\chapter{Quantum field theory and lattice QCD}

\section{Basic philosophical idea}
\label{sec:philosophy}

Quantum field theory claims to describe the basic constitution of the
world, providing the principles underlying all matter and forces ---
the `basic matter' and `basic force'.  Although its current
incarnation, the Standard Model, is not a final theory (it must be
supplemented with general relativity, for one thing), most improvements
and suggestions for further developments of the Standard Model are
formulated within the framework of quantum field theory.  The entities
it concerns itself with are remote from ordinary life --- their typical
scales are 15 levels of magnitude down, at the subatomic and subnuclear
level, and most of them are only `observed' in big, complex detectors
in big accelerators.  Much of lattice QCD concerns itself with bridging
the first part of the gap with the macroscopic world -- constructing
hadrons and hadronic physics from the world of quark and gluon fields.

The essence of the world as described by quantum field theory has some
notable features, distinguishing it from previous fundamental theories,
which describe the world at more `normal' scales.

Matter and force are treated on the same footing, so the `basic matter'
and the `basic force' are much the same kind of entity.  There is a
rough identification of matter with fermions and of force with (gauge)
bosons, but even that is no longer the case in supersymmetric theories.

The basic entities of the theory are fields, derived from a
mathematical language originally used for describing and explaining
forces.  More precisely, the basic entities are operator fields, which
create and destroy particle --- `matter' --- states.  This may be the
first time in fundamental physics since Aristotle that forces have
taken priority over matter.  

Rather than individual particles, it is the species of particles that is
considered primary, since each field corresponds to one particle species
--- undermining the individuality that has traditionally been seen as a
defining property of matter.

The entities that are considered fundamental by the theory are not those
that can be `observed' by experiment.  Since it is not possible to
separate a particle or field from its interactions, the fields and
parameters must be renormalised so that everything is described in terms
of the `observable' fields which are dressed with self-interactions. 
Much of this thesis concerns itself with this aspect of the theory.

%The condition of microcausality leads to the spin--statistics theorem,
%which states that all scalar, vector or tensor fields (with integer
%spin) commute and therefore have bosonic statistics, while spinor fields
%(with half-integer spin) anticommute, leading to fermionic statistics.

\section{Functional integral formalism}
\label{sec:func-integral}

The dynamics of the fields (and the corresponding particles) are
determined by the action $S$, which is a functional of the fields.  The
action must satisfy the conditions of locality, microcausality and
Poincar\'e invariance.  Specifically, the requirement of locality
implies that the action can be written as an integral $S = \int d^4 x
{\cal{L}}(x)$ of the Lagrangian $\cal{L}$, which is a function of the
fields and their derivatives. 

There are two formalisms commonly used for quantising field theories. 
The canonical operator formalism treats the fields as operators which
create and destroy particles, and imposes commutation relations between
them, like the equal time commutator between fields and canonical momenta, or
Heisenberg's equation of motion.  The functional integral formalism
treats the fields as ordinary variables (or Grassmann variables in the
case of fermionic fields) with a weight for each field configuration
$[\Phi]$ given by $\exp(iS[\Phi])$.  By
integrating over all possible field configurations one can obtain the
Green functions or vacuum expectation values of any product of field
variables 
%
\begin{eqnarray}
\langle \Phi(x_1)\Phi(x_2)\cdots\Phi(x_n)\rangle & \equiv &
\langle 0|T(\Phi(x_1)\Phi(x_2)\cdots\Phi(x_n)|0\rangle \nonumber \\
 & = &\int{\cal{D}}\Phi \Phi(x_1)\Phi(x_2)\cdots\Phi(x_n) e^{iS[\Phi]}
\end{eqnarray}
%
where $\Phi$ refers to any field --- scalar, spinor, vector or tensor
--- and $x_1,x_2,\ldots$ are general coordinates, referring to both
space--time and internal indices.

The Green functions contain all the physical information of the theory.
 In particular, they are related to transition amplitudes between
external $n$-particle states through the Lehmann--Symanzik--Zimmermann
reduction formula
%
\begin{eqnarray}
\lefteqn{\langle p_1 p_2 \ldots p_n, {\rm out}| q_1 q_2\ldots q_m,{\rm
in}\rangle} \nonumber \\
 & = & (i Z_\Phi^{-1/2})^{n+m+1} \int d^4y_1\cdots d^4x_m \exp(\sum ip_k y_k
-\sum iq_k x_k) \nonumber\\
 & \times & (\Box_{y_1}+m^2)\cdots(\Box_{x_m}+m^2)
\langle\Phi(y_1)\cdots\Phi(x_m)\rangle \\
 & + & \mbox{ disconnected terms}\nonumber
\end{eqnarray}
%
For fermionic fields, $(\Box+m^2)$ is replaced by the fermion operator
$(i\dslash+m)$. 

The Green functions can all be expressed in terms of the generating
functional
%
\beq
Z[J] = \int{\cal D}\Phi e^{iS[\Phi] + i\int d^4x J(x)\cdot \Phi(x)}
\eeq
%
which gives
\begin{eqnarray}
{\cal G}^{(1)}(x) \equiv \langle\Phi(x)\rangle & = & \left.
\frac{\delta}{i\delta J(x)} Z[J]\right |_{J=0} \\
{\cal G}^{(2)}(x_1,x_2) \equiv \langle\Phi(x_1)\Phi(x_2)\rangle & =
& \left.\frac{\delta}{i\delta J(x_1)}\frac{\delta}{i\delta J(x_2)}
Z[J] \right |_{J=0} \\
 & {\rm etc.} & \nonumber
\end{eqnarray}

%${\cal G}_C^{(n)}(x_1,\ldots,x_n)$
We are normally interested only in the {\em connected} Green functions,
which can be derived from the generating functional $W[J]$ where
\beq Z[J] = e^{W[J]} \eeq

The one-particle irreducible (proper) Green functions
$\Gamma^{(n)}(x_1,\ldots,x_n)$ can be defined in terms of their
corresponding Feynman diagrams:  They are truncated (all external
propagator legs are removed) and cannot be divided into subdiagrams by
cutting one internal line.  All Feynman diagrams and Green functions can
be written in terms of proper functions.  They can also be derived from
the effective action $\Gamma[\Phi]$, which is arrived at by a Legendre
transformation of $W[J]$
%
\beq
i\Gamma[\Phi] = W[J] - i\int d^4x J(x)\Phi(x)
\eeq
where
\begin{eqnarray}
\Phi(x) & = & \frac{\delta}{i\delta J(x)}W[J] \\
\Rightarrow J(x) & = & -\frac{\delta\Gamma}{\delta\Phi(x)}
\end{eqnarray}
We then have
\beq
\Gamma^{(n)}(x_1,\ldots,x_n) = \left. \frac{\delta}{\delta\Phi(x_1)}\cdots
\frac{\delta}{\delta\Phi(x_n)}\Gamma[\Phi] \right|_{\Phi=\Phi_c}
\label{eq:properfuncs}
\eeq
where \( \Phi_c=\bra\Phi\ket \) are the classical fields.
These proper functions are the building blocks of perturbation theory
and the renormalisation programme.

\section{Gauge theories}

\subsection{The gauge principle}
\label{sec:gauge-principle}

In addition to the external space--time degrees of freedom (including
Dirac and Lorentz indices for spinors and vectors/tensors), the matter
fields have internal degrees of freedom, which can be expressed as one
or more complex phases $\Phi = \Phi_0 e^{-igt^a\theta^a}$.  The physics of
the system should be independent of how these phases are chosen, so it
should be invariant under local changes of phase, or local gauge
transformations.  This leads to the introduction of gauge fields
$A_\mu^a$ and covariant derivatives $D_\mu = \partial_\mu + igt^a
A_\mu^a$.  The transformation of the matter fields 
\beq
\Phi(x) \rightarrow e^{-igt^a\theta^a(x)}\Phi(x)
\eeq
is compensated by the transformation of the gauge fields
\beq
A_\mu(x) \rightarrow
e^{-igt^a\theta^a(x)}(A_\mu(x)-\frac{i}{g}\partial_\mu)e^{igt^a\theta^a(x)}, 
\eeq
where $A_\mu(x) = t^aA_\mu^a(x)$.  Then the covariant derivative of the
matter field will transform in the same way as the matter field itself:
%
\beq
D_\mu\Phi(x) \to e^{-igt^a\theta^a(x)}D_\mu\Phi(x)
\eeq

Alternatively, this can be understood as parallel transport in a general
coordinate space.  In this picture, the matter fields are vectors (fibre
bundles) in the curved space of gauge coordinates.  The connection
between two adjacent fibre bundles is provided by the gauge field
$A_\mu$ --- when a vector is parallel transported from $x$ to $x+dx$, it
picks up a phase \(igA_\mu(x)dx_\mu\):
\beq
\Phi(x+dx) = \Phi(x) + igA_\mu(x)\Phi(x)dx_\mu
\eeq
or, non-infinitesimally,
\beq
\Phi(x') = {\cal P}e^{ig\int_x^{x'}A_\mu(z)dz_\mu}\Phi(x)
\label{parallel-transport} 
\eeq
where path-ordering $\cal P$ means that the later points in the path
come to the left.  In this picture, gauge invariance can be seen to be
an expression of translation invariance in this space.

The dynamics of the gauge fields themselves is given by the Yang--Mills
action 
%
\beq
\lagr^{YM} = - \frac{1}{4}\Tr F_{\mu\nu}F^{\mu\nu} \label{eq:YM-def}
\eeq
where $F_{\mu\nu} = -\frac{i}{g}[D_\mu,D_\nu]$ is the field tensor (curvature
tensor, in the geometric picture), which transforms as
\beq
F_{\mu\nu}(x) \to e^{-igt^a\theta^a(x)}F_{\mu\nu}(x)e^{igt^a\theta^a(x)},
\eeq
so $\lagr^{YM}$ is gauge invariant, as it should be.

\subsection{Quantising gauge theories}
\label{sec:gauge-quantise}

The extra, unphysical gauge degrees of freedom lead to problems when one
tries to quantise gauge theories, since the naive path integral will
contain an infinite number of contributions from gauge equivalent
configurations.  This manifests itself in the propagator
$D^{ab}_{\mu\nu}(x-y) = \langle A^a_\mu(x) A^b_\nu(y)\rangle$ being
ill-defined, or its inverse $\Gamma^{AA}(x,y)$ singular.\footnote{This
is only really a problem when one tries to perform a saddle point
approximation to the functional integral, ie in perturbation theory.  If
we instead, as in lattice simulations, evaluate the functional integral
directly, the problem does not arise.}  To avoid this, we factor out the
integration over the gauge group by inserting
%
\beq
1 = \Delta_{FP}[A_\mu(x)] \int{\cal D}g(x) \delta(F[A^g_\mu(x)]),
\label{eq:FP-def}
\eeq
%
where $F$ is a gauge non-invariant functional of $A_\mu$, into the path
integral.  After reordering, the full partition function for a pure
gauge theory becomes
%
\beq
Z = \int\cald g\int\cald A \Delta_{FP}[A]\delta(F[A]) e^{iS[A]}
\eeq

Both the gauge fixing term $\delta(F[A])$ and the Faddeev--Popov term
$\Delta_{FP}[A]$ can be expressed as exponentials of a local action.  In
particular, for a covariant gauge, $F^a(x) = \partial^\mu A^a_\mu(x) -
B^a(x)$, we can write
%
\beq
\int\cald B(x) e^{-\frac{i}{2\xi}\int d^4 x
B^2(x)}\prod_{x,a}\delta(F^a(x)) = e^{-\frac{i}{2\xi}\int 
d^4 x(\partial^\mu A^a_\mu(x))^2} \label{eq:covgauge}
\eeq
while
\beq
\Delta_{FP}[A] \equiv \det\frac{\delta F}{\delta\alpha} = \det{\cal M} =
\int\cald\bar{\eta}\cald\eta e^{-i\int
d^4xd^4y\bar{\eta}_a(y)\frac{\delta F^a(y)}{\delta\alpha_b(x)}\eta_b(x)}
\eeq
where $\alpha^a(x)$ is the parameter for an infinitesimal gauge
transformation, and $\bar{\eta}, \eta$ are scalar anticommuting
variables or ghosts.  For the covariant gauge, this can be rewritten
%
\beqa
\delta F^a(x) & = & \partial^{\mu}(\delta A_\mu)^a(x) = 
\partial^{\mu}D_{\mu}^{ab}\delta\alpha^b(x) \nonumber \\
\Rightarrow \frac{\delta F^a}{\delta\alpha_b} & = &
\partial^{\mu}D_{\mu}^{ab}
\eeqa

Including $N_f$ flavours of fermion fields, the full Lagrangian for a
gauge theory like QCD in a covariant gauge is
\begin{eqnarray}
\lagr(x) & = & \lagr^{YM}(x) + \lagr^{F}(x) + \lagr^{GF}(x) +
\lagr^{FP}(x) \label{eq:QCD} \\
\mbox{where} \nonumber \\
\lagr^{F}(x) & = & \psibar(x)(i\Dslash - m)\psi(x)
\label{eq:fermi-lagr} \\
\lagr^{GF}(x) & = & -\frac{1}{2\xi} (\partial^\mu A_\mu)^2
\label{eq:gfix-lagr} \\
\lagr^{FP}(x) & = & \partial_\mu\bar{\eta}(x)D^\mu\eta(x)
\label{eq:fp-lagr} 
\end{eqnarray}
and $\lagr^{YM}$ is given by (\ref{eq:YM-def})

\section{Euclidean quantum field theory}
\label{sec:euclid}

The functional integral in Minkowski space is not well-defined and is
ill-suited for numerical studies.  Therefore, most numerical studies,
both those using Dyson--Schwinger equations and lattice simulations, are
carried out in Euclidean space.

The Euclidean action and functional integral are obtained from the
Minkowski space equivalents by the substitutions
%
\begin{eqnarray}
\int d^4x & \rightarrow & -i \int d^4x^E \label{eq:eucl1} \\
\gamma^\mu\partial_\mu & \rightarrow & i\gamma^E\cdot\partial^E \\
\gamma^\mu A_\mu & \rightarrow & -i\gamma^E\cdot A^E \\
A^\mu B_\mu & \rightarrow & -A^E\cdot B^E
\end{eqnarray}
%
where $a\cdot b = \sum_{i=1}^4 a_i b_i$, and the $\gamma^E_\mu$'s satisfy
\beq
\{\gamma_\mu^E,\gamma_\nu^E\} = 2\delta_{\mu\nu}
\mbox{\hspace{1cm}and\hspace{1cm}} 
\gamma_5^E = -\gamma_1^E\gamma_2^E\gamma_3^E\gamma_4^E \label{eq:eucl5}
\eeq

Under this transformation, the proper Green functions are directly
carried over to the Euclidean space, while the full, connected Green
functions $G_C^{(n)}$ pick up a prefactor $i (-i)^n$.

This is equivalent to an analytic continuation (Wick rotation) $x^0
\rightarrow -ix_4$ of the time variable in the complex plane.  This
continuation is valid provided the Green functions have no singularities
in the third and first quadrant of the complex plane.  This is usually
true in perturbation theory, but not necessarily non-perturbatively. 
This problem may be overcome by {\em defining} the theory in Euclidean
space and solving for the Euclidean position-space Green functions
(Schwinger functions).  These may then be continued back to Minkowski
space -- though quantities like masses and decay constants are invariant
under the Wick rotation, so the analytic continuation to Minkowski space
is not needed if this is all we are interested in.

A necessary condition for the Euclidean theory to be well-defined (or
the analytic continuation to Minkowski space to be possible) is
reflection positivity, which means that
\beq
\langle(\Theta F)F\rangle \geq 0
\eeq
where 
\beq
\Theta F(\vec{x},t) = \bar{F(\vec{x},-t)}
\eeq
for any functional $F$ which depends only on the fields at positive
times.  This is specifically a statement about Schwinger functions, and
is the Euclidean-space version of the spectral condition and requirement
that the scalar product in Hilbert space is positive.

The Euclidean path integral is
\beq
  Z = \int\cald\Phi^E e^{-S^E[\Phi^E]}
\eeq
where $S^E[\Phi^E]$ is arrived at using the transcription rules
(\ref{eq:eucl1})--(\ref{eq:eucl5}), and is real.  The theory thus takes
the form of a statistical mechanics problem, with a weight
$e^{-S^E[\Phi]}$ given to each configuration $[\Phi]$, and lends itself
to simulations using Monte Carlo methods.  Hereafter, everything will be
in Euclidean space unless explicitly stated otherwise, and the
superscript ${}^E$ will be dropped.

\section{Lattice QCD}
\label{sec:lat-qcd}

In order to simulate the theory on a computer, with a finite number of
variables, we discretise space--time to form a finite lattice,
replacing $\int dx$ with $a\sum_x$ and derivatives with finite
differences. The lattice fields $\Phi^L(x)$ are taken to be some
average of the continuum fields over the space near $x$.

\subsection{Lattice gauge fields}
\label{sec:lat-gauge}

The gauge fields are represented by link fields $U_\mu(x)$
which belong to the gauge group $G$ itself, rather than by the
fields $A_\mu(x)$ which belong to the gauge algebra.  The
link fields are the parallel transporters
%
\beq
U(x,x+\muhat) = U_\mu(x) = e^{-igaA_\mu(x+\hat{\mu}/2)} 
\approx e^{ig\int^{x}_{x+\hat{\mu}} A_\mu(x) dx}
\eeq
which transform a matter field at $x+\muhat$ to one at $x$, cf.\ section
\ref{sec:gauge-principle}. 
Under a local gauge transformation $\Lambda(x)$ these will transform as
\beq
 U^\Lambda_\mu(x) = \Lambda(x)U_\mu(x)\Lambda^{-1}(x+\muhat)
\label{eq:gtrans-gauge} 
\eeq

The continuum functional integral $\int\cald A_\mu$ is replaced by
\beq
\int\cald U = \prod_{x,\mu}\int dU_\mu(x)  \equiv 
\prod_{\stackrel{\rm links}{b}} 
\int dU(b)
\eeq

This is finite and well-defined for any compact group if for $dU$ we use
the invariant group measure (Haar measure), which obeys
\beq
\int_G f(U)dU = \int_G f(VU)dU = \int_G f(UV)dU \hspace{1cm}
\forall\, V \in G
\eeq
\beq
\int_G dU = 1
\eeq

Since the functional integral is already well-defined, there is no need
to fix the gauge if we want to compute gauge invariant quantities.

The gauge invariant quantities that can be constructed in this theory
are
\[ \Phi^\dagger(x)U({\cal C}_{x,y})\Phi(y) \mbox{\hspace{1cm} and \hspace{1cm}}
U({\cal C}_{x,x}) \]
%
where $\Phi$ is a matter (scalar or spinor) field which transforms as
\beq 
\Phi^\Lambda = \Lambda\Phi \hspace{2cm} (\Phi^\dagger)^\Lambda =
\Phi^\dagger\Lambda^{-1} \label{eq:gtrans-matter}
\eeq 
and \(U({\cal C}_{x,y})\) is a parallel transporter 
\(U(x,z_1)U(z_1,z_2)\cdots U(z_n,y)\) along any path 
\({\cal C} = (x,z_1,z_2,\ldots,z_n,y)\) joining
$x$ and $y$.  The covariant forward derivative can be constructed as
\beq
D^L_\mu\Phi(x) = \frac{1}{a}[U_\mu(x)\Phi(x+\hat{\mu})-\Phi(x)]
\eeq
so that 
\beq
\Phi^\dagger(x)D^L_\mu\Phi(x) =
\frac{1}{a}[\Phi^\dagger(x)U_\mu(x)\Phi(x+\hat{\mu}) -
\Phi^\dagger(x)\Phi(x)]
\eeq
is obviously gauge invariant.

The pure gauge action is expressed in terms of closed loops, with the
simplest being the Wilson action \cite{Wilson-gauge} (for a SU($N$)
gauge theory)
%
\beqa
S_\Box & = & \frac{1}{g^2}\sum_x(12N - \re\sum_{\mu<\nu}\Tr
U^{\mu\nu}_\Box(x)) \nonumber\\
& \equiv &
\beta\sum_x\sum_{\mu<\nu}\left(1-\frac{1}{N}\re\Tr U^{\mu\nu}_{\Box}(x)\right)
\label{eq:wilson-gauge} \\
 &= & S^{YM} + {\cal O}(a^2) \nonumber
\eeqa
where $\beta=2N/g^2$ and $U^{\mu\nu}_\Box$ is the plaquette
\beq
U^{\mu\nu}_\Box(x) \equiv
U_\mu(x)U_\nu(x+\hat{\mu})U^\dagger_\mu(x+\hat{\nu})U^\dagger_\nu(x)
\eeq

\subsection{Lattice fermions}
\label{sec:lat-fermion}

Fermions are represented on the lattice by anticommuting spinors $\psi$
on the lattice sites.  The fermion action can be discretised in a
straightforward way:

\begin{eqnarray}
S^F & = &\sum_x\bigg[ \sum_\mu\left(\psibar(x)\gamma_\mu
U_\mu(x)\psi(x+\muhat)
 - \psibar(x+\muhat)U^\dagger_\mu(x)\gamma_\mu\psi(x)\right) \nonumber\\ 
& & + m\psibar(x)\psi(x) \bigg] \label{eq:naive-fermion}
\end{eqnarray}

This is called the {\em naive fermion action}.  It suffers from
the presence of {\em doublers} ---
unphysical zero modes of the fermion matrix.  This can be seen
from a simple inspection of the fermion matrix in momentum space:
%
\begin{eqnarray}
M_{xy} & = & \sum_\mu \gamma_\mu(\delta_{y,x+\muhat}-\delta_{xy}) +
m\delta_{xy} \\
M(p) & = & \sum_x e^{-ip(x-y)}M_{xy} = \sum_\mu\gamma_\mu \sin(p_\mu) + m
\end{eqnarray}
%
This is 0 for all $p_\mu \in \{0,\pi\}$.  Since all zeros in the fermion
matrix correspond to a particle (which can, in an interacting theory, be
created in a pair production process), there will be not one, but 16
fermion species.  Nielsen and Ninomiya \cite{no-go} have shown that any
lattice theory fulfilling the conditions of
\begin{itemize}
\item translational invariance
\item locality 
\item hermiticity of the Hamiltonian
\end{itemize}
and with a set of locally defined, discretely valued conserved charges
(fermion numbers) which are bilinear in the fermion fields will have an
equal number of left- and right-handed particles for each set of
quantum numbers.  This means that neutrinos cannot be simulated on the
lattice, and chiral symmetry cannot be preserved in lattice QCD.

Wilson \cite{Wilson-fermion} proposed a remedy for this problem by
adding an extra irrelevant operator to the action
\beq
S^W = \frac{r}{a}\sum_x\left[\psibar(x)\psi(x) -
\half\sum_\mu\left(\psibar(x+\muhat)U^\dagger_\mu(x)\psi(x)
+\psibar(x)U_\mu(x)\psi(x+\muhat)\right) \right]
\eeq
which gives the doublers a mass on the scale of the cutoff $a^{-1}$. 
$r$ is an arbitrary parameter, which is normally chosen to be equal to
1.  This term explicitly breaks chiral symmetry.

The Wilson fermion action is often rewritten in terms of the hopping
parameter $\kappa$ as
%
\begin{eqnarray}
S^{WF} & = & \sum_x\bigg\{ \psibar(x)\psi(x) 
 +  \kappa\sum_\mu\psibar(x)\big[U_\mu(x)(\gamma_\mu-r)\psi(x+\muhat)
 \label{eq:wilson-fermion} \\
& & -U^\dagger(x-\muhat)(\gamma_\mu+r)
\psi(x-\muhat)\big]\bigg\}
\nonumber 
\end{eqnarray}
where
\beq
m = \frac{1-8\kappa r}{2\kappa}
\eeq

This gives the momentum space free propagator
\beq
S(p) = \frac{-i\sum_\mu\gamma_\mu \sin(p_\mu a) + r\sum_\mu(1-\cos(p_\mu
a)) + m}{\sum_\mu\sin^2(p_\mu a)+[r\sum_\mu(1-\cos(p_\mu a)+m]^2 } 
\label{eq:free-wilson}
\eeq

Since there is no chiral symmetry to prevent the fermions from acquring
a dynamical mass, they will typically do so.  This means that the
massless limit can only be determined a posteriori as the value of
$\kappa$ where the pseudoscalar meson is massless. 

\subsection{The quenched approximation}
\label{sec:quenched}

It is not possible to simulate Grassmann variables numerically in a
direct way.  But provided the action is bilinear in the fermion fields,
they can be integrated out analytically. From the properties of
Grassmann variables,
\beq
\int\psi d\psi = \frac{d}{d\psi}\psi = 1 \hspace{1cm} 
\psi_i\psi_j = -\psi_j\psi_i
\eeq
it follows that
\beq
\int\prod_{i}d\psibar_{i}d\psi_{i}e^{-\psibar_{j}M_{jk}\psi_k} = \det M
\eeq

This means the effective gauge action is given by
\beqa
Z & = & \int\cald U\det M[U]e^{-S_G[U]} 
= \int\cald U e^{-S_G[U]+\ln\det M[U]} \nonumber\\
& = & \int\cald U e^{-S_G^{eff}[U]}
\eeqa

There exist several algorithms for simulating the fermion determinant,
but these are computationally expensive, so most lattice simulations up
to now have been performed in the {\em quenched approximation}, where
one sets $\det M$=1.  This is equivalent to ignoring the effect of
fermions int the gauge distribution, ie ignoring vacuum polarisation
effects from fermion loops.  The quenched theory can be treated
analytically as a theory with $N_f=0$.

One would expect that setting $\det M$, which numerical simulations
have shown to be a wildly fluctuating quantity, equal to 1, would be a
very bad approximation.  However, it turns out that most results
obtained using this approximation agree quite well with experiment,
suggesting that fermion loops only have marginal effects for most
quantities.

\section{Gauge fixing on the lattice}
\label{sec:lat-gfix}

Although all physical quantities are gauge invariant and most of them
can be computed on the lattice in a gauge invariant way, there are
still many cases where gauge fixing is desirable or necessary.  Most
important among these are gauge dependent wave functions, and quark and
gluon correlation functions (or anything involving matrix elements
between quark and gluon states rather than hadronic states).  The gauge
can be fixed by applying a gauge transformation
(\ref{eq:gtrans-gauge})on each link so that the resulting link
variables conform to our gauge fixing prescription.

The most commonly used gauges in lattice studies are the Coulomb gauge
$\nabla\cdot\vec{A}=0$ and the Landau gauge $\partial_\mu A_\mu=0$,
corresponding to $\xi=0$ in (\ref{eq:gfix-lagr}).  The
Landau gauge can be obtained %following an idea from Gribov
by finding an extremal value for
%
\beq
F(\Lambda) = \| A^\Lambda \|^2 =
\int\Tr(A^\Lambda_\mu(x)A^\Lambda_\mu(x))d^4x
\eeq
since, if we write \(\Lambda(x)=\exp(it^a\theta^a(x))\), then
\beq
\frac{\delta F}{\delta\theta^a} = 0 \Rightarrow \partial_\mu A^a_\mu = 0
\eeq
%
This can be carried over to the lattice by replacing $\|A\|^2$ with
\beq
\|A\|^{2,LAT} = \sum_{x,\mu} \re\Tr U_\mu(x)-1
\eeq

However, this will not give a unique solution.  Gribov \cite{Gribov}
showed that in the continuum theory, there will be several non-trivial
solutions satisfying this condition, with different $\re\Tr U =
F(\Lambda_{min})$, and this is also the case in the lattice theory
\cite{lat-gribov}.  The problem can be reduced, but not eliminated, by
imposing additional conditions, eg.\ for the Faddeev--Popov matrix
$|{\cal M}|>0$.

More general gauges can be obtained by letting the gauge transformation
be a stochastic distribution rather than one definite transformation per
gauge configuration.  Specifically, the family of covariant gauges in
(\ref{eq:covgauge}) can be simulated by generating random matrices
$B(x)$ with a distribution \(\exp(-\frac{1}{2\xi}\Tr B(x)) \) and fixing
\(\partial_\mu A_\mu(x)-B(x)\) in the same way as \(\partial_\mu
A_\mu(x)\) is fixed in the Landau gauge \cite{Giusti}.

Another family of covariant gauges \cite{softgauge} is
obtained by taking the gauge fixing functional
%
\beq
F[A] = e^{-S_{GF}} \hspace{1cm} S_{GF} = \frac{\beta}{2}M^2\int
d^4x(A_\mu^a(x))^2
\eeq
to replace \(\delta(F[A])\) in (\ref{eq:FP-def}).  This bypasses the
problem of Gribov copies, and if $M\to\infty$, the point on the gauge
orbit where \((A_\mu^a(x))^2\) reaches its global minimum is selected. 
This is equivalent to the Landau gauge \(\partial_\mu A_\mu^a(x)=0\),
but with no copies.  As $M\to 0$, the theory approaches a theory without
gauge fixing.

This can be carried over to lattice simulations \cite{lat-softcov},
replacing $A^2$ with $\re\Tr U$, and evaluating the functional integral
in two stages.  For each configuration, a set of gauge transformations
are generated with the weight $e^{-S_{GF}}$ and the (gauge dependent)
operator we are interested in is averaged over those transformations,
before the configuration average is taken.

It is also possible to transcribe the continuum gauge fixed action
(\ref{eq:QCD}) to the lattice, and this is necessary when one wishes to
perform perturbative lattice calculations.  However, this does involve
significant complications in computing the Faddeev--Popov determinant,
and is not feasible for practical numerical simulations.  Some
numerical studies of ghost fields \cite{suman-ghost} have, however,
been carried out.

\section{Improved actions}
\label{sec:improve}

Although all effects of discretisation should disappear as $a\to 0$, in
any simulation using the Wilson actions (\ref{eq:wilson-gauge}) and
(\ref{eq:wilson-fermion}) at realistic values of $\beta$, the $\Op(a)$
and $\Op(a^2)$ discretisation errors may be significant.  There are
also lattice artefacts that are small at tree level, but may become
large non-perturbatively, or at higher-loop order.  Both these kinds of
effects may give uncertainties of up to 20--30\% in estimates of
physical quantities, for the simulations that are possible today.  It
is therefore a great advantage if one can construct actions which
reduce or avoid these problems. 

\subsection{Symanzik improvement}
\label{sec:symanzik}

The idea of this programme \cite{Symanzik} is to add higher-dimensional
terms to the action to cancel order-by-order the discretisation
effects.  In the continuum (or, equivalently, on a much finer lattice),
the lattice action, which is non-local, can be written as a local
effective action
%
\beq
S = S_0 + aS_1 + a^2S_2 + \ldots = 
\int d^4x[\lagr_0(x) + a \lagr_1(x) + \ldots]
\eeq
%
Similarly, the renormalised composite lattice operator $\Op_R$ can be
represented by local effective fields
%
\beq
\Op_R(x) = \Op_0(x) + a\Op_1(x) + a^2\Op_2(x) + \ldots
\eeq
%
where $S_0$ and $\Op_0$ are the continuum action and operators
respectively.

The operators that enter into $\lagr_1, \Op_1$ etc can be 
constructed by considering their dimensionality and the symmetries they
must obey, such as parity, charge conjugation, and Euclidean or lattice
symmetries.  In the improved action, counterterms with the same
structure are added to cancel the contributions from $S_1, \Op_1$ etc to
the Green functions.  Their coefficients can be computed perturbatively
from the zero-momentum proper Green functions, or using a
non-perturbative prescription \cite{alpha}.

Wilson's pure gauge action is already $\Op(a)$-improved, while the
Wilson fermion action has $\Op(a)$ errors.  We can write down the
operators that enter into $S_1^{WF}$
\begin{eqnarray}
\Op^1 & = & \psibar\sigma_{\mu\nu}F_{\mu\nu}\psi \\
\Op^2 & = & \psibar \stackrel{\leftrightarrow}{D^2}\psi \\
\Op^3 & = & m\Tr F_{\mu\nu}F_{\mu\nu} \\
\Op^4 & = & m(\psibar\Dslash\psi-\psibar\Dslashbak\psi) \\
\Op^5 & = & m^2\psibar\psi
\end{eqnarray}

Of these, $\Op^2$ and $\Op^4$ can be eliminated using the equations of
motion (provided we are only considering on-shell quantities), while
$\Op^3$ and $\Op^5$ imply a redefinition of the mass and coupling
constant, respectively.  So we are left with the Sheikholeslami--Wohlert
\cite{SW} action
\beq
S^{SW} = S^{WF} + c_{SW}\int
d^4x\psibar(x)\sigma_{\mu\nu}P_{\mu\nu}(x)\psi(x) \label{eq:SW}
\eeq
where $P_{\mu\nu}$ is some lattice representation of the field tensor
$F_{\mu\nu}$.  One possible choice, which makes the improved action
depend only on nearest-neighbour couplings, is the cloverleaf term
\beq
P_{\mu\nu} = U_\Box^{\mu\nu} - U_\Box^{-\nu,-\mu} + U_\Box^{\nu,-\mu} -
U_\Box^{\mu,-\nu}
\eeq
The coefficient $c_{SW}=1$ at tree level, but non-perturbative
calculations \cite{alpha-SW} have shown it to be significantly different
from this.

\subsection{Tadpole improvement}
\label{sec:tadpole}

Lattice operators, including those entering into the action, are usually
constructed from or matched to continuum operators by expanding
\beq
U_\mu(x) = e^{iagA_\mu(x)} \to 1+iagA_\mu(x)
\eeq

However, the higher order terms in this expansion include `tadpole
diagrams' which generate power divergences in $1/a$, cancelling
the powers of $a$ in the expansion. This means that the vacuum
expectation value of $U_\mu(x)$ (in some gauge) is smaller than 1 ---
the main contribution is from $-\langle A^2_\mu(x)\rangle < 0$ --- and 
a better expansion would be
\beq
U_\mu(x) = u_0(1+iagA_\mu(x))
\eeq
where $u_0$ represents the mean value of $U_\mu$.  Gauge invariance
requires that $u_0$ is a constant.  It can be defined in several ways,
either as the expectation value of $U_\mu$ in the Landau (or some other)
gauge, or by using the mean value of the plaquette,
\beq
u_0 = \left\langle\frac{1}{3}\Tr U_\Box\right\rangle^{1/4}.
\eeq
All these definitons give similar results \cite{LM}.

The mean-field tadpole improvement prescription \cite{LM} simply
replaces $U_\mu(x)$ with $U_\mu(x)/u_0$ everywhere in the action and in
operators.  For Wilson's gauge and fermion actions, this simply amounts
to a rescaling of the parameters $\beta$ and $\kappa$, but for improved
actions, and when calculating certain matrix elements, it is
non-trivial. 

