|
|
|
|
Monday,
August 24: The Lorentz group
Recall the definition of a group.
Exercise: Prove that the ÒLorentz groupÓ is a
group
Note the difference between R3 as a vector space
and R3 as a manifold: the symmetry of vector length is the orthogonal
group, while the symmetry group of the Euclidean line element includes
translations as well. The same distinction holds for spacetime, giving the
Lorentz group as the transformations preserving 4-vector length, and the
PoincarŽ group for the Minkowski line element.
Deriving a quadratic algebraic condition on Lorentz
transformation matrices from the invariance of 4-vector length.
Linearize by finding those transformations infinitesimally
close to the identity. This gives the infinitesimal generators:
Develop some simplified notation:
Take the limit of infinitely many infinitesimal
transformations to get a finite transformation. The result is the exponential
of the generator. Carry out the exponentiation of the Lorentz generators
(boosts only) to find the form of a Lorentz transformation. Check for a boost
along the x direction.
Exercise: Write out the matrix for a general
boost, apply it to a position 4-vector, and prove directly that the new
4-vector has the same proper length as the original 4-vector.
The Lie algebra is made up of the commutators of the
generators.
Exercise: Derive the commutators of the Ji
and Ki generators.
Wednesday,
August 26: Lie groups; SU(2)
Units in field theory; some nomenclature for Lie groups.
SU(2) is the Lie group of 2x2, unit determinant, unitary
matrices. They may be written using the Pauli matrices.
Properties of the Pauli matrices
Exercise: Check the product relations for the
Pauli matrices
Comparing SO(3) and SU(2). Constructively show that SU(2)
can produce rotations on 3-vectors.
Exercise: Let XÕ = AXA . Show that XÕ is traceless if and only if A is
unitary.
There are 2 SU(2) matrices for each SO(3) rotation
Expand the exponential of a generator
Use the expanded form to explicitly compute the effect of an
SU(2) similarity transformation on a 3-vector
The geometry of the result: a rotation about n through
an angle j.
Exercise: Let the unit vector n point in the z direction, so n = k. Show
that under the SU(2) rotation exp( ij/2
n.s )
a general vector x is rotated about the z axis through an angle j.
Monday,
August 31: Spin groups; SU(2), SL(2,C)
Lecture:
Spin groups; SU(2), SL(2,C)
A strange man sneaks in and erases some comments on the
metric in gauge theory. The blurred right hand indicates motion near light
speed.
Similarity transformations. MÕ = AMA-1 preserves
the form of vector equations when M is a mapping on a vector space M: V ˆ V. When M
is doubly covariant or contravariant, the transpose or adjoint is required
instead.
Spinors for rotations
The general construction of Spin(p,q); Clifford algebras. Exercises.
Exercise: Find the commutator of the sigma
matrices, where sigma is the commutator of two gamma matrices.
Exercise: Show that the usual rotation algebra in
3-dim takes the same form if we use the Levi-Civita tensor.
Problem: Carry out the entire construction of
Spin(3) starting from SO(3).
The sigmas generate SO(p,q) pseudo-rotations:
SL(2,C) is the (2:1 cover of the) Lorentz group
Representations of SL(2,C):
Wednesday,
September 2:
Clarification of the rotation of a 3-vector using SU(2):
Generators and Lie algebra of SL(2,C):
Exercise: Explore the form of some group elements.
The action of SL(2,C) on a 4-vector; representation of
states using commuting SU(2) generators:
Exercise: Perform a boost using SL(2,C)
Features of the su(2) x su(2) representation:
Wednesday,
September 9: Lie groups
Finite groups; Lie groups
Representations; irreducible representations; infinitesimal
generators
The Lie algebra from the Lie group; closure of commutation
relations; structure constants
Jacobi identity; Casimir operators:
Exercise: Show that [J2, Ji]
= 0, so that J2 is a Casimir operator for SO(3).
Casimir operators of the PoincarŽ group:
Parity of J+ and J- operators;
rotations of left and right handed Weyl
spinors:
Exercise: Prove that the 4-vector formed from a
single 2-spinor is null.
Monday,
September 14:
Lecture:
Representations of the Lorentz group
SO(3,1) and SL(2,C) transformations; generators; action on
vectors, tensors
Transpose relationship between an SO(3,1) transformation and
its inverse
SL(2,C) tensors as products of spinors
Irreducible parts of a doubly contravariant SO(3,1) tensor
Irreducible parts of a rank 2 SL(2,C) tensor
The adjoint representation of a Lie algebra
Decomposition of SO(3,1) tensors into SO(3) covariant parts
Representation of SL(2,C) tensors in terms of left- and
right-handed representations, su(2)L and su(2)R
Complex conjugation of Pauli matrices using Pauli matrices;
interchanging left and right transformations; definition of charge conjugation
Representation of 4-vectors using left- and right- spinors.
Vectors, cont.
Field representation of Lorentz generators. Scalar fields; orbital
angular momentum
Topology of SO(3)
Topology of SU(2)
Friday,
September 18:
Field representation of the Lorentz generators on scalars;
spinors
Review of Lagrangian mechanics
Review of Hamiltonian mechanics
If it werenÕt for classes, when would astrophysicists sleep?
Hamiltonian mechanics: the symplectic form
Hamiltonian mechanics: the Poisson bracket
Lagrangian formulation of field theory: the Lagrangian
density
Klein-Gordon scalar field
Free Maxwell action
(Reading right to left) Adding the Maxwell source currents;
conservation of current and gauge invariance; conserved charge
The gauge transformation of the solder form, first as part
of the (local) PoincarŽ group, then under local Lorentz transformations only.
Monday,
September 21:
Lecture:
Lagrangian and Hamiltonian field theory
Checking the form of a Lorentz transformation from the
problems:
Comments on the relationship between SU(2) and SO(3)
Lagrangian field theory
Example: Scalar field
Hamiltonian field theory, with example
Friday,
October 2:
Lecture:
Noether's theorem in mechanics and field theory
Definition of a symmetry of the action.
Examples: translation, rotation, time translation
NoetherÕs theorem in mechanics
Examples: translational symmetry leads to conservation of
momentum; rotational symmetry leads to conservation of angular momentum
NoetherÕs theorem in field theory. Conserved currents and
conserved charge.
The surface integral
The variation in detail
Example: translations (There are many (nearly!) duplicate
slides here)
The problem was mostly getting the notation to agree with
Maggiore. We also had to digress to figure out the variation of the volume
element under a coordinate transformation.
Five versions of the translational currents:
Five versions of the variation:
Monday,
October 5:
Lecture:
Real Klein-Gordon (scalar) fields
The action functional for the Klein-Gordon field; vary the
action to find the field equation:
Show in detail that the surface term vanishes
Solve the Klein-Gordon equation using a Fourier integral.
Use reality of the field to relate the coefficients
A note on using exponentials for solutions with circular
functions
Compute the conjugate momentum and the Hamiltonian; show
that the Hamiltonian is the T00 component of the energy-momentum
tensor.
Exercise: Find Pi
Find the conserved Noether currents associated with Lorentz
transformations. These include angular momentum.
Define the inner product of two Klein-Gordon fields; show
that it is conserved as long as the masses are equal.
Show that the inner product is the conserved Noether charge
of a field rotation:
"Integrate the function of all things so that it equals
done."
Dr. James Wheeler, early morning of 10/07
Friday,
October 9: Complex scalar fields, begin Dirac fields
Lecture:
Complex Klein-Gordon (scalar) fields
The action functional for the Klein-Gordon field; vary the
action to find the field equation. We get the same result whether we vary real
and imaginary parts, or vary the field and its conjugate:
The solution to the free field equation as a Fourier
integral. The U(1) current of the complex scalar field.
The Dirac field: first, we need the covering group of the Lorentz
group, SO(3,1). The covering group is Spin(3,1), which is isomorphic to
SL(2,C). An identical calculation holds for the covering (spin) group of any
SO(p,q):
Properties of the Dirac (gamma) matrices
Exercise 1: Show that (g5)2
= 1
Exercise 2: Prove that g5
anti-commutes with the other basis g-matrices,
{g5ga} =
0
Exercise 3: Find 4 matrices satisfying the basic
relation of a Clifford algebra, {ga gb} =
hab
Exercise 4: In terms of your answer to Exercise 3,
find the matrices
GA = {1,
ga, sab =
[ga ,gb], g5ga ,g5
}
and show that the trace of all but the identity is zero:
tr(1) = 4
tr(ga) =
tr(sab ) =
tr(g5ga) =
tr(g5)
= 0
Exercise 5: Find the trace of all products of
pairs of GAs. In particular, show that
tr(GAGB)
= 0 for all A
B
tr((GA)2) = lA
0 for A
and find the values of lA
in each case.
Exercise 6: Show that the 16 matrices GA
are independent.
Hint: Let M be an arbitrary linear combination of the GA:
M = wAGA =
a1 + baga + wab sab
+ cag5ga +
dg5
Show that M = 0 implies wA = 0 for
all A.
Hint: Multiply M by any given one of the GA
and take the trace.
Question: How do we solve for the torsion in PoincarŽ gauge
theory?
Question: How do we derive the formula for a delta function
of a function?
Monday,
October 12: Lorentz transformations of spinor fields
Lecture:
Lorentz transformations of spinors
Question: Is the complex scalar field equivalent to two real
fields.
Answer: Yes, as long as they have the same mass:
HereÕs what goes wrong if the action is not real:
The Dirac action; Lorentz transformations of spinors; begin
example. Finding a particular set of Dirac matrices for the example
QFT09 Lecture
notes 10/12e (duplicate)
Writing a 4-vector in terms of a pair of spinors, including
a clever trick for writing the adjoint of the Dirac matrices
QFT09 Lecture
notes 10/12h
QFT09 Lecture
notes 10/12j (duplicate)
Acting with an infinitesimal boost on a general spinor
Inducing the same transformation on a vector built from
spinors
Some comments on covering groups
Thursday,
October 15: Solutions to the Dirac equation
Lecture:
Solving the Dirac equation
The Dirac equation; some comments on the gamma matrices
Alternate forms of the Dirac action
Variation of the action. Solutions to the Dirac equation
satisfy the Klein-Gordon equation
Solve for one Fourier mode
One pair of components is much less than the other pair at
nonrelativistic velocities
Lorentz transformation of the solutions
A complementary pair of projection operators. These break
the spinor into left- and right- handed parts
We can also form positive and negative energy projections
Spinors in 10-dimensions and octonions
Monday,
October 19: Dirac energy-momentum tensor
Lecture:
Energy-momentum tensor for Dirac fields
An intensely interested student:
A more typical student:
Some comments on octonions and representations of the
PoincarŽ and conformal groups
Exercise 1: Derive 3.103 and 3.107 in Maggiore by
boosting the rest frame solution
Exercise 2: Derive 3.103 and 3.107 directly from
the Dirac equation
Exercises, some notation, and a convenient rewriting of the
energy factor
The energy-momentum tensor of the Dirac field,
The conserved momentum
Exercise 2: Show that the energy-momentum tensor
is divergence-free, hence conserved
The symmetry of the energy-momentum tensor
Finding the term we need to symmetrize the energy-momentum
tensor. We require the existence of a totally antisymmetric 3-tensor with
divergence equal to the antisymmetric part of the asymmetric energy-momentum
tensor
Try another approach: vary the metric. The usual
construction of a symmetric energy-momentum tensor by varying the metric fails
– the result is identically zero!
The integrability condition required to symmetrize the Dirac
energy-momentum tensor. Satisfying the condition requires the vanishing of the
term we are trying to get rid of. The argument does not allow us to symmetrize
A comment on supersymmetry
Studying the conserved energy of the Dirac solutions. The
energy is a sum over energies of the various modesÕ number operators
QFT09 Lecture
notes 10/19q (the sign should be plus)
Checking the sign above. We need to change integration
variable in order to get the solution in a symmetric form
Summary of the decomposition of the Dirac solution into
positive and negative energy parts, and left- and right-handed parts. Checking
the rotation properties.
Friday,
October 23: Vector fields
Review of scalar, complex scalar and spinor actions. A
Klein-Gordon type action for a vector field
Three types of vector field action: Klein-Gordon type, with mass; Maxwell
action, with gauge invariance; Proca action, with mass
Conserved current in combined (but uncoupled) Klein-Gordon,
Dirac and Maxwell action
QFT09 Lecture
notes 10/23e (duplicate)
Gauging to couple the three fields, the conserved U(1)
currents provide a source for the Maxwell equations
QFT09 Lecture
notes 10/23h (duplicate)
QFT09 Lecture
notes 10/23i (duplicate)
Fixing the Dirac/Coulomb gauge for the Maxwell field.
Because of gauge invariance, we have some nonphysical fields. This shows up
when we compute the conjugate momentum:
We can fix the problem by choosing a gauge which leaves no
unphysical freedom
The energy-momentum tensor of the Maxwell field
The conserved energy is the usual electromagnetic field
energy; the conserved momentum is the Poynting vector
Monday,
October 26: Interacting fields in quantum field theory – overview
Lecture:
Interacting quantum fields - overview
Hyperfine corrections to atomic levels, and the interactions
of photons that lead to them
Possible tensor operators constructed from spinors and the
gamma matrices; the form of the corresponding interactions
The dance of the diagrams. Possible Feynman diagrams arising
from minimal coupling for Abelian and non-Abelian gauge fields
Expanding the Dirac equation for low velocity, to second
order. This gives first order relativistic corrections to the Schršdinger
equation. Higher order corrections require QED
Moment of inertia of rectangular doughnuts and hot dogs
Friday,
October 30: Quantization of the Klein-Gordon field, sort of
Lecture:
Quantizing fields. The real scalar field
The Stern-Gerlach experiment and the functioning of a Cesium
clock
Quantization of the Klein-Gordon field. Find the conjugate
momentum and impose canonical commutation relations [x,p] = i. Writing the
result in terms of momentum density gives a Dirac delta function
Invert the Fourier integral and solve for the commutation
relations of the mode amplitudes. (See the next lecture for more on this)
Monday,
November 2: Quantization of the Klein-Gordon field
Lecture:
Quantization of the Klein-Gordon field
Write the Fourier expansion of the field and its conjugate
momentum density. Check the dimensions of the fields
Finish checking units. Impose canonical commutation
relations. The relationships will be just like those of the quantum simple
harmonic oscillator, only we have many of them
Invert the Fourier transforms to solve for the mode
amplitudes
Compute the commutator of the mode amplitude (annihilation
operator, lowering operator) and its adjoint (creation operator, raising
operator)
Recall the canonical commutator of the field and its
momentum density. WeÕre still dealing with densities here
Now compute the Hamiltonian in terms of the mode amplitudes.
Find one term at a time then add:
The final Hamiltonian operator, in terms of the mode
amplitudes. The combination ata acts as a number operator, counting
states of a given momentum and energy. This combination is multiplied by the
energy of that mode, then summed over all momenta. The term arising from the
commutation of a and its adjoint diverges, and we need a new quantization rule,
normal ordering, to avoid the infinity. This has its origin in the difference
between quantum and classical dynamical variables. Since quantum variables do
not commute, we need to be specific about what order we write operators in. We
choose the order in the unique way that avoids divergent quantities.
Rewrite the Hamiltonian using normal ordering, and the
divergent term never occurs
Friday,
November 6: Klein-Gordon states. Quantization of the complex Klein-Gordon field
Lecture:
Quantization of the complex Klein-Gordon field
A short review of Hamiltonian mechanics
From the form we found last time for the normal-ordered
Hamiltonian operator, we can define a complete set of states. Positivity of the
inner product shows us that there is a lower bound to eigenvalues of the
Hamiltonian, leading to the existence of the vacuum state, |0>. From this we
build up by acting with arbitrary numbers of creation operators:
Quantizing the complex Klein-Gordon field. Starting from the
action, write the full solution for the field and its complex conjugate, and
their conjugate momenta, as a Fourier integrals. Invert the Fourier integrals
to solve for the two mode amplitudes and their adjoints.
It is now straightforward to check the commutation relations
of the mode operators:
Exercise: Check the commutation relations for [at,b]
and [b,bt].
Monday,
November 9: Complex Klein-Gordon field: Hamiltonian and charge
Lecture:
Quantum Hamiltonian and charge for complex Klein-Gordon fields
Results from last time: quantized field and momentum,
creation and annihilation operators.
The form of the Hamiltonian
Computing the Hamiltonian operator
Interpreting energy eigenstates
The U(1) symmetry of the complex Klein-Gordon field, and its
conserved (electric) charge
Computing the electric charge
Other symmetries
Exercise: Find the conserved momentum operator.
Comment on interactions
(Answer to a question on classical gravitational energy QFT09 Lecture
notes 11/09l )
Friday,
November 13: Quantization of the free Dirac field
Lecture:
Quantization of the free Dirac field
(Answer to questions: the Green function for the wave
equation QFT09
Lecture notes 11/13a
and the completeness relation for Fourier integrals QFT09 Lecture
notes 11/13b )
Review of the solution to the Dirac equation
Exercise: Prove equations 3.109 – 3.117 in
Maggiore. These are simple relationships between the basis spinors.
Fourier integral for the Dirac field; Hamiltonian for the
Dirac field
Hamiltonian and conjugate momentum; inverting to find the
creation and annihilation operators
Basis spinors
Anticommutation relations for the mode amplitudes
Exercise: Find the anticommutator of the remaining
mode operator and its adjoint, {b,bt}
(Notes on oscillation rate (or expansion rate in a conformal
model) of a Schršdinger particle at rest. Removal of this oscillation
(expansion) by a gauge transformation:
Monday,
November 16: Hamiltonian and charge of the free Dirac field
Lecture:
The Miraculous Dirac Hamiltonian
(Answer to questions:
Detail of the
gradient-squared term of the Hamiltonian calculation: QFT09 Lecture
notes 11/16a
Equality of
energies when multiplied by a Dirac delta function: QFT09 Lecture
notes 11/16b)
Review of the solution to the Dirac field operators and the
form of the Hamiltonian
Computing the Hamiltonian operator in terms of the mode
amplitudes. First part:
Second part. Putting the two together:
Checking signs – somethingÕs awry here:
Checking the orthogonality of two of the basis spinors
A miracle occurs . . . and we have the form of the final
Hamiltonian. (See my QFT book for the calculation without magical invocation)
The form of the charge operator:
Friday,
November 20: Quantization of the electromagnetic field
Lecture:
Hamiltonian formulation of electromagnetism
Review of the solution to the Dirac field operators for the
Hamiltonian and charge; interpretation of antiparticles
Why we vary the potential instead of the fields (to vary the
fields, we must use Lagrange multipliers); the two relativistic invariants of
the electromagnetic field:
The action; the Hamiltonian. The vanishing of one of the
conjugate momenta leads us to fix the gauge to eliminate the extra degrees of
freedom:
Invariance of the fields under gauge transformation (lower
left)
The Lagrange density, Hamiltonian density and Hamiltonian in
the new gauge. HamiltonÕs equations for fields as functional derivatives
Computing the functional derivatives to find the wave
equation for the vector potential
Solution to the field equations. Note that the final
commutator in the last picture is not yet correct – it does not satisfy
the gauge constraints. We will fix this in the next lectures.
Monday,
November 23: Dirac states; Quantization of the electromagnetic field
Lecture:
Quantization of the electromagnetic field, continued
Dirac states; Fermi-Dirac statistics
Canonical commutation relations for the electromagnetic
potential and momentum must satisfy the transversality condition, and will
therefore give a projection operator orthogonal to the wave vector, k. To show
this, we start from the commutator for the mode amplitudes and work back to
find the commutators of the fields.
The projection operator arises from the products of the
polarization vectors. We therefore seek a set of algebraic conditions which
determine the allowed polarization vectors. Since these may be complex, we find
that the orthonormality conditions are insufficient to determine all inner
products of the polarization vectors. Continued Friday.
Friday,
November 30: Dirac monopoles; spin of the photon; polarization vectors for EM
waves
Lecture:
Dirac monopoles and photon spin
Dirac
monopole
The Dirac monopole as a non-trivial U(1) bundle
Are monopoles observable, or can they be eliminated by a
redefinition of the electromagnetic field? The equations of electric-magnetic
symmetric field theory allow free rotation between the electric and magnetic
fields and charges. This allows us to eliminate all magnetic monopole sources
if and only if all particles have the same ratio of magnetic charge to electric
charge:
Charge quantization from quantization of angular momentum as
an electron passes a magnetic monopole.
Spin
of the photon; polarization vectors
We want to work with eigenstates of the angular momentum and
z-component of angular momentum. To find these eigenstates, we first write the
EM polarization vectors as rank 2 tensors under SU(2). In this form, we seek
linear combinations of the two polarization vectors that commute with Jz.
(We know immediately, since the polarization vectors are rank 2, that the total
angular momentum is l = 1). We find, because of the transverse constraint from
gauge invariance, that only two of the possible 2l+1 = 3 values of Jz
occur. The polarization eigenstates correspond to left and right circular
polarization, so we will quantize choosing a basis for circularly polarized
states. Classically, these are waves in which the electric and magnetic fields
at a point each rotate in a circle as the wave passes.
Finding the class of allowed polarization vectors. Since the
complex orthonormality condition is invariant under phase changes, we may mod
out an arbitrary phase from each of the two polarization vectors. This leaves
us with a 2-parameter class of suitable polarizations. We choose the phases so
that both linearly and circularly polarized light are special cases of the same
parameterization.
We choose a convenient notation for the momentum dependence
of a pair of circularly polarized basis vectors. In adapted coordinates, it is
easy to see that their outer product is a projection operator.
Friday,
December 4: Time dependence of a wave packet. Phase and group velocity
Lecture: Phase and group
velocity
When we take superpositions of waves to form a wave packet,
the wave speed may differ from the velocity of the wave packet.
The effect is already present in the superposition of two
waves: the frequency of beats differs greatly from the average wave frequency:
Consider a continuous superposition, with a Gaussian
envelope. Perform the k-integral
by completing the square.
Rewrite the final wave as a moving Gaussian. The Gaussian
moves to the right with velocity equal to the particle velocity; the packet
also spreads with time.
The velocity of the packet is dw/dk, while the
velocity of each wave is w/k.
Comments on the Òsquare rootÓ of the biconformal wave
equation. Squaring the Dirac operator brings in a curvature scalar.
|
|
|
|