Formal development of fermionic PI
based 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 propagator
inverse 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
We define a function of
with
.
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
Similarly, let’s define the left derivative of
as:
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 over
Let’s define the left derivative of
completely antisymmetric on
exchange of any two indices
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:
matrix of commuting numbers
then we have:
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
antisymmetric
matrix of (complex)
commuting numbers
we have:
expanding the exponential:
we find:
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For larger (even) n we can bring a complex antisymmetric matrix to a
block-diagonal form:
we will later need:
a unitary matrix
taking:
we have:
represents 2x2 blocks
we drop primes
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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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Let’s define complex Grassmann variables:
we can invert this to get:
thus we have:
also since
determinant =
we have:
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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:
(the integral doesn’t care whether
not important
is the complex conjugate of
)
we have
we want to evaluate:
a general complex matrix
can 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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