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Lattice regularizations are pivotal in the non-perturbative quantization of gauge field theories. Wilson's proposal to employ group-valued link fields simplifies the regularization of gauge fields in principal fiber bundles, preserving gauge symmetry within the discretized lattice theory. Maintaining gauge symmetry is desirable as its violation can introduce unwanted degrees of freedom. However, not all theories with gauge symmetries admit gauge-invariant lattice regularizations, as observed in general relativity where the diffeomorphism group serves as the gauge symmetry. In such cases, gauge symmetry-breaking regularizations become necessary. In this paper, we argue that a broken lattice gauge symmetry is acceptable as long as gauge symmetry is restored in the continuum limit. We propose a method to construct the continuum limit for a class of lattice-regularized Hamiltonian field theories, where the regularization breaks the Lie algebra of first-class constraints. Additionally, we offer an approach to represent the exact gauge group on the Hilbert space of the continuum theory. The considered class of theories is limited to those with first-class constraints linear in momenta, excluding the entire gauge group of general relativity but encompassing its subgroup of spatial diffeomorphisms. We discuss potential techniques for extending this quantization to the full gauge group.

In this series of papers, we present a set of methods to revive quantum geometrodynamics which encountered numerous mathematical and conceptual challenges in its original form promoted by Wheeler and De Witt. In this paper, we introduce the regularization scheme on which we base the subsequent quantization and continuum limit of the theory. Specifically, we employ the set of piecewise constant fields as the phase space of classical geometrodynamics, resulting in a theory with finitely many degrees of freedom of the spatial metric field. As this representation effectively corresponds to a lattice theory, we can utilize well-known techniques to depict the constraints and their algebra on the lattice. We are able to compute the lattice corrections to the constraint algebra. This model can now be quantized using the usual methods of finite--dimensional quantum mechanics, as we demonstrate in the following paper. The application of the continuum limit is the subject of a future publication.

This paper represents the second in a series of works aimed at reinvigorating the quantum geometrodynamics program. Our approach introduces a lattice regularization of the hypersurface deformation algebra, such that each lattice site carries a set of canonical variables given by the components of the spatial metric and the corresponding conjugate momenta. In order to quantize this theory, we describe a representation of the canonical commutation relations that enforces the positivity of the operators $\hat q_{ab} s^a s^b$ for all choices of $s$. Moreover, symmetry of $\hat q_{ab}$ and $\hat p^{ab}$ is ensured. This reflects the physical requirement that the spatial metric should be a positive definite, symmetric tensor. To achieve this end, we resort to the Cholesky decomposition of the spatial metric into upper triangular matrices with positive diagonal entries. Moreover, our Hilbert space also carries a representation of the vielbein fields and naturally separates the physical and gauge degrees of freedom. Finally, we introduce a generalization of the Weyl quantization for our representation. We want to emphasize that our proposed methodology is amenable to applications in other fields of physics, particularly in scenarios where the configuration space is restricted by complicated relationships among the degrees of freedom.

In the companion paper we motivated a renormalisation flow on Osterwalder-Schrader data (OS-data) consisting of 1. a Hilbert space, 2. a cyclic vacuum and 3. a Hamiltonian annihilating that vacuum. As the name suggests, the motivation was via the OS reconstruction theorem which allows to reconstruct the OS data from an OS measure satisfying (a subset of) the OS axioms, in particular reflection positivity. The guiding principle was to map the usual Wilsonian path integral renormalisation flow onto a flow of the corresponding OS data. We showed that this induced flow on the OS data has an unwanted feature which disqualifies the associated coarse grained Hamiltonians from being the projections of a continuum Hamiltonian onto vectors in the coarse grained Hilbert space. This motivated the definition of a direct Hamiltonian renormalisation flow which follows the guiding principle but does not suffer from the afore mentioned caveat. In order to test our proposal, we apply it to the only known completely solvable model, namely the case of free scalar quantum fields. In this paper we focus on the Klein Gordon field in two spacetime dimensions and illustrate the difference between the path integral induced and direct Hamiltonian flow. Generalisations to more general models in higher dimensions will be discussed in our companion papers., Comment: 30 pages

In this article we extend the test of Hamiltonian Renormalisation proposed in this series of articles to the D-dimensional case using a massive free scalar field. The concepts we introduce are explicitly computed for the D=2 case but transfer immediately to higher dimensions. In this article we define and verify a criterion that monitors, at finite resolution defined by a cubic lattice, whether the flow approaches a rotationally invariant fixed point., Comment: 28 pages

A possible avenue towards a non-perturbative Quantum Field Theory (QFT) on Minkowski space is the constructive approach which employs the Euclidian path integral formulation, in the presence of both ultraviolet (UV) and infrared (IR) regulators, as starting point. The UV regulator is to be taken away by renormalisation group techniques which in case of success leads to a measure on the space of generalised Euclidian fields in finite volume. The IR regulator corresponds to the thermodynamic limit of the system in the statistical physics sense. If the resulting measure obeys the Osterwalder-Schrader axioms, the actual QFT on Minkowski space is then obtained by Osterwalder-Schrader reconstruction. In this work we study the question whether it is possible to reformulate the renormalisation group non-perturbatively directly at the operator (Hamiltonian) level. Hamiltonian renormalisation would be the natural route to follow if one had easier access to an interacting Hamiltonian operator rather than to a path integral, at least in the presence of UV and/or IR cut-off, which is generically the case in complicated gauge theories such as General Relativity. Our guiding principle for the definition of the direct Hamiltonian renormalisation group is that it results in the same continuum theory as the covariant (path integral) renormalisation group. In order to achieve this, we invert the Osterwalder-Schrader reconstruction, which may be called Osterwalder-Schrader construction of a Wiener measure from the underlying Hamiltonian theory. The resulting correspondence between reflection positive measures and Osterwalder-Schrader data consisting of a Hilbert space, a Hamiltonian and a ground state vector allows us to monitor the effect of the renormalisation flow of measures in terms of their Osterwalder-Schrader data which motivates a natural direct Hamiltonian renormalisation scheme., Comment: 36 pages

This is the third paper in a series of four in which a renormalisation flow is introduced which acts directly on the Osterwalder-Schrader data (OS data) without recourse to a path integral. Here the OS data consist of a Hilbert space, a cyclic vacuum vector therein and a Hamiltonian annihilating the vacuum which can be obtained from an OS measure, that is a measure respecting (a subset of) the OS axioms. In the previous paper we successfully tested our proposal for the two-dimensional massive Klein-Gordon model, that is, we could confirm that our framework finds the correct fixed point starting from a natural initial naive discretisation of the finite resolution Hamiltonians, in particular the underlying Laplacian on a lattice, and a natural coarse graining map that drives the renormalisation flow. However, several questions remained unanswered. How generic can the initial discretisation and the coarse graining map be in order that the fixed point is not changed or is at least not lost, in other words, how universal is the fixed point structure? Is the fix point in fact stable, that is, is the fixed point actually a limit of the renormalisation sequence? We will address these questions in the present paper., Comment: 27 pages