Refined Algebraic Quantization

As we are all well aware, the whole quantization programme of QFT is based on the so-called Wightman Axioms. We can loosely summarize this procedure into four steps: fixing a background metric $\eta_{\mu\nu}$, finding its corresponding symmetric group $\mathcal{P}$, formalizing the commutation relations and finally defining the vacuum state. From this, one can construct the full Fock space, which is the space of physical states. However, if one is to generalize this procedure to General Relativity, one will immediately run into trouble for failing to fix a background. For this and many other reasons, we must search for another way to quantize General Relativity.

Luckily, GR can be formulated as a constrained Hamiltonian system with first class constraints. The quantization of such a system had been explored by Dirac and many others and is known by the name of Refined Algebraic Quantization. This note is written in the aim of refining my understanding on this topic.

For any system, we start with a phase space $\mathcal{M}$. It has a set of constraints $C_I$ (In the case of GR, these constraints reflects the degrees of freedom in foliation and choosing spatial cordinates on hypersurfaces.) and a Hamiltonian $\mathcal{H}$.

Here let's stop to consider what a phase space with constraints will look like. $C_I$ limits $\mathcal{M}$ to a hypersurface $\bar{\mathcal{M}}$. It is also the generator of gauge transformations that leaves $\bar{\mathcal{M}}$ invariant. Therefore if we consider a point $m \in \bar{\mathcal{M}}$, a gauge $C$ generates an orbit $[m]$ on $\bar{\mathcal{M}}$. We can now assert that the physical phase space is the equivalence class of $[m]$.

For simplicity, we choose a polarization on $\mathcal{M}$ such that $\mathcal{M}$ is a cotangent bundle, and the configuration space $\mathcal{C}$ as its base manifold: $\mathcal{M} = T^\ast \mathcal{C}$. We will also assume there exist a lagrangian $\mathcal{L}$ and vanishing poisson brackets on $\mathcal{C}$. With these assumptions, we can always consider the fiber of $\mathcal{M}$, that is the momentum space. Then we compute the poisson bracket between the momentum and a function $f(q) \in C^\infty(\mathcal{C})$: $\{p,f\}(q)$. These brackets have the same structure as a deriviative of $f$: $\{p,\cdot\}(f)(q)$. It is easy to recognize this object as a vector on configuration space $v\in V^\infty(\mathcal{C})$, therefore we denote $v_p[f] := \{p,f\}(q)$, a vector field generated by $p$. This vector field no doubt carries the physical dynamics of the system and preserves the sympletric structure. We can also consider the Lie algebra $C^\infty(\mathcal{C})\times V^\infty(\mathcal{C})$ defined by $[(f,v),(f^\prime,v^\prime)] = (v[f^\prime] - v^\prime[f], [v,v^\prime])$, choosing the preferred vector field $v_p$ gives us a subalgebra $(f, v_p)$ denoted by $\mathcal{B}$.

Equipped with this algebra $\mathcal{B}$, we can construct the corresponding Hilbert space. This step is similar to the process of constructing Fock space with Poincare group in QFT, but here we won't result in a space of physical states, rather a kinematical space. We need a irreducible *-representation $\pi:\mathcal{B}\to L(\mathcal{H}_{kin})$. That is to say, $\pi$ is a linear operator on $\mathcal{H}_{kin}$ such that *-operations are adjoints and $\forall a,b\in \mathcal{B}$,

As stated by the Groenewald-van Hove Theorem, it is impossible to faithfully represent the full poisson algebra.

One can usually describe a Hilbert space in the form $L_2(\bar{\mathcal{C}},\mathrm{d}\mu)$, where $\bar{\mathcal{C}}$ is an extension of the configuration space, and $\mathrm{d}\mu$ a measure.

Even though Groenewald-van Hove Theorem limits the numbers of representations we can take, there is still plenty of reps for us to choose from. We want our reps to also represent the constraints and Hamiltonians into operators. But these corresponding operators are often unbounded and therefore not well-defined on all of $\mathcal{H}_{kin}$. We need a dense subspace which is close to Hamiltonians and constraints and can avoid infinities. We accomplish this by choosing a space of rapid decrease smooth functions $\mathcal{D}_{kin}$. By this, we mean $\mathcal{D}_{kin}$ has a finer topology than $\mathcal{H}_{kin}$, thus a topological inclusion

We now impose our constraints. The quantum version of the constraint equation $\hat{C}_Il = 0$ is basicly requiring $l$ to be invariant under gauge transformations. Because of this, when we compute the total probability of a physical state $l$, we should integrate $l$ along its gauge orbit:

which obviously diverges. This means physical states, namely states that solve the constraint equation cannot lie in $\mathcal{H}_{kin}$. Observe that $\hat{C}_I$ has continuous spectra; for self-adjoint operators like this, Gel'fand-Maurin Theorem guides us to find its eigenvectors in the algebraic dual space of a nuclear space of the Hilbert space, which is exactly $\mathcal{D}^\ast_{kin}$. This by definition is a space of linear functionals, and has pointwise divergence. Again $\mathcal{H}_{kin}$ has a finer topology then $\mathcal{D}^\ast_{kin}$. We now have the Gel'fand triple:

We now define the dual of the constraint operator as

where $l \in \mathcal{D}^\ast_{kin}$ and $f \in \mathcal{D}_{kin}$. It is a natural extension of the following relation in $\mathcal{H}_{kin}$:

Now the constraint equation becomes

elements of $\mathcal{D}^\ast_{kin}$ that solve this equation form a subspace $\mathcal{D}^\ast_{phys}\subset \mathcal{D}^\ast_{kin}$.

Quantum theory comes with anomalies. Constraints come with an algebra and a structural function $\{C_I, C_J\} = f_{IJ}^KC_K$. After quantizing, the algebra becomes $[\hat{C_I}, \hat{C_J}] = \mathrm{i}\hbar\hat{f_{IJ}^K}\hat{C_K} = \mathrm{i}\hbar\{[\hat{C}_K, \hat{f_{IJ}^K}] + \hat{f_{IJ}^K}\hat{C_K}\}$. Any physical state would now also solve the equation $[\hat{C}_K, \hat{f_{IJ}^K}]l = 0$, which itself may not be a constraint. Therefore the quantized theory may have less physical degrees of freedom than the classical theory.

As we've discussed, $\mathcal{H}_{kin} \cap\mathcal{D}^\ast_{phys} = \emptyset$, which is annoying because we can't equip $\mathcal{D}^\ast_{phys}$ with a scalar product. To resolve this, we introduce Dirac Observables. A strong dirac observable is an operator on $\mathcal{H}_{kin}$ that commutes with all constraints and are densely defined on $\mathcal{H}_{kin}$. A weak dirac observable only commutes with the constraint on the constraint hypersurface: $[\hat{O},\hat{C}_{I}]_{|C_{J}=0}=0$. These observables also have their natural dual $\hat{O}^prime$ on $\mathcal{D}^\ast_{kin}$, then a weak dirac observable is an operator that leaves the physical space invariant: $\hat{O}^\prime\mathcal{D}^\ast_{phys}\subset \mathcal{D}^\ast_{phys}$.

Now we can finally define a inner product on a subset $\mathcal{H}_{phys}\subset\mathcal{D}^\ast_{phys}$ as a positive definite sesquilinear form with respect to which $\hat{O}^\prime$ becomes self-adjoint operators, that is $\hat{O}^\prime = (\hat{O}^\prime)^\ast$. We also have $[\hat{O}_1^\prime, \hat{O}_2^\prime] = ([\hat{O}_1, \hat{O}_2])^\prime$. The commutator on the RHS is defined on $\mathcal{H}_{kin}$, but the LHS is defined on $\mathcal{H}_{phys}$. This is a nice relation because the commutation relations on $\mathcal{H}_{kin}$ are automatically tranferred to $\mathcal{H}_{phys}$. Here we still need a dense subspace $\mathcal{D}_{phys}\subset\mathcal{H}_{phys}$ where we have well defined operators for reasons explained above. We have finally resulted in a second Gel'fand triple:

This is where the physics happen.

The logic here is that when we take our dirac observables and force them to be self-adjoint, we hope these restrictions are powerful enough to limit the form of the inner product.

In very fortunate cases where constraints are mutually self-adjointing operators, namely $[\hat{C}_I, \hat{C}_J] = 0$, all constraints share the same eigenstate, therfore by spectrum theorem we can decompose $\mathcal{H}_{kin}$ into

over the spectrum $S$ of the constraint algebra. Then it's obvious that $\mathcal{H}_{phys} = \mathcal{H}_0$.

The whole process is basicly $\mathcal{H}_{kin}\to \mathcal{D}_{kin}\to \mathcal{D}^\ast_{kin} \to \mathcal{D}^\ast_{phys}\to \mathcal{H}_{phys}\to \mathcal{D}_{phys}$.

The last step is to make sure this quantum theory actually has the correct classical limit. In other words, we have to make sure that

This means the dirac observables have correct expectation values and their relative fluctuations are small. Only then we are sure we have constructed a quantized theory that admits the correct classical limit.

In the next post, we will discuss how these abstract mathematical procedures can be used to really QUANTIZE a theory with constraints.

Reference:

• arXiv
  /0210094v1