P
Grassmann Variables
In the path integral representation of quantum amplitudes (chapter 16) the
fields are regarded as classical functions. Matrix elements of time-ordered
products of bosonic operators could be satisfactorily represented (see the discussion following (16.79)). But something new is needed to represent, for
example, the time-ordered product of two fermionic operators: there must
be a sign difference between the two orderings, since the fermionic operators
anticommute. Thus it seems that to represent amplitudes involving fermionic
operators by path integrals we must think in terms of ‘classical’ anticommuting variables.
Fortunately, the necessary mathematics was developed by Grassmann in
1855, and applied to quantum amplitudes by Berezin (1966). Any two Grassmann numbers θ 1 , θ 2 satisfy the fundamental relation
θ 1 θ 2 + θ 2 θ 1 = 0,
(P.1)
and of course
θ
2
θ
2
= 2 = 0.
(P.2)
1
Grassmann numbers can be added and subtracted in the ordinary way, and
muliplied by ordinary numbers. For our application, the essential thing we
need to be able to do with Grassmann numbers is to integrate over them.
It is natural to think that, as with ordinary numbers and functions, integration would be some kind of inverse of differentiation. So let us begin with
differentiation.
We define
∂(aθ) = a,
(P.3)
∂θ
where a is any ordinary number, and
∂ (θ 1 θ 2 ) = θ 2 ;
( P . 4 )
∂θ 1
then necessarily
∂ (θ 1 θ 2 ) = −θ 1 .
(P.5)
∂θ 2
Consider now a function of one such variable, f (θ). An expansion of f in
powers of θ terminates after only two terms because of the property (P.2):
f (θ) = a + bθ.
(P.6)
467
Grassmann Variables
In the path integral representation of quantum amplitudes (chapter 16) the
fields are regarded as classical functions. Matrix elements of time-ordered
products of bosonic operators could be satisfactorily represented (see the discussion following (16.79)). But something new is needed to represent, for
example, the time-ordered product of two fermionic operators: there must
be a sign difference between the two orderings, since the fermionic operators
anticommute. Thus it seems that to represent amplitudes involving fermionic
operators by path integrals we must think in terms of ‘classical’ anticommuting variables.
Fortunately, the necessary mathematics was developed by Grassmann in
1855, and applied to quantum amplitudes by Berezin (1966). Any two Grassmann numbers θ 1 , θ 2 satisfy the fundamental relation
θ 1 θ 2 + θ 2 θ 1 = 0,
(P.1)
and of course
θ
2
θ
2
= 2 = 0.
(P.2)
1
Grassmann numbers can be added and subtracted in the ordinary way, and
muliplied by ordinary numbers. For our application, the essential thing we
need to be able to do with Grassmann numbers is to integrate over them.
It is natural to think that, as with ordinary numbers and functions, integration would be some kind of inverse of differentiation. So let us begin with
differentiation.
We define
∂(aθ) = a,
(P.3)
∂θ
where a is any ordinary number, and
∂ (θ 1 θ 2 ) = θ 2 ;
( P . 4 )
∂θ 1
then necessarily
∂ (θ 1 θ 2 ) = −θ 1 .
(P.5)
∂θ 2
Consider now a function of one such variable, f (θ). An expansion of f in
powers of θ terminates after only two terms because of the property (P.2):
f (θ) = a + bθ.
(P.6)
467