158
16. Lattice Field Theory, and the Renormalization Group Revisited
The symmetry under (16.26)–(16.27), which is proportional to the infinitesimal version of (12.152) as a → 0, provides a lattice theory with all the fundamental symmetry properties of continuum chiral gauge theories (Hasenfratz
et al. 1998). Finding an operator which satisfies (16.25) is, however, not so
easy – but that problem has now been solved, indeed in three different ways:
Kaplan’s ‘domain wall’ fermions (Kaplan 1992); ‘classically perfect fermions’
(Hasenfratz and Niedermayer 1994); and overlap fermions (Narayanan and
Neuberger 1993a, b, 1994, 1995). Unfortunately all these proposals are computationally more expensive than the Wilson or staggered fermion alternatives.
16.2.3 Gauge fields
Having explored the discretization of actions for free scalars and Dirac fermions,
we must now think about how to implement gauge invariance on the lattice. In
the usual (continuum) case, we saw in chapter 13 how this was implemented by
replacing ordinary derivatives by covariant derivatives, the geometrical significance of which (in terms of parallel transport) is discussed in appendix N. It
is very instructive to see how the same ideas arise naturally in the lattice case.
We illustrate the idea in the simple case of the Abelian U(1) theory, QED.
Consider, for example, a charged scalar field φ(x), with charge e. To construct
a gauge-invariant current, for example, we replaced φ
† ∂ μ φ by φ
† (∂ μ + ieA μ )φ,
so we ask: what is the discrete analogue of this? The term φ
† (x)
∂ φ(x)
∂x
becomes, as we have seen,
φ
† (n 1 )
1 [φ(n 1 + 1) − φ(n 1 )a]
(16.28)
a
in one dimension. We do not expect (16.28) by itself to be gauge invariant,
and it is easy to check that it is not. Under a gauge transformation for the
continuous case, we have
dθ(x)
φ(x) → e
ieθ(x) φ(x), A(x) → A(x) + dx
;
(16.29)
then φ
† (x)φ(y) transforms by
φ
† (x)φ(y) → e
−ie[θ(x)−θ(y)] φ
† (x)φ(y),
(16.30)
and is clearly not invariant. The essential reason is that this operator involves
the fields at two different points, and so the term φ
† (n 1 )φ(n 1 + 1) in (16.28)
will not be gauge invariant either. The discussion in appendix N prepares us
for this: we are trying to compare two ‘vectors’ (here, fields) at two different
points, when the ‘coordinate axes’ are changing as we move about. We need
to parallel transport one field to the same point as the other, before they can
be properly compared. The solution (N.18) shows us how to do this. Consider
the quantity
∫ x
O(x, y) = φ
† (x)exp[ie
Adx
′ ]φ(y).
(16.31)
y
16. Lattice Field Theory, and the Renormalization Group Revisited
The symmetry under (16.26)–(16.27), which is proportional to the infinitesimal version of (12.152) as a → 0, provides a lattice theory with all the fundamental symmetry properties of continuum chiral gauge theories (Hasenfratz
et al. 1998). Finding an operator which satisfies (16.25) is, however, not so
easy – but that problem has now been solved, indeed in three different ways:
Kaplan’s ‘domain wall’ fermions (Kaplan 1992); ‘classically perfect fermions’
(Hasenfratz and Niedermayer 1994); and overlap fermions (Narayanan and
Neuberger 1993a, b, 1994, 1995). Unfortunately all these proposals are computationally more expensive than the Wilson or staggered fermion alternatives.
16.2.3 Gauge fields
Having explored the discretization of actions for free scalars and Dirac fermions,
we must now think about how to implement gauge invariance on the lattice. In
the usual (continuum) case, we saw in chapter 13 how this was implemented by
replacing ordinary derivatives by covariant derivatives, the geometrical significance of which (in terms of parallel transport) is discussed in appendix N. It
is very instructive to see how the same ideas arise naturally in the lattice case.
We illustrate the idea in the simple case of the Abelian U(1) theory, QED.
Consider, for example, a charged scalar field φ(x), with charge e. To construct
a gauge-invariant current, for example, we replaced φ
† ∂ μ φ by φ
† (∂ μ + ieA μ )φ,
so we ask: what is the discrete analogue of this? The term φ
† (x)
∂ φ(x)
∂x
becomes, as we have seen,
φ
† (n 1 )
1 [φ(n 1 + 1) − φ(n 1 )a]
(16.28)
a
in one dimension. We do not expect (16.28) by itself to be gauge invariant,
and it is easy to check that it is not. Under a gauge transformation for the
continuous case, we have
dθ(x)
φ(x) → e
ieθ(x) φ(x), A(x) → A(x) + dx
;
(16.29)
then φ
† (x)φ(y) transforms by
φ
† (x)φ(y) → e
−ie[θ(x)−θ(y)] φ
† (x)φ(y),
(16.30)
and is clearly not invariant. The essential reason is that this operator involves
the fields at two different points, and so the term φ
† (n 1 )φ(n 1 + 1) in (16.28)
will not be gauge invariant either. The discussion in appendix N prepares us
for this: we are trying to compare two ‘vectors’ (here, fields) at two different
points, when the ‘coordinate axes’ are changing as we move about. We need
to parallel transport one field to the same point as the other, before they can
be properly compared. The solution (N.18) shows us how to do this. Consider
the quantity
∫ x
O(x, y) = φ
† (x)exp[ie
Adx
′ ]φ(y).
(16.31)
y