Download Maxwell`s equations

Maxwell’s equations
based on S-54
Our next task is to find a quantum field theory description of spin-1
particles, e.g. photons.
The first two Maxwell’s equations can be written as:
Classical electrodynamics is governed by Maxwell’s equations:
charge-current density 4-vector
electric field
charge density
taking the four-divergence:
current density
magnetic field
can be solved by writing fields in terms of
a scalar potential and a vector potential
we find that the electromagnetic current is conserved:
The last two Maxwell’s equations can be written as:
automatically satisfied!
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The gauge transformation in four-vector notation:
The potentials uniquely determine the fields, but the fields do not uniquely
determine the potentials, e.g.
arbitrary function of spacetime
The field strength transforms as:
result in the same electric and magnetic fields.
More elegant relativistic notation:
gauge transformation
(a change of potentials that
does not change the fields)
four-vector potential, or gauge field
the field strength
= 0 (derivatives commute)
the field strength is gauge invariant!
Next we want to find an action that results in Maxwell’s equations as the
equations of motion; it should be Lorentz invariant, gauge invariant, parity
and time-reversal invariant and no more than second order in derivatives;
the only candidate is:
in components:
we will treat the current as an external source
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We can impose the Coulomb gauge by acting with a projection operator:
obviously gauge invariant
in the momentum space it corresponds to
multiplying
by the matrix
,
that projects out the longitudinal component.
(also known as transverse gauge)
total divergence
In terms of the gauge field:
total divergence
the lagrangian in terms of scalar and vector potentials:
equations of motion:
equivalent to the first two Maxwell’s equations!
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Electrodynamics in Coulomb gauge
based on S-55
Next step is to construct the hamiltonian and quantize the electromagnetic
field ...
integration by parts
integration by parts
0
Which
should we quantize?
too much freedom due to gauge invariance
There is no time derivative of
and so this field has no conjugate
momentum (and no dynamics).
To eliminate the gauge freedom we choose a gauge, e.g.
0
equation of motion
integration by parts
Poisson’s equation
unique solution:
Coulomb gauge
an example of a manifestly relativistic gauge is Lorentz gauge:
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we get the lagrangian:
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we impose the canonical commutation relations:
the equation of motion for a free field (
):
with the projection operator
massless Klein-Gordon equation
the general solution:
polarization vectors (orthogonal to k)
these correspond to the canonical commutation relations for creation and
annihilation operators:
(the same procedure as for the scalar field)
we can choose the polarization vectors to correspond to right- and left-handed
circular polarizations:
in general:
creation and annihilation operators for
photons with helicity +1 (right-circular
polarization) and -1 (left-circular polarization)
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now we can write the hamiltonian in terms of creation and annihilation
operators:
following the procedure used for a scalar field we can express the
operators in terms of fields:
(the same procedure as for the scalar field)
to find the hamiltonian we start with the conjugate momenta:
2-times the zero-point
energy of a scalar field
the hamiltonian density is then
this form of the hamiltonian of electrodynamics is used in calculations of
atomic transition rates, .... in particle physics the hamiltonian doesn’t play a
special role; we start with the lagrangian with specific interactions, calculate
correlation functions, plug them into LSZ to get transition amplitudes ...
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LSZ reduction for photons
based on S-56
Next step is to get the LSZ formula for the photon. The derivation closely follows the
scalar field case; the only difference is due to the presence of polarization vectors:
For a scalar field we found that in order to obtain a transition amplitude we simply
replace the creation and annihilation operators in the transition amplitude by:
Now we want to calculate correlation functions (the derivation again
closely follows the scalar field case).
the propagator for a free field theory:
similarly, for an incoming and outgoing photon we simply replace:
correlation functions of more fields given in terms of propagators...
Next we want to calculate the path integral for the free EM field:
we will treat the current as an external source
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the LSZ formula is then valid if the field is normalized according to the free
field formulae:
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In the Coulomb gauge we integrate over those field configurations that
satisfy
; in addition the zero’s component is not dynamical we can
replace it by the solution of the equation of motion
where a single photon state is normalized according to:
and for the rest of the path integral we will guess the result based on the
result we got for a scalar field:
and the renormalization of fields results in the Z-factors in the lagrangian:
we will discuss this next semester...
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propagator
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we can make it look better:
this looks better but we can simplify the propagator further...
where
the momentum can be replaced
by the derivative with respect to .
acting on the exponential, and then integrate
by parts to obtain
which vanishes.
and the Coulomb term is reproduced thanks to:
and we get:
=0
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We obtained a very simple formula for the photon propagator:
We can simplify the propagator further...
Let’s define:
and
as a unit vector in the
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direction:
Feynman gauge
(it would still be in the Coulomb gauge if we
had kept the terms proportional to momenta)
now we can replace:
and thus we get:
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The path integral for photons
based on S-57
We will discuss the path integral for photons and the photon propagator
more carefully using the Lorentz gauge:
We can decompose the gauge field
into components aligned along a
set of linearly independent four-vectors, one of which is
and then this
component does not contribute to the quadratic term because
as in the case of scalar field we Fourier-transform to the momentum space:
and it doesn’t even contribute to the linear term because
and so there is no reason to integrate over it; we define the path integral
as integral over the remaining three basis vector; these are given by
we shift integration variables so that mixed terms disappear...
which is equivalent to
Problem: the matrix
has zero eigenvalue and cannot be inverted.
Lorentz gauge
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Within the subspace orthogonal to
the projection matrix is simply the
identity matrix and the inverse is straightforward; thus we get:
To see this, note:
where
is a projection matrix
and so the only allowed eigenvalues are 0 and +1
going back to the
position space
Since
propagator in the Lorentz gauge (Landau gauge)
we can again neglect the term with momenta because the current is
conserved and we obtain the propagator in the Feynman gauge:
it has one 0 and three +1 eigenvalues.
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Quantum electrodynamics (QED)
based on S-58
Quantum electrodynamics is a theory of photons interacting with the
electrons and positrons of a Dirac field:
We can also define the transformation rule for D:
then
Noether current of the
lagrangian for a free Dirac field
as required.
we want the current to be conserved and so we need to enlarge the gauge
transformation also to the Dirac field:
Now we can express the field strength in terms of D’s:
global symmetry is
promoted into local
symmetry of the lagrangian and so the current is
conserved no matter if equations of motion are satisfied
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We can write the QED lagrangian as:
Then we simply see:
covariant derivative
(the covariant derivative of a field transforms as the field itself)
and so the lagrangian is manifestly gauge invariant!
Proof:
no derivatives act on
exponentials
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the field strength is gauge invariant as we already knew
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To get Feynman rules we follow the usual procedure of writing the
interacting lagrangian as a function of functional derivatives, ...
the arrow for the photon can point both ways
vertex
We have to make more precise statement over which field
configurations we integrate because now also the Dirac fields
transform under the gauge transformation (next semester).
one arrow in and one out
draw all topologically inequivalent diagrams
for internal lines assign momenta so that momentum is conserved in each vertex
(the four-momentum is flowing along the arrows)
propagators
Imposing
we can write it as:
for each internal photon
for each internal fermion
sum of connected Feynman diagrams with sources!
(no tadpoles)
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Feynman rules to calculate
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:
vertex and the rest of the diagram
external lines:
spinor indices are contracted by starting at the end of the fermion line that has
the arrow pointing away from the vertex, write
or
; follow the
fermion line, write factors associated with vertices and propagators and end up
with spinors
or
.
follow arrows backwards!
The vector index on each vertex is contracted with the vector index on either
the photon propagator or the photon polarization vector.
incoming electron
outgoing electron
incoming positron
assign proper relative signs to different diagrams
draw all fermion lines horizontally with arrows from left to right; with left end points labeled in
the same way for all diagrams; if the ordering of the labels on the right endpoints is an even
(odd) permutation of an arbitrarily chosen ordering then the sign of that diagram is positive
(negative).
outgoing positron
incoming photon
sum over all the diagrams and get
outgoing photon
additional rules for counterterms and loops
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Scattering in QED
based on S-59
we also want to sum over the final photon polarizations:
Let’s calculate the scattering amplitude for a simple process:
in the Coulomb gauge we found the polarization sum to be:
doesn’t contribute: the scattering amplitude should be
invariant under a gauge transformation and so we should have:
in addition,
and so we find:
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We follow the same procedure as before:
Ward identity
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for the spin averaged/summed amplitude we get:
and the amplitude squared is:
averaging over the initial electron and positron spins we get:
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we can plug the result to the formulae for differential cross section...
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