Loop quantum gravity

General relativity is an example of a constrained system. In the Hamiltonian formulation of ordinary classical mechanics the Poisson bracket is an important concept. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson-bracket relations,

${\displaystyle \{q_{i},p_{j}\}=\delta _{ij}}$

where the Poisson bracket is given by

${\displaystyle \{f,g\}=\sum _{i=1}^{N}\left({\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right).}$

for arbitrary phase space functions ${\displaystyle f(q_{i},p_{j})}$ and ${\displaystyle g(q_{i},p_{j})}$. With the use of Poisson brackets, the Hamilton's equations can be rewritten as,

${\displaystyle {\dot {q}}_{i}=\{q_{i},H\},}$

${\displaystyle {\dot {p}}_{i}=\{p_{i},H\}.}$

These equations describe a "flow" or orbit in phase space generated by the Hamiltonian ${\displaystyle H}$. Given any phase space function ${\displaystyle F(q,p)}$, yields

${\displaystyle {\frac {\operatorname {d} }{\operatorname {d} t}}F(q_{i},p_{i})=\{F,H\}.}$

In a similar way the Poisson bracket between a constraint and the phase space variables generates a flow along an orbit in (the unconstrained) phase space generated by the constraint. There are three types of constraints in Ashtekar's reformulation of classical general relativity:

The Gauss constraints

${\displaystyle G_{j}(x)=0.}$

This represents an infinite number of constraints one for each value of ${\displaystyle x}$. These come about from re-expressing General relativity as an ${\displaystyle \mathrm {SU} (2)}$ Yang–Mills type gauge theory (Yang–Mills is a generalization of Maxwell's theory where the gauge field transforms as a vector under Gauss transformations, that is, the Gauge field is of the form ${\displaystyle A_{a}^{i}(x)}$ where ${\displaystyle i}$ is an internal index. See Ashtekar variables). These infinite number of Gauss gauge constraints can be "smeared" by test fields with internal indices, ${\displaystyle \lambda ^{j}(x)}$,

${\displaystyle G(\lambda )=\int G_{j}(x)\lambda ^{j}(x)\,\operatorname {d} ^{3}\!x.}$

which must vanish for any such function. These smeared constraints defined with respect to a suitable space of smearing functions give an equivalent description to the original constraints.

Ashtekar's formulation may be thought of as ordinary ${\displaystyle \mathrm {SU} (2)}$ Yang–Mills theory together with the following special constraints, resulting from diffeomorphism invariance, and a Hamiltonian that vanishes. The dynamics of such a theory are thus very different from that of ordinary Yang–Mills theory.

The spatial diffeomorphism constraints

${\displaystyle C_{a}(x)=0}$

can be smeared by the so-called shift functions ${\displaystyle {\vec {N}}(x)}$ to give an equivalent set of smeared spatial diffeomorphism constraints,

${\displaystyle C({\vec {N}})=\int C_{a}(x)\,N^{a}(x)\,\operatorname {d} ^{3}\!x.}$

These generate spatial diffeomorphisms along orbits defined by the shift function ${\displaystyle N^{a}(x)}$.

The Hamiltonian

${\displaystyle H(x)=0}$

can be smeared by the so-called lapse functions ${\displaystyle N(x)}$ to give an equivalent set of smeared Hamiltonian constraints,

${\displaystyle H(N)=\int H(x)N(x)\,\operatorname {d} ^{3}\!x}$.

These generate time diffeomorphisms along orbits defined by the lapse function ${\displaystyle N(x)}$.

In Ashtekar formulation the gauge field ${\displaystyle A_{a}^{i}(x)}$ is the configuration variable (the configuration variable being analogous to ${\displaystyle q}$ in ordinary mechanics) and its conjugate momentum is the (densitized) triad (electrical field) ${\displaystyle {\tilde {E}}_{i}^{a}(x)}$. The constraints are certain functions of these phase space variables.

An important aspect of the action of the constraints on arbitrary phase space functions is the Lie derivative, ${\displaystyle {\mathcal {L}}_{V}}$, which is basically a derivative operation that infinitesimally "shifts" functions along some orbit with tangent vector ${\displaystyle V}$.

The constraints define a constraint surface in the original phase space. The gauge motions of the constraints apply to all phase space but have the feature that they leave the constraint surface where it is, and thus the orbit of a point in the hypersurface under gauge transformations will be an orbit entirely within it. Dirac observables are defined as phase space functions, ${\displaystyle O}$, that Poisson commute with all the constraints when the constraint equations are imposed,

${\displaystyle \{G_{j},O\}_{G_{j}=C_{a}=H=0}=\{C_{a},O\}_{G_{j}=C_{a}=H=0}=\{H,O\}_{G_{j}=C_{a}=H=0}=0}$,

that is, they are quantities defined on the constraint surface that are invariant under the gauge transformations of the theory.

Then, solving only the constraint ${\displaystyle G_{j}=0}$ and determining the Dirac observables with respect to it leads us back to the Arnowitt–Deser–Misner (ADM) phase space with constraints ${\displaystyle H,C_{a}}$. The dynamics of general relativity is generated by the constraints, it can be shown that six Einstein equations describing time evolution (really a gauge transformation) can be obtained by calculating the Poisson brackets of the three-metric and its conjugate momentum with a linear combination of the spatial diffeomorphism and Hamiltonian constraint. The vanishing of the constraints, giving the physical phase space, are the four other Einstein equations.[11]

Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to quantum operators because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekar's new variables. Ashtekar variables describe canonical general relativity in terms of a new pair of canonical variables closer to those of gauge theories. The first step consists of using densitized triads ${\displaystyle {\tilde {E}}_{i}^{a}}$ (a triad ${\displaystyle E_{i}^{a}}$ is simply three orthogonal vector fields labeled by ${\displaystyle i=1,2,3}$ and the densitized triad is defined by ${\displaystyle {\tilde {E}}_{i}^{a}={\sqrt {\det(q)}}E_{i}^{a}}$) to encode information about the spatial metric,

${\displaystyle \det(q)q^{ab}={\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b}\delta ^{ij}}$.

(where ${\displaystyle \delta ^{ij}}$ is the flat space metric, and the above equation expresses that ${\displaystyle q^{ab}}$, when written in terms of the basis ${\displaystyle E_{i}^{a}}$, is locally flat). (Formulating general relativity with triads instead of metrics was not new.) The densitized triads are not unique, and in fact one can perform a local in space rotation with respect to the internal indices ${\displaystyle i}$. The canonically conjugate variable is related to the extrinsic curvature by ${\displaystyle K_{a}^{i}=K_{ab}{\tilde {E}}^{ai}/{\sqrt {\det(q)}}}$. But problems similar to using the metric formulation arise when one tries to quantize the theory. Ashtekar's new insight was to introduce a new configuration variable,

${\displaystyle A_{a}^{i}=\Gamma _{a}^{i}-iK_{a}^{i}}$

that behaves as a complex ${\displaystyle \operatorname {SU} (2)}$ connection where ${\displaystyle \Gamma _{a}^{i}}$ is related to the so-called spin connection via ${\displaystyle \Gamma _{a}^{i}=\Gamma _{ajk}\epsilon ^{jki}}$. Here ${\displaystyle A_{a}^{i}}$ is called the chiral spin connection. It defines a covariant derivative ${\displaystyle {\mathcal {D}}_{a}}$. It turns out that ${\displaystyle {\tilde {E}}_{i}^{a}}$ is the conjugate momentum of ${\displaystyle A_{a}^{i}}$, and together these form Ashtekar's new variables.

The expressions for the constraints in Ashtekar variables; the Gauss's law, the spatial diffeomorphism constraint and the (densitized) Hamiltonian constraint then read:

${\displaystyle G^{i}={\mathcal {D}}_{a}{\tilde {E}}_{i}^{a}=0}$

${\displaystyle C_{a}={\tilde {E}}_{i}^{b}F_{ab}^{i}-A_{a}^{i}({\mathcal {D}}_{b}{\tilde {E}}_{i}^{b})=V_{a}-A_{a}^{i}G^{i}=0}$,

${\displaystyle {\tilde {H}}=\epsilon _{ijk}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b}F_{ab}^{k}=0}$

respectively, where ${\displaystyle F_{ab}^{i}}$ is the field strength tensor of the connection ${\displaystyle A_{a}^{i}}$ and where ${\displaystyle V_{a}}$ is referred to as the vector constraint. The above-mentioned local in space rotational invariance is the original of the ${\displaystyle \operatorname {SU} (2)}$ gauge invariance here expressed by the Gauss law. Note that these constraints are polynomial in the fundamental variables, unlike as with the constraints in the metric formulation. This dramatic simplification seemed to open up the way to quantizing the constraints. (See the article Self-dual Palatini action for a derivation of Ashtekar's formalism).

With Ashtekar's new variables, given the configuration variable ${\displaystyle A_{a}^{i}}$, it is natural to consider wavefunctions ${\displaystyle \Psi (A_{a}^{i})}$. This is the connection representation. It is analogous to ordinary quantum mechanics with configuration variable ${\displaystyle q}$ and wavefunctions ${\displaystyle \psi (q)}$. The configuration variable gets promoted to a quantum operator via:

${\displaystyle {\hat {A}}_{a}^{i}\Psi (A)=A_{a}^{i}\Psi (A),}$

(analogous to ${\displaystyle {\hat {q}}\psi (q)=q\psi (q)}$) and the triads are (functional) derivatives,

${\displaystyle {\hat {\tilde {E_{i}^{a}}}}\Psi (A)=-i{\delta \Psi (A) \over \delta A_{a}^{i}}}$.

(analogous to ${\displaystyle {\hat {p}}\psi (q)=-i\hbar d\psi (q)/dq}$). In passing over to the quantum theory the constraints become operators on a kinematic Hilbert space (the unconstrained ${\displaystyle \operatorname {SU} (2)}$ Yang–Mills Hilbert space). Note that different ordering of the ${\displaystyle A}$'s and ${\displaystyle {\tilde {E}}}$'s when replacing the ${\displaystyle {\tilde {E}}}$'s with derivatives give rise to different operators – the choice made is called the factor ordering and should be chosen via physical reasoning. Formally they read

${\displaystyle {\hat {G}}_{j}\vert \psi \rangle =0}$

${\displaystyle {\hat {C}}_{a}\vert \psi \rangle =0}$

${\displaystyle {\hat {\tilde {H}}}\vert \psi \rangle =0}$.

There are still problems in properly defining all these equations and solving them. For example, the Hamiltonian constraint Ashtekar worked with was the densitized version instead of the original Hamiltonian, that is, he worked with ${\displaystyle {\tilde {H}}={\sqrt {\det(q)}}H}$. There were serious difficulties in promoting this quantity to a quantum operator. Moreover, although Ashtekar variables had the virtue of simplifying the Hamiltonian, they are complex. When one quantizes the theory, it is difficult to ensure that one recovers real general relativity as opposed to complex general relativity.

The classical result of the Poisson bracket of the smeared Gauss' law ${\displaystyle G(\lambda )=\int d^{3}x\lambda ^{j}(D_{a}E^{a})^{j}}$ with the connections is

${\displaystyle \{G(\lambda ),A_{a}^{i}\}=\partial _{a}\lambda ^{i}+g\epsilon ^{ijk}A_{a}^{j}\lambda ^{k}=(D_{a}\lambda )^{i}.}$

The quantum Gauss' law reads

${\displaystyle {\hat {G}}_{j}\Psi (A)=-iD_{a}{\delta \lambda \Psi [A] \over \delta A_{a}^{j}}=0.}$

If one smears the quantum Gauss' law and study its action on the quantum state one finds that the action of the constraint on the quantum state is equivalent to shifting the argument of ${\displaystyle \Psi }$ by an infinitesimal (in the sense of the parameter ${\displaystyle \lambda }$ small) gauge transformation,

${\displaystyle \left[1+\int d^{3}x\lambda ^{j}(x){\hat {G}}_{j}\right]\Psi (A)=\Psi [A+D\lambda ]=\Psi [A],}$

and the last identity comes from the fact that the constraint annihilates the state. So the constraint, as a quantum operator, is imposing the same symmetry that its vanishing imposed classically: it is telling us that the functions ${\displaystyle \Psi [A]}$ have to be gauge invariant functions of the connection. The same idea is true for the other constraints.

Therefore, the two step process in the classical theory of solving the constraints ${\displaystyle C_{I}=0}$ (equivalent to solving the admissibility conditions for the initial data) and looking for the gauge orbits (solving the 'evolution' equations) is replaced by a one step process in the quantum theory, namely looking for solutions ${\displaystyle \Psi }$ of the quantum equations ${\displaystyle {\hat {C}}_{I}\Psi =0}$. This is because it obviously solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because ${\displaystyle {\hat {C}}_{I}}$ is the quantum generator of gauge transformations (gauge invariant functions are constant along the gauge orbits and thus characterize them).[12] Recall that, at the classical level, solving the admissibility conditions and evolution equations was equivalent to solving all of Einstein's field equations, this underlines the central role of the quantum constraint equations in canonical quantum gravity.

It was in particular the inability to have good control over the space of solutions to Gauss's law and spatial diffeomorphism constraints that led Rovelli and Smolin to consider the loop representation in gauge theories and quantum gravity.[13]

LQG includes the concept 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

${\displaystyle h_{\gamma }[A]}$.

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

${\displaystyle (h'_{e})_{\alpha \beta }=U_{\alpha \gamma }^{-1}(x)(h_{e})_{\gamma \sigma }U_{\sigma \beta }(y)}$.

For a closed loop ${\displaystyle x=y}$ and assuming ${\displaystyle \alpha =\beta }$, yields

${\displaystyle (h'_{e})_{\alpha \alpha }=U_{\alpha \gamma }^{-1}(x)(h_{e})_{\gamma \sigma }U_{\sigma \alpha }(x)=[U_{\sigma \alpha }(x)U_{\alpha \gamma }^{-1}(x)](h_{e})_{\gamma \sigma }=\delta _{\sigma \gamma }(h_{e})_{\gamma \sigma }=(h_{e})_{\gamma \gamma }}$

or

${\displaystyle \operatorname {Tr} h'_{\gamma }=\operatorname {Tr} h_{\gamma }.}$.

The trace of an holonomy around a closed loop is written

${\displaystyle W_{\gamma }[A]}$

and is called a Wilson loop. Thus Wilson loops are gauge invariant. The explicit form of the Holonomy is

${\displaystyle h_{\gamma }[A]={\mathcal {P}}\exp \left\{-\int _{\gamma _{0}}^{\gamma _{1}}ds{\dot {\gamma }}^{a}A_{a}^{i}(\gamma (s))T_{i}\right\}}$

where ${\displaystyle \gamma }$ is the curve along which the holonomy is evaluated, and ${\displaystyle s}$ is a parameter along the curve, ${\displaystyle {\mathcal {P}}}$ denotes path ordering meaning factors for smaller values of ${\displaystyle s}$ appear to the left, and ${\displaystyle T_{i}}$ are matrices that satisfy the ${\displaystyle \operatorname {SU} (2)}$ algebra

${\displaystyle [T^{i},T^{j}]=2i\epsilon ^{ijk}T_{k}}$.

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 ${\displaystyle (N+1)\times (N+1)}$ matrices with ${\displaystyle N=1,2,3,\dots }$, 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 ${\displaystyle \operatorname {SU} (2)}$ algebra. The most fundamental representation being the Pauli matrices. The holonomy is labelled by a half integer ${\displaystyle N/2}$ according to the irreducible representation used.

The use of Wilson loops explicitly solves the Gauss gauge constraint. Loop representation is required handle the spatial diffeomorphism constraint. With Wilson loops as a basis, any Gauss gauge invariant function expands as,

${\displaystyle \Psi [A]=\sum _{\gamma }\Psi [\gamma ]W_{\gamma }[A].}$

This is called the loop transform and is analogous to the momentum representation in quantum mechanics (see Position and momentum space). The QM representation has a basis of states ${\displaystyle \exp(ikx)}$ labelled by a number ${\displaystyle k}$ and expands as

${\displaystyle \psi [x]=\int dk\psi (k)\exp(ikx)}$.

and works with the coefficients of the expansion ${\displaystyle \psi (k).}$

The inverse loop transform is defined by

${\displaystyle \Psi [\gamma ]=\int [dA]\Psi [A]W_{\gamma }[A]}$.

This defines the loop representation. Given an operator ${\displaystyle {\hat {O}}}$ in the connection representation,

${\displaystyle \Phi [A]={\hat {O}}\Psi [A]\qquad Eq\;1,}$

one should define the corresponding operator ${\displaystyle {\hat {O}}'}$ on ${\displaystyle \Psi [\gamma ]}$ in the loop representation via,

${\displaystyle \Phi [\gamma ]={\hat {O}}'\Psi [\gamma ]\qquad Eq\;2,}$

where ${\displaystyle \Phi [\gamma ]}$ is defined by the usual inverse loop transform,

${\displaystyle \Phi [\gamma ]=\int [dA]\Phi [A]W_{\gamma }[A]\qquad Eq\;3.}$

A transformation formula giving the action of the operator ${\displaystyle {\hat {O}}'}$ on ${\displaystyle \Psi [\gamma ]}$ in terms of the action of the operator ${\displaystyle {\hat {O}}}$ on ${\displaystyle \Psi [A]}$ is then obtained by equating the R.H.S. of ${\displaystyle Eq\;2}$ with the R.H.S. of ${\displaystyle Eq\;3}$ with ${\displaystyle Eq\;1}$ substituted into ${\displaystyle Eq\;3}$, namely

${\displaystyle {\hat {O}}'\Psi [\gamma ]=\int [dA]W_{\gamma }[A]{\hat {O}}\Psi [A]}$,

or

${\displaystyle {\hat {O}}'\Psi [\gamma ]=\int [dA]({\hat {O}}^{\dagger }W_{\gamma }[A])\Psi [A]}$,

where ${\displaystyle {\hat {O}}^{\dagger }}$ means the operator ${\displaystyle {\hat {O}}}$ but with the reverse factor ordering (remember from simple quantum mechanics where the product of operators is reversed under conjugation). The action of this operator on the Wilson loop is evaluated as a calculation in the connection representation and the result is rearranged purely as a manipulation in terms of loops (with regard to the action on the Wilson loop, the chosen transformed operator is the one with the opposite factor ordering compared to the one used for its action on wavefunctions ${\displaystyle \Psi [A]}$). This gives the physical meaning of the operator ${\displaystyle {\hat {O}}'}$. For example, if ${\displaystyle {\hat {O}}^{\dagger }}$ corresponded to a spatial diffeomorphism, then this can be thought of as keeping the connection field ${\displaystyle A}$ of ${\displaystyle W_{\gamma }[A]}$ where it is while performing a spatial diffeomorphism on ${\displaystyle \gamma }$ instead. Therefore, the meaning of ${\displaystyle {\hat {O}}'}$ is a spatial diffeomorphism on ${\displaystyle \gamma }$, the argument of ${\displaystyle \Psi [\gamma ]}$.

In the loop representation, the spatial diffeomorphism constraint is solved by considering functions of loops ${\displaystyle \Psi [\gamma ]}$ that are invariant under spatial diffeomorphisms of the loop ${\displaystyle \gamma }$. That is, knot invariants are used. This opens up an unexpected connection between knot theory and quantum gravity.

Any collection of non-intersecting Wilson loops satisfy Ashtekar's quantum Hamiltonian constraint. Using a particular ordering of terms and replacing ${\displaystyle {\tilde {E}}_{i}^{a}}$ by a derivative, the action of the quantum Hamiltonian constraint on a Wilson loop is

${\displaystyle {\hat {\tilde {H}}}^{\dagger }W_{\gamma }[A]=-\epsilon _{ijk}{\hat {F}}_{ab}^{k}{\frac {\delta }{\delta A_{a}^{i}}}{\frac {\delta }{\delta A_{b}^{j}}}W_{\gamma }[A]}$.

When a derivative is taken it brings down the tangent vector, ${\displaystyle {\dot {\gamma }}^{a}}$, of the loop, ${\displaystyle \gamma }$. So,

${\displaystyle {\hat {F}}_{ab}^{i}{\dot {\gamma }}^{a}{\dot {\gamma }}^{b}}$.

However, as ${\displaystyle F_{ab}^{i}}$ is anti-symmetric in the indices ${\displaystyle a}$ and ${\displaystyle b}$ this vanishes (this assumes that ${\displaystyle \gamma }$ is not discontinuous anywhere and so the tangent vector is unique).

With regard to loop representation, the wavefunctions ${\displaystyle \Psi [\gamma ]}$ vanish when the loop has discontinuities and are knot invariants. Such functions solve the Gauss law, the spatial diffeomorphism constraint and (formally) the Hamiltonian constraint. This yields an infinite set of exact (if only formal) solutions to all the equations of quantum general relativity![13] This generated a lot of interest in the approach and eventually led to LQG.

The easiest geometric quantity is the area. Let us choose coordinates so that the surface ${\displaystyle \Sigma }$ is characterized by ${\displaystyle x^{3}=0}$. The area of small parallelogram of the surface ${\displaystyle \Sigma }$ is the product of length of each side times ${\displaystyle \sin \theta }$ where ${\displaystyle \theta }$ is the angle between the sides. Say one edge is given by the vector ${\displaystyle {\vec {u}}}$ and the other by ${\displaystyle {\vec {v}}}$ then,

${\displaystyle A=\|{\vec {u}}\|\|{\vec {v}}\|\sin \theta ={\sqrt {\|{\vec {u}}\|^{2}\|{\vec {v}}\|^{2}(1-\cos ^{2}\theta )}}={\sqrt {\|{\vec {u}}\|^{2}\|{\vec {v}}\|^{2}-({\vec {u}}\cdot {\vec {v}})^{2}}}}$

In the space spanned by ${\displaystyle x^{1}}$ and ${\displaystyle x^{2}}$ there is an infinitesimal parallelogram described by ${\displaystyle {\vec {u}}={\vec {e}}_{1}dx^{1}}$ and ${\displaystyle {\vec {v}}={\vec {e}}_{2}dx^{2}}$. Using ${\displaystyle q_{AB}^{(2)}={\vec {e}}_{A}\cdot {\vec {e}}_{B}}$ (where the indices ${\displaystyle A}$ and ${\displaystyle B}$ run from 1 to 2), yields the area of the surface ${\displaystyle \Sigma }$ given by

${\displaystyle A_{\Sigma }=\int _{\Sigma }dx^{1}dx^{2}{\sqrt {\det(q^{(2)})}}}$

where ${\displaystyle \det(q^{(2)})=q_{11}q_{22}-q_{12}^{2}}$ and is the determinant of the metric induced on ${\displaystyle \Sigma }$. The latter can be rewritten ${\displaystyle \det(q^{(2)})=\epsilon ^{AB}\epsilon ^{CD}q_{AC}q_{BD}/2}$ where the indices ${\displaystyle A\dots D}$ go from 1 to 2. This can be further rewritten as

${\displaystyle \det(q^{(2)})={\epsilon ^{3ab}\epsilon ^{3cd}q_{ac}q_{bc} \over 2}}$.

The standard formula for an inverse matrix is

${\displaystyle q^{ab}={\epsilon ^{bcd}\epsilon ^{aef}q_{ce}q_{df} \over 2!\det(q)}}$.

There is a similarity between this and the expression for ${\displaystyle \det(q^{(2)})}$. But in Ashtekar variables, ${\displaystyle {\tilde {E}}_{i}^{a}{\tilde {E}}^{bi}=\det(q)q^{ab}}$. Therefore,

${\displaystyle A_{\Sigma }=\int _{\Sigma }dx^{1}dx^{2}{\sqrt {{\tilde {E}}_{i}^{3}{\tilde {E}}^{3i}}}}$.

According to the rules of canonical quantization the triads ${\displaystyle {\tilde {E}}_{i}^{3}}$ should be promoted to quantum operators,

${\displaystyle {\hat {\tilde {E}}}_{i}^{3}\sim {\delta \over \delta A_{3}^{i}}}$.

The area ${\displaystyle A_{\Sigma }}$ can be promoted to a well defined quantum operator despite the fact that it contains a product of two functional derivatives and a square-root.[14] Putting ${\displaystyle N=2J}$ (${\displaystyle J}$-th representation),

${\displaystyle \sum _{i}T^{i}T^{i}=J(J+1)1}$.

This quantity is important in the final formula for the area spectrum. The result is

${\displaystyle {\hat {A}}_{\Sigma }W_{\gamma }[A]=8\pi \ell _{\text{Planck}}^{2}\beta \sum _{I}{\sqrt {j_{I}(j_{I}+1)}}W_{\gamma }[A]}$

where the sum is over all edges ${\displaystyle I}$ of the Wilson loop that pierce the surface ${\displaystyle \Sigma }$.

The formula for the volume of a region ${\displaystyle R}$ is given by

${\displaystyle V=\int _{R}d^{3}x{\sqrt {\det(q)}}=\int _{R}dx^{3}{\sqrt {{\frac {1}{3!}}\epsilon _{abc}\epsilon ^{ijk}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b}{\tilde {E}}_{k}^{c}}}}$.

The quantization of the volume proceeds the same way as with the area. Each time the derivative is taken, it brings down the tangent vector ${\displaystyle {\dot {\gamma }}^{a}}$, and when the volume operator acts on non-intersecting Wilson loops the result vanishes. Quantum states with non-zero volume must therefore involve intersections. Given that the anti-symmetric summation is taken over in the formula for the volume, it needs intersections with at least three non-coplanar lines. At least four-valent vertices are needed for the volume operator to be non-vanishing.

Assuming the real representation where the gauge group is ${\displaystyle \operatorname {SU} (2)}$, Wilson loops are an over complete basis as there are identities relating different Wilson loops. These occur because Wilson loops are based on matrices (the holonomy) and these matrices satisfy identities. Given any two ${\displaystyle \operatorname {SU} (2)}$ matrices ${\displaystyle \mathbb {A} }$ and ${\displaystyle \mathbb {B} }$,

${\displaystyle \operatorname {Tr} (\mathbb {A} )\operatorname {Tr} (\mathbb {B} )=\operatorname {Tr} (\mathbb {A} \mathbb {B} )+\operatorname {Tr} (\mathbb {A} \mathbb {B} ^{-1})}$.

This implies that given two loops ${\displaystyle \gamma }$ and ${\displaystyle \eta }$ that intersect,

${\displaystyle W_{\gamma }[A]W_{\eta }[A]=W_{\gamma \circ \eta }[A]+W_{\gamma \circ \eta ^{-1}}[A]}$

where by ${\displaystyle \eta ^{-1}}$ we mean the loop ${\displaystyle \eta }$ traversed in the opposite direction and ${\displaystyle \gamma \circ \eta }$ means the loop obtained by going around the loop ${\displaystyle \gamma }$ and then along ${\displaystyle \eta }$. See figure below. Given that the matrices are unitary one has that ${\displaystyle W_{\gamma }[A]=W_{\gamma ^{-1}}[A]}$. Also given the cyclic property of the matrix traces (i.e. ${\displaystyle \operatorname {Tr} (\mathbb {A} \mathbb {B} )=\operatorname {Tr} (\mathbb {B} \mathbb {A} )}$) one has that ${\displaystyle W_{\gamma \circ \eta }[A]=W_{\eta \circ \gamma }[A]}$. These identities can be combined with each other into further identities of increasing complexity adding more loops. These identities are the so-called Mandelstam identities. Spin networks certain are linear combinations of intersecting Wilson loops designed to address the over-completeness introduced by the Mandelstam identities (for trivalent intersections they eliminate the over-completeness entirely) and actually constitute a basis for all gauge invariant functions.

Graphical representation of the simplest non-trivial Mandelstam identity relating different Wilson loops.

As mentioned above the holonomy tells one how to propagate test spin half particles. A spin network state assigns an amplitude to a set of spin half particles tracing out a path in space, merging and splitting. These are described by spin networks ${\displaystyle \gamma }$: the edges are labelled by spins together with 'intertwiners' at the vertices which are prescription for how to sum over different ways the spins are rerouted. The sum over rerouting are chosen as such to make the form of the intertwiner invariant under Gauss gauge transformations.

Let us go into more detail about the technical difficulties associated with using Ashtekar's variables:

With Ashtekar's variables one uses a complex connection and so the relevant gauge group is actually ${\displaystyle \operatorname {SL} (2,\mathbb {C} )}$ and not ${\displaystyle \operatorname {SU} (2)}$. As ${\displaystyle \operatorname {SL} (2,\mathbb {C} )}$ is non-compact it creates serious problems for the rigorous construction of the necessary mathematical machinery. The group ${\displaystyle \operatorname {SU} (2)}$, on the other hand, is compact and the needed constructions have been developed.

As mentioned above, because Ashtekar's variables are complex the resulting general relativity is complex. To recover the real theory, one has to impose what are known as the "reality conditions." These require that the densitized triad be real and that the real part of the Ashtekar connection equals the compatible spin connection (the compatibility condition being ${\displaystyle \nabla _{a}e_{b}^{I}=0}$) determined by the densitized triad. The expression for compatible connection ${\displaystyle \Gamma _{a}^{i}}$ is rather complicated and as such non-polynomial formula enters through the back door.

Given that a tensor density of weight ${\displaystyle W}$ transforms like an ordinary tensor except that the ${\displaystyle W}$th power of the Jacobian,

${\displaystyle J=\left|{\frac {\partial x^{a}}{\partial x^{'b}}}\right|}$

also appears as a factor, i.e.

${\displaystyle {T'}_{b\dots }^{a\dots }=J^{W}{\frac {\partial x^{'a}}{\partial x^{c}}}\dots {\frac {\partial x^{d}}{\partial x^{'b}}}T_{d\dots }^{c\dots }.}$

It is impossible, on general grounds, to construct a UV-finite, diffeomorphism non-violating operator corresponding to ${\displaystyle {\sqrt {\det(q)}}H}$. The reason is that the rescaled Hamiltonian constraint is a scalar density of weight two while it can be shown that only scalar densities of weight one have a chance to result in a well defined operator. Thus, one is forced to work with the original unrescaled, density one-valued, Hamiltonian constraint. However, this is non-polynomial and the whole virtue of the complex variables is questioned. In fact, all the solutions constructed for Ashtekar's Hamiltonian constraint only vanished for finite regularization, however, this violates spatial diffeomorphism invariance.

Without the implementation and solution of the Hamiltonian constraint no progress can be made and no reliable predictions are possible.

To overcome the first problem one works with the configuration variable

${\displaystyle A_{a}^{i}=\Gamma _{a}^{i}+\beta K_{a}^{i}}$

where ${\displaystyle \beta }$ is real (as pointed out by Barbero, who introduced real variables some time after Ashtekar's variables[15][16]). The Gauss' law and the spatial diffeomorphism constraints are the same. In real Ashtekar variables the Hamiltonian is

${\displaystyle H={\epsilon _{ijk}F_{ab}^{k}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b} \over {\sqrt {\det(q)}}}+2{\beta ^{2}+1 \over \beta ^{2}}{({\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b}-{\tilde {E}}_{j}^{a}{\tilde {E}}_{i}^{b}) \over {\sqrt {\det(q)}}}(A_{a}^{i}-\Gamma _{a}^{i})(A_{b}^{j}-\Gamma _{b}^{j})=H_{E}+H'}$.

The complicated relationship between ${\displaystyle \Gamma _{a}^{i}}$ and the desitized triads causes serious problems upon quantization. It is with the choice ${\displaystyle \beta =\pm i}$ that the second more complicated term is made to vanish. However, as mentioned above ${\displaystyle \Gamma _{a}^{i}}$ reappears in the reality conditions. There is still the problem of the ${\displaystyle 1/{\sqrt {\det(q)}}}$ factor.

Thiemann was able to make it work for real ${\displaystyle \beta }$. First he could simplify the troublesome ${\displaystyle 1/{\sqrt {\det(q)}}}$ by using the identity

${\displaystyle \{A_{c}^{k},V\}={1 \over 4}{\epsilon _{abc}\epsilon ^{ijk}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b} \over {\sqrt {\det(q)}}}}$

where ${\displaystyle V}$ is the volume. Combining this identity with the simple identity

${\displaystyle \epsilon ^{abc}\epsilon _{a'b'c}=\delta _{a'}^{a}\delta _{b'}^{b}-\delta _{b'}^{a}\delta _{a'}^{b}}$

yields,

${\displaystyle 2\epsilon ^{abc}\{A_{c}^{k},V\}={\epsilon ^{ijk}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b} \over {\sqrt {\det(q)}}}.}$

Contracting both sides with ${\displaystyle F_{ab}^{k}}$ gives

${\displaystyle 2\epsilon ^{abc}F_{ab}^{k}\{A_{c}^{k},V\}={\epsilon ^{ijk}F_{ab}^{k}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b} \over {\sqrt {\det(q)}}}.}$

The smeared Euclidean Hamiltonian constraint functional can then be written (${\displaystyle N}$ is the lapse function)

${\displaystyle H_{E}[N]=2\int _{\Sigma }d^{3}xN(x)\epsilon ^{abc}F_{ab}^{k}\{A_{c}^{k},V\}.}$

The ${\displaystyle A_{c}^{k},F_{ab}^{K}}$, and ${\displaystyle V}$ can be promoted to well defined operators in the loop representation and the Poisson bracket is replaced by a commutator upon quantization; this takes care of the first term. It turns out that a similar trick can be used to treat the second term. One introduces the quantity

${\displaystyle K=\int d^{3}xK_{a}^{i}{\tilde {E}}_{i}^{a}}$

and notes that

${\displaystyle K_{a}^{i}=\{A_{a}^{i},K\}}$.

so,

${\displaystyle A_{a}^{i}-\Gamma _{a}^{i}=\beta K_{a}^{i}=\beta \{A_{a}^{i},K\}}$.

The reason the quantity ${\displaystyle K}$ is easier to work with at the time of quantization is that it can be written as

${\displaystyle K=-\left\{V,\int d^{3}xH_{E}\right\}}$

where we have used that the integrated densitized trace of the extrinsic curvature, ${\displaystyle K}$, is the "time derivative of the volume".

In the long history of canonical quantum gravity formulating the Hamiltonian constraint as a quantum operator (Wheeler–DeWitt equation) in a mathematically rigorous manner has been a formidable problem. It was in the loop representation that a mathematically well defined Hamiltonian constraint was finally formulated in 1996.[10] We leave more details of its construction to the article Hamiltonian constraint of LQG. This together with the quantum versions of the Gauss law and spatial diffeomorphism constrains written in the loop representation are the central equations of LQG (modern canonical quantum General relativity).

Finding the states that are annihilated by these constraints (the physical states), and finding the corresponding physical inner product, and observables is the main goal of the technical side of LQG.

A very important aspect of the Hamiltonian operator is that it only acts at vertices (a consequence of this is that Thiemann's Hamiltonian operator, like Ashtekar's operator, annihilates non-intersecting loops except now it is not just formal and has rigorous mathematical meaning). More precisely, its action is non-zero on at least vertices of valence three and greater and results in a linear combination of new spin networks where the original graph has been modified by the addition of lines at each vertex together and a change in the labels of the adjacent links of the vertex.

We solve, at least approximately, all the quantum constraint equations and for the physical inner product to make physical predictions.

Before we move on to the constraints of LQG, lets us consider certain cases. We start with a kinematic Hilbert space ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$ as so is equipped with an inner product—the kinematic inner product ${\displaystyle \langle \phi ,\psi \rangle _{\text{Kin}}}$.

i) Say we have constraints ${\displaystyle {\hat {C}}_{I}}$ whose zero eigenvalues lie in their discrete spectrum. Solutions of the first constraint, ${\displaystyle {\hat {C}}_{1}}$, correspond to a subspace of the kinematic Hilbert space, ${\displaystyle {\mathcal {H}}_{1}\subset {\mathcal {H}}_{\text{Kin}}}$. There will be a projection operator ${\displaystyle P_{1}}$ mapping ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$ onto ${\displaystyle {\mathcal {H}}_{1}}$. The kinematic inner product structure is easily employed to provide the inner product structure after solving this first constraint; the new inner product is simply

${\displaystyle \langle \phi ,\psi \rangle _{1}=\langle P\phi ,P\psi \rangle _{\text{Kin}}}$

They are based on the same inner product and are states normalizable with respect to it.

ii) The zero point is not contained in the point spectrum of all the ${\displaystyle {\hat {C}}_{I}}$, there is then no non-trivial solution ${\displaystyle \Psi \in {\mathcal {H}}_{\text{Kin}}}$ to the system of quantum constraint equations ${\displaystyle {\hat {C}}_{I}\Psi =0}$ for all ${\displaystyle I}$.

For example, the zero eigenvalue of the operator

${\displaystyle {\hat {C}}=\left(i{d \over dx}-k\right)}$

on ${\displaystyle L_{2}(\mathbb {R} ,dx)}$ lies in the continuous spectrum ${\displaystyle \mathbb {R} }$ but the formal "eigenstate" ${\displaystyle \exp(-ikx)}$ is not normalizable in the kinematic inner product,

${\displaystyle \int _{-\infty }^{\infty }dx\psi ^{*}(x)\psi (x)=\int _{-\infty }^{\infty }dxe^{ikx}e^{-ikx}=\int _{-\infty }^{\infty }dx=\infty }$

and so does not belong to the kinematic Hilbert space ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$. In these cases we take a dense subset ${\displaystyle {\mathcal {S}}}$ of ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$ (intuitively this means either any point in ${\displaystyle {\mathcal {S}}}$ is either in ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$ or arbitrarily close to a point in ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$) with very good convergence properties and consider its dual space ${\displaystyle {\mathcal {S}}'}$ (intuitively these map elements of ${\displaystyle {\mathcal {S}}}$ onto finite complex numbers in a linear manner), then ${\displaystyle {\mathcal {S}}\subset {\mathcal {H}}_{\text{Kin}}\subset {\mathcal {S}}'}$ (as ${\displaystyle {\mathcal {S}}'}$ contains distributional functions). The constraint operator is then implemented on this larger dual space, which contains distributional functions, under the adjoint action on the operator. One looks for solutions on this larger space. This comes at the price that the solutions must be given a new Hilbert space inner product with respect to which they are normalizable (see article on rigged Hilbert space). In this case we have a generalized projection operator on the new space of states. We cannot use the above formula for the new inner product as it diverges, instead the new inner product is given by the simply modification of the above,

${\displaystyle \langle \phi ,\psi \rangle _{1}=\langle P\phi ,\psi \rangle _{\text{Kin}}.}$

The generalized projector ${\displaystyle P}$ is known as a rigging map.

Implementation and solution the quantum constraints of LQG.

Let us move to LQG, additional complications will arise from that one cannot define an operator for the quantum spatial diffeomorphism constraint as the infinitesimal generator of finite diffeomorphism transformations and the fact the constraint algebra is not a Lie algebra due to the bracket between two Hamiltonian constraints.

Implementation and solution the Gauss constraint:

One does not actually need to promote the Gauss constraint to an operator since we can work directly with Gauss-gauge-invariant functions (that is, one solves the constraint classically and quantizes only the phase space reduced with respect to the Gauss constraint). The Gauss law is solved by the use of spin network states. They provide a basis for the Kinematic Hilbert space ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$.

Implementation of the quantum spatial diffeomorphism constraint:

It turns out that one cannot define an operator for the quantum spatial diffeomorphism constraint as the infinitesimal generator of finite diffeomorphism transformations, represented on ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$. The representation of finite diffeomorphisms is a family of unitary operators ${\displaystyle {\hat {U}}_{\varphi }}$ acting on a spin-network state ${\displaystyle \psi _{\gamma }}$ by

${\displaystyle {\hat {U}}_{\varphi }\psi _{\gamma }:=\psi _{\varphi \circ \gamma },}$

for any spatial diffeomorphism ${\displaystyle \varphi }$ on ${\displaystyle \Sigma }$. To understand why one cannot define an operator for the quantum spatial diffeomorphism constraint consider what is called a 1-parameter subgroup ${\displaystyle \varphi _{t}}$ in the group of spatial diffeomorphisms, this is then represented as a 1-parameter unitary group ${\displaystyle {\hat {U}}_{\varphi _{t}}}$ on ${\displaystyle {\mathcal {H}}_{\text{Kin}}}$. However, ${\displaystyle {\hat {U}}_{\varphi _{t}}}$ is not weakly continuous since the subspace ${\displaystyle \psi _{\varphi _{t}\circ \gamma }}$ belongs to and the subspace ${\displaystyle \psi _{\gamma }}$ belongs to are orthogonal to each other no matter how small the parameter ${\displaystyle t}$ is. So one always has

${\displaystyle |<\psi _{\gamma }|{\hat {U}}_{\varphi _{t}}|\psi _{\gamma }>_{Kin}-<\psi _{\gamma }|\psi _{\gamma }>_{Kin}|=<\psi _{\gamma }|\psi _{\gamma }>_{Kin}\not =0,}$

even in the limit when ${\displaystyle t}$ goes to zero. Therefore, the infinitesimal generator of ${\displaystyle {\hat {U}}_{\varphi _{t}}}$ does not exist.

Solution of the spatial diffeomorphism constraint.

The spatial diffeomorphism constraint has been solved. The induced inner product ${\displaystyle <\cdot ,\cdot >_{Diff}}$ on ${\displaystyle {\mathcal {H}}_{\text{Diff}}}$ (we do not pursue the details) has a very simple description in terms of spin network states; given two spin networks ${\displaystyle s}$ and ${\displaystyle s'}$, with associated spin network states ${\displaystyle \psi _{s}}$ and ${\displaystyle \psi _{s'}}$, the inner product is 1 if ${\displaystyle s}$ and ${\displaystyle s'}$ are related to each other by a spatial diffeomorphism and zero otherwise.

We have provided a description of the implemented and complete solution of the kinematic constraints, the Gauss and spatial diffeomorphisms constraints which will be the same for any background-independent gauge field theory. The feature that distinguishes such different theories is the Hamiltonian constraint which is the only one that depends on the Lagrangian of the classical theory.

Problem arising from the Hamiltonian constraint.

Details of the implementation the quantum Hamiltonian constraint and solutions are treated in a different article Hamiltonian constraint of LQG. However, in this article we introduce an approximation scheme for the formal solution of the Hamiltonian constraint operator given in the section below on spinfoams. Here we just mention issues that arises with the Hamiltonian constraint.

The Hamiltonian constraint maps diffeomorphism invariant states onto non-diffeomorphism invariant states as so does not preserve the diffeomorphism Hilbert space ${\displaystyle {\mathcal {H}}_{\text{Diff}}}$. This is an unavoidable consequence of the operator algebra, in particular the commutator:

${\displaystyle [{\hat {C}}({\vec {N}}),{\hat {H}}(M)]\propto {\hat {H}}({\mathcal {L}}_{\vec {N}}M)}$

as can be seen by applying this to ${\displaystyle \psi _{s}\in {\mathcal {H}}_{Diff}}$,

${\displaystyle ({\vec {C}}({\vec {N}}){\hat {H}}(M)-{\hat {H}}(M){\vec {C}}({\vec {N}}))\psi _{s}\propto {\hat {H}}({\mathcal {L}}_{\vec {N}}M)\psi _{s}}$

and using ${\displaystyle {\vec {C}}({\vec {N}})\psi _{s}=0}$ to obtain

${\displaystyle {\vec {C}}({\vec {N}})[{\hat {H}}(M)\psi _{s}]\propto {\hat {H}}({\mathcal {L}}_{\vec {N}}M)\psi _{s}\not =0}$

and so ${\displaystyle {\hat {H}}(M)\psi _{s}}$ is not in ${\displaystyle {\mathcal {H}}_{Diff}}$.

This means that one cannot just solve the spatial diffeomorphism constraint and then the Hamiltonian constraint. This problem can be circumvented by the introduction of the master constraint, with its trivial operator algebra, one is then able in principle to construct the physical inner product from ${\displaystyle {\mathcal {H}}_{\text{Diff}}}$.

The Hamiltonian constraint generates 'time' evolution. Solving the Hamiltonian constraint should tell us how quantum states evolve in 'time' from an initial spin network state to a final spin network state. One approach to solving the Hamiltonian constraint starts with what is called the Dirac delta function. This is a rather singular function of the real line, denoted ${\displaystyle \delta (x)}$, that is zero everywhere except at ${\displaystyle x=0}$ but whose integral is finite and nonzero. It can be represented as a Fourier integral,

${\displaystyle \delta (x)=\int e^{ikx}dk}$.

One can employ the idea of the delta function to impose the condition that the Hamiltonian constraint should vanish.

${\displaystyle \prod _{x\in \Sigma }\delta ({\hat {H}}(x))}$

is non-zero only when ${\displaystyle {\hat {H}}(x)=0}$ for all ${\displaystyle x}$ in ${\displaystyle \Sigma }$. Using this we can 'project' out solutions to the Hamiltonian constraint. With analogy to the Fourier integral given above, this (generalized) projector can formally be written as

${\displaystyle \int [dN]e^{i\int d^{3}xN(x){\hat {H}}(x)}}$.

This is formally spatially diffeomorphism-invariant. As such it can be applied at the spatially diffeomorphism-invariant level. Using this the physical inner product is formally given by

${\displaystyle \left\langle \int [dN]e^{i\int d^{3}xN(x){\hat {H}}(x)}s_{\text{int}}s_{\text{fin}}\right\rangle _{\text{Diff}}}$

where ${\displaystyle s_{\text{int}}}$ are the initial spin network and ${\displaystyle s_{\text{fin}}}$ is the final spin network.

The exponential can be expanded

${\displaystyle \left\langle \int [dN]\left(1+i\int d^{3}xN(x){\hat {H}}(x)+{\frac {i^{2}}{2!}}\left[\int d^{3}xN(x){\hat {H}}(x)\right]\left[\int d^{3}x'N(x'){\hat {H}}(x')\right]+\cdots \right)s_{\text{int}},s_{\text{fin}}\right\rangle _{\text{Diff}}}$

and each time a Hamiltonian operator acts it does so by adding a new edge at the vertex. The summation over different sequences of actions of ${\displaystyle {\hat {H}}}$ can be visualized as a summation over different histories of 'interaction vertices' in the 'time' evolution sending the initial spin network to the final spin network. This then naturally gives rise to the two-complex (a combinatorial set of faces that join along edges, which in turn join on vertices) underlying the spin foam description; we evolve forward an initial spin network sweeping out a surface, the action of the Hamiltonian constraint operator is to produce a new planar surface starting at the vertex. We are able to use the action of the Hamiltonian constraint on the vertex of a spin network state to associate an amplitude to each "interaction" (in analogy to Feynman diagrams). See figure below. This opens up a way of trying to directly link canonical LQG to a path integral description. Now just as a spin networks describe quantum space, each configuration contributing to these path integrals, or sums over history, describe 'quantum space-time'. Because of their resemblance to soap foams and the way they are labeled John Baez gave these 'quantum space-times' the name 'spin foams'.

The action of the Hamiltonian constraint translated to the path integral or so-called spin foam description. A single node splits into three nodes, creating a spin foam vertex. ${\displaystyle N(x_{n})}$ is the value of ${\displaystyle N}$ at the vertex and ${\displaystyle H_{nop}}$ are the matrix elements of the Hamiltonian constraint ${\displaystyle {\hat {H}}}$.

There are however severe difficulties with this particular approach, for example the Hamiltonian operator is not self-adjoint, in fact it is not even a normal operator (i.e. the operator does not commute with its adjoint) and so the spectral theorem cannot be used to define the exponential in general. The most serious problem is that the ${\displaystyle {\hat {H}}(x)}$'s are not mutually commuting, it can then be shown the formal quantity ${\displaystyle \int [dN]e^{i\int d^{3}xN(x){\hat {H}}(x)}}$ cannot even define a (generalized) projector. The master constraint (see below) does not suffer from these problems and as such offers a way of connecting the canonical theory to the path integral formulation.

It turns out there are alternative routes to formulating the path integral, however their connection to the Hamiltonian formalism is less clear. One way is to start with the BF theory. This is a simpler theory than general relativity, it has no local degrees of freedom and as such depends only on topological aspects of the fields. BF theory is what is known as a topological field theory. Surprisingly, it turns out that general relativity can be obtained from BF theory by imposing a constraint,[17] BF theory involves a field ${\displaystyle B_{ab}^{IJ}}$ and if one chooses the field ${\displaystyle B}$ to be the (anti-symmetric) product of two tetrads

${\displaystyle B_{ab}^{IJ}={1 \over 2}(E_{a}^{I}E_{b}^{J}-E_{b}^{I}E_{a}^{J})}$

(tetrads are like triads but in four spacetime dimensions), one recovers general relativity. The condition that the ${\displaystyle B}$ field be given by the product of two tetrads is called the simplicity constraint. The spin foam dynamics of the topological field theory is well understood. Given the spin foam 'interaction' amplitudes for this simple theory, one then tries to implement the simplicity conditions to obtain a path integral for general relativity. The non-trivial task of constructing a spin foam model is then reduced to the question of how this simplicity constraint should be imposed in the quantum theory. The first attempt at this was the famous Barrett–Crane model.[18] However this model was shown to be problematic, for example there did not seem to be enough degrees of freedom to ensure the correct classical limit.[19] It has been argued that the simplicity constraint was imposed too strongly at the quantum level and should only be imposed in the sense of expectation values just as with the Lorenz gauge condition ${\displaystyle \partial _{\mu }{\hat {A}}^{\mu }}$ in the Gupta–Bleuler formalism of quantum electrodynamics. New models have now been put forward, sometimes motivated by imposing the simplicity conditions in a weaker sense.