2)-tensors under theSU(2) representation transformation.

• W 3µ is simply one particular component of the SU(2) tensor (W 1

µ,W2µ,W

3µ).

• αAµ + βW 3µ will have no meaning if we perform a transformation.

The combination αAµ+βW 3µ 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 embeddingU(1)× SU(2)× SU(3) into a larger Lie group.

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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)

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By PRI, the Lagrangian action coupling the four fundamental interactions isnaturally given by

L =

∫M

[LEH + LEM + LW + LS + LD]√−gdx,

LEH = R+8πG

c4S,

LEM = −1

4FµνF

µν,

LW = −1

4GwabW

aµνW

µνb,

LS = −1

4GsklS

kµνS

µνl,

LD = Ψ(iγµDµ −m)Ψ.

29

Unified field model (Ma-Wang, 2012):

Rµν −1

2gµνR+

8πG

c4Tµν =

[∇µ +

eαE

~cAµ +

gwαwa

~cW aµ +

gsαsk

~cSkµ

]Φν,(30)

∂νFνµ − eψγµψ =

[∇µ +

eαE

~cAµ +

gwαwa

~cW aµ +

gsαsk

~cSkµ

]φE,(31)

Gwab

[∂νW b

νµ −gw~cλbcdg

αβW cαµW

dβ

]− gwLγµσaL(32)

=

[∇µ −

1

4k2wxµ +

eαE

~cAµ +

gwαwb

~cW bµ +

gsαsk

~cSkµ

]φwa ,

Gskj

[∂νSjνµ −

gs~c

ΛjcdgαβScαµS

dβ

]− gsqγµτkq(33)

=

[∇µ −

1

4k2sxµ +

eαE

~cAµ +

gwαwa

~cW aµ +

gsαsj

~cSjµ

]φsk,[

iγµDµ −mc

~

]Ψ = 0.(34)

30

Conclusions and Predictions of the Unified Field Model

1. Duality: The unified field model induces a natural duality:

(35)

gµν (massless graviton) ←→ Φµ,

Aµ (photon) ←→ φE,

W aµ (massive bosons W± & Z) ←→ φwa for a = 1, 2, 3,

Skµ (massless gluons) ←→ φsk for k = 1, · · · , 8.

2. Decoupling and Unification: An important characteristics is that the unifiedmodel can be easily decoupled. Namely, Both PID and PRI can be applied directlyto individual interactions.

For gravity alone, we have derived modified Einstein equations, leading to a unifiedtheory for dark matter and dark energy.

31

3. Spontaneous symmetry breaking and mass generation mechanism: Weobtained a much simpler mechanism for mass generation and energy creation,completely different from the classical Higgs mechanism. This new mechanismoffers new insights on the origin of mass.

4. Weak and Strong interaction charges and potentials: The two SU(2)and SU(3) constant vectors αwa and αsk, containing 11 parameters, representthe portions distributed to the gauge potentials by the weak charge gw and strongcharge gs. We define, e.g., the total potential Sµ:

Sµ = αskSkµ =S0, S1, S2, S3

=strong charge potential, strong rotational potential.

32

5. Strong and weak interaction potentials: For the first time, we derive Layeredstrong interactions potentials (Ma-Wang, 2012 & 13):

(36) Φs = Ngs(ρ)

[1

r− Aρ

(1 + kr)e−kr], gs(ρ) =

(ρwρ

)3

gs,

where ρw is the radius of the elementary particle (i.e. the w∗ weakton), ρ is theparticle radius, k > 0 is a constant with k−1 being the strong interaction attractionradius of this particle, and A is the strong interaction constant, which depends onthe type of particles.

These potentials match very well with experimental data.

Again, strong interaction demonstrates both attraction and repelling behaviors.

33

rr

!

34

Quark confinement: With these potentials, the binding energy of quarks can beestimated as

(37) Eq ∼(ρnρq

)7

En ∼ 1020En ∼ 1018GeV,

where En ∼ 10−2GeV 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 ratiobetween strong force and electromagnetic attraction force is

FaFe∼

10−50 at the atomic level,

10−56 at the molecular level.

This demonstrates the short-range nature of strong interaction.

35

Weak Interaction potential:

W = Ngw

(ρwρ

)3

e−r/r0[

1

r− Bρ

(1 +2r

r0)e−r/r0

]where r0 = 10−16, N is the number of weak charges, ρ is the radius of the particle,ρw 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 fewfundamental 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), andthe principle of symmetry-breaking (PSB) for unification (Ma-Wang, 12-14).

• PID takes the variation of the action under the energy-momentum conservationconstraints, and is required by the dark matter and dark energy phenomena forgravity), by the quark confinements (for strong interaction), and by the Higgsfield (for the weak interaction).

37

• PRI requires that the gauge theory be independent of the choices of therepresentation generators. These representation generators play the same roleas coordinates, and in this sense, PRI is a coordinate-free invariance/covariance,reminiscent of the Einstein principle of general relativity. In other words, PRI ispurely a logic requirement for the gauge theory.

• PSB offers an entirely different route of unification from the Einstein unificationroute which uses large symmetry group. The three sets of symmetries — thegeneral relativistic invariance, the Lorentz and gauge invariances, as well as theGalileo invariance — are mutually independent and dictate in part the physicallaws in different levels of Nature. For a system coupling different levels ofphysical laws, part of these symmetries must be broken.

38

• These three new principles have profound physical consequences, and, inparticular, 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 thegauge 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 oflongstanding mysteries in modern theoretical physics, including e.g. 1) darkmatter and dark energy phenomena, 2) the structure of back holes, 3) thestructure and origin of our Universe, 4) the unified field theory, 5) quarkconfinement, and 6) mechanism of supernovae explosion and active galacticnucleus jets.

39