Presentation on theme: "Physics 357/457 Spring 2014 Summary The elementary particles Relativistic formulation Lagrangians Principle of least action QED and field operators."— Presentation transcript:
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 e ) all have spin = ½ they are fermions that’s it!
T he forces (each force is associated with an exchanged particle) electromagnetic (photon) weak ( W + W - Z 0 ) strong (8 gluons) gravitational ( graviton not yet observed) all have spin = 1 (or 2 for graviton) they are bosons
Review: Special Relativity Speed of light = C = |r 2 – r 1 | / (t 2 –t 1 ) = |r 2 ’ – r 1 ’ | / (t 2 ’ –t 1 ‘ ) = |dr/dt| = |dr’/dt’| Einstein’s assumption: the speed of light is independent of the (constant ) velocity, v, of the observer. It forms the basis for special relativity.
4-dimensional vector component notation x µ ( x 0, x 1, x 2, x 3 ) µ=0,1,2,3 = ( ct, x, y, z ) = (ct, r) x µ ( x 0, x 1, x 2, x 3 ) µ=0,1,2,3 = ( ct, -x, -y, -z ) = (ct, -r) contravariant components covariant components
Invariant dot products using 4-component notation x µ x µ = µ=0,1,2,3 x µ x µ (repeated index one up, one down) summation) x µ x µ = (ct) 2 -x 2 -y 2 -z 2 Einstein summation notation covariant contravariant
Any four vector dot product has the same value in all frames moving with constant velocity w.r.t. each other. Examples: x µ x µ p µ x µ p µ p µ µ µ p µ µ µ A µ
Suppose we consider the four-vector: ( E/c, p x, p y, p z ) (E/c) 2 – ( p x ) 2 – ( p y ) 2 – ( p z ) 2 is also invariant. In the center of mass of a particle this is equal to (mc 2 / c) 2 – (0) 2 – (0) 2 – (0) 2 = m 2 c 2 So, for a particle (in any frame) (E/c) 2 – ( p x ) 2 – ( p y ) 2 – ( p z ) 2 = m 2 c 2
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 / x 2 + 2 / y 2 + 2 / z 2 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
1. We have seen how Maxwell’s equations can be cast into a single wave equation for the electromagnetic 4-vector, A µ. This A µ now represents the E and B of the EM field … and something else: the photon! Solutions to 2.If A µ 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:
Spin vector Spin vector annihilation operator creation operator 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 a k ± 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 field operator for a neutral, spin =0, particle is creates a single particle with momentum p= k and p 0 = k 0 at (r,t) Destroys a single particle with momentum p= k and p 0 = k 0 at (r,t)
In quantum field theory, the Euler-Lagrange equations give the particle wave function! Lagrangians and the Lagrangian Density Recall that, and the Euler-Lagrange equations give F = ma
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
and the Euler-Lagrange equation From the Lagrangian density we can derive the wave equation Particles with Charge: two fields , and *
creates positively charged particle with momentum p= k and p 0 = k 0 at (r,t) destroys negatively charged particle with momentum p= k and p 0 = k 0 at (r,t) creates negatively charged particle with momentum p= k and p 0 = k 0 at (r,t) destroys negatively charged particle with momentum p= k and p 0 = k 0 at (r,t)
Gauge Invariance and Conserved Quantities “Noether's theorem” was proven by German mathematician, Emmy Noether in 1915 and published in 1918. 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 1882-1935
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: p incoming outgoing incoming particle outgoing particle integrate over all space integrate over time S 0 (t) One obtains a number! the charge operator.
The time disappears! Q is time independent. The integration over k’ is done with the Dirac delta function from the d 3 x 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: 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. 4x4 matrix4x4 unit matrix 4-row column matrix
The Dirac equation in full matrix form 00 11 22 33 spin dependence space-time dependence
p along z and spin = +1/2 p along z and spin = -1/2 Spinors for the particle with p along z direction
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: p µ p µ = m 2 c 2
Lagrangian Density for Spin 1/2 Fermions 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! Comments: 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 1.Gauge transformations are like “rotations” 2.How do functions transform under “rotations”? 3.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!
Gauge transformations are like the “rotations” we have just been considering Real function of space and time can be an operator -- as we have just seen. one has to find a Lagrangian which is invariant under this transformation.
How are Lagrangians made invariant under these “rotations”? It won’t work!
Constructing a gauge invariant Lagrangian: 1. Begin with the “old Lagrangian”: 2. Replace 3. “old” Lagrangian the interaction term. called the “covariant derivative” A µ is the gauge boson (exchange particle) field!
Showing L is invariant transformed L A µ = A µ - (1/e) transformed 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 Lagrangianinteraction Lagrangian This use of the covariant derivative will be applied to all the interaction terms of the Standard Model. If the coupling, e, is turned off, L reverts to the free electron L.
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. Comments: 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 1 + 3 + 8 spin = 1 force carrying gauge particles.
n = 2 3 components 3 gauge particles SU(2) and SU(n) dot product The are called the generators of the group. Pauli spin matrix functions of x,y,z,t
“rotated” flavor stateoriginal flavor state SU(2): rotations in Flavor Space These are the Pauli spin matrices, 1 2 3 local depends on x, y, z, and t.
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. 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.
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 The final invariant L is given by: generators of SU(2) interaction term coupling constant generator of SU(2) covariant derivative rotations in flavor space! 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”: red green blue = d red green 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. quark field operators
A general “rotation” in color space can be written as a local, (non-abelian) SU(3) gauge transformation local generators of SU(3 ) Since the a don’t commute, the SU(3) gauge transformations are non-abelian. a = 1,2,3,…8 red green blue
The generators of SU(3): eight 3x3 a matrices (a = 1,2,3…8) All 3x3 matrix elements of SU(3) can be written as a linear combination of these 8 a plus the identity matrix. (n 2 – 1) = 3 2 - 1 = 8 generators 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 =
1 f 123 = 1 = - f 213 = f 231 12 3 = 2 i f 123 3 [ 1, 2 ] the a don’t commute Likewise one can show: (for the graduate students) f abc = -f bac = f bca f 458 = f 678 = 3 /2, f 147 = f 516 = f 246 = f 257 = f 345 = f 637 = ½ … all the rest = 0.
SU(3) gauge invariance in the Standard Model The invariant Lagrangian density is given by: interaction term generators of SU(3)
The Lagrangian density with the U(1), SU(2) and SU(3) gauge particle interactions neutral vector boson heavy vectors bosons (W , W 3 ) 8 gluons Y
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.
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 = d R
The following is the interaction Lagrangian density for the first generation of particles with the left and right handed parts shown explicitly. BB BB BB BB BB BB W1W1 W1W1 W2+W2+ W2+W2+ W3W3 W3W3 a G a sum over a = 1,2,…8 a G a
Weinberg’s decomposition of the B and W : sin 2 W 0.23 -- to be determined experimentally! W = Weinberg angle
Next steps: rewriting interaction Lagrangian density so that interactions with the photon are identified. The neutrino has zero charge and can’t interact with the photon. 1. 2.2. 3.3.
After substituting the expressions for B and W 0 (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, g 2 and g 1. One finds that: g 2 = e / sin W g 1 = e / cos W Y L = -1 Y R = 2 Y L Also one defines: T 3 f = + 1/2 for the u L = - 1/2 for the d L = 0 for u R = 0 for d R
(E & M) QED interactions weak neutral current interactions weak flavor changing interactions QCD color interactions + + The Standard Model Interaction Lagrangian for the 1 st generation
The U(1) and SU(2) interaction terms AA Z+Z+ZZ W+W+ W-W- weak neutral current interactions (E & M) QED interactions weak flavor changing interactions g2g2 g2g2 e g 2 = e / sin W
The following values for the constants gives the correct charge for all the particles.
Weak neutral current interactions Z0Z0 Z0Z0 Z0Z0 Z0Z0
Feynman rules applied to a 2-vertex electron positron scattering diagram left vertex functionright vertex function M fi = spin time propagator metric tensor 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. coupling constant – one for each vertex Note that each vertex is generated by the interaction Lagrangian density.
Confinement of quarks free quark termsfree gluon termsquark- 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 quark loop gluon loop NfNf NcNc N f = # flavorsN c = # colors normal free gluon term 3-gluon vertex Note sign
momentum squared of exchanged gluon NfNf NcNc NcNc NfNf In QED one has no terms corresponding to the number of colors (the 3-gluon) vertex. This term aslo has a negative sign. -7 M 2 quark
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
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
Grand Unified Theory - SU(5) 8 gluons (W 0 +B) W+W+ W-W- 24 Gauge bosons SU(5) m x 10 15 GeV SU(5) gau For symmetry under SU(5), the x and y particles must be massless! Quarks & leptons in same multiplet Left handed d red d green d blue L Georgii & Glashow, Phys. Rev Lett. 32, 438 (1974). ; Gauge invariance e -i (x,y,t) L SU(5) is invariant under e - - rgbrgb
> 10 34 years 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! D = - i g 5 /2 j=1,24 j X i where X i = the 24 gauge bosons SU(5) generators and covariant derivative a) q up = 2/3 ; q d = -1/3 b) sin 2 W -.23 c) the proton decays! d) baryon number not conserved e) only one coupling constant, g 5 (g 1, g 2, and g 3, are related) Predictions: The 5 2 -1 = 24 generators of SU(5) are the i 24 components : i (x,y,t) = i (x,y,t) has all real, continuous functions 5x5 matrices which do not commute. SU(5) is a non-abelian local gauge theory. This includes the Standard Model covariant derivative ( couplings are different).
+ Note: one coupling constant, g 5 quark to lepton, no color change 3-color vertex 3-color vertex = 1,2,3 Q = - 4/3 Q= - 1/3 + Hermitian Conjugate (contains X + and Y + terms) XX Y Y Charge conjugation operator T transpose - -
d red u green u blue X + red e+e+ d red - X red - anti-up blue green blue green X +red proton 0 Decay of proton in SU(5) 3-color vertex
SUPER SYMMETRIC (SUSY) THEORIES: http://www.pha.jhu.edu/~gbruhn/IntroSUSY.html Supergravity 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
Supersymmetric String Theories Elementary particles are one-dimensional strings: closed strings open strings L = 2 r L = 10 -33 cm. = Planck Length M planck 10 19 GeV/c 2 or See Schwarz, Physics Today, November 1987, p. 33 “Superstrings”.no free parameters The Planck Mass is approximately that mass whose gravitational potential is the same strength as the strong QCD force at r 10 -15 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.
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 T he 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 GUT electroweak Electro- Weak Symmetry breaking Planck region SU(3) Electro weak unification GeV Super- symmetry
Inflation and Spontaneous Symmetry Breaking. At about 10 –36 seconds and an average thermal energy kT 10 15 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.
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 10 15 GeV). The Higgs keeps its mass ( the thermal energy of the universe, kT 10 15 GeV). This Higgs particle has too large a mass to be seen in accelerators. Higgs Mechanism 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.
Schematic of Inflation T (GeV/k) R(t) m T=2.7K 10 -43 10 -34 10 -31 10 19 10 10 14 R t 1/2 T t -1/2 R e Ht R t 2/3 T t -2/3 10 -13 time (sec)
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. GUTSU(2) x U(1) symmetry Planck Era gravity decouples SUSY Supergravity inflation X,Y take on mass W , Z 0 take on mass.. all forces unified bosons fermions quarks leptons all particles massless
..... COBE data 2.7K Standard Model W , Z 0 take on mass.. n, p formed nuclei formed atoms formed 100Gev only gluons and photons are massless
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 two points Begin with the metric tensor for the 4- dimensional space: General Relativity. scale factor
Rather than the relativistic red shift, the Cosmological red shift is now used in interpreting the Hubble constant: 1 + z = R(t now )/R(t then ) 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 S = 0 Einstein ‘s equations … use Noether’s theorem.
The = = 0 component of Einstein’s equations gives Hubble’s Law: https://www.youtube.com/watch?v=EIpEzZqkd9c dE = -pdV
Some comments on Inflation: potential form. possible tunneling long slow “roll” into minimum absolute minimum steep asymmetric rise Energetic coherent oscillations about 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.