# Physics 357/457 Spring 2014 Summary

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Physics 357/457 Spring 2014 Summary
The elementary particles Relativistic formulation Lagrangians Principle of least action QED and field operators the models how can we understand it?

the elementary particles (as far as we know at this time)
six quarks (u d c s t b) six leptons (e ne m nm t nt) all have spin = ½  they are fermions that’s it!

The forces (each force is associated with an exchanged particle)
electromagnetic (photon) weak ( W+ W- Z0) strong (8 gluons) gravitational ( graviton not yet observed) all have spin = 1 (or 2 for graviton)  they are bosons

Review: Special Relativity
Einstein’s assumption: the speed of light is independent of the (constant ) velocity, v, of the observer. It forms the basis for special relativity. Speed of light = C = |r2 – r1| / (t2 –t1) = |r2’ – r1’ | / (t2’ –t1‘) = |dr/dt| = |dr’/dt’|

4-dimensional vector component notation
xµ  ( x0, x1 , x2, x3 ) µ=0,1,2,3 = ( ct, x, y, z ) = (ct, r) xµ  ( x0 , x1 , x2 , x3 ) µ=0,1,2,3 = ( ct, -x, -y, -z ) = (ct, -r) contravariant components covariant components

Invariant dot products using 4-component notation
contravariant xµ xµ = µ=0,1,2,3 xµ xµ (repeated index one up, one down)  summation) xµ xµ = (ct)2 -x2 -y2 -z2 covariant Einstein summation notation

Any four vector dot product has the same value in all frames moving with constant velocity w.r.t. each other. Examples: xµxµ pµxµ pµpµ µµ pµµ µAµ

(E/c)2 – (px)2– (py)2 – (pz)2 is also invariant.
Suppose we consider the four-vector: (E/c, px , py , pz ) (E/c)2 – (px)2– (py)2 – (pz)2 is also invariant. In the center of mass of a particle this is equal to (mc2 /c)2 – (0)2– (0)2 – (0)2 = m2 c2 So, for a particle (in any frame) (E/c)2 – (px)2– (py)2 – (pz)2 = m2 c2

Invariant dot products using 4-component notation
µµ = µ=0,1,2,3 µµ (repeated index summation ) = 2/(ct)2 - 2 2 = 2/x2 + 2/y2 + 2/z2 Einstein summation notation

Lorentz Invariance Lorentz invariance of the laws of physics
is satisfied if the laws are cast in terms of four-vector dot products! Four vector dot products are said to be “Lorentz scalars”. In the relativistic field theories, we must use “Lorentz scalars” to express the interactions.

Standard Model Requires Treatment of Particles as Fields
Hamiltonian, H=E, is not Lorentz invariant. QM not a relativistic theory. Lagrangian, T-V, used in particle physics. Creation and annihilation must be described. Relativistic Quantum Field theory!

Motivation for Lagrangians and the Law of Least Action

What is a Particle Field?
A good example of a particle field is the electromagnetic field. It can be represented by the field function, A = ((r,t), A(r,t)). Classically  and A are related to E and B. The zero mass particles, photons, can be created and destroyed and represent the “quantization” of the field.

The wave equations for A and 
can be put into 4-vector form: Lorentz gauge! A+ (1/c) /(ct) = 0

Solutions to 1. We have seen how Maxwell’s equations can be cast into a single wave equation for the electromagnetic 4-vector, Aµ . This Aµ now represents the E and B of the EM field … and something else: the photon! If Aµ is to represent a photon – we want it to be able to represent any photon. That is, we want the most general solution to the equation:

This “Fourier expansion” of the photon operator is called
creation operator annihilation operator Spin vector Spin vector This “Fourier expansion” of the photon operator is called “second quantization”. Note that the solution to the wave equation consists of a sum over an infinite number of “photon” creation and annihilation terms. Once the ak± are interpreted as operators, the A becomes an operator.

creation & annihilation operators
The procedure by which quantum fields are constructed from individual particles was introduced by Dirac, and is (for historical reasons) known as second quantization. Second quantization refers to expressing a field in terms of creation and annihilation operators, which act on single particle states: |0> = vacuum, no particle |p> = one particle with momentum vector p

From Quantum Mechanics to Lagrangian Densities
Just as there is no “derivation” of quantum mechanics from classical mechanics, there is no derivation of relativistic field theory from quantum mechanics. The “route” from one to the other is based on physically reasonable postulates and the imposition of Lorentz invariance and relativistic kinematics. The final “theory” is a model whose survival depends absolutely on its success in producing “numbers” which agree with experiment.

Note that *(r,t) (r,t) does not represent the probability
per unit volume density of the particle being at (r,t).

The “wave equation”:

The field operator for a neutral, spin =0, particle is
creates a single particle with momentum p= k and p0 =  k0 at (r,t) Destroys a single particle with momentum p= k and p0 =  k0 at (r,t)

and the Euler-Lagrange equations give F = ma
Lagrangians and the Lagrangian Density Recall that, and the Euler-Lagrange equations give F = ma In quantum field theory, the Euler-Lagrange equations give the particle wave function!

This calls for a different kind of “Lagrangian” -- not like the one used
in classical or quantum mechanics. So, we have another postulate, defining what is meant by a “Lagrangian” – called a Lagrangian density. d/dt in the classical theory

Summary for neutral (Q=0) scalar (spin = 0) particle, , with mass, m.
Lagrangian density wave equation field operator

Particles with Charge: two fields , and *
From the Lagrangian density and the Euler-Lagrange equation we can derive the wave equation

creates positively charged particle with momentum
p= k and p0 =  k0 at (r,t) destroys negatively charged particle with momentum p= k and p0 =  k0 at (r,t) destroys negatively charged particle with momentum p= k and p0 =  k0 at (r,t) creates negatively charged particle with momentum p= k and p0 =  k0 at (r,t)

Gauge Invariance and Conserved Quantities
“Noether's theorem” was proven by German mathematician, Emmy Noether in 1915 and published in Noether's theorem has become a fundamental tool of quantum field theory – and has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". Amalie Emmy Noether

’  An astounding result: we can vary the (complex) phase
of the field operator, , everywhere in space by any continuous amount and not affect the “laws of physics” (that is the L) which govern the system! Note that everywhere in space the phase changes by the same . This is called a global symmetry. ’ Remember Emmy Noether!

With the help of Emmy Noether, we can prove that
charge is conserved!

Deriving the conserved current and the conserved charge:
Euler-Lagrange equation conserved current But our Lagrangian density also contains a *, so we obtain additional terms like the above, with  replaced by *. In each case the Euler –Lagrange equations are satisfied. So, the remaining term is as follows:

Now we evaluate . The great advantage of  being a continuous constant is that there are an infinite number of very small  which carry with them all the physics of the “continuity”. That is, with no loss of rigor we can assume  is small!

The value of the charge is calculated from:
the charge operator. integrate over time S0(t) incoming particle outgoing particle integrate over all space One obtains a number! p incoming outgoing

Calculation of Charge

Note Dirac delta function in k and p

Note: + and/or – must be together.
The time disappears! Q is time independent. The integration over k’ is done with the Dirac delta function from the d3x integration. The remaining integration over k will be done with the Dirac delta functions from the commutation relations. Note: + and/or – must be together.

Message: calculating charge
is a lot of work -- but can be done!

Fermions and the Dirac Equation
In 1928 Dirac proposed the following form for the electron wave equation: 4-row column matrix 4x4 matrix 4x4 unit matrix The four µ matrices form a Lorentz 4-vector, with components, µ. That is, they transform like a 4-vector under Lorentz transformations between moving frames. Each µ is a 4x4 matrix.

The Dirac equation in full matrix form
0 1 2 3 spin dependence space-time dependence

Spinors for the particle with p along z direction
p along z and spin = +1/2 p along z and spin = -1/2

Field operator for the spin ½ fermion
Spinor for antiparticle with momentum p and spin s Creates antiparticle with momentum p and spin s Note: pµ pµ = m2 c2

Lagrangian Density for Spin 1/2 Fermions
Comments: 1. This Lagrangian density is used for all the quarks and leptons – only the masses will be different! 2. The Euler Lagrange equations, when applied to this Lagrangian density, give the Dirac Equation! 3. Note that L is a Lorentz scalar.

Lagrangian Density for Spin 1/2 Quarks and Leptons
Now we are ready to talk about the gauge invariance that leads to the Standard Model and all its interactions. Remember a “gauge invariance” is the invariance of the above Lagrangian under transformations like   e i . The physics is in the  -- which can be a matrix operator and depend on x,y, z and t.

Local Gauge Invariance and Existence of the Gauge Particles
Gauge transformations are like “rotations” How do functions transform under “rotations”? How can we generalize to rotations in “strange” spaces (spin space, , flavor space, color space)? 4. How are Lagrangians made invariant under these “rotations”? (Lagrangians  “laws of physics” for particles interactions.) 5. Invariance of L requires the existence of the gauge boson!

momentum operator x component momentum operator

The angular momentum operator, generates rotations in x,y,z space!

Gauge transformations are like the “rotations” we have just been considering
Real function of space and time one has to find a Lagrangian which is invariant under this transformation.  can be an operator -- as we have just seen.

How are Lagrangians made invariant under these “rotations”?
It won’t work!

Constructing a gauge invariant Lagrangian:
1. Begin with the “old Lagrangian”: called the “covariant derivative” 2. Replace Aµ is the gauge boson (exchange particle) field! 3. “old” Lagrangian the interaction term.

Showing L is invariant A µ = Aµ - (1/e)  transformed L
transformed A Maxwell’s equations are invariant under this!

Summary of Local gauge symmetry
Real function of space and time covariant derivative The final invariant L is given by:

The correct, invariant Lagrangian density, includes the interaction between the electron (fermion) and the photon (the gauge particle). free electron Lagrangian interaction Lagrangian If the coupling, e, is turned off, L reverts to the free electron L. This use of the covariant derivative will be applied to all the interaction terms of the Standard Model.

Comments: 1. There is no difference between changing the phase of the field operator of the fermion (by (r,t) at every point in space) and the effects of a gauge transformation [ -(1/e)µ (r,t) ] on the photon field! 2. Maxwell’s equations are invariant under A µ A µ - (1/e)µ (r,t) -- and, in particular, the gauge transformation has no effect on the free photon. 3. It is only because (r,t) depends on r and t that the above is possible. This is called a local gauge transformation. 4. Note that a global gauge transformation would require that  is a constant!

Lecture 10: Standard Model Lagrangian
The Standard Model Lagrangian is obtained by imposing three local gauge invariances on the quark and lepton field operators: symmetry: gauge boson U(1) “QED-like”  neutral gauge boson SU(2) weak  3 heavy vector bosons SU(3) color  8 gluons This gives rise to spin = 1 force carrying gauge particles.

SU(2) and SU(n) n = 2  3 components  3 gauge particles dot product
Pauli spin matrix functions of x,y,z,t The  are called the generators of the group. n = 2  3 components  3 gauge particles

SU(2): rotations in Flavor Space
“rotated” flavor state original flavor state These are the Pauli spin matrices, 1 2 3 local  depends on x, y, z, and t.

Flavor Space Flavor space is used to describe an intrinsic property of a particle. While this is not (x,y,z,t) space we can use the same mathematical tools to describe it. Flavor space can be thought of as a three dimensional space. The particle eigenstates we know about (quarks and leptons) are “doublets” with flavor up or down – along the “3” axis.

Summary: QED local gauge symmetry
Real function of space and time covariant derivative The final invariant L is given by:

SU(2) local gauge symmetry
generator of SU(2) rotations in flavor space! covariant derivative coupling constant generators of SU(2) The final invariant L is given by: interaction term interaction term

Rotations (on quark states) in color space: SU(3)
The quarks are assumed to carry an additional property called color. So, for the down quark, d, we have the “down quark color triplet”: quark field operators = d red red green green blue blue There is a color triplet for each quark: u, d, c, s, t, and b, but, for now we won’t need the t and b.

A general “rotation” in color space can be written as a local,
(non-abelian) SU(3) gauge transformation generators of SU(3) local red green blue a = 1,2,3,…8 Since the a don’t commute, the SU(3) gauge transformations are non-abelian.

The generators of SU(3): eight 3x3 a matrices (a = 1,2,3…8)
1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = (n2 – 1) = = 8 generators All 3x3 matrix elements of SU(3) can be written as a linear combination of these 8 a plus the identity matrix.

[ 1 , 2 ] 1 2 the a don’t commute = 2i f123 3 1 3  f123 = 1 = - f213 = f231 Likewise one can show: (for the graduate students) fabc = -fbac = fbca f458 = f678 = 3 /2, f147 = f516 = f246 = f257 = f345 = f637 = ½ … all the rest = 0.

SU(3) gauge invariance in the Standard Model
generators of SU(3) generators of SU(3) The invariant Lagrangian density is given by: interaction term

The Lagrangian density with the U(1), SU(2) and SU(3)
gauge particle interactions Y neutral vector boson heavy vectors bosons (W, W3) 8 gluons

Standard Model Lagrangian with Electro-Weak Unification
The Standard Model assumes that the mass of the neutrino is zero and that it is “left handed” -- travelling with its spin pointing opposite to its direction of motion. Since in this case there would be no “right handed” neutrino, the “flavor” partner of the neutrino must be a “left handed” electron. This changes the structure of the Standard Model Lagrangian – which is assumed to treat only left handed flavor doublets.

These are the only spinors allowed for a zero mass neutrino!
positive helicity negative helicity The neutrino, if it has a zero mass can only have its spin pointing along (or opposite to) it’s momentum.

Non-conservation of parity: Wu 1957

Each term in the SM Lagrangian density containing quarks and the leptons can be rewritten using the following expression. For the neutrinos, however, only the left handed term exists. In the following slide we use the notation d R = dR

of particles with the left and right handed parts shown explicitly.
The following is the interaction Lagrangian density for the first generation of particles with the left and right handed parts shown explicitly. B B B B B B W3 W1 W2+ W1 W2+ W3 sum over a = 1,2,…8   aGa   aGa

sin2W  0.23 W = Weinberg angle -- to be determined experimentally!
Weinberg’s decomposition of the B and W: W = Weinberg angle -- to be determined experimentally! sin2W  0.23

Next steps: rewriting interaction Lagrangian density so
that interactions with the photon are identified. 1. 2. 3. The neutrino has zero charge and can’t interact with the photon.

g 2 = e / sinW T 3f = + 1/2 for the uL YL = -1 YR = 2 YL
After substituting the expressions for B and W0 (which takes some work), one can identify factors which equal e, the electronic charge, or the up quark charge, etc. This permits one to find relationships between sinW , cos W , e, g2 and g1. One finds that: g 2 = e / sinW g 1 = e / cosW YL = -1 YR = 2 YL T 3f = + 1/2 for the uL = - 1/2 for the dL = 0 for uR = 0 for dR Also one defines:

The Standard Model Interaction Lagrangian for the 1st generation
(E & M) QED interactions weak neutral current interactions + weak flavor changing interactions + QCD color interactions

e g 2 = e / sinW A g2 g2 The U(1) and SU(2) interaction terms W+
(E & M) QED interactions g2 Z+ Z weak neutral current interactions g2 W+ W- weak flavor changing interactions

The following values for the constants
gives the correct charge for all the particles.

Weak neutral current interactions
Z0 Z0 Z0 Z0

Weak charged flavor changing interactions
quarks g2 leptons

Quantum Chromodynamics (QCD): color forces
Only non-zero components of  contribute.

To find the final form of the QCD terms, we rewrite the above sum,
collecting similar quark “color” combinations.

The QCD interaction Lagrangian density

grg - - ggb Note that there are only 8 possibilities: r g
The red, anti-green gluon - ggb The green, anti-blue gluon

At any time the proton is color neutral. That is,
The gluon forces hold the proton together At any time the proton is color neutral. That is, it contains one red, one blue and one green quark. proton

beta decay u u d d d u W- proton neutron W doesn’t see color

- W production from - p p p d u p u W+ - d - u - p p - u

n p p n The nuclear force u u d d d u u W- u d d d u u
Note that W-  d + u = - In older theories, one would consider rather the exchange of a - between the n and p. -

Cross sections and Feynman diagrams
everything happens here transition probability amplitude must sum over all possible Feynman diagram amplitudes with the same initial and final states .

Feynman rules applied to a 2-vertex electron positron scattering diagram
Note that each vertex is generated by the interaction Lagrangian density. time spin spin metric tensor Mfi = left vertex function right vertex function coupling constant – one for each vertex propagator The next steps are to do the sum over  and  and carry out the matrix multiplications. Note that  is a 4x4 matrix and the spinors are 4-component vectors. The result is a a function of the momenta only, and the four spin (helicity) states.

Confinement of quarks free quark terms free gluon terms
quark- gluon interactions The free gluon terms have products of 2, 3 and 4 gluon field operators. These terms lead to the interaction of gluons with other gluons.

G G Nf= # flavors Nc= # colors Nc Nf Note sign
normal free gluon term 3-gluon vertex Nf= # flavors Nc= # colors Nc Nf quark loop gluon loop

Nf Nc Nf Nc -7 momentum squared of exchanged gluon  M2quark
In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex. This term aslo has a negative sign.

Quark confinement arises from the increasing strength of the interaction at
long range. At short range the gluon force is weak; at long range it is strong. This confinement arises from the SU(3) symmetry – with it’s non-commuting (non-abelian) group elements. This non-commuting property generates terms in the Lagrangian density which produce 3-gluon vertices – and gluon loops in the exchanged gluon “propagator”.

Grand Unified Theory, Running Coupling Constants and the Story of our Universe
These next theories are in a less rigorous state and we shall talk about them, keeping in mind that they are at the ‘”edge” of what is understood today. Nevertheless, they represent a qualitative view of our universe, from the perspective of particle physics and cosmology. GUT -- Grand Unified Theories – symmetry between quarks and leptons; decay of the proton. Running coupling constants: it’s possible that at one time in the development of the universe all the forces had the same strength The Early Universe: a big bang, cooling and expanding, phase transitions and broken symmetries

Grand Unified Theory (GUT)
We have incorporated into the Lagrangian density invariance under rotations in U(1)XSU(2)flavor space and SU(3)color space, but these were not really unified. That is, the gauge bosons, (photon, W, and Z, and gluons) were not manifestations of the same force field. If one were to “unify” these fields, how might it occur? The attempts to do so are called Grand Unified Theories. Grand Unified Theory (GUT) GUT includes invariance under U(1) X SU(2)flavor space and SU(3)color and invariance under the following transformations: quarks  leptons leptons  quarks

e-i(x,y,t) ; Grand Unified Theory - SU(5) d red dgreen d blue e- -
Georgii & Glashow, Phys. Rev Lett. 32, 438 (1974). mx  1015GeV Quarks & leptons in same multiplet 8 gluons d red rgb dgreen 24 Gauge bosons ; L d blue (W 0+B) W+ e- - W- (W 0 +B) Left handed L SU(5) is invariant under SU(5) gau Gauge invariance e-i(x,y,t) For symmetry under SU(5), the x and y particles must be massless!

i D = - i g5/2j=1,24jXi where Xi = the 24 gauge bosons
SU(5) generators and covariant derivative i The = generators of SU(5) are the 5x5 matrices which i(x,y,t) has do not commute. SU(5) is a non-abelian local gauge theory. 24 components: i(x,y,t) = all real, continuous functions D = - i g5/2j=1,24jXi where Xi = the 24 gauge bosons This includes the Standard Model covariant derivative (couplings are different). Predictions: a) qup = 2/3 ; qd = -1/3 b) sin2W  -.23 c) the proton decays! > years d) baryon number not conserved e) only one coupling constant, g5 (g1, g2, and g3, are related) So far, there is no evidence that the proton decays. But note that the lifetime of the universe is 14 billion years. The probability of detecting a decaying proton depends a large sample of protons!

- - X +  = 1,2,3 Q = - 4/3 Y  Q= - 1/3  = 1,2,3
3-color vertex quark to lepton, no color change  = 1,2,3 Q = - 4/3 - Y  3-color vertex quark to lepton, no color change Q= - 1/3  = 1,2,3 + Hermitian Conjugate (contains X+ and Y+ terms) Note: one coupling constant, g5 T  transpose Charge conjugation operator

charge dred X-4/3red e+ e+

- e+ Decay of proton in SU(5) - Xred proton X +red d red d red u green
anti-up - u green d red - green Xred u blue X+ red blue e+ 3-color vertex proton X +red green blue

Supergravity SUPER SYMMETRIC (SUSY) THEORIES:
SUSYs contain invariance of the Lagrangian density under operations which change bosons (spin = 01,2,..) fermions (spin = ½, 3/2 …). SUSY  unifies E&M, weak, strong (SU(3) and gravity fields. usually includes invariance under local transformations Supergravity

Supersymmetric String Theories
Elementary particles are one-dimensional strings: open strings closed strings .no free parameters or L = 2r L = cm. = Planck Length Mplanck  GeV/c2 See Schwarz, Physics Today, November 1987, p. 33 “Superstrings” The Planck Mass is approximately that mass whose gravitational potential is the same strength as the strong QCD force at r  cm. An alternate definition is the mass of the Planck Particle, a hypothetical miniscule black hole whose Schwarzchild radius is equal to the Planck Length.

Up to 10 – 43 seconds after the Big Bang
Particle Physics and the Development of the Universe Very early universe All ideas concerning the very early universe are speculative. As of early today, no accelerator experiments probe energies of sufficient magnitude to provide any experimental insight into the behavior of matter at the energy levels that prevailed during this period. Planck epoch Up to 10 – 43 seconds after the Big Bang At the energy levels that prevailed during the Planck epoch the four fundamental forces— electromagnetism U(1) , gravitation, weak SU(2), and the strong SU(3) color — are assumed to all have the same strength, and “unified” in one fundamental force. Little is known about this epoch. Theories of supergravity/ supersymmetry, such as string theory, are candidates for describing this era.

Grand unification epoch: GUT
Between 10–43 seconds and 10–36 seconds after the Big Bang The universe expands and cools from the Planck epoch. After about 10–43 seconds the gravitational interactions are no longer unified with the electromagnetic U(1) , weak SU(2), and the strong SU(3) color interactions. Supersymmetry/Supergravity symmetires are roken. After 10–43 seconds the universe enters the Grand Unified Theory (GUT) epoch. A candidate for GUT is SU(5) symmetry. In this realm the proton can decay, quarks are changed into leptons and all the gauge particles (X,Y, W, Z, gluons and photons), quarks and leptons are massless. The strong, weak and electromagnetic fields are unified.

Running Coupling Constants
Electro weak unification Planck region Electro- Weak Symmetry breaking Super- symmetry SU(3) GUT electroweak GeV

Inflation and Spontaneous Symmetry Breaking.
At about 10–36 seconds and an average thermal energy kT  1015 GeV, a phase transition is believed to have taken place. In this phase transition, the vacuum state undergoes spontaneous symmetry breaking. Spontaneous symmetry breaking: Consider a system in which all the spins can be up, or all can be down – with each configuration having the same energy. There is perfect symmetry between the two states and one could, in theory, transform the system from one state to the other without altering the energy. But, when the system actually selects a configuration where all the spins are up, the symmetry is “spontaneously” broken.

Higgs Mechanism When the phase transition takes place the vacuum state transforms into a Higgs particle (with mass) and so-called Goldstone bosons with no mass. The Goldstone bosons “give up” their mass to the gauge particles (X and Y gain masses 1015 GeV). The Higgs keeps its mass ( the thermal energy of the universe, kT 1015 GeV). This Higgs particle has too large a mass to be seen in accelerators. What causes the inflation? The universe “falls into” a low energy state, oscillates about the minimum (giving rise to the masses) and then expands rapidly. When the phase transition takes place, latent heat (energy) is released. The X and Y decay into ordinary particles, giving off energy. It is this rapid expansion that results in the inflation and gives rise to the “flat” and homogeneous universe we observe today. The expansion is exponential in time.

R eHt Schematic of Inflation R(t) m T (GeV/k) Rt2/3 1019 Rt1/2
T t-1/2 R eHt 1014 T t-1/2 Rt1/2 Tt-2/3 T=2.7K 10-13 10-43 10-34 10-31 time (sec) 10

Electroweak epoch Between 10–36 seconds and 10–12 seconds after the Big Bang The SU(3) color force is no longer unified with the U(1)x SU(2) weak force. The only surviving symmetries are: SU(3) separately, and U(1)X SU(2). The W and Z are massless. A second phase transition takes place at about 10–12 seconds at kT = 100 GeV. In this phase transition, a second Higgs particle is generated with mass close to 100 GeV; the Goldstone bosons give up their mass to the W, Z and the particles (quarks and leptons). It is the search for this second Higgs particle that is taking place in the particle accelerators at the present time.

After the Big Bang: the first 10-6 Seconds
W , Z0 take on mass inflation X,Y take on mass gravity decouples Planck Era . SUSY Supergravity GUT SU(2) x U(1) symmetry . all forces unified bosons  fermions . quarks  leptons all particles massless

W , Z0 take on mass 2.7K Standard Model . . atoms formed . .
COBE data . 2.7K Standard Model 100Gev . . . only gluons and photons are massless . atoms formed n, p formed nuclei formed .

Dark Matter cannot be seen directly with telescopes; it neither emits nor absorbs light; estimated to constitute 84.5% of the total matter in the universe – and 26.8% of the total mass/energy of the universe; its existence is inferred from gravitational effects on visible matter and gravitational lensing of background radiation;

Rotational curves for a typical galaxy indicate that the
mass of the galaxy is not concentrated in its center. Our own galaxy is predicted to have a spherical halo of dark matter.

Visualization of dark matter halo for spiral galaxy

Candidates for nonbaryonic dark matter
Axions (0 spin, 0 charge, small mass, Goldstone bosons) Supersymmetric particles (partners in SUZY) – not been seen yet Neutrinos (small fraction ) Weakly interacting massive particles .. so far none have been detected.

Dark Energy The size and the smoothness of the Universe can be explained by very rapid expansion—inflation. However, there is not enough observable matter to generate stars or galaxies. The force of gravity from observable matter is too weak. This is one of a number of reasons we need dark matter. Finally, to explain the acceleration of the expansion of the Universe, we need dark energy; ideally, that would explain both early inflation and today's inflation.

ds is measure of distance between
Begin with the metric tensor for the 4-dimensional space: General Relativity. scale factor ds is measure of distance between two points

Rather than the relativistic red shift, the Cosmological
red shift is now used in interpreting the Hubble constant: 1 + z = R(tnow)/R(tthen) 1 + z = observed/ emitted z = (observed - emitted)/ emitted Hubble’s Law: v = H d v = recessional speed H = Hubble’s constant d = distance

Acceleration of the expansion of the observable
universe is at this point too small to affect the “measured” value of the Hubble constant. But one can see from the following expression that an increase in H must follow from a term not yet included in the equation of state. missing terms – due to dark energy?

Einstein’s Equations and Hubble Law Derivation
… use Noether’s theorem.  S = 0  Einstein ‘s equations

The  =  = 0 component of Einstein’s equations gives Hubble’s Law:

Some comments on Inflation: potential form.
Energetic coherent oscillations about minimum long slow “roll” into minimum possible tunneling steep asymmetric rise absolute minimum

Difference between polarization characteristic
of density fluctuations and gravitational waves:

Difference between polarization characteristic
of density fluctuations and gravitational waves:

On March 17, 2014 scientists announced the first direct detection of the cosmic inflation behind the rapid expansion of the universe just a tiny fraction of a second after the Big Bang 13.8 billion years ago. A key piece of the discovery is the evidence of gravitational waves, a long-sought cosmic phenomenon that has eluded astronomers until now. https://www.youtube.com/watch?v=PCxOEyyzmvQ

With classical Newtonian mechanics and electrodynamics
we probed large scale phenomena in our solar system. We looked outward to study the stars and galaxies, and the expanding space of our observable universe. Atomic scale theories of quantum mechanical phenomena and relativistic formalisms generated technology which made these studies possible. Probing deeper into nuclei, quarks, leptons, symmetry generated gauge bosons, and general relativity we are at the brink of understanding the story of our universe.