Properties of the Hamiltonian Renormalisation and its application to quantum mechanics on the circle

We consider the Hamiltonian renormalisation group flow of discretised one-dimensional physical theories. In particular, we investigate the influence the choice of different embedding maps has on the RG flow and the resulting continuum limit, and show in which sense they are, and in which sense they are not equivalent as physical theories. We are furthermore elucidating the interplay of the RG flow and the algebras operators satisfy, both on the discrete and the continuum. Further, we propose preferred renormalisation prescriptions for operator algebras guaranteeing to arrive at preferred algebraic relations in the continuum, if suitable extension properties are assumed. Finally, we introduce a weaker form of distributional equivalence, and show how unitarily inequivalent continuum limits, which arise due to a choice of different embedding maps, can still be weakly equivalent in that sense.


I. INTRODUCTION AND MOTIVATION
Upon constructing a quantum theory of a given system one is often faced with many ambiguities. A common way to attack those is by starting to demand the behaviour of the quantum system at a certain (coarse) resolution. Afterwards, the theory at other scales can be determined by implementing suitable compatibility criteria, e.g. cylindrical consistency.
This method is known as renormalisation [1,2] in the context of the covariant path integral quantisation and let to many prominent applications [3,4]. There are many formulations of renormalisation, but the philosophy employed throughout this paper comes closest to the block spin transformations of lattice gauge theories [5][6][7]. While numerical investigations have proven succesful under various approximations, there are still open questions remaining on the conceptual side, e.g. the chocie of how to relate coarse degrees of freedom with those on finer scales (to which we continue to refeer to as embedding map). These issues become paramount when there is no comparison to experiments yet, e.g. if turning towards avenues for quantum gravity [8][9][10][11][12][13].
On the Hamiltonian side, utilising the renormalisation group (RG) for constructing quantum field theories (QFT) is best enunciated in the language of inductive limits [14][15][16][17][18]. As of today the Hamiltonian renormalisation is less developed then its covariant counterpart and thus many conceptual questions remain unanswered as well (such as the role of embedding maps, the final interpretation of the limit Hilbert space etc.). In this paper, we address some of these problems and demonstrate the consequences exemplary for the case of 1-particle quantum mechanics on the circle. * benjamin.bahr@desy.de † klaus.liegener@desy.de In section II we will discuss the properties of the Hamiltonian RG on the general level, using the formulation of inductive limits: an inductive family is a family of Hilbert spaces endowed with suitable embedding maps. From the physical point of view, we may interpret each Hilbert space as the collection of those states which can be fully described at some coarse resolution M .
Here the resolution M serves as a generalisation of a UV-cutoff scale, in the sense that it specifies, up to which resolution information of the system is accessible. In QFT, this is the equivalent of e.g. lattice spacing, while in quantum mechanics with finitely many degrees of freedom, this resolution specifies a finite-dimensional subspace of the full Hilbert space. The advantage of this notion of resolution and coarse graining lies in the fact that the mathematical framework does not necessarily require a background metric, which allows for a notion of background-independent renormalisation scheme, which is specifically useful for approaches to quantum gravity, see e.g. [11,12,[18][19][20][21][22][23][24][25][26][27][28][29][30][31].
The fact that there exists also finer resolutions M ′ by which those states can be described without losing information is encapsulated via the embedding maps which embed the Hilbert space of resolution M into the one of resolution M ′ . For an inductive family the embedding maps are suitably chosen, such that they finally allow the restoration of a inductive limit or continuum Hilbert space, i.e. the collection of states at all scales.
The choice of embedding map encapsulates significant physical information, and can make the renormalisation of the system harder or easier, depending on whether it fits together well with the dynamics of the system. Indeed, the central point of approaches like MERA or TNR (see e.g. [32][33][34]) is to construct the correct embedding maps for a given Hamiltonian. In this case, the embedding maps contain all the information about the continuum vacuum state.
Assuming that such an inductive family of Hilbert spaces is given, we focus our attention on observables of the continuum Hilbert space. We recall in section II A that a family of operators on the Hilbert spaces of finite resolution which obeys weak (/strong) cylindrical consistency can be promoted to a bilinear form (/operator) on the inductive limit Hilbert space. A typical application of the RG is constructing such a cylindrical consistent family starting from some initial ad-hoc choice, i.e. a discretisation.
Every fixed point of the RG flow yields a cylindrical consistent family, however the precise form of the family is dependent on both the choice of embedding map as well as the initial discretisation. However, when being interested in the whole set of possible fixed points, we show in section II C that actually one of those data is redundant: one may either fix the embedding map once and for all and study the RG flow of all possible initial discretisations (or vice versa) without loosing any cylindrical consistent families.
In general, one is not interested in a cylindrically consistent family for a single observable, but usually a whole algebra thereof (constituting the set of possible questions to ask to the system). Such an algebra is characterised by the algebraic relations between its elements (e.g. commutator relations) and we investigate in section II D how the RG flow of algebraic relations between operators at discrete level translates to operators on the continuum Hilbert space (hereby we often need to make strong assumptions that the bilinear forms are extendible). In subsection II E we reformulate our findings in a constructive criterion to obtain cyl. cons. algebras obeying given algebraic relations.
Finally, in section II F we ask about the interpretation of observables in the continuum Hilbert space as part of the space of distributions D ′ . Closable operators can be understood as suitable restrictions of maps from A ′ : D ′ → D ′ . Simultaneously, different inequivalent fixed point theories may be embeddable into D ′ with the help of suitable faithful embeddings. In case that these embeddings approximate different restrictions of the same map A ′ , we call them weakly equivalent.
In order to show-case our findings, section III considers discretised quantum mechanics on the circle. We study two embedding maps and their RG flow starting from a common initial discretisation leading to two inequivalent fixed points. However, we show that ultimately both fixed points are weakly equivalent as both approximate the continuum.
In section IV we conclude with an outlook for further research directions.

A. Inductive limits
We recall basic facts from the theory of inductive limits and streamline the notation. We omit the corresponding proofs and refer to [14] or the appendix of [18] for further details.
The starting point of each inductive family are partial Hilbert spaces H M , which we take to be finitedimensional (if not specified otherwise). The label M belongs to a partially ordered, directed set I, i.e. there is a relation M ≤ M ′ , such that for every two M, M ′ there is a M ′′ such that M ≤ M ′′ and M ′ ≤ M ′′ . We define embedding maps for M ≤ M ′ as which are isometries and satisfy as well as I M→M = id HM . Given this data, we can define where ψ ∼ I M→M ′ φ for φ ∈ H M and ψ ∈ H M ′ . The completion of the pre-Hilbert space D ∞ is the continuum Hilbert space Continuum embedding maps I M : H M → H ∞ are then given by Then, if M ≤ M ′ it is clear that On Hilbert spaces, we denote operators byÂ. Each operatorÂ determines a bilinear 1 form Note the opposite is not true: Assume that A : D × D → C, is a bilinear form, where D ⊂ H is any dense domain containing an orthonormal basis {e n } n . Then there is an operatorÂ : for all ψ ∈ D. We also use the notation for the kernel with respect to an ONB {e n }. This means that iff n |A(m, n)| 2 < ∞ for all m, thenÂ can be defined on at least the finite linear span of the e n .
In the following we consider constructing bilinear forms and operators on H ∞ , if those on H M are given.

Definition 1 A family of bilinear forms {Â
∀ψ, φ ∈ H M . Given a cylindrically consistent family of bilinear forms, we can define a bilinear form on the continuum in a straightforward way. Some families allow directly to define operators on the continuum level: we say that a family of operators Note that the bilinear forms defined by strongly cylindrically consistent operators are weakly cylindrically consistent, but not the other way round. We expect interesting physics to be encoded in operators on H ∞ which are the extension of weakly consistent forms but not satisfying (12). This is due to the fact that (12) impliesÂ M to preserve subspace of resolution M , which is typically violated in applications.
We make some further assumption about the label set I, which are not strictly necessary, but which make our life much easier, and comply with most interesting physical situations: Firstly, we assume that there exists a smallest element M 0 ∈ I. This will correspond to the maximally possible coarse graining, and often appears in situations in which no infra-red divergences appear, e.g. with compact spatial manifolds. Secondly, we assume that there are amenable subsequences, i.e. sequences I 0 ⊂ I such that for every M ∈ I there is an M ′ ∈ I 0 with M ≤ M ′ . If both assumptions are true, one can replace I by I 0 ≃ N, and obtain equivalent Hilbert spaces, operators, etc. In particular, it will make the RG flow equations simpler, which is why we will assume we can work with the label set from now on, without loss of generality.

B. The RG flow equations
The RG flow is a tool to construct (weakly) cylindrically consistent families of bilinear forms 2 .
In practice, one starts with a sequence of bilinear forms (o) A M , which are not necessarily cylindrically consistent. Then, we define where ψ, φ ∈ H M . Technically, one can define the flow iteratively 3 (15) and consider the limit of the (n) A M for large n. If this limit exists, the resulting A M define a cylindrically consistent family of bilinear forms.
Note, that for the RG flow of bilinear forms, it is never guaranteed that the limit converges to a meaningful form a priori. This is highly dependent on the choice of initial discretisation and embedding map. A counter-example is presented in the paragraph on the momentum operator in III D, where an initial discretisation together with certain choice of embedding map gives a good limit, while another choice of embedding map sends the same initial discretisation to zero.

C. Unitary Equivalence of Embedding maps
Under quite general assumptions, different discretisations are equivalent on the level of Hilbert spaces.
To be precise, we assume that we have given two families of partial Hilbert spaces H Proof. First, we show that for all M there exist bijections ξ M : H 2 Indeed, given a family of partial Hilbert spaces H M and maps I M →M ′ which are not necessarily isometries, one can use the RG flow to modify the inner products on each H M in order to arrive at a cylindrically consistent family of Hilbert spaces of which one can take the continuum limit as described above. See [18] for details. In this article however, we always assume that the maps I M →M ′ are isometries. 3 The flow is defined generally, the iterative version exists only if the label set has an amenable subsequence.
We build ξ M by recursion over M = 2 L , starting from ξ M=1 = 1. For 2M = 2 L+1 we define the unitary isomorphism on the image of embedding maps as: M . To be precise, we can choose some orthonormal bases {e M and finish the construction of ξ 2M by defining: This gives a sequence of ξ M obviously satisfying Thus, this defines an isometry ξ ∞ : D ∞ , which can be completed to the respective continuum Hilbert spaces H This finishes the proof for the finite-dimensional case. In case of infinite-dimensional separable H M , the proof runs similarly, with one additional condition.
Then, as all infinite-dimensional, separable Hilbert spaces are isomorphic to ℓ 2 by using Zorn's Lemma, there also exists an isomorphism ξ on the inductive limit Hilbert spaces.
A corollary of those lemmas is that each cylindrical consistent family of operators on H (1) M can be unitarily mapped to a cylindrical consistent family on H (2) M such that the inductive limit operator families are equivalent as well. Further, also the algebraic relation between the operators in the continuum will be the same. That is, the resulting Hilbert spaces and operators in the continuum are unitarily equivalent and thus the precise choice of embedding map does not matter at least conceptually. Hence, while in principle one has to choose an initial discretisation and an embedding map in order to study some RG flow, conceptually it suffices to restrict the attention to one initial discretisation and several different embedding maps (as will be done in section III) or vice versa. However, it must be noted that in practice, one typically needs to do some approximations while computing an involved RG flow. Now, different embedding maps will more or less susceptible to different approximation procedures, thus the map should be chosen such that possible errors during the approximations process do not influence the resulting physics. For further discussion on this subject see [35] and references therein.

D. Algebraic relations of operators
Physical observables are encoded as operators on the continuum Hilbert space. The RG flow constructs bilinear forms, which can be turned into operators if criterion (8) is satisfied. However, whether the correct physics is implemented, is governed by the algebra satisfied by the respective operators.
Here, one encounters an interesting tension between the continuum operatorsÂ and the partial operatorsÂ M defined at a specific coarse graining scale. Namely, the partial and continuum operators satisfy the same algebra if the partial operators are strongly cylindrically consistent, i.e. satisfy (12). However, in practice the operators on the continuum Hilbert space are described by only weakly cylindrically consistent operators, i.e. whose bilinear forms satisfy (10), and for which the correct algebra does not have to hold both on H ∞ and H M . I.e. while the continuum physics is correctly represented, at finite discretisation scale there will be anomalies. These can be interpreted as discretisation artefacts. An interesting example will be given in section III D. Nevertheless, given partial operatorsÂ M ,B M , . . . for which the limit (14) exists, one can make certain statements about the RG flow of their algebraic relations.
Then, the RG flow of (o)Ĉ M :=Â MBM -defined via (14) -converges, and gives rise to a quadratic form C ∞ on D ∞ which exists as operatorĈ ∞ on D ∞ satisfyinĝ Proof. See appendix C. The proof is indeed given for an arbitrary monomial.
There is an interesting consequence of this theorem, which alludes to our earlier statement: Even though theÂ M ,B M are cylindrically consistent, the (o)Ĉ M = A MBM are not. However, as a starting point for the RG flow, they flow toĈ M which are the partial operators of C ∞ =Â ∞B∞ . If theÂ M ,B M are not strongly consistent, however, the RG flow is nontrivial, and even though (19) holds, one hasĈ This impacts the operator algebras defined on the continuum Hilbert space, since the above statement extends to arbitrary monomials, and commutators. In particular, even though the correct algebra might be satisfied at the continuum level, at the level of the partial Hilbert spaces, anomalies can arise. Thus, one should not check whether a set of weakly consistent operators satisfies certain algebraic relations, but could instead test whether the algebraic relations are satisfied in the limit of the RG flow in order to guarantee the correct continuum physics.
It should be noted that this feature is conceptually related, but different from the breaking of symmetries by discretisations, discussed in [10,36]. In particular, thê A M ,B M andĈ M are the perfect discretisations from the continuum, but their algebra is still anomalous due to discretisation arefacts.

E. Modified RG flow
Typically, one is interested in the quantisation of some specific algebra of observables A. Can one use the RG flow to enforce its algebraic relations for the continuum operators?
By virtue of its definition, every algebra is endowed with a map A × A → A, capturing its defining algebraic relations. As example, consider for the bilinear product "·" and assume that for all A, A ′ ∈ A there exist finitely many z a ∈ C such that where α is some labelling of the elements in A.
Given the set-up of section II B, one might attempt to implement a quantisation of A in the following way: first, take some initial choice of operators (o)Â M on H M . For each, construct the weak cylindrical consistent fam-iliesÂ M following (14) and collect those to obtain the candidate-quantisation set A M . Even under the assumption that the bilinear forms on D ∞ turn out to be extendible to operators, one must finally check, whether those operators fulfil the algebraic relations (21) of A. In general, nothing guarantees that the algebra closes. However, if the resulting bilinear forms are extendible to operators, let us note that a sufficient criterion for obtaining the correct algebra comes in a slight variation of then the correct algebraic relations are obtained for the operators in the continuum. 4 Ultimately, it would be advantageous to have a constructive prescription which returns operators satisfying the correct algebraic relations in the continuum. In some situations, this can be indeed achieved: Theorem 2 Given an algebra A whose algebraic relation can be brought into the following form: Starting from some initial data (o) A M , the fixed points of the modified RG flow resurrect the correct algebra in the continuum if being extendible to operators on D ∞ , i.e. satisfy (8).
In situations where this theorem applies, starting from some suitable initial discretisation one may study the RG flow of (24) under a given embedding map. It will lead either (i) to a trivial fixed point, (ii) not converge, or (iii) to a theory which obeys the correct algebra for M → ∞ however not necessarily for finite M .

F. Distributional Embedding
Finally, although not all possible fixed pointed will algebras agree in their properties, often one may interpret them as approximating the same continuum theory via distributional embeddings.
First, we introduce a suitable space of distributions: Assume that we have a (finite-dimensional) manifold M with non-degenerate metric, and the space which can be given a nuclear topology, yielding the Gelfand triple with the Hilbert space H = L 2 (M, dvol), and D ′ being the topological dual of (25), the members of which can be regarded as distributions over M. In the following, we write ϕ, f to denote the pairing of the distribution ϕ ∈ D ′ and the function f ∈ D.
Next we assume that we have a collection of (finitedimensional) Hilbert space H M and embedding maps I M→2M , which are used to define the continuum Hilbert space as in (4). A priori the two Hilbert spaces H and H ∞ do not have anything to do with one another. To connect them, we define the notion of a faithful embedding: By construction, a faithful embedding defines a map which is a linear isomorphism onto its image. We also call this map the embedding, if no confusion arises.
Note that a faithful embedding satisfies that for all In general, comparison of the two continuum theories on H and H ∞ is straightforward if φ ∞ can be extended to H ∞ : for all M and all ψ ∈ H M . To see (i), consider a Cauchy sequence {ψ n } n in D ∞ . Then, for any function f ∈ D one has, by regularity, that For (ii), let ψ n be a Cauchy sequence in D ∞ converging to ψ ∈ H ∞ . Assume φ ∞ (ψ) = 0, then Corollary 1 A faithful, regular, separating embedding lets us realise H ∞ as a subspace of D ′ , with its own inner product.
It can happen that the thus realised φ ∞ (H ∞ ) and the original H have only the zero in common. Still, a closable operatorÂ densely defined on D ⊂ H can be carried over to a linear mapÂ via: • a faithful embedding {φ M } M , such that: • The bilinear form A ∞ is extendible to an operator • The faithful embedding φ ∞ can be extended to φ ∞ : Then, the discretised system is said to approximate the continuum if there is a densely defined, closable operator A few remarks are in order: The above notion of weak equivalence states that there is a way, in which the partial operators A M approximate the continuum operatorÂ. However, since they live on potentially different Hilbert spaces, this approximation has to be established carefully. In particular, the continuum bilinear form A ∞ might exist as an operator on D ∞ , but might not leave it invariant. It can happen that it has to be extended to some larger space D = D A + D ∞ , and only its restriction to D A is a closable operator with invariant domain of definition. 6 On the other hand, in order to make contact with distributions, the faithful embedding φ ∞ needs to be extendible to D as well. Therefore, this notion of weak equivalence is a subtle condition on the interplay of the operators A M and the φ M . We will see an example for this in section III F. The denseness of φ ∞ (D A ) ⊂ D ′ , together with the given denseness of D ⊂ D ′ [37], means that states in either space can be approximated by states from the other space in the sense of distributions. From a physical point of view, this means that by performing measurements by taking inner products with elements in D with finite precision, there is no way of distinguishing whether a state is in either D or φ ∞ (D A ). Of course, if one has access to the respective spectra of the operators with infinite precision, one might distinguish which system one is working with, since the systems (H,Â) and (H ∞ ,Â ∞ ) do not have to be unitarily equivalent. However, one can argue that that this knowledge is unattainable if one is working with only finite measurement precision.
Finally, we call two discretisations weakly equivalent, if both approximate the continuum. In fact, in that case they also approximate each other in the above sense.

III. DISCRETISED QUANTUM MECHANICS ON THE CIRCLE
This section, puts the lessons learned from the previous section to an explicit test: Let us consider the Weyl algebra of exponentiated operators on the unit circle. In some regards, this is mathematically much more convenient than position and momentum operators for quantum mechanics on the circle. 7 To showcase the equivalence properties, we consider inductive families with two different embedding maps I (i) M→M ′ .

A. Continuum theory
We begin with the original continuum Hilbert space H = L 2 (S 1 ) ≃ L 2 ([0, 1)). In the following, we represent states in H as functions with the inner product On H, we define operatorsÛ andT (l) with l ∈ [0, 1) One can think of these asÛ = exp(2πix) andT (l) = exp(ilp/ ). They satisfy the relationŝ Later on, in subsection III D, we will consider to a discretisation of the operatorsÛ ,T (l), on two different families of inductive Hilbert spaces H (i) M , which we construct in the following subsection.

B. Discretisations of H
We introduce two different ways to discretise the continuum Hilbert space, by representing it as the inductive limit of two different families {H From these follow arbitrary I (i) 2 L →2 L ′ for L ′ > L, and consequently result in an inductive limit Hilbert space The dense subspace is endowed with the equivalence relations for either i = 1, 2 generated by for any f ∈ M (i) and all M . We also denote the canonical unitary embedding maps (52)

C. Unitary equivalence
It is clear that H ∞ and H ∞ are unitarily equivalent, since both are separable Hilbert spaces. Even further though, there is a unitary equivalence which is consistent with the two discretisations, in the sense that it intertwines the embedding maps (see subsection II C): Claim: There exists a family of unitary maps which satisfy Proof. To construct these unitary maps, we first note that there is a unitary isomorphism given by for m = 1, . . . , M − 1, m ′ = 0, 1. In what follows, we will suppress χ M , for legibility, and implicitly use the isomorphism (55). This will allow to compute the unitary maps by induction over M . First, it is clear that ξ 1 : C → C is given by the identity map. From this we can see that (up to a complex number), the unique ξ 2 satisfying (54) for M = 1 is given by where (57) is the matrix representation with respect to the orthonormal basis elements (2) δ m , m = 0, 1. Furthermore, we define for M = 2 L , using the isomorphism (55). Since, with respect to this isomorphism, the embedding maps can be written as This shows that (58) defines an intertwining map, as can be checked by direct computation. For instance, the unitary maps for M = 4 and M = 8 can be written as where each matrix is written with respect to the ONB consisting of which satisfies with the canonical embedding maps (52).

D. Discretisation of operators & RG flow
At this point, we consider a discretisation of the fundamental operators (40) for both cases i = 1, 2 and investigate their RG flow. For completeness (and to highlight why we are interested in the Weyl operators) we will also discuss at the end the RG fixed points of the non-exponentiated momentum operator.
For i = 1, 2, we define discrete versions of the basic operators via where l can take on the discrete values Note that the discretised operators satisfŷ which mimics the continuum algebra (41), the difference being that l can take on only discrete values. Indeed, each of the H (i) M thus becomes a representation space of a subalgebra of the continuum algebra.

Case i = 1
It can be shown straightforwardly that both operatorŝ U   which is not all of [0, 1), but lies densely in it. These continuum operators also straightforwardly satisfy the alge-braÛ The continuum limit therefore appears as a representation of a subalgebra of the original algebra (41), in that only l ∈ P ⊂ [0, 1) are allowed.

Case i = 2
The situation is more involved in the case of i = 2. Notably, the operatorsT (2) M are also strongly cylindrically consistent with respect to the embedding maps I Hence, theÛ M are not even weakly cylindrically consistent in the sense of bilinear forms. Therefore, they do not define a continuum bilinear form on H (2) ∞ . However, we can use the RG flow to construct a proper continuum operator: Claim: The RG flow, as defined in (14), of embedding map I and hence, iterating this: This allows us to explicitly compute the p-th step of the RG flow as Hence, the RG flow has a fixed point, and the resulting limit partial operators define a continuum bilinear form U For this example, we are in the advantageous position that U ∞ already satisfy the correct algebraic relations (41). For the sake of completeness, we shall also test the prescription of subsection II E which guarantees a priori that its fixed points will restore the correct Weyl-relation in the continuum. For this purpose, we start with (o) U M and (o) T M and study the flow of (24). For (o) T M , one checks easily that it is already at its fixed point. For (o) U M the modified RG equations read explicitly: After one step, we obtain: e 2πi(m+2 −N s)/M δ mn = e 2πi m+1/2 M δ mn sinc(π/M ) as was already shown for (69). Indeed, this is already a fixed point of (24), since (1) U ∞ (m, n) = (o) U ∞ (m, n) for all m, n, i.e. the bilinear forms in the continuum agree and only those enter the flow of (24).

Momentum operators
Additionally to the translation operatorsT (l), on the continuum Hilbert space L 2 ([0, 1], dx) the infinitesimal version, i.e. the momentum operator exists. This operator is self-adjoint an unbounded, i.e. it is only defined on a dense domain of H, which, however, includes D [38]. The momentum operator is essentially a differentiation w.r.t. x, and neither discretisation i = 1, 2 straightforwardly allows such a differentiation, since x can take only discrete values. Still, one can construct a finite difference operator and investigate its properties. On the Hilbert spaces H So this series of sesquilinear forms ist not cylindrically consistent. Moreover, by the above calculation we have confirmed that, in the RG flow, the family of forms becomes the zero form, after only one RG step.
Turning to i = 2 next, one can show that M (e M m , e M n ) Hence, this family of sesquilinear forms is already cylindrically consistent, and therefore straightforwardly defines a sesquilinear form For f, g ∈ H (2) M , one has that and hence ∂ M is anti-Hermitean on H M , due to our periodic boundary conditions (74). Therefore, also ∂ for ψ, φ ∈ D = C ∞ (S 1 ). Indeed, ∂ ∞ can be extended continuously (in the nuclear topology on D) to a sesquilinear form on the restriction of which on D then agrees with (77). In that sense, the ∂ M are the correct discretisation of the continuum derivation operator. This shows that, even though the collection of partial Hilbert spaces correctly approximate the continuum, and certain physical observables can be defined properly on the continuum, the issue of finding the correct dense domain of definition for the resulting operator is not necessarily addressed or solved by the RG flow procedure. In the specific example of the momentum operator, this has to do with the fact that the discretisation and some aspects of the continuum operators (such as differentiation) are at odds with one another. This can happen in a strong sense, as with i = 1, where the naive discretisation of the continuum operators immediately flows to the zero operator, or in the more subtle case i = 2, where the naive discretisation leads to a continuum form, for which the correct dense domain of definition still has to be found, in order to make it into an operator on the continuum Hilbert space.

E. Comparison of continuum limits
So far, we have considered two different discretisations of quantum mechanics on the circle, and considered their respective continuum limits. In both cases we obtained continuum Hilbert spaces H and therefore extend to an isometry ξ ∞ between the two continuum Hilbert spaces. However, as a quick calculation shows, these isometries do not intertwine the discrete operatorsÛ 8 As a result, one has e.g.
ξ ∞Û (1) Indeed, there is no unitary equivalence between the two continuum Hilbert spaces and the respective operators. This can be seen easily e.g. by observing that every e M m is a normalisable Eigenvector ofÛ (1) ∞ , whileÛ (2) ∞ has U (1) as its continuous spectrum. Also, since H (2) ∞ ≃ L 2 (S 1 ) with the standard quantisation ofÛ andT (l) (see section III A), the family (2)T ∞ (l) can be extended to e 2πil ∈ U (1), which is a strongly continuous 1-parameter family. On the other hand, while l n := 2 −n converges to 0, the vector (1)T ∞ (l n ) e 0 1 is orthogonal to e 0 1 , therefore the family (2)T ∞ (l) can not be extended continuously to real l. In particular, there is no self-adjointp such that It should be noted that this makes this type of quantisation conceptually very similar to Loop Quantum Gravity, and (if one replaces the circle by the real line), goes by the name Polymer Quantum Mechanics (see [39]).
The fact that both quantisations feature operators with different spectra, highlights the fact that in general several unitary inequivalent representations of the same algebra exist. And the method of inductive limit can not necessarily serve as a procedure to physically distinguish them. However, it can help in classifying them, as they can all be obtained as the fixed points of the RG flow for one and the same choice of embedding map and different suitable initial starting points. Here we show that although the two discretisations for i = 1, 2 are unitarily inequivalent, they are at least 8 In the case of theÛ (i) M this can easily be seen by the fact that one of the two is unitary, the other is not. It is, however, also true for theT (l) weakly equivalent in the sense that they approximate the same continuum physics.
To show this, we construct the embedding maps φ where δ x denotes the delta-distribution at x ∈ [0, 1), and χ [a,b) the characteristic function on the interval [a, b). It turns out that we cannot compare both H (i) ∞ directly, as only one embedding allows to be extendible to the corresponding Hilbert space: Claim: For both discretisations, the maps φ M are faithful embeddings maps. Also, the case i = 2 is regular and separating, while i = 1 is not regular. Proof.
(i) We need to verify (29) and since I Hence, both φ and hence, by the Cauchy-Schwartz inequality: To see that φ ∞ can not be regular, it is enough to consider the Cauchy sequence where {e k } denotes an ONB lying in D ∞ such that e k = e m M for some M, m, i.e.
with x k being a sequence in P without any element being hit more than once. It is clear that φ (1) ∞ (ψ n ) does not converge in D ′ , e.g. by considering its action on the constant function: which diverges, while ψ n converges in H ∞ . From this it follows that for both discretisation i = 1, 2 there exist faithful embeddings φ It is also straightforward to show that the extension of φ (2) ∞ to H (2) ∞ is separating, i.e. no element in H (2) ∞ gets mapped to the zero distribution.
Thus, we can try to establish at least weak equivalency. We will do this now, by first confirm the conditions of definition 4.
Claim: Both faithful embeddings φ Proof. For i = 2 this is easy to show, since the image φ ∞ ) is actually the space of (regular 9 ) piecewise constant functions on [0, 1), which are well-known to be dense in C 0 ([0, 1)) in the uniform topology, which is stronger than the weak- * -topology, and C 0 ([0, 1)) is dense in D ′ in the weak- * -topology, so we are done. Indeed, we have shown that one can canonically identify For i = 1, it is enough to show that it is dense 10 in D.
Hence, let f ∈ D. It is now enough to show that there is a sequence ψ n in φ (1) ∞ ) which converges to f in the weak- * -topology, i.e. that, for every g ∈ D we have lim n→∞ ψ n , g = f, g = As one can see from (83), each ψ n is in φ Since Riemann sums approximate L 2 -integrals, this clearly converges to f, g as n → ∞, and we are done. Note that the sequence of the ψ n also converges in the stronger L 2 -topology, albeit always to the zero vector in φ (1) It remains to show that not only the Hilbert spaces are weakly equivalent, but also the operators U Proof: In order to show their weak equivalence, one needs to show that they are both weakly equivalent to the original continuum physics, i.e. one has is an invariant domain of definition of bothÛ for some x ∈ P . Thus by anti-linearity, we get Thus we have which shows the claim forÛ ∞ the claim follows directly from the translation invariance of the integral.
For i = 2, we note that the operatorsÛ  ∞ , which is also true for φ (2) ∞ , since the faithful embedding is regular and separating. Therefore, we are free to choose any dense invariant domain. Since φ  ∞ ) −1 (D) coincides with the pull-back of the continuum operatorÛ . The proof is given in appendix B. The same can be trivially shown for the translation operator. Thus, the claim is shown.

IV. CONCLUSION
In this article we have studied different properties of the Hamiltonian Renormalisation. We showed that there is (at least conceptually) a redundancy when looking at embedding maps and initial discretisations. One of both may be fixed without loosing information. This could in principle motivate to pick once and for all an embedding map for which one has control over the numerical approximations and look at the fixed points obtained from several initial discretisations. Of course, different initial discretisations will in general flow into different fixed points and we did not establish criteria to identify whether those turn out to be trivial or physically interesting -this remains to be checked a posteriori just as in the path-integral renormalisation framework. However, in contrast to there, we could formulate a modified RG flow, which (in suitable situations) will drive initially discretised algebras of operators 11 to those fixed points which restore the correct continuum commutation relations (given that the fixed point theory is not trivial or divergent). This improves over the situation in covariant RG. We checked these claims explicitly for the case of quantum mechanics on the circle: we proposed an initial discretisation and two embedding maps such that two relevant fixed points were found. In both cases exponentiated momentum and position operator fulfil the Weyl algebra in the continuum. Importantly, both fixed point theories are such that the operators are unitarily inequivalent, e.g. they have different spectra. This highlights that indeed there exist fixed points with different physical properties and further input is required to single a preferred one out.
Finally, we introduced the notion of "weak equivalence" between inductive limit-theories.
Taking advantage of usually being interested in the distributions D ′ over the Hilbert space H ⊂ D ′ as well, we established a condition under which even unitary inequivalent theories can be embedded into the same D ′ such that they both approximate the theory on H. This was again verified for the afore-mentioned test case. Thus, although the fixed point theories were different, in the sense of weak equivalence both are valid descriptions of the same system. This hints at even more hidden redundancy in the choice of initial data for investigations of the RG flow.
We hope that further research in this direction helps in identifying physically interesting theories for cases where the continuum QFT effects are yet unknown such as quantum gravity. Hence, we have shown that the partial bilinear forms of the product of the continuum operators indeed coincide with the limit of the RG flow of the product of the partial operators. This finishes the proof.
It should be noted that it is easy to see that this proof straightforwardly extends to a product of N operators. The crucial point is the appearance of multiple limits by introducing several resolutions of unity, which however clearly commute.
We point out the necessity of both operators leaving the inductive-limit domain D ∞ invariant. This restric-tion is indeed unavoidable and cannot easily be lifted as one convinces himself with the following example: consider the momentum operatorp from paragraph 3 of subsection III D whose domain of definition D does not agree with the inductive-limit domain D (2) ∞ on which a weakly cylindrical consistent bilinear form was obtained (see 78). Now, forÂ =B :=p albeit their product AB =p 2 exists on D it does not exist on D (2) ∞ and fittingly also the RG flow of C M := (∂ M ) 2 diverges, invalidating the generality of theorem 1 for cases where D = D ∞ .