In physics, lattice gauge theory is the study
of gauge theories on a spacetime that has
been discretized into a lattice.
Gauge theories are important in particle physics,
and include the prevailing theories of elementary
particles: quantum electrodynamics, quantum
chromodynamics (QCD) and particle physics'
Standard Model. Non-perturbative gauge theory
calculations in continuous spacetime formally
involve evaluating an infinite-dimensional
path integral, which is computationally intractable.
By working on a discrete spacetime, the path
integral becomes finite-dimensional, and can
be evaluated by stochastic simulation techniques
such as the Monte Carlo method. When the size
of the lattice is taken infinitely large and
its sites infinitesimally close to each other,
the continuum gauge theory is recovered.
== Basics ==
In lattice gauge theory, the spacetime is
Wick rotated into Euclidean space and discretized
into a lattice with sites separated by distance
a
{\displaystyle a}
and connected by links. In the most commonly
considered cases, such as lattice QCD, fermion
fields are defined at lattice sites (which
leads to fermion doubling), while the gauge
fields are defined on the links. That is,
an element U of the compact Lie group G (not
algebra) is assigned to each link. Hence,
to simulate QCD with Lie group SU(3), a 3×3
unitary matrix is defined on each link. The
link is assigned an orientation, with the
inverse element corresponding to the same
link with the opposite orientation. And each
node is given a value in ℂ3 (a color 3-vector,
the space on which the fundamental representation
of SU(3) acts), a bispinor (Dirac 4-spinor),
an nf vector, and a Grassmann variable.
Thus, the composition of links' SU(3) elements
along a path (i.e. the ordered multiplication
of their matrices) approximates a path-ordered
exponential (geometric integral), from which
Wilson loop values can be calculated for closed
paths.
== Yang–Mills action ==
The Yang–Mills action is written on the
lattice using Wilson loops (named after Kenneth
G. Wilson), so that the limit
a
→
0
{\displaystyle a\to 0}
formally reproduces the original continuum
action. Given a faithful irreducible representation
ρ of G, the lattice Yang-Mills action is
the sum over all lattice sites of the (real
component of the) trace over the n links e1,
..., en in the Wilson loop,
S
=
∑
F
−
ℜ
{
χ
(
ρ
)
(
U
(
e
1
)
⋯
U
(
e
n
)
)
}
.
{\displaystyle S=\sum _{F}-\Re \{\chi ^{(\rho
)}(U(e_{1})\cdots U(e_{n}))\}.}
Here, χ is the character. If ρ is a real
(or pseudoreal) representation, taking the
real component is redundant, because even
if the orientation of a Wilson loop is flipped,
its contribution to the action remains unchanged.
There are many possible lattice Yang-Mills
actions, depending on which Wilson loops are
used in the action. The simplest "Wilson action"
uses only the 1×1 Wilson loop, and differs
from the continuum action by "lattice artifacts"
proportional to the small lattice spacing
a
{\displaystyle a}
. By using more complicated Wilson loops to
construct "improved actions", lattice artifacts
can be reduced to be proportional to
a
2
{\displaystyle a^{2}}
, making computations more accurate.
== Measurements and calculations ==
Quantities such as particle masses are stochastically
calculated using techniques such as the Monte
Carlo method. Gauge field configurations are
generated with probabilities proportional
to
e
−
β
S
{\displaystyle e^{-\beta S}}
, where
S
{\displaystyle S}
is the lattice action and
β
{\displaystyle \beta }
is related to the lattice spacing
a
{\displaystyle a}
. The quantity of interest is calculated for
each configuration, and averaged. Calculations
are often repeated at different lattice spacings
a
{\displaystyle a}
so that the result can be extrapolated to
the continuum,
a
→
0
{\displaystyle a\to 0}
.
Such calculations are often extremely computationally
intensive, and can require the use of the
largest available supercomputers. To reduce
the computational burden, the so-called quenched
approximation can be used, in which the fermionic
fields are treated as non-dynamic "frozen"
variables. While this was common in early
lattice QCD calculations, "dynamical" fermions
are now standard. These simulations typically
utilize algorithms based upon molecular dynamics
or microcanonical ensemble algorithms.The
results of lattice QCD computations show e.g.
that in a meson not only the particles (quarks
and antiquarks), but also the "fluxtubes"
of the gluon fields are important.
== Quantum triviality ==
Lattice gauge theory is also important for
the study of quantum triviality by the real-space
renormalization group. The most important
information in the RG flow are what's called
the fixed points.
The possible macroscopic states of the system,
at a large scale, are given by this set of
fixed points. If these fixed points correspond
to a free field theory, the theory is said
to be trivial or noninteracting. Numerous
fixed points appear in the study of lattice
Higgs theories, but the nature of the quantum
field theories associated with these remains
an open question.Triviality has yet to be
proven rigorously, but lattice computations
have provided strong evidence for this. This
fact is important as quantum triviality can
be used to bound or even predict parameters
such as the mass of Higgs boson.
== Other applications ==
Originally, solvable two-dimensional lattice
gauge theories had already been introduced
in 1971 as models with interesting statistical
properties by the theorist Franz Wegner, who
worked in the field of phase transitions.When
only 1×1 Wilson loops appear in the action,
Lattice gauge theory can be shown to be exactly
dual to spin foam models.
== See also ==
Hamiltonian lattice gauge theory
Lattice field theory
Lattice QCD
Quantum triviality
