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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