Crossed Products, Conditional Expectations and Constraint Quantization

Recent work has highlighted the importance of crossed products in correctly elucidating the operator algebraic approach to quantum field theories. In the gravitational context, the crossed product simultaneously promotes von Neumann algebras associated with subregions in diffeomorphism covariant quantum field theories from type III to type II, and provides the necessary ingredients to gravitationally dress operators, thereby enforcing the constraints of the theory. In this note we enhance the crossed product construction to the context of general gauge theories with arbitrary combinations of internal and spacetime local symmetries. This is done by leveraging the correspondence between the crossed product and the extended phase space. We then undertake a detailed study of constraint quantization from the perspective of a generic crossed product algebra. We study and compare four distinct approaches to constraint quantization from this point of view: refined algebraic quantization, BRST quantization, path integral quantization, and the commutation theorem for crossed products. Far from simply reproducing existing analyses, the operator algebraic viewpoint sheds new light on old problems by reformulating the dressing of operators in terms of conditional expectations and other closely related projection maps. We conclude by applying our approach to the constraint quantization of three distinct gauge theories including a discussion of gravity on null hypersurfaces.


Introduction
In recent work [1] we have argued that the operator algebra associated with a subregion in a diffeomorphism covariant quantum field theory is more well-behaved if one includes degrees of freedom associated with diffeomorphisms of the interface between the given subregion and its complement.By more well-behaved we mean that the von Neumann algebra associated with quantum fields in the subregion goes from type III to type II.This implies that, in contrast to the naive subregion algebra, the enhanced algebra possesses a faithful semi-finite trace, admits density operators, and allows for a meaningful notion of entanglement entropy to be assigned to states in the theory (up to an additive constant).The algebraic mechanism underlying the improvement to the subregion theory is the implementation of a crossed product between the naive subregion algebra and its modular automorphism group [2][3][4].The relationship between the modular automorphism group and the diffeomorphism covariance of the theory is explicated in [1] by appealing to the correspondence between modular Hamiltonians for subregion quantum field theories and diffeomorphisms of entanglement cuts therein [5,6].
An alternative approach to realizing the algebra appropriate to a subregion in the presence of semiclassical gravity has been undertaken in [7][8][9][10] building on [11][12][13] in which large N von Neumann algebras were studied in the context of the AdS/CFT correspondence. 1 In these papers the crossed product algebra is motivated by introducing an "observer" who provides a point of reference through which observables of the theory can be "gravitationally dressed".By dressing one means a procedure that alters observables so as to ensure that they commute with a set of operators that represent constraints of the theory; in the gravitational context these constraints correspond to the imposition of diffeomorphism invariance.The role of the observer to this end is quite clear -mandating that operators always be defined with respect to the worldline of the observer turns the measurement of such operators into a relative object.Qualitatively, one may therefore regard the observer as defining a "clock" relative to which meaningful experiments can be conducted.From this point of view the crossed product is instrumental in accomplishing two ends -firstly, as was the case above, it enhances the algebra from type III to type II by incorporating the generators of the modular automorphism, and secondly it provides the necessary ingredients for implementing the constraints of the theory.
The introduction of an observer may be regarded as a resolution of the dilemma that a theory is not complete if it does not furnish the tools necessary to distinguish physical from unphysical degrees of freedom.Put differently, a theory is made meaningful only once it is equipped with a list of items which it can be used to measure.With that being said, the resolution offered by the observer to this problem is incomplete in a different way.Its most glaring defect, as acknowledged by the authors, is that the observer is introduced in a fashion that is distinctly separate from the rest of the relevant features of the theory.In a fully satisfactory theory both the system that is being measured and the tools being used to make measurements should be made up of the same "stuff".A desire to rectify this problem was largely the motivation behind [1].A central result from that note is that the same crossed product algebra that is implicated by the introduction of the observer can be realized by carefully ensuring that all the symmetries of a given system are equivariantly realized in an appropriate prequantum phase space.
The aforementioned fact is a specific example of a more general correspondence between the crossed product of a von Neumann algebra with a group of automorphisms, and the extension of a symplectic phase space to account for a group of automorphisms acting on its Poisson algebra.We refer to the latter construction as the extended phase space [19,20].The formal correspondence between the crossed product and the extended phase space is summarized in the commutative diagram: Here, EP S and CP are, respectively, the extension of a phase space by the group2 G and the crossed product of an algebra by the group G, and Q is a valid quantization procedure which takes as input the data of a symplectic manifold and outputs an algebra of quantum observables.From the point of view of the extended phase space, the notion of an observer emerges automatically as a consequence of the fact that the (gauge) symmetries are represented equivariantly.We will explain this structure formally as well as through a series of examples later in the paper.
In this note we aim to extend this previous work in two specific directions.First, we will demonstrate how the correspondence (1) allows for the formulation of a complete algebra of operators in arbitrary quantum gauge theories.Secondly, we undertake a detailed study of constraint quantization from the operator algebraic perspective highlighting the central role of the extended phase space and its associated crossed product algebra in guiding and unifying various approaches to implementing constraints quantum mechanically.As mentioned, our previous work largely emphasized the role of the crossed product in enhancing subregion algebras from type III to type II as part of an effort to understand how geometry naturally regularizes entanglement entropy.We did not provide there an equally complete discussion of how the crossed product, when constructed via the correspondence (1) rather than through the inclusion of an observer, allows for the implementation of constraints in the vein of [7][8][9][10].Thus, in this note we concentrate our attention on this facet of the crossed product.It is worth mentioning that, in recent work by one of the authors [21], the symplectic analysis of the Raychaudhuri equation on null hypersurfaces was given in the context of an arbitrary gravity theory coupled to matter.One of the main insights of that work is an explicit realization of the role of extended degrees of freedom in defining observer like variables that allow for the satisfaction of constraints.We will comment further on this example later in the paper.
The paper is organized into two major technical sections, along with a section that presents several examples.In Section 2 we review the geometric approach to gauge theories in terms of Atiyah Lie algebroids, construct their prequantum phase space, and describe their subsequent quantization (albeit only qualitatively). 3The first major advantage of formulating the theory in this way is that it provides a framework in which both internal symmetries and diffeomorphisms are treated on an equal footing as gauge symmetries, and, moreover, in which these gauge symmetries are all faithfully realized in the extended phase space of the theory.A different way of phrasing the previous result is that the extended phase space of the gauge theory constructs an equivariant moment map into its Poisson algebra from a Lie algebroid whose elements correspond to generically field-dependent spacetime local gauge transformations and diffeomorphisms.The existence of such a moment map is a critical feature because it implies that approaches such as geometric quantization can, in principle, be applied to the quantization of the (unconstrained) gauge theory.
The second major advantage of this approach is that it allows for a distinction to be made between symmetries which are pure redundancies of a given theory and symmetries which are physically realized in the sense that they map physically distinct states to physically distinct states.In this regard, we can treat the extended phase space of a gauge theory as possessing two copies of the group of gauge transformations and diffeomorphisms: one which furnishes a set of constraints that will need to be enforced and one which merely introduces an additional set of physical charges, a structure that is implied by Noether's second theorem.As has been a major theme in recent work [22][23][24][25][26][27][28][29][30][31][32][33][34], the physical charges in a gauge theory are intimately related with the presence of codimension two submanifolds in spacetime which are often referred to as corners.
After having given an overview of the prequantum phase space associated with a general gauge theory, we employ the correspondence introduced in [1] to identify the quantized algebra of operators as the result of a pair of crossed products which realize unitary operators associated with both constraints and physical charges.We refer to these extensions respectively as the extension at codimension one and the extension at codimension two.Organizing the extended phase space in terms of this pair of group automorphisms sheds some light on the multifaceted role of the crossed product previously mentioned.The extension at codimension two, which was explored in depth in [1], ensures that corner supported charges are realized as Hamiltonian functions in the prequantum theory and quantized as physical operators in the quantum theory.An example of such an operator is the modular Hamiltonian associated with corner supported diffeomorphisms.Thus, the extension at codimension two is responsible for ensuring that the quantized algebra is always semi-finite.By contrast, the extension at codimension one furnishes a unitary representation of the constraint group which is critical for the implementation of constraints in the quantum theory.
This brings us to Section 3 which is the main technical thrust of the paper.The algebra realized by quantizing the full extended phase space has not yet been subjected to the constraints required by gauge symmetry.The problem of quantizing theories with constraints, or constraint quantization, was largely initiated by Dirac in [35].In the intervening years constraint quantization has become a deep subject of research which lies at the heart of efforts to understand gauge theories and gravity.In this note, we will concentrate our attention on four apparently distinct approaches to constraint quantization.The first three of these approaches are hopefully quite familiar to a physics audience, they are: refined algebraic quantization (RAQ) [36][37][38][39][40] discussed in Section 3.1, BRST quantization [41][42][43][44][45][46][47][48] discussed in Section 3.2, and path integral quantization [49][50][51][52] discussed in Section 3.3.The existing literature on each of these approaches is vast, and the references we have included are obviously not exhaustive.Our unique contribution to this literature will be to investigate these techniques entirely from the point of view of a generic crossed product algebra M ⋊ G, and to demonstrate in what sense M ⋊ G possesses all of the tools necessary to impose G as a group of constraints in the style of any of the aforementioned procedures.This point of view is underscored in Section 3.4 in which we discuss a fourth approach to constraint quantization by means of the commutation theorem [53][54][55].The commutation theorem is a powerful result from the theory of operator algebras which ensures that the full crossed product algebra can be regarded as an invariant algebra under a suitably defined set of constraints.To connect the commutation theorem with the previous approaches we provide an interpretation for it within the framework of the correspondence (1).
In summary, by formulating our analysis from an operator algebraic perspective we are able to shed new light on each of the aforementioned strategies, suggesting new remedies to overcome existing shortcomings and, perhaps most excitingly, allowing for a clear exposition on how all four forms of constraint quantization may be understood as facets of a common fundamental approach.The crucial new ingredient to this end is the structure of a (generalized) conditional expectation on a von Neumann algebra which projects from the crossed product down to the algebra of gauge invariant operators.In this regard, we recognize how the crossed product obtained by quantizing the extended phase space of a generic gauge theory allows for the implementation of a dressing procedure that generalizes [7][8][9][10].
Although Section 2 and Section 3 are conceptually linked, they are largely self-consistent and can each be read on their own.Recognizing that much of our discussion is quite formal, in Section 4 we provide a series of worked examples in which our approach to constraint quantization is applied to well-known gauge theories.We hope that these examples are familiar to the reader, as our primary motivation in this section is to illustrate the pertinent features of our formalism and the extended phase space perspective.We begin in Section 4.1 with a quantum mechanics system, the symmetric D-dimensional harmonic oscillator and consider gauging the SO(D) subgroup of its global symmetry, SU (D).This is a useful example, because we can understand its quantization from either the Hilbert space perspective, the operator algebra perspective or the path integral perspective, and we discuss it carefully from the point of view of extended phase space.Since this is a quantum mechanics problem, it lacks one important feature, which is codimension-2 structure.In Section 4.2, we therefore revisit Yang-Mills theory from the extended phase space perspective, and explain in detail how the conditional expectation interpretation can be understood through the familiar Fadeev-Popov procedure.What may not be familiar to all readers here is that we present the path integral directly in phase space, and we take pains to explain how the path integral effectively solves constraints while leaving gauge charges generally undetermined.In Section 4.3 we discuss Einstein-Hilbert gravity on a null hypersurface using our conditional expectation oriented approach.Here our main focus is to elucidate the relationship between the symplectic analysis undertaken in [21] and the extended phase space, a useful groundwork for future exploration of quantization.We conclude in Section 5 with a discussion which summarizes our results and suggests some directions for future work.

Gauge Theory
In this section we provide an outline of the operator algebra that is realized in a gauge quantum field theory starting from the extended phase space introduced in [19,20].
In Appendix A we provide an overview on how the field theoretic data of a gauge theory can be encoded in an Atiyah Lie algebroid, A. In this subsection, we turn our attention to the construction of the extended phase space associated with such a gauge theory.We use the notation (A, ω) to refer to a Lie algebroid with connection, drawing attention to the fact that the vector bundle A endowed with different choices for the connection ω correspond to different Lie algebroids. 4 We denote by the set of all Atiyah Lie algebroids based on the common vector bundle A. More concretely, we regard Φ ∈ as corresponding to the set of data specifying all possible configurations of fields in Ω • (A; E) including the connection ω and, if appropriate, a solder form e.
Next, we introduce the set consisting of all Lie algebroid isomorphisms between elements in .That is, an element ϕ ∈ is a map ϕ : (A, ω) → (A, ω ′ ) which is a vector space automorphism of A, and moreover which preserves the brackets defined on (A, ω) and (A, ω ′ ), respectively.In [20], it is demonstrated that such maps correspond precisely with the combined set of local gauge transformations and diffeomorphisms of the base M .This can be understood via the fact that ϕ preserves brackets which implies that ω and ω ′ share the same curvature and are therefore gauge equivalent in the conventional sense.The map ϕ also induces an isomorphism g (E) ϕ : E → E for all vector bundles associated to A via a Lie algebroid representation induced by the connection reform ω [20].Given a generic element ψ ∈ Ω p (A; E) we represent the action of ϕ ∈ via the Lie algebroid pullback Although it is written in a very formal way, when pulled apart (2) reduces to the usual equations describing gauge transformations and diffeomorphisms of the relevant fields, as was described in detail in [20,56].These together combine into a section of the Lie algebroid, and in the following we will refer to such as generalized gauge transformations.The sets and provide the necessary kinematical data for constructing the extended phase space of an arbitrary gauge theory.The field content of such a theory is given by the configuration data Φ = (Φ 1 , ..., Φ N ), where each individual field Φ i ∈ Ω p i (A; E i ) is a p i form taking values in a representation E i .The fields Φ have momentum Π = (Π 1 , ..., Π N ) where Π i ∈ Ω d−1−p i (A; E * i ) takes values in the representation dual to their conjugate field.To complete the specification of the theory we require a codimension one Cauchy surface Σ ⊂ M which is embedded into M via a map ϕ (1) : Σ ֒→ M .More completely, we extend ϕ (1) to the Lie algebroid isomorphism ϕ (1) : A Σ → A, where A Σ is a Lie algebroid over a tubular neighborhood of Σ inside M , M Σ .We regard (Φ, Π) as defining coordinates for a symplectic manifold X, endowed with the canonical symplectic potential density: giving rise to the symplectic potential [20]: To complete the specification of the theory we must also dictate dynamical data in the form of a Hamiltonian function which generates time translations.Let H : X → Ω d−1 (A) be a (d − 1)-form functional of the fields which we refer to as the Hamiltonian density.Integrating, we obtain the Hamiltonian -regarded as a function on the phase space of the theory, H ∈ C ∞ (X).Given a curve γ : [0, 1] → X we can define the action functional in the usual way as = In (6) we have defined the Lagrangian density L ∈ Ω d (A Σ ).
As was introduced in [19,20], in theories with gravity (3) must be augmented to properly account for the fact that geometrical data, such as the embedding ϕ (1) , become part of the phase space.The extended configuration space is the principal bundle È → X, with extended symplectic potential (density)5 Here, ̟ ∈ A * ⊗ A is the Maurer-Cartan form associated with with the understanding that A exponentiates to [20].The additional term in (8) arises from the non-invariance of the Lagrangian under gauge transformations (including diffeomorphisms).Let a : × X → X denote the action of on the phase space.We'll write this action as Alternatively, recognizing that locally È ≃ × X, we can regard this map as a projection from È with local coordinates (ϕ, Φ, Π) into X with coordinates (Φ, Π).
Using (9) we can compute: where δ X Φ = îX δ ̟ Φ is the variation of Φ under a generalized gauge transformation generated by X ∈ Γ(A).One may interpret the terms appearing in (10) as quantifying the free and gauge variations of the field degrees of freedom.Plugging (10) into (8) we obtain a new formulation of the extended symplectic potential: Essentially, we have just pulled apart the variation of Φ into its two aforementioned pieces.The notation L means the Lagrangian evaluated at ( Φ, Π).The symplectic potential has now been divided into two pieces -a free variational term and a variational term: We note that the extended symplectic potential contains a term, eq. ( 12), of the unextended phase space, written in the tilded variables, plus terms (13) that correspond to the extension.This general decomposition of the extended symplectic potential is a central property of gauge theories, as we will see, and facilitates the implementation of constraints and gauge fixing.A significant feature of ( 13) is that if we contract it with a vector field on phase space, we obtain the Noether current density.In the notation of [20], this is the statement that is precisely the Noether current density.Here − (X) is the vertical vector field in the extended phase space that reproduces the action of X ∈ A (containing locally a spacetime vector field and a Lie algebra element), with A regarded as the isotropy bundle of the configuration algebroid.The Noether currents are conserved in the sense that dJ X = 0 where d is the algebroid exterior derivative.The current moreover splits into two pieces: where Å X ∈ H d−1 (A) and É X ∈ Ω d−2 (A).By Noether's second theorem Å X must vanish when the constraints of the theory are implemented.With that being said, É X are left unconstrained and source non-zero charges supported on surfaces of codimension 2 relative to bulk spacetime.These charges generate physical transformations and thus must not be removed in the process of constraint quantization/symplectic reduction.
The structure of eq. ( 15) suggests that, given (14), we can rewrite the symplectic potential by defining Maurer-Cartan forms ̟ (1) , ̟ (2) , viz where Å M , etc., are components relative to a local basis {E M } for A.Here we regard ̟ (1) and ̟ (2) as two separate copies of the Maurer-Cartan form which generate codimension one and codimension two supported transformations, respectively.In the symplectic sense, we read (16) as dictating that the infinitesimal generators ̟ M (1) and ̟ M (2) are canonically dual to the currents Å M and É M .Then, the extended symplectic potential takes the final form Or, once integrated over the hypersurface: At the level of the extended phase space, the symplectic form induced by ( 18) is non-degenerate.In the context of the gauge theory, as was mentioned below (15), Å M will be interpreted as constraints; setting these to zero in some way renders the symplectic form degenerate. Gauge fixing will then reduce to a symplectic form that is non-degenerate on an appropriate quotient space.

Quantization of the Extended Phase Space
In this note, we take the point of view that the procedure of quantization entails the promotion of the Poisson algebra associated with a classical symplectic/Poisson geometry into a von Neumann algebra.In this section, we recall the insights of [1] to contextualize the operators which belong to the quantization of the extended phase space.In particular, we propose that the resulting von Neumann algebra be interpreted as the result of a pair of crossed products which introduce, respectively, codimension one operators that allow for the implementation of constraints and codimension two operators implicated by the presence of a bounded subregion.
To preface this discussion, let us first recall the correspondence between the extended phase space and the crossed product in general terms.Let (X, Ω) denote a (pre)-symplectic manifold which admits a G-action a : G × X → X.We work in the context that the G-action is presymplectic, meaning that it preserves the symplectic form An immediate corollary of (19) is that a induces an automorphism of the Poisson algebra, M pq X , associated with (X, Ω). 6 Nonetheless, it is not manifest that the generators of the G-action can themselves be represented by elements of the Poisson algebra, rather it is merely true that G acts on the Poisson algebra as an automorphism.For this reason, it is not immediately clear that the generators of G can be quantized using standard methods.Conversely, if the algebra of generators of the G action are faithfully encoded in the Poisson algebra it is termed equivariant.In other words, there exists an algebra homomorphism called a moment map.The quantization of such a moment map is well understood.Thus, lifting a presymplectic action to an equivariant one is a useful step in completing the quantization of a theory admitting a group action.
In [1] we demonstrated that, starting from a (pre)-symplectic manifold (X, Ω) with presymplectic Gaction a, there exists a principal G-bundle, X ext → X, for which the presymplectic G-action can be extended to an equivariant G-action.(In the last section, the role of X ext was played by È).In other words, the Poisson algebra M pq Xext always admits a moment map À : g → M pq Xext .The usefulness of X ext is that it provides a prequantum phase space complete with the generators of the group G which can, at least in principle, be quantized.
The crux of the connection between the extended phase space and the crossed product is that the above procedure can be understood as a (pre)-symplectic analog of the following scenario in the context of von Neumann algebras.Let M be a von Neumann algebra acted upon by the G-automorphism α : G×M → M.Moreover, let π : M → B(H) denote a faithful representation of the algebra on a Hilbert space H which also carries a covariant representation of the triple (M, G, α).Then, there exists a unitary representation The G-automorphism α is termed inner if and only if im(G) ⊂ im(π); i.e., if and only if the group elements can be understood as operators inside of the intrsinsic algebra M. Otherwise, the automorphism is termed outer.The situation of having an outer automorphism is akin to having a presymplectic but not equivariant G-action at the prequantum level.As the extended phase space provides a mechanism for promoting presymplectic actions to equivariant actions, the crossed product provides a mechanism for promoting outer automorphisms to inner ones.Explicitly, the crossed product can be understood as the von Neumann algebra generated by combining the images of the maps π and U : The basic argument of [1] is that a suitable quantization of the Poisson algebra of the extended phase space results in a von Neumann algebra that is isomorphic to the crossed product of M pq X and the group G with respect to an appropriate automorphism induced by the G-action a.
Having set the stage, we can now describe the role of the extended phase space/crossed product construction as it pertains to the quantization of gauge theories on subregions.The starting point, as we have discussed, is the symplectic geometry induced by the extended phase space: (È, Ω ext ).This geometry is acted upon equivariantly by the groupoid of morphisms , which corresponds to the set of all local gauge transformations and diffeomorphisms.In particular, this implies that there exists an equivariant moment map À and thus, at least in principle, this symplectic geometry can be quantized.This quantization promotes each of the Hamiltonian charge functions to operators which we signify by the addition of a hat: À → À.We denote by M È the von Neumann algebra obtained by quantizing the Poisson algebra associated with È.
Recall that È is a principal -bundle over X, the symplectic manifold associated with the field data alone.Let us dwell for a moment on X.As is well known [57], X itself has the structure of a principal -bundle over the quotient space X/ .In other words, we may regard the space of all field configurations as the set of all gauge inequivalent field configurations along with their -orbits.To be explicit, the copy of which appears in X is generated by codimension one currents Å X as in (15).In other words, the extended phase space has the following schematic structure: We interpret (23) as identifying the phase space È as the result of two extensions -one with respect to the group of constraints (1) and one with respect to the group of physical charges (2) .Reiterating an earlier discussion, each charge splits into two pieces: Hereafter we will refer to the former as constraint charges, and to the latter as the physical charges.We regard À (1) : A → Ω 0 (È) and À (2) : A → Ω 0 (È) as generating two separate representations of the gauge algebra A in the Poisson algebra of È.Let H aux be a Hilbert space for which the algebra M È possesses a faithful representation π : M È → B(H aux ).Then, at the level of H aux , (24) implies that we have a pair of unitary representations: which give rise to two separate von Neumann algebra automorphisms The von Neumann algebra resulting from the quantization of the extended phase space is therefore of the form Eqn. ( 28) is the von Neumann algebra analog of (23).
The conclusion of the preceding paragraph could have qualitatively been obtained by observing the symplectic potential (18), which designates the classical variables that should be quantized to operators.From that perspective, the complete quantum theory should have operators associated with three sets of conjugate pairs: ( Φ, Π) corresponding to (dressed) field configuration data, (̟ (1) , Å) corresponding to generators and charges associated with pure gauge transformations, and (̟ (2) , É) corresponding to generators and charges associated with physical transformations supported on corners.One may therefore regard the operator algebra realized by quantizing the extended phase space as To summarize this section, we have now recognized how the extended phase space prepares a quantum theory in which all of the relevant operators are present including operators which correspond with constraints.In the remainder of this note we will discuss the naturally related problem of implementing these aforementioned constraints in a manner which is quantum mechanically consistent.

Constraints Four Ways
In this section we will demonstrate how the extended phase space figures naturally in the implementation of constraints by considering four distinct methods: (i) Refined Algebraic Quantization, (ii) BRST Quantization, (iii) Path Integral Quantization, and (iv) the commutation theorem.In this respect, we will also seek to accomplish the goal of unifying these distinct approaches to quantizing systems with constraints under the umbrella of a single geometric framework.As we will demonstrate, all four approaches to constraint quantization are linked in the sense that they construct projection maps originating from the crossed product algebra.
In what follows we will concentrate on generic algebras of the form We may regard M cp as arising from the quantization of an extended phase space X ext → X, where (X, Ω) is a symplectic manifold acted upon at least presymplectically by the group G. 7 In this regard it will also be beneficial to regard M as the algebra obtained by quantizing the non-extended space X.Our goal will be to describe how the machinery of the crossed product allows for the seamless implementation of constraints associated with the group G. Let us emphasize, in light of Section 2.1, that when we have a gauge theory in mind this does not entail constraining the physical charges in any way.One may always think of the gauge theory algebra as being of the form where the constraint quantization procedures described below are performed for (1) while treating M X ⋊ α (2) (2) as its own algebra.This observation plays out in the examples discussed in Section 4.
Before moving into the analysis of this section, it will be prudent to first review different kinds of maps accessible in the operator algebraic context which share properties we associate with conditional expectations.For the purpose of this discussion, let M be a von Neumann algebra and N ⊂ M a von Neumann subalgebra.We will also use the notation M to denote the set of operators affiliated with a von Neumann algebra M .Given a faithful representation π : An operator-valued weight [60,61] is a map T : M + → N which is linear and satisfies the bi-module property It is worth noting that, in the case that N = ½ ⊂ M , the definition of an operator-valued weight reduces to that of an ordinary weight.As is the case for ordinary weights, the map T defining an operator-valued weight can be extended uniquely to a map T : M → N by decomposing elements in M into the difference of its positive and negative parts.If an operator-valued weight T is unital, i.e.T (½ M ) = ½ N , it is called a conditional expectation.In the case that N = ½, a conditional expectation defines what is typically referred to as a state.Given a state ϕ on M , we say that the conditional expectation E : M → N preserves the state if This is the non-commutative analog of factorizing a probability density as a product of a marginal and a conditional density.A theorem by Takesaki [62] states that a state preserving conditional expectation will only exist if N is invariant under the modular automorphism generated by ϕ.
Alternatively to operator-valued weights one has generalized conditional expectations [63].To introduce the generalized conditional expectation, let us first recall the Petz dual of a quantum channel.Given a state ϕ on M the KMS inner product is defined (up to a possible quotient by its kernel) by In (34) σ ϕ : Ê → Aut(M ) is the modular automorphism induced by ϕ.Next, let α : M → N be a normal, completely positive, unital map between two von Neumann algebras (for now N need not be a subalgebra of M ).Such a map is called a quantum channel.Let ϕ 0 be a normal state on N such that ϕ ≡ ϕ 0 • α is a normal and faithful state on M .Then, we define the Petz dual [64] of the channel α with respect to the state ϕ 0 as the map α † ϕ 0 : N → M which is formally adjoint to α with respect to the intertwining inner products g KM S N,ϕ 0 and g KM S M,ϕ 0 •α .That is, It can be shown that α † ϕ 0 is also a quantum channel.In the case that N ⊂ M is a von Neumann subalgebra, the inclusion i : N ֒→ M is a normal, completely positive, unital map and given any state ϕ on M the state ϕ • i is normal and faithful on N .Thus, there exists a canonical quantum channel E N ϕ : M → N obtained from the Petz dual of i with respect to the state ϕ.The map E N ϕ is called the generalized conditional expectation induced by the pair (N, ϕ).The map E N ϕ is always unital and preserves the state ϕ in the sense that However, the generalized conditional expectation does not automatically possess the bi-module property of an operator-valued weight.This fact is expressed in the following theorem [63]: Given n ∈ N the following are all equivalent: 1.
In other words, the generalized conditional expectation obtains the bi-module property provided the modular automorphism of ϕ, restricted to N , is the same as the modular automorphism of ϕ| N .In the event that E N ϕ possesses the bi-module property for all n ∈ N , it can be shown that the generalized conditional expectation is equivalent to the unique ϕ-preserving conditional expectation from M to N .
In summary, the strict definition of a conditional expectation is a linear, unital map E : M → N satisfying the bi-module property (32).In general the existence of a conditional expectation between an algebra and a given subalgebra is not guaranteed.However, one can always realize conditional expectation 'like' maps by relaxing either the unital condition -in which case one obtains an operator-valued weightor the bi-module condition -in which one obtains a generalized conditional expectation.In the following we will find need to use these relaxations of the conditional expectation to construct projection maps which implement constraints.

Refined Algebraic Quantization
The starting point of Refined Algebraic Quantization (RAQ) is a Hilbert space H aux upon which the group of constraints G are realized as unitary operators.One way of thinking about H aux is that it corresponds to the quantization of the unconstrained system associated with the constrained system we are interested in studying.As was alluded to in Section 2, this is precisely the role of the extended phase space.
More explicitly, starting from the crossed product (30) there is a canonical Hilbert space which can play the role of H aux . 8Let H be any Hilbert space for which there exists a faithful representation π : M → B(H).Then, as is established in [2], we can realize the crossed product algebra as operators acting on an extended Hilbert space equivalent to the set of square integrable functions on the group G with values in the Hilbert space H, H aux = L 2 (H; G, µ) ≃ L 2 (G) ⊗ H.Given two such elements, ξ, ξ ′ ∈ H aux , we define the inner product where here µ(g) is the left-invariant Haar measure on G, and h : H × H → is an inner product on H. 9We may subsequently define the explicit representations: The representation U : G → B(H aux ) is unitary by construction due to the left invariance of (38).Occasionally it may be necessary to notate the representation of an arbitrary element x ∈ M cp without specifying if it is in M or G.In these cases we write ρ(x) and let the explicit representation be dictated by the nature of the element x.
From the quantum mechanical perspective implementing the constraints of the theory corresponds to identifying a Hilbert space of elements which are left invariant by the constraint operators -the so-called physical Hilbert space H phys .RAQ provides a direct construction of this space.The impetus for RAQ was an observation that Dirac's approach to constraint quantization [35] is ill-fated because physical states typically don't lie inside of the Hilbert space one starts with under the naive quantization of a constrained system [36][37][38][39]65].Rather, it is necessary to consider an enlarged Hilbert space in which the physical states are "filled back in".
To accomplish this aim, we let H * aux denote the algebraic dual of H aux endowed with the topology of pointwise convergence. 10The representation (39) descends to a representation on H * aux via the dual action More generally, the dual action of M cp on H * aux is given by We can now construct a set of states in H * aux which are invariant under the action (40).Such states are realized by defining a so-called rigging map [40] η : H aux → H * aux satisfying the following properties: Condition (42) implies that the image of η are invariant under the dual action of G. Conditions ( 43) and ( 44) together imply that defines an inner product on η(H aux ) ⊂ H * aux ; the first implying that ( 45) is sesquilinear and the second implying that it is positive definite.Thus, conditions ( 42)-( 44) imply that we can define a physical Hilbert space by taking the closure of η(H aux ) ⊂ H * aux under the topology induced by the inner product ( 45): In [37,38] an explicit form for a rigging map satisfying the aforementioned properties was given in terms of a group-averaging procedure.Moreover, it was shown that whenever such a procedure is well-defined, the resulting map is the unique rigging map up to a multiplicative constant.For simplicity, we assume that G is a unimodular group meaning it possesses a unique bi-invariant Haar measure µ. 12 Then, we define the rigging map It is straightforward to show that (46) satisfies conditions ( 42)- (44).Most crucially, the invariance of η(ξ) follows immediately from the invariance of the Haar measure: In the preceding discussion, we have stressed a Hilbert space oriented approach to implementing constraints, as is typical in the literature.However, as we will now describe, there is a very natural alternative approach to the problem of constructing the physical space of states which leads instead with the algebra.To begin, let us define by the set of G-invariant operators in M cp .Moreover, let ω : M inv → denote a weight on M inv .Given this data, we can form the GNS representation of M inv , H inv with inner product g ω . 13The GNS construction is characterized by the maps η ω : M inv → H inv and π ω : M inv → B(H inv ) along with a vector ψ ω ∈ H inv such that Notice that a corollary of ( 49) is that which is the standard result that ω(x) computes the expectation value of x in ψ ω .
The group G is realized on H inv in terms of a unitary representation U ω : G → U (H inv ) which is compatible with the automorphism α in the sense that π ω • α g (x) = U ω (g)π ω (x)U ω (g) −1 .Thus, by (48) At the same time, using (50) along with (48) we can write which implies that Combining ( 51) and ( 53), we can easily show that each element in the GNS Hilbert space is invariant under the action of the group: Thus, we conclude that H inv is an alternative construction of a physical Hilbert space.
The connection between the standard approach to RAQ and the algebraic approach based on the GNS construction arises when we consider the remaining loose end, which is an identification of the algebra (48).
To do so we define the map Eq. ( 55) is nothing but the group averaging applied directly to the algebra.It is easy enough to check that Thus, we may take M inv ≡ Π d (M cp ).More to the point, if we extend ω to a weight on the full algebra M cp then we can construct a GNS representation for the full algebra which we denote by H cp along with its corresponding set of data.In the context of the GNS Hilbert space H cp we can construct the map which is precisely the group averaging map in H cp .
There are some subtleties that we should address in order to ensure the validity of (57).We have tacitly assumed that ω is invariant under the action of G: This was true trivially when restricted to the subalgebra M inv , but it is not immediately clear that (58) holds when applied to the full algebra.One way of interpreting this state of affairs is that we have reduced the problem of finding a complete Hilbert space of group invariant elements down to the problem of finding just one such element.Once this initial element, ψ ω , is identified all other such states can be constructed via (57) which we interpret as the vector created by acting on ψ ω with G-invariant operators.It is at this point that we arrive at a very important insight -if M cp possesses a tracial weight τ : M cp → , then it can always play the role of our desired G-invariant element.Indeed, τ • α g = τ follows immediately from the cyclicity property of the trace, provided the automorphism α is inner unitarily implemented which the crossed product also guarantees.As was discussed in great detail in [1], M cp will possess such a trace if and only if it is semi-finite.The semi-finite nature of a von Neumann algebra is intimately related with other desirable properties such as the existence of density operators and entropies.In the aforementioned paper and in other related work [7][8][9][10], it has been suggested that a physical theory which possesses diffeomorphism covariance should always been semi-finite.This can be understood as an immediate consequence of the relationship between the extended phase space and the crossed product, in addition to the role of corner supported diffeomorphisms in specifying the modular automorphism group of a subregion algebra [5,6].Thus, once again, we see that the crossed product saves the day: so long as M cp is a semi-finite von Neumann algebra, we can construct the GNS Hilbert space associated with the tracial weight and then perform the group averaging procedure implicated by (57) in order to construct a complete set of physical states.
One might be inclined to think of the tracial weight τ and its associated Hilbert space representation, ψ τ , as playing the role of the G-invariant vacuum.Then, the content of ( 57) is simply that all of the physical states of the theory can be obtained by perturbing the vacuum by gauge invariant operators.This is closely related to the conclusions reached in [66,67] in which the authors construct a Hilbert space of states in de Sitter quantum gravity.This is unsurprising, as [66,67] is a modification on a classic result by Higuchi [68,69] which produces a space of de Sitter invariant states in the non-gravitational limit precisely by group averaging over the symmetries of the background.The more recent papers refine this approach by making use of the Fadeev-Popov method for implementing constraints in the path integral sense.We will revisit the map (55) again in Section 3.3 where it appears naturally in precisely this context.
As we have alluded to, there is one further subtlety which warrants attention as it concerns the use of the map (55).In the case that the group G is compact, ( 55) is a conditional expectation as defined in Section 3. In general, however, the integral over the group will diverge when applied to the identity element, and thus the map Π d cannot be unital. 14In this case Π d is only an operator-valued weight, rather than a conditional expectation.This is an issue that has plagued the RAQ approach to constraint quantization since its inception, since an operator-valued weight cannot be used to construct normalizable states.In particular, this seems to put a damper on the usefulness of Π d as a tool for implementing constraints in gravitational theories wherein the group of symmetries is not compact.In this regard, however, passing from the Hilbert space oriented approach to the operator algebraic approach seems serendipitous.Although an explicit form in terms of group averaging is not valid in this case, we can in principle upgrade the map Π d by appealing to Accardi and Cecchini's construction of the generalized conditional expectation [63].This map, while lacking the bi-module property of an operator-valued weight, is manifestly unital and so may be used to construct normalizable states.Then, although the form of the maps ( 55) and ( 57) may be different, the ideas of the preceding discussion can be enhanced to make sense even in this case.In future work we plan to address the problem of constructing such a generalized conditional expectation in detail.

BRST Quantization
The next approach to implementing constraints that we would like to consider is the BRST quantization scheme.As was the case for the Refined Algebraic Quantization scheme, BRST quantization begins with an enlarged Hilbert space which includes states that will ultimately be regarded as unphysical.In particular, one introduces by hand a tensor factor to the space of states involving ghosts.In this subsection, we will demonstrate how this Hilbert space arises naturally in the crossed product algebra as a simple extension of H aux introduced in Section 3.1.Harnessing the relationship between the crossed product and the extended phase space, the presence of ghost degrees of freedom come as no surprise.Indeed, one of the main motivations for constructing the extended phase space was to take advantage of the fact that ghost degrees of freedom manifest as differential forms in the vertical subbundle of an Atiyah Lie algebroid [56].
In the following we will concentrate our discussion of BRST from the point of the view of the crossed product algebra M ⋊ α G.We will now generalize the construction (39) to realize the crossed product in terms of operators acting on the Hilbert space Here Ω • (G; H) is the vector space of differential forms on G taking values in H.The aforementioned vector space is promoted to a Hilbert space by means of the following inner product: where ⋆ is the Hodge dual induced by a left invariant metric on G and ξ, η ∈ H ext . 15Notice that (59) reduces to (38) in the case that ξ, η ∈ Ω 0 (G; H).We can similarly extend the representations (39) as 16 Again, (63) reduces to (39) when ξ ∈ Ω 0 (G; H). 17Working in the Hilbert space H ext it is now straightforward to realize the BRST complex.To begin, we identify T e G ≃ g where g is the Lie algebra integrating to G.Then, we can introduce the following (potentially unbounded) operators which act on H ext through their action on the tensor factor Ω • (G): Clearly, b(µ) : Ω p (G; H) → Ω p−1 (G; H) and c(β) : Ω p (G; H) → Ω p+1 (G; H).These operators satisfy the relations where {•, •} is the anticommutator with respect to the composition of maps.Eq. ( 65) are precisely the anticommutation relations which characterize the BRST complex.We can present these equations in a more familiar form if we introduce dual bases for g A=1 , and define the corresponding indexed operators b A ≡ b(t A ), and c A ≡ c(t A ). Then we can rewrite (65) as In tandem with the maps b and c, we can also introduce a Lie algebra representation on the non-extended Hilbert space H.To realize such a map, we regard the unitary group representation U : G → B(H) as arising from the exponentiation of a Lie algebra representation v H : g → L(H) such that U exp(µ) = exp • v H (µ). 18 We can combine the maps v H , b, and c to form a representation v ext : g → L(H ext ) as 15 The notation in (59) should be read as with ξ = ξ a ⊗ e a and η = η b ⊗ e b coordinatizations of ξ and η with respect to a basis for H, and each ξ a , η b ∈ Ω • (G).For the metric on G we take where ̟ ∈ Ω 1 (G; g) is the left invariant Maurer-Cartan form.In the case that G is semi-simple, G reduces to the Killing metric. 16Here is the left action of G on itself.
where f AB C are the structure constants of the algebra.To understand (67) it is instructive to consider its action on an element α = α A ⊗ t A ∈ Ω 1 (G; H): The first term in (68) corresponds to the action of the non-extended representation on the components of the form α which are each elements of H.The second term in ( 68) is precisely the co-adjoint action of g on its dual g * .Recall, the co-adjoint representation is a map v g * : g → End(g * ) which is defined through its duality with the adjoint action as Thus, we can read (68) as Eq. ( 70) makes clear the role of the extra terms appearing in (67) relative to the Lie algebra representation v H for the Hilbert space alone: these terms carry the action of the Lie algebra on the dual Lie algebra elements.As this action is implemented co-adjointly it is moreover clear that ( 67) is a proper representation.On higher order forms the second term in (67) will simply act via the co-adjoint action on each of the generators of the Lie algebra dual.
To study the BRST cohomology we introduce the operator The reader may recognize (71) as precisely the coboundary operator generating Lie algebra cohomology as discussed, for example, in [56,57].As a consequence of the fact that v ext is a Lie algebra representation and that the c operators anticommute, Q is nilpotent: Q 2 = 0. Thus we can introduce the sets and identify the p th cohomology class with the quotient H p (G; H) ≡ Z p (G; H)/B p (G; H).The significance of the cohomology classes H p (G; H) arise from the following observation: In fact, ( 73) is nothing but the Hilbert space version of the familiar Cartan relation between the exterior derivative, the vector contraction, and the Lie derivative: This is not an analogy: ( 73) is precisely equivalent to (74) in the context of Lie algebra cohomology with values in the representation space H. Using (73) we can deduce that cohomology classes are preserved under the extended action of the group.For ξ ∈ H p (G; H), under the action of G generated infinitesimally by the element µ ∈ g we have: In the first equality we have used (73), and the final equality is up to cohomology.An alternative view point on (75) is that the cohomology classes of ( 71) correspond to G-orbits in H ext .By definition, if one takes a point in H ext and averages it over the group an element will be obtained that remains in the same G-orbit.Thus, the group averaging approach discussed in Section 3.1 may be regarded as a specific approach to identifying BRST cohomology classes.Both are sufficient approaches to implementing the G-constraints.

Path Integral Quantization
As we have seen, both the Refined Algebraic and BRST quantization procedures may be interpreted as implementing constraints by imposing an equivalence relation identifying points along orbits of the action of the constraint group.In this section, we will argue that the path integral can also be viewed in this way.
Our exposition is related to the usual Faddeev-Popov formalism, but leverages the geometric formulation inherent in the extended phase space to give it a new interpretation which, as we show, fits very nicely with the other quantization structures.In particular, we provide an outline of how the Faddeev-Popov procedure can be interpreted from the operator algebra perspective as a (generalized) conditional expectation related to a group averaging map.Again we consider a symplectic geometry (X, Ω) acted upon by a group G, denoting by M pq X the Poisson algebra associated with (X, Ω).For the moment, let us ignore the G-action on X entirely.Instead, we wish to concentrate on providing an operator algebraic interpretation for the path integral.Generally speaking, a path integral can be regarded as a map, ϕ : M pq X → , which assigns to each element of the Poisson algebra a number that we interpret as its expectation value.Conventionally, we regard this mapping to be of the form: with x : I ⊂ Ê → X specifying a parameterized curve in phase space.For example, one might like to think of where S(x) is a functional which one identifies with the phase space action as defined in (6), and is the standard path integral measure on phase space.For ease of notation, we will hereafter denote (76) simply by with the understanding that x refers to a generally "time-dependent" element of X, while the operator insertion is localized at points along a curve γ in X. Regardless of the fine-grained form of (76), it is natural to regard ϕ as defining a weight on the Poisson algebra M pq X .We denote the normalization of ( 76) Having established our desired interpretation for the path integral as defining a weight on the prequantum algebra of observables, let us now address the problem of implementing the constraints.In particular, we resurrect the G action19 a : G × X → X.Given a point x ∈ X, the a-orbit of x corresponds to the set of points that can be reached by x upon applying the action a: The quotient space X/G can be understood as the set of such orbits.At present we are interested in the case where a preserves the measure of (80): In other words, from the perspective of the path integral (80), all points along an orbit of the action a are identical.
In the usual approach, one would motivate Faddeev-Popov gauge fixing at this point by arguing that the invariance of the path integrand under the group G implies that (80) overcounts the number of distinct configurations.In keeping with the theme of our operator centric approach, we take a different but ultimately equivalent perspective.Consider the following observation: generic functions in the Poisson algebra M pq X needn't share the symmetries (82).The weight (76) does not disallow the computation of expectation values associated with these operators and thus there is an immediate risk of violating the constraints of the theory.To remedy this problem we must modify the weight ϕ in such a way that it automatically enforces the symmetry constraints for any observable that we insert into it.From this point of view it is abundantly clear that the role of the Faddeev-Popov insertion in the path integral is precisely the same as the procedures outlined in Sections 3.1 and 3.2, namely to project generic functions down to the algebra of G-invariant functions.Thus the quantization procedure leads directly to properly dressed operators, from the canonical perspective.
In following with the standard Faddeev-Popov approach, the simplest way to implement the desired symmetry constraint is to construct a unique representative of each group orbit; this defines a projection from the phase space X to the quotient space X/G: Composing an arbitrary element f ∈ M pq X with the projection π we realize a function which is, by construction, G-invariant: In other words, the mapping takes an arbitrary element of M pq X and turns it into a gauge-fixed version satisfying the G-symmetry constraint.A particularly nice way of realizing the projection (83) -as utilized in the standard Faddeev-Popov procedure -is to introduce a group non-invariant map F : X → g whose kernel intersects each gauge orbit exactly once.That is, We subsequently take [x] to be the representative of the orbit a G (x), thereby defining a projection (83).Equivalently, (86) defines a section20 z : X → G of the extended phase space X ext ≃ X × G, such that The appearance of the section z provides an indication of what our next step should be -namely we pass to X ext , the extended phase space obtained from the symplectic manifold (X, Ω) with respect to the G-action defined by a. X ext. is a principal bundle but locally a point in X ext corresponds to the pair (x, g) where x ∈ X and g ∈ G.In this regard we can view the Poisson algebra of X ext as the set of maps from the group G into the Poisson algebra of X, or what is the same, the set of functions on the product space X × G.In particular, to each f ∈ M pq X we can associate an "extended" function: The function i(f ) contains the information of f evaluated along each of its gauge orbits.
From the Poisson algebra perspective, i is a morphism -following from the fact that the action a preserves the symplectic form.This means that {i(f ), i(g)} Xext. is i({f, g} X ).Equivalently the symplectic form Ω ext.pulls back to Ω = i * Ω ext., and Hamiltonian vector fields on X ext.are push forwards V i(f ) = i * V f , where V f is a Hamiltonian vector field on X.These observations fit into the larger view of M pq Xext as a "pre"-crossed product algebra comprised of two kinds of elements.First, there are the extended functions comprising the image of (89) which furnish a copy of the non-extended Poisson algebra M pq X .Second, there are functions representing each infinitesimal generator of G, furnishing a copy of its Lie algebra.The complete Poisson algebra M pq Xext therefore contains a copy of the non-extended algebra, a copy of the Lie algebra of G, and cross terms associated with the infinitesimal action of G on M pq X .For a more formal derivation of this algebra, see [1].
Let us now define the Faddeev-Popov determinant: Here µ is the left invariant Haar measure of the group, and F is the gauge fixing functional. 21It is straightforward to show that (90) is invariant under the action of G: Together, the FP determinant and the delta function define a conditional probability distribution characterized by the following properties: Here we have used the fact that for each x the unique element g that solves F • a g (x) = 0 is g = z(x), and . By extension, we can use Φ(g | x) to define a conditional expectation As is evident from (93) the composition of ( 89) and (94) turns out to be precisely the gauge fixing map we are looking for: We are now prepared to define an extended path integral construction which properly accounts for the group of constraints.In particular, we define a "dual weight" φ : M pq Xext → given explicitly by Acting on an element of the non-extended Poisson algebra we have22 Here ν ext ϕ is the path integral measure obtained by promoting ν ϕ to the extended phase space.The second equality in (97) is obtained by first performing the integration over the group to activate the gauge fixing, with ν X/G ϕ the path integral measure on the quotient space obtained from pushing forward ν ext ϕ by the projection (83) in the measure theoretic sense -including a factor of the FP determinant.
At this point we should pause to make an observation.The constraint quantization defined by ( 97) is distinct from the usual Faddeev-Popov formalism in the following way -rather than mandating that the insertion f be invariant we have allowed for the insertion of arbitrary "dressed" functions, and then used the fact that (95) renders such elements G-equivariant.In this respect it is not true that the expectation value of a non-invariant insertion is zero in (86).Rather, (97) computes the expectation value of the gauge-invariant projection of any given insertion.Here the gauge invariant projection of a generic operator can be identified with the gauge orbit to which it belongs.These ideas will become more manifest in the examples discussed in Section 4.
It is straightforward to show that the new state obtained by promoting ϕ → φ • i is invariant under the action of the group.In particular, To move from the first to the second line we have changed variables, y = a g ′ (x), and used (82) to change the measure ν ϕ with impunity.Recognizing f • a g ′ = a * g ′ f and using the definition of the Lie derivative, (98) implies: where ξ µ is the tangent vector generating the action a g ′ : X → X as its integral curves, with µ ∈ g the Lie algebra element integrating to g ′ ∈ G.As discussed in [1], in the extended phase space , where H µ is the Hamiltonian function associated with the group action generated by µ and {, } ext. is the Poisson bracket.Thus, (99) further implies that Eqn. (100) indicates that the Hamiltonian functions H µ act trivially inside of expectation values for physical states.

Dual Weights and Constraint Quantization
Having outlined how the Faddeev-Popov gauge fixing procedure works in the path integral sense, let us conclude this section by discussing how this approach can be connected to the crossed product M ⋊ α G via the dual weight theorem.To make this comparison we employ the correspondence (1).Let M be a von Neumann algebra acted upon by the group G via the automorphism α : G × M → M. One should view M as the quantized version of the non-extended Poisson algebra M pq X .We denote by M ext the set of strongly continuous, compactly supported maps from the group G into the von Neumann algebra M.This is inspired by the definition (88) for the Poisson algebra of the extended phase space.As we shall see, the von Neumann algebra associated with M ext is equivalent to the crossed product M ⋊ α G, and may therefore be regarded as the quantization of M pq Xext .We will denote elements of M ext by X, Y, etc. with the understanding that these objects are maps i.e., X(g) ∈ M for each g ∈ G.
To explicitly study M ext as a C * algebra, and to formulate its associated von Neumann algebra we follow the construction of Haagerup [70,71].Recall that integration on the group G is defined in terms of a left invariant Haar measure, µ, along with its module function 23 δ : G → which tracks the failure of µ to be right invariant.The set M ext can be made into an involutive Banach algebra by endowing it with the following operations which define a product and an involution, respectively: Given the representation (39), we can define a ⋆-representation of M ext acting24 on H aux : In [70] it is shown that im(ρ) is dense in M ⋊ α G in the weak operator topology.Thus, the weak closure of M ext is equivalent to the crossed product algebra.We now come to the crucial topic of dual weights.To begin, we define the map: Here e ∈ G is the identity element.In other words, (105) is nothing other than a projection of M ext down to M, as in (93).As demonstrated by Haagerup in [54,70], this projection can be used to define an operator-valued weight from the crossed product to the algebra M. A faithful, semi-finite, normal weight ϕ on M corresponds to a quantum state, and therefore plays the same role as the path integral (76).
Combining such a weight with the map (105) we obtain a weight on the extended algebra: Of course, this is totally analogous to (96).
We would now like to make use of (106) in order to construct G-invariant weights on M. To do so, we take motivation from (89) and specify an explicit embedding of the algebra M inside of M ext .Since M 23 The module function is a group homomorphism, δ(g)δ(h) = δ(gh), defined by the property is linearly spanned by the set of positive elements therein, we define this procedure on positive operators x * x ∈ M and then extend the definition appropriately. 25The relevant embedding map is of the form: In (107) the notation α(a) corresponds to the element α(a) : G → M in M ext such that α(a) (g) = α g (a).
Using (102) and (103) it is easy to compute:26 which is precisely (55) corresponding to the dressing of the operator x.Thus, as was the case in (95), composing the maps T and i results in our desired mapping from M to the invariant subalgebra M inv .Notice that in this case rather than realizing a gauge fixing condition, (108) results in the group averaging map from Section 3.1.In conclusion, we have shown that Hopefully the analogy between this analysis and the path integral analysis is clear.The algebra M ext plays the role of the extended Poisson algebra.It is explicitly obtained by considering maps from the group G into the algebra M, as the algebra M pq Xext can be thought of as the set of maps from G into M pq X (88).The projection (105) is analogous to the conditional expectation defined by the Faddeev-Popov procedure (93), which localizes maps from G down to the zero section of an appropriate fibration.In both cases, the composition of T with a weight ϕ of the non-extended algebra results in a dual weight, ( 96) and ( 106), respectively.To compute the value of the dual weight applied to elements in the non-extended algebra we need a lifting map that embeds the non-extended algebra in the extended one.In the path integral context this map is supplied by (89), while in the algebraic context this map is supplied by (107).In both cases these maps can be interpreted as dressing the non-extended operators by acting on them with the relevant group automorphism.The projection map and the embedding then conspire to perform the gauge fixing, as can be seen in ( 95) and (108).Thus, when applied to these dressed operators, the aforementioned dual weights are manifestly G-invariant -(97) and (110).A summary of this correspondence can be found in Table 1.

Path Integral
Crossed Product Non-extended Algebra: Table 1: Overview of the correspondence between the path integral approach to constraint quantization, and the same approach using the crossed product and dual weights.This can be regarded as an extension of the correspondence (1).

Commutation Theorem for Crossed Products
This section details an alternative approach to implementing constraints via the crossed product algebra by augmenting the action of the gauge group.As we shall see, this approach is equivalent to the constraint quantization procedures we have introduced, and fits naturally into the extended phase space formalism.The discussion in this section may be regarded as further explicating the connection between the conditional expectation oriented approach to implementing constraints, and the related approach of designating a socalled observer.
Let (M, G, α) denote a covariant system; that is M is a von Neumann algebra acted upon via the G-automorphism α : G × M → M. Let π : M → B(H) be a Hilbert space representation of M, and denote by H aux ≡ L 2 (H; G, µ) ≃ H ⊗ L 2 (G, µ) the canonical covariant representation space of the system (M, G, α).As always, µ is a left invariant Haar measure on G.The covariant representation of (M, G, α) on H aux is specified in terms of the pair of maps where here and is left translation.The crossed product algebra M ⋊ α G is the von Neumann algebra generated by π α (x) and λ(g) closed in the weak operator topology induced by H aux .
It can be shown that M⋊ α G is realized as a subalgebra of M⊗B(L 2 (G; µ)). 27From this point of view, we can now state the commutation theorem, which may be taken as an alternative definition of the crossed product algebra.Let 28 Then, M ⋊ α G is equivalent to the invariant subalgebra of M ⊗ B(L 2 (G; µ)) with respect to the action θ: For a proof of this theorem, see [55].In fact, the proof proceeds by a group averaging argument which is very similar to the one outlined in Section 3.1.
Notice that the commutation theorem does not imply that the crossed product algebra is invariant under the action of the original automorphism α.Instead, the invariance of the algebra has been realized in this case by augmenting the action α → θ.In the literature, e.g.[7][8][9][10], this augmentation is achieved through the introduction of an 'observer' whose Hamiltonian implements the right translation.In the current note, we have avoided appealing to the notion of an observer in favor of a geometric presentation of the crossed product.By consequence, we can now see exactly how these two approaches are related.In our presentation, the role of the right action is played by the conditional expectation.This is especially clear in Section 3.3, in which the conditional expectation projects points along generic gauge orbits to a reference point singled out by the choice of gauge fixing functional.This point will also be addressed in Section 4, whereupon it is observed that, after dressing the fields, a gauge transformation can be absorbed entirely into a right translation of the group element g introduced into the extended phase space.
At the same time, we can provide a compatible interpretation for the extension of the action α → θ = α • Ad ρ through the symplectic analysis of the extended phase space.Let (X, Ω) be a symplectic manifold with G-action a : G × X → X quantizing to the von Neumann algebra M with G-automorphism α : G × M → M. The action a is generated by symplectic (but not necessarily Hamiltonian) vector fields ξ µ ∈ T X, such that That is, for each Lie algebra element µ ∈ g, we have a vector field ξ µ whose integral curves generate the action a.When we move from X to the extended phase space X → X ext the action a is promoted to an action Φ which is automatically Hamiltonian (and moreover equivariant).Infinitesimally, this is accomplished by mapping where we have regarded T X ext ≃ T X ⊕ g. 29 Eqn.( 117) is the symplectic analog of the algebraic promotion α → α • Ad ρ .
In fact, (117) generate the right action which is implicated in the structure of the extended phase space viewed as a principal G-bundle over X [1].
Recalling the map (89) we can now show that dressed observables i(f ) ∈ M pq Xext are automatically invariant under the extended action: This is the symplectic precursor to the observation of the commutation theorem, namely that dressed operators π α (x) ∈ M ⋊ α G are invariant under the extended action θ.It is also immediately clear that observables obtained via (95) will be invariant under the extended action: Here, we have made the identification M pq X ⊂ M pq Xext and used the fact that Thus, we have now demonstrated that dressing (119) and gauge fixing (120) provide two different but compatible approaches to implementing constraints in the context of the extended phase space/crossed product algebra.In summary, one interpretation for the projection map that implements constraints is as a right action which compensates the gauge transformation α thereby rendering generic observables gauge invariant.Denoting the projection by E, one may therefore regard α ′ ≡ α • E as a modified action to be compared 29 More rigorously, this is done by passing to the Atiyah Lie algebroid associated with the principal bundle Xext.See Section 4 of [20] for a detailed analysis.
with θ = α • Ad r .In other words, the projection should be viewed as generating an 'emergent observer', or, equivalently, the observer viewed as introducing a projection.This point of view is consistent with the correspondence between the extended phase space and the crossed product, and provides additional support to the claim that the extension provides a geometrization of the observer.We note in passing that this observation bears a close resemblance to Relational Quantum Dynamics and the Page-Wootters formalism.We refer the reader to [72] for a recent work which provides a good entrance point to these fields.

Examples
In this section we work through a series of three examples to draw out relevant features of the general theory presented in Sections 2 and 3.The examples are presented in order of increasing sophistication, culminating in a discussion of gravitational theories quantized on null hypersurfaces, in relation to the recent work [21].

The Symmetric Harmonic Oscillator
As a simple first example, we consider gauging in a quantum mechanical system, the D-dimensional symmetric harmonic oscillator.Since this is a quantum mechanics system, it can be regarded as a field theory in (0 + 1) spacetime dimensions, and consequently has some simplicity compared to the general case (for example, there are no codimension-2 structures).The phase space of the ungauged parent system is coordinatized by q, p, the Hamiltonian generating time translations is and we take the symplectic form to be Ω = δp i ∧δq i .It is well-known that this system possesses Hamiltonian vector fields generating SU (D), and the states at the n th energy level come in the n-index symmetric tensor representation, for n = 0, 1, ....The operators generating SU (D) are linear combinations of â † i âj , where â † j = 1 2 (p j + iq j ) are a set of ladder operators.The SO(D) ⊂ SU (D) symmetry acts as and is interpreted as spatial rotation.
In this section we consider gauging the SO(D) subgroup.At the level of the Hilbert space it is clear that this requires the SO(D) generators to annihilate physical states.Since the states of the ungauged theory come in particular irreps of SU (D), it is a simple matter to extract the states of the gauged theory as they correspond to singlets that may occur in the decomposition of the SU (D) irreps into SO(D) representations.Such singlets occur for even n, as the trace is the invariant.Thus all states of the gauged theory are generated from the vacuum by Â † := 1 2 j â † j â † j which indeed is an SO(D)-invariant.From the operator algebra point of view, the ungauged theory has operators generated by â † j , âj , forming the algebra M = [â † j , âj ]/∼ where the equivalence refers to [â i , â † j ] = δ i j Î.This is a von Neumann algebra, with Hilbert space representation on, for example, L 2 (Ê D ).Gauging the SO(D) symmetry in the manner outlined above yields the von Neumann algebra M inv.= A({ Î, Ĥ, Â † , Â}). 30 In particular, there is a (spectrum generating) SL( 2) algebra 30 Here, A(•) denotes the von Neumann algebra generated by the indicated elements.
where Ĥ = 1 2 j â † j âj + D 4 Î.The spectrum consists of a single unitary lowest weight representation beginning with energy E 0 = D/4.
We now consider the path integral, along with a Faddeev-Popov insertion which we would like to interpret as a conditional expectation.Starting from the phase space X = { q, p}, the path integral of the ungauged theory uses Hamilton's principle function evaluated along a curve γ in phase space The path integral is then of the form where t makes reference to the intrinsic length of a path in phase space.Here f is a compact notation for a possible series of insertions along the time contour.For simplicity, we hereafter restrict our attention to the D = 2 case, as it already possesses the features that we wish to display.The SO(2) symmetry is generated on phase space by the vector field Gauging the SO(2) ⊂ SU (2) symmetry corresponds to introducing an SO(2) connection on the phase space X that promotes the symmetry to be local in time.That is, we build on X a principal SO(2) bundle with connection, which is X ext. .An SO(2) connection includes two pieces.The first is the familiar gauge field which covariantizes time derivatives, with D t q = ∂ t q − A 0 ε • q and thus L z = p • ε • q.We therefore recognize L z as the Noether current for SO(2) and, since we are in (0 + 1)-dimensions, this coincides with the gauge constraint.
The second piece of the SO(2) connection is a vertical component which can be thought of as χ = δφ, for φ an element of the Lie algebra so(2).The extended phase space is coordinatized locally by X ext.= { q, p, φ} with φ thought of as a local fibre coordinate (that is, an exponential coordinate for the group SO(2)).The group action is given by a : X × G → X, where a : ( q, p, φ) → a φ ( q, p) ≡ ( q, p).More specifically, a φ is of the form with R(φ) ∈ SO(2) a rotation through angle φ.When the extended phase space is pulled back to a bundle over a curve parameterized by t, we should in addition include A 0 = Ã0 + φ in order to ensure the covariance of (128).We interpret (129) as defining tilded variables, ( q, p), which are invariant under the SO(2) action in the sense that a transformation of the untilded variables can be absorbed into a shift of the angle φ, see (129).In this regard ( q, p) represent a single point on a gauge orbit.We can extend this statement to an arbitrary function f ( q, p) on X by introducing the map (89) which passes f to a function on the extended phase space Similarly, using (129) the symplectic potential can be rewritten The term δφ is the Maurer-Cartan form on the extended space of fields.This perspective is consistent with the aforementioned observation that the pair (A 0 , φ) specifies a full connection on the extended phase space regarded as a principal bundle over X, with χ being the vertical part.As was observed in Section 2, the Maurer-Cartan form and the constraint Lz form a symplectic pair.
From the algebroid perspective, what we have done in (132) is to extract a vertical, or group variational term as was introduced in (13).For the associated transformation of the symplectic structure to be canonical the count of degrees of freedom should remain the same, and thus we should project θ( q, p) to a form on X/G.This is analogous to introducing a choice of gauge fixing, corresponding to the addition into the path integral of a conditional probability measure: for some gauge fixing functional F : X → so(2).The measure (133) defines a conditional expectation from functions of { q, p, φ} to functions of { q, p} as For example, we may take F = i(f ) with f ∈ Ω 0 (X), in which case the conditional expectation (134) maps arbitrary functions on the phase space to gauge invariant observables as in (95).
To see how this works in this example, we will compose the path integral (126) with the conditional expectation (134) to obtain a path integral on the full extended phase space.A nice choice for F is In this case the F-P determinant is unity.Writing for brevity we obtain To proceed, we change variables on phase space (x, ỹ, px , py ) → (r, pr , θ, pθ ).The Jacobian is trivial, and recognizing Lz = pθ , we obtain We have also included in (137) an integral over the gauge field A 0 which serves to implement the constraint pθ = 0.The bracketed elements in (137) can be written explicitly, in this gauge, as which are representatives of each gauge equivalence class as determined by the gauge fixing condition (135).One should interpret f ([ q], [ p]) as a dressed operator insertion.The path integral (137) reproduces the algebra of gauge invariant observables expected from the operator based analysis (124).A small remaining subtlety here is that the coordinate r is positive in (138); this can be treated by gauging spatial parity of an ordinary oscillator of frequency ω.This gauging then removes the states of energy E − E 0 = n for odd n.

Yang Mills
As a second example, we consider Yang-Mills theory in 4d.In contrast to the quantum mechanics example of the last subsection, there is an important conceptual feature present here which is the existence of a gauge charge having support on surfaces of codimension two relative to spacetime.We will take care to keep track of this.We consider quantizing on a spatial hypersurface Σ; so the theory has a symplectic potential and a Hamiltonian31 Both ( 140) and ( 141) are invariant under the hypersurface gauge transformation where V is regarded as an element of some compact group G.The symplectic form is non-degenerate off-shell, however going on-shell invokes the Gauss constraint leading to a degeneracy and necessitating a corresponding gauge fixing.
Although one could apply the techniques of RAQ or BRST to construct gauge invariant states and dressed operators, in this section we will discuss the path integral quantization more explicitly.To this effect we consider a t-parameterized curve through phase space and begin by promoting the hypersurface gauge invariance to be local along such curves.In particular, this means that the phase space action (or Hamilton's principle function) takes the form As in the previous example, A 0 covariantizes the action with respect to t-dependent gauge transformations, here in a way that is consistent with the additional hypersurface gauge invariance.That is, the field A 0 has a hypersurface covariant derivative We note in passing that the resulting theory (143) possesses a global Lorentz symmetry.Expanding (143) we obtain the action Up to a total derivative, A 0 plays the role of a Lagrange multiplier 32 for the Gauss constraint G = D j π j = ∂ j π j + [A j , π j ].More precisely, A 0 is a Lagrange multiplier in the bulk of the hypersurface.In standard analyses this is often taken to be the end of the story, but we are interested in evaluating a path integral for a generic subregion of the hypersurface Σ.To do so, we must introduce an embedding map φ (2) : S → Σ for a corner, regarded here as the boundary of the subregion. 33Then (145) becomes where a 0 is A 0 • φ (2) and É = φ * (2) ( * 3 π), the pullback of π j to the corner.As we shall see, the inclusion of these corner degrees of freedom has important implications for the physics of the subregion.
The naive unextended phase space of the Yang-Mills theory is coordinatized by {A i , π i }.In the absence of a boundary, we would define an extended phase space as a principal bundle over the naive phase space and we would manage the corresponding theory through the usual Faddeev-Popov procedure.On the other hand, in the presence of a boundary, physical fields will emerge with support on the boundary that survive the gauge-fixing procedure.This corresponds to the existence of a residual global symmetry at codimension-2, with gauge-fixed fields continuing to transform in non-trivial representations of the resulting global symmetry group.In Section 2, we described this in general through the idea of separate codimension-1 and codimension-2 extensions to the phase space.We will describe how this idea plays out in the specific case of the Yang-Mills theory.
To begin we consider the group action on the space of fields, The principle function satisfies S[A i , π i , A 0 ] = S[ Ãi , πi , Ã0 ] ≡ S, independent of U .As in the previous example, we can interpret the tilded variables as defining a gauge orbit, and in that sense, the gauge symmetry can be thought of as a right action on U alone with Consequently we refer to ( Ãj , πj ) as dressed fields.One can check that other quantities, such as the magnetic field, are similarly dressed.Using (147) the symplectic potential density becomes where G = ∂ j πj + [ Ãj , πj ] is the dressed Gauss law constraint.Integrating over Σ, we then find the symplectic potential In (150) we can recognize the Maurer-Cartan form ̟ (1) = δU U −1 .In fact, (150) has the same form as the general expression (16) with the constraint Å = − * 3 G.Moreover, in the presence of a boundary the charge É appears in the symplectic potential conjugate to the boundary value of the Maurer-Cartan form ̟ (2) = δuu −1 .We interpret this to mean that there is a corner symplectic pair present.The charge É acts as the generator of the emergent global symmetry supported on the boundary, and the u fields are subsequently in a representation of this global symmetry.
Eqn. ( 150) implies that we should regard the physical phase space to include the fields34 X = {A i , π i , u, É}.In the language of Section 2.1 this constitutes what we referred to as the extension at codimension-2.Hereafter we introduce the notation γ = (A i , π i , u, É) to refer collectively to these degrees of freedom.The gauge group acts on X via the map a (1) : G × X → X defined by (147) in the bulk along with a trivial action on u, É.We have included a superscript (1) to distinguish the gauge symmetry a (1)   from the global symmetry generated by the corner charge.In the following we will refer to the latter by a (2) .For an arbitrary function f (γ) on X, we can define the map (89) which passes f to a function on the extended phase space The codimension-1 piece of (149), tr( G̟), resembles a pure gauge symplectic pair.In order for the above analysis to correspond to a canonical transformation for fields in the bulk of the hypersurface, we introduce a gauge fixing to project tr(π j δ Ãj ) to X/G.In the path integral this corresponds to the introduction of a conditional probability distribution for some F : X → g.The measure (152) defines a conditional expectation from functions of {A i , π i , u, É, U } to functions of {A i , π i , u, É} as with µ(U (t)) a left-invariant Haar measure on G in the bulk.A representative gauge choice is F = D j A j .At least in the Abelian case, the A i , π i are then reduced to their transverse components.
We now have all the pieces we need to construct a path integral for the Yang-Mills theory.In particular 35 , The action S0 is defined through the equation that is we have separated the bulk Lagrange multiplier away from the physical part of the action.The integral (154) can be evaluated in a series of steps.First, integrating over A 0 generates a delta functional which sets the constraint current G = 0. Second, integrating over U activates the gauge fixing.In the end, we are left with an integral of the form where [γ] is the representative of the gauge orbit which is selected by the chosen gauge fixing.An important observation that should be made is that the gauge fixed function f ([γ]) may still depend on data derived from the corner fields, (u, É).In other words, gauge fixed functions remain in the representation space of the global symmetry a (2) .Thus, while implementing the constraints of the theory removes the redundancy introduced by gauge symmetry, the algebra of gauge invariant operators retains a crossed product structure.Here, the 'extension' is provided by corner supported transformations which act as global symmetries in the gauge fixed theory.Formally, one may therefore interpret the path integral (154) as including a sum over corner supported Lie algebroid isomorphisms parameterized by the field u.
In the gravitational context the analogous isomorphism will correspond to the corner embedding [19,22,24] as was implemented in a quantum context in recent work [73].We will have more to say about this in the next section.

4d Gravity on a Null Hypersurface
In this penultimate section, we consider the phase space of a gravitational theory (in particular, the Einstein-Hilbert theory) formulated on a null hypersurface N .This will serve as an example of a diffeomorphisminvariant theory.The intrinsic geometry of such a null hypersurface is given by a Carroll structure and a Carrollian connection.A Carroll structure is determined by a codimension-1 metric q ab and a null vector field ℓ a .An Ehresmann connection k a yields a rigging, defining a horizontal section of the tangent bundle and a corresponding projector q a b = δ a b − k a ℓ b , ℓ a being vertical.As reviewed recently in [21], the symplectic potential [74] (see also [75,76]) can be written as where Here we are using the same notation as [21], and all of the details and associated discussion can be found there.Our goal in this section is to draw out an interpretation of the analysis of [21] that coincides with the general picture presented in this paper.As in [21], we will focus on two of the gauge symmetries, namely an internal boost symmetry and the diffeomorphisms that reparameterize the null coordinate, leaving the general case to future work.
To begin, we can transform the fields of the theory by a boost parameterized by a function α such that δℓ a = −δαℓ a − e −α δu a with k a δu a = 0.Under this transformation, one finds that the canonical symplectic potential can be written Here we have separated out the determinant of the metric, q ab = Ωq ab with det q = 1, and Ω = √ det q.The final term in (159) is a corner contribution with Ω = Ω • φ (2) , where φ (2) is an embedding of the corner C in the hypersurface.The sub/superscripts (α) in (159) are meant to remind the reader that the fields appearing therein have been transformed by a local boost transformation. 36In fact, the result (159) can be thought of in terms of extracting the internal boost parameter as a canonical variable, with the rest of the fields being interpreted as defining a point on the boost orbit, the analogue of (129) in the harmonic oscillator or (147) in Yang-Mills.Given that interpretation, we expect to see the boost constraint appear in the bulk of the hypersurface and the boost charge on the corner.In this specific case, it happens that the boost constraint vanishes identically off-shell, and so the boost current appears entirely as the boost charge on the corner.One sees from (159) that the boost corner charge is given by the area density Ω.So (α, Ω) can be considered as a non-trivial symplectic pair with support on a corner, that is, an edge mode pair.
Next, we consider a reparameterization of the null fibre.We will do so in such a way that extracting the corresponding gauge transformation does not impact the previous boost symmetry analysis.To this end, we consider a combination of a diffeomorphism and a boost; this can be thought of as 'block-diagonalizing' the gauge transformations, and in [21] was referred to as a primed diffeomorphism.The primed diffeomorphism is quantified by a standard diffeomorphism v → V (v, σ) (referred to as the dressing time) together with a compensating boost.It is convenient at this point to introduce U := ∂ v V , which can be thought of as the boost group element associated with the primed diffeomorphism.In particular for an infinitesimal diffeomorphism V (v, σ) ≃ v + φ(v, σ) where φ is to be regarded as the vector field generating the diffeo, we have U ≃ 1 + ∂ v φ(v, σ), see Eqn. (148) in [21] for more detail.
As was the case in the previous examples, our intention will be to interpret the action of the primed diffeomorphisms as a canonical transformation on the space of fields.To this end, we introduce the bulk action a V which transforms the fields by a combination of the pullback by V and a boost U .In particular, By extension, we have for a phase space function F the dressing In Eqn. ( 160) and (161) we have introduced tilded variables with an interpretation analogous to that which was described below Eqn. ( 129) and (147).
Applying the transformation (160) to the symplectic potential (simplifying notation somewhat), we find where in the last line we dropped a total variation and we have defined We recognize this as the Raychaudhuri constraint, written in the tilde variables.As established in [21], the Raychaudhuri constraint C is conjugate to V -it is the constraint appearing in the bulk term of the primed diffeomorphism Noether current.We moreover see that the transformation (160) has brought to light two additional terms in the corner piece of the symplectic potential.We now interpret the first two terms in the corner term of (162) as corresponding to the charge of the primed diffeomorphism, while the third term is related to the boost charge.Notice that these terms are of the expected form tr(É̟ (2) ), for appropriate Maurer-Cartan forms ̟ (2) , U −1 δU for the boost part of the primed diffeomorphism, δV • V −1 for the standard diffeomorphism, and δα for the internal boost.
To interpret (160) as a canonical transformation a gauge-fixing condition is required.The transformed canonical symplectic potential θcan contains the spin-0 term −μδ Ω.In [21], the (classical) condition μ = 0 was chosen; here we can promote this to a Fadeev-Popov gauge fixing procedure in a quantum theory by choosing an appropriate gauge-fixing functional F (both for boosts and primed diffeomorphisms).
Since we are considering the theory from the extended phase space point of view, the algebra of constraints and the algebra of charges are expected to close.As explained in [19,22], this comes about via the proper inclusion of embedding maps (or Lie algebroid morphisms in the more general presentation of [20]) in the phase space.In [21], this was shown explicitly for the constraint algebra, where it became clear that the spin-0 sector can be interpreted as codimension-1 extended degrees of freedom.This will also be true for the spin-1 sector, associated with the Damour constraint.For the charge algebra, however, a codimension-2 extension is required, and the above analysis can be interpreted in those terms.In fact the Maurer-Cartan forms seen above can be recognized formally as the vertical part of a connection on the configuration algebroid associated with boost and diffeomorphism gauge symmetries. 37The diffeomorphism V can in fact be thought of as an embedding of the corner inside the null hypersurface -it defines a cut or section of the Carroll structure.While V is defined in the bulk of the hypersurface, it is pure gauge in the sense that it is conjugate to a constraint; there is a residual part of V that has support on a corner.This generates a subgroup of the universal corner symmetry.
Because the extended phase space construction yields integrability of charges, we expect that contracting the symplectic potential with the phase space vector field generating symmetries yields the corresponding charge.Working on-shell, in the case of the boost this immediately yields where we used I ′ λδα = −λ, while for the primed diffeomorphism where we used I ′ f δV = f and I ′ f U −1 δU = ∂ v f .These expressions coincide with the charges found in [21] by the more standard approach of constructing Hamiltonian functions relative to the symplectic form.
The analysis of this section should be regarded as providing the skeleton for the quantization of gravity on null hypersurfaces along with an actionable prescription for implementing the requisite diffeomorphism constraints (although here and in [21] only the Raychaudhuri constraint has been considered).We emphasize that this has been done without recourse to an observer, with the spin-0 sector of the naive phase space playing the role of a clock degree of freedom relative to which generic observables can be gravitationally dressed.As an additional point of emphasis, we should note that the appearance of the area as a Noether charge for corner supported boosts seems to signal a connection with the generalized entropy.We plan to address this in detail in future work.
We have stopped short of writing down the path integral in this section in recognition of the fact that there are important technical challenges which warrant a more complete discussion to be presented in future work.For example, integrating over the group of gauge transformations in this gravitational context raises the sticky issue of the non-locally compact nature of the group of diffeomorphisms.Nevertheless, as presented in Appendix E of [1], it seems possible that this difficulty may be circumvented by treating the set of diffeomorphisms as the pair groupoid and using the fact that every Lie groupoid is locally compact as a topological groupoid.Then one can appeal to the theory of left invariant Haar systems for locally compact groupoids which provides a suitable generalization to the theory of left invariant Haar measures in the context of a locally compact group.This observation should also prove instrumental in making sense of integrating over the set of all corner embeddings, as implicated by the codimension-2 aspect of the null hypersurface story discussed in this section.This is the analog of integrating over the field u in the Yang-Mills example.

Discussion
In this note we have made use of the crossed product to study the quantization of generic gauge theories, as well as the closely associated problem of implementing constraints.A crucial ingredient in this undertaking has been the extended phase space, which figures into the quantization of gauge theories through the correspondence (1).Using this correspondence and the geometry of Atiyah Lie algebroids we are able to formulate an integrable symplectic structure in which charges associated with internal gauge symmetries and diffeomorphisms are treated on the same footing.
In Section 2 we provided a sketch of the operator algebra which arises if one can adequately quantize the aforementioned symplectic geometry.An important insight that is drawn from this analysis is that this algebra can be regarded as the result of a pair of crossed products associated with automorphisms generated by charges that live on codimension one and codimension two submanifolds of spacetime, respectively.The codimension one charges generate gauge transformations and therefore must be implemented as constraints in the physical theory.By contrast, the codimension two charges are unconstrained and therefore generate physical transformations of states in the quantum theory.These charges are a central ingredient in the corner proposal [22][23][24][25][26][27][28][29][30][31][32][33][34], and play a key role in regulating entanglement observables in diffeomorphism covariant theories [1].
In Section 3 we have undertaken a general study of implementing constraints for a quantum theory in which a von Neumann algebra M is acted upon by an automorphism α : G × M → M regarded as a gauge symmetry.Our main goal in this section was to emphasize that the crossed product M ⋊ α G is the correct algebraic setting in which to accomplish this.In Subsections 3.1, 3.2, 3.3, and 3.4 we have provided an overview of four independent methods for implementing constraints in a quantum theory.In each subsection, we have highlighted precisely how the crossed product and the extended phase space figure naturally into the specific methodology at hand.For convenience, we review these observations now.
In the Hilbert space approach to Refined Algebraic Quantization, the role of the crossed product is to construct an auxiliary Hilbert space in which the constraints are automatically realized as unitary operators (39).From this point it is straightforward to follow the standard approach to obtain a physical Hilbert space in which all states are invariant under the gauge group.Alternatively, from the operator algebraic perspective the crossed product admits a projection to a subalgebra of G-invariant operators (55).The GNS Hilbert space associated with this subalgebra can play the role of a physical Hilbert space.The relationship between these approaches is summarized by the commutative diagram: where here Π d is the group averaging map in the algebra, and η is the rigging map.Of central importance in the algebraic approach to RAQ is the existence of a tracial weight in the crossed product algebra which plays the role of an invariant vacuum.This emphasizes the interplay between the two roles of the crossed product that we have stressed, in order to allow for the implementation of constraints it is necessary for the algebra to first be rendered semi-finite.In the BRST quantization scheme, the role of the crossed product is to furnish a representation of the BRST algebra complete with ghost and antighost degrees of freedom (64).This is made possible by constructing a covariant representation of the crossed product on the tensor product space Ω • (G) ⊗ H where H is a standard representation of the algebra and Ω • (G) is the exterior algebra of the group G.A Lie algebra representation on this Hilbert space is a tensor product between a standard representation on H and the co-adjoint representation acting on the exterior algebra generators (67).The BRST differential is obtained naturally by acting on elements of the Hilbert space with the extended representation of the ghost field c = c A ⊗ t A (67).Using the Cartan algebra inherited from the exterior algebra (73), it is easy to show that elements of the BRST cohomology are invariant under the extended action of the group and thus distinct BRST cohomology classes may play the role of a physical Hilbert space (75).
From the phase space perspective, a path integral is a weight on the Poisson algebra of a symplectic manifold (76).The problem of implementing constraints is motivated by the observation that generic observables (elements of the Poisson algebra) are not invariant under the action of the group G on X.To remedy this, we pass to the extended phase space, X ext ≃ X × G, and introduce a conditional expectation which projects from the Poisson algebra of X ext to the Poisson algebra of X (94).When this conditional expectation is paired with a dressing procedure (89) the result is a map from the Poisson algebra of X into the G-invariant sector therein (95).Using this data the naive path integral can be promoted to a G-invariant path integral by including a gauge fixing delta function analogous to the standard approach of Faddev-Popov.Utilizing the correspondence (1), the Faddeev-Popov gauge fixing procedure can be implemented in a completely algebraic fashion.This correspondence is made particularly transparent by appealing to Haagerup's construction of a C * algebra whose weak closure yields the crossed product [70].From this perspective, the gauge fixed path integral is analogous to a dual weight composed with a dressing procedure appropriate to Haagerup's algebra (107).
Finally, we have the commutation theorem.From an algebraic point of view the commutation theorem may be regarded as an alternative definition of the crossed product.It states that, given a covariant system (M, G, α), the associated crossed product, M⋊ α G, is precisely the invariant subalgebra of M⊗B(L 2 (G; µ) under the extended automorphism θ ≡ α•Ad r .Here r : G → U (L 2 (G; µ)) is the right regular representation of G on L 2 (G; µ).From the point of view of the correspondence (1), the extended action θ is parallel to the complete right action of the group G on the extended phase space (118) which combines the G-action on the non-extended phase space, a, with the standard right action of G on itself.The commutation theorem follows essentially from the fact that operators in the crossed product are 'dressed'.When acting on dressed operators the original automorphism α can be absorbed entirely into a right translation of the dressing group elements.This translation is subsequently canceled by the explicit right action of the group on the dressed operator.In this way the two actions on the crossed product algebra conspire to leave all of its elements invariant (119).The explicit right action of the group may be interpreted in analogy with the gauge fixing projection appearing in the path integral approach (120).Indeed, the role of this projection is precisely to offset any gauge variation with a compensating translation that solders each operator to a chosen reference point along its gauge orbit.
The algebraic perspective on constraint quantization makes it clear that each of these four approaches are merely devices for constructing conditional expectation-like objects which map operators in the crossed product to a G-invariant subalgebra.The need to relax some of the properties of conditional expectations as they are strictly defined in this context is useful because it provides some indication of how one may overcome pitfalls that have plagued constraint quantization, particularly in the gravitational context.For example, the group averaging construction yields a conditional expectation only when the group in question is compact, else it diverges and results in an operator-valued weight.However, as was discussed in Section 3.1, the group averaging map is just one possible choice for the requisite projection.A more flexible construction known as the generalized conditional expectation presents a possible alternative which always exists given a von Neumann algebra M with subalgebra M 0 and faithful semi-finite normal weight ϕ on M. Every von Neumann algebra possesses a faithful semi-finite normal weight [4], thus the only remaining question is whether the G-invariant subspace of M is a bona-fide von Neumann subalgebra.Nevertheless, while this would establish the existence of such a generalized conditional expectation it leaves the formidable problem of constructing it.What's more, as we have discussed in Section 3 generalized conditional expectations generically fail to possess the bi-module property of an operator-valued weight.The repercussions of this failure deserves careful consideration relative to the problem of constructing and interpreting physical states/algebras.We plan to address this point in future work.
A second problem that is encountered in constraint quantization for gauge theories and gravity is the fact that the constraints are not formulated in terms of a global group, but rather are local in spacetime.This problem is also addressed by our construction in passing the geometric structure of the constraints into the form of a groupoid.Again, however, there are technical subtleties which must be addressed in relation to, for example, integrating over Lie groupoids. 38We intend to explore these issues in forthcoming publications.

B Rigged Hilbert Spaces
In this short appendix we remind the reader of some relevant details about rigged Hilbert spaces (RHS).RHS were introduced by Gelfand in [78] as a mathematical formalization of Dirac's bra-ket notation for quantum mechanics.More broadly, RHS provide a formalism in which elements of the continuous spectrum of a self adjoint operator can be associated with generalized eigenvectors, as elements of the discrete spectrum are associated with genuine eigenvectors.In constraint quantization, as discussed in Section 3.1, we seek solutions to the equation U (ψ) = ψ, (B.9) where U = exp(Q) is a unitary operator on H obtained by exponentiating the self adjoint, but possibly unbounded, operator Q.The connection to RHS therefore arises by reformulating (B.9) as Because Q generically has a continuous spectrum, solutions to (B.10) are not always contained in the Hilbert space.Nevertheless, there are solutions to (B.10) if we pass to an appropriate RHS in which ψ can be interpreted as a generalized eigenvector with eigenvalue zero.
To begin, let us define the RHS.Our discussion follows closely that of [79].Let H be an infinite dimensional, separable Hilbert space.A RHS associated with H is a triple 41  Here, Φ ⊂ H is a dense subspace of H, and Φ * is the set of all continuous complex-linear functionals on Φ.As a subspace, the embedding of Φ in H is obtained trivially, while the embedding of H in Φ * is obtained by recognizing that any ψ ∈ H can be promoted to a complex-linear map on Φ using the pairing of vectors defined by the inner product g on H.It is important to recognize that Φ * contains more than just the metric dual of H, which will be central importance moving forward.Let A : Φ → H be a linear map with adjoint A † : H → Φ.If We now have all of the tools we require to state our main result which is the so-called Gelfand-Maurin theorem [78, 80].Let A be any self adjoint operator on H and denote by σ(A) ⊆ Ê + its spectrum.Then, there exists a RHS Φ ֒→ H ֒→ Φ * such that the following are true: Here µ L is the standard Lebesgue measure on Ê + .
To conclude this Appendix, we can shed some light on the very formal results discussed by considering the most quintessential application of RHS: the spectrum of the position operator in quantum mechanics.Let H = L 2 (Ê, dx), and consider the operator In standard quantum mechanics we are taught that the operator A has eigenvectors denoted by |x which are meant to be represented by the "functions" |x ∼ δ(x).However, delta functions are formally distributions or generalized functions and do not belong to the Hilbert space H. Nevertheless, they do make sense as functionals of H. Let us initiate the following notation: δ : Ê + → H * , δ x (ψ) = ψ(x).