Observables in the General Boundary Formulation
Abstract
We develop a notion of quantum observable for the general boundary formulation of quantum theory. This notion is adapted to spacetime regions rather than to hypersurfaces and naturally fits into the topological quantum field theory like axiomatic structure of the general boundary formulation. We also provide a proposal for a generalized concept of expectation value adapted to this type of observable. We show how the standard notion of quantum observable arises as a special case together with the usual expectation values. We proceed to introduce various quantization schemes to obtain such quantum observables including path integral quantization (yielding the timeordered product), BerezinToeplitz (antinormal ordered) quantization and normal ordered quantization, and discuss some of their properties.
Observables in the General Boundary Formulation]
Observables in the
General Boundary Formulation
Robert Oeckl]Robert Oeckl
Primary 81P15; Secondary 81P16, 81T70, 53D50, 81S40, 81R30
1 Motivation: Commutation relations and quantum field theory
In standard quantum theory one is used to think of observables as encoded in operators on the Hilbert space of states. The algebra formed by these is then seen as encoding fundamental structure of the quantum theory. Moreover, this algebra often constitutes the primary target of quantization schemes that aim to produce a quantum theory from a classical theory. Commutation relations in this algebra then provide a key ingredient of correspondence principles between a classical theory and its quantization.
We shall argue in the following that while this point of view is natural in a nonrelativistic setting, it is less compelling in a special relativistic setting and becomes questionable in a general relativistic setting.
In nonrelativistic quantum mechanics (certain) operators
correspond to measurements that can be applied at any given time, meaning that the
measurement is performed at that time. Let us say we consider the measurement of two
quantities, one associated with the operator and another associated with the
operator . In particular, we can then also say which operator is associated with
the consecutive measurement of both quantities. If we first measure and then
the operator is the product , and if we first measure and then the
operator is the product .
The treatment of operators representing observables is different in quantum field theory. Here, such operators are labeled with the time at which they are applied. For example, we write for a field operator at time . Hence, if we want to combine the measurement processes associated with operators and say, there is only one operationally meaningful way to do so. The operator associated with the combined process is the timeordered product of the two operators, . Of course, this timeordered product is commutative since the information about the temporal ordering of the processes associated with the operators is already contained in their labels. Nevertheless, in traditional treatments of quantum field theory one first constructs a noncommutative algebra of field operators starting with equaltime commutation relations. Since the concept of equaltime hypersurface is not Poincaré invariant, one then goes on to generalize these commutation relations to field operators at different times. In particular, one finds that for two localized operators and , the commutator obeys
(1) 
which is indeed a Poincaré invariant condition. The timeordered product is usually treated as a concept that is derived from the noncommutative operator product. From this point of view, condition (1) serves to make sure that it is well defined and does not depend on the inertial frame. Nevertheless, it is the former and not the latter that has a direct operational meaning. Indeed, essentially all the predictive power of quantum field theory derives from the amplitudes and the Smatrix which are defined entirely in terms of timeordered products. On the other hand, the noncommutative operator product can be recovered from the timeordered product. Equaltime commutation relations can be obtained as the limit,
The property (1) can then be seen as arising from the transformation properties of this limit and its nonequal time generalization.
We conclude that in a special relativistic setting, there are good reasons to regard the timeordered product of observables as more fundamental than the noncommutative operator product. This suggests to try to formulate the theory of observables in terms of the former rather than the latter. In a (quantum) general relativistic setting with no predefined background metric a condition such as (1) makes no longer sense, making the postulation of a noncommutative algebra structure for observables even more questionable.
In this paper we shall consider a proposal for a concept of quantum observable that takes these concerns into account. The wider framework in which we embed this is the general boundary formulation of quantum theory (GBF) [1]. We start in Section 2 with a short review of the relevant ingredients of the GBF. In Section 3 we introduce a concept of quantum observable in an axiomatic way, provide a suitably general notion of expectation value and show how standard concepts of quantum observable and expectation values arise as special cases. In Section 4 we consider different quantization prescriptions of classical observables that produce such quantum observables, mainly in a field theoretic context.
2 Short review of the general boundary formulation
2.1 Core axioms
The basic data of a general boundary quantum field theory consists of two types: geometric objects that encode a basic structure of spacetime and algebraic objects that encode notions of quantum states and amplitudes. The algebraic objects are assigned to the geometric objects in such a way that the core axioms of the general boundary formulation are satisfied. These may be viewed as a special variant of the axioms of a topological quantum field theory [2]. They have been elaborated, with increasing level of precision, in [3, 1, 4, 5]. In order for this article to be reasonably selfcontained, we repeat below the version from [5].
The geometric objects are of two kinds:
 Regions

These are (certain) oriented manifolds of dimension (the spacetime dimension), usually with boundary.
 Hypersurfaces

These are (certain) oriented manifolds of dimension , here assumed without boundary.
^{3}
Depending on the theory to be modeled, the manifolds may carry additional structure such as that of a Lorentzian metric in the case of quantum field theory. For more details see the references mentioned above. The core axioms may be stated as follows:

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

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

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

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

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

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

Let and be regions and be their disjoint union. Then is also a disjoint union and . Then, for all and ,
(2) 
Let be a region with its boundary decomposing as a disjoint union , where is a copy of . Let denote the gluing of with itself along and suppose that is a region. Note . Then, for all and . Moreover, for any orthonormal basis of , we have for all ,
(3) where is called the gluing anomaly factor and depends only on the geometric data.
As in [5] we omit in the following the explicit mention of the maps .
2.2 Amplitudes and probabilities
In standard quantum theory transition amplitudes can be used to encode measurements. The setup, in its simplest form, involves an initial state and a final state . The initial state encodes a preparation of or knowledge about the measurement, while the final state encodes a question about or observation of the system. The modulus square , where is the timeevolution operator between initial and final time, is then the probability for the answer to the question to be affirmative. (We assume states to be normalized.) This is a conditional probability , namely the probability to observe given that was prepared.
In the GBF this type of measurement setup generalizes considerably.
(4) 
Here and are the orthogonal projectors onto the subspaces and respectively. is the linear map given by the composition of the amplitude map with the projector . A requirement for (4) to make sense is that this composed map is continuous, but does not vanish. (The amplitude map is generically not continuous.) That is, must be neither “too large” nor “too small”. Physically this means that must on the one hand be sufficiently restrictive while on the other hand not imposing an impossibility. The continuity of means that it is an element in the dual Hilbert space . The norm in is denoted in formula (4) by . With an analogous explanation for the numerator the mathematical meaning of (4) is thus clear.
In [1], where this probability interpretation of the GBF was originally proposed, the additional assumption was made, and with good reason. Physically speaking, this condition enforces that we only ask questions in a way that takes into account fully what we already know. Since it is of relevance in the following, we remark that formula (4) might be rewritten in this case as follows:
(5) 
Here the inner product is the inner product of the dual Hilbert space . Indeed, whenever and commute, (5) coincides with (4).
2.3 Recovery of standard transition amplitudes and probabilities
We briefly recall in the following how standard transition amplitudes are recovered from amplitude functions. Similarly, we recall how standard transition probabilities arise as special cases of the formula (4). Say is some initial time and some final time, and we consider the spacetime region in Minkowski space. is the disjoint union of hypersurfaces of constant and respectively. We have chosen the orientation of here to be opposite to that induced by , but equal (under timetranslation) to that of . Due to axioms (T2) and (T1b), we can identify the Hilbert space with the tensor product . The amplitude map associated with can thus be viewed as a linear map .
In the standard formalism, we have on the other hand a single Hilbert space of states and a unitary timeevolution map . To relate the two settings we should think of , and as really identical (due to timetranslation being an isometry). Then, for any , the amplitude map and the operator are related as
(6) 
Consider now a measurement in the same spacetime region, where an initial state is prepared at time and a final state is tested at time . The standard formalism tells us that the probability for this is (assuming normalized states):
(7) 
In the GBF, the preparation of and observation of are encoded in the following subspaces of :
(8) 
Using (6) one can easily show that then , i.e., the expressions (4) and (7) coincide. This remains true if, alternatively, we define disregarding the knowledge encoded in , i.e., as
(9) 
3 A conceptual framework for observables
3.1 Axiomatics
Taking account of the fact that realistic measurements are extended both in space as well as in time, it appears sensible to locate also the mathematical objects that represent observables in spacetime regions. This is familiar for example from algebraic quantum field theory, while being in contrast to idealizing measurements as happening at instants of time as in the standard formulation of quantum theory.
Mathematically, we model an observable associated with a given spacetime region as a replacement of the corresponding amplitude map . That is, an observable in is a linear map , where is the dense subspace of appearing in core axiom (T4). Not any such map needs to be an observable though. Which map exactly qualifies as an observable may generally depend on the theory under consideration.

Associated to each spacetime region is a real vector space of linear maps , called observable maps. In particular, .
The most important operation that can be performed with observables is that of composition. This composition is performed exactly in the same way as prescribed for amplitude maps in core axioms (T5a) and (T5b). This leads to an additional condition on the spaces of observables, namely that they be closed under composition.

Let and be regions as in (T5a) and and . Then, there is such that for all and ,
(10) 
Let be a region with its boundary decomposing as a disjoint union and given as in (T5b) and . Then, there exists such that for any orthonormal basis of and for all ,
(11)
We generally refer to the gluing operations of observables of the types described in (O2a) and (O2b) as well as their iterations and combinations as compositions of observables. Physically, the composition is meant to represent the combination of measurements. Combination is here to be understood as in classical physics, when the product of observables is taken.
3.2 Expectation values
As in the standard formulation of quantum theory, the expectation value of an observable depends on a preparation of or knowledge about a system. As recalled in Section 2.2, this is encoded for a region in a closed subspace of the boundary Hilbert space . Given an observable and a closed subspace , the expectation value of with respect to is defined as
(12) 
We use notation here from Section 2.2. Also, as there we need to be continuous and different from zero for the expectation value to make sense.
We proceed to make some remarks about the motivation for postulating the expression (12). Clearly, the expectation value must be linear in the observable. Another important requirement is that we would like probabilities in the sense of Section 2.2 to arise as a special case of expectation values. Indeed, given a closed subspace of and setting we see that expression (12) reproduces exactly expression (5). At least in the case where the condition is met, this coincides with expression (4) and represents the conditional probability to observe given .
3.3 Recovery of standard observables and expectation values
Of course, it is essential that the present proposal for implementing observables in the GBF can reproduce observables and their expectation values as occurring in the standard formulation of quantum theory. There observables are associated to instants of time, i.e., equaltime hypersurfaces. To model these we use “infinitesimally thin” regions, also called empty regions, which geometrically speaking are really hypersurfaces, but are treated as regions.
Concretely, consider the equaltime hypersurface at time in Minkowski space, i.e., . We denote the empty region defined by the hypersurface as . The relation between an observable map and the corresponding operator is then analogous to the relation between the amplitude map and the timeevolution operator as expressed in equation (6). By definition, is equal to the disjoint union so that . The Hilbert space is identified with the conventional Hilbert space and for we require
(13) 
Note that we can glue two copies of together, yielding again a copy of . The induced composition of observable maps then translates via (13) precisely to the composition of the corresponding operators. In this way we recover the usual operator product for observables of the standard formulation.
Consider now a normalized state encoding a preparation. This translates in the GBF language to the subspace of as reviewed in Section 2.3. The amplitude map can be identified with the inner product of due to core axiom (T3x). Thus, , where is the orthogonal projector in onto the subspace spanned by . This makes it straightforward to evaluate the denominator of (12). Let be an orthonormal basis of , which moreover we choose for convenience such that . Then,
(14) 
For the numerator of (12) we observe
(15) 
Hence, the GBF formula (12) recovers in this case the conventional expectation value of with respect to the state .
4 Quantization
We turn in this section to the problem of the quantization of classical observables. On the one hand, we consider the question of how specific quantization schemes that produce Hilbert spaces and amplitude functions satisfying the core axioms can be extended to produce observables. On the other hand, we discuss general features of quantization schemes for observables and the relation to conventional schemes.
4.1 SchrödingerFeynman quantization
Combining the Schrödinger representation with the Feynman path integral yields a quantization scheme that produces Hilbert spaces for hypersurfaces and amplitude maps for regions in a way that “obviously” satisfies the core axioms [3, 7, 8]. We shall see that it is quite straightforward to include observables into this scheme. Moreover, the resulting quantization can be seen to be in complete agreement with the results of standard approaches to quantum field theory.
We recall that in this scheme states on a hypersurface arise as wave functions on the space space of field configurations on . These form a Hilbert space of squareintegrable functions with respect to a (fictitious) translation invariant measure :
(16) 
The amplitude map for a region arises as the Feynman path integral,
(17) 
where is the action evaluated in , and is the space of field configurations in .
The Feynman path integral is of course famous for resisting a rigorous definition and it is a highly nontrivial task to make sense of expressions (17) or even (16) in general. Nevertheless, much of textbook quantum field theory relies on the Feynman path integral and can be carried over to the present context relatively easily for equaltime hypersurfaces in Minkowski space and regions bounded by such. Moreover, for other special types of regions and hypersurfaces this quantization program has also been successfully carried through for linear or perturbative quantum field theories. Notably, this includes timelike hypersurfaces [7, 8] and has led to a widening of the concept of an asymptotic Smatrix [9, 10].
We proceed to incorporate observables into the quantization scheme. To this end, a classical observable in a region is modeled as a real (or complex) valued function on . According to Section 3.1 the quantization of , which we denote here by , must be a linear map . We define it as
(18) 
Before we proceed to interpret this formula in terms of textbook quantum field theory language, we emphasize a key property of this quantization prescription. Suppose we have disjoint, but adjacent spacetime regions and supporting classical observables and respectively. Applying first the operation of (O2a) and then that of (O2b), we can compose the corresponding quantum observables and to a new observable, which we shall denote , supported on the spacetime region . On the other hand, the classical observables and can be extended trivially to the spacetime region and there be multiplied to a classical observable . The composition property of the Feynman path integral now implies the identity
(19) 
That is, there is a direct correspondence between the product of classical observables and the spacetime composition of quantum observables. This composition correspondence, as we shall call it, is not to be confused with what is usually meant with the term “correspondence principle” such as a relation between the commutator of operators and the Poisson bracket of classical observables that these are representing. Indeed, at a careless glance these concepts might even seem to be in contradiction.
Consider now in Minkowski space a region , where . Then, with notation as in Section 2.3. Consider a classical observable that encodes an point function,
(20) 
where . Given an initial state at time and a final state at time , the quantization of according to formula (18) can be written in the more familiar form
(21) 
where are the usual quantizations of the classical observables , is the usual quantization of the Hamiltonian operator at time and signifies timeordering. Thus, in familiar situations the prescription (18) really is the “usual” quantization performed in quantum field theory, but with timeordering of operators. From formula (21) the correspondence property (19) is also clear, although in the more limited context of temporal composition. We realize thus the goal, mentioned in the introduction, of implementing the timeordered product as more fundamental than the noncommutative operator product.
For a linear field theory, it turns out that the quantization prescription encoded in (18) exhibits an interesting factorization property with respect to coherent states. We consider the simple setting of a massive free scalar field theory in Minkowski space with equaltime hypersurfaces. Recall ([10] equation (26)) that a coherent state in the Schrödinger representation at time can be written as
(22) 
where is a complex function on momentum space encoding a solution of the KleinGordon equation. is the vacuum wave function and is a normalization constant. Consider as above an initial time , a final time and the region in Minkowski space. Let represent a classical observable. Evaluating the quantized observable map on an initial coherent state encoded by and a final coherent state encoded by yields,
(23) 
Here, denote the restrictions of the configuration to time . To obtain the second equality we have shifted the integration variable by
(24) 
and used the conventions of [10]. Note that is a complexified classical solution in determined by and . We have supposed that naturally extends to a function on the complexified configuration space . Viewing the function as a new observable , the remaining integral in (23) can be interpreted in terms of (18) and we obtain the factorization identity
(25) 
That is, the quantum observable map evaluated on a pair of coherent states factorizes into the plain amplitude for the same pair of states and the quantum observable map for a shifted observable evaluated on the vacuum. Note that the second term on the right hand side here is a vacuum expectation value.
It turns out that factorization identities analogous to (25) are generic rather than special to the types of hypersurfaces and regions considered here. We will come back to this issue in the next section, where also the role of the complex classical solution will become clearer from the point of view of holomorphic quantization. For the moment let us consider the particularly simple case where is a linear observable. In this case and the second term on the right hand side of (25) decomposes into a sum of two terms,
(26) 
The first term on the right hand side is a onepoint function which vanishes in the present case of a linear field theory. ( is antisymmetric under exchange of and , while the other expressions in (18) are symmetric.) The second factor in the second term is the amplitude of the vacuum and hence equal to unity. Thus, in the case of a linear observable (25) simplifies to
(27) 
4.2 Holomorphic quantization
A more rigorous quantization scheme that produces a GBF from a classical field theory is the holomorphic quantization scheme introduced in [5]. It is based on ideas from geometric quantization and its Hilbert spaces are versions of “Fock representations”. An advantage of this scheme is that taking an axiomatically described classical field theory as input, it produces a GBF as output that can be rigorously proved to satisfy the core axioms of Section 2.1. A shortcoming so far is that only the case of linear field theory has been worked out.
Concretely, the classical field theory is to be provided in the form of a real vector space of (germs of) solutions near each hypersurface . Moreover, for each region there is to be given a subspace of the space of solutions on the boundary of . This space has the interpretation of being the space of solutions in the interior of (restricted to the boundary). Also, the spaces carry nondegenerate symplectic structures as well as complex structures . Moreover, for each hypersurface , the symplectic and complex structures combine to a complete real inner product and to a complete complex inner product . Another important condition is that the subspace is Lagrangian with respect to the symplectic structure .
The Hilbert space associated with a hypersurface is the space of holomorphic squareintegrable functions on with respect to a Gaussian measure .
(28) 
Heuristically, the measure can be understood as
(29) 
where is a fictitious translation invariant measure on . The space is essentially the Fock space constructed from viewed as a 1particle space with the inner product .
The amplitude map associated with a region is given by the integral formula
(30) 
The integration here is over the space of solutions in with the measure , which heuristically can be understood as
(31) 
where again is a fictitious translation invariant measure on .
Particularly useful in the holomorphic quantization scheme turn out to be the coherent states that are associated to classical solutions near the corresponding hypersurface. On a hypersurface the coherent state associated to is given by the wave function
(32) 
The natural vacuum, which we denote by , is the constant wave function of unit value. Note that .
Creation and annihilation operators
Oneparticle states on a hypersurface are represented by nonzero continuous complexlinear maps , where complexlinearity here implies . By the Riesz Representation Theorem such maps are thus in onetoone correspondence with nonzero elements of . Concretely, for a nonzero element the corresponding oneparticle state is represented by the wave function given by
(33) 
The normalization is chosen here such that . Physically distinct oneparticle states thus correspond to the distinct rays in , viewed as a complex Hilbert space. An particle state is represented by a (possibly infinite) linear combination of the product of wave functions of this type. The creation operator for a particle state corresponding to is given by multiplication,
(34) 
The corresponding annihilation operator is the adjoint. Using the reproducing property of the coherent states we can write it as,
(35) 
Note in particular, that the action of an annihilation operator on a coherent state is by multiplication,
(36) 
For the commutation relations are, as usual,
(37) 
BerezinToeplitz quantization
A natural way to include observables into this quantization scheme seems to be the following. We model a classical observable on a spacetime region as a map (or ) and define the associated quantized observable map via
(38) 
To bring this into a more familiar form, we consider, as in Section 3.3, the special case of an empty region , given geometrically by a hypersurface . Then, for encoding “initial” and “final” state we have
(39) 
We can interpret this formula as follows: The wave function is multiplied by the function . The resulting function is an element of the Hilbert space (supposing to be essentially bounded), but not of the subspace of holomorphic functions. We thus project back onto this subspace and finally take the inner product with the state . This is precisely accomplished by the integral. We may recognize this as a version of BerezinToeplitz quantization, where in the language of Berezin [11] the function is the contravariant symbol of the operator that is related to by formula (13). That is,
(40) 
In the following we shall refer to the prescription encoded in (38) simply as BerezinToeplitz quantization.
Note that any complex valued continuous reallinear observable can be decomposed into its holomorphic (complex linear) and antiholomorphic (complex conjugate linear) part
(41) 
If we consider real valued observables only, we can parametrize them by elements of due to the Riesz Representation Theorem. (In the complex valued case the parametrization is by elements of , the complexification of , instead.) If we associate to the real linear observable given by
(42)  
(43) 
Using the results of Section 4.2.1, we see that quantized according to the prescription (39) and expressed in terms of operators as in (40) yields
(44) 
Consider now reallinear observables . We shall be interested in the antinormal ordered product of the corresponding operators , which we denote by . To evaluate matrix elements of this antinormal ordered product we decompose the observables according to (41) into holomorphic and antiholomorphic parts, corresponding to creation operators and annihilation operators respectively. The creation operators then act on wave functions by multiplication with according to (34). Converting the annihilation operators into creation operators by moving them to the lefthand side of the inner product, we see that these correspondingly contribute factors in the inner product (28). We obtain,
Setting this coincides precisely with the righthand side of (39). Thus, in the case of a hypersurface (empty region) the BerezinToeplitz quantization precisely implements antinormal ordering.
Remarkably, the BerezinToeplitz quantization shares with the SchrödingerFeynman quantization the factorization property exhibited in equation (25). In fact, it is in the present context of holomorphic quantization that this property attains a strikingly simple form. In order to state it rigorously, we need a bit of technical language. For a map and an element we denote by the translated map . We say that is analytic iff for each pair the map is real analytic. We denote the induced extension also by , where is the complexification of . We say that is analytic and sufficiently integrable iff for any the map is integrable in . We recall (Lemma 4.1 of [5]) that elements of decompose uniquely as , where are elements of .
Proposition 4.1 (Coherent Factorization Property).
Let be analytic and sufficiently integrable. Then, for any we have
(45) 
where is given by .
Proof.
Recall that for we can rewrite the wave function of the coherent state as follows,
(46) 
We restrict first to the special case , i.e., . Translating the integrand by (using Proposition 3.11 of [5]) we find