Affine holomorphic quantization

We present a rigorous and functorial quantization scheme for affine field theories, i.e., field theories where local spaces of solutions are affine spaces. The target framework for the quantization is the general boundary formulation, allowing to implement manifest locality without the necessity for metric or causal background structures. The quantization combines the holomorphic version of geometric quantization for state spaces with the Feynman path integral quantization for amplitudes. We also develop an adapted notion of coherent states, discuss vacuum states, and consider observables and their Berezin-Toeplitz quantization. Moreover, we derive a factorization identity for the amplitude in the special case of a linear field theory modified by a source-like term and comment on its use as a generating functional for a generalized S-matrix.

1 Introduction

Ever since its inception, efforts have been made to put quantum field theory on an axiomatic basis. There are multiple objectives behind such undertakings. Conceptually, one would like to have a better understanding of what quantum field theory “really is” (and what it is not), possibly including an elucidation of aspects of the meaning or interpretation of quantum theory itself. Mathematically, an axiomatic system offers a rigorous definition and a context to make mathematically precise statements about certain quantum field theories or quantum field theory as such. Finally, an axiomatic formulation may help to indicate how quantum field theories can be extended to realms where they have not previously been experimentally tested. An important example for the latter is the extension from Minkowski space to more general curved spacetime.

An axiomatic approach that has proven particularly useful in this latter respect is algebraic quantum field theory (AQFT) [1]. In AQFT the causal structure of spacetime is intimately entwined with the algebraic structure of the objects of the quantum theory. This has advantages and disadvantages. Most notably, this leads to a very concise way of encoding local physics in a spacetime region, with just one core mathematical structure (a von Neumann or algebra) per spacetime region. Moreover, in quantization prescriptions this structure is directly linked to the classical observables in that spacetime region. This conciseness combined with mathematical rigor has justifiably fascinated physicists and mathematicians over the decades, making it today the best developed axiomatic approach to quantum field theory.

On the other hand, the central role played by causality in the core structure of AQFT makes it indispensable as a fixed ingredient of spacetime. This precludes the direct applicability of AQFT to situations where such a structure is not a priori given.

This limitation, which is even more stringent in most other approaches to quantum field theory, has motivated a new axiomatic approach, called the general boundary formulation (GBF). The GBF has been put forward with the express aim of disentangling the elementary mathematical objects of a theory (in this case states, amplitudes, observables) and their basic physical interpretation, from the metric or causal structure of spacetime. This is achieved on the one hand by explicitly localizing states on hypersurfaces and amplitudes in spacetime regions [2] in the spirit of topological quantum field theory [3]. On the other hand this requires an extension of the probability postulates of quantum theory for amplitudes [4] and observables [5]. While still considerably less developed than, say, AQFT, the GBF offers the perspective of further extending the realm of quantum field theory to contexts where spacetime is not equipped with a predetermined metric or causal background structure. It is widely expected that a quantum theory of gravity should live precisely in such a “background independent” context.

Most realistic quantum field theories are obtained or at least motivated through a process of quantization starting with a classical field theory. It is thus important for the usefulness of a given axiomatic approach that there be quantization prescriptions that produce the elementary objects which are the subject of the axioms starting from data encoding a classical field theory. In the case of the GBF the quantization prescription most straightforwardly adapted from well known tools of quantum (field) theory is Schrödinger-Feynman quantization [4, 6], which combines the Schrödinger representation [7] for state spaces with the Feynman path integral [8] for amplitudes. This quantization prescription has been successfully applied in various contexts including a non-perturbative integrable model [9], a generalization of the perturbative S-matrix [10], and in curved spacetime [11, 12]. Even though many of these applications lead to structures that rigorously satisfy the axioms, the quantization prescription itself is not rigorously formulated, at least not in its present form.

Ideally, quantization should not only be rigorous, but should provide something like a functor from a category of classical theories to a category of quantum theories. For the GBF such a functorial quantization scheme has indeed been described recently for the case of linear field theory [13]. There, the concept of a linear classical field theory is axiomatized and a construction is given that produces from the elementary objects of such a classical theory the elementary objects of a quantum field theory in the framework of the GBF. In particular, it is proven that the objects of the quantum theory obtained in this way do indeed satisfy the axioms of the GBF. Moreover, although it is not made explicit there, this construction is functorial, and in many ways so. For example, for a given system of spacetime hypersurfaces and regions we obtain a functor if we take the categories of classical and quantum field theories with morphisms given by the respective notion of “subtheory”: On the classical side a “subtheory” is obtained by restricting the local spaces of solutions consistently to subspaces, while on the quantum side a “subtheory” is obtained by decomposing the local Hilbert spaces of states into tensor products and selecting one component in a consistent way. Other possibilities for choices of categories include ones where each object carries its own system of hypersurfaces and regions etc.

A classical linear field theory is formalized in [13] as follows: For each region in spacetime we are given a real vector space of solutions of the field equations. Also, for each hypersurface in spacetime we are given a real vector space of germs of solutions. The latter spaces are moreover equipped with non-degenerate symplectic forms. Then, the natural maps from the former spaces to the latter (restricting solutions in regions to neighborhoods of the boundary) have to yield Lagrangian subspaces with respect to these symplectic forms. Although perhaps not obviously so, these conditions are well motivated from Lagrangian field theory. An additional ingredient which might be seen as structure already pertaining to the quantum realm is a compatible complex structure on the solution space for each hypersurface. This summarizes the axioms given in [13] for a classical linear field theory in an informal language.

The quantization prescription consists then of a combination of a version of geometric quantization for hypersurfaces and a certain integral quantization for regions. For each hypersurface, the construction of the associated Hilbert space of states is equivalent to the usual Fock space construction, where the phase space (here really the space of germs of classical solutions in a neighborhood of the hypersurface) with additional symplectic and complex structure is seen as the (dual of the) 1-particle Hilbert space. However, it is realized concretely as a space of holomorphic functions in the spirit of Bargmann. From the point of view of geometric quantization this is really the space of Kähler polarized sections of the prequantum bundle. For each region, the quantization prescription in [13] is given by a seemingly ad hoc integral prescription, although verified by providing the “right” results in certain examples.

In the present paper we consider affine field theory, as a first case of a rigorous and functorial quantization prescription targeting the GBF beyond linear field theory. By affine field theory we mean here field theory with affine spaces of local solutions and such that the natural symplectic forms associated to hypersurfaces are invariant with respect to the affine structure in addition to being non-degenerate. In many ways this can be seen as a generalization of the linear case and its treatment in [13]. For hypersurfaces, this requires a refinement of the geometric quantization prescription (Section 2.3), clarifying the role of the prequantum bundle and its relevant trivializations. For regions, we motivate the quantization as a variant of the Feynman path integral prescription (Section 2.4), thus justifying at the same time the origin of the prescription given in [13] as a special case of this.

Based on a suitable geometric setting for spacetime (Section 3.1), the axioms for classical field theory (Section 3.2) are a relatively straightforward generalization of those for linear field theory given in [13]. However, they involve additional structural elements from Lagrangian field theory (see Sections 2.1 and 2.2), notably the action and the symplectic potential. Also, they are considerably more extensive as both local spaces of solutions and their tangent spaces need to be kept track of separately since they are no longer canonically identified.

The central part of this paper is Section 4 where the quantization prescription is specified rigorously and the validity of the GBF core axioms (listed in Section 3.3) is proven. As in [13] the Hilbert spaces of states associated to hypersurfaces are realized concretely as spaces of functions (Section 4.1). However, the domain spaces (or rather their extensions) for these functions do not directly carry measures as in [13]. Rather, any choice of base point gives rise to an identification with a space of holomorphic functions with a measure on (an extension of) the domain space. This is then used to obtain the inner product, which turns out to be independent of the base point. In terms of geometric quantization these different function spaces arise from different trivializations of the prequantum bundle.

In Section 4.2, coherent states are defined. These are called affine coherent states to distinguish them from the usual coherent states (used in [13]). While the latter can also be “imported” into the affine setting, their definition and properties are base point dependent and therefore less convenient than the manifestly base point independent affine coherent states. In Section 4.3 the amplitude functions are defined and some of their elementary properties considered. In particular, an explicit formula for the amplitude of coherent states is obtained, generalizing the corresponding result from the case of linear field theory. Section 4.4 provides a proof of the gluing axiom, with the other GBF core axioms already proven in the previous sections. As in [13] the proof of this last axiom requires an additional integrability condition on the classical data.

In Section 5 some further aspects of the proposed quantization prescription are considered: Section 5.1 discusses some aspects of the picture that emerges if we choose to focus on amplitudes that may be viewed as “transition” amplitudes in a context of “evolution” between hypersurfaces. In Section 5.2 vacua in the sense of [4] are discussed. Unsurprisingly, there is no longer a preferred vacuum in the affine theory as there is in the linear theory. Nevertheless the finding in [13] that each global solution of the classical theory gives rise to a vacuum remains true in the affine setting. The relation between the linear and the affine setting on the quantum level is clarified in Section 5.3. Observables in the sense of [5] are discussed in Section 5.4. In particular, the Berezin-Toeplitz quantization of observables given in [5] for the linear setting is generalized to the affine setting, including a generalization of the coherent factorization property.

Finally, in Section 6 we consider in some detail a special case of particular interest. A linear field theory is given in a spacetime region. In the interior of that region a linear term is added to the action making the theory there affine. We are then interested in describing this affine theory in terms of the original linear theory. This turns out to lead to a remarkable factorization of the amplitude of the affine theory (Section 6.1). An important example for this setting is the case where the linear addition to the action is a source term (Section 6.2) in which case the resulting amplitude may be seen as leading to a generator of the perturbative S-matrix. In an evolution picture (Section 6.3) one recovers a generalization of the well known particle creation from the vacuum through a source.

Section 7 presents a brief outlook.

Coming back to issues mentioned at the beginning of this section, we stress that all constructions and results of this paper (except where explicitly stated otherwise) apply to field theory understood in a rather abstract and general sense. In particular, nowhere do we need to assume a particular spacetime metric or causal structure or even the existence of such a structure.

Various constructions in Section 4 as well as most proofs in this paper build on results of [13], to which we refer the interested reader for those details.

2 Motivation of quantization scheme

The quantization scheme put forward in this paper may be seen as a combination of a geometric quantization (for state spaces) with a Feynman path integral quantization (for amplitudes). We proceed to explain this in the present section.

2.1 Ingredients from classical field theory

We recall certain elementary ingredients of Lagrangian field theory here, relying on the conventions and notation in [13]. Thus, we suppose a classical field theory to be defined on a smooth spacetime manifold of dimension and determined by a first order Lagrangian density with values in -forms on . Here denotes a point in spacetime, a field configuration at a point and the spacetime derivative at a point of a field configuration. We shall assume that the configurations are sections of a trivial vector bundle over . We shall also assume in the following that all fields decay sufficiently rapidly at infinity where required (i.e., where regions or hypersurfaces are non-compact).

Given a spacetime region and a field configuration in its action is given by

(1)

is usually viewed as a real valued function on the space of field configurations on . However, in the following we will often be interested only in the value of on the space of solutions of the Euler-Lagrange equations in . Given a hypersurface we denote by the space of (germs of) solutions of the Euler-Lagrange equations in a neighborhood of . The symplectic potential is then the one-form on defined as

(2)

Here while is a tangent vector to , i.e., an element of the space of solutions linearized around . The restriction of solutions in the interior of a region to its boundary induces a map . Given this induces a map between linearized solutions . The symplectic potential is then related to the exterior derivative of the action via

(3)

For a hypersurface , the symplectic form is the two-form on given by the exterior derivative of the symplectic potential,

(4)

We shall assume that the symplectic structure is always non-degenerate.

Note that a change of orientation of the hypersurface changes the sign of the symplectic potential in (2) and consequently that of the symplectic form in (4). In quantization schemes that consider a global space of solutions in , the orientation of the (then usually spacelike) hypersurface has no particular importance and the sign of the symplectic potential and the symplectic form is chosen in a manner convenient for the quantization. Indeed, in text books the formulas (2) and (4) are often presented with the opposite sign. For our purposes, however, the choice of sign turns out to be uniquely determined by the interplay between geometric quantization on hypersurfaces and Feynman quantization on regions. The key is here the relative sign in equation (3). We will come back to this issue in Section 4.3.

Recall that given a region and a solution , the space of solutions linearized around is an isotropic subspace of as follows by taking the exterior derivative on both sides of (3) and noticing that . In many cases of interest this subspace is also coisotropic and hence Lagrangian, see [13] for further remarks on this point.

2.2 Specializing to affine field theory

We specialize now to the type of field theory of principal interest in the present paper: affine field theory. We proceed to explain exactly what we mean by this. Firstly, we suppose that the spaces of solutions for regions and for hypersurfaces are affine spaces. That is, there exist corresponding real vector spaces and with transitive and free abelian group actions and respectively, written as addition “”. This allows to identify canonically all the tangent spaces with and with respectively. On hypersurfaces, the symplectic potential may then be seen as a map , linear in the second argument. We shall switch from here onwards to the notation instead of the previous notation . Our second key assumption is that the symplectic potential is equivariant with respect to the affine structure in the following sense: There exists a bilinear form such that

(5)

This implies in turn that the symplectic structure is independent of the base point and may be viewed as an anti-symmetric bilinear map given in terms of the symplectic potential as follows,

(6)

For a region , the relation (3) can then be integrated to determine the action in terms of the symplectic potential , up to a constant,

(7)

For simplicity of notation we have omitted writing explicitly the composition with the map in the arguments of the symplectic potential.

2.3 Ingredients from geometric quantization

In order to construct the quantum state spaces associated to hypersurfaces we will use ingredients from geometric quantization. We thus proceed to give a lightning review of geometric quantization with special attention to the relevant case of holomorphic or Kähler quantization. We warn the reader that the following account is mostly based on thinking of phase space as a finite-dimensional manifold. Moreover, it is highly simplified and inaccurate in various respects. Nevertheless, it will suffice for our purposes. For a proper appreciation of geometric quantization we refer to standard text books such as [14].

Geometric quantization of a classical phase space with symplectic two-form proceeds in two steps: A hermitian line bundle , the prequantum bundle is constructed over , equipped with a connection whose curvature is given by the symplectic form . The prequantized Hilbert space is then given by square-integrable sections of with respect to a measure that is invariant under symplectic transformations. The inner product between sections is thus,

(8)

where denotes the hermitian inner product on the fiber over . Note that a symplectic potential, i.e., a one-form over such that gives rise to a trivialization of the bundle through the choice of a special section that satisfies

(9)

for all vector fields on . Any other section of can then be obtained as , where is a complex valued function on . We then have

(10)

Moreover, by adjusting the overall normalization of if necessary we can arrange

(11)

The inner product (8) may then be written as,

(12)

While is usually “too large”, the “true” Hilbert space of states is then obtained by a suitable restriction of through a polarization. This is the second step. A polarization consists roughly of a choice of Lagrangian subspace of the complexified tangent space for each point . One then defines polarized sections of to be those satisfying

(13)

where is a complex vector field valued at each point in the polarized subspace . The restriction of to the polarized sections yields the Hilbert space .

In the holomorphic case, the polarization is induced by a complex structure on the tangent spaces , which is at the same time a symplectic transformation. Then, projects onto the polarized subspace . At least locally, there exists then a Kähler potential and an adapted complex symplectic potential such that

(14)

where are local holomorphic coordinates with respect to the complex structure . We can choose a (local) section of satisfying (9) with respect to the complex one-form to trivialize . Then, general sections of can be obtained as with a complex valued function on . They satisfy the analogue of equation (10) with replaced by and replaced by . The point is that the subspace of polarized sections admits a simple description in terms of this trivialization. Namely, the polarized sections are now precisely the sections , where is a holomorphic function on . Since is complex, the section cannot be normalized in analogy to (11). However, it can be related to the section that satisfies (9) with respect to a given real symplectic potential . Indeed, let be the complex function on such that . Then, we can use (12) to write the inner product on as follows,

(15)

As already mentioned the above account of geometric quantization is inaccurate in various respects. Nevertheless it is good enough to motivate our further discussion which will be limited to the case of affine field theory. Thus, we seek to quantize the space of solutions associated to a hypersurface . The key additional ingredient apart from the classical data already described is a complex structure on the tangent spaces of . Since these tangent spaces are all canonically identified with and the symplectic structure is independent of the base point it will suffice to consider a single complex structure on as was the case in the treatment of linear field theory in [13]. Thus, the complex structure is a linear map satisfying and . This gives rise to the symmetric bilinear form by

(16)

We shall assume that this form is positive definite. The next step is to complete to a real Hilbert space with the inner product . (We will continue to write for this completion.) It is then true that the sesquilinear form

(17)

makes into a complex Hilbert space, where multiplication with is given by applying .

As discussed above, the complex structure defines a polarization and implies the existence of a Kähler potential. The Kähler potential is not unique, but a choice of base point gives a natural definition of via

(18)

The adapted symplectic potential is then,

(19)

Recall that on the other hand we have the real symplectic potential . Suppose and are sections of the prequantum bundle over satisfying (9) with respect to and respectively. Moreover suppose that is normalized in the sense of (11). Let be such that . Then, it follows from (10) applied with on the one hand and with on the other that,

(20)

This determines up to a constant factor, which is unimportant as it can be reabsorbed into the normalization of . We set

(21)

We would then like to define the Hilbert space to consist of sections of that can be written as with a holomorphic function on . The inner product would be given by formula (15), where is a probability measure on invariant under translations by elements of . Indeed, if is finite-dimensional this immediately yields a nicely defined Hilbert space. However, in the more interesting case that is infinite-dimensional no such measure exists. Thinking of the factor in (15) as being part of a measure improves the situation. Still, no such measure exists on . However, thinking of as living on rather than on and suitably extending to a larger space does yield a well-defined measure. Such measures are well known, see e.g. [15] and an explicit construction suitable for the present setting was provided in [13]. The latter will be used in Section 4 to give a properly defined analogue of (15).

2.4 Ingredients from Feynman quantization

Attempts to construct the amplitude maps associated to spacetime regions via quantization schemes that describe (time-)evolution through infinitesimal generators meet considerable difficulties. (Recall for example the difficulties in making the Tomonaga-Schwinger approach [16, 17] well defined.) In contrast, the Feynman path integral provides a conceptually much more satisfying approach to amplitudes. Of course, it comes with its own difficulties, but these do not show up in the simple setting of affine field theory considered here.

Recall in particular, that the combination of the Feynman path integral with the Schrödinger representation yields a rather direct construction of amplitude maps [4, 6]. To put this into the present context we recall that the Schrödinger representation may be seen as a particular case of geometric quantization with a real polarization. In the language of Section 2.3, given a point in the space of solutions , the polarized subspace of the complexified tangent space arises as the complexification of a real subspace of the real tangent space . More specifically, in the case of the Schrödinger representation is the subspace generated by the “momenta” (in the notation of Section 2.1). Fix to be the section of the prequantum bundle over satisfying (9) with respect to the symplectic potential (2) as well as (11). Then, the polarized sections of are those that take the form , where is a complex function on that depends only on “position” coordinates .

If is a spacetime region and a state in the Schrödinger polarized boundary Hilbert space, its amplitude is given heuristically by the Feynman path integral via

(22)

where is the space of field configurations in and is a measure on it that is invariant under symplectic transformations. Of course, no such measure exists and even the precise definition of the space may be unclear. As a first step to improve the situation we assume that there is a correspondence between field configuration data on the boundary and solutions in the interior, i.e., splits additively into , where is the space of solutions in while is the space of field configurations in that vanish on the boundary. Then, (22) may be rewritten as

(23)

To further improve the situation we switch to the special case of affine field theory. The action is thus a polynomial of degree two on and by the variational principle we obtain for and with some function. (In the case where is quadratic .) This allows to factorize the inner integrand in (23) and, discarding a normalization factor that only depends on , to arrive at the expression

(24)

This is still ill-defined, but it turns out (Section 4.3) that the problem with the definition of the measure may be resolved in a manner similar to that indicated in the previous Section.

In the present paper, however, we take (24) as a motivation for defining amplitudes by the same (rigorous equivalent of) formula (24), but with the Schrödinger representation replaced by the holomorphic representation, discussed in the previous Section. That is, instead of being a function on field configurations it is taken to have the form , where is a holomorphic function on . ( is a base point, the choice of which is irrelevant at this point.) As will be shown elsewhere, this replacement step can be justified rigorously. For purposes of the present paper we merely offer the partial justification that in both cases (Schrödinger and holomorphic) we are interpreting formula (24) at least with respect to the very same trivialization of the prequantum bundle.

3 Axioms for classical and quantum field theory

3.1 Geometric data

In the previous section we have referred to regions and hypersurfaces in some fixed global spacetime. In contrast, from now on we will use a notion of spacetime in the spirit of topological quantum field theory, which is more abstract, but also more flexible. Nevertheless, a precise meaning is given to the concepts of region and hypersurface. While the setting we use is identical to that of [13] we recall it briefly here for completeness.

Concretely, our geometric setting is the following: There is a fixed positive integer , the dimension of spacetime. We are given a collection of oriented topological manifolds of dimension , possibly with boundary, that we call regions. Furthermore, there is a collection of oriented topological manifolds without boundary of dimension that we call hypersurfaces. All manifolds may only have finitely many connected components. When we want to emphasize explicitly that a given manifold is in one of those collections we also use the attribute admissible. These collections satisfy the following requirements:

  • Any connected component of a region or hypersurface is admissible.

  • Any finite disjoint union of regions or of hypersurfaces is admissible.

  • Any boundary of a region is an admissible hypersurface.

  • If is a hypersurface, then , denoting the same manifold with opposite orientation, is admissible.

It will turn out to be convenient to also introduce empty regions. An empty region is topologically simply a hypersurface, but thought of as an infinitesimally thin region. Concretely, the empty region associated with a hypersurface will be denoted by and its boundary is defined to be the disjoint union . There is one empty region for each hypersurface (forgetting its orientation). When an explicit distinction is desirable we refer to the previously defined regions as regular regions.

There is also a notion of gluing of regions. Suppose we are given a region with its boundary a disjoint union , where is a copy of . ( may be empty.) Then, we may obtain a new manifold by gluing to itself along . That is, we identify the points of with corresponding points of to obtain . The resulting manifold might be inadmissible, in which case the gluing is not allowed.

Depending on the theory one wants to model, the manifolds may carry additional structure such as for example a differentiable structure or a metric. This has to be taken into account in the gluing and will modify the procedure as well as its possibility in the first place. Our description above is merely meant as a minimal one. Moreover, there might be important information present in different ways of identifying the boundary hypersurfaces that are glued. Such a case can be incorporated into our present setting by encoding this information explicitly through suitable additional structure on the manifolds.

For brevity we shall refer to a collection of regions and hypersurfaces with the properties given above as a spacetime system. A spacetime system can be induced from a global spacetime manifold by taking suitable submanifolds. (This setting was termed a global background in [4].) On the other hand, a spacetime system may arise by considering regions as independent pieces of spacetime that are not a priori embedded into any global manifold. Indeed, depending on the context, it might be physically undesirable to assume knowledge of, or even existence of, a fixed global spacetime structure.

3.2 Classical data

Given a spacetime system, the considerations of Section 2 motivate the following axiomatic definition of a classical affine field theory. At the same time these axioms provide a natural generalization of the respective axioms presented in [13] for the case of linear field theory.

  • Associated to each hypersurface is a complex separable Hilbert space and an affine space over with the induced topology. The latter means that there is a transitive and free abelian group action which we denote by . The inner product in is denoted by . We also define and and denote by the scalar multiplication with in . Moreover we suppose there are continuous maps and such that is real linear in the second argument, is real bilinear, and both structures are compatible via

    (25)

    Finally we require

    (26)
  • Associated to each hypersurface there is a homeomorphic involution and a compatible conjugate linear involution under which the inner product is complex conjugated. We will not write these maps explicitly, but rather think of as identified with and as identified with . Then, and we also require and for all and .

  • Suppose the hypersurface decomposes into a disjoint union of hypersurfaces . Then, there is a homeomorphism and a compatible isometric isomorphism of complex Hilbert spaces . Moreover, these maps satisfy obvious associativity conditions. We will not write these maps explicitly, but rather think of them as identifications. Also, and .

  • Associated to each region is a real vector space and an affine space over . Also, there is a map .

  • Associated to each region there is a map and a compatible linear map of real vector spaces . We denote by the image of under and by the image of under . is a closed Lagrangian subspace of the real Hilbert space with respect to the symplectic form . We often omit the explicit mention of the maps and . We also require if , and

    (27)
  • Let and be regions and be their disjoint union. Then, there is a bijection and a compatible isomorphism of real vector spaces such that and . Moreover, these maps satisfy obvious associativity conditions. Hence, we can think of them as identifications and omit their explicit mention in the following. We also require .

  • Let be a region with its boundary decomposing as a disjoint union , where is a copy of . Let denote the gluing of to itself along and suppose that is a region. Note . Then, there is an injective map and a compatible injective linear map such that

    (28)

    are exact sequences. Here, for the first sequence, the arrows on the right hand side are compositions of the map with the projections of to and respectively (the latter identified with ). For the second sequence the arrows on the right hand side are compositions of the map with the projections of to and respectively (the latter identified with ). We also require . Moreover, the following diagrams commute, where the bottom arrows are the projections.

    (29)

In the spirit of Section 2.2, the spaces and should be thought of as spaces of classical solutions in or near . Correspondingly the spaces and should be thought of as their tangent spaces. Since and are affine we can naturally identify the tangent spaces at different points so that we do not need to distinguish them. Moreover, we assume the Hilbert space structure on the tangent spaces to be invariant under this identification. Thus, compared to the setting in [13], where the spaces of solutions where assumed vector spaces, each axiom contains now corresponding statements for both types of spaces, and , as well as a statement of their compatibility. The latter is always supposed to mean that given the commuting diagrams expressing a certain property for and separately, combining these diagrams with the action diagrams yields a commuting diagram. Note also that forgetting the spaces as well as the structures , and , the axioms (C1)–(C7) strictly reduce to those given in [13]. One may also remark that the axioms present quite some redundancy. For example certain properties of the spaces together with compatibility imply certain properties of the spaces and vice versa. However, the explicit form of the axioms was motivated more by conceptual simplicity and comparability with [13] rather than by minimality.

We recall the following basic fact from [13]:

Lemma 3.1.

Let be a region. Then, understood as a real Hilbert space decomposes into an orthogonal direct sum .

We will use the notation for this decomposition, where , . There is a similar decomposition for elements of given by the following Lemma.

Lemma 3.2.

Let be a region. Then, decomposes into a generalized direct sum .

Proof.

Let . We first show that there exists a decomposition with and and then proceed to show its uniqueness. Fix . Then is element of and thus decomposes as with according to Lemma 3.1. It is then easy to see that setting and yields the desired decomposition. Suppose we are given two decompositions of the required form, . Their difference is . But and so the latter amounts to a decomposition of in the sense of Lemma 3.1. Uniqueness implies then and . ∎

3.3 Core axioms of the GBF

A quantum (field) theory is encoded in the GBF by assigning “algebraic” data to the geometric data of a spacetime system, again in the spirit of topological quantum field theory. More concretely, Hilbert spaces are assigned to hypersurfaces and amplitude maps to regions. This is made precise in the following list of core axioms. This list is essentially identical to that given in [13] and included here for completeness. We refer to the cited paper for further explanations. For brevity we call a theory satisfying these axioms for a given spacetime system a general boundary quantum field theory on the spacetime system.

  • Associated to each hypersurface is a complex separable Hilbert space , called the state space of . We denote its inner product by .

  • Associated to each hypersurface is a conjugate linear isometry . This map is an involution in the sense that is the identity on .

  • Suppose the hypersurface decomposes into a disjoint union of hypersurfaces . Then, there is an isometric isomorphism of Hilbert spaces . The composition of the maps associated with two consecutive decompositions is identical to the map associated to the resulting decomposition.

  • The involution is compatible with the above decomposition. That is, .

  • Associated with each region is a linear map from a dense subspace of the state space of its boundary (which carries the induced orientation) to the complex numbers, . This is called the amplitude map.

  • Let be a hypersurface. The boundary of the associated empty region decomposes into the disjoint union , where denotes a second copy of . Then, . Moreover, restricts to a bilinear pairing such that .

  • Let and be regions and be their disjoint union. Then is also a disjoint union and . Then, for all and ,

    (30)
  • Let be a region with its boundary decomposing as a disjoint union , where is a copy of . Let denote the gluing of with itself along and suppose that is a region. Note . Then, for all and . Moreover, for any ON-basis of , we have for all ,

    (31)

    where is called the gluing anomaly factor and depends only on the geometric data.

4 Quantization

In this section we describe a quantization prescription that produces for a given classical affine field theory (satisfying the axioms of Section 3.2) on a spacetime system a general boundary quantum field theory on the same spacetime system. In particular, we rigorously prove that the produced theory satisfies the core axioms of the GBF as presented in Section 3.3.

4.1 State Spaces

As explained in [13], the inner product on the complex Hilbert space defines a Gaussian measure on the space . Here, is the algebraic dual of the topological dual of so that there is a natural inclusion . Recall furthermore that the square-integrable holomorphic functions on form a separable complex Hilbert space , whose elements are uniquely determined by their values on the subspace (Theorem 3.18 of [13]). We denote the complex vector space of functions on that arise as restrictions of elements in by . Obviously, inherits the inner product of , making it naturally isomorphic to that space as a complex Hilbert space. Note that the elements of are in particular continuous functions on .

Denote the algebra of complex valued continuous functions on by . We define the Hilbert space associated to the hypersurface as a certain subspace of as follows. Fix a base point and define the following element of , motivated by (21),

(32)

Now, define as the subspace of of elements that take the form

(33)

where . Moreover, we define the inner product on as follows,222Here and in the following elements of appearing in an integral should be thought of as representing the respective elements of .

(34)

Clearly, becomes a complex separable Hilbert space in this way, which is moreover naturally isomorphic to . Moreover, it turns out that the definition is independent of the choice of base point.

Lemma 4.1.

The above definition of is independent of the choice of base point.

Proof.

Fix . Straightforward computation yields,

(35)

Thus, suppose we have decomposed as in (33) with respect to the base point . We equate this to a decomposition with respect to the base point ,

(36)

Using (35) we obtain,

(37)

Note that the inner product is holomorphic in its second argument, so the exponential expression in (37) is holomorphic in . On the other hand is holomorphic by assumption and so is thus the composition of with a translation. Thus is holomorphic, being the product of holomorphic functions. Proposition 3.11 of [13] with , , and , yields that the extension of

(38)

is square-integrable on . So this function is in . On the other hand (37) and (38) differ only by a constant factor, so the extension of is also in . That is, . This already shows that as a subspace of is independent of the choice of base point.

It remains to show that the inner product (34) is also invariant under choice of base point. For two elements decompose as above with respect to two different base points . Then,

The second equality here is another consequence of Proposition 3.11 of [13], applied as above. This completes the proof. ∎

Heuristically, the definition of the inner product (34) is motivated by the manifestly base point independent expression

(39)

where stands for a (non-existent) translation invariant measure on , recall expression (12) in Section 2.3.

We shall refer to the elements of also as wave functions. Note that a function that is holomorphic on is anti-holomorphic on and vice versa. Also, . Thus, complex conjugation of wave functions yields a conjugate linear isomorphism . For disjoint unions of hypersurfaces and ,