Fock Space: A Model of Linear Exponential Types. School of Computer Science. R.A.G. Seely z. Montreal, Canada H3A 2K6.

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1 Fock Sace: A Model of Linear Exonential Tyes R.F. Blute Deartment of Mathematics University of Ottawa Ottawa, Canada K1N 6N5 rblute@mathstat.uottawa.ca Prakash Panangaden y School of Comuter Science McGill University Montreal, Canada H3A 2A7 rakash@cs.mcgill.ca R.A.G. Seely z Deartment of Mathematics McGill University Montreal, Canada H3A 2K6 rags@math.mcgill.ca Abstract It has been observed by several eole that, in certain contexts, the free symmetric algebra construction can rovide a model of the linear modality!. This construction arose indeendently in quantum hysics, where it is considered as a canonical model of quantum eld theory. In this context, the construction is known as (bosonic) Fock sace. Fock sace is used to analyze such quantum henomena as the annihilation and creation of articles. There is a strong intuitive connection to the rincile of renewable resource, which is the hilosohical interretation of the linear modalities. In this aer, we examine Fock sace in several categories of vector saces. We rst consider vector saces, where the Fock construction induces a model of the ; &;! fragment in the category of symmetric algebras. When considering Banach saces, the Fock construction rovides a model of a weakening cotrile in the sense of Jacobs. While the models so obtained model a smaller fragment, it is closer in ractice to the structures considered by hysicists. In this case, Fock sace has a natural interretation as a sace of holomorhic functions. This suggests that the \nonlinear" functions we arrive at via Fock sace are not merely continuous but analytic. Finally, we also consider fermionic Fock sace, which corresonds algebraically to skew symmetric algebras. By considering fermionic Fock sace on the category of nite-dimensional vector saces, we obtain a model of full roositional linear logic, although the model is somewhat degenerate in that certain connectives are equated. 1 Introduction Linear logic was introduced by Girard [G87] as a consequence of his decomosition of the traditional connectives of logic into more rimitive connectives. The resulting logic is more resource sensitive; this is achieved by lacing strict control over the structural rules of contraction and weakening, Research suorted in art by FCAR of Quebec. y Research suorted in art by an NSERC Research Grant. z Research suorted in art by an NSERC Research Grant and by FCAR of Quebec. 1

2 introducing a new \modal" oerator of course (denoted! ) to indicate when a formula may be used in a resource-insensitive manner i.e. when a resource is renewable. Without the! oerator, the essence of linear logic is carried by the multilicative connectives; at its most basic level, linear logic is a logic of monoidal-closed categories (in much the same way that intuitionistic logic is a logic of cartesian-closed categories). In modelling linear logic, one begins with a monoidal-closed category, and then adds aroriate structure to model linear logic's additional features. To model linear negation, one asses to the -autonomous categories of Barr [B79]. To model the additive connectives, one then adds roducts and coroducts. Finally, to model the exonentials, and so regain the exressive strength of traditional logic, one adds a trile and cotrile, satisfying roerties to be outlined below. This rogram was rst outlined by Seely in [Se89]. Linear logic bears strong resemblance to linear algebra (from which it derives its name), but one signicant dierence is the diculty in modelling!. The category of vector saces over an arbitrary eld is a symmetric monoidal closed category, indeed in some sense the rototyical monoidal category, and as such rovides a model of the intuitionistic variant of multilicative linear logic. Furthermore, this category has nite roducts and coroducts with which to model the additive connectives. It thus makes sense to look for models of various fragments of linear logic in categories of vector saces. However, modelling the exonentials is more roblematic. It is the rimary urose of this aer to resent methods of modelling exonential tyes in categories arising from linear algebra. We study models of the exonential connectives in categories of linear saces which have monoidal (but generally not monoidal-closed) structure. (We shall also include a model in nite-dimensional vector saces which is closed.) The construction which will be used to model the exonential formulas, although standard in algebra, arose indeendently in quantum eld theory, and is known as Fock sace. It was designed as a framework in which to consider many-article states. The key oint of dearture for quantum eld theory was the realization that elementary articles are created and destoyed in hysical rocesses and that the mathematical formalism of ordinary quantum mechanics needs to be revised to take this into account. The hysical intuitions behind the Fock construction will also be familiar to mathematicians in that it corresonds to the free symmetric algebra on a sace. As such, it induces a air of adjoint functors, and hence a cotrile in the algebra category. It is this cotrile which will be used to model!. It should be noted that this category of algebras inherits the monoidal structure from the underlying category of saces, but there is no hoe that this category could have a monoidal-closed structure. While Fock sace has an abstract reresentation in terms of an innite direct sum, hysicists such as Ashtekar, Bargmann, Segal and others, see [AM-A80, Ba61, S62] have analyzed concrete reresentations of Fock sace as certain classes of holomorhic functions on the base sace. Thus, these models further the intuition that the exonentials corresond to the analytic roerties of the sace. In fact, there is a clear sense in which morhisms in the Kleisli category for the cotrile can be viewed as generalized holomorhic functions. Thus, there should be an analogy to coherence saces where the Kleisli category corresonds to the stable mas. Fock sace also has two additional features which corresond to additional structure, not exressible in the syntax of linear logic. These are the annihilation and creation oerators, which are used to model the annihilation and creation of articles in a eld. These may give a tighter control of resources not exressible in the ure linear logic. Thus, these models may be closer to the bounded linear logic of Girard, Scedrov and Scott [GSS91]. A ossible alication of this work is that the rened connectives of linear logic may lend insight into certain asects of quantum eld theory. For examle, there are two distinct methods of combining article states. One can suerimose two states onto a single article, or one can have two articles coexisting. The former seems to corresond to additive conjunction and the latter 2

3 to the multilicative. This hysical imagery is missing in quantum mechanics, which was secially designed to handle a single article; it only shows u in quantum eld theory. In this aer, we begin by reviewing the categorical structure necessary to model linear logic, and secically exonential tyes. We then describe the Fock construction on vector saces and exlain the roerties of the resulting model. We next consider Fock sace on normed vector saces. While the model so obtained has weaker roerties, this case is closer to that considered by hysicists. In fact, in this case the Fock construction gives a model of a weakening cotrile in the sense of Jacobs [J93]. We next describe the interretation of Fock sace as a sace of Holomorhic functions. Finally, the hysical meaning of the Fock construction is discussed. We wish to oint out that this aer corrects an error in an earlier draft [BPS]. This is discussed in section 6. 2 Linear Logic and Monoidal Categories We shall begin with a few reliminaries concerning linear logic. We shall not reroduce the formal syntax of linear logic, nor the usual discussion of its intuitive interretation or utility for this the reader is referred to the standard references, such as [G87]. We do recall [Se89] that a categorical semantics for linear logic may be based on Barr's notion of -autonomous categories [B79]. If only to establish notation, here is the denition. Denition 1 A category C is -autonomous if it satises the following: 1. C is symmetric monoidal closed; that is, C has a tensor roduct A B and an internal hom A? B which is adjoint to the tensor in the second variable Hom(A B; C) = Hom(B; A? C) 2. C has a dualizing object?; that is, the functor ( )? : C o?! C dened by A? = A?? is an involution ( viz. the canonical morhism A?! ((A??)??) is an isomorhism). In addition various coherence conditions must hold a good account of these may be found in [M-OM89]. Coherence theorems may be found in [BCST, Bl92]. An equivalent characterization of -autonomous categories is given in [CS91], based on the notion of weakly distributive categories. That characterization is useful in contexts where it is easier to see how to model the tensor, the \ar". and linear negation, and the coherence conditions may be exressed in terms of those oerations. The structure of a -autonomous category models the evident eonymous structure of linear logic: the categorical tensor is the linear multilicative and the internal hom? is linear imlication. The dualizing object? is the unit for linear \ar"., or equivalently, is the dual of the unit I for the tensor 1. There are a number of variants of linear logic whose categorical semantics is based on this. First is full \classical" linear logic, which includes the additive oerations. These corresond to requiring that the category C have roducts and coroducts. (If C is -autonomous, one of these will imly the other by de Morgan duality.) There is also Girard's notion of \intuitionistic" linear 1 In other aers we have used the notation > for the unit for, and instead of.. Here we shall try to avoid controversy by using notation traditional in the context of Banach saces, and by generally ignoring the \ar". So in this aer, means direct sum, which coincides with Girard's notation. We use for cartesian roduct, corresonding to Girard's &. And we shall use the usual notation for the aroriate saces when referring to the units. 3

4 logic [GL87], which omits linear negation and \ar" this corresonds to merely requiring that C be autonomous, that is to say, symmetric monoidal closed (with or without roducts and coroducts, deending on whether or not the additives are wanted). There is an intermediate notion, \full intuitionistic linear logic" due to de Paiva [dp89], in which the morhism A?! A?? need not be an isomorhism. And as mentioned above, there is the notion of weakly distributive category [CS91, BCST], where negation and internal hom are not required. One imortant class of -autonomous categories are the comact categories [KL80] where the tensor is self-dual: (AB)? = A? B?. These categories form the basis for Abramsky's interaction categories [Abr]. In this aer we shall model various fragments of linear logic; we shall describe the fragments in terms of the categorical structure resent, without exlicitly identifying the fragments. Finally, in order to be able to recature the full strength of classical (or intuitionistic) logic, one must add the \exonential"! (and its de Morgan dual? ). (All our structures will model!.) We saw in [Se89] that this amounts to the following. Denition 2 A monoidal category C with nite roducts admits (Girard) storage if there is a cotrile! : C?! C (with the usual structure mas A A?! A A??!!! A), satisfying the following: 1. for each object A 2 C,! A carries (naturally) the structure of a (cocommutative) -comonoid > e A??! A d A??!! A! A (and the coalgebra mas are comonoid mas), and 2. there are natural comonoidal isomorhisms I?!! 1 and! A! B?!! (A B). Some remarks: First, it is not hard to see that the rst condition above is redundant, the comonoidal structure on! A being induced by the isomorhisms of the second condition. However, the rst condition is really the key oint here, as may be seen from several generalizations of this denition, to the intuitionistic case without nite roducts in [BBPH], and to the weakly distributive case, again without nite roducts, [BCS93]. The main oint here is that without roducts one relaces the second condition with the requirement that the cotrile! (and the natural transformations ; ) be comonoidal. And second, one ought not drown in the categorical terminology terms like \comonoidal" in essence refer to various coherence (or commutativity) conditions which may be looked u when needed. Readers not interested in coherence questions can follow the discussion by just noting the existence of aroriate mas, and believe that all the \right" diagrams will commute. They can regard it as somebody else's business to ensure that this is indeed the case. (In this vein we ought to cite [Bi94], where the denition above is imroved by requiring that the induced adjunction between C and C! is monoidal, in order to guarantee the soundness of term equalities.) In the mid-1980's, Girard studied coherence saces as a model of system F, and realized the following fact, which led directly to the creation of linear logic. Of course Girard did not ut the matter in these categorical terms at the time, but the essential content remains the same ordinary imlication factors through linear imlication via the cotrile!. (Another way of exressing this is to say that a model of full classical linear logic induces an interretation of the tyed -calculus.) Theorem 1 If C is a -autonomous category with nite roducts admitting Girard storage!, then the Kleisli category C! is cartesian closed. 4

5 This result is virtually folklore, but a roof may be found in [Se89]. One of the roblems with nding models of linear logic comes from the diculty of nding wellbehaved (in the above sense) cotriles on -autonomous categories. For examle, one of the main roblems with vector saces as a model of linear logic is the lack of any natural interretation of!. (We shall soon return to this oint, and indeed, in a sense this is the main oint of this aer.) This question seems closely bound u with questions of comleteness. Barr [B91] has shown how in certain cases one can get aroriate cotriles (via cofree coalgebras) from a subcategory of the Chu construction [B79]. One case where this route works out fairly naturally is if the -autonomous category is comact. The following is roved in [B91]. Theorem 2 Given a comlete comact closed category, one can construct cofree coalgebras by the formula! A = > A (A s A) (A s A s A) (where the tensors s are the symmetric tensor owers discussed below). We observe that a comact category which is comlete is also cocomlete, by self-duality. This theorem is the basis for Abramsky's modeling of the exonentials in interaction categories [Abr]. 3 Fock Sace In this section, we describe the basic construction of Fock sace; the exosition follows [Ge85] closely. Fock sace is one of the crucial constructions of quantum eld theory, and is designed to treat quantum systems of many identical, noninteracting articles. One of the crucial notions of quantum eld theory is that articles may be created or annihilated, and Fock sace will be equied with canonical oerators to model this henomenon. We should observe that generally hysicists consider Fock sace on Hilbert saces, but for the uroses of this discussion, vector saces are sucient. The states of a quantum system form a comlex vector sace. Given two such systems, they may be combined via the oerator. So if the rst system is in state v 1 and the second in state v 2, then the combined system would be in state v 1 v 2. Note that when we say we are combining the systems, we are only viewing them as a single system. We are not allowing interaction. So, if V reresents a one-article system, then V V reresents a two-article system. To model quantum eld theory, one wishes to consider a system of many articles. A natural candidate would be: C V (V V ) (V V V ) : : : However, this is not quite correct. We wish the articles of the system to be indistinguishable. This leads us to relace the abve tensor with either the symmetrized or antisymmetrized tensor. 3.1 Symmetric and Antisymmetric Tensors First, we introduce the symmetric tensor roduct of a vector sace with itself. Denition 3 Let A be a vector sace. coequalizer: Note that is the twist ma, a b 7! b a. The vector sace A s A is dened to be the following id A A??!??! A A?! A s A 5

6 This is the general denition of symmetrized tensor. It turns out that in categories of vector saces, this quotient is canonically isomorhic to the equalizer of these two mas, and that this equalizer is slit by the ma: a b 7! 1 (a b + b a) 2 We will frequently use this reresentation in the sequel. The n th symmetric ower is dened analogously. The vector sace Nn A has n! canonical endomorhisms, and the vector sace N n s A is the coequalizer of all of these. Again, it is isomorhic to the equalizer, and there is a slitting, as above. A good way to view the symmetrized tensor is to observe that the symmetric grou acts on the sace Nn A, and that the symmetrized tensor is the invariant subsace. As such, an aroriate notation for the symmetrized tensor is: N n A n! We will also freely use this reresentation, as well. The antisymmetric tensor will be dened in a similar fashion. Again, we rst dene the antisymmetric tensor of a vector sace B with itself. It will be denoted B A B. It is the coequalizer of the following diagram: id B B???!???! B B?! B A B? Here,? is the ma a b 7!?b a. Members of this sace can canonically be viewed as elements of the ordinary tensor roduct, of the form: x = 1 (a b? b a) 2 The n th antisymmetric ower is dened analogously, and is denoted N n A. 3.2 Bosonic and Fermionic Fock Sace Denition 4 Let B be a comlex vector sace. The symmetric Fock sace of B is the in- nite direct sum of the saces N n s B, where, when n is zero we use the comlex numbers. The antisymmetric Fock sace of B is the innite direct sum of the saces N n A B. F(B) = C B N n s B F A (B) = C B N n A B The articles of symmetric Fock sace are called bosons. Examles of such are hotons. Particles of antisymmetric Fock sace are called fermions. Examles of such are electrons and neutrinos. An interesting roerty of fermions is revealed in the above construction. Suose one had a system of two fermions, each in the same state v. This system would be reresented in fermionic Fock sace by the following exression: (0; 0; 1 (v v? v v); 0; 0; : : :) 2 This exression is clearly 0. This leads to the observation that one may not have two fermions existing in the same state. This is known as the Pauli exclusion rincile. 6

7 Given the nature of innite direct sums of vector saces, it is reasonable to think of elements of Fock sace as olynomials. Symmetrizing the tensor ensures that the variables commute. In the fermionic case, we get anticommuting variables. When we consider categories of normed vector saces, this analogy becomes even clearer. Polynomials are relaced by convergent ower series. We will show that the bosonic Fock sace of a Banach sace has a canonical reresentation as a sace of holomorhic functions. 3.3 Annihilation and Creation of Particles For ease of exosition, we consider the unsymmetrized Fock sace: U = C V (V V ) (V V V ) : : : The oerators we discuss are easily extended to the bosonic and fermionic cases. Given an arbitrary nonzero v 2 V, we dene a ma C v : U! U C v ((v 0 ; v 1 ; v 2 ; : : :)) = (0; v 0 v; 2v v 1 ; 3v v 2 ; : : :) In the above exression, v n 2 V n. This oerator is thought of as \creating a article in state v". Similarly one may annihilate articles. Choose an element of the dual sace V, and dene: A ((v 0 ; v 1 ; v 2 v 3 ; v 4 v 5 v 6 ; : : :)) = ((v 1 ); 2(v 2 )v 3 ; 3(v 4 )v 5 v 6 ; : : :) This oeration takes an n article state to an n? 1 article state and so on. The square roots in the above two exressions are \normalization" factors, and are added to make the desired equations hold. In this exression, each v i is an element of V. The equations exressing the interaction of the annihilation and creation oerators are to be found in [Ge85]. A more comlete discussion of the hysical meaning of Fock sace is contained in the enultimate section. 4 Fock Sace as a Model of Storage Now we check that the Fock sace actually satises all the roerties that need to be satised by an exonential tye, i.e. satises the roerties of [Se89], discussed in Section 2. This consists of two arts, verifying that Fock saces form a cotrile on the category of symmetric algebras and verifying the so-called exonential law, viz.! (A B) =! A! B. We check the former by dislaying a suitable adjunction in the next subsection. Note that in the category of vector saces, we have =. Proosition 3 Let A and B be vector saces. F(A B) = F(A) F(B) 7

8 Proof { For the uroses of this roof only, we will denote the n-th symmetric tensor by S n. Let V and W be vector saces. We construct a morhism S n (V ) S m (W )! S n+m (V W ) (v 1 s v 2 : : :) (w 1 s w 2 : : :) 7! (v 1 s v 2 : : :) s (w 1 s w 2 : : :) On the righthand side of the above exression, we are viewing each v i and w i as an element of V W. This lifts to an isomorhism: S n (V W ) = nm a=0 S a (V ) S n?a (W ) The inverse ma is dened as follows. We now denote vectors in V or W, when considered in V W, by (v i ; 0) and (0; w i ) resectively. (Remember that elements of this form generate V W.) (v 1 ; 0) s (v 2 ; 0) s : : : s (v i ; 0) s (0; w i+1 ) s : : : s (0; w n ) 7! (v 1 s v 2 s : : : s v i ) (w i+1 s : : : s w n ) The naturality of these mas, and the fact that they are inverse are left to the reader. We note at this oint that the symmetrization of the tensors is crucial for establishing that this is an isomorhism, as it was necessary to rearrange terms. Finally, it is straightforward to verify that this extends to an isomorhism of the desired form. Note that F(A B) is generated by ure tensors, i.e. exressions which are nonzero only on one term of the direct sum. We also note that the exression L n a=0 S a (V ) S n?a (W ) corresonds to the nite rank art of F(A) F(B). By the denition of the countable direct sum of vector saces, all elements of F(A) F(B) are contained in a nite rank iece. The above roof follows [FH], Aendix B, closely. In fact, this argument can be carried out at a categorical level, as is clear from the reviously mentioned theorem of Barr [B91]. The above is also roved in [BSZ92] for Hilbert saces. In the next section, we will see that Fock sace corresonds to the free symmetric algebra. It is also straightforward to verify that the isomorhism constructed in the above roof is in fact an algebra homomorhism. Now we consider the antisymmetrized Fock sace. We will show that one gets a model of the exonential tyes in the category of nite-dimensional vector saces using the antisymmetrized Fock sace. Proosition 4 If V is a nite-dimensional vector sace of dimension n, then F A (V ) is also a nite-dimensional vector sace with dimension 2 n. N Proof { Consider the vector sace A V with > n. We claim N that this sace is the zero vector sace. Since is adjoint to internal hom in VEC fd, the sace A V is isomorhic to the sace of comletely antisymmetric -linear mas from V to the scalars. Let f denote such a ma. Since V is only n-dimensional one cannot have linearly indeendent arguments to such mas. Thus one of the arguments must be a linear combination of the others. Thus on any arguments f becomes 8

9 a combination of terms of the form f(: : : ; u; : : :; u; : : :) where two arguments must be N equal. But antisymmetry makes such a term zero. Thus f is the zero vector and the vector sace A V is the one-oint sace. Thus the innite direct sum becomes a nite direct sum. Now consider n. It is clear that one can only choose N C n sets of linearly indeendent vectors given a basis. Thus the dimensionality of the sace A V is Cn and hence, adding the dimensions to get the dimension of the direct sum, we conclude that the dimension of F A (V ) is 2 n. The exonential law for the antisymmetric case can be argued similarly to the symmetric case. The detailed verication can be found in [BSZ92] in Section 3.2 on exonential laws, or in [FH] in aendix B. 4.1 Categories of algebras In this section we shall review some basic facts about categories of algebras, and see in articular how these t into the current context. (See [M71] for a review of the basic categorical facts, and [L65] for the basic algebra, for instance.) For reference, we do give the following denition here. Denition 5 A trile consists of a functor F : B?! B, together with natural transformations : id?! F and : F F?! F, such that F = F = id and F = F. One simle oint to recall is that categories of algebras and of coalgebras are closely connected to the existence of triles and cotriles. Given a trile F : B?! B, (with structure morhisms ; ), an F -algebra is an object B and a morhism h: F (B)?! B (subject to two commutativity conditions, corresonding to the associative and unit laws). (This notion can be generalized to arbitrary functors.) There is a canonical category of such algebras, the Eilenberg-Moore category C F, and an adjunction C?!? C F. Any adjunction canonically induces a trile, and this one canonically induces the original trile. The category of free F -algebras is the Kleisli category C F of the trile; again, there is a canonical adjunction C?!? C F which induces the original trile. Of course this dualizes for cotriles, with the corresonding notion of coalgebras. (We shall avoid the unleasant use of terms like \coeilenberg-moore" and \cokleisli".) Usually mathematicians have been more interested in the Eilenberg-Moore category of a trile (or cotrile) than in the Kleisli category; although there has been some interest in Kleisli categories recently (for instance in the context of linear logic, as mentioned earlier in this aer), we shall follow this tradition and shall work in Eilenberg-Moore categories. Indeed, it is there that we shall nd some of our models. One reason for this is quite ractical: it is often simler to recognize the category of algebras and so derive the trile (similarly, once one has a candidate for a trile, it is often simler to construct the category of algebras and verify the adjunction than to directly show the original functor is a trile). But there is another reason: we want to show that the Fock sace functor is a cotrile (so as to model! ), but on the categories of saces we consider, this is not the case rather it is a trile. By assing to the algebras, we can x this, because of the following fact: Fact Given an adjunction C F??!?? the comosite F U is a cotrile on D [BW]. So we obtain our model of! on the category of algebras. U D, F a U, the comosite UF is a trile on C, and so (dually) 9

10 4.1.1 Algebras for the symmetric (bosonic) Fock sace construction We begin with a more traditional notion of algebra; the connection between these comes via the trile induced by the adjunction given by the free algebra construction, as outlined above. In other words, the category of (traditional) algebras is equivalent to the category of U F algebras. Denition 6 An algebra A is a sace A equied with morhisms satisfying m: A A?! A and i: C?! A A A A m id - A A id m?? A A m - A C A i id - A A id i A C H HHHH = H m HHHHj =? A Here we are suosing the base eld to be C; otherwise relace C with the base eld k. If in addition the following diagram commutes, then the algebra A is said to be symmetric or commutative. ( is the canonical \twist" morhism.) m A A A An examle of such an algebra comes from the Fock sace, the multilication m is dened by \multilication of olynomials" in an evident manner. The use of the symmetrized tensor in the denition of Fock sace guarantees that this will indeed be a symmetric algebra, and it is standard that this descrition gives the free such algebra. In other words, we have the following roosition. Proosition 5 Given a vector sace B, the Fock sace F(B) canonically carries an algebra structure, and indeed is the free symmetric algebra generated by B. It follows from this that we have a cotrile on the category SALG of symmetric algebras, given by taking the Fock algebra on the underlying sace of an algebra. As the details of this are both standard and similar to the case of the antisymmetric Fock sace construction, which we shall discuss in more detail next, we shall leave the details here to the reader. 10

11 4.2 Algebras for the antisymmetric (fermionic) Fock sace construction Recall that we work in the context of nite dimensional vector saces VEC fd when considering the antisymmetric Fock construction. This category is self-dual, and is comact with biroducts: the roduct and coroduct coincide. This duality also imlies that a trile is also a cotrile, so we can model! in the category of saces. However, to show that the Fock sace construction denes a trile (or cotrile), it is again simler to consider the category of algebras. Although we are not familiar with any revious consideration of this category of algebras as such, the context is familiar: the antisymmetric Fock sace construction is usually called (when thought of as an algebra) the Grassman algebra, or the \alternating" or \exterior" algebra; the multilication dened on it is called the \wedge roduct" (a term derived from the usual notation for this roduct). Denition 7 An alternating algebra A is a graded algebra A (with unit) whose multilication ma satises the roerty that, if x; y are of degree m; n resectively, then xy = (?1) nm yx (which by the grading must be of degree n + m). Note that the unit must be of degree 0. Morhisms of alternating algebras are just homomorhisms as algebras. Proosition 6 There is a canonical alternating algebra structure on F A (V ), for any nite dimensional vector sace V. The antisymmetric Fock construction is left adjoint to the forgetful functor F A U: VEC fd???!??? U AALG, where AALG is the category of alternating algebras. As a consequence, F A denes a trile (and so cotrile) on VEC fd. Proof { (Sketch) The multilication on F A (V ) is the standard \wedge" roduct [L65], which to elements x 1 A : : : A x n ; y 1 A : : : A y m gives the roduct x 1 A : : : A x n A y 1 A : : : A y m. Here x A y means the equivalence class of x y in A A A. (Essentially this is the same \multilication of ower series" we had in the symmetric case, with the alternating roduct used in lace of the usual tensor.) For a vector sace V, dene : V?! UF A (V ) as the canonical injection. Given an alternating algebra A, dene : F A (UA)?! A by \adding the terms of the series": hx 0 ; x 1 ; x 1 2 A x 2 2; : : :i 7! i(x 0 ) + x 1 + m(x 1 2; x 2 2) +, where i; m are the algebra mas. To verify that we have an adjunction we must show the following commute: F A (V ) F A () - F A (UF A @@R? F A (V ) FA UA U - UF UA 11

12 The second diagram is obvious; to verify the rst, notice that F A ((x)) mas hx 0 ; x 1 ; x 1 2 A x 2; : : :i 2 7! hx 0; h0; x 1 ; 0; : : :i; h0; 0; x 1; 0; : : :i 2 A h0; 0; x 2 2 ; 0; : : :i; and it is clear that \adding u this series" just returns the original term.. i It now follows that we can model! in VEC fd with F A, via the formula! V = (F A (V? ))?. In summary, we have the following theorem: Theorem 7 In the category of symmetric algebras, we have a model of the fragment! ; ; &. In the category VEC fd, we have a model of full roositional classical linear logic. We mention that while VEC fd models the full roositional calculus, it is somewhat degenerate being a comact category. We now consider Fock sace on normed vector saces. While we lose some of the exressive ower in such models, (in articular, we are no longer able to model contraction) Fock sace is articularly interesting in such categories. We will see that it has a canonical reresentation as a sace of holomorhic functions. 5 Normed Vector Saces Vector saces are, in some sense, intrinsically nitary structures. Every vector is a nite sum of multiles of basis vectors, and one is only allowed to take nite sums of arbitrary vectors. It seems reasonable that to model! and?, one should be able to take innite sums of vectors, thereby caturing the idea of innitely renewable resource. However, to do this, one needs a notion of convergence. And to dene convergence, one needs a notion of norm. Once a sace is normed, then it is ossible to dene limits and Cauchy sequences, and so on. Normed vector saces, which are the rincial objects of study in functional analysis, should be considered as the meeting ground of concets from linear algebra and analysis. They are also an ideal lace to model linear logic. We will now briey review the basic concets of the subject. In this aer, we will focus on Banach saces. Much of the discussion easily lifts to Hilbert saces. We will introduce Banach saces in detail, and refer the reader to [KR83] for a discussion of Hilbert saces. Henceforth all vector saces are assumed to be over the comlex numbers and are allowed to be innite-dimensional. We will use Greek letters for comlex numbers and lower-case Latin letters from the end of the alhabet for vectors. Denition 8 A norm on a vector sace V is a function, usually written k k, from V to R, the real numbers, which satises 1. k v k 0 for all v 2 V, 2. k v k = 0 if and only if v = 0, 3. k v k = j jk v k, 4. k v + w k k v k + k w k. 12

13 For nite dimensional vector saces the norm usually used is the familiar Euclidean norm. As soon as one has a norm one obtains a metric by the equation d(u; v) = k u? v k. It turns out that the saces that are comlete with resect to this metric lay a central role in functional analysis. Denition 9 A Banach sace is a comlete, normed vector sace. Examle 1 Consider the sace of sequences of comlex numbers. We write a for such a sequence, a = fa n g 1 n=1 and we write k a k 1 for the suremum of the j a i j. l 1 = fa : k a k 1 < 1g This is a Banach sace with k a k 1 as the norm. Another norm is obtained on sequences as follows. Dene: Then let: More generally, if 0, we may dene: k a k 1 = 1 n=1j a i j l 1 = fa : k a k 1 < 1g l = fa : k a k ( 1 n=1j a j ) 1= < 1g All of these will be examles of Banach saces. Furthermore, these can be dened not only for sequences of comlex numbers, but for sequences obtained from any Banach sace. The following theorem shows one common way in which Banach saces arise. First we need a denition. Denition 10 Suose that B 1 ; B 2 are Banach saces and that T is a linear ma from B 1 to B 2. k T x k We say that T is bounded if su x6=0 exists. We dene the norm of T, written k T k, to k x k be this number. If T is indeed bounded, then a standard argument [KR83], establishes Lemma 8 su jjxjj=1 k T x k = k T k: Thus one can use vectors of unit norm to calculate the norm of a linear function rather than having to look for the su over all nonzero vectors. Linear mas from a Banach sace to itself are traditionally called oerators, and the norm of such mas is called the oerator norm. Since a Banach sace is also a metric sace under the induced metric described above, one wishes to characterize which linear mas are also continuous. In this regard, we have the following result. Lemma 9 A linear ma from f : A! B is continuous if and only if it is bounded. The following theorem shows that the category of Banach saces and bounded linear mas is enriched over itself. Theorem 10 If A is a normed vector sace and B is a Banach sace then the sace of bounded linear mas with the norm above is a Banach sace. 13

14 We will denote this sace A? B. There are several ossible categories of interest with Banach saces as the objects. The most obvious one is the category with bounded linear mas as the morhisms. However, it turns out that the category with contractive mas 2 is of greater interest and has nicer categorical roerties. Denition 11 A contractive ma, T, from A to B is a bounded linear ma satisfying the condition, k T x k k x k. Equivalently, the contractive mas are those of norm less than or equal to 1. We will write BAN CON for the category of Banach saces and contractive mas. 5.1 Monoidal Structure of BAN CON We rst oint out that BAN CON has a canonical symmetric monoidal closed structure. We begin by constructing a tensor roduct. Let A and B be objects in BAN CON : form the tensor of A and B, A C B, as comlex vector saces. We rst dene a artial norm for elements of the form a b by the equation: k a b k = k a kk b k We would like to extend this artial norm to a norm on all of A C B. Such a norm is called a cross norm. It turns out that there are many such cross norms, a number of which were discovered by Grothendieck. The one we will use in this aer is called the rojective cross norm. It is in some sense the least such. A detailed discussion of these issues is contained in [T79]. The rojective cross norm is dened for an arbitrary element, x, of A C B by the following formula: k x k = inffk a kk b k such that x = a bg One can verify that this is in fact a cross norm on A C B. Now, the resulting normed sace will not be comlete in general, so one obtains a Banach sace by comleting it. This will act as the tensor roduct in the category BAN CON. It will be denoted simly by A B. Furthermore, we have the following adjunction. Lemma 11 The functor B ( ) is left adjoint to B? ( ). Corollary 12 BAN CON is a symmetric monoidal closed category. As such, BAN CON is a model of (at least) the multilicative fragment of intuitionistic linear logic. 5.2 Comleteness Proerties of BAN CON We begin by constructing coroducts. Denition 12 Let A and B be Banach saces. The direct sum, A B, is the Cartesian roduct equied with the norm k a b k = k a k + k b k. Then we have the distributivity roerty of over. 2 Strictly seaking, they should be called \non-exansive" mas. 14

15 Proosition 13 A (B B 0 ) = (A B) (A B 0 ). We now discuss nite roducts. Denition 13 The roduct of two Banach saces, A B, has as its underlying sace A B, but now with norm given by: k a b k = maxfk a k; k b kg As a category of vector saces, BAN CON is fairly unique in this resect. While most such categories model the additive fragment of linear logic, they invariably equate the two connectives, since nite roducts and coroducts coincide. In other words, BAN CON does not share the familiar roerty of being an additive category. We now resent countably innite roducts and coroducts. Denition 14 Let fa i g 1 be a sequence of Banach saces. Dene (A i=1 i) to be those sequences which converge in the l 1 norm, i.e. bounded sequences equied with the obvious norm. Dene (A i ) to be all sequences which converge in the l 1 norm. This gives countable roducts and coroducts in BAN CON. Similar constructions can be alied for uncountable roducts and coroducts. Equalizers in BAN CON corresond to equalizers in the underlying category of vector saces. The fact that bounded mas are continuous imlies that the subsace will be comlete. Coequalizers are obtained as a quotient, with the induced norm being the inmum of the norms of the elements of the equivalence class. See [C90] for a discussion of quotients of Banach saces. Theorem 14 BAN CON is comlete and cocomlete. 6 BAN CON as a Model of Weakening In the next section, we will show that the Fock sace of a Banach sace has a canonical interretation as a sace of holomorhic functions. In this section, we exlore the nature of BAN CON as a model of linear logic. We obtain a somewhat weaker model for the following reason. In this category, as we have reviously observed, roducts and coroducts do not coincide. Thus, the isomorhism:! A! B =! (A B) is not useful for modelling the additive fragment. We obtain instead a weakening cotrile in the sense of Jacobs [J93]. Jacobs denotes a weakening cotrile by! w. Such a cotrile satises all of the axioms of modelling!, excet that the coalgebras will not have the comonoid structure necessary to model contraction. Thus, we have syntax of the following form:?; ` A?;! w B ` A W e?; B ` A!?;! w B ` A Der w? ` A!? `! A Sto w w One models these roof rules as in linear logic, following [Se89]. We oint out that the ma: 15

16 ! A! B?!! (A B) necessary to model storage is indeed a contraction. This ma is obtained as! A! B =! (A B)!!! (A B) =! (! A! B)!! (A B) It is shown in an aendix that this ma is contractive. Theorem 15 In the category BAN CON, we obtain a model of the fragment! w ; ; &. Remark 16 We wish to oint out an error in an earlier draft of this aer [BPS]. In that aer, it is stated that the Fock construction is functorial on the larger category of Banach saces and bounded linear mas. In fact, when one alies the Fock construction to a ma of norm greater than 1, one might obtain a divergent exression. Thus, we are forced to work in the smaller category of contractions. 7 The Holomorhic-Function Reresentation of Fock Sace It has been observed by a number of eole that the free symmetric algebra rovides a model of the exonential tye. But the observation that this construction corresonds to bosonic Fock sace allows us to relate results in quantum hysics to the model theory of linear logic. In this section, we resent one such relationshi. The symmetrized Fock sace on a Banach sace B, turns out to be a sace of holomorhic functions (analytic functions) on B, roerly dened. This hints at ossible deeer connections between analyticity and comutability which need to be exlored. The ideas here stem from early work by Bargmann [Ba61] on Hilbert saces of analytic functions in quantum mechanics. This was extended by Segal [S62, BSZ92] to quantum eld theory and Segal's extension was used by Ashtekar and Magnon [AM-A80] to develo quantum eld theory in curved sacetimes. (A brief summary of the ideas is contained in an aendix to [P80] and in [P79].) The latter work involved making sense of the familiar Cauchy-Riemann conditions on innite-dimensional saces. We quickly recaitulate the basic notion of analytic function in terms of one comlex variable before resenting the innite-dimensional case. A very good elementary reference is Comlex Analysis by Ahlfors [Ah66]. Given the comlex lane, C, one can dene functions from C to C. Let z be a comlex variable; we can think of it as x + iy and thus one can think of functions from C to C as functions from R 2 to R 2. An analytic or holomorhic function is one that is everywhere dierentiable. In the notion of dierentiation, the limit being comuted, viz. f(x + h)? f(x) lim h!0 h allows h to be an arbitrary comlex number and hence this limit is required to exist no matter in what direction h aroaches 0. This much more stringent requirement makes comlex dierentiability much stronger than the usual notion of dierentiability. If a comlex function is dierentiable at a oint it can be reresented by a convergent ower series in a suitable oen region about the 16

17 oint. If one uses the fact that h can aroach zero along either axis one can derive the Cauchy- Riemann equations for a comlex valued function f = u(x; y) + iv(x; y) of the comlex variable z = x @x : What is remarkable about comlex functions is that this denition of analyticity yields the result that a comlex-analytic function can be exressed by a convergent ower-series in a region of the comlex lane. This is remarkable because only one derivative is involved in the Cauchy- Riemann equations whereas the statement that a ower-series reresentation exists is stronger, for real-valued functions, even than requiring innite dierentiability. In real analysis one has examles of functions that are innitely dierentiable at a oint, but do not have a ower series reresentation in any neighbourhood of that oint. A function may have a ower series reresentation that is valid everywhere, a so-called entire holomorhic function; the comlex exonential function is an examle. There is a formal ersective, due to Wierstrass, that is rather more illuminating. Think of a comlex variable z = x + iy and its conjugate z = x? iy as being, formally, indeendent variables. A function could deend on z and on its comlex conjugate, z, for examle, the function that mas each z to zz + izz. An analytic or holomorhic function is one which has no deendance on z. This is exressed formally by df=dz = 0. When exressed in terms of the real and imaginary arts of f and z, this equation becomes the familiar Cauchy-Riemann equations. Thus this reinforces the view that a holomorhic function is roerly thought of as a single comlex-valued function of a single variable rather than as two real-valued functions of two real variables. The theory of functions of nitely many comlex variables is a nontrivial extension of the theory of functions of a single comlex variable. Entirely new henomena occur, which have no analogues in the theory of a single comlex variable. An excellent recent text is the three volume treatise by Gunning [Gu90]. For our uroses we need only the barest beginnings of the theory. Given C n, we can have functions from C n to C. One can introduce comlex coordinates on C n, z 1 ; : : :; z n. One can dene a holomorhic function here as one having a convergent ower-series exansion in z 1 ; : : :; z n. The key lemma that allows one to mimic some of the results of the one-dimensional case is Osgood's lemma 3. Lemma 17 If a comlex-valued function is continuous in an oen subset D of C n and is holomorhic in each variable seerately, then it is holomorhic in D. From this one can conclude that a holomorhic function in n variables satises the i = 0. One is free to take either one of (a) satisfying Cauchy-Riemann equations or (b) having convergent ower-series reresentations as the denition of holomorhicity. Now we describe how to dene holomorhic functions on innite-dimensional, comlex, Banach saces. The basic intuition may be summarized thus. One starts with subsaces of nite codimension. Thus the quotient saces are isomorhic to some C n. One can dene what is meant by a holomorhic function on these quotient saces as in the receding aragrah. By comosing a holomorhic function with the canonical surjection from the original Banach sace to the quotient sace we get a function on the original Banach sace. These functions can all be taken to be holomorhic. 3 There is a considerably harder theorem, called Hartog's theorem, which dros the requirement of continuity. 17

18 B? B= = C n - R C Intuitively these are the functions that are constant along all but nitely many directions, and holomorhic in the directions along which they do vary. These functions are called cylindric holomorhic functions. Because the sequence of coecients of a ower-series is absolutely convergent, we can dene an l 1 norm on these functions in terms of the ower-series. Finally the collection of all holomorhic funcitons is dened by taking the l 1 -norm comletion of the cylindric holomorhic functions. Given a Banach sace B, let U be a subsace with nite codimension n, i.e. the quotient sace B=U is an n comlex-dimensional vector sace. The sace B=U is isomorhic to C n. Let : B=U! C n be an isomorhism; such a ma denes a choice of comlex coordinates on B=U. Let U be the canonical surjection from B to B=U. Denition 1 A cylindric holomorhic function on B is a function of the form f U, where U; U and are as above and f is a holomorhic function from C n to C. We need to argue that the choice of coordinates does not make a real dierence. Of course which functions get called holomorhic does deend on the choice of coordinates, but the sace of holomorhic functions has the same structure 4. Suose that U and V are both subsaces of B and that U is included in V. Suose that both these saces are saces of nite codimension, say n and m resectively. Clearly n m. Now we have a linear ma UV : B=U! B=V given by x + U 7! x + V ; clearly this is a surjection. Now given coordinate functions : B=U! C n and : B=V! C m we can dene a function : C n! C m, given by UV?1, which makes the diagram commute. Thus we do not have to imose \coherence" conditions on the choice of coordinates, we can always translate back and forth between dierent coordinate systems. We will suress these translation functions in what follows and assume that the coordinates have been serendiitously chosen to make the form of the functions simle. In other words, we can x a family of subsaces fw n jn 2 Ng with W n having codimension n and W n+1 W n. The coordinates can be chosen so that the sace B=W n has coordinates z 1 ; : : :; z n. Suose that f is a cylindric holomorhic function on B. This means that there is a nitecodimensional subsace W, and a holomorhic function f W, from W to C, such that f = f W W. The function f W regarded as a function of n comlex variables has a ower-series reresentation f W (z 1 ; : : :; z n ) = a i1 :::i k z i 1 1 : : : zi k k and furthermore we have the following convergence condition ja i1 :::i k j < 1: Thus with each such cylindric holomorhic function we can dene the sum of the absolute values of the coecients in the ower-series exansion as the norm of the function. Viewing the sequences of coecients as the elements of a comlex vector sace, we have an l 1 norm. We write k f k for this norm of a cylindric holomorhic function. 4 This haens even in the one dimensional case. The function z is considered anti-holomorhic traditionally, but one could have called it holomorhic by interchanging the role of z and z. 18

19 Denition 2 An l 1 -holomorhic function on B is the limit of a sequence of cylindric holomorhic function in the above norm. The l 1 emhasizes that the holomorhic functions are obtained by a articular norm comletion. In the corresonding theory of holomorhic functions on Hilbert saces, one uses the inner-roduct to dene olynomials and then erform a comletion in the L 2 norm. A key dierence is that our norm is dened on the sequence of coecients whereas in the Hilbert sace case, one uses the L 2 norm which is dened in terms of integration. In the resulting Banach sace there are several formal entities that were adjoined as art of the norm-comletion rocess. We need to discuss in what sense these formally-dened entities can be regarded as bona-de functions. Let W 1 ; : : :; W r ; : : : be an innite sequence of subsaces of B, each embedded in the revious. Assume, in addition, that all these saces have nite codimension. Now assume that there is a sequence of cylindric holomorhic functions, f n, on B obtained from a holomorhic function, f (n) on each of the quotient saces B=W i. Finally, assume that the sequence k f n k of (real) numbers is convergent. Such a sequence of cylindric holomorhic functions denes a holomorhic function on B. We call this function f. We need to exhibit f as a ma from B to C. Accordingly, let x be a oint of B. For each of the functions f n we have jf n (x)j k f n k. Since the sequence of norms converges we have the sequence f n (x) converges absolutely and hence converges. Thus the function f qua function is given at each x of B by lim n!1 f n (x). However, in order to use the word \function" we need to show that the ower-series has a domain of convergence. Unfortunately, it may not have a non-trivial domain of convergence but, in a sense to be made recise, it comes close to having a non-trivial domain of convergence. The ower-series reresentation of the function f is given as follows. It deends, in general, on innitely many variables but each term in the ower series will be a monomial in nitely many variables. Consider the coecient of z j 1 i 1 : : :z j k ik in the exansion of f. In all but nitely many of the f n all the indicated variables will aear in their ower-series exansions. Consider the coecients of this term in each ower series; this forms a sequence of comlex numbers n where n is 0 if there is no such term in the exansion of f n. Since j n j k f n k the sequence n converges absolutely and hence converges to, say,. This is the coecient of z j 1 i 1 : : : z j k i k in the ower-series exansion of f. Consider the coordinates z 1 ; : : :; z n. This denes an n-dimensional subsace of the Banach sace, which we call U n. Now consider the ower-series for f. It denes a family of holomorhic functions f n where f n is dened on the subsace U n and is obtained by retaining only those terms in the ower-series exansion of f which involve variables among z 1 ; : : :; z n. These are analytic functions on the U n and, as such, have non-trivial domains of convergence. However, as n increases the radii of convergence could tend to 0. So we have the slightly weaker statement than the usual nite-dimensional notion; instead of having a non-zero radius of convergence in the Banach sace we have a non-zero radius of convergence on every nite-dimensional subsace. If one uses entire functions, rather than analytic functions, at the starting oint of the construction, then one can show that the resulting functions are entire; see age 67, theorem 1.13, of the book by Baez, Segal and Zhou [BSZ92]. Unfortunately when using the reresentation of elements of Fock sace one may carry out simle oerations that do not roduce entire functions so we cannot just choose to work with entire functions. Nevertheless, many common functions, most notably the exonential, are entire. Given a bona de holomorhic function one can exress it as a ower series. The coecients are calculated in the usual way, viz. by using Taylor's theorem f = n i1 +:::+i k =n 1 i 1! : : :i k! i 1 z1 : : :@ i k zk z i 1 1 : : :zi k k