Free Fermi and Bose Fields in TQFT and GBF

We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from the standard formulation of quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case.


Introduction
Free field theories and their quantization are usually the first examples treated in text books of quantum field theory. Apart from being simple toy examples they also serve to illustrate foundations of the formalism, its elementary objects and elucidate the physical interpretation of quantum field theory at its most basic level. Moreover, they serve to exhibit quantization schemes which are then refined for the treatment of more complicated theories. They also play a crucial role as the basis for perturbation theory and S-matrix theory. Finally, they are part of the small class of quantum field theories that have been defined mathematically rigorously.
It is natural to expect an equally important role for free field theories in approaches to quantum theory that aim to go beyond the context of spacetime with a fixed background metric. Here we shall be interested in the general boundary formulation (GBF), an axiomatic approach to quantum theory with precisely that aim [7]. In the GBF, spacetime is modeled through manifolds that need not carry any structure in addition to a topological one. Its basic objects are that of a specific version of topological quantum field theory (TQFT) [1]. The other fundamental ingredient of the GBF consists of rules that allow to extract predictions of measurement outcomes from these basic objects.
In quantum field theory in Minkowski spacetime a comprehensive and universal description of free field theory can be given, see e.g. [2]. By universal we mean here that it applies not just to specific examples, but to free field theories as a class. For the GBF such a universal description of free, i.e., linear bosonic field theory has been presented in [9]. More specifically, motivated by properties of Lagrangian field theory, an axiomatic definition of classical linear bosonic field theories as a class was given. Then a functorial quantization scheme was exhibited that produces a corresponding quantum theory in the sense of the GBF. In particular, it was proven rigorously that the quantum theory so produced satisfies the axioms of the GBF. It was also shown that the employed quantization is "correct" for certain examples of quantum field theories. Later, even the formal equivalence of the quantization to the Feynman path integral (where applicable) was established [6,12].
Our main interest in the present work shall be a functorial quantization scheme for linear fermionic field theory in the GBF, paralleling that of the linear bosonic theory in [9]. That is, we start with an axiomatization of classical field theory, exhibit a quantization prescription, and show that the quantum theory obtained satisfies the axioms of the GBF. We supplement this with an explicit example of a realistic quantum field theory, in this case the Dirac field in Minkowski spacetime. This serves the dual purpose of demonstrating on the one hand that standard quantum field theories fit into the framework, while on the other hand exhibiting features of the latter not accessible via the standard formulation. Concretely, we provide here the consistent implementation of states on certain timelike hypersurfaces.
We recall that the quantization scheme in [9] used the holomorphic representation and relied heavily on coherent states. While there exist notions of coherent states also in the fermionic case, these have rather different properties and are not amenable to an analogous treatment. We thus opt for a Fock space quantization. While this seems simple enough, it turns out to complicate considerably the demonstration that the quantized theory satisfies the GBF axioms. On the other hand, it makes possible a treatment of the bosonic theory alongside the fermionic one in a unified fashion and with little additional effort. In this way we also gain new insight into the bosonic theory from a perspective rather different than the one employed in [9].
Even though we work within the specific framework of the GBF, much of our treatment should be applicable to a large class of TQFTs. In particular, it is designed to be universal also in the sense of being independent of, but compatible with various types of additional structure on the manifolds modeling spacetime, such as a metric or conformal structure. We also note that for other specific types of TQFT frameworks there is a well developed understanding of the notion of free field theory. This is notably so in the TQFT approach to conformal field theory due to Segal [17,18]. It is beyond the scope of the present work, however, to explore the potential connections to this approach.
An early and intriguing result of the present work is the necessity, in the fermionic case, to abandon Hilbert spaces as (generalized) state spaces of the quantum theory in favor of Krein spaces. Apart from the justifications for this laid out in the main body of this work, certain fermionic examples with Krein spaces in [18] served as an additional motivation. The appearance of indefinite inner product spaces in quantization prescriptions in quantum field theory is not at all new, a famous example is the Gupta-Bleuler quantization of the electromagnetic field [3,5]. What is new here, however, is that such spaces not only appear at intermediate stages of the quantization process, but as the final physical state spaces. The appearance of Krein spaces might lead to concerns about the feasibility of a consistent probability interpretation. We remind the reader, however, that the state spaces of the GBF can not in general be identified with state spaces of the standard formulation of quantum theory, where this would indeed be a problem. What is more, we show in this work explicitly (in Section 11) that a consistent probability interpretation is possible.
Another intriguing aspect of linear fermionic field theories, present already at the classical level, is something that may by called an emergent notion of time in the dynamics. We refer the reader to Section 4.3 for details.
Finally, we remark on the integrability condition that was necessary to impose in [9] for a classical linear bosonic field theory to possess a quantization satisfying all the axioms of the GBF. This integrability condition arises due the presence of the gluing anomaly factor. Roughly speaking the integrability condition amounts to requiring that this anomaly factor be finite. In the present work we exhibit the same gluing anomaly factor. However, in the fermionic case we are able to carry out a renormalization procedure so that even if the gluing anomaly factor is "infinite", a renormalized version of the relevant gluing axiom still holds. Thus the quantization of any linear fermionic field theory satisfying the axioms of the classical theory satisfies the renormalized version of the GBF axioms, without exception.
In Section 2 we provide motivation for the axiomatization of the classical theory and aspects of the quantization scheme in the fermionic case. In Section 3 we give a short review of some necessary mathematical ingredients, notably Krein spaces and the associated Fock spaces. The axiomatization of classical linear field theory is discussed in Section 4. The example of the free Dirac field is treated in Section 5, both with spacelike and timelike hypersurfaces. The axioms of the GBF are reviewed and generalized to include the fermionic case in Section 6. The basics of the quantization scheme are described in Section 7. In Section 8 relations between amplitude and inner product are examined. Section 9 treats the gluing axioms and exhibits their proofs, while concentrating the most difficult part into a lemma. Due to its length, the proof of this lemma is delegated to Appendix A. The gluing anomaly and its renormalization are the subjects of Section 10. The probability interpretation in the context of Krein spaces is elaborated on in Section 11. In Section 12 a more detailed comparison to the holomorphic approach of [9] is given in the bosonic case. Finally, a brief outlook is presented in Section 13.

Motivation
In this section we shall provide some motivation for concrete aspects of the way we axiomatize classical field theory, and also for certain aspects of the quantization scheme we describe in the following.

Schematic view of quantization
In the following we recall in a very abbreviated and schematic fashion the first steps in the quantization of free fields from the point of view of geometric quantization [19].
In both the bosonic and the fermionic case the complex Hilbert space H of states for a free field is most conveniently constructed starting from another complex Hilbert space L. This space L may be thought of as the space of solutions to the equations of motion of the underlying classical system to be quantized. Most commonly (and also in the present work), H is then constructed as the Fock space over L. However, there are other, but equivalent ways to construct H from L which may more convenient depending on the context. We shall remark on this in Section 12.
Taking the point of view of geometric quantization, the space L is obtained in the bosonic case as follows: At first we take L to be the real vector space of solutions of the equations of motion with suitable regularity properties. This space becomes a symplectic vector space by equipping it with the symplectic form ω : L × L → R that is derived from the Lagrangian of the theory in the standard way. The next step is then to introduce a complex structure J : L → L that leaves ω invariant and is compatible with the dynamics. The latter usually means something along the lines that J leads to a useful notion of "positive energy" versus "negative energy" solutions and respects relevant symmetries, such as Lorentz transformations. The details of this are not important here. Combining the complex structure with the symplectic structure leads to a real inner product g : L × L → R, which is usually required to be positive definite (possibly requiring taking a quotient), g(φ , φ) := 2ω(φ , Jφ).
In the fermionic case the construction of L is slightly more complicated. The symplectic structureω : L × L → R obtained from the Lagrangian does not in itself serve to define the final complex inner product. Rather, it has to be combined with a complex structureJ : L → L which is usually a natural complex structure on the spinor bundle that would serve to define the classical system. This yields the real inner product g : L × L → R, Another complex structure J : L → L, related to the dynamics as in the bosonic case, then yields the relevant symplectic structure The real inner product and the symplectic structure are then combined as in the bosonic case via equation (2.1).

Quantization in the GBF
For the convenience of the reader we have described quantization so far as if we were interested in a single Hilbert space describing our quantum system. This is not the case. Rather, in the GBF we need one such Hilbert space for each (admissible) hypersurface in spacetime. It turns out that the schematic procedure described above is already naturally adapted to this [9]. Given a hypersurface Σ we think of L Σ as the space of solutions in a neighborhood of Σ (or more precisely as the space of germs of solutions). Consider the bosonic case first. The symplectic structure ω derived from the Lagrangian is actually not necessarily a structure on a global space of solutions. Rather it arises as the integral of a (d − 1)-form (where d is the spacetime dimension) over a hypersurface. Moreover, the dependence of the (d − 1)-form on the field is completely local so that the symplectic structure depends only on the field and its derivatives on the hypersurface. Taking the hypersurface to be Σ, it is thus naturally a structure on L Σ and we denote it by ω Σ . The complex structure by contrast is generally somewhat non-local, but can nevertheless be specified as a complex structure on L Σ . We denote it thus by J Σ . We write the resulting complex inner product as {·, ·} Σ . Looking more closely at the above description of the space L Σ and associated structures ω Σ , g Σ and {·, ·} Σ it turns out that the latter not only depend on the choice of hypersurface Σ itself, but also on its orientation. We shall write Σ to denote the same hypersurface Σ, but with opposite orientation. Now, recall that in the bosonic case the symplectic structure ω Σ arises as the integral of a (d − 1)-form over the hypersurface Σ. But reversing the orientation changes the sign of this integral. The complex structure J Σ , encoding in some sense the distinction between "positive energy" and "negative energy" solutions, also changes sign under change of orientation R. Oeckl of Σ. As a consequence, the real inner product is invariant under orientation change while the complex inner product is complex conjugated. In summary, Precisely these ingredients (the spaces L Σ with the given structures) have been taken as the "kinematical" part of the data of an axiomatic formalization of classical linear bosonic field theory in [9]. Moreover, they have been shown there to integrate nicely into a functorial quantization scheme which moreover reproduces and extends basic examples from quantum field theory. We consider that this confirms the "correctness" of the above schematic description.
In the fermionic case it is the symplectic formω that arises as the integral of a field-local (d − 1)-form on a hypersurface. The complex structureJ is also completely local in its field dependence. So, the real inner product g is the integral of a field-local (d − 1)-form on the hypersurface. For a hypersurface Σ we denote it by g Σ . As in the bosonic case the complex structure J Σ introduces a certain non-locality which is thus inherited by the symplectic structure ω Σ . {·, ·} Σ denotes the resulting complex inner product. The symplectic formω Σ depends on the orientation of Σ as in the bosonic case.J Σ on the other hand does not depend on the orientation while g Σ does, coming from a field-local (d − 1)-form. J Σ does depend on the orientation as in the bosonic case and for the same reason. So ω Σ does not depend on the orientation. In summary, This is in precise analogy to the bosonic case (2.3) if we think of the difference between bosons and fermions as due to the interchange of symmetric with anti-symmetric structures.
A closer look at the relations (2.4) reveals a profound and surprising implication. In contrast to the bosonic case, it cannot be consistently required that all the spaces L Σ have positivedefinite inner product. Indeed, suppose g Σ was positive-definite for some hypersurface Σ. Then g Σ would necessarily be negative-definite. Moreover, if L Σ carries an indefinite inner product then so does the Fock space H Σ based on it. We are thus forced to give up the insistence that state spaces should be Hilbert spaces.
While this narrative of the fermionic case appears compelling in view of its tight analogy to the more established bosonic case, the abandonment of Hilbert spaces calls for the presentation of strong evidence in its favor. The purpose of the present work is in part to provide just that, by embedding it into a functorial quantization scheme that successfully reproduces known quantum field theories. We shall also alleviate fears (in Section 11) that this may destroy a consistent probability interpretation. Moreover, we will explain why the usual formulation of quantum theory gets along nicely without "seeing" these indefinite inner product spaces in fermionic quantum field theories (see Section 8.3).

Dynamical aspects
We have so far exclusively concentrated on the "kinematical" aspects of the classical theory and its quantization. We proceed to discuss aspects of the dynamics. For linear bosonic field theory this was formalized in [9] as follows. Given a spacetime region M , the "dynamics" in M is given by a real vector space L M of solutions of the equations of motion. This is a subspace of the space L ∂M of (germs of) solutions on the boundary ∂M of M . Indeed, the key is that it is a Lagrangian subspace with respect to the symplectic form ω ∂M on L ∂M . This was motivated extensively in [9] from Lagrangian field theory and confirmed in the sense that it fits precisely into the quantization scheme given there. Moreover, it also fits with the examples of known quantum field theories considered there.
In view of the analogy between fermions and bosons we should thus expect the dynamics in a "classical" fermionic field theory to be encoded in a subspace L M ⊆ L ∂M that is "Lagrangian" with respect to the symmetric bilinear form g ∂M . That is the subspace L M should be a neutral subspace which is maximal in a certain sense. It turns out that the right notion is that of a hypermaximal neutral subspace.
The quantum dynamics for a region M is encoded in the amplitude map ρ M : H ∂M → C. This was given for the bosonic theory in [9] in terms of a certain integral of the holomorphic wave function on the boundary over the space of solutions in the interior, L M . It was later shown [6,12] that this is precisely equivalent to the usual Feynman path integral and thus "correct" from a quantum field theory point of view. This same quantization rule, translated to the Fock space setting is the basis for the quantization of the dynamics in the bosonic case also here. (We shall have much more to say about this in Section 12.) The quantization rule in the fermionic case is deduced by analogy. Again, our overall results may be seen to "confirm" it.

Krein space
Let V be an indefinite inner product space with an orthogonal direct sum decomposition V = V + ⊕ V − such that V + is positive definite and V − is negative definite. Let V − denote the space V − with the sign of its inner product inverted. Then V − is positive definite and V is canonically isomorphic as a vector space to the positive inner product space V + ⊕ V − . If V + ⊕ V − is a Hilbert space then we say that V is a Krein space and equip it with the Hilbert space topology.
In general there are many ways to decompose a Krein space into an orthogonal direct sum V = V + ⊕ V − with the described properties. We shall be interested, however, exclusively in Krein spaces that come with a canonical such decomposition. In order not to complicate notation, Krein space in the following always refers to this strict version of Krein space.
We call V + the positive part and V − the negative part of V . We also view the decomposition as a Z 2 -grading with the notation Let W be a subset of the Krein space V . Then the subspace is called the orthogonal companion of W . A subspace W of a Krein space V is called neutral iff the inner product of each element of W with itself vanishes. This is equivalent to the property that the real part of the inner product vanishes on W . W is called maximal neutral iff moreover, W is not a proper subspace of any neutral subspace of V . Since the closure of a neutral subspace is neutral, a maximal neutral subspace is necessarily closed. A subspace W is called hypermaximal neutral iff W = W ⊥ . This implies in particular that W is maximal neutral. In case of a complex Krein space we use in addition the adjective real if the subspace in question is a real subspace only and its property refers to the real part of the inner product only.
A linear map a : V 1 → V 2 is called an isometry iff a preserves the inner product. It is called an adapted isometry if in addition a respects the decompositions of V 1 and V 2 by mapping V + 1 to V + 2 and V − 1 to V − 2 . A linear map a : V 1 → V 2 is called a antiisometry iff a reverses the sign of the inner product. It is called an adapted anti-isometry if in addition a respects the decompositions of V 1 and V 2 by mapping V + 1 to V − 2 and V − 1 to V + 2 . Note that an adapted isometry and an adapted anti-isometry are both isometries for the canonical Hilbert space structures of V 1 and V 2 . In particular, they are continuous. In case of a complex Krein space we use in addition the adjective real if the map in question is only real linear and is an isometry or anti-isometry only with respect to the real part of the inner product.
Lemma 3.1. Let V = V + ⊕ V − be a real Krein space. Then there is a natural one-to-one correspondence between involutive adapted anti-isometries u : V → V and hypermaximal neutral subspaces W ⊆ V . Moreover, this correspondence is such that W is the fixed point set of u.
Proof . Let u be an involutive adapted anti-isometry u : V → V . Define W to be the fixed point set of u. Then W is obviously a closed subspace of V . Moreover, we must have That is, W is neutral. Since we are in the real case this implies Combining this with the above result yields W = W ⊥ , i.e., W is hypermaximal neutral.
Conversely, suppose that W is a hypermaximal neutral subspace of V . Let M ⊆ V + be the subset of vectors v ∈ V + such that there exists w ∈ V − with v + w ∈ W . It is easy to see that M is a subspace of V + and that it is closed (due to W being closed). In particular, M has an orthogonal complement N in the Hilbert space V + . Let n ∈ N and v ∈ W . Decompose v = v + + v − with v + ∈ V + and v − ∈ V − . Then v + ∈ M , so n is orthogonal both to v + and to v − . In particular, n is orthogonal to v. We see that any element of N is orthogonal to any element of W . Since W is hypermaximal neutral this implies N ⊆ W . On the other hand On the other hand given v ∈ V + let w, w ∈ V − be such that v + w and v + w are elements of W . Thus we have both w − w ∈ W and w − w ∈ V − . But W ∩ V − = {0}, so w = w . Suppose now there was an involutive adapted anti-isometry u : V → V with fixed point set W . Then by adaptedness we would need to have u(v) ∈ V − while also having v + u(v) ∈ W by involutiveness. Thus, we would necessarily have u(v) = w. We take this here as the definition of a map u : V + → V − and define analogously u : V − → V + . It is easy to see that the so defined map u : V → V is indeed adapted and involutive. To see that it is an anti-isomorphism take v, w ∈ V with canonical decompositions v = v + + v − and w = w + + w − . Then This completes the proof.
Remark 3.2. The lemma shows in particular, that not every real Krein space admits a hypermaximal neutral subspace. Indeed, the existence of such a subspace is equivalent to the existence of an involutive adapted anti-isometry. The latter is equivalent to an anti-isometry between the positive and negative parts of the Krein space. Taking the Hilbert space inner product this is just an ordinary isometry between Hilbert spaces. Such an isometry thus exists precisely when both parts have the same cardinality.
Let V be a complex Krein space. Then V is also a real Krein space by forgetting the imaginary part of the inner product. Suppose V has a real hypermaximal neutral subspace W . Denote the complex structure of V by J : V → V . We say that J and W are compatible iff the induced involutive adapted real anti-isometry u : V → V is conjugate linear, i.e., u • J = −J • u. Lemma 3.3. Let V be a complex Krein space with a compatible real hypermaximal neutral subspace W . Then V as a real vector space decomposes as a direct sum V = W ⊕JW . Moreover, JW is the eigenspace of the induced adapted real anti-isometry u : V → V with eigenvalue −1. Also, JW is a real hypermaximal neutral subspace of V . Furthermore, W and JW are symplectic complements with respect to the symplectic form given by the imaginary part of the inner product of V .
We leave the straightforward proof to the reader. Lemma 3.4. Let V be a complex Krein space and W be a closed Lagrangian subspace with respect to the imaginary part of the inner product. Then JW is also a closed Lagrangian subspace and V as a real Krein space has the orthogonal direct sum decomposition V = W ⊕ JW . Moreover, the real linear map u : V → V defined as the identity on W and minus the identity on JW is a conjugate linear involutive real isometry of V . Also, if W is adapted then so are JW and u.
This lemma is essentially a straightforward generalization of Lemma 4.1 in [9] to the Krein space case. We leave the proof to the reader. In the following it will be useful to view the structures appearing in Lemmas 3.1 and 3.3 on the one hand and in Lemma 3.4 on the other hand from a common perspective. To this end let κ = −1 in the former and κ = 1 in the latter case. The map u is then a conjugate linear involution with the property Moreover the map u has precisely the structure of a complex conjugation or real structure on the Krein space V . Let V be a Krein space with decomposition V = V + ⊕ V − . We say that a subset B ⊆ V is an orthonormal basis (short: ON-basis) of V iff B is the union of subsets B + ⊆ V + and B − ⊆ V − such that B + is an ON-basis of V + as a Hilbert space and B − is an ON-basis of V − .

Fock space
We recall in the following some elementary facts about Fock space and introduce our notation. It will be convenient to treat both the bosonic and the fermionic case at once. To this end we introduce the constant κ, defined as κ := 1 in the bosonic case, κ := −1 in the fermionic case.
The fermionic or bosonic Fock space F is a Krein space which arises as the completion of an N 0 -graded Krein space, To denote the degree we write We refer to this grading as the Fock grading. Another important grading is the fermionic grading or f-grading. This is a Z 2 -grading defined as follows. All elements of a bosonic Fock space are assigned degree zero. Elements of a fermionic Fock space have degree zero if their Fock degree is even and degree one if their Fock degree is odd. The space F 0 is the canonical one-dimensional Hilbert space over C. We write it also as F 0 = 1C with 1, 1 = 1. If L is the Krein space that generates F, then F n is defined to be the space of continuous n-linear maps from the n-fold direct product of L with itself to C that are either symmetric (bosonic case) or anti-symmetric (fermionic case). Thus F n := ψ : L × · · · × L → C, n-linear continuous : Here S n denotes the symmetric group acting on L × · · · × L, while |σ| equals 0 or 1 depending on whether σ ∈ S n is even or odd.
Let ξ 1 , . . . , ξ n ∈ L. We define a corresponding element ψ[ξ 1 , . . . , ξ n ] ∈ F n as follows Here, {·, ·} denotes the inner product in L. Note that this expression is conjugate linear in the variables ξ 1 , . . . , ξ n . We define the inner product between these elements in F n as This makes F n into a Krein space with the subspace generated by elements of the form ψ[ξ 1 , . . . , ξ n ] dense. We shall refer to these special states as generating states. Let L = L + ⊕ L − be the canonical decomposition of L as a Krein space. We define subspaces of F n , Then F + n is a complete positive-definite inner product space (a Hilbert space) while F − n is a complete negative-definite inner product space. Moreover, F n is the orthogonal direct sum F n = F + n ⊕ F − n . This yields the Krein space structure of F n . Note also that The induced canonical decomposition of F as a Krein space we denote by F + ⊕ F − . Lemma 3.5. Let L be a Krein space. Suppose that u : L → L is a conjugate linear involutive adapted real isometry in the bosonic case or anti-isometry in the fermionic case. Define U n : F n → F n by (U n ψ)(ξ 1 , . . . , ξ n ) := ψ(u(ξ n ), . . . , u(ξ 1 )).

(3.3)
Then U n is a conjugate linear involutive adapted real anti-isometry in the fermionic case if n is odd and an isometry otherwise. Combining the maps U n to a map U : F → F yields a conjugate linear involutive adapted real f -graded isometry in the sense, U ψ , U ψ = κ |ψ| ψ , ψ .
We leave the straightforward proof to the reader. It also straightforward to check the action of U on generating states, U ψ[ξ 1 , . . . , ξ n ] = κ n ψ[u(ξ n ), . . . , u(ξ 1 )] ∀ ξ 1 , . . . , ξ n ∈ L. (3.4) Note also that U has the structure of a complex conjugation or real structure for the Fock space F.

Classical data
In this section we provide an axiomatic description of a linear classical field theory, either fermionic or bosonic. Here the attribute "classical" has to be taken with a grain of salt as we already include certain data that belongs more properly into the quantum realm, notably complex structures.

Geometric data
We recall briefly the formalization of the notion of spacetime in terms of a spacetime system in the GBF. The presentation here is a refined version of previous presentations such as the one in [9]. There is a fixed positive integer d ∈ N, the dimension of spacetime. We are given a collection M c 0 of connected oriented topological manifolds of dimension d, possibly with boundary, R. Oeckl that we call connected regular regions. Furthermore, there is a collection M c 1 of connected oriented topological manifolds without boundary of dimension d − 1 that we call hypersurfaces. The manifolds are either abstract manifolds or they are all concrete submanifolds of a given fixed spacetime manifold. In the former case we call the spacetime system local, in the latter we call it global.
There is an operation of union both for regular regions and for hypersurfaces. This leads to the collection M 0 , of all formal finite unions of elements of M c 0 , and to the collection M 1 , of all formal finite unions of elements of M c 1 . In case the spacetime system is global, only unions with members who are disjoint are allowed in M 1 and only unions with members whose interiors are disjoint are allowed in M 0 . For simplicity, and to capture the intuitive meaning of these unions we use the term disjoint unions uniformly. Note that in the global case the elements of M 1 are actual submanifolds of the spacetime manifold. This is not the case for all elements of M 0 as overlaps on boundaries may occur.
The collection M 1 has to be closed under orientation reversal. That is, given Σ ∈ M 1 , the hypersurface Σ which is identical to Σ, but with reversed orientation is also in M 1 . Also, any boundary of a regular region in M 0 is a hypersurface in M 1 . That is, taking the boundary defines a map ∂ : M 0 → M 1 . When we want to emphasize explicitly that a given manifold is in one of those collections we also use the attribute admissible.
It is convenient to also introduce the concept of slice regions 1 . A slice region is topologically simply a hypersurface, but thought of as an infinitesimally thin region. Concretely, the slice region associated to a hypersurface Σ will be denoted byΣ and its boundary is defined to be the disjoint union ∂Σ = Σ ∪ Σ. There is one slice region for each hypersurface (forgetting its orientation). The reason for the terminology is that slice regions can be treated in certain respects in the same way as regular regions. Thus, we refer to regular regions and slice regions collectively as regions.
There is also a notion of gluing of regions. Suppose we are given a region M with its boundary a disjoint union ∂M = Σ 1 ∪ Σ ∪ Σ , where Σ is a copy of Σ (Σ 1 may be empty). Then we may obtain a new region M 1 by gluing M to itself along Σ, Σ . That is, we identify the points of Σ with corresponding points of Σ to obtain M 1 . The resulting region M 1 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 admissibility 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.

Axioms of the classical theory
Given a spacetime system, we axiomatize a linear classical theory on the spacetime system as follows. As previously, we set κ = −1 in the fermionic case and κ = 1 in the bosonic case. In the bosonic case these axioms coincide with those previously proposed in [9] up to the small difference that we allow for Krein spaces here rather than only for Hilbert spaces.
(C2) Associated to each hypersurface Σ there is a conjugate linear involution L Σ → L Σ , written as an identification, under which the inner product is transformed as follows (C3) Suppose the hypersurface Σ decomposes into a disjoint union of hypersurfaces Σ = Σ 1 ∪ · · · ∪ Σ n . Then there is an isometric isomorphism of complex Krein spaces L Σ 1 ⊕ · · · ⊕ L Σn → L Σ . We will not write this map explicitly, but rather think of it as an identification.
(C7) Let M be a region with its boundary decomposing as a disjoint union ∂M = Σ 1 ∪ Σ ∪ Σ , where Σ is a copy of Σ. Let M 1 denote the gluing of M to itself along Σ, Σ and suppose that M 1 is a region. Note ∂M 1 = Σ 1 . Then there is an injective linear map r M ;Σ,Σ : is an exact sequence. Here the arrows on the right hand side are compositions of the map r M with the orthogonal projections of L ∂M to L Σ and L Σ respectively (the latter identified with L Σ ). Moreover, the following diagram commutes, where the bottom arrow is the orthogonal projection, We add the following observations: g Σ is a real symmetric non-degenerate bilinear form making L Σ into a real Krein space. ω Σ is a real anti-symmetric non-degenerate bilinear form making L Σ into a symplectic vector space. Also Moreover, for all φ, φ ∈ L Σ , The structures coming from Lemma 3.3 in the fermionic case or Lemma 3.4 in the bosonic case for a region M will play an important role in the following. In the fermionic case the real subspace LM ⊆ L ∂M is a compatible real hypermaximal neutral subspace and we have the real direct sum decomposition L ∂M = LM ⊕ JLM . Moreover, LM and JLM are both hypermaximal neutral subspaces with respect to g ∂M and they are symplectic complements with respect to ω ∂M . In the bosonic case the real subspace LM ⊆ L ∂M is a closed adapted Lagrangian subspace and we have the real orthogonal direct sum decomposition L ∂M = LM ⊕ JLM . Moreover, LM and JLM are both closed adapted Lagrangian subspaces with respect to ω ∂M . In both cases, we denote the associated complex conjugation by u M : L ∂M → L ∂M .

A notion of evolution
For a region M the physical interpretation of the space L M is that of the space of solutions of the equations of motions inside M . It is as such naturally a subspace of the space L ∂M of germs of solutions on the boundary of M . It is in this sense that we may think of the spaces L M as encoding the "dynamics" of a theory and of the spaces L ∂M the "kinematics". In order to connect this to a more traditional view of "dynamics" versus "kinematics" we make use of the map u M . Lemma 4.1. Suppose that L 2 ⊕ L 1 = L ∂M is a decomposition as an orthogonal direct sum of complex Krein spaces such that u M (L 1 ) = L 2 . Then we have the identity The proof is immediate from the properties of u M exhibited in Section 3.1. This statement admits the following interpretation: u M (restricted to L 1 ) describes the "evolution" of the "initial data" encoded in L 1 to the "final data" encoded in L 2 . Of course, the role of L 1 and L 2 may be interchanged at will, since u M is involutive. Moreover, there may be many such decompositions of L ∂M , leading to different such notions of "evolution".
Such an algebraic notion of "evolution" of classical data may be seen to have a geometric underpinning when the underlying decomposition of the space L ∂M arises from a corresponding decomposition of the boundary hypersurface ∂M .
Proof . Preservation of the inner product follows from the combination of property (3.1) of u M with property (4.1) associated to hypersurface orientation change according to axiom (C2). Complex linearity comes from the fact that both u M and the map implementing orientation change are complex conjugate linear.
In this context the "initial data" L Σ 1 and the "final data" L Σ 2 are really localized on separate hypersurfaces, giving stronger justification to the attributes "initial" and "final".
To make the link with a traditional notion of dynamics even more pertinent consider the following setting. Suppose there is a fixed spacetime manifold with a metric. Moreover, suppose the metric is Lorentzian and the spacetime is globally hyperbolic. Now consider only hypersurfaces that are spacelike (and possibly have some additional nice properties). Consider regions that are submanifolds of spacetime, bounded by pairs of such hypersurfaces. Adding finite disjoint unions yields a spacetime system. Suppose that a classical field theory is given by hyperbolic partial differential equations of motion so that all the admissible connected hypersurfaces are Cauchy. Supposing the theory is Lagrangian, proceeding roughly along the lines sketched in Sections 2.1 and 2.2, we should obtain a model satisfying the axioms of Section 4.2. Crucially, we are assuming here that in addition to the purely classical information, sensible complex structures exist on the hypersurfaces. Suppose moreover that the complex structures for the different hypersurfaces are chosen in a compatible way (e.g., if they come from a global complex structure). We would then have the assumption u M (L Σ 1 ) = L Σ 2 of Proposition 4.2 satisfied for any connected admissible region M with initial boundary component Σ 1 and final boundary component Σ 2 . (Here we take all admissible connected hypersurfaces to be oriented as past boundaries of an admissible region lying in the future.) Thenũ M literally is the timeevolution map from the initial data L Σ 1 on Σ 1 to the final data L Σ 2 on Σ 2 . For future reference we shall refer to this setting as the standard globally hyperbolic setting.
So far we have treated here the bosonic and fermionic case on the same footing. However, the axioms of Section 4.2 imply a remarkable asymmetry between the two cases with respect to the (algebraic) notion of "evolution". In contrast to the bosonic case the fermionic case exhibits a preferred decomposition of L ∂M of the required type, inducing thus a preferred (algebraic) notion of "evolution". This is the decomposition into positive and negative parts L ∂M = L + ∂M ⊕ L − ∂M , which satisfies naturally u M (L + ∂M ) = L − ∂M . Moreover, for standard examples of free fermionic field theories in globally hyperbolic spacetimes formalized along the lines sketched in the previous paragraph, this decomposition is precisely the one induced by the geometric decomposition of ∂M into initial and final hypersurface. More formally we have L + In Section 5 we shall exhibit this concretely for the example of the Dirac field in Minkowski spacetime. On the other hand, the preferred algebraic notion of "evolution" is present even if the spacetime system does not come equipped with a notion of time. Moreover, it behaves very much like time-evolution in that it is directed and compatible with composition of regions etc. This suggests to speak of an emergent notion of time inherent in fermionic theories satisfying the axioms of Section 4.2.

The free Dirac f ield in Minkowski spacetime
In this section we review the free Dirac field in Minkowski spacetime and show explicitly how it fits into the axiomatic system of Section 4.2.

The real inner product
We consider the free Dirac field in Minkowski spacetime. We use the γ-matrices and other notation of high energy physics such as in [15]. The field X representing the Dirac spinor is a section of a trivial vector bundle over Minkowski spacetime with fiber isomorphic to C 4 . The Lagrangian is Given a hypersurface Σ the space L Σ is essentially the space of spinors restricted to Σ. The Lagrangian yields on it the symplectic structure 3 Here d 3 x denotes the 3-form induced from the metric in the hypersurface and n denotes the normal vector to the hypersurface. This is to be understood in the sense where the right-hand side denotes contraction of the partial derivative ∂ µ as a vector field with the volume 4-form d 4 x of Minkowski spacetime. To fix signs on the left-hand side we introduce the additional convention that the integral over d 3 x is locally always positive so that the orientation information of the hypersurface resides exclusively in the sign of n.
Using the complex structure of the fiber,J is simply the natural multiplication with i of the spinor X understood as a complex vector field. This yields the symmetric real bilinear form on L Σ , 2 We have fixed here an overall choice of sign. The other choice would interchange L + ∂M and L − ∂M . 3 We use the sign conventions for the symplectic structure as in [6].
In order to analyze its properties it will be convenient to rewrite it as follows is a complex 4 × 4-matrix valued function on Σ. We note that P (x) is self-adjoint, since γ 0 γ µ is self-adjoint and n µ (x) is real. In particular, we may decompose P (x) as P ( is non-degenerate the decomposition is unique with P + (x) strictly positive and P − (x) strictly negative. As we will see this is the case if n is not light-like. For simplicity we shall only consider the cases that Σ is either completely spacelike or completely timelike. We start with the spacelike case. If Σ is in particular an equal-time hypersurface we have depending on the orientation of Σ. Since γ 0 γ 0 = 1 this implies P (x) = 1 in the first case and P (x) = −1 in the second 4 . With the positivity of the integral this means that the real inner product g Σ given by (5.1) is either positive-definite or negative-definite. Suppose now that Σ is an arbitrary spacelike hypersurface in Minkowski spacetime. Picking a point x ∈ Σ we can always use a Lorentz transformation locally to have n(x) aligned with the time axis (i.e. of the form (5.2)). Since the rank of P (x) must change continuously under Lorentz transformations, but is a discrete quantity, it remains constant. So P (x) is strictly positive or strictly negative. Moreover, P (x) must have the same sign for all x ∈ Σ. Thus g Σ is positive-definite or negative definite depending on the orientation of Σ. For spacelike hypersurfaces there is a global notion of orientation since we may talk about the normal vector n(x) being future or past pointing. In particular, if we orient them all in the same way the inner products g Σ will be all positive-definite or all negative-definite. Since the inner product associated to spacelike hypersurfaces with a fixed orientation is the starting point for conventional quantization prescriptions, it is natural there to consider only positive-definite inner products.
We proceed to consider the case that the hypersurface Σ is timelike. Proceeding in a manner similar to that of the spacelike case it will be sufficient for our purposes to consider just one special normal vector. All others are related to this by a Lorentz transformation, due to their spacelike nature. We choose n(x) = (0, 0, 0, 1).
With either the standard or the chiral representation of the γ-matrices [15] this yields We read off immediately that P (x) has one positive eigenvalue, 1, and one negative eigenvalue, −1, each with an associated two-dimensional eigenspace. By the same argument as in the spacelike case we may conclude that a decomposition of P (x) into a positive and a negative part P (x) = P + (x) + P − (x), each of rank 2, holds everywhere on the hypersurface. Moreover, it holds for any timelike hypersurface. (Note the symmetry of the statement under orientation reversal.) The decomposition of P induces a corresponding decomposition of the space L Σ of sections of the spinor bundle on Σ induced by the eigenspace decomposition of each fiber. That is, we have the orthogonal decomposition

Plane waves
As already mentioned, the complex structure is in general not local in the field, making its discussion in the abstract setting we have used so far difficult. It is convenient, and for our motivational purposes here sufficient, to consider a global space of solutions of the Dirac equation in Minkowski spacetime in terms of plane wave spinors. Moreover, we shall consider only two special choices of hypersurface, one spacelike and one timelike. In the treatment of spinors we follow mostly text book style conventions, in particular those of [15].
Recall the Dirac equation We expand its solutions in Minkowski spacetime in terms of plane waves in the standard way as Here u s and v s with s ∈ {1, 2} are certain spinors in momentum space. That is, they are complex 4-dimensional vector valued functions on momentum space R 3 , forming a convenient basis of the solutions of the equations They satisfy moreover the following properties Let Σ be an equal-time hypersurface located at time t. We shall denote the corresponding quantities with a subscript t rather than a subscript Σ. For the space L t we simply take the global solution space parametrized by (5.3). The inner product g t is given by (5.1) with P (x) = 1. This yields The example of a timelike hypersurface we shall consider is the hypersurface Σ located at a constant value of the coordinate x 3 , which we shall denote by z for simplicity. It is then convenient to parametrize the plane wave solutions in a sightly different (but strictly equivalent) way compared to (5.3). We base the parametrization on Fourier modes on the hypersurface Σ which we shall denote (E,k) where the "energy" E can be positive or negative whilek = (k 1 , k 2 ). We restrict the integral over modes to |E| ≥ m, which means that we only consider the propagating wave modes contained in (5.3) and exclude evanescent waves, Herex is a shorthand forx = (x 1 , x 2 ) and k 3 = E 2 −k 2 − m 2 is chosen positive. The momentum space spinorsũ,ṽ are defined in terms of the spinors u, v as follows They satisfy the equations analogous to (5.4). They have the following important properties analogous to (5.5), (5.6): We denote quantities related to the hypersurface Σ of constant z with a subscript z. The inner product g z is given by (5.1) with P as in (5.1). Using the identities (5.8)-(5.11), we obtain We see that the space L z of spinors on the hypersurface splits as a direct sum L z = L + z ⊕ L − z into a part L + z with E > 0 where g z is positive-definite and a part L − z where g z is negative-definite. It is also instructive to express this inner product in terms of the original parametrization (5.3) of the global solution space. The relation between the two parametrizations is the following We obtain In this parametrization the subspaces L + z and L − z are distinguished by the direction of the momentum component k 3 that is perpendicular to the hypersurface.

The complex structure
We proceed to consider the complex structure J Σ which, together with the real inner product, gives rise to the complex inner product on L Σ . As already mentioned this encodes the distinction between "positive energy" and "negative energy" solutions and is not a field-local object.
In the case where the hypersurface Σ is a hypersurface of constant time t the standard complex structure J t is given by Looking at the parametrization (5.3) this means that solutions with a temporal dependence of the form e −iEt are multiplied by i, while solutions with a temporal dependence of the form e iEt are multiplied by −i. (Here E > 0 by convention.) The symplectic form according to (2.2) results to be Combining real inner product and symplectic form according to (2.1) yields This is indeed the usual complex inner product for the 1-particle space of the Dirac field. We proceed to the timelike case with Σ being the hypersurface of constant coordinate value z = x 3 . The analogue of "positive energy" versus "negative energy" solutions is now given by the distinction between solutions with a dependence in z-direction of the form e ik 3 versus e −ik 3 in the parametrization (5.7). (Here k 3 > 0 by convention.) That is, the complex structure J z is given by With (2.2) we obtain the symplectic form Combining real inner product and symplectic form according to (2.1) yields the complex inner product Remarkably, expressing the symplectic form ω z in terms of the standard field parametrization (5.3) yields identical to (5.12) for ω t . This suggests that the symplectic structure here is the same not only for all reasonable spacelike hypersurfaces (which is known), but also for a class of timelike hypersurfaces.

Dynamics and axiomatic form
We proceed to integrate the structures derived so far in terms of the axiomatic system of Section 4.2, including the already implicit dynamics. We start with the case of constant-time hypersurfaces. Thus, admissible hypersurfaces are constant-time hypersurfaces and their finite disjoint unions. Admissible regions are regions bounded by pairs of such hypersurfaces and their finite disjoint unions 5 . For a time t ∈ R the space L t associated to the hypersurface Σ t of constant time t is a copy of the Hilbert space L. This in turn is the L 2 -space of equivalence classes of square-integrable functions X s a,b (k) with the inner product (5.13). Here, we take Σ t to have the standard orientation by which we mean that it is oriented as the past boundary of an admissible region. The admissible regions all have the same orientation inherited from some global standard orientation of Minkowski space.
This yields the data of axiom (C1) for connected admissible hypersurfaces with standard orientation. For a connected admissible hypersurface Σ t with the opposite orientation we define the associated space L t simply as identical to L t as a real vector space, but with opposite complex structure and inner product modified according to relation (4.1). Note in particular that the space L t is thus negative-definite. This yields the data of axioms (C1) and (C2) for all connected admissible hypersurfaces. Next, we define the space associated to a finite disjoint union of connected hypersurfaces to be the direct sum of the spaces associated to the individual hypersurfaces. This yields the complete data for axioms (C1), (C2) and (C3).
Consider now a connected admissible region M . M has the structure of a time-interval [t 1 , t 2 ] times all of space R 3 . We index associated structures with [t 1 , t 2 ]. The space L [t 1 ,t 2 ] of solutions associated to this region is a copy of the global space of solutions L, understood merely as a real vector space without additional structure. For admissible regions that are disjoint unions of connected regions the associated real vector space is taken to be the direct sum of the vector spaces associated to the components. This yields the data for axiom (C4).
For axiom (C5) it will be sufficient to consider a connected admissible region. Say the region is determined by the time interval [t 1 , t 2 ]. We define the map recalling that all the involved spaces are copies of L. It is also easy to verify that the image L of this map is a real hypermaximal subspace of L t 1 ⊕ L t 2 since the real inner product on the latter is given by So if φ 1 = φ 2 and η 1 = η 2 the result vanishes while for φ 1 = φ 2 we can always find some η 1 = η 2 giving a non-vanishing result. The according to Lemma 3.1 induced involutive antiisometry u [t 1 ,t 2 ] takes the form In particular, it anti-commutes with the complex structure since the latter takes opposite signs on the hypersurfaces at t 1 and at t 2 , due to opposite orientation. Thus, the compatibility condition with the complex structure is satisfied and so is axiom (C5) and also (C6).
With respect to the discussion in Section 4.3 note that the mapũ [t 1 ,t 2 ] : L t 1 → L t 2 as defined in Proposition 4.2 and encoding time-evolution is here simply the identity between different copies of the same Hilbert space L. In particular, the preferred algebraic notion of evolution discussed in Section 4.3 coincides with the time-evolution. Moreover, given the fixed standard orientation all the connected admissible hypersurfaces L t are Hilbert spaces.
For axiom (C7) it is sufficient to consider the case of a region M that is the disjoint union of regions [t 1 , t 2 ] and [t 2 , t 3 ] for t 1 < t 2 < t 3 . Gluing amounts to producing the region [t 1 , t 3 ]. It is then straightforward to check the details of the axiom, which we leave to the reader. This completes the construction of our model of the axioms encoding the Dirac field theory in Minkowski space with constant-time hypersurfaces.
We proceed to consider the setting based on hypersurfaces of constant z. The construction is very much analogous to the one just performed, with the role of the time coordinate t now played by the space coordinate z. We will therefore limit ourselves to highlighting the differences. We equip the global solution space L to this end with the inner product (5.14). More precisely, we take L + to be the L 2 -space of equivalence classes of square-integrable functionsX s a,b (E,k) with E > 0 and the inner product (5.14). We take L − to be the L 2 -space of equivalence classes of square-integrable functionsX s a,b (E,k) with E < 0 and the inner product the negative of (5.14). The two Hilbert spaces L + and L − are then combined to the Krein space L = L + ⊕ L − with positive part L + and negative part L − . Apart from this difference the implementation of the axioms (C1)-(C7) is analogous to the setting with constant-time hypersurfaces.
Concerning the discussion in Section 4.3 the induced evolution mapũ [z 1 ,z 2 ] : L z 1 → L z 2 from the hypersurface Σ z 1 to the hypersurface Σ z 2 is again the identity, this time between copies of the same Krein space (which is not a Hilbert space). However, this notion of spatial evolution is distinct from the preferred algebraic notion of evolution since the decomposition of the boundary solution space L ∂[z 1 ,z 2 ] into positive part L + ∂[z 1 ,z 2 ] and negative part L − ∂[z 1 ,z 2 ] now does not coincide with its decomposition into the hypersurface solution spaces L z 1 and L z 2 . Indeed, the preferred algebraic notion of evolution continues to agree with time-evolution.
6 Axioms of the quantum theory 6.1 Core axioms of the GBF The core axioms of the GBF are generalized here to include fermionic theories in addition to bosonic ones. Firstly, this is reflected in the appearance of a Z 2 -grading on state spaces. Secondly, this is manifest in the fact that state spaces may be Krein spaces rather than merely Hilbert spaces. In the special case that all state spaces are purely of degree 0 and are moreover Hilbert spaces we recover the purely bosonic core axioms in the version put forward in [9].
(T1) Associated to each hypersurface Σ is a complex separable Z 2 -graded Krein space H Σ , called the state space of Σ. We denote its indefinite inner product by ·, · Σ .
(T1b) Associated to each hypersurface Σ is a conjugate linear adapted Z 2 -graded isometry ι Σ : H Σ → H Σ . This map is an involution in the sense that ι Σ • ι Σ is the identity on H Σ .

Vacuum axioms
In addition to the core axioms we shall impose the vacuum axioms, which we recall in the following. These formalize the notion of a vacuum state, suitably adapted to the GBF. Their present form is a slight adaption from their form in [14, Section 5.1] 6 .
(V1) For each hypersurface Σ there is a distinguished state ψ Σ,0 ∈ H Σ of degree 0, called the vacuum state.
(V5) The amplitude of the vacuum state is unity. That is, for any region M , ρ M (ψ ∂M,0 ) = 1.

Quantization
We proceed to specify in this section the quantization scheme that yields, starting from classical data satisfying the axioms of Section 4, the ingredients of a corresponding quantum theory. We shall also show here the validity of some of the core axioms of Section 6.1 for these data, namely (T1), (T1b), (T2), (T2b) and (T4). Moreover, we show that all of the vacuum axioms hold.
As seen there this is an isometric isomorphism of Krein spaces. Due to associativity, which is easily verified, this extends to a prescription for decompositions of hypersurfaces with an arbitrary finite number of components. We thus satisfy axioms (T2) and (V3). In particular, the graded transposition property comes from equation (3.7). It is also straightforward to verify axiom (T2b). Given η 1 , . . . , η m ∈ L Σ 1 and ξ 1 , . . . , ξ m ∈ L Σ 2 we also note, using (3.6),

Amplitudes
Let M be a region. Given ξ ∈ L ∂M we decompose it as ξ = ξ R + J ∂M ξ I , where ξ R , ξ I ∈ LM . We define ξ ∈ L C M ⊆ L C ∂M as ξ := ξ R − iξ I .
Here L C M and L C ∂M are complexifications of the real vector spaces LM and L ∂M , where we denote the new complex structure by i. The real bilinear forms {·, ·} ∂M , g ∂M , ω ∂M all extend to L C ∂M as complex bilinear forms. This is understood in the following.
We define the subspace H • ∂M ⊆ H ∂M on which ρ M will be defined to be the space of linear combinations of vectors ψ[ξ 1 , . . . , ξ n ] for n ∈ N and ξ 1 , . . . , ξ n ∈ L ∂M together with the vector 1.
If n ∈ N is odd and ξ 1 , . . . , ξ n ∈ L ∂M are arbitrary we define the amplitude map to vanish, Note that the right-hand side is symmetric in the bosonic and anti-symmetric in the fermionic case under interchange of any two elements ξ i , ξ j with i = j. In particular, in the fermionic case R. Oeckl we note that the real parts of the inner products {·, ·} ∂M appearing on the right-hand side vanish since the real subspace LM and its complexification L C M is neutral with respect to g ∂M . Thus, we could equally well replace {·, ·} ∂M with the symplectic form 2iω ∂M (·, ·) in (7.5), making the anti-symmetry manifest. In the bosonic case the imaginary parts of the inner products {·, ·} ∂M vanish due to the isotropic nature of the subspace L C M ⊆ L C ∂M . Thus we could replace these inner products with their real parts g ∂M , making the symmetry manifest. This makes ρ M well defined and completes its definition on H • ∂M , satisfying (T4).
8 Amplitude and inner product 8.
Proof . Let η 1 , . . . , η m ∈ L Σ and ξ 1 , . . . , ξ n ∈ L Σ . We have The step from (8.4) to (8.5) consists in the application of (7.1), while the step to (8.6) arises from (7.2). In applying the amplitude map (7.5), many permutations do not contribute due to (8.1) and the remaining ones can be reorganized leading to (8.7). Applying (8.2) and moving the permutations from the ξ-variables to the η-variables yields (8.8). This expression coincides with the inner product on H Σ , see (3.2). This shows the equality (8.3) on a dense subspace. Since the right hand side extends continuously to the whole Krein space this defines the left hand side there also. This implies the claimed well definedness of ρΣ on τ Σ,Σ ;∂Σ (H Σ ⊗ H Σ ), completing the proof.

Complex conjugation
Let M be a region. Then the subspace LM ⊆ L ∂M gives rise to a conjugate linear involutive adapted real (anti-)isometry u M : L ∂M → L ∂M due to axiom (C5) as explained in Section 4.
Recall also that u M is a complex conjugation on L ∂M . With the conventions of Section 7.2 we have the identity Using this we may rewrite formula (7.5) for the amplitude map as Given u M , Lemma 3.5 gives rise to a conjugate linear involutive adapted real f -graded isometry U M : H ∂M → H ∂M defined by expression (3.3). Recall from Section 3.2 that the map U M may also be seen as a complex conjugation. It turns out that the amplitude map commutes with this complex conjugation in the following sense.
It is sufficient to carry out the proof for generating states. Moreover, it is trivial for states of odd Fock degree so we consider states of even Fock degree only. Let ξ 1 , . . . , ξ 2n ∈ L ∂M . Then Here we have used the definition of U M given by expression (3.3), the amplitude map in the form (8.9) and the involutiveness of u M .

Quantum evolution
A quite different perspective on the roles of the maps u M and U M is afforded by comparing formula (8.9) for the amplitude to formula (3.2) for the inner product. As we shall see this suggest to think of (a restricted version of) U M as an "evolution" operator, providing a quantum analogue of the role of u M in the classical theory as discussed in Section 4.3.
The following is in a sense the quantum analogue of Lemma 4.1.
Lemma 8.3. Suppose that L 2 ⊕ L 1 = L ∂M is a decomposition as an orthogonal direct sum of complex Krein spaces such that u M (L 1 ) = L 2 . Then U M (F(L 1 )) = F(L 2 ) with F(L 1 ) and F(L 2 ) understood as subspaces of H ∂M = F(L ∂M ). With the isometric isomorphism of Fock spaces F(L 2 )⊗ F(L 1 ) → F(L ∂M ) given by expression (3.5) we have the following identity, where ·, · 2 denotes the inner product of H ∂M = F(L ∂M ) restricted to F(L 2 ).
Proof . The claimed property of U M follows from the definition (3.3). For the proof of equation (8.10) it is enough to consider generating states. Thus, let ξ 1 , . . . , ξ n ∈ L 1 and η 1 , . . . , η m ∈ L 2 . Then Analogous to the classical case, such an algebraic notion of "evolution" underlying the amplitude map may be seen to arise from a geometric notion of "evolution" when the underlying decomposition of the space L ∂M arises from a corresponding decomposition of the hypersurface ∂M . This yields the following quantum analogue of Proposition 4.2.
Proof . This follows rather straightforwardly from Lemma 8.3. Setting L 1 = L Σ 1 and L 2 = L Σ 2 we have for ψ ∈ L Σ 1 and ψ ∈ L Σ 2 , This suggests to view M as a cobordism from Σ 1 to Σ 2 , inducing the unitary "evolution" mapŨ M between the associated state spaces. This would then be in line with the often preferred categorial view of TQFT, in the sense of assigning to the morphism M : In particular, if M as a cobordism takes the role of time-evolution,Ũ M is the associated unitary quantum operator. We also note that in the special case of a slice region with Σ 1 ≈ Σ 2 , we recover Proposition 8.1. The "evolution" operatorŨ M is then simply the identity.
Recall from Section 4.3 that in the fermionic case (but not the bosonic one) there is a preferred algebraic notion of evolution, coming from the canonical decomposition L ∂M = L + Recall now the standard globally hyperbolic setting described in Section 4.3. As already stated in that Section and as exhibited explicitly in the example of the Dirac field theory (Section 5), for standard fermionic field theories in this setting the algebraic notion of evolution coincides with the usual notion of time-evolution. That is, the decomposition of the boundary of a region M into initial and final component induces precisely the decomposition of the Krein space L ∂M into its positive and negative part. Applying Proposition 8.4, we obtain a description of the dynamics of the quantum theory in terms of the unitary mapsŨ M between Hilbert spaces associated to these hypersurfaces (all oriented in the same way). In particular, no indefinite inner product space appears in this way of describing the dynamics. This clarifies the absence of indefinite inner product spaces in the standard description of conventional fermionic quantum field theories.

Disjoint gluing
We proceed to consider the gluing axioms (T5a) and (T5b). We start with the proof of (T5a), describing the disjoint gluing of regions.
Proof . Let η 1 , . . . , η m ∈ L ∂M 1 and ξ 1 , . . . , ξ n ∈ L ∂M 2 . Then with (7.2) we get If n + m is odd, then (7.3) applies here (i.e., to the left hand side of (9.1)) and also to one of the two factors on the right hand side of (7.3), confirming the equality. We may thus suppose that n + m is even and (7.5) applies. Since mixed terms of the type R. Oeckl is odd we do not have any contributing term and the result vanishes, confirming (9.1) in this case also. It remains to consider the case where both m and n are even.

Gluing along a hypersurface
Axiom (T5b) is considerably more delicate. Consider a region M with boundary decomposing as a disjoint union ∂M = Σ 1 ∪ Σ ∪ Σ where Σ is a copy of Σ. We suppose that M can be glued to itself along Σ/Σ to form the new region M 1 with boundary Σ 1 . In view of the definition (7.4) of the amplitude on the vacuum state the composition identity (6.1) would imply the following identity for the gluing anomaly, given any ON-basis {ζ i } i∈I of H Σ . However, if the dimension of H Σ is infinite, it is not even clear whether the sum converges, given some particular ordered basis. Indeed, it is easy to construct simple models of a spacetime system together with data satisfying the axioms of Section 4 such that for a particular gluing this sum never converges. For the core axioms to be valid we thus need to introduce an additional assumption, not contained in the axioms of Section 4. This is precisely the assumption that the gluing anomaly c(M ; Σ, Σ ) is well defined for all admissible gluings. In the bosonic case this was discussed in [9] where it was shown that this is equivalent to an integrability condition in the holomorphic representation. Here we need to deal with this from a Fock space point of view. Given the Fock space structure of H Σ there is a natural family of ON-basis arising from ON-basis of the underlying space L Σ . It will be convenient to formulate the well-definedness criterion for the gluing anomaly using these special basis only. Let {ξ i } i∈N be an ON-basis of L Σ , where N = {1, . . . , dim L Σ } if L Σ is finite-dimensional and N = N otherwise. Using the ON-basis (3.9) of Fock space in the bosonic case we can then write the sum appearing in the identities (6.1) and (9.5) as follows ∞ m=0 a 1 ≤···≤am∈N In fact, this very same formula serves also in the fermionic case since the fermionic ON-basis (3.8) corresponds to a subset of the bosonic one and the extraneous terms then simply vanish due to antisymmetry. Moreover, we can simplify the formula by eliminating the combinatorial factors K a 1 ,...,am in exchange for an unrestricted sum of the indices over N m while dividing by a factor m!. This yields This continues to apply to both the bosonic and the fermionic case. Suppose now that the Krein space L Σ is finite-dimensional and N thus finite. The inner sum in (9.7) is then a well defined element of the tensor product H Σ ⊗ H Σ . Moreover, it does not depend on the choice of basis of L Σ . This follows from the conjugate linearity of the states ψ[ξ a 1 , . . . , ξ am ] in terms of the variables ξ a 1 , . . . , ξ am combined with the conjugate linearity of ι Σ . To see this explicitly note that any other ON-basis of L Σ is of the form {Λξ i } i∈N for a unitary map Λ : L Σ → L Σ . The sum for the modified basis reduces to that of the original basis as follows a 1 ,...,am∈N ...,am∈N b 1 ,...,bm∈N d 1 In the fermionic case the outer sum in (9.7) is then also finite, since terms with m > dim L Σ vanish.
Replacing in (9.5) expression (9.6) with expression (9.7) yields for the gluing anomaly, If L Σ is finite-dimensional, the gluing anomaly is automatically well defined in the fermionic case, as can also be seen from the fact that H Σ is then finite-dimensional. In the bosonic case, H Σ is infinite-dimensional even when L Σ is finite-dimensional. However, only the outer sum in (9.8) may diverge then. We declare c(M ; Σ, Σ ) well defined whenever this sum converges absolutely.
If L Σ is infinite-dimensional the situation is more complicated. Consider the projective system {L Σ,α } α∈A of all finite-dimensional complex subspaces of L Σ . (Here A denotes a suitable index set.) The projective limit of {L Σ,α } α∈A is L Σ . Associated to each subspace L Σ,α ⊆ L Σ we define the corresponding restricted anomaly factor c α (M ; Σ, Σ ) by the evaluation of the right hand side of (9.8) if it exists. We then say that the anomaly factor is well defined if the projective limit exists, More explicitly, this may be formulated as follows. In the fermionic case, every c α (M ; Σ, Σ ) is well defined. We say then that the limit exists and is equal to a quantity c ∈ C, i.e., lim − →α c α (M ; Σ, Σ ) = c if c has the following property: For any > 0 there exists β ∈ A such that for any γ ∈ A with γ ≥ β we have In the bosonic case the definition is the same with the additional requirement that c β (M ; Σ, Σ ) as well as c γ (M ; Σ, Σ ) for any γ ≥ β must be well defined.
Our definition of the anomaly factor really amounts to saying that equation (9.5) has to make sense, but only for certain particular choices of ON-basis of H Σ and certain particular choices of orderings of the sum. We apply this same definition to the right-hand side of (6.1). Thus, our main theorem shall consist of a proof of equation (6.1) and thus of axiom (T5b), but with the slight modification of restricting the ON-basis {ζ i } i∈I and their orderings.
Theorem 9.2. Assume the geometric context of axiom (T5b). Moreover, assume that the anomaly factor c(M ; Σ, Σ ) is well defined and non-zero. Then the composition identity (6.1) holds in the sense discussed and thus axiom (T5b) holds (for a restricted choice of ON-basis and orderings).
Proof . By our definition of the subspace H • ∂M 1 ⊆ H ∂M 1 it is sufficient to prove the composition identity (6.1) for states of the form ψ[φ 1 , . . . , φ n ] for φ 1 , . . . , φ n ∈ L ∂M 1 . Consider a finitedimensional complex Krein subspace L Σ,α of L Σ , let N α = {1, . . . , dim L Σ,α } and let {ξ a } a∈Nα be an ON-basis of L Σ,α as a complex Krein space. We define if the outer sum converges absolutely. The version of the identity (6.1) we need to prove amounts to showing the following limit, More explicitly this means that given > 0 we need to show the existence of β ∈ A such that for all γ ≥ β we have Instead we are going to show the stronger statement that there exists β ∈ A such that for all γ ≥ β we have To simplify the proof it is convenient to exploit complex conjugate linearity of the states ψ[φ 1 , . . . , φ n ] in terms of the variables φ 1 , . . . , φ n in conjunction with the decomposition (−i) l 1 +···+ln ψ φ l 1 1 , . . . , φ ln n .
Since the amplitude map is complex linear we can apply this decomposition to both sides of (9.9). It is thus sufficient to restrict to the special case φ 1 , . . . , φ n ∈ LM 1 , as we shall do in the following. Given φ ∈ LM 1 we observe that due to the exact sequence (4.2) of axiom (C7) there is a uniqueφ ∈ L Σ such that (φ,φ,φ) ∈ LM ⊆ L Σ 1 × L Σ × L Σ . With this notation let β ∈ A be determined such that L Σ,β is the complex Krein subspace of L Σ generated by {φ 1 , . . . ,φ n }. On the other hand let β ∈ A be such that for any γ ≥ β , c γ (M ; Σ, Σ ) is well defined. Such a β must exist since by assumption c(M ; Σ, Σ ) is well defined. Now let β ∈ A such that β ≥ β and β ≥ β . We claim that with this choice of β we satisfy (9.9) for all γ ≥ β. This is precisely the content of the following Lemma 9.3. Lemma 9.3. Assume the geometric context of axiom (T5b). Let φ 1 , . . . , φ n ∈ LM 1 . Let L Σ,α be a complex Krein subspace of L Σ such thatφ 1 , . . . ,φ n ∈ L Σ,α with the notation as above. Suppose that c α (M ; Σ, Σ ) is well defined. Define N α := {1, . . . , dim L Σ,α } and let {ξ a } a∈Nα be an ON-basis of L Σ,α . Then (9.10) The rather lengthy proof of this lemma is contained in Appendix A.
To summarize: Given a spacetime system and a model satisfying the axioms of Section 4, we need an additional assumption, namely the well definedness of the gluing anomaly for any admissible gluing. If this assumption is satisfied, the quantization described in this section yields a general boundary quantum field theory satisfying the core axioms for a slightly modified version of axiom (T5b).

.1 An important special case
Although it is not obvious in general how to ensure the well definedness of the gluing anomaly and thus the validity of all of the core axioms, there is an important special case where well definedness can be demonstrated easily. This is the case where two regions are glued together, one of which can be seen as a cobordism with an associated notion of evolution in the sense of Sections 4.3 and 8.3.
Proposition 10.1. Let P be a region with boundary decomposing disjointly as ∂P = Σ 1 ∪ Σ 2 and Q a region with boundary decomposing disjointly as ∂Q = Σ 2 ∪ Σ 3 , with Σ 2 a copy of Σ 2 . Suppose furthermore that u P (L Σ 1 ) = L Σ 2 . Let M be the disjoint union of P and Q and consider the gluing of M to itself along Σ 2 . Then the associated gluing anomaly is well defined and equal to 1, that is c(M ; Σ 2 , Σ 2 ) = 1.
In particular, this applies to the standard globally hyperbolic setting described in Section 4.3. The quantization of this setting thus proceeds without any obstruction from a potential illdefinedness of the gluing anomaly.

Renormalization
The necessity of an additional integrability condition, here in the form of the existence of the gluing anomaly factor, in order for the quantum theory to be well defined is somewhat unsatisfactory. This is especially so given that we know that allowing for both, non-trivial topologies and infinite-dimensional state spaces, will likely lead to a violation of this condition. Clearly, throwing out axiom (T5b) is not an option as this forms a corner stone of the coherence of the whole edifice of TQFT and consequently of the GBF. It is therefore natural to look for some kind of "renormalization" of the gluing anomaly. That is, one would "regularize" the quantities on both sides of relation (6.1) by introducing some "cut-off". Then the equation would have to hold in the "limit" that the "cut-off" is taken away.
In the fermionic case, to which we restrict in the following, this informal description of renormalization can be realized in a precise way. Moreover, the hard work for this has already been done in Section 9.2 and Appendix A. The role of the "cut-off" will be played by restricting the space L Σ associated to the gluing hypersurface Σ to the subspace L Σ,α . The role of the "limit" is taken by the projective limit in the system {L Σ,α } α∈A .
Define the orthogonal projector P α as the projector onto the Fock subspace F(L Σ,α ) of the Fock space F(L Σ ). The way we have treated the right hand side of relation (6.1) in Theorem 9.2 then amounts to saying that it is well defined and coincides with the limit in α of the expression whenever this limit exists. Here {ζ i } i∈I is an arbitrary ON-basis of H Σ = F(L Σ ) whose choice is irrelevant, as we saw already in Section 9.2. Expression (10.1) will play the role of the regularized version of the right hand side of relation (6.1). Our definition of the regularized gluing anomaly factor from Section 9.2, essentially equation (9.8), can be rewritten in a similar way as follows Again, the choice of ON-basis is irrelevant. Note that P α projects onto a finite-dimensional subspace, so expressions (10.1) and (10.2) are always well defined. We are now ready to state the renormalized version of axiom (T5b).

Remarks on the probability interpretation
In the standard formulation of quantum theory an indefinite inner product on a state space would lead to serious problems with the probability interpretation. The situation turns out to be more nuanced in the GBF. A key fact to keep in mind is that the concept of state space is considerably more general in the GBF. Only state spaces associated with certain rather special hypersurfaces may be compared in their role to the state spaces of the standard formulation. In particular, as explained in Section 8.3, in the standard globally hyperbolic setting, restricting ourselves to state spaces that play this role leaves us with only Hilbert spaces, even in fermionic theories.
In the following we shall consider this issue from an intrinsic GBF perspective. Recall to this end the probability interpretation for amplitudes in the GBF in the setting of Hilbert spaces [7]. That is, we have a region M and a boundary Hilbert space H ∂M . To define a measurement we need to specify a closed subspace S ⊆ H ∂M that encodes "preparation" and another closed subspace A ⊆ S that encodes the "question" we are asking in the measurement. The probability for an affirmative outcome of the measurement (if defined) is then given by the formula (11.1) Here {ξ i } i∈I is an orthonormal basis of S that restricts to an orthonormal basis {ξ i } i∈J of A with J ⊆ I. The quantity (11.1) has then all the right properties of a probability, reduces to the usual probabilities in the standard formulation, admits a notion of probability conservation in spacetime etc. [7,11]. So what changes if we wish to generalize from Hilbert spaces to Krein spaces? Apparently not much as the quantity (11.1) still has all the right properties and all the relevant arguments in [7,11] still apply. However, there is a crucial difference. For expression (11.1) to be defined in the first place the subspaces A and S have to be Krein spaces in the sense of Section 3.1. This amounts to A and S being decomposable as direct sums in terms of positive and negative parts. Concretely, given the decomposition The Krein space case is thus more restrictive than the Hilbert space case in the following sense. Recall that H ∂M is also a Hilbert space by taking the inner product of H + ∂M ⊕ H − ∂M . All the Krein subspaces of H ∂M are Hilbert subspaces of H + ∂M ⊕ H − ∂M , but there are many more Hilbert subspaces. One potential way to make physical sense of this limitation in the Krein space case would be to think of it as originating from a kind of "superselection rule". This superselection rule would here amount to saying that superpositions between states in the subspace H + ∂M and states in the subspace H − ∂M do not make physical sense. An undesirable feature of the setting as described so far is its apparent dependence on conventions. Consider a general boundary quantum field theory given in terms of the axioms of Section 6.1. If we invert the global orientation of spacetime, i.e., the orientation of all regions and hypersurfaces of the spacetime system, we obtain again a theory satisfying the axioms. Indeed, physically the new theory should really be equivalent to the old theory. However, given a region M , the decomposition of its boundary Krein space H ∂M into a direct sum of positive and negative parts is generally different from the corresponding decomposition of H ∂M = ι ∂M (H ∂M ) (the boundary Krein space H ∂M in the orientation-reversed theory). This would seem to be an indication against a physical significance of this decomposition.
However, there is another superselection rule at play here: The amplitude of a state with odd fermionic degree must vanish. This is stipulated in our formulation in axiom (T4), via the Z 2 -gradedness of the amplitude map. It is of course a basic and long established fact of quantum field theory, interacting or not 8 . Let us denote by H ∂M,0 the part of H ∂M with Z 2degree 0 (equivalent to even fermion number). By axiom (T1b) the restriction of ι ∂M to H ∂M,0 is an isometry. In particular, the decomposition into positive and negative part of H ∂M,0 is the same as that of ι ∂M (H ∂M,0 ). So, if we also enforce this superselection rule in the choice of the subspaces S and A, the new rule behaves well. Concretely, this enforcement consists of restricting the subspaces S and A to be subspaces of H ∂M,0 . 9 Let us emphasize that the new superselection rule does not show in predictions that can also be made within the standard formalism for conventional bosonic or fermionic field theories. Indeed, for bosonic theories there does not seem to be any realistic example theory that would require genuine Krein spaces instead of Hilbert spaces. For fermionic theories, Krein spaces do necessarily appear in a GBF setting as we have seen. However, the predictions of measurement outcomes that can be made in the standard formalism are tied to transition amplitudes. In these, S encodes a condition on the initial hypersurface and A an additional condition on the final hypersurface. However, due to the coincidence of preferred algebraic and geometric notion of time in the standard globally hyperbolic setting (recall Sections 4.3 and 8.3), this implies that S and A then automatically satisfy the new superselection rule, supposing that we explicitly enforce the usual fermionic Z 2 -superselection rule.

A brief comparison to holomorphic quantization
The previous treatment of the free bose field and its quantization in the GBF in [9] differs from the present one principally in the quantization method used. While we employ a Fock space construction here, a holomorphic quantization was performed in [9]. Nevertheless, the results are equivalent. We shall provide a basic dictionary between the two approaches here, restricted to the Hilbert space case.
We first recall the quantization of state spaces in the holomorphic approach. Let Σ be a hypersurface. Denote byL Σ the algebraic dual of the topological dual of L Σ , equipped with the weak * topology. Note that there is a natural inclusion L Σ ⊆L Σ . The inner product of L Σ induces a Gaussian Borel measure ν Σ onL Σ . The square integrable holomorphic functions onL Σ form a separable Hilbert space H 2 (L Σ , ν Σ ) with inner product It turns out that elements of H 2 (L Σ , ν Σ ) are uniquely determined by their values on L Σ . That is, we may think of H 2 (L Σ , ν Σ ) as a space of holomorphic functions on L Σ rather than onL Σ . This defines the state space H Σ in the holomorphic approach. The Fock space F(L Σ ), which is the model for H Σ in the present work, is related to it by an isometric isomorphism T : F(L Σ ) → H 2 (L Σ , ν Σ ) with the following characteristics. T is the sum of isometries T n : F n (L Σ ) → H 2 (L Σ , ν Σ ) given by In particular, a generating state ψ[ξ 1 , . . . , ξ n ] with ξ 1 , . . . , ξ n ∈ L Σ is mapped to the wave function Of particular importance in the holomorphic setting are the coherent states K ξ which are associated to elements ξ ∈ L Σ . As elements of H 2 (L Σ , ν Σ ) these take the form Expanding the exponential and comparing to formula (12.1), we see that the coherent state K ξ may be written in terms of generating states as the sum Conversely, we can recover a generating state ψ[ξ 1 , . . . , ξ n ] for ξ 1 , . . . , ξ n ∈ L Σ from a coherent state by using partial derivatives, T n (ψ[ξ 1 , . . . , ξ n ]) = 2 n ∂ ∂λ 1 · · · ∂ ∂λ n K λ 1 ξ 1 +···+λnξn λ 1 =0,...,λn=0 .

(12.2)
We proceed to the comparison of the amplitude maps ρ M : H ∂M → C. This was given in the holomorphic approach through an integral ρ M (ψ) = LM ψ(ξ) dνM (ξ). (12.3) The integral is over the subspaceLM ⊆L ∂M that carries an induced Gaussian measure νM . This, however, was shown to lead for coherent states to the remarkably simple expression where we are using the conventions of Section 7.2. It is then straightforward, using formula (12.2) to extract the amplitude for a generating state. The result is precisely given by the definitions (7.4), (7.3) and (7.5) of Section 7.2. Thus, not only the state spaces, but also the amplitudes are equivalent in both approaches. In fact, this is actually the way in which we obtained the bosonic versions of formulas (7.4), (7.3) and (7.5).
Its was shown in [6,12] that the quantization rule (12.3) is precisely equivalent to the usual Feynman path integral. So this applies also to the quantization scheme put forward in the present work.

Outlook
An obvious question concerns the generalization to interacting field theories. Neither the core axioms (Section 6.1) nor the probability interpretation (Section 11) are specific to the linear setting. What needs to be generalized is the encoding of the classical theory and the quantization prescription. In the bosonic case a perturbative approach to interactions was briefly discussed in [6]. This basically amounts to the standard techniques of quantum field theory motivated through the Feynman path integral, involving generating functions and leading to Feynman diagrams. It should not be difficult to adapt this to the fermionic case. In the bosonic case the holomorphic quantization approach followed in [9] is suggestive of non-perturbative generalizations in view of its motivation from geometric quantization. This was exploited in [6] in a mild way by generalizing from linear to affine theories. Due to the large body of work on geometric quantization of non-linear systems (although mostly limited to systems with finitely many degrees of freedom) [19] one should expect that this line of investigation could be pushed much further. In the purely fermionic case this looks much less promising, but one would expect interesting fermionic systems to contain also bosonic degrees of freedom.
Apart from state spaces and amplitudes, the GBF also naturally accommodates a notion of observable [10]. For linear bosonic field theory this was comprehensively developed in [12]. An essentially parallel treatment should be possible for linear fermionic field theory. A likely limitation from the axiomatic point of view would be the restriction for observables to have fermionic degree 0. In the light of the remarks in Section 11 this would likely also ensure compatibility with the probability interpretation and thus a consistent notion of expectation value.
As shown in Section 4.3, classical linear fermionic field theory, axiomatized according to Section 4.2, turns out to exhibit an "emergent" (algebraic) notion of time. As explained in Section 8.3, the quantized theory inherits this notion. Moreover, for theories with Lorentzian metric backgrounds (such as the Dirac theory considered in Section 5) this notion seems to agree precisely with the usual (geometric) notion of time evolution. This suggests to examine linear fermionic theories that do not come with a (Lorentzian) metric background to analyze the physical meaning of this emergent notion of time. Going beyond the linear case, the core axioms of the quantum theory (Section 6.1) alone do not seem to induce any such notion. This begs the question whether this phenomenon is strictly limited to the linear theory or has a more general counterpart.
The generalization of state spaces from Hilbert spaces to Krein spaces is a serious step in view of the justified reservations against giving up Hilbert spaces in quantum theory. However, as we have shown in Section 11, taking this step in the GBF is not at all the same thing as taking it in the standard formulation of quantum theory. Indeed, as we have seen there, a consistent probability interpretation with Krein spaces in the GBF is perfectly possible. Even though the necessity for Krein spaces only arises in the fermionic case, this suggests to permit them also in bosonic theories. Indeed, throughout this work, we have allowed for this theoretical possibility. On the other hand, the known realistic bosonic field theories work fine with Hilbert spaces. This might change though when we consider them on non-standard hypersurfaces. (However, in the specific examples considered in [4,8,13] there seems to be no indication to this effect.) Also, Krein spaces might become useful when abandoning Lorentzian background metrics.
A subject we have touched upon neither in [9] nor here is that of corners, i.e., the admission of hypersurfaces with boundaries. The problem is not that it would be difficult to write down axioms for this and have them satisfied. Indeed, we could easily include corners as follows. First we modify the axioms for a spacetime system to include hypersurfaces with boundaries. The notion of a disjoint decomposition of a hypersurface would be replaced by a more general notion that allows the components to have boundaries. (Precisely this was done in [14].) We could then proceed to modify both the classical axioms of Section 4.2 and the core axioms of Section 6.1 by simply replacing the disjoint notion of decomposition of hypersurfaces with the more general one. This would not affect at all the "algebraic" side of things. Everything would still work, including the quantization scheme and the proofs. The problem with this is that we would no longer be able to capture standard examples of quantum field theories. Generically, there would not be a suitable Hilbert or Krein space to be associated with a hypersurface with boundary, even in the classical linear theory. This is related to the non-locality of the complex structure J Σ associated with the hypersurface Σ, compare Sections 2.2 and 5.3. More generally, this problem can be seen in the light of the Reeh-Schlieder theorem [16]. This essentially implies that state spaces cannot be localized in parts of hypersurfaces. We remark that, on the other hand, in the case of 2-dimensional Yang-Mills theory a certain modification of the core axioms was shown to correctly implement corners [14].

A Proof of the main lemma
To prepare for the proof of Lemma 9.3 we note some useful identities. We assume the geometric context of axiom (T5b) or equivalently that of (T5b*). Consider φ ∈ L M 1 . Due to the exact sequence (4.2) of axiom (C7) there is thenφ ∈ L Σ such that (φ,φ,φ) ∈ L M ⊆ L Σ 1 × L Σ × L Σ . 10 With this notation we have the following identities. We omit the straightforward proofs.
Lemma A.1. Let ξ ∈ L Σ and φ ∈ L M 1 . Then Then Proof of Lemma 9.3. We first note that the left hand side and the right hand side of expression (9.10) are both necessarily zero if n is odd, due to definition (7.3). We may thus assume that n is even. To simplify notation we replace n in the following by 2n, where the new variable n is not necessarily even. We start by evaluating the right hand side of expression (9.10), which we shall denote by R (in order to avoid sub-indices we shall abbreviate ξ a i by ξ i throughout this proof) 0, 0) {l,k,a,b,c,d:2n=2l+a+b, m=2c+b+k=2d+a+k} { (0, 0, ξ ν(a+k+i) ), (0, 0, ξ ν(a+k+d+i) )} ∂M . (A.6) The step from expression (A.4) to (A.5) consists of evaluating the maps ι Σ and τ Σ 1 ,Σ,Σ ;∂M . The next step consists of applying the amplitude formula (7.5). This generates all possible pairings of 1-particle states. In order to organize these comprehensively, we introduce the variables l, k, a, b, c, d. These are non-negative integers that obey certain relations as indicated. They count the different types of pairing that appear. This reorganization and the associated particular summation over certain permutations leads to combinatorial factors as indicated in the first line of expression (A.6). At the same time, reorderings that occur lead in the fermionic case to a sign factor, which is written as a power of κ at the end of the first line and in the second line of expression (A.6). Note in particular, that implicit use was made of the fact that the summation over the indices a 1 , . . . , a m is symmetrical. This allowed to remove a summation over permutations of the factors ξ 1 , . . . , ξ m and thus a cancellation of the factor 1/(m!) that appears in (A.5). We proceed to perform the substitution of the identities (0, 0, ξ ν(e+f +g+k+i) ), (0, 0, ξ ν(e+f +g+k+d+i) ) ∂M .
Next, we contract the factors of the form {ξ ν(i) ,φ σ(2l+i) } Σ with corresponding pairs, by viewing the summation in the participating ξ variables as sums over complete basis. To this end we use the identity for η ∈ L Σ . Note that this leads to a factor κ e . The contraction involves a distribution of factors over the different types of matching pairs. This makes necessary the introduction of additional counting variables p, q, r, s, t, u, v, w, x, y, z. At the same time the variables e, h, j, o, k, c are eliminated. The relation between old and new variables is the following e = p + q + r + s + t + 2u, h = p + v, j = q + w, o = r + x, k = s + y, c = z + t + u.
Returning to expression (A.8) for R 1 , we observe that it may be simplified by identifying the various factors as arising from expanding the left hand side of identity (A.3) of Lemma A.2 in terms of the right hand side. This eliminates the variables l, a, d, p, r, leading to the much simpler expression But this coincides precisely with the definition (7.5) for an amplitude in M 1 , So, summarizing, we obtain ρ M 1 (ψ[φ 1 , . . . , φ 2n ]) · c α (M ; Σ, Σ ) = R.
This is expression (9.10) of Lemma 9.3 (with our redefinition of the variable n as 2n). This completes the proof. As a final remark, observe that absolute convergence of expression (A.9) is ensured by the assumption of well definedness of c α (M ; Σ, Σ ). This then implies absolute convergence of the sums appearing in the preceding expressions for R. This is the case in spite of the fact that some reorderings are taking place (recall the substitutions of the variable m), since the reorderings are bounded in range by the fixed constant 2n.