Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces

TL;DR

The paper develops a systematic quantum framework for variational discrete systems with temporally evolving discretizations, where dynamics are generated by propagators constructed from the action $S_{n+1}$ and enforced by group-averaging constraints. It shows how evolving Hilbert spaces and cylindrical consistency arise when discretizations change in time, and how constrained move composition leads to both finite state sums and potential divergences that are regularized by gauge fixing. A key insight is that physical Hilbert spaces and Dirac observables become move-dependent under coarse-graining or refining evolution, with non-unitary projections capturing irreversible changes in information content. The approach unifies covariant state-sum and canonical pictures, frames the path integral as a projector onto constraint-satisfying states, and connects to a discrete general boundary formulation relevant for quantum gravity models.

Abstract

A temporally varying discretization often features in discrete gravitational systems and appears in lattice field theory models subject to a coarse graining or refining dynamics. To better understand such discretization changing dynamics in the quantum theory, an according formalism for constrained variational discrete systems is constructed. While the present manuscript focuses on global evolution moves and, for simplicity, restricts to Euclidean configuration spaces, a companion article discusses local evolution moves. In order to link the covariant and canonical picture, the dynamics of the quantum states is generated by propagators which satisfy the canonical constraints and are constructed using the action and group averaging projectors. This projector formalism offers a systematic method for tracing and regularizing divergences in the resulting state sums. Non-trivial coarse graining evolution moves lead to non-unitary, and thus irreversible, projections of physical Hilbert spaces and Dirac observables such that these concepts become evolution move dependent on temporally varying discretizations. The formalism is illustrated in a toy model mimicking a `creation from nothing'. Subtleties arising when applying such a formalism to quantum gravity models are discussed.

Figures (5)

• Figure 1: Schematic illustration of two evolution moves $0\rightarrow1$ and $1\rightarrow2$. In a space-time context, the individual moves correspond to pieces of space-time. Each piece comes with its own set of pre-- and post--momenta and pre-- and post--constraints. The composition of two moves corresponds to a gluing of the two pieces and to solving the equations of motion or, equivalently, matching momenta at $n=1$, ${}^+p^1={}^-p^1$.
• Figure 2: Composition of the moves $0\rightarrow1$ and $1\rightarrow2$ to an effective move $0\rightarrow2$. The composition can be done by (i) matching the pre-- and post--physical Hilbert spaces at $n=1$ and subsequent regularization, or, equivalently, by (ii) directly convoluting the two propagators without any state matching. If cases (b1) and (b2) occur, the pre-- and post--physical Hilbert spaces ${}^-\tilde{\mathcal{H}}^{\rm phys}_0$ and ${}^+\tilde{\mathcal{H}}^{\rm phys}_2$ of the move $0\rightarrow2$ are non-unitary projections of the original ${}^-\mathcal{H}^{\rm phys}_0$ of the move $0\rightarrow1$ and ${}^+\mathcal{H}^{\rm phys}_2$ of the move $1\rightarrow2$, respectively (see section \ref{['sec_effcon']})).
• Figure 3: In a space-time context, different global evolution moves correspond to different regions in space-time. Here we associate the (variables in the) boundary surface $\Sigma_n$ to time step $n$. The move $0\rightarrow1$ is associated to the region $R_1$ with boundary hypersurface $\Sigma_0\cup\Sigma_1$ and boundary Hilbert space $\mathcal{H}^{\rm phys}_{0\rightarrow1}={}^-\mathcal{H}^{\rm phys}_0\otimes({}^+\mathcal{H}^{\rm phys}_1)^*$. The move $1\rightarrow2$ is associated to the region $R_2$ with boundary hypersurface $\Sigma_1\cup\Sigma_2$ and boundary Hilbert space $\mathcal{H}^{\rm phys}_{1\rightarrow2}={}^-\mathcal{H}^{\rm phys}_1\otimes({}^+\mathcal{H}^{\rm phys}_2)^*$. Gluing these regions along $\Sigma_1$ to produce the effective move $0\rightarrow2$ yields the new region $R$ with boundary $\Sigma_0\cup\Sigma_2$. The new region is associated to a new boundary Hilbert space $\mathcal{H}^{\rm phys}_{0\rightarrow2}={}^-\tilde{\mathcal{H}}^{\rm phys}_0\otimes({}^+\tilde{\mathcal{H}}^{\rm phys}_2)^*$.
• Figure 4: Schematic illustration of a discrete version of the 'no boundary' proposal. (a) A move $0\rightarrow n$ from an empty triangulation to some spherical hypersurface is devoid of pre--observables at $0$ and post--observables at $n$. (b) A further move $n\rightarrow n+x$, however, may feature pre--observables at $n$.
• Figure 5: The toy model of a scalar field on the vertices of a 2D lattice for a 'creation from nothing'. (a) The move $0\rightarrow1$ starts from an empty set and introduces one vertex with one field variable $\varphi_1$ at $n=1$. The move $1\rightarrow2$ maps to a field configuration on two vertices at $n=2$. (b) The move $1\rightarrow2$ is a singular move which corresponds to gluing a triangle onto a single vertex.

Theorems & Definitions (13)

• Remark
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Example 5.1
• Remark
• Example 6.1