3 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 fermionsthat’s it!
4 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
6 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’|
8 Invariant dot products using 4-component notation contravariantxµ xµ = µ=0,1,2,3 xµ xµ(repeated index one up, one down) summation)xµ xµ = (ct)2 -x2 -y2 -z2covariantEinstein summation notation
9 Any four vector dot product has the same value in all frames moving with constant velocity w.r.t. each other.Examples:xµxµ pµxµpµpµ µµpµµ µAµ
10 (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 c2So, for a particle (in any frame)(E/c)2 – (px)2– (py)2 – (pz)2 = m2 c2
12 Lorentz Invariance Lorentz invariance of the laws of physics is satisfied if the laws are cast interms of four-vector dot products!Four vector dot products are said to be“Lorentz scalars”.In the relativistic field theories, we mustuse “Lorentz scalars” to express theinteractions.
13 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!
14 Motivation for Lagrangians and the Law of Least Action
15 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.
16 The wave equations for A and can be put into 4-vector form:Lorentz gauge!A+ (1/c) /(ct) = 0
17 Solutions to1. We have seen how Maxwell’s equations can be cast into a single waveequation for the electromagnetic 4-vector, Aµ . This Aµ now representsthe 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 representany photon. That is, we want the most general solution to the equation:
18 This “Fourier expansion” of the photon operator is called creationoperatorannihilationoperatorSpinvectorSpinvectorThis “Fourier expansion” of the photon operator is called“second quantization”. Note that the solution to thewave equation consists of a sum over an infinite numberof “photon” creation and annihilation terms. Once the ak±are interpreted as operators, the A becomes an operator.
19 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
20 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.
21 Note that *(r,t) (r,t) does not represent the probability per unit volume density of the particle being at (r,t).
23 The field operator for a neutral, spin =0, particle is creates a singleparticle withmomentump= k andp0 = k0at (r,t)Destroys a singleparticle withmomentump= k andp0 = k0at (r,t)
24 and the Euler-Lagrange equations give F = ma Lagrangians and the Lagrangian DensityRecall that,and the Euler-Lagrange equations give F = maIn quantum field theory, the Euler-Lagrangeequations give the particle wave function!
25 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
26 Summary for neutral (Q=0) scalar (spin = 0) particle, , with mass, m. Lagrangian densitywave equationfield operator
27 Particles with Charge: two fields , and * From the Lagrangian densityand the Euler-Lagrange equationwe can derive the wave equation
28 creates positively charged particle with momentum p= k andp0 = k0 at (r,t)destroys negatively charged particle with momentump= k andp0 = k0 at (r,t)destroys negatively charged particle with momentump= k andp0 = k0 at (r,t)creates negatively charged particle with momentump= k andp0 = k0 at (r,t)
29 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
30 ’ An astounding result: we can vary the (complex) phase of the field operator, , everywhere in space by any continuous amountand 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!
31 With the help of Emmy Noether, we can prove that charge is conserved!
32 Deriving the conserved current and the conserved charge: Euler-Lagrange equationconserved currentBut our Lagrangian density also contains a *, so we obtain additional termslike the above, with replaced by *. In each case the Euler –Lagrange equationsare satisfied. So, the remaining term is as follows:
33 Now we evaluate .The great advantage of being a continuous constant is that thereare an infinite number of very small which carry with them all thephysics of the “continuity”. That is, with no loss of rigor we canassume is small!
34 The value of the charge is calculated from: the charge operator.integrateover timeS0(t)incomingparticleoutgoingparticleintegrate over all spaceOne obtains a number!pincomingoutgoing
37 Note: + and/or – must be together. The time disappears! Q is time independent.The integration over k’ is donewith the Dirac delta functionfrom the d3x integration.The remaining integration over k will be done with the Dirac delta functionsfrom the commutation relations.Note: + and/or – must be together.
38 Message: calculating charge is a lot of work -- but can be done!
39 Fermions and the Dirac Equation In 1928 Dirac proposed the following form for the electron wave equation:4-row column matrix4x4 matrix4x4 unit matrixThe four µ matrices form a Lorentz 4-vector, withcomponents, µ. That is, they transform like a 4-vectorunder Lorentz transformations between movingframes. Each µ is a 4x4 matrix.
40 The Dirac equation in full matrix form 0123spin dependencespace-timedependence
41 Spinors for the particle with p along z direction p along z and spin = +1/2p along z and spin = -1/2
42 Field operator for the spin ½ fermion Spinor for antiparticlewith momentum p and spin sCreates antiparticle withmomentum p and spin sNote: pµ pµ = m2 c2
43 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 Lagrangiandensity, give the Dirac Equation!3. Note that L is a Lorentz scalar.
44 Lagrangian Density for Spin 1/2 Quarks and Leptons Now we are ready to talk about the gauge invariance that leads tothe Standard Model and all its interactions. Remember a “gaugeinvariance” is the invariance of the above Lagrangian undertransformations like e i . The physics is in the -- which can be a matrix operator and depend on x,y, z and t.
45 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!
50 Gauge transformations are like the “rotations” we have just been considering Real function of space and timeone has to find a Lagrangian which is invariant under this transformation. can be an operator -- as we have just seen.
51 How are Lagrangians made invariant under these “rotations”? It won’t work!
52 Constructing a gauge invariant Lagrangian: 1. Begin with the “old Lagrangian”:called the “covariant derivative”2. ReplaceAµ is the gauge boson(exchange particle) field!3.“old” Lagrangianthe interaction term.
53 Showing L is invariant A µ = Aµ - (1/e) transformed L transformed AMaxwell’s equationsare invariant under this!
54 Summary of Local gauge symmetry Real function of space and timecovariant derivativeThe final invariant L is given by:
55 The correct, invariant Lagrangian density, includes the interaction between the electron (fermion) and the photon (the gauge particle).free electron Lagrangianinteraction LagrangianIf the coupling, e, is turned off, L reverts to the free electron L.This use of the covariant derivative will be applied toall the interaction terms of the Standard Model.
56 Comments:1. There is no difference between changing the phaseof the field operator of the fermion (by (r,t) atevery point in space) and the effects of a gaugetransformation [ -(1/e)µ (r,t) ] on the photon field!2. Maxwell’s equations are invariant underA µ A µ - (1/e)µ (r,t) -- and, in particular, the gaugetransformation has no effect on the free photon.3. It is only because (r,t) depends on r and t thatthe above is possible. This is called a local gaugetransformation.4. Note that a global gauge transformation would require that is a constant!
57 Lecture 10: Standard Model Lagrangian The Standard Model Lagrangian is obtained by imposingthree local gauge invariances on the quark andlepton field operators:symmetry: gauge bosonU(1) “QED-like” neutral gauge bosonSU(2) weak 3 heavy vector bosonsSU(3) color 8 gluonsThis gives rise to spin = 1 force carrying gauge particles.
58 SU(2) and SU(n) n = 2 3 components 3 gauge particles dot product Pauli spin matrixfunctions of x,y,z,tThe are called thegenerators of the group.n = 2 3 components 3 gauge particles
59 SU(2): rotations in Flavor Space “rotated” flavor stateoriginal flavor stateThese are the Pauli spin matrices, 1 2 3local depends on x, y, z, and t.
60 Flavor SpaceFlavor 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.
61 Summary: QED local gauge symmetry Real function of space and timecovariant derivativeThe final invariant L is given by:
62 SU(2) local gauge symmetry generator of SU(2)rotations inflavor space!covariant derivativecoupling constantgenerators of SU(2)The final invariant L is given by:interaction terminteraction term
63 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= dredredgreengreenblueblueThere 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.
64 A general “rotation” in color space can be written as a local, (non-abelian) SU(3) gauge transformationgenerators of SU(3)localredgreenbluea = 1,2,3,…8Since the a don’t commute, the SU(3) gauge transformationsare non-abelian.
65 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 generatorsAll 3x3 matrix elements of SU(3) can be written as a linear combinationof these 8 a plus the identity matrix.
67 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
68 The Lagrangian density with the U(1), SU(2) and SU(3) gauge particle interactionsYneutral vector bosonheavy vectors bosons (W, W3)8 gluons
69 Standard Model Lagrangian with Electro-Weak Unification The Standard Model assumes that the mass of the neutrino is zero and thatit is “left handed” -- travelling with its spin pointing opposite to its directionof 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 changesthe structure of the Standard Model Lagrangian – which is assumed to treatonly left handed flavor doublets.
70 These are the only spinors allowed for a zero mass neutrino! positive helicitynegative helicityThe neutrino, if it has a zero mass can only have its spinpointing along (or opposite to) it’s momentum.
72 Each term in the SM Lagrangian density containing quarks and the leptons can be rewritten using the following expression. Forthe neutrinos, however, only the left handed term exists.In the following slide we use the notation d R = dR
73 of particles with the left and right handed parts shown explicitly. The following is the interaction Lagrangian density for the first generationof particles with the left and right handed parts shown explicitly.BBBBBBW3W1W2+W1W2+W3sum over a = 1,2,…8 aGa aGa
74 sin2W 0.23 W = Weinberg angle -- to be determined experimentally! Weinberg’s decomposition of the B and W:W = Weinberg angle-- to be determined experimentally!sin2W 0.23
75 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.
76 g 2 = e / sinW 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 sinW , cos W , e, g2 and g1.One finds that:g 2 = e / sinWg 1 = e / cosWYL = -1YR = 2 YLT 3f = + 1/2 for the uL= - 1/2 for the dL= 0 for uR= 0 for dRAlso one defines:
77 The Standard Model Interaction Lagrangian for the 1st generation (E & M) QED interactionsweak neutral current interactions+weak flavor changing interactions+QCD color interactions
78 e g 2 = e / sinW A g2 g2 The U(1) and SU(2) interaction terms W+ (E & M) QED interactionsg2Z+Zweak neutral current interactionsg2W+W-weak flavor changing interactions
79 The following values for the constants gives the correct charge for all the particles.
92 Feynman rules applied to a 2-vertex electron positron scattering diagram Note that each vertex isgenerated by the interactionLagrangian density.timespinspinmetric tensorMfi =left vertex functionright vertex functioncoupling constant –one for each vertexpropagatorThe 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 aa function of the momenta only, and the four spin (helicity) states.
93 Confinement of quarks free quark terms free gluon terms quark- gluon interactionsThe free gluon terms have products of 2, 3 and 4 gluon field operators. Theseterms lead to the interaction of gluons with other gluons.
95 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.
96 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 generatesterms in the Lagrangian density which produce 3-gluon vertices – and gluonloops in the exchanged gluon “propagator”.
97 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 aboutthem, keeping in mind that they are at the ‘”edge” of what is understoodtoday. 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; decayof the proton.Running coupling constants: it’s possible that at one time in the developmentof the universe all the forces had the same strengthThe Early Universe: a big bang, cooling and expanding, phase transitionsand broken symmetries
98 Grand Unified Theory (GUT) We have incorporated into the Lagrangian density invariance underrotations in U(1)XSU(2)flavor space and SU(3)color space, but these werenot really unified. That is, the gauge bosons, (photon, W, and Z, andgluons) were not manifestations of the same force field. If one wereto “unify” these fields, how might it occur? The attempts to do so arecalled Grand Unified Theories.Grand Unified Theory (GUT)GUT includes invariance under U(1) X SU(2)flavor space and SU(3)colorand invariance under the following transformations:quarks leptonsleptons quarks
99 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 1015GeVQuarks& leptonsin samemultiplet8gluonsd redrgbdgreen24Gaugebosons;Ld blue(W 0+B)W+e--W-(W 0 +B)Left handedL SU(5) is invariant underSU(5) gauGauge invariancee-i(x,y,t)For symmetry under SU(5), the x and y particles must be massless!
100 i D = - i g5/2j=1,24jXi where Xi = the 24 gauge bosons SU(5) generators and covariant derivativeiThe = generators of SU(5) are the5x5 matrices whichi(x,y,t) hasdo not commute. SU(5) is a non-abelian local gauge theory.24 components: i(x,y,t) =all real, continuous functionsD = - i g5/2j=1,24jXi where Xi = the 24 gauge bosonsThis includes the Standard Model covariant derivative (couplings are different).Predictions:a) qup = 2/3 ; qd = -1/3b) sin2W -.23c) the proton decays!> yearsd) baryon number not conservede) only one coupling constant, g5 (g1, g2, and g3, are related)So far, there is no evidence that the proton decays. But note that thelifetime of the universe is 14 billion years. The probability of detectinga decaying proton depends a large sample of protons!
101 - - X + = 1,2,3 Q = - 4/3 Y Q= - 1/3 = 1,2,3 3-colorvertexquark to lepton, no color change = 1,2,3Q = - 4/3-Y 3-colorvertexquark to lepton, no color changeQ= - 1/3 = 1,2,3+ Hermitian Conjugate (contains X+ and Y+ terms)Note: one coupling constant, g5T transposeCharge conjugationoperator
103 - e+ Decay of proton in SU(5) - Xred proton X +red d red d red u green anti-up-u greend red-greenXredu blueX+ redbluee+3-colorvertexprotonX +redgreenblue
104 Supergravity SUPER SYMMETRIC (SUSY) THEORIES: SUSYs contain invariance of the Lagrangian density under operations which changebosons (spin = 01,2,..) fermions (spin = ½, 3/2 …).SUSY unifies E&M, weak, strong (SU(3) and gravity fields.usually includes invariance under local transformationsSupergravity
105 Supersymmetric String Theories Elementary particles are one-dimensional strings:open stringsclosed strings.no freeparametersorL = 2rL = cm. = Planck LengthMplanck GeV/c2See Schwarz, Physics Today, November 1987, p. 33“Superstrings”The Planck Mass is approximately that mass whose gravitational potential is thesame strength as the strong QCD force at r cm.An alternate definition is the mass of the Planck Particle, a hypothetical minisculeblack hole whose Schwarzchild radius is equal to the Planck Length.
106 Up to 10 – 43 seconds after the Big Bang Particle Physics and the Development of the UniverseVery early universeAll 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 epochUp to 10 – 43 seconds after the Big BangAt 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.
107 Grand unification epoch: GUT Between 10–43 seconds and 10–36 seconds after the Big BangThe 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.
109 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.
110 Higgs MechanismWhen the phase transition takes place the vacuum state transformsinto 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.
111 R eHt Schematic of Inflation R(t) m T (GeV/k) Rt2/3 1019 Rt1/2 T t-1/2R eHt1014T t-1/2Rt1/2Tt-2/3T=2.7K10-1310-4310-3410-31time (sec)10
112 Electroweak epochBetween 10–36 seconds and 10–12 seconds after the Big BangThe 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.
113 After the Big Bang: the first 10-6 Seconds W , Z0 take on massinflationX,Y take on massgravitydecouplesPlanck Era.SUSYSupergravityGUTSU(2) x U(1) symmetry.all forces unifiedbosons fermions.quarks leptonsall particles massless
114 W , Z0 take on mass 2.7K Standard Model . . atoms formed . . COBE data.2.7KStandard Model100Gev...only gluons and photons are massless.atoms formedn, p formed nuclei formed.
116 Dark Mattercannot 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 thetotal mass/energy of the universe;its existence is inferred from gravitational effects on visible matter and gravitational lensing of background radiation;
117 Rotational curves for a typical galaxy indicate that the mass of the galaxy is not concentrated in its center. Ourown galaxy is predicted to have a spherical halo ofdark matter.
118 Visualization of dark matter halo for spiral galaxy
119 Candidates for nonbaryonic dark matter Axions (0 spin, 0 charge, small mass, Goldstone bosons)Supersymmetric particles (partners in SUZY) – not been seen yetNeutrinos (small fraction )Weakly interacting massive particles.. so far none have been detected.
120 Dark EnergyThe 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.
121 ds is measure of distance between Begin with the metric tensor for the 4-dimensional space: General Relativity.scale factords is measure of distance betweentwo points
122 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/ emittedz = (observed - emitted)/ emittedHubble’s Law:v = H dv = recessional speedH = Hubble’s constantd = distance
124 Acceleration of the expansion of the observable universe is at this point too small to affect the“measured” value of the Hubble constant. Butone can see from the following expressionthat an increase in H must follow from a termnot yet included in the equation of state.missing terms – due to dark energy?
125 Einstein’s Equations and Hubble Law Derivation … use Noether’s theorem. S = 0 Einstein ‘s equations
126 The = = 0 component of Einstein’s equations gives Hubble’s Law: https://www.youtube.com/watch?v=EIpEzZqkd9cdE = -pdV
127 Some comments on Inflation: potential form. Energetic coherentoscillationsabout minimumlong slow“roll” intominimumpossibletunnelingsteepasymmetricriseabsoluteminimum
129 Difference between polarization characteristic of density fluctuations and gravitational waves:
130 Difference between polarization characteristic of density fluctuations and gravitational waves:
131 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
132 With classical Newtonian mechanics and electrodynamics we probed large scale phenomenain our solar system.We looked outward to study the starsand galaxies, and the expanding spaceof our observable universe.Atomic scale theories of quantum mechanicalphenomena and relativistic formalismsgenerated technology whichmade these studies possible.Probing deeper into nuclei, quarks, leptons,symmetry generated gauge bosons, andgeneral relativity we are at thebrink of understanding thestory of our universe.