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Monday, January 11:
Squaring the spacetime Dirac equation
Lecture: Squaring
the Dirac equation in curved spacetime
Scaling properties of the
Einstein-Hilbert action and the scalar field action. The gravity action is not
dimensionless.
Combine the scalar
curvature and a scalar field to make a gravity action which is scale invariant.
Dirac used this action in his Large Numbers papers, suggesting that the scalar
field be interpreted as a dynamical gravitational constant, G(t).
The resulting wave
equation contains the scalar curvature.
The Dirac action
generalizes to curved spacetime by changing the derivative to a covariant
derivative, and modifying the volume element. We must also generalize the Dirac
matrices. (See Brill & Wheeler on neutrinos)
Squaring the covariant Dirac
operator now leads to both symmetric and antisymmetric pieces. The symmetric
piece gives the covariant dÕAlembertian, while the antisymmetric piece give
Spin(3,1) Lorentz generators.
The antisymmetric generators
induce a commutator on the covariant derivatives, which we resolve using the
Ricci identity.
We actually need the Ricci
identity in the spinor representation of the Lorentz generators. This takes
some work:
Inserting the Ricci identity
for the commutator of covariant derivatives gives a pair of Spin(3,1)
generators contracted with the curvature tensor.
Since the symmetric part of
any product of two Dirac matrices may be replaced by a metric, the product of 4
Dirac matrices must be a linear combination of the identity, the generators,
and g5 . (That is, 0, 2 or 4 Dirac matrices,
antisymmetrized.) We can find the coefficients by taking traces, but we only
need one. The other terms, contracted with the curvature tensor, will vanish by
symmetry.
We are left with exactly the
scale-invariant field equation used by Dirac, acting on each component of the
spinor.
It is not unreasonable to
assume, since this is the wave operator that the Dirac equation squares to,
that this is the right way to think about scalar waves in spacetime.
A similar calculation using
the U(1) connection does not yield a scalar wave equation, since one remaining
generator still mixes components of the spinor:
Wednesday, January 13:
Transition amplitudes for scattering
Lecture: Transition
amplitudes for scattering in quantum field theory
Scattering experiments
have initial and final states that are essentially two different free field
solutions. We write the Hamiltonian as a free Hamiltonian plus an interaction
Hamiltonian. The two asymptotic, free states differ due to the effects of the
interaction part of the Hamiltonian, which typically has a perturbative effect.
The scattering matrix, S,
connects initial and final states. Unitarity of the S matrix preserves overall
probability.
We expand S = 1 + iT, and
find it convenient to compute T.
Recall that the space of
physical states is found by acting with creation operators on the vacuum. The
vacuum is the state annihilated by all destruction operators.
It is convenient to use
time-independent, Heisenberg states instead of time-dependent Schrdinger
states.
In the Heisenberg
representation, the operators become time dependent. This is convenient for
field theory, where the spacetime-dependent fields become the operators.
Recall our solution for
the creation and annihilation operators for scalar fields.
Define ŅinÓ and ŅoutÓ
fields as free fields, with an additional normalization constant. This will let
us take into account the scattered part of the state more easily.
Write the normalization
the other way.
Consider a transition
amplitude between an initial state with m particles and a final state with n
particles. We want to rewrite this as a product of field operators between
vacuum states.
WeÕll need a trick or two.
Write the spatial integral as a spacetime integral of a time derivative.
Now we can replace the
second derivatives with respect to time using the field equation. We can then
combine everything as a complete wave operator in a spacetime integral
Substitute into the
transition amplitude. We have eliminate one of the incoming momenta from the
state, replacing it by a spacetime integral and a wave operator
Friday, January 15:
Transition amplitudes in terms of vacuum expectation values
Lecture: Transition
amplitudes, continued. Amplitudes in terms of vacuum expectation values
(Answer to a question on
solving for the connection in GR: QFT II Jan
15b )
Last time, we described
transition amplitudes and the S and T matrices. We expressed the transition
amplitude in terms of m initial and n final momenta, and showed how to
eliminate one momentum from the initial state, replacing it with a combination
of creation and annihilation operators.
One of these operators
gives zero:
We will need the time
ordering operator, T. Acting on a product of fields, it re-orders them with the
latest times to the left. We can see the need for this if we consider a
perturbative expansion of the time dependence of a quantum state. At second
order, we rewrite the sequenced integrals, t1 t2 t3 as
half the time-ordered product of two integrals over the full time range, t1
t3 . Repeating the argument at all orders
leads to a time-ordered exponential.
The time-ordering
operator:
As shown last time, we can
replace one of the incoming momenta with raising and lowering operators;
Near duplicate: QFT II Jan
15h
Taking the adjoint gives
the same effect on an outgoing momentum. Iterating for all incoming and
outgoing momenta gives a vacuum expectation:
Note that, since we assume
no final momentum equals any incoming momentum, we have actually computed the T
matrix
The vacuum expectation
value is an n+m point Green function. We write its Fourier transform.
The final expression:
We have now written a
general transition amplitude in terms of a vacuum expectation value.
Wednesday, January 20:
Vacuum expectation values in terms of free fields
Lecture: Using
the interaction representation to express transition amplitudes in terms of
free fields
We can write the raising
and lowering operators in terms of field integrals. The form here is a little
more compact than what we wrote last semester, but equivalent. Check the
equivalence.
(Duplicate: QFT II Jan
20d )
The full time evolution of
the system is given by an exponential of the Hamiltonian, but for the
interaction picture, we define operators evolved by the free Hamiltonian only.
Inserting an identity (written as the free field evolution operator times its
inverse) into the full time evolution, we identify the operator U(t,tÕ) as the
thing to be computed.
U(t,tÕ) satisfies a simple
differential equation
The solution to the
equation is as expected, except that it requires time ordering
The full vacuum expectation
value, written with interaction picture fields and U(t,tÕ) operators:
We show that the product
U(t, tÕ) U(tÕ, tÕÕ) = U(t, tÕÕ), that is, these operators obey a simple product
rule
Using this product rule,
we eliminate almost all of the U(t, tÕ) operators, leaving a product of
interaction picture fields. Remember that these interaction picture fields are
computed from the free field, and the free field Hamiltonian. The only effect
of the interactions is in the U(t, tÕ) operators. These, in turn, are computed
from the interaction terms of the Hamiltonian.
Choosing the starting and finishing times judiciously gives
a simple expression.
Next time we will reduce this further.
Friday, January 22: WickÕs
theorem and Feynman propagators
Lecture: Wick's
theorem and Feynman propagators
Recall the definition of
the time ordering operator, T
Last time, we expressed
the transition amplitude in terms of vacuum expectation values of time
ordered products of fields, together
with time evolution operators, U(t,tÕ). By a convenient choice of the initial
and final times we eliminate all but one such factor, U(-°, +°). An additional
phase shows up as a normalization.
Now study the time ordered
product of two fields. We can rearrange the fields using the commutation
relations of the creation and annihilation operators, with the one
non-vanishing commutator, D, expressed as a Fourier integral
The simplest example of
WickÕs theorem: we rewrite the time ordered product in terms of a normal
ordered product and a commutator. The commutator is simply a function.
Substituting these expressions
into the vacuum expectation, the normal ordered terms vanish. The final result,
D(x-y), is the Green function for the original wave equation. It is called the
Feynman propagator; it acts like a heat kernel, evolving a distribution from
one time to another.
Explicit calculation of
the Feynman propagator. Contour integration is used, closing in the lower half
plane for x0 > y0 and the upper half plane for y0 > x0.
QFT II Jan
22f (In lower left, a quick
review of how a Green function is used)
QFT II Jan
22h (We will frequently use
the Fourier transform of the propagator)
Monday, January 25: An
example: 0th and 1st order perturbation of the scalar
field interaction
Lecture: Perturbation
theory and Feynman diagrams
Exercise: Prove
Eq. 5.86:
<0| T{f1 f2 f3 f4 } |0> = D12
D34 + D13 D24 +
D14 D23
An exercise, a statement
of a little more general version of WickÕs theorem, and the reason that vacuum
expectation values of normal order products of fields always vanishes:
Feynman diagrams (far
right) for two non-interacting particles
Example: 1st
and 2nd order calculations of the scalar field scattering matrix
QFT II Jan
25c (see close-ups below)
Proof that if the
Hamiltonian is polynomial in the fields, then the interaction Hamiltonian is
the same polynomial of the interacting fields
Writing out the transition
amplitude in terms of propagators at zeroth order in lambda
A simple change of
coordinates lets us compute the integrals explicitly
Putting it all together,
we see that the final result for the iT matrix vanishes when we set p2
– m2 = 0. This is
expected, because at zeroth order there is no scattering.
Moving to order l, we get
non-vanishing contributions only for those propagators that relate the original
fields to fields in the interaction Hamiltonian. Otherwise, the terms vanish as
above. This time, the integrals separate nicely and with a trivial change of
variable each integral gives a phase times a momentum space propagator.
Performing all integrals gives the same product of propagators that we need to
cancel, together with a conservation-of-momentum Dirac delta function and some
factors.
Cancelling the common
propagator factors, we are left with a coupling strength, factors of 2¹, and
the conservation of momentum delta function. We draw the Feynman diagram
corresponding to this term of the calculation.
Wednesday, January 27: 2nd
order perturbation of the cubic scalar field interaction; loops
Lecture: Second
order perturbation of the scalar field
The scattering matrix for
2nd order perturbation of the scalar field with cubic interaction
and the corresponding Feynman diagrams.
Further 2nd
order diagrams and the calculation for diagram 1.
The integrals simplify
quickly and we are again left with a factor, the propagator poles that cancel,
and the conservation of momentum delta function.
The delta function
integral: the definition of the matrix Mfi. We will develop general
rules for writing down Mfi directly.
Where the factor of -i comes from (lower left).
There should have been a
factor of (-i)2 in our original equation (note this
factor in the upper right).
At second order with the
quartic interaction it is possible to form loops:
The calculation of a loop
diagram leaves us with an extra integral
We will look at this
integral next time.
Friday, January 29: Loops
and tadpoles
Lecture: Loops
and tadpoles
Recall from last time how
a loop diagram leads to an extra integral over a pair of propagators.
Three similar diagrams all
contribute at this order. It is simple to write down the contributions from the
remaining two.
This remaining integral
diverges as the magnitude of the integrated momentum tends to infinity. To
calculate the form of the divergent part, we can drop p2 relative to
q2. It is most convenient to turn the integral into a Euclidean
integral, and avoid the integrate directly. We can do this because the residue
theorem applies to any loop containing a given pole – we can rotate our
contour 90 degrees counterclockwise in the complex plane. This turns the
denominator into a positive definite expression as we integrate up the
imaginary energy axis.
To do the integral, we
write it in 4-dimensional spherical coordinates. The integral over angles gives
2¹2. It is the radial part which diverges. We need to do a little
bit of geometry to find the volume element.
QFT II Jan
29e (repeat)
All of the integrals are
now easy. To see the behavior of the divergence, we put a cut-off at L instead of
letting the integral run all the way to infinity. The divergence is
logarithmic. Since we set p2
to zero, we canÕt immediately tell what the finite part is, though we could do
it if necessary.
We also find loops on the
propagator. The divergence is easy to integrate, and now has a quadratic piece
in addition to a logarithm.
We can sum all such
ŅtadpoleÓ graphs. The effect is to shift the mass in the propagator by an
amount dependent upon the cutoff.
Monday, February 1:
Feynman rules and the Dirac propagator
Before writing the Feynman
rules, we look at some research related questions.
Lecture: Feynman rules and
the Dirac propagator
Discussion of various
questions:
In biconformal space,
there is an induced Killing metric and a natural symplectic form. These allow
us to find canonically conjugate, orthogonal, metric submanifolds of twistor
space. Here is a quick review of how to tell when fields, or basis forms, are canonically
conjugate
A loop diagram
Simplifying a double arrow
expression
Now, back to field theory.
Why the time ordered
product of fields equals the Green function of the field equation
Feynman rules for lf4 interactions of scalar fields. We can
use these rules to write Mfi directly.
Diagrams at various orders
in perturbation theory.
Exercise: Draw all 3rd order diagrams.
The Dirac propagator. Find
the Green function for the Dirac equation by Fourier transform.
The final momentum space
expression for the Dirac propagator:
Wednesday, February 3:
Feynman rules for QED; renormalization
Lecture: Feynman
rules for QED; renormalization
Expanding the Schrdinger
equation as a time-ordered exponential.
Rewriting the double
integral so both have full time range brings in the factor of ½ for the exponential
The QED (U(1) gauge
theory) action with a variable gauge parameter. The divergence of the vector
potential is taken to vanish on physical states.
We choose the simplest
form of the photon propagator. The field expansions tell us what the
interactions can do.
The interaction term is a
product of all three fields. This gives eight possible interactions. However,
conservation of momentum rules out two of them: there must be at least one
particle in the initial state, and at least one in the final state. (Upper right: photons can interact!)
The form of the coupling
gives a gamma matrix, a unit charge, and the usual -i. Each propagator will carry either a basis spinor or
polarization vector as well.
Renormalization.
Start from the action with
fields which are now allowed to depend on the cutoff. This gives a class of
theories, one for each value of the cutoff. We are free to choose the cutoff
dependence of the fields and parameters, and we make the choice so that the
limit of infinite cutoff has the usual measured values of the parameters (such
as mass and charge; we are also free to choose a normalization of the field).
At the 2-loop level, we can define the dependence of mass on cutoff so that the
combination of all of the tadpoles and the cutoff dependent mass equal the
usual measured mass.
Higher orders require
sterner measures. Renormalizing the field can fix the divergences from loop
amplitudes.
If we fix the parameters
at one value of momentum, the fix holds for other momenta – the
difference is finite.
Friday, February 5:
Asymptotic series (Jeff); superficial divergence
Lecture: Convergent
versus asymptotic series; superficial divergence
Asymptotic
series
A convergent series
converges; asymptotic series may or may not converge:
The exponential integral,
integrated by parts to produce a series for large x, is asymptotic:
The exact integral is
bounded:
At fixed x, the series
starts to diverge for N > x. By that value of N, the series is often
extremely accurate.
Superficial
divergence
Some Feynman diagrams. Can
we tell when higher order diagrams will introduce new divergences?
Each loop adds four
factors of the momentum; each propagator subtracts two. We therefore need to
count the number of internal lines and loops for any given number of vertices.
We can express the
divergence in terms of the number of vertices and the number of external lines.
The final result can be
expressed in terms of the dimension, as a power of length, of the coupling
constant(s). Positive powers of length are generally non-renormalizable.
The cosmological constant
problem. Though renormalizable, the cosmological constant requires extremely
fine tuning. This seems odd.
Monday, February 8: Decay
rates and differential cross sections
Lecture: Decay
rates and differential cross sections
Leptons, mesons
(quark-antiquark pairs) and baryons (quark triples). The production of jets in
proton-antiproton colliders.
Recall the general form of
the scattering matrix. We consider the decay of a single particle first, then
the scattering of a pair of particles. Because the probability of interaction
goes as the square of the matrix element, giving us a delta function squared,
we normalize in a finite box. This allows us to replace one of the delta
functions with the volume of the box. The volume will cancel before we are
done.
Compare state
normalizations in non-relativistic and relativistic scattering.
The box normalization; the
decay rate; the phase space volume element
QFT II Feb
08e (overview slide)
QFT II Feb
08f (normalization)
QFT II Feb
08g (decay rate and phase volume)
The probability of
scattering depends on the number densities of the two particles and their
relative velocity. We find relativistic generalizations of these quantities.
Pulling it all together:
Wednesday, February 10:
Mandelstam variables
Lecture: Mandelstam
variables
Friday, February 12:
Virtual and real particles
Lecture: Virtual
and real particles
Wednesday, February 17:
Scattering and CoulombÕs law
Lecture: Scattering,
decay rates, and Coulomb's law
Monday, February 22: QED
example: Electron positron annihilation into muon antimuon
The
complete calculation (with some minor corrections since the lecture): Muon
photoproduction
Midterm Exam: Study Compton scattering. The initial
state contains one photon and one electron, and the final state contains one
electron and one photon. Average over the initial spin and polarization; sum
over the final spin and polarization.
a)
Why is there is no first-order diagram?
b) Draw the two second-order diagrams.
c) Compute the differential cross-section (at second order) as a
function of the angle between the incident photon and the scattered photon.
Work in the rest frame of the initial electron.
d) The result is the Klein-Nishina formula. Consider the low energy
limit to derive the Thompson (total) cross-section.
Wednesday, March 3: QED
example: Electron-positron (Bhabha) scattering
Friday, March 5: QED
example: Electron-positron (Bhabha) scattering
Monday, March 8: QED
example: Electron-positron (Bhabha) scattering
The complete calculation: Electron-positron scattering. The Bhabha cross section.
Monday, March 22: Questions
on the midterm
Midterm Exam: Study Compton scattering. The initial
state contains one photon and one electron, and the final state contains one
electron and one photon. Average over the initial spin and polarization; sum
over the final spin and polarization.
TodayÕs class was a
student-led discussion of setting up the Compton scattering problem.
First: Draw and label the
Feynman diagrams. Also, various forms of the fermion propagator.
Write the matrix element
for each diagram. Add the matrix elements for each diagram and square. Where
the vertex contribution comes from.
There are various ways to
write the sum over a pair of polarization vectors. The result is a projection
operator, into the two polarization directions. It is probably easiest, in this
problem, to leave the sum until the kinematic stage of the problem.
Rearranging and reducing
the squared matrix element to a trace (first term only)
Wednesday, March 24
Question on the derivation
of the Feynman path integral.
Begin: The
Standard Model
This
pdf (last update April 29, 2010) contains detailed notes on the whole
series of lectures below.
These notes
are now essentially complete.
Wednesday, March 24:
Yang-Mills gauge theory
We want to build an SU(n)
gauge theory over flat spacetime. Start with the product of SU(n) with the
Poincar group and take the quotient by the product of SU(n) with the Lorentz
group. This gives SU(n) and Lorentz fibers over a 4-dim manifold.
Write the Lie algebra of
these groups. The Lie algebra for SU(n) can be written using a basis of nxn
Hermitian matrices. For the symmetric, off-diagonal ones, we should have
subtracted a trace.
More SU(n) generators; the
Poincar Lie algebra
Generators from the
different sub-algebras commute. Introduce connection 1-forms dual to the generators
and write the Maurer-Cartan equations.
The resulting
Maurer-Cartan equations. The fiber bundle. After modifying the connection
1-forms, we have the Cartan equations, with horizontal curvature or field
strength 2-forms.
Restrict to flat,
torsion-free space and solve the spacetime structure equations. The spin
connection may (with Cartesian coordinates) be chosen to be zero, and the
components of the solder form may be written as differentials of the Cartesian
coordinates. Our base manifold is now Minkowski space. There remains only one
field – the SU(n) field strength, defined by the structure equation.
Expand the field strength
and find the components. Listing
the available tensors for writing the action. Some index conventions.
Lorentz tensors which we
can construct from the Dirac matrices and spinors. The Yang-Mills action. The
covariant derivative.
Friday, March 26:
Yang-Mills action; background on the electroweak interaction
Lecture: Action
functional for SU(n) Yang-Mills
First, some history of the
weak interaction: the 4-Fermi interaction. The need for left-handed spinors in
weak interactions.
The diagonal generators of
SU(n) commute:
Now, the Yang-Mills action
Monday, March 29: Tensors,
covariant derivative, conserved currents
Begin the Standard Model
Lecture: Covariant
derivative, conserved currents. The Standard Model
SU(n) and Lorentz tensors
The covariant derivative.
A derivation must be linear and Leibnitz. We check these, and check covariance.
Non-Abelian covariant derivatives require one connection term for each rank of
the differentiated tensor; Abelian derivations require an additive weight.
The Yang-Mills action.
Apply NoetherÕs theorem to find the conserved current.
The adjoint representation
of a semi-simple Lie group.
The Standard Model:
Gauging the symmetry group
The Standard Model: A
trial Lagrangian and its various problems
Wednesday, March 31: QCD
Lecture: Quantum
Chromodynamics
The antisymmetry of the
proton wave function suggests the need for a new symmetry. It must be at least
3 dimensional in order to make three-quark systems into singlet states. We
invent SU(3) colors: red, blue, green.
The QCD action and the
Gell-Mann form of the su(3) generators.
Interactions between the
gauge particles and the quarks arise from the covariant derivative. Each term
annihilates one color and creates another.
Cubic and quartic
interactions of the gluons.
Friday, April 2:
Spontaneous symmetry breaking and the Higgs mechanism
Lecture: The
Higgs Mechanism
Spontaneous
symmetry breaking.
The most general
renormalizable potential for a scalar field is quartic.
We can remove some terms
by field redefinitions. The resulting potential, if it has the right signs,
gives a ŅMexican hatÓ potential, with a peak in the center and symmetric minima
At low energy, the field
settles near a minimum, breaking the symmetry
We may expand the field around
this minimum.
The Higgs
Mechanism
Consider the trial action
for the electroweak symmetry, SU(2) x U(1). Introduce a complex scalar doublet.
As a doublet, it will be coupled to the gauge fields
Pick a gauge to simplify
the form of the Higgs doublet
QFT II Apr
02h (duplicate)
QFT II Apr
02i (duplicate)
Expand the action around
the minimum – the vacuum expectation value of the Higgs. There are terms
in the action which are quadratic in the gauge fields, that are now multiplied
by the vacuum expectation value of the Higgs. These give an effective mass to
the gauge particles. Three of the four gauge fields acquire mass, one (the
photon) remains massless.
QFT II Apr
02l (same, except the eraser moved)
Monday, April 5: Parity
violation and the Yukawa potential
Lecture: Parity
violation and the Yukawa potential
Answer to question: Proof
that the curvature/field strength of a gauged theory is a tensor
The gauge fields couple to
vector currents. The simplest way to violate parity is to take a linear
combination with a pseudovector.
The V – A
interaction is just right. Left and right projection operators
With a left lepton
doublet, a right lepton singlet, and the Higgs doublet we can build the Yukawa
potential. When the Higgs takes on its vacuum expectation value (vev), we get
mass terms for the fermions
An improved action for the
electroweak interaction.
Yukawa terms for the
quarks. In order for the quarks to change flavor in electroweak decays, we need
mixing between the different families. This was first done with the Cabibbo
angle for the s and d quarks. Later, Kobayashi and Maskawa predicted a third
generation of quarks when they proved that two families is insufficient to give
the observed CP violation
Wednesday, April 21: Mass
Matrices
Lecture: Mass
Matrices: The Cabibbo angle, the Kobayashi-Maskawa (CKM) Matrix, and the PMNS
Matrix
First, two puzzles!
I. A computer memory chip
of dimensions x by 2x is filled with a checkerboard of squares of side w. The
alternate, empty squares provide barriers between bits. We wish to calculate
the bit size, w, in terms of x and the total number of bits on the chip, N. We
do this calculation in two ways:
1. Write the area A
of the chip in two ways:
A = 2x2 =
2N w2
This gives w as x over the square root of N.
2. Let b be the
number of bits along x. Then there
are 2b bits along the 2x side. Then the number of bits must be 2N2,
and we have 2bw = x. This results in the previous formula, divided by the
square root of 2.
WhatÕs wrong?
II. A simple calculation
showing that x to any power y equals 1. WhatÕs wrong?
The CKM matrix allows us
to give different masses to the different quarks. A similar matrix is required
for the leptons and neutrinos.
We now know that there
exist mass differences between the neutrinos because electron neutrinos
produced by fusion in the sun oscillate into muon and tau neutrinos by the time
they reach Earth. It is a simple quantum problem to show that this occurs only
if the neutrinos have different masses.
For the quarks, it was
realized early on that the strange quark decays into the up quark. But so far,
our model only allows such interactions between two quarks in the same weak
doublet.
Cabibbo solved the problem
by building the (u,d) doublet using a linear combination of d and s instead of
just the d.
When CP violation was seen
in quark decays, Kobayashi and Maskawa showed that CP violation cannot occur
even with a full mixing matrix between the u,d,s and c quarks that were then
known. They correctly suggested the existence of a third pair of quarks. It is
easier to accommodate the mixing in the Yukawa mass terms than in the
interaction term of the Lagrangian.
We can now write a fairly
complete Lagrangian for the standard model. We include a second mixing matrix,
shown necessary by Portocorvo and introduced by Maki, Nakagawa and Sakata, for
the leptons.
The number of free
parameters of the standard model is largely determined by the CKM and PMNS
matrices. There are about 23 parameters:
Additional terms can be
added to the model. For example, parity violating terms such as Joe Slansky
described are possible for the quarks. Also, the theory can be
supersymmetrized. Frequently, experiments test predictions of the MSSM –
the Minimal Supersymmetric Standard Model.
Monday, April 26: The
Feynman Path Integral
Lecture: Feynman
Path Integral
Recall Juan TrujilloÕs
lecture from last week, deriving the phase space path integral as the
transition amplitude for a particle at (x,t) to arrive at (xÕ,tÕ).
Question: Sorting out
details of the phase factors:
The form of the phase
space path integral. The volume form becomes an infinite product of integrals
in the limit as the time spacing goes to zero.
We consider an action of
the general form, L = T – V.
To evaluate the momentum
integrals, we back up a step. With the standard form of the kinetic energy, the
momentum integrals are all Gaussian integrals. We complete the square, and the
term we add and subtract is the kinetic energy in terms of the velocity.
The result is the path
integral in configuration space, together with a divergent normalization. This
peculiar factor cancels with a similar factor from the configuration space
terms. The phase is the usual action functional.
Now restrict attention to
a free particle. We can check that the divergent terms cancel.
Again we have Gaussians.
Each integral involves a pair of terms. The only contribution is a
normalization factor, while the neighboring terms coalesce into a similar term
with doubled time interval. The factors cancel and the transition amplitude is
a pure phase given by the action evaluated along the classical (straight line)
path, that is, the phase is HamiltonÕs principal function.
The path integral for
fields becomes a functional integral over all field configurations, often
called a Ņsum over histories.Ó
Wednesday, April 28: Path
integrals in field theory
Lecture: Feynman
Path Integral
Lecture presentation by
Jeff Hazboun.
For field Lagrangians, the
momentum typically enters quadratically, just as in the particle case.
Integration over the momentum gives the usual spacetime action.
We consider a field action
with minimal coupling to a current density as source. We can decouple the
source from the fields by first expanding the fields in Fourier series, then
making a field redefinition.
The source terms come out
of the path integral as a real phase factor, leaving the free particle action
in the path integral. Inverting the Fourier transform of the currents, we find
the propagator of the field.
For gauge field theories,
we must avoid counting the non-physical modes of the field corresponding to
gauge transformations. By writing the functional integral as an unconstrained
field integral times a delta function that imposes a general, unspecified gauge
condition, we can extract a factor equal to the volume of the gauge group.
A change of variable gives
a factor of the Jacobian – the determinant of the derivative of the gauge
with respect to the gauge constraint.
Friday, April 30: Mass
estimates from lattice computations
A very polished powerpoint
presentation by Nathan Piepgrass.
Thanks (alphabetically) to
Jeff Hazboun, Joe Slansky, Juan Fernandez, Juan Trujillo, and Nathan Piepgrass!
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