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:

based on S-67,68

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A comment on functional derivative:

!"b(y) = !ba!4(y ! x)!"a(x)

!"b(y) =!

a

"dx!ba!4(y ! x)!"a(x)

!F ("a(x))!"b(y)

=#F

#"a(x)!"a(x)!"b(y)

=#F

#"a(x)!ba!4(y ! x)

!F ("a(x)) =!

b

"dy4 #F

#"a(x)!ba!4(y ! x)!"b(y) =

#F

#"a(x)!"a(x)

Similarly:

or, for the action:

!S =!

dx4 !S

!"a(x)!"a(x)

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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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Ward identities in QEDbased on S-67,68

When discussing QED we used the result that a scattering amplitude for a process that includes an external photon with momentum should satisfy (as a result of Ward identity):

now we are going to prove it!

we used it, for example, to obtain a simple formula for the photon polarization sum that we needed for calculations of cross sections:

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To simplify the discussion let’s treat all particles in the LSZ formula as outgoing (incoming particles have ):

it can be also written as:

we do not fix

must include an overall energy-momentum delta function

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(for simplicity we work with scalar fields, but the same applies to fields of any spin)

near we can write:

multivariable pole

residue of the pole

contributions that do not have this multivariable pole do not contribute to !

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near we can write:

multivariable pole

residue of the pole

this means that in Schwinger-Dyson equations:

contributions that do not have this multivariable pole do not contribute to !

contact terms in a correlation function F do not contribute to the scattering amplitude!

classical eq. of motion

contact terms:

in the momentum space a contact term is a function of ; it doesn’t have the

right pole structure to contribute!

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Let’s consider a scattering process in QED with an external photon:with momentum .

the LSZ formula:

the classical equation of motion in the Lorentz gauge:

thus we find:

contact terms cannot generate the proper singularities and thus these do not contribute to the scattering amplitude!

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replace by

integrate by parts

now we use Ward-Takahashi identity:

contact terms do not contribute to the scattering amplitude!

and thus we find:

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Finally, we will derive another consequence of Ward identity that we used:

consider the correlation function:

adds a vertex

in the momentum space:

exact propagator

exact 1PI vertex functionlater we will use:

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integrate by parts

The Ward identity:

becomes:

multiply by

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before we found:

thus we get:

or:

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finite

in the scheme

since the divergent pieces must cancel, and there are only divergent pieces

In the OS scheme, near , , we have:

and so we get !

The Ward identity means that the kinetic term and the interaction term are renormalized in the same way and so they can be combined into covariant derivative term (which we could have guessed from gauge invariance)!

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Formal development of fermionic PIbased on S-44

We want to derive the fermionic path integral formula (that we previously postulated by analogy with the path integral for a scalar field):

Feynman propagatorinverse of the Dirac wave operator

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Let’s define a set of anticommuting numbers or Grassmann variables:

for we have just one number with .

We define a function of by a Taylor expansion:

the order is important!if f itself is commuting then b has to be an anticommuting number:

and we have:

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Let’s define the left derivative of with respect to as:

Similarly, let’s define the left derivative of with respect to as:

We define the definite integral with the same properties as those of an integral over a real variable; namely linearity and invariance under shifts:

The only possible nontrivial definition (up to an overall numerical factor) is:

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Let’s generalize this to , we have:

all indices summed overcompletely antisymmetric on exchange of any two indices

Let’s define the left derivative of with respect to as:

and similarly for the right derivative...

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To define (linear and shift invariant) integral note that:

just a number(if n is even)

Levi-Civita symbol,

the only consistent definition of the integral is:

alternatively we could write the differential in terms of individual differentials:

and use

to derive the result above.

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Consider a linear change of variable:

then we have:

matrix of commuting numbers

integrating over we get and thus:

Recall, for integrals over real numbers with we have:

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We are interested in gaussian integrals of the form:

for we have:

expanding the exponential:

antisymmetric matrix of (complex) commuting numbers

we find:

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For larger (even) n we can bring a complex antisymmetric matrix to a block-diagonal form:

a unitary matrix

taking:

we have:

represents 2x2 blocks

we drop primes

we will later need:

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using the result for n= 2 we get:

we finally get:

Recall, for integrals over real numbers we have:

a complex symmetric matrix

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also since we have:

Let’s define complex Grassmann variables:

we can invert this to get:

thus we have:determinant =

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A function can be again defined by a Taylor expansion:

the integral is:

in particular:

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Let’s consider n complex Grassmann variables and their conjugates:

define:

under a change of variable:

we have

not important

(the integral doesn’t care whether is the complex conjugate of )

we want to evaluate:

a general complex matrixcan be brought to a diagonal form with all entries positive by a bi-unitary transformation

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under such a change of variable we get:

positive

we drop primes

Analogous integral for commuting complex variable

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using shift invariance of integrals:

we get:

generalization for continuous spacetime argument and spin index

the determinant does not depend on fields or sources and can be absorbed into the overall normalization of the path integral

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Functional determinantsbased on S-53

We are going to discuss situations where a functional determinant depends on some other field and so it cannot be absorbed into the overall normalization of the path integral.

Consider a theory of a complex scalar field:

a real scalar background field(a fixed function of spacetime)

we define the path integral:

we fix

for the gaussian of n complex variables we found:

in the continuum limit the “matrix” in our case is:

we need to calculate the determinant of this “matrix”

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we can write it as a product of two matrices:

where

now we can calculate the determinant:

independent of the background field(can be absorbed into the overall normalization)

thus:

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the path integral is given by:

using we get:

thus we find:

where

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we found:

where

It represents a connected diagram with n insertions of a vertex:

cyclic symmetry

and the path integral is given by

sum of all connected diagrams

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Consider now a theory of a Dirac field:

a real scalar background field(a fixed function of spacetime)

we define the path integral:

we fix

for the gaussian of n complex Grassmann variables we found:

in the continuum limit the “matrix” in this case is:

we need to calculate the determinant of this “matrix”

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we can write it as a product of two matrices:

where

now we can calculate the determinant:

independent of the background field(can be absorbed into the overall normalization)

thus:

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the path integral is given by:

using we get:

thus we find:

where

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we found:

where

It represents a connected diagram with n insertions of a vertex:

cyclic symmetry

and the path integral is given by

sum of all connected diagrams

Note the trace and

-1 for fermion loop!

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