Download Physical Laws of Nature vs Fundamental First Principles

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, gµν ) with gµν 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:
Z √
8πG
−gdx.
(1)
LEH ({gµν }) =
R+ 4 S
c
M
• The Einstein field equations are the Euler-Lagrangian equations of LEH :
(2)
1
8πG
Rµν − gµν R = − 4 Tµν ,
2
c
∇µTµν = 0
3
Dark Matter: Theoretically for galactic rotating stars, one should have
2
vtheoretical
Mobserved G
.
=
r
r2
In 1970s, Vera Rubin observed the discrepancy between the observed velocity and
theoretical 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 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 Tµν may no longer be conserved: ∇µTµν 6= 0.
Consequently, the variation of LEH 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 LEH (gµν + λXµν ) = 0 ∀X = {Xµν } with ∇µXµν = 0.
dλ λ=0
Then we derive a new set of gravitational field equations
(4)
1
8πG
Rµν − gµν R = − 4 Tµν − ∇µΦν ,
2
c
8πG
∇µ
Tµν + ∇µΦν = 0.
4
c
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 action
density, 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 and
mass M . It is known that the metric of the central field for r > r0 can be written
in the form
(5)
ds2 = −euc2dt2 + ev dr2 + r2(dθ2 + sin2 θdϕ2),
u = u(r), v = v(r).
Then the field equations (4) take the form
(6)
1
r
v 0 + (ev − 1) = − u0φ0,
r
2
1
1
u0 − (ev − 1) = r(φ00 − v 0φ0),
r
2
1 0 1
2
u00 +
u +
(u0 − v 0) = − φ0.
2
r
r
7
We have derived in (Ma & Wang, 2012) an approximate gravitational force formula:
2
(7)
F =
1
mc u
e − (ev − 1) − rφ00 .
2
r
This can be further simplified as
(8)
1
k0
F = mM G − 2 −
+ k1 r
r
r
k0 = 4 × 10−18Km−1,
for r > r0,
k1 = 10−57Km−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):
def
s 0
s
v(es )
s 0
s
Let x(s) = (x1(s), x2(s), x3(s)) = e u (e ), e
− 1, e φ (e ) . Then the nonautonomous gravitational field equations (6) become an autonomous system:
(9)
1
1
1
1
x01 = −x2 + 2x3 − x21 − x1x3 − x1x2 − x21x3,
2
2
2
4
1
1
x02 = −x2 − x1x3 − x22 − x1x2x3,
2
2
1
1
x03 = x1 − x2 + x3 − x2x3 − x1x23,
2
4
(x1, x2, x3)(s0) = (α1, α2, α3)
with r0 = es0 .
• 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)
1
1
1
1
x3 = h(x1, x2) = − x1 + x2 + x21 − x22 + O(|x|3).
2
2
16
16
The field equations (9) are reduced to a two-dimensional dynamical system:
1 2 1 2 3
= −x1 − x1 − x2 − x1x2 + O(|x|3),
8
8
4
1
1
x02 = −x2 + x21 − x22 − x1x2 + O(|x|3),
4
4
(x1, x2)(s0) = (α1, α2),
x01
(11)
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 central
gravitational field, the following assertions hold true:
1) The gravitational force F is given by
mc2 u 0
F =−
e u,
2
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 force
F 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, all
physically meaningful central fields satisfy the condition (note that a black hole
is enclosed by a huge quantity of matter with radius r0 2M G/c2).
In fact, the Schwarzschild initial values are as
(14)
δ
x1(r0) = x2(r0) =
,
1−δ
2M G
δ= 2 .
c r0
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, gµν }, with
the 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 determine
gravitational potential {gµν } and its dual vector field Φµ = ∇µφ;
4. Gravity can display both attractive and repulsive effect, caused by the duality
between the attracting gravitational field {gµν } and the repulsive dual vector
field {Φµ}, together with their nonlinear interactions governed by the field
equations (4).
5. It is the nonlinear interaction between {gµν } and the dual field Φµ 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, δLEH (gµν ) = −∇µΨν , 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
Z
(16)
√
L({gµν }, A, ψ) −gdx
L({gµν }, A, ψ) =
M
where gµν 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 (gµν , A, ψ) satisfy the variation equations of L with
divA-free constraints:
δ
b
L({g
},
A,
ψ)
=
(∇
+
α
A
ij
µ
b
µ )Φν ,
δg µν
δ
(a) b
a
L({g
},
A,
ψ)
=
(∇
+
β
A
)φ
,
ij
µ
µ
b
δAaµ
δ
L({gij }, A, ψ) = 0.
δψ
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 gµν → eα(x)gµν leading to conformal
connection.
• 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 0, ψ 0 =
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)
~
1
∂
H
~ =−
curl E
,
c ∂t
~ 4π
1
∂
E
~ =
~
curl H
+ J,
c ∂t
c
~ = 0,
div H
~ = 4πρ,
div E
~ H
~ are the electric and magnetic fields, and J~ is the current density, and
where E,
ρ is the charge density.
~ such that H
~ = curl A,
~ and there is an A0 such that
By (17), there is a vector A
~ = − 1 ∂ A~ + ∇A0.
E
c ∂t
~ Then the Maxwell equations can be written as
Take Aµ = (A0, A).
(19)
∂ µFµν =
4π
Jν
c
with
~
Fµν = ∂µAν − ∂ν Aµ, Jµ = (−cρ, J).
18
Question: What is the dynamic motion law of a charged electron?
Dirac equations:
[iγ µ∂µ − m] ψ = 0
(20)
γ µ = (γ 0, γ 1, γ 2, γ 3),
σ1 =
0 1
,
1 0
Special solutions:
 
1
 
−mc2 t/~ 0
e
0 ,
0
ψ = (ψ1, ψ2, ψ3, ψ4)T
I 0
,
0 −I
0 −i
σ2 =
,
i 0
γ0 =
 
0
 
−mc2 t/~ 1
e
0 ,
0
γk =
σ3 =
 
0
 
+mc2 t/~ 0
e
1 ,
0
0
−σk
1 0
0 −1
σk
0
 
0
 
+mc2 t/~ 0
e
0
1
19
µ
Experimentally, one cannot distinguish the two states ψe = eiθ(x )ψ and ψ.
Mathematically, the phenomenon says that the Dirac equations are covariant under
e µψe = eiθ Dµψ):
the U (1) gauge transformation (i.e., D
(21)
eA
eµ) =
(ψ,
µ
1
eiθ(x )ψ, Aµ − ∂µθ .
e
The U (1) gauge symmetry and the Lorentz symmetry dictate the action:
Z
1
(22)
L(Aµ, ψ) =
− Fµν F µν + ψ̄ [i(γ µ∂µ + ieAµ) − m] ψdM,
4
M
whose variation is the quantum electrodynamics (QED) field equations:
(23)
∂ µ(∂µAν − ∂ν Aµ) − eψ̄γν ψ = 0
(24)
[iγ µ(∂µ + ieAµ) − 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 = (ψ1, · · · , ψN )T
def
• K = N 2 − 1 gauge fields, representing the interacting potentials between these
N particles:
for a = 1, · · · , K.
Gaµ = (Ga0 , Ga1 , Ga2 , Ga3 )
21
The Dirac equations for the N fermions are:
(25)
[iγ µDµ − m] Ψ = 0,
Dµ = ∂µ + igGaµτa.
Consider the SU (N ) gauge transformation
(26)
e = ΩΨ
Ψ
∀Ω = eiθ
a
τa
∈ SU (N ),
where {τ1, · · · , τK } is a basis of the set of traceless Hermitian matrices with λjkl
being the structure constants.
e µΨ
e = ΩDµΨ, which implies
The covariance of Dirac eqs (25) is equivalent to D
that
(27)
e aτa = GaµΩτaΩ−1 + i (∂µΩ)Ω−1.
G
g
22
Hence the SU (N ) gauge transformation takes the following form:
e G
e aµτa, m)
(Ψ,
e =
i
ΩΨ, GaµΩτaΩ−1 + (∂µΩ)Ω−1, ΩmΩ−1 .
g
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 (C4)N
• gauge fields as connections: Dµ = ∂µ + igGaµτa
• field strength as curvature:
i
i
i
a
a
[Dµ, Dν ] = (∂µ + igGµτa)(∂ν + igGν τa) − (∂ν + igGaν τa)(∂µ + igGaµτa)
g
g
g
a
a
a
b
=(∂µGν − ∂ν Gµ)τa − ig Gµτa, Gν τb
def
def
a
τa = Fµν
=(∂µGaν − ∂ν Gaµ + gλabcGbµGcν )τa = Fµν
• Gauge invariance and and Lorentz invariance dictate the Yang-Mills action:
Z 1
a
(28)
LY M (Gaµ, Ψ) =
− GabFµν
F µνb + Ψ̄ (iγ µDµ − m) Ψ dx.
4
M
24
Consider the following representation transformation of SU (N ):
(29)
τea = xbaτb,
X = (xba) is a nondegenerate complex matrix.
Then it is easy to verify that θa, Aaµ, Gab = 12 Tr(τaτb†) and λcab are SU (N )-tensors
under the representation transformation (29).
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)
ν b
1
c
Gab ∂ Fνµ − gλbcdg αβ Fαµ
Gdβ − g Ψ̄γµτaΨ = ∂µ − kg2xµ + αbGbµ φa,
4
(iγ µDµ − 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, 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 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 gauge
groups are prohibited by PRI.
For example, the term αAµ + βWµ3 in the electroweak theory violates PRI:
a
w
• The physical quantities Wµa, Wµν
, λabc and Gab
are SU (2)-tensors under the
SU (2) representation transformation.
• Wµ3 is simply one particular component of the SU (2) tensor (Wµ1, Wµ2, Wµ3).
• αAµ + βWµ3 will have no meaning if we perform a transformation.
The combination αAµ +βWµ3 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. JonaLasinio 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
Z
L=
√
[LEH + LEM + LW + LS + LD ] −gdx,
M
8πG
LEH = R + 4 S,
c
1
LEM = − Fµν F µν ,
4
1
a
µνb
LW = − G w
,
ab Wµν W
4
1
k
LS = − GsklSµν
S µνl,
4
LD = Ψ̄(iγ µDµ − m)Ψ.
29
Unified field model (Ma-Wang, 2012):
(30)
(31)
(32)
(33)
(34)
8πG
eαE
gw αaw a gsαks k
1
Aµ +
Wµ +
S Φν ,
Rµν − gµν R + 4 Tµν = ∇µ +
2
c
~c
~c
~c µ
E
w
s
gw αa a gsαk k E
eα
Aµ +
Wµ +
Sµ φ ,
∂ ν Fνµ − eψ̄γµψ = ∇µ +
~c
~c
~c
h
i
g
w
ν
b
c
Gw
λbcdg αβ Wαµ
Wβd − gw L̄γµσaL
ab ∂ Wνµ −
~c
E
w
s
eα
gw αb b gsαk k w
1 2
xµ +
Aµ +
Wµ +
Sµ φa ,
= ∇ µ − kw
4
~c
~c
~c
h
i
g
s
j
j
c
Gskj ∂ ν Sνµ
− Λcdg αβ Sαµ
Sβd − gsq̄γµτk q
~c
1 2
eαE
gw αaw a gsαjs j s
= ∇ µ − ks x µ +
Aµ +
Wµ +
S φ ,
4
~c
~c
~c µ k
h
mc i
µ
iγ Dµ −
Ψ = 0.
~
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 ,
Wµa
(massive bosons W ± & Z) ←→ φw
a
Sµk
(massless gluons)
←→ φsk
for a = 1, 2, 3,
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 {αaw } and {αks }, containing 11 parameters, represent
the portions distributed to the gauge potentials by the weak charge gw and strong
charge gs. We define, e.g., the total potential Sµ:
Sµ = αks Sµk ={S0,
={strong charge potential,
S1, S2, S3}
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)
Φs = N gs(ρ)
1 A
− (1 + kr)e−kr ,
r ρ
gs(ρ) =
ρw
ρ
3
gs,
where ρw 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
r̄
r
34
Quark confinement: With these potentials, the binding energy of quarks can be
estimated as
7
ρn
(37)
Eq ∼
En ∼ 1020En ∼ 1018GeV,
ρq
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 ratio
between strong force and electromagnetic attraction force is
( −50
10
at the atomic level,
Fa
∼
Fe
10−56 at the molecular level.
This demonstrates the short-range nature of strong interaction.
35
Weak Interaction potential:
3
1
B
2r
ρw
e−r/r0
W = N gw
− (1 + )e−r/r0
ρ
r
ρ
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 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