Gravitational Hilbert spaces: invariant and co-invariant states, inner products, gauge-fixing, and BRST

Abstract

Hilbert spaces in theories of gravity are notoriously subtle due to the Hamiltonian constraints, particularly regarding the inner product. To demystify this subject, we review and extend a collection of ideas in canonical gravity, and connect to the sum-over-histories approach by clarifying the Hilbert space interpretation of various gravitational path integrals. We use one-dimensional (or mini-superspace) models as the simplest context to exemplify the conceptual ideas. We emphasise that a physical Hilbert space can be defined either by requiring states to be annihilated by constraint operators (e.g., the Wheeler-DeWitt equation) or by equivalence relations between wavefunctions, and explain that these two approaches are related by an inner product. We advocate that the group averaging procedure constructs the correct physical inner product. The Klein-Gordon inner product is not positive-definite, which we explain as arising from a bad gauge choice; nonetheless, it agrees with group averaging when such a problem is absent. These concepts are all embedded in the BRST/BFV formalism, which provides a systematic way to construct these and other physically equivalent inner products (e.g., from maximal-volume gauge and Gaussian averaged gauges). Finally we discuss the application of these ideas in the semi-classical approximation, including non-perturbative gravitational effects.

Figures (4)

• Figure 1: A summary of the relationships between the spaces of invariant and co-invariant states ($\mathcal{H}_\mathrm{inv}$ and $\mathcal{H}_\mathrm{co}$), inner products on these spaces, and maps between them.
• Figure 2: The generalised inverse relation $\eta\kappa\eta=\eta$ between group averaging and gauge-fixing maps has a geometric interpretation in terms of cutting and gluing spacetimes in the path integral. On the left, we integrate over all geometries $g_1$ and $g_2$, glued together with an operator $\kappa$ which implements a gauge condition fixing the red cutting surface, giving matrix elements of $\eta\kappa\eta$. If this is a good gauge (selecting a unique intermediate Cauchy surface in every geometry), thn the result is equivalent to the integral over all geometries $g$ which computes $\eta$.
• Figure 3: Pairings between various types of states in one-dimensional gravity arise naturally from path integrals over different sorts of intervals. A path integral over a finite interval (left) gives a pairing between co-invariants; fixing to constant lapse gauge leaves a single modulus, and integrating this over $\mathbb{R}$ gives the group-average $\eta$. A path integral over an interval which is infinite at both ends (middle) naturally describes a pairing between invariants (described by asymptotic boundary conditions); there is a residual time-translation symmetry which must be gauge-fixed by insertion of a $\kappa$ map. Finally, a semi-infinite integral has neither a modulus nor a residual symmetry, and it leads to the simple canonical pairing between one invariant and one co-invariant state.
• Figure 4: The complex $t$ plane for the group averaging integral computing $\llangle q|q'\rrangle$, where $q,q'$ are separated by a potential barrier. Red dots indicate saddle points $E=0$. Steepest descent contours are shown in black, with the red arrows indicating the descent direction (i.e., the direction of increasing $\operatorname{Im} S$). The figure is periodically repeated in the imaginary direction. The original real $t$ contour of integrating is in blue. We can deform this to the green contour, since $\operatorname{Im} S$ is large and positive as $\operatorname{Im} t\to +\infty$.