Transcript Lecture.16.Summary.Reviewx

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
cs
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
contravariant
components
•
xµ
covariant
components
 ( x0, x1 , x2, x3 )
= ( ct, x, y, z )
= (ct, r)
µ=0,1,2,3
 ( x0 , x1 , x2 , x3 )
= ( ct, -x, -y, -z )
= (ct, -r)
µ=0,1,2,3
Invariant dot products using
4-component notation
contravariant
covariant
xµ xµ = µ=0,1,2,3 xµ xµ
Einstein summation notation
(repeated index one up, one down)  summation)
xµ xµ
= (ct)2 -x2 -y2 -z2
Any four vector dot product has the same value
in all frames moving with constant
velocity w.r.t. each other.
Examples:
xµxµ
pµpµ
pµxµ
pµµ
 µAµ
 µ µ
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
Einstein summation notation
(repeated index summation )
= 2/(ct)2 - 2
2
= 2/x2
+ 2/y2 + 2/z2
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, Am = ((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!
2. 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:
creation
operator
Spin
vector
annihilation
operator
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)
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 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:
the charge operator.
integrate
over time
S0(t)
outgoing
particle
integrate over all space
One obtains a number!
p
incoming
outgoing
incoming
particle
Calculation of Charge
Note Dirac delta function in k and p
‘
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
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!
momentum operator
x component
momentum operator
angular 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
transformed L
transformed 
A µ = Aµ - (1/e) m
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:
U(1) “QED-like”
SU(2) weak
SU(3) color
gauge boson
 neutral gauge boson
 3 heavy vector bosons
 8 gluons
This gives rise to 1 + 3 + 8 spin = 1 force carrying gauge particles.
SU(2) and SU(n)
dot product
Pauli spin matrix
functions of x,y,z,t
The t 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
The final invariant L is given by:
coupling constant
generators of SU(2)
interaction
interaction term
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
red
= d red
green
= d green
blue
= d 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
local
generators of SU(3)
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) = 32 - 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.
[ 11, 22]
the a don’t commute
= 2i f1123 3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
The following is the interaction Lagrangian density for the first generation
of particles with the left and right handed parts shown explicitly.
Bm
Bm
Bm
Bm
Bm
W1m
W1m
W2m+
W2m+
Bm
W3m
W3m
sum over a = 1,2,…8
 m aGam
 m aGam
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.
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
Also one defines:
T 3f
YL = -1
YR = 2 YL
= + 1/2 for the uL
= - 1/2 for the dL
= 0 for uR
= 0 for dR
The Standard Model Interaction Lagrangian for the 1st generation
(E & M) QED interactions
weak neutral current interactions
+
weak flavor changing interactions
+
QCD color interactions
The U(1) and SU(2) interaction terms
g2
= e / sinW
e
Am
(E & M) QED interactions
g2
Zm+
Zm
weak neutral current interactions
g2
W+m
weak flavor changing interactions
W-m
The following values for the constants
gives the correct charge for all the particles.
Weak neutral current interactions
Z0m
Z0
m
Z0
Z0
Weak charged flavor changing interactions
quarks
leptons
g2
g2
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
Note that there are only 8 possibilities:
grrg-g
ggb-
The red, anti-green gluon
The green, anti-blue gluon
The gluon forces hold the
proton together
proton
At any time the proton
is color neutral. That is,
it contains one red, one
blue and one green
quark.
beta decay
u
d
u
d
d
u
W-
neutron
W doesn’t see color
proton
W production from
d
p
u
u
p-
-d
-u
-u
W+
ppp--
The nuclear force
u
n
u
d
d
d
u
u
p
W-
p
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.
n
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
coupling constant –
one for each vertex
right vertex function
propagator
The next steps are to do the sum over m and  and carry out the matrix multiplications.
Note that m 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.
Gm
normal free gluon term
Nf= # flavors
Gm
Note sign
3-gluon vertex
Nc= # colors
Nf
quark
loop
Nc
gluon
loop
momentum squared of exchanged gluon
Nf
Nc
 M2quark
Nf
Nc
-7
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
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)
SU(5)
Georgii & Glashow, Phys. Rev Lett. 32, 438 (1974).
d red
dgreen
d blue
e-
rgb
L
-
SU(5)
gau
Quarks
& leptons
in same
multiplet
;
mx  1015GeV
8
gluons
24
Gauge
bosons
(W 0+B)
W-
Left handed
Gauge invariance
For symmetry under SU(5), the
L SU(5)
W+
(W 0 +B)
is invariant under
e-i(x,y,t)
x and y particles must be massless!
SU(5) generators and covariant derivative
The
52 -1 = 24 generators of SU(5) are the
i
do not commute. SU(5) is a non-abelian local gauge theory.
24 components: i(x,y,t) =
5x5 matrices which
i(x,y,t) has
all real, continuous functions
Dm = m- i g5/2 jXi
where
j=1,24
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!
> 1034 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!
-
Xm
+
 = 1,2,3
quark to lepton, no color change
3-color
vertex
Q = - 4/3
Ym-
 = 1,2,3
quark to lepton, no color change
3-color
vertex
Q= - 1/3
+ Hermitian Conjugate (contains Xm+ and Ym+ terms)
Charge conjugation
operator
T  transpose
Note: one coupling constant,
g5
charge
X-4/3red
e+
dred
e+
Decay of proton in SU(5)
d red
d red
-
u green
d red
u blue
proton
-
green
Xred
X+ red
3-color
vertex
anti-up
0
blue
e+
X +red
green
blue
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
http://www.pha.jhu.edu/~gbruhn/IntroSUSY.html
Supergravity
Supersymmetric String Theories
Elementary particles are one-dimensional strings:
open strings
closed strings
or
.no free
parameters
L = 2r
L = 10-33 cm. = Planck Length
Mplanck  1019 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  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
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
ElectroWeak
Symmetry
breaking
Planck
region
Supersymmetry
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.
Schematic of Inflation
1019
R(t) m
Rt2/3
T (GeV/k)
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
Planck Era
SUSY
Supergravity
inflation
gravity
.X,Y take
decouples on mass
GUT
W , Z0
take on
mass
SU(2) x U(1) symmetry
.
all forces unified
bosons  fermions
quarks  leptons
all particles massless
.
.
W , Z0
take on
mass
COBE data
.
2.7K
Standard Model
100Gev
.
.
.
.
.
only gluons and photons are massless
n, p formed nuclei formed
atoms 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.
Begin with the metric tensor for the 4dimensional space: General Relativity.
ds is measure
of distance
between
two points
scale factor
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=Hd
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 m = n = 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
Energetic coherent
oscillations
about minimum
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.