169
16.3. Representation of quantum amplitudes
some constraint, such as the Lorentz gauge condition. Such constraints can
be imposed on the corresponding path integral, and indeed this was the route
followed by Faddeev and Popov (1967) in first obtaining the Feynman rules
for non-Abelian gauge theories, as mentioned in section 13.5.3.
In the discrete case, the appropriate integration variables are the link variables U (l i ) where l i is the i
th link. They are elements of the relevant gauge
group – for example U (n 1 , n 1 + 1) of (16.3.1) is an element of U (1). In the
case of the unitary groups, such elements typically have the form (cf (12.35))
∼ exp(i Hermitean matrix), where the ‘Hermitean matrix’ can be parametrized
in some convenient way – for example, as in (12.31) for SU (2). In all these
cases, the variables in the parametrization of U vary over some bounded domain (they are essentially ‘angle-type’ variables, as in the simple U (1) case),
and so, with a finite number of lattice points, the integral over the link variables is well-defined without gauge-fixing. The integration measure for the
link variables can be chosen so as to be gauge invariant, and hence provided
the action is gauge invariant, the formalism provides well-defined expressions,
independently of perturbation theory, for vevs of gauge invariant quantities.
There remains one more conceptual problem to be addressed in this approach: namely, how are we to deal with fermions? It seems that we must
introduce new variables which, though not quantum field operators, must
nevertheless anticommute with each other. Such ‘classical’ anticommuting
variables are called Grassmann variables, and are briefly described in appendix P. Further details are contained in Ryder (1985) and in Peskin and
Schroeder (1995) section 9.5). For our purposes, the important point is that
the fermion Lagrangian is bilinear in the (Grassmann) fermion fields ψ, the
fermionic action for one flavour having the form
∫
d
4
¯
S ψ f =
x E ψ f M f (U )ψ f ,
(16.89)
where M f is a matrix representing the Dirac operator i /
D−m f in its discretized
and Euclideanized form. This means that in a typical fermionic amplitude of
the form (cf the denominator of (16.87))
∫
Z ψ f = Dψ ¯ f Dψ f exp[−S ψ f ] ,
(16.90)
one has essentially an integral of Gaussian type (albeit with Grassmann variables), which can actually be performed analytically
2 . The result is simply
det [M f (U )], the determinant of the Dirac operator matrix. For N f flavours,
this easily generalizes to
N f
∏
detM f (U ).
(16.91)
f=1
2 See appendix P.
16.3. Representation of quantum amplitudes
some constraint, such as the Lorentz gauge condition. Such constraints can
be imposed on the corresponding path integral, and indeed this was the route
followed by Faddeev and Popov (1967) in first obtaining the Feynman rules
for non-Abelian gauge theories, as mentioned in section 13.5.3.
In the discrete case, the appropriate integration variables are the link variables U (l i ) where l i is the i
th link. They are elements of the relevant gauge
group – for example U (n 1 , n 1 + 1) of (16.3.1) is an element of U (1). In the
case of the unitary groups, such elements typically have the form (cf (12.35))
∼ exp(i Hermitean matrix), where the ‘Hermitean matrix’ can be parametrized
in some convenient way – for example, as in (12.31) for SU (2). In all these
cases, the variables in the parametrization of U vary over some bounded domain (they are essentially ‘angle-type’ variables, as in the simple U (1) case),
and so, with a finite number of lattice points, the integral over the link variables is well-defined without gauge-fixing. The integration measure for the
link variables can be chosen so as to be gauge invariant, and hence provided
the action is gauge invariant, the formalism provides well-defined expressions,
independently of perturbation theory, for vevs of gauge invariant quantities.
There remains one more conceptual problem to be addressed in this approach: namely, how are we to deal with fermions? It seems that we must
introduce new variables which, though not quantum field operators, must
nevertheless anticommute with each other. Such ‘classical’ anticommuting
variables are called Grassmann variables, and are briefly described in appendix P. Further details are contained in Ryder (1985) and in Peskin and
Schroeder (1995) section 9.5). For our purposes, the important point is that
the fermion Lagrangian is bilinear in the (Grassmann) fermion fields ψ, the
fermionic action for one flavour having the form
∫
d
4
¯
S ψ f =
x E ψ f M f (U )ψ f ,
(16.89)
where M f is a matrix representing the Dirac operator i /
D−m f in its discretized
and Euclideanized form. This means that in a typical fermionic amplitude of
the form (cf the denominator of (16.87))
∫
Z ψ f = Dψ ¯ f Dψ f exp[−S ψ f ] ,
(16.90)
one has essentially an integral of Gaussian type (albeit with Grassmann variables), which can actually be performed analytically
2 . The result is simply
det [M f (U )], the determinant of the Dirac operator matrix. For N f flavours,
this easily generalizes to
N f
∏
detM f (U ).
(16.91)
f=1
2 See appendix P.