Higuchi's Modified norm, Asymptotic WDW states and Refined Algebraic Quantization

 In a review I am writing with a collaborator, I am reviewing a bit of Higuchi's proposal (introduced in this paper) for a modified norm. A couple of papers into inspire, I found a very nice set of papers (see this, this and this) that indicated that there was a little more to the story of the group averaging proposal. It was first initiated by Moncrief in his paper ``Space-Time Symmetries and Linearization Stability of the Einstein Equations. 2", where it was conjectured that the infinite norm of a Hilbert space could be modified into a finite norm by dividing by the volume of the group. In the example of the Hilbert space proposal by Suvrat et al [2303.16315], where the asymptotic solutions to the Wheeler-DeWitt equation are of the form

$$\Psi [g, \chi ]=e^{iS[g, \chi ]}\mathcal{Z}[g, \chi ]\;,$$

one has to take into account of the Gauss law, which requires that these WDW states are invariant under the de Sitter isometry group $G_{dS}$. These are of the form (see the paper for a nice discussion about this and a more proper description of how these states are found to be invariant)

• Translations: $\tilde{x}^{a}=x^{a}+\epsilon ^{a}$,
• Rotations: $\tilde{x}^{a}=R^{a}_{b}x^{b}$,
• Dilatations: $\tilde{x}^{a}=\lambda x^{a}$ and 
• Special conformal transformations: $\tilde{x}^{a}=\frac{x^{a}-\beta ^{a}|x|^{a}}{1-2(\beta \cdot x)+|\beta |^{2}|x|^{2}}$.

For the $t$ coordinate, the above transformations can be found correspondingly. However, if one defines a ``seed" state built on top of the Euclidean vacuum state $|0\rangle$ (called the Bunch-Davies state) by

$$|\mathrm{seed}\rangle =\int dx_{1}\dots dx_{n}\psi (x_{1}\dots x_{n})\chi (x_{1})\dots \chi (x_{n})|0\rangle \;,$$

where $\psi (x_{n})$ is a ``smearing" function with a compact support and $\chi (x_{n})$ are a collection of $n$ massive scalar fields, we see that the only state invariant under $G_{dS}$ (imposed as a constraint) is that of $|0\rangle$. Due to the smearing function, and the nature of the group, the norm $\langle \Psi , \Psi \rangle$ is infinite. This can be interpreted as forming a one-state subspace $\mathcal{H}^{G_{dS}}$ of $G_{dS}$-invariant states from the full Hilbert space $\mathcal{H}$. To correct this, one uses Higuchi's proposal, which proved Moncrief's conjecture, and uses group averaging to define a modified norm

$$\langle \langle \Psi , \Psi \rangle \rangle =\frac{1}{\mathrm{vol}(SO(1, D+1))}\langle \Psi , \Psi \rangle \;,$$

where $\langle \langle \cdot , \cdot \rangle \rangle$ denotes the modified norm (this is somewhat against the usual convention of $(\cdot , \cdot )$, but I will be using this in a different sense below). What one is doing here is defining the space of states formed by group averaging under $G_{dS}$:

$$|\Psi \rangle =\int _{g\in G_{dS}}dg \; U|\mathrm{seed}\rangle \;,$$

where $dg$ denotes the Haar measure and $U(g)$ is a unitary operator. One then has a finite norm set of states which are invariant under $G_{dS}$ isometry group.

In the sense of Refined Algebraic Quantization (abbreviated to RAQ), the whole consideration above reduces into a mathematical problem: given an auxiliary Hilbert space $\mathcal{H}_{aux}$ and a corresponding set of $*$-algebra of observables $\mathcal{A}_{obs}$, one has to find the ``physical" Hilbert space $\mathcal{H}_{phys}$ such that the inner product satisfies the finiteness constraint, and so that the inner product on $\mathcal{H}_{aux}$ is related to $\mathcal{H}_{phys}$. The constraints in the above construction are that the states are invariant under $G_{dS}$, and essentially what we have done is to find the physical Hilbert space, which is a subspace of the initial (in the RAQ terminology, auxiliary) Hilbert space. In the sense of an $\eta$-mapping as discussed in Marolf's paper, one would find a measure so that the following modified norm converges:

$$(\Psi , \Psi )_{\eta }=\int _{g\in G_{dS}} dg\; \langle \Psi |U|\Psi \rangle \;,$$

where $(\cdot , \cdot )_{\eta }$ refers to the modified inner product defined in terms of $\eta$. Group averaging in the above sense of WDW states becomes the norm $(\cdot , \cdot )_{\eta }\to \langle \langle \cdot , \cdot \rangle \rangle$ of $\mathcal{H}_{phys}\to \mathcal{H}^{G_{dS}}$.