Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states (Gauss gauge invariant states). The idea is well known in the context of lattice Yang–Mills theory (see lattice gauge theory). Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations.
The introduction by Ashtekar of a new set of variables (Ashtekar variables) cast general relativity in the same language as gauge theories and allowed one to apply loop techniques as a natural nonperturbative description of Einstein's theory. In canonical quantum gravity the difficulties in using the continuous loop representation are cured by the spatial diffeomorphism invariance of general relativity. The loop representation also provides a natural solution of the spatial diffeomorphism constraint, making a connection between canonical quantum gravity and knot theory. Surprisingly there were a class of loop states that provided exact (if only formal) solutions to Ashtekar's original (ill-defined) Wheeler–DeWitt equation. Hence an infinite set of exact (if only formal) solutions had been identified for all the equations of canonical quantum general gravity in this representation! This generated a lot of interest in the approach and eventually led to loop quantum gravity (LQG).
The loop representation has found application in mathematics. If topological quantum field theories are formulated in terms of loops, the resulting quantities should be what are known as knot invariants. Topological field theories only involve a finite number of degrees of freedom and so are exactly solvable. As a result, they provide concrete computable expressions that are invariants of knots. This was precisely the insight of Edward Witten who noticed that computing loop dependent quantities in Chern–Simons and other three-dimensional topological quantum field theories one could come up with explicit, analytic expressions for knot invariants. For his work in this, in 1990 he was awarded the Fields Medal. He is the first and so far the only physicist to be awarded the Fields Medal, often viewed as the greatest honour in mathematics.
Gauge invariance of Maxwell's theory
The idea of gauge symmetries was introduced in Maxwell's theory. Maxwell's equations are
where is the charge density and the current density. The last two equations can be solved by writing fields in terms of a scalar potential, , and a vector potential, :
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The potentials uniquely determine the fields, but the fields do not uniquely determine the potentials - we can make the changes:
without affecting the electric and magnetic fields, where is an arbitrary function of space-time. These are called gauge transformations. There is an elegant relativistic notation: the gauge field is
and the above gauge transformations read,
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The so-called field strength tensor is introduced,
which is easily shown to be invariant under gauge transformations. In components,
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Maxwell's source-free action is given by:
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The ability to vary the gauge potential at different points in space and time (by changing ) without changing the physics is called a local invariance. Electromagnetic theory possess the simplest kind of local gauge symmetry called (see unitary group). A theory that displays local gauge invariance is called a gauge theory. In order to formulate other gauge theories we turn the above reasoning inside out. This is the subject of the next section.
The connection and gauges theories
The connection and Maxwell's theory
We know from quantum mechanics that if we replace the wave-function, , describing the electron field by
that it leaves physical predictions unchanged. We consider the imposition of local invariance on the phase of the electron field,
The problem is that derivatives of are not covariant under this transformation:
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In order to cancel out the second unwanted term, one introduces a new derivative operator that is covariant. To construct , one introduces a new field, the connection :
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Then
The term is precisely cancelled out by requiring the connection field transforms as
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We then have that
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Note that is equivalent to
which looks the same as a gauge transformation of the gauge potential of Maxwell's theory. It is possible to construct an invariant action for the connection field itself. We want an action that only has two derivatives (since actions with higher derivatives are not unitary). Define the quantity:
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The unique action with only two derivatives is given by:
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Therefore, one can derive electromagnetic theory from arguments based solely on symmetry.
The connection and Yang-Mills gauge theory
We now generalize the above reasoning to general gauge groups. One begins with the generators of some Lie algebra:
Let there be a fermion field that transforms as
Again the derivatives of are not covariant under this transformation. We introduce a covariant derivative
with connection field given by
We require that transforms as:
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We define the field strength operator
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As is covariant, this means that the tensor is also covariant:
Note that is only invariant under gauge transformations if is a scalar, that is, only in the case of electromagnetism.
We can now construct an invariant action out of this tensor. Again we want an action that only has two derivatives. The simplest choice is the trace of the commutator:
The unique action with only two derivatives is given by:
This is the action for Yang-mills theory.
The loop representation of the Maxwell theory
We consider a change of representation in the quantum Maxwell gauge theory. The idea is to introduce a basis of states labeled by loops whose inner product with the connection states is given by
The loop functional is the Wilson loop for the abelian case.
The loop representation of Yang–Mills theory
We consider for simplicity (and because later we will see this is the relevant gauge group in LQG) an Yang–Mills theory in four dimensions. The field variable of the continuous theory is an connection (or gauge potential) , where is an index in the Lie algebra of . We can write for this field
where are the generators, that is the Pauli matrices multiplied by . note that unlike with Maxwell's theory, the connections are matrix-valued and don't commute, that is they are non-Abelian gauge theories. We must take this into account when defining the corresponding version of the holonomy for Yang–Mills theory.
We first describe the quantum theory in terms of connection variable.
The connection representation
In the connection representation the configuration variable is and its conjugate momentum is the (densitized) triad . It is most natural to consider wavefunctions . This is known as the connection representation. The canonical variables get promoted to quantum operators:
(analogous to the position representation ) and the triads are functional derivatives,
(analogous to )
The holonomy and Wilson loop
Let us return to the classical Yang–Mills theory. It is possible to encode the gauge invariant information of the theory in terms of `loop-like' variables.
We need the notion of a holonomy. A holonomy is a measure of how much the initial and final values of a spinor or vector differ after parallel transport around a closed loop ; it is denoted
Knowledge of the holonomies is equivalent to knowledge of the connection, up to gauge equivalence. Holonomies can also be associated with an edge; under a Gauss Law these transform as
For a closed loop if we take the trace of this, that is, putting and summing we obtain
or
Thus the trace of an holonomy around a closed loop is gauge invariant. It is denoted
and is called a Wilson loop. The explicit form of the holonomy is
where is the curve along which the holonomy is evaluated, and is a parameter along the curve, denotes path ordering meaning factors for smaller values of appear to the left, and are matrices that satisfy the algebra
The Pauli matrices satisfy the above relation. It turns out that there are infinitely many more examples of sets of matrices that satisfy these relations, where each set comprises matrices with , and where none of these can be thought to `decompose' into two or more examples of lower dimension. They are called different irreducible representations of the algebra. The most fundamental representation being the Pauli matrices. The holonomy is labelled by a half integer according to the irreducible representation used.
Giles' Reconstruction theorem of gauge potentials from Wilson loops
An important theorem about Yang–Mills gauge theories is Giles' theorem, according to which if one gives the trace of the holonomy of a connection for all possible loops on a manifold one can, in principle, reconstruct all the gauge invariant information of the connection. That is, Wilson loops constitute a basis of gauge invariant functions of the connection. This key result is the basis for the loop representation for gauge theories and gravity.
The loop transform and the loop representation
The use of Wilson loops explicitly solves the Gauss gauge constraint. As Wilson loops form a basis we can formally expand any Gauss gauge invariant function as,
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This is called the loop transform. We can see the analogy with going to the momentum representation in quantum mechanics. There one has a basis of states labelled by a number and one expands
and works with the coefficients of the expansion .
The inverse loop transform is defined by
This defines the loop representation. Given an operator in the connection representation,
one should define the corresponding operator on in the loop representation via,
where is defined by the usual inverse loop transform,
A transformation formula giving the action of the operator on in terms of the action of the operator on is then obtained by equating the R.H.S. of with the R.H.S. of with substituted into , namely
or
where by we mean the operator but with the reverse factor ordering (remember from simple quantum mechanics where the product of operators is reversed under conjugation). We evaluate the action of this operator on the Wilson loop as a calculation in the connection representation and rearranging the result as a manipulation purely in terms of loops (one should remember that when considering the action on the Wilson loop one should choose the operator one wishes to transform with the opposite factor ordering to the one chosen for its action on wavefunctions ).
The loop representation of quantum gravity
Ashtekar–Barbero variables of canonical quantum gravity
The introduction of Ashtekar variables cast general relativity in the same language as gauge theories. It was in particular the inability to have good control over the space of solutions to the Gauss' law and spatial diffeomorphism constraints that led Rovelli and Smolin to consider a new representation – the loop representation.