contin uous symmetries and conser ved cur rents - iu bdermisek/qft_08/qft-i-11-1p.pdf · contin...
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setting we would get equations of motion
Consider a set of scalar fields , and a lagrangian density
Continuous symmetries and conserved currentsbased on S-22
let’s make an infinitesimal change:
variation of the action:
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if a set of infinitesimal transformations leaves the lagrangian unchanged, invariant, , the Noether current is conserved!
this is called Noether current; now we have:
thus we find:
= 0 if eqs. of motion are satisfied
current densitycharge density
Emmy Noether
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Consider a theory of a complex scalar field:
in terms of two real scalar fields we get:
clearly is left invariant by:
and the U(1) transformation above is equivalent to:
U(1) transformation (transformation by a unitary 1x1 matrix)
SO(2) transformation (transformation by an orthogonal 2x2 matrix with determinant = +1)
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infinitesimal form of is:
and the current is:
we treat and as independent fields
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repeating the same for the SO(2) transformation:
the Noether current is:
which is equivalent to
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Let’s define the Noether charge:
integrating over ,using Gauss’s law to write the volume integral of as
a surface integral and assuming on that surface
we find:
Q is constant in time!
using free field expansions, we get:
for an interacting theory these formulas as valid at any given time
counts the number of a particles minus the number of b particles;it is time independent and so the scattering amplitudes do not change the value of Q; in Feynman diagrams Q is conserved in every vertex.
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Schwinger-Dyson equations:
The path integral
doesn’t change if we change variables
thus we have:assuming the measure is invariant under the change of variables
taking n functional derivatives with respect to and setting we get:
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since is arbitrary, we can drop it together with the integral over .
since the path integral computes vacuum expectation values of T-ordered products, we have:
Schwinger-Dyson equations
for a free field theory of one scalar field we have:
for a free field theory of one scalar field we have:
SD eq. for n=1:
is a Green’s function for the Klein-Gordon operator as we already know
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in general Schwinger-Dyson equations imply
thus the classical equation of motion is satisfied by a quantum field inside a correlation function, as far as its spacetime argument differs from those of all other fields.
if this is not the case we get extra contact terms
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For a theory with a continuous symmetry we can consider transformations that result in :
Ward-Takahashi identity
Ward-Takahashi identity:
summing over a and dropping the integral over
thus, conservation of the Noether current holds in the quantum theory, with the current inside a correlation function, up to contact terms (that depend on the infinitesimal transformation).
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there is still a conserved current:
Consider a transformation of fields that change the lagrangian density by a total divergence:
Another use of Noether current:
e.g. space-time translations:
we get:
stress-energy or energy-momentum tensor
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for a theory of a set of real scalar fields:
we get:
in particular:
hamiltonian density
then by Lorentz symmetry the momentum density must be:
plugging in the field expansions, we get:
as expected
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The energy-momentum four-vector is:
Recall, we defined the space-time translation operator
so that
we can easily verify it; for an infinitesimal transformation it becomes:
it is straightforward to verify this by using the canonical commutation relations for and .
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The same procedure can be repeated for Lorentz transformations:
the resulting conserved current is:
antisymmetric in the last two indices as a result of
being antisymmetric
the conserved charges associated with this current are:
generators of the Lorentz group
again, one can check all the commutators...
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