Fo ck Space A Mo del of Linear Exp onential Typ es

y

Prakash Panangaden RF Blute

Scho ol of Computer Science Department of Mathematics

McGill University University of Ottawa

Montreal Canada HA A Ottawa Canada KN N

prakashcsmcgillca rblutemathstatuottawaca

z

RAG Seely

Department of Mathematics

McGill University

Montreal Canada HA K

ragsmathmcgillca

Abstract

It has b een observed by several p eople that in certain contexts the free symmetric algebra

construction can provide a mo del of the linear mo dality This construction arose indep en

dently in quantum physics where it is considered as a canonical mo del of quantum eld theory

In this context the construction is known as bosonic Fock space Fo ck space is used to analyze

such quantum phenomena as the annihilation and creation of particles There is a strong intu

itive connection to the principle of renewable resource which is the philosophical interpretation

of the linear mo dalities

In this pap er we examine Fo ck space in several categories of vector spaces We rst consider

vector spaces where the Fo ck construction induces a mo del of the ; ; fragment in the cat

egory of symmetric algebras When considering Banach spaces the Fo ck construction provides

a mo del of a weakening cotriple in the sense of Jacobs While the mo dels so obtained mo del a

smaller fragment it is closer in practice to the structures considered by physicists In this case

Fo ck space has a natural interpretation as a space of holomorphic functions This suggests that

the nonlinear functions we arrive at via Fo ck space are not merely continuous but analytic

Finally we also consider fermionic Fo ck space which corresp onds algebraically to skew sym

metric algebras By considering fermionic Fo ck space on the category of nitedimensional vec

tor spaces we obtain a mo del of full prop ositional linear logic although the mo del is somewhat

degenerate in that certain connectives are equated

Intro duction

Linear logic was intro duced by Girard G as a consequence of his decomp osition of the traditional

connectives of logic into more primitive connectives The resulting logic is more resource sensitive

this is achieved by placing strict control over the structural rules of contraction and weakening

Research supp orted in part by FCAR of Queb ec

y

Research supp orted in part by an NSERC Research Grant

z

Research supp orted in part by an NSERC Research Grant and by FCAR of Queb ec

intro ducing a new mo dal op erator of course denoted to indicate when a formula may b e

used in a resourceinsensitive mannerie when a resource is renewable Without the op erator

the essence of linear logic is carried by the multiplicative connectives at its most basic level linear

logic is a logic of monoidalclosed categories in much the same way that intuitionistic logic is a

logic of cartesianclosed categories In mo delling linear logic one b egins with a monoidalclosed

category and then adds appropriate structure to mo del linear logics additional features To mo del

linear negation one passes to the autonomous categories of Barr B To mo del the additive

connectives one then adds pro ducts and copro ducts Finally to mo del the exp onentials and

so regain the expressive strength of traditional logic one adds a triple and cotriple satisfying

prop erties to b e outlined b elow This program was rst outlined by Seely in Se

Linear logic b ears strong resemblance to linear algebra from which it derives its name but

one signicant dierence is the diculty in mo delling The category of vector spaces over an

arbitrary eld is a symmetric monoidal closed category indeed in some sense the prototypical

monoidal category and as such provides a mo del of the intuitionistic variant of multiplicative

linear logic Furthermore this category has nite pro ducts and copro ducts with which to mo del

the additive connectives It thus makes sense to lo ok for mo dels of various fragments of linear

logic in categories of vector spaces However mo delling the exp onentials is more problematic It is

the primary purp ose of this pap er to present metho ds of mo delling exp onential typ es in categories

arising from linear algebra We study mo dels of the exp onential connectives in categories of linear

spaces which have monoidal but generally not monoidalclosed structure We shall also include

a mo del in nitedimensional vector spaces which is closed

The construction which will b e used to mo del the exp onential formulas although standard in

algebra arose indep endently in quantum eld theory and is known as Fock space It was designed

as a framework in which to consider manyparticle states The key p oint of departure for quantum

eld theory was the realization that elementary particles are created and destoyed in physical

pro cesses and that the mathematical formalism of ordinary quantum mechanics needs to b e revised

to take this into account The physical intuitions b ehind the Fo ck construction will also b e familiar

to mathematicians in that it corresp onds to the free symmetric algebra on a space As such it

induces a pair of adjoint functors and hence a cotriple in the algebra category It is this cotriple

which will b e used to mo del It should b e noted that this category of algebras inherits the

monoidal structure from the underlying category of spaces but there is no hop e that this category

could have a monoidalclosed structure

While Fo ck space has an abstract representation in terms of an innite direct sum physicists

such as Ashtekar Bargmann Segal and others see AMA Ba S have analyzed concrete

representations of Fo ck space as certain classes of holomorphic functions on the base space Thus

these mo dels further the intuition that the exp onentials corresp ond to the analytic prop erties of

the space In fact there is a clear sense in which morphisms in the Kleisli category for the cotriple

can b e viewed as generalized holomorphic functions Thus there should b e an analogy to coherence

spaces where the Kleisli category corresp onds to the stable maps

Fo ck space also has two additional features which corresp ond to additional structure not ex

pressible in the syntax of linear logic These are the annihilation and creation op erators which

are used to mo del the annihilation and creation of particles in a eld These may give a tighter

control of resources not expressible in the pure linear logic Thus these mo dels may b e closer to

the b ounded linear logic of Girard Scedrov and Scott GSS

A p ossible application of this work is that the rened connectives of linear logic may lend

insight into certain asp ects of quantum eld theory For example there are two distinct metho ds of

combining particle states One can sup erimp ose two states onto a single particle or one can have

two particles co existing The former seems to corresp ond to additive conjunction and the latter

to the multiplicative This physical imagery is missing in quantum mechanics which was sp ecially

designed to handle a single particle it only shows up in quantum eld theory

In this pap er we b egin by reviewing the categorical structure necessary to mo del linear logic

and sp ecically exp onential typ es We then describ e the Fo ck construction on vector spaces and

explain the prop erties of the resulting mo del We next consider Fo ck space on normed vector

spaces While the mo del so obtained has weaker prop erties this case is closer to that considered by

physicists In fact in this case the Fo ck construction gives a mo del of a weakening cotriple in the

sense of Jacobs J We next describ e the interpretation of Fo ck space as a space of Holomorphic

functions Finally the physical meaning of the Fo ck construction is discussed

We wish to p oint out that this pap er corrects an error in an earlier draft BPS This is discussed

in section

Linear Logic and Monoidal Categories

We shall b egin with a few preliminaries concerning linear logic We shall not repro duce the formal

syntax of linear logic nor the usual discussion of its intuitive interpretation or utilityfor this the

reader is referred to the standard references such as G We do recall Se that a categorical

semantics for linear logic may b e based on Barrs notion of autonomous categories B If only

to establish notation here is the denition

Denition A category C is autonomous if it satises the fol lowing

C is symmetric monoidal closed that is C has a tensor product 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

op

C has a dualizing object that is the functor C C dened by A A is an

involution viz the canonical morphism A A is an isomorphism

In addition various coherence conditions must holda go o d account of these may b e found in

MOM Coherence theorems may b e found in BCST Bl An equivalent characterization of

autonomous categories is given in CS based on the notion of weakly distributive categories

That characterization is useful in contexts where it is easier to see how to mo del the tensor

...

..

..

.. . .

.. . .

. . . .

. .

. . .

. . .

. . .

. .

.

. . .

. . .

. . .

. .

. . .

. . .

. .

. .

......

. .

. .

. .

. .

. .

.

.

..

and linear negation and the coherence conditions may b e expressed in terms of those .. the par

.

op erations

The structure of a autonomous category mo dels the evident ep onymous structure of linear

logic the categorical tensor is the linear multiplicative and the internal hom is linear

..

..

..

... . .

.. . .

. . . .

. . .

. .

. .

. . .

. . .

. . .

.

. . .

. . .

. .

. .

. . .

. . .

......

. .

.

. .

. .

. .

. .

.

.

..

or equivalently is the dual of ... implication The dualizing ob ject is the unit for linear par

.

the unit I for the tensor

There are a numb er of variants of linear logic whose categorical semantics is based on this

First is full classical linear logic which includes the additive op erations These corresp ond to

requiring that the category C have pro ducts and copro ducts If C is autonomous one of these

will imply the other by de Morgan duality There is also Girards notion of intuitionistic linear

..

..

..

.. . .

.. . .

. . . .

. . .

. .

. .

. . .

. . .

. . .

.

. . . 1

. . .

. .

. .

. . .

. . .

......

. .

.

. .

. .

. .

. .

.

.

..

Here we shall try to .. In other pap ers we have used the notation > for the unit for and  instead of

.

avoid controversy by using notation traditional in the context of Banach spaces and by generally ignoring the par

So in this pap er  means direct sum which coincides with Girards notation We use  for cartesian pro duct

corresp onding to Girards And we shall use the usual notation for the appropriate spaces when referring to the

units

logic GL which omits linear negation and parthis corresp onds to merely requiring that C b e

autonomous that is to say symmetric monoidal closed with or without pro ducts and copro ducts

dep ending on whether or not the additives are wanted There is an intermediate notion full

intuitionistic linear logic due to de Paiva dP in which the morphism A A need not

b e an isomorphism And as mentioned ab ove there is the notion of weakly distributive category

CS BCST where negation and internal hom are not required

One imp ortant class of autonomous categories are the compact categories KL where the

A B These categories form the basis for Abramskys interaction tensor is selfdual A B

categories Abr

In this pap er we shall mo del various fragments of linear logic we shall describ e the fragments

in terms of the categorical structure present without explicitly identifying the fragments

Finally in order to b e able to recapture the full strength of classical or intuitionistic logic

one must add the exp onential and its de Morgan dual All our structures will mo del

We saw in Se that this amounts to the following

Denition A monoidal category C with nite products admits Girard storage if there is a

A A

A satisfying the fol lowing A cotriple C C with the usual structure maps A

for each object A C A carries natural ly the structure of a cocommutative comonoid

e d

A A

A A A and the coalgebra maps are comonoid maps and

there are natural comonoidal isomorphisms

I and A B A B

Some remarks First it is not hard to see that the rst condition ab ove is redundant the

comonoidal structure on A b eing induced by the isomorphisms of the second condition However

the rst condition is really the key p oint here as may b e seen from several generalizations of

this denition to the intuitionistic case without nite pro ducts in BBPH and to the weakly

distributive case again without nite pro ducts BCS The main p oint here is that without

pro ducts one replaces the second condition with the requirement that the cotriple and the

natural transformations b e comonoidal And second one ought not drown in the categorical

terminologyterms like comonoidal in essence refer to various coherence or commutativity

conditions which may b e lo oked up when needed Readers not interested in coherence questions

can follow the discussion by just noting the existence of appropriate maps and b elieve that all the

right diagrams will commute They can regard it as someb o dy elses business to ensure that this

is indeed the case

In this vein we ought to cite Bi where the denition ab ove is improved by requiring that

the induced adjunction b etween C and C is monoidal in order to guarantee the soundness of term

equalities

In the mids Girard studied coherence spaces as a mo del of system F and realized the

following fact which led directly to the creation of linear logic Of course Girard did not put the

matter in these categorical terms at the time but the essential content remains the sameordinary

implication factors through linear implication via the cotriple Another way of expressing this

is to say that a mo del of full classical linear logic induces an interpretation of the typ ed calculus

Theorem If C is a autonomous category with nite products admitting Girard storage then

the Kleisli category C is cartesian closed

This result is virtually folklore but a pro of may b e found in Se

One of the problems with nding mo dels of linear logic comes from the diculty of nding well

b ehaved in the ab ove sense cotriples on autonomous categories For example one of the main

problems with vector spaces as a mo del of linear logic is the lack of any natural interpretation of

We shall so on return to this p oint and indeed in a sense this is the main p oint of this pap er

This question seems closely b ound up with questions of completeness Barr B has shown how

in certain cases one can get appropriate cotriples via cofree coalgebras from a sub category of the

Chu construction B One case where this route works out fairly naturally is if the autonomous

category is compact The following is proved in B

Theorem Given a complete compact closed category one can construct cofree coalgebras by the

formula

A A A A A A A

s s s

where the tensors are the symmetric tensor powers discussed below

s

We observe that a compact category which is complete is also co complete by selfduality This

theorem is the basis for Abramskys mo deling of the exp onentials in interaction categories Abr

Fo ck Space

In this section we describ e the basic construction of Fo ck space the exp osition follows Ge

closely Fo ck space is one of the crucial constructions of quantum eld theory and is designed

to treat quantum systems of many identical noninteracting particles One of the crucial notions

of quantum eld theory is that particles may b e created or annihilated and Fo ck space will b e

equipp ed with canonical op erators to mo del this phenomenon We should observe that generally

physicists consider Fo ck space on Hilbert spaces but for the purp oses of this discussion vector

spaces are sucient

The states of a quantum system form a complex vector space Given two such systems they

may b e combined via the op erator So if the rst system is in state v and the second in state

v then the combined system would b e in state v v 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

represents a oneparticle system then V V represents a twoparticle system To mo del quantum

eld theory one wishes to consider a system of many particles A natural candidate would b e

C V V V V V V

However this is not quite correct We wish the particles of the system to b e indistinguishable

This leads us to replace the abve tensor with either the symmetrized or antisymmetrized tensor

Symmetric and Antisymmetric Tensors

First we intro duce the symmetric tensor pro duct of a vector space with itself

Denition Let A be a vector space The vector space A A is dened to be the fol lowing

s

coequalizer

id

A A A A A A

s

Note that is the twist map a b b a

This is the general denition of symmetrized tensor It turns out that in categories of vector

spaces this quotient is canonically isomorphic to the equalizer of these two maps and that this

equalizer is split by the map

a b b a a b

We will frequently use this representation in the sequel

N

n

th

The n symmetric p ower is dened analogously The vector space A has n canonical

N

n

endomorphisms and the vector space A is the co equalizer of all of these Again it is isomorphic

s

to the equalizer and there is a splitting as ab ove A go o d way to view the symmetrized tensor is

N

n

to observe that the symmetric group acts on the space A and that the symmetrized tensor is

the invariant subspace As such an appropriate notation for the symmetrized tensor is

N

n

A

n

We will also freely use this representation as well

The antisymmetric tensor will b e dened in a similar fashion Again we rst dene the anti

symmetric tensor of a vector space B with itself It will b e denoted B B It is the co equalizer

A

of the following diagram

id

B B B B B B

A

Here is the map a b b a

Memb ers of this space can canonically b e viewed as elements of the ordinary tensor pro duct of

the form

x a b b a

N

n

th

The n antisymmetric p ower is dened analogously and is denoted

A

Bosonic and Fermionic Fo ck Space

Denition Let B be a complex vector space The symmetric Fo ck space of B is the in

N

n

B where when n is zero we use the complex numbers The nite direct sum of the spaces

s

N

n

antisymmetric Fo ck space of B is the innite direct sum of the spaces B

A

N

n

F B C B B

s

N

n

F B C B B

A

A

The particles of symmetric Fo ck space are called bosons Examples of such are photons Particles

of antisymmetric Fo ck space are called fermions Examples of such are electrons and neutrinos

An interesting prop erty of fermions is revealed in the ab ove construction Supp ose one had a

system of two fermions each in the same state v This system would b e represented in fermionic

Fo ck space by the following expression

v v v v

This expression is clearly This leads to the observation that one may not have two fermions

existing in the same state This is known as the Pauli exclusion principle

Given the nature of innite direct sums of vector spaces it is reasonable to think of elements

of Fo ck space as p olynomials Symmetrizing the tensor ensures that the variables commute In

the fermionic case we get anticommuting variables When we consider categories of normed vector

spaces this analogy b ecomes even clearer Polynomials are replaced by convergent p ower series

We will show that the b osonic Fo ck space of a Banach space has a canonical representation as a

space of holomorphic functions

Annihilation and Creation of Particles

For ease of exp osition we consider the unsymmetrized Fo ck space

U C V V V V V V

The op erators we discuss are easily extended to the b osonic and fermionic cases Given an

arbitrary nonzero v V we dene a map

C U U

v

p p

C v v v v v v v v v

v

n

In the ab ove expression v V This op erator is thought of as creating a particle in state

n

v

Similarly one may annihilate particles Cho ose an element of the dual space V and dene

p p

v v v A v v v v v v v v v v

This op eration takes an n particle state to an n particle state and so on The square ro ots in

the ab ove two expressions are normalization factors and are added to make the desired equations

hold In this expression each v is an element of V The equations expressing the interaction of

i

the annihilation and creation op erators are to b e found in Ge A more complete discussion of

the physical meaning of Fo ck space is contained in the p enultimate section

Fo ck Space as a Mo del of Storage

Now we check that the Fo ck space actually satises all the prop erties that need to b e satised by

an exp onential typ e ie satises the prop erties of Se discussed in Section This consists

of two parts verifying that Fo ck spaces form a cotriple on the category of symmetric algebras

and verifying the socalled exp onential law viz A B A B We check the former by

displaying a suitable adjunction in the next subsection Note that in the category of vector spaces

we have

Prop osition Let A and B be vector spaces

F A B F A F B

Pro of

n

For the purp oses of this pro of only we will denote the nth symmetric tensor by S Let V and

W b e vector spaces We construct a morphism

n m nm

S V S W S V W

v v w w v v w w

s s s s s

On the righthand side of the ab ove expression we are viewing each v and w as an element of

i i

V W

This lifts to an isomorphism

n

M

a na n

S V S W S V W

a

The inverse map is dened as follows We now denote vectors in V or W when considered in

V W by v and w resp ectively Rememb er that elements of this form generate V W

i i

v v v w w

s s s i s i s s n

v v v w w

s s s i i s s n

The naturality of these maps and the fact that they are inverse are left to the reader We

note at this p oint that the symmetrization of the tensors is crucial for establishing that this is an

isomorphism as it was necessary to rearrange terms Finally it is straightforward to verify that

this extends to an isomorphism of the desired form Note that F A B is generated by pure

tensors ie expressions which are nonzero only on one term of the direct sum We also note that

L

n

a na

the expression S V S W corresp onds to the nite rank part of F A F B By the

a

denition of the countable direct sum of vector spaces all elements of F A F B are contained

in a nite rank piece

The ab ove pro of follows FH App endix B closely In fact this argument can b e carried out

at a categorical level as is clear from the previously mentioned theorem of Barr B The ab ove

is also proved in BSZ for Hilb ert spaces

In the next section we will see that Fo ck space corresp onds to the free symmetric algebra It

is also straightforward to verify that the isomorphism constructed in the ab ove pro of is in fact an

algebra homomorphism

Now we consider the antisymmetrized Fo ck space We will show that one gets a mo del of the

exp onential typ es in the category of nitedimensional vector spaces using the antisymmetrized

Fo ck space

Prop osition If V is a nitedimensional vector space of dimension n then F V is also a

A

n

nitedimensional vector space with dimension

N

p

V with p n We claim that this space is the zero vector Pro of Consider the vector space

A

N

p

V is isomorphic to the space of space Since is adjoint to internal hom in VEC the space

f d

A

completely antisymmetric plinear maps from V to the scalars Let f denote such a map Since V

is only ndimensional one cannot have p linearly indep endent arguments to such maps Thus one

of the arguments must b e a linear combination of the others Thus on any arguments f b ecomes

a combination of terms of the form f u u where two arguments must b e equal But

N

p

antisymmetry makes such a term zero Thus f is the zero vector and the vector space V is the

A

onep oint space Thus the innite direct sum b ecomes a nite direct sum Now consider p n It

n

is clear that one can only cho ose C sets of p linearly indep endent vectors given a basis Thus the

p

N

p

n

V is C dimensionality of the space and hence adding the dimensions to get the dimension of

p

A

n

the direct sum we conclude that the dimension of F V is

A

The exp onential law for the antisymmetric case can b e argued similarly to the symmetric case

The detailed verication can b e found in BSZ in Section on exp onential laws or in FH in

app endix B

Categories of algebras

In this section we shall review some basic facts ab out categories of algebras and see in particular

how these t into the current context See M for a review of the basic categorical facts and

L for the basic algebra for instance For reference we do give the following denition here

Denition A triple consists of a functor F B B together with natural transformations

id F and FF F such that F F id and F F

One simple p oint to recall is that categories of algebras and of coalgebras are closely connected to

the existence of triples and cotriples Given a triple F B B with structure morphisms an

F algebra is an ob ject B and a morphism h F B B sub ject to two commutativity conditions

corresp onding to the asso ciative and unit laws This notion can b e generalized to arbitrary

F

functors There is a canonical category of such algebras the Eilenb ergMo ore category C and an

F

adjunction C C Any adjunction canonically induces a triple and this one canonically induces

the original triple The category of free F algebras is the Kleisli category C of the triple again

F

there is a canonical adjunction C C which induces the original triple Of course this dualizes

F

for cotriples with the corresp onding notion of coalgebras We shall avoid the unpleasant use of

terms like coEilenb ergMo ore and coKleisli

Usually mathematicians have b een more interested in the Eilenb ergMo ore category of a triple

or cotriple than in the Kleisli category although there has b een some interest in Kleisli categories

recently for instance in the context of linear logic as mentioned earlier in this pap er we shall

follow this tradition and shall work in Eilenb ergMo ore categories Indeed it is there that we shall

nd some of our mo dels One reason for this is quite practical it is often simpler to recognize the

category of algebras and so derive the triple similarly once one has a candidate for a triple it is

often simpler to construct the category of algebras and verify the adjunction than to directly show

the original functor is a triple But there is another reason we want to show that the Fo ck space

functor is a cotriple so as to mo del but on the categories of spaces we consider this is not the

caserather it is a triple By passing to the algebras we can x this b ecause of the following fact

F

D F a U the comp osite UF is a triple on C and so dually Fact Given an adjunction C

U

the comp osite FU is a cotriple on D BW

So we obtain our mo del of on the category of algebras

Algebras for the symmetric b osonic Fo ck space construction

We b egin with a more traditional notion of algebra the connection b etween these comes via the

triple induced by the adjunction given by the free algebra construction as outlined ab ove In other

words the category of traditional algebras is equivalent to the category of UF algebras

Denition An algebra A is a space A equipped with morphisms

m A A A and i C A

satisfying

m id

A A A A A

id m m

m

A A

A

i id id i

C A A A A C

H

H

H

H

m

H

H

H

H

H

Hj

A

Here we are supp osing the base eld to b e C otherwise replace C with the base eld k If in addition

the following diagram commutes then the algebra A is said to b e symmetric or commutative

is the canonical twist morphism

A A A A

m

m

R

A

An example of such an algebra comes from the Fo ck space the multiplication m is dened by

multiplication of p olynomials in an evident manner The use of the symmetrized tensor in the

denition of Fo ck space guarantees that this will indeed b e a symmetric algebra and it is standard

that this description gives the free such algebra In other words we have the following prop osition

Prop osition Given a vector space B the Fock space F B canonical ly carries an algebra struc

ture and indeed is the free symmetric algebra generated by B

It follows from this that we have a cotriple on the category S ALG of symmetric algebras given

by taking the Fo ck algebra on the underlying space of an algebra As the details of this are b oth

standard and similar to the case of the antisymmetric Fo ck space construction which we shall

discuss in more detail next we shall leave the details here to the reader

Algebras for the antisymmetric fermionic Fo ck space construction

Recall that we work in the context of nite dimensional vector spaces VEC when considering the

f d

antisymmetric Fo ck construction This category is selfdual and is compact with bipro ducts the

pro duct and copro duct coincide This duality also implies that a triple is also a cotriple so we can

mo del in the category of spaces However to show that the Fo ck space construction denes a

triple or cotriple it is again simpler to consider the category of algebras Although we are not

familiar with any previous consideration of this category of algebras as such the context is familiar

the antisymmetric Fo ck space construction is usually called when thought of as an algebra the

Grassman algebra or the alternating or exterior algebra the multiplication dened on it is

called the wedge pro duct a term derived from the usual notation for this pro duct

Denition An alternating algebra A is a graded algebra A with unit whose multiplication map

nm

satises the property that if x y are of degree m n respectively then xy y x which by the

grading must be of degree n m

Note that the unit must b e of degree Morphisms of alternating algebras are just homomorphisms

as algebras

Prop osition There is a canonical alternating algebra structure on F V for any nite dimen

A

sional vector space V The antisymmetric Fock construction is left adjoint to the forgetful functor

F

A

AALG where AALG is the category of alternating algebras As a consequence F U VEC

A f d

U

denes a triple and so cotriple on VEC

f d

Pro of Sketch The multiplication on F V is the standard wedge pro duct L which to

A

elements x x y y gives the pro duct x x y y Here

A A n A A m A A n A A A m

x y means the equivalence class of x y in A A Essentially this is the same multiplication

A A

of p ower series we had in the symmetric case with the alternating pro duct used in place of

the usual tensor For a vector space V dene V U F V as the canonical injection

A

Given an alternating algebra A dene F UA A by adding the terms of the series

A

hx x x x i ix x mx x where i m are the algebra maps

A

To verify that we have an adjunction we must show the following commute

F

A

F V F U F V

A A A

F

A

R

F V

A

U

U F UA

UA

A

U

R

UA

The second diagram is obvious to verify the rst notice that F x maps

A

hx x x x i hx

A

h x i

h x i h x i

A

i

and it is clear that adding up this series just returns the original term

It now follows that we can mo del in VEC with F via the formula V F V In

f d A A

summary we have the following theorem

Theorem In the category of symmetric algebras we have a model of the fragment In

the category VEC we have a model of ful l propositional classical linear logic

f d

We mention that while VEC mo dels the full prop ositional calculus it is somewhat degenerate

f d

b eing a compact category We now consider Fo ck space on normed vector spaces While we

lose some of the expressive p ower in such mo dels in particular we are no longer able to mo del

contraction Fo ck space is particularly interesting in such categories We will see that it has a

canonical representation as a space of holomorphic functions

Normed Vector Spaces

Vector spaces are in some sense intrinsically nitary structures Every vector is a nite sum of

multiples of basis vectors and one is only allowed to take nite sums of arbitrary vectors It seems

reasonable that to mo del and one should b e able to take innite sums of vectors thereby

capturing 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 space is normed then

it is p ossible to dene limits and Cauchy sequences and so on Normed vector spaces which are

the principal ob jects of study in functional analysis should b e considered as the meeting ground

of concepts from linear algebra and analysis They are also an ideal place to mo del linear logic

We will now briey review the basic concepts of the sub ject In this pap er we will fo cus on

Banach spaces Much of the discussion easily lifts to Hilb ert spaces We will intro duce Banach

spaces in detail and refer the reader to KR for a discussion of Hilb ert spaces

Henceforth all vector spaces are assumed to b e over the complex numb ers and are allowed to b e

innitedimensi onal We will use Greek letters for complex numb ers and lowercase Latin letters

from the end of the alphab et for vectors

Denition A norm on a vector space V is a function usual ly written k k from V to R the

real numbers which satises

k v k for al l v V

k v k if and only if v

k v k j jk v k

k v w k k v k k w k

For nite dimensional vector spaces the norm usually used is the familiar Euclidean norm As

so on as one has a norm one obtains a metric by the equation du v k u v k It turns out that

the spaces that are complete with resp ect to this metric play a central role in functional analysis

Denition A Banach space is a complete normed vector space

Example Consider the space of sequences of complex numb ers We write a for such a sequence

a fa g and we write k a k for the supremum of the j a j

n i

n

l fa k a k g

This is a Banach space with k a k as the norm

Another norm is obtained on sequences as follows Dene

k a k j a j

i

n

Then let

l fa k a k g

More generally if p we may dene

p

p

l fa k a k j a j g

p

n

p

All of these will b e examples of Banach spaces Furthermore these can b e dened not only for

sequences of complex numb ers but for sequences obtained from any Banach space

The following theorem shows one common way in which Banach spaces arise First we need a

denition

Denition Suppose that B B are Banach spaces and that T is a linear map from B to B

k T x k

exists We dene the norm of T written k T k to We say that T is b ounded if sup

x

k x k

be this number

If T is indeed b ounded then a standard argument KR establishes

Lemma sup k T x k k T k

jjxjj

Thus one can use vectors of unit norm to calculate the norm of a linear function rather than

having to lo ok for the sup over all nonzero vectors Linear maps from a Banach space to itself are

traditionally called operators and the norm of such maps is called the operator norm

Since a Banach space is also a metric space under the induced metric describ ed ab ove one

wishes to characterize which linear maps are also continuous In this regard we have the following

result

Lemma A linear map from f A B is continuous if and only if it is bounded

The following theorem shows that the category of Banach spaces and b ounded linear maps is

enriched over itself

Theorem If A is a normed vector space and B is a Banach space then the space of bounded

linear maps with the norm above is a Banach space

We will denote this space A B

There are several p ossible categories of interest with Banach spaces as the ob jects The most

obvious one is the category with b ounded linear maps as the morphisms However it turns out

that the category with contractive maps is of greater interest and has nicer categorical prop erties

Denition A contractive map T from A to B is a bounded linear map satisfying the con

dition k T x k k x k Equivalently the contractive maps are those of norm less than or equal to

We will write B AN C O N for the category of Banach spaces and contractive maps

Monoidal Structure of B AN C O N

We rst p oint out that B AN C O N has a canonical symmetric monoidal closed structure We b egin

by constructing a tensor pro duct Let A and B b e ob jects in B AN C O N form the tensor of A and

B A B as complex vector spaces We rst dene a partial norm for elements of the form a b

C

by the equation

k a b k k a kk b k

We would like to extend this partial norm to a norm on all of A B Such a norm is called a

C

cross norm It turns out that there are many such cross norms a numb er of which were discovered

by Grothendieck The one we will use in this pap er is called the projective cross norm It is in some

sense the least such A detailed discussion of these issues is contained in T The pro jective

cross norm is dened for an arbitrary element x of A B by the following formula

C

k x k inf fk a kk b k such that x a bg

One can verify that this is in fact a cross norm on A B Now the resulting normed space

C

will not b e complete in general so one obtains a Banach space by completing it This will act as

the tensor pro duct in the category B AN C O N It will b e denoted simply by A B Furthermore

we have the following adjunction

Lemma The functor B is left adjoint to B

Corollary B AN C O N is a symmetric monoidal closed category

As such B AN C O N is a mo del of at least the multiplicative fragment of intuitionistic linear

logic

Completeness Prop erties of B AN C O N

We b egin by constructing copro ducts

Denition Let A and B be Banach spaces The direct sum A B is the Cartesian product

equipped with the norm k a b k k a k k b k

Then we have the distributivity prop erty of over

2

Strictly sp eaking they should b e called nonexpansive maps

Prop osition A B B A B A B

We now discuss nite pro ducts

Denition The product of two Banach spaces A B has as its underlying space A B but

now with norm given by

k a b k maxfk a k k b kg

As a category of vector spaces B AN C O N is fairly unique in this resp ect While most such

categories mo del the additive fragment of linear logic they invariably equate the two connectives

since nite pro ducts and copro ducts coincide In other words B AN C O N do es not share the

familiar prop erty of b eing an additive category

We now present countably innite pro ducts and copro ducts

Denition Let fA g be a sequence of Banach spaces Dene A to be those sequences

i i

i

which converge in the l norm ie bounded sequences equipped with the obvious norm

Dene A to be al l sequences which converge in the l norm

i

This gives countable products and coproducts in B AN C O N Similar constructions can be applied

for uncountable products and coproducts

Equalizers in B AN C O N corresp ond to equalizers in the underlying category of vector spaces

The fact that b ounded maps are continuous implies that the subspace will b e complete Co equal

izers are obtained as a quotient with the induced norm b eing the inmum of the norms of the

elements of the equivalence class See C for a discussion of quotients of Banach spaces

Theorem B AN C O N is complete and cocomplete

B AN C O N as a Mo del of Weakening

In the next section we will show that the Fo ck space of a Banach space has a canonical interpretation

as a space of holomorphic functions In this section we explore the nature of B AN C O N as a mo del

of linear logic We obtain a somewhat weaker mo del for the following reason In this category as

we have previously observed pro ducts and copro ducts do not coincide Thus the isomorphism

A B A B

is not useful for mo delling the additive fragment We obtain instead a weakening cotriple in the

sense of Jacobs J

Jacobs denotes a weakening cotriple by Such a cotriple satises all of the axioms of mo delling

w

except that the coalgebras will not have the comonoid structure necessary to mo del contraction

Thus we have syntax of the following form

A

A B A

w

S to W e D er

A B A B A

w w w w

One mo dels these pro of rules as in linear logic following Se

We p oint out that the map

A B A B

necessary to mo del storage is indeed a contraction This map is obtained as

A B A B A B A B A B

It is shown in an app endix that this map is contractive

Theorem In the category B AN C O N we obtain a model of the fragment

w

Remark We wish to point out an error in an earlier draft of this paper BPS In that paper

it is stated that the Fock construction is functorial on the larger category of Banach spaces and

bounded linear maps In fact when one applies the Fock construction to a map of norm greater

than one might obtain a divergent expression Thus we are forced to work in the smal ler category

of contractions

The HolomorphicFunction Representation

of Fo ck Space

It has b een observed by a numb er of p eople that the free symmetric algebra provides a mo del of

the exp onential typ e But the observation that this construction corresp onds to b osonic Fo ck space

allows us to relate results in quantum physics to the mo del theory of linear logic In this section

we present one such relationship The symmetrized Fo ck space on a Banach space B turns out

to b e a space of holomorphic functions analytic functions on B prop erly dened This hints at

p ossible deep er connections b etween analyticity and computability which need to b e explored

The ideas here stem from early work by Bargmann Ba on Hilb ert spaces of analytic functions

in quantum mechanics This was extended by Segal S BSZ to quantum eld theory and

Segals extension was used by Ashtekar and Magnon AMA to develop quantum eld theory

in curved spacetimes A brief summary of the ideas is contained in an app endix to P and

in P The latter work involved making sense of the familiar CauchyRiemann conditions on

innitedimensi onal spaces

We quickly recapitulate the basic notion of analytic function in terms of one complex variable

b efore presenting the innitedimensional case A very go o d elementary reference is Complex Anal

ysis by Ahlfors Ah Given the complex plane C one can dene functions from C to C Let

z b e a complex variable we can think of it as x iy and thus one can think of functions from C

to C as functions from R to R An analytic or holomorphic function is one that is everywhere

dierentiable In the notion of dierentiation the limit b eing computed viz

f x h f x

lim

h

h

allows h to b e an arbitrary complex numb er and hence this limit is required to exist no matter in

what direction h approaches This much more stringent requirement makes complex dierentia

bility much stronger than the usual notion of dierentiability If a complex function is dierentiable

at a p oint it can b e represented by a convergent p ower series in a suitable op en region ab out the

p oint If one uses the fact that h can approach zero along either axis one can derive the Cauchy

Riemann equations for a complex valued function f ux y iv x y of the complex variable

z x iy

u v u v

x y y x

What is remarkable ab out complex functions is that this denition of analyticity yields the

result that a complexanalytic function can b e expressed by a convergent p owerseries in a region

of the complex plane This is remarkable b ecause only one derivative is involved in the Cauchy

Riemann equations whereas the statement that a p owerseries representation exists is stronger for

realvalued functions even than requiring innite dierentiability In real analysis one has examples

of functions that are innitely dierentiable at a p oint but do not have a p ower series representation

in any neighb ourho o d of that p oint A function may have a p ower series representation that is

valid everywhere a socalled entire holomorphic function the complex exp onential function is an

example

There is a formal p ersp ective due to Wierstrass that is rather more illuminating Think of a

complex variable z x iy and its conjugate z x iy as b eing formally indep endent variables

z for example the function that maps A function could dep end on z and on its complex conjugate

z iz z An analytic or holomorphic function is one which has no dep endance on z This each z to z

is expressed formally by df d z When expressed in terms of the real and imaginary parts of

f and z this equation b ecomes the familiar CauchyRiemann equations Thus this reinforces the

view that a holomorphic function is prop erly thought of as a single complexvalued function of a

single variable rather than as two realvalued functions of two real variables

The theory of functions of nitely many complex variables is a nontrivial extension of the theory

of functions of a single complex variable Entirely new phenomena o ccur which have no analogues

in the theory of a single complex variable An excellent recent text is the three volume treatise

by Gunning Gu For our purp oses we need only the barest b eginnings of the theory Given

n n n

C we can have functions from C to C One can intro duce complex co ordinates on C z z

n

One can dene a holomorphic function here as one having a convergent p owerseries expansion in

z z The key lemma that allows one to mimic some of the results of the onedimensional case

n

is Osgo o ds lemma

n

Lemma If a complexvalued function is continuous in an open subset D of C and is holomor

phic in each variable seperately then it is holomorphic in D

From this one can conclude that a holomorphic function in n variables satises the CauchyRiemann

f

equations One is free to take either one of a satisfying CauchyRiemann equations or

z

i

b having convergent p owerseries representations as the denition of holomorphicity

Now we describ e how to dene holomorphic functions on innitedimensi onal complex Banach

spaces The basic intuition may b e summarized thus One starts with subspaces of nite codi

n

mension Thus the quotient spaces are isomorphic to some C One can dene what is meant by

a holomorphic function on these quotient spaces as in the preceding paragraph By comp osing a

holomorphic function with the canonical surjection from the original Banach space to the quotient

space we get a function on the original Banach space These functions can all b e taken to b e

holomorphic

3

There is a considerably harder theorem called Hartogs theorem which drops the requirement of continuity

B

p

p

p

p

p

p

p

p

p

p

p

p

p

R

n

B

C C

Intuitively these are the functions that are constant along all but nitely many directions and

holomorphic in the directions along which they do vary These functions are called cylindric holo

morphic functions Because the sequence of co ecients of a p owerseries is absolutely convergent

we can dene an l norm on these functions in terms of the p owerseries Finally the collection of

all holomorphic funcitons is dened by taking the l norm completion of the cylindric holomorphic

functions

Given a Banach space B let U b e a subspace with nite co dimension n ie the quotient

n

space B U is an n complexdimensional vector space The space B U is isomorphic to C Let

n

B U C b e an isomorphism such a map denes a choice of complex co ordinates on B U

Let b e the canonical surjection from B to B U

U

Denition A cylindric holomorphic function on B is a function of the form f where

U

n

U and are as above and f is a holomorphic function from C to C

U

We need to argue that the choice of co ordinates do es not make a real dierence Of course which

functions get called holomorphic do es dep end on the choice of co ordinates but the space of holo

morphic functions has the same structure Supp ose that U and V are b oth subspaces of B and

that U is included in V Supp ose that b oth these spaces are spaces of nite co dimension say n

and m resp ectively Clearly n m Now we have a linear map B U B V given by

UV

n

x U x V clearly this is a surjection Now given co ordinate functions B U C and

m n m

B V C we can dene a function C C given by which makes

UV

the diagram commute Thus we do not have to imp ose coherence conditions on the choice of

co ordinates we can always translate back and forth b etween dierent co ordinate systems

We will suppress these translation functions in what follows and assume that the co ordinates

have b een serendipitously chosen to make the form of the functions simple In other words we

can x a family of subspaces fW jn N g with W having co dimension n and W W The

n n n n

co ordinates can b e chosen so that the space B W has co ordinates z z

n n

Supp ose that f is a cylindric holomorphic function on B This means that there is a nite

co dimensional subspace W and a holomorphic function f from W to C such that f f

W W W

The function f regarded as a function of n complex variables has a p owerseries representation

W

i

i

1 k

z f z z a z

W n i i

1

k

k

and furthermore we have the following convergence condition

ja j

i i

1

k

Thus with each such cylindric holomorphic function we can dene the sum of the absolute values

of the co ecients in the p owerseries expansion as the norm of the function Viewing the sequences

of co ecients as the elements of a complex vector space we have an l norm We write k f k for

this norm of a cylindric holomorphic function

4

z is considered antiholomorphi c traditional ly but This happ ens even in the one dimensional case The function

z one could have called it holomorphic by interchanging the role of z and

Denition An l holomorphic function on B is the limit of a sequence of cylindric holomorphic

function in the above norm

The l emphasizes that the holomorphic functions are obtained by a particular norm completion

In the corresp onding theory of holomorphic functions on Hilb ert spaces one uses the innerpro duct

to dene p olynomials and then p erform a completion in the L norm A key dierence is that our

norm is dened on the sequence of co ecients whereas in the Hilb ert space case one uses the L

norm which is dened in terms of integration

In the resulting Banach space there are several formal entities that were adjoined as part of

the normcompletion pro cess We need to discuss in what sense these formallydened entities can

b e regarded as b onade functions Let W W b e an innite sequence of subspaces of B

r

each emb edded in the previous Assume in addition that all these spaces have nite co dimension

Now assume that there is a sequence of cylindric holomorphic functions f on B obtained from a

n

n

holomorphic function f on each of the quotient spaces B W Finally assume that the sequence

i

k f k of real numb ers is convergent Such a sequence of cylindric holomorphic functions denes

n

a holomorphic function on B We call this function f We need to exhibit f as a map from B to C

Accordingly let x b e a p oint of B For each of the functions f we have jf xj k f k Since the

n n n

sequence of norms converges we have the sequence f x converges absolutely and hence converges

n

Thus the function f qua function is given at each x of B by lim f x However in order

n n

to use the word function we need to show that the p owerseries has a domain of convergence

Unfortunately it may not have a nontrivial domain of convergence but in a sense to b e made

precise it comes close to having a nontrivial domain of convergence

The p owerseries representation of the function f is given as follows It dep ends in general

on innitely many variables but each term in the p ower series will b e a monomial in nitely many

j j

1

k

in the expansion of f In all but nitely many of the z variables Consider the co ecient of z

i i

1

k

f all the indicated variables will app ear in their p owerseries expansions Consider the co ecients

n

of this term in each p ower series this forms a sequence of complex numb ers where is if there

n n

is no such term in the expansion of f Since j j k f k the sequence converges absolutely

n n n n

j j

1

k

in the p owerseries expansion of z and hence converges to say This is the co ecient of z

i i

1

k

f

Consider the co ordinates z z This denes an ndimensional subspace of the Banach

n

space which we call U Now consider the p owerseries for f It denes a family of holomorphic

n

n n

functions f where f is dened on the subspace U and is obtained by retaining only those terms

n

in the p owerseries expansion of f which involve variables among z z These are analytic

n

functions on the U and as such have nontrivial domains of convergence However as n increases

n

the radii of convergence could tend to So we have the slightly weaker statement than the usual

nitedimensional notion instead of having a nonzero radius of convergence in the Banach space

we have a nonzero radius of convergence on every nitedimensional subspace If one uses entire

functions rather than analytic functions at the starting p oint of the construction then one can

show that the resulting functions are entire see page theorem of the b o ok by Baez Segal

and Zhou BSZ Unfortunately when using the representation of elements of Fo ck space one

may carry out simple op erations that do not pro duce entire functions so we cannot just cho ose to

work with entire functions Nevertheless many common functions most notably the exp onential

are entire

Given a b ona de holomorphic function one can express it as a p ower series The co ecients

are calculated in the usual way viz by using Taylors theorem

n

i

i

k 1

f z z

n i i n

1

k

k

i i

1

k

i i z z

k k

Since the mixed partial derivatives commute the functions are holomorphic and hence certainly

dierentiable enough the partial derivatives are concretely sp eaking symmetric arrays Abstractly

sp eaking this just means that they are elements of the symmetrized tensor pro duct

We can write this as follows

Theorem A holomorphic function can be represented by its powerseries expansion where the

th th

n term in the powerseries expansion is a symmetrized n derivative

k

f k D f

k th

where the notation D f means symmetrized k derivative of f

The symmetrized derivatives live in the symmetrized tensor pro ducts of B with itself One

thus has a corresp ondence with the standard Fo ck representation and the notion of holomorphic

function since in each case one has a string of symmetrized vectors

The Physical Origin of Fo ck Space

The Fo ck space constructions describ ed in the previous sections were indep endently invented by

physicists and mathematicians The symmetric Fo ck space called the b osonic Fo ck space by physi

cists is well known to mathematicians as the symmetric tensor algebra whereas the antisymmetric

Fo ck space fermionic Fo ck space was invented by Grassman at least in the nitedimensional

case under the name of exterior algebra or alternating algebra In this section we describ e the

role of Fo ck space in quantum eld theory In order to prevent intolerable regress in denitions we

assume that the reader has an at least intuitive grasp of dierential equations the denition of a

smo oth manifold and asso ciated concepts like that of a smo oth vector eld

We b egin with a brief discussion of quantum mechanics and classical mechanics In classical

mechanics one has systems which vary in time The role of theory is to describ e the temp oral

evolution of systems Such temp oral evolution is governed by a dierential equation The fact that

one uses dierential equations says something fundamental ab out the lo cal nature of the dynamics of

physical systems at least according to conventional classical mechanics In dealing with dierential

equations one has to distinguish b etween quantities that are determined and quantities that may b e

freely sp ecied the so called initial conditions Exp eriment tells one that systems are describ ed

by secondorder dierential equations and hence that the functions b eing describ ed and their rst

derivatives at a given p oint of time are part of the initial conditions The space of all p ossible

initial conditions is called the space of p ossible states or phase space and is the kinematical arena

on which dynamical evolution o ccurs The p oints of phase space are called states If the system

is a collection of say particles the states will corresp ond to the numb ers required to sp ecify

the p ositions and the velo cities of each of the particles in threedimensional space

Through each p oint in phase space is a vector giving rise to a smo oth vector eld called the

Hamiltonian vector eld One can draw a family of curves such that at every p oint there is exactly

one curve passing through that p oint and the Hamiltonian is tangent to the curve at that p oint

Roughly sp eaking the vector eld denes a dierential equation and the curves represent the

family of solutions where each p oint represents a p ossible sp ecication of initial conditions An

observable is a physical quantity that is determined by the state As such it corresp onds to a

realvalued function on phase space A typical example is the total energy of a system Most of

exp erimental mechanics is aimed at determining the Hamiltonian In the formal development of

analytical mechanics there is a sp ecial antisymmetric form called the symplectic form which plays

a fundamental mathematical role but is hard to describ e in an intuitive or purely physical way

In quantum mechanics the ab ove picture changes in the following fundamental ways The

observables b ecome the fundamental physical entities These are dened to form a particular

subalgebra of an algebraic structure called a C algebra The key p oint is that this algebra is not

commutative unlike the algebra of smo oth functions on a manifold Furthermore the failure of

commutativity is directly linked to the symplectic form this was Diracs contribution to the theory

of quantum mechanics Thus structures available at the classical level provide guidance as to what

the correct C algebra should b e

There is a representation of this algebra as the algebra of op erators on a Hilb ert space The

space of states acquires the structure of a Hilb ert space and b ecomes the carrier of the repre

sentation of the C algebra One presentation of this abstract Hilb ert space is as the space of

squareintegrable complexvalued functions on a suitable underlying space for example the space

of p ossible congurations of a system The space of states has acquired linear structure this means

that one can add states reecting the intuition that in quantum mechanics a system can b e in the

sup erp osition of two or more states The inner pro duct measures the extent to which two states

resemble each other Finally the fact that one has complex functions is strongly suggested by the

observation of interference phenomena in nature

An observable is a selfadjoint op erator The link b etween the mathematics and exp eriment is

the following If one attempts to measure the observable O for a system in state one will obtain

an eigenvalue of O Selfadjoint op erators have real eigenvalues so we will get a realvalued result

If is an eigenvector with eigenvalue then with no indeterminacy or uncertainty one will obtain

the value If is not an eigenvector one can express as a linear combination of eigenvectors in

the form a where the are assumed to b e eigenvectors with eigenvalues The result

i i i i

of measuring O will b e with probability ja j It is imp ortant to keep in mind that the absolute

i i

squares of the a corresp ond to probabilities but it is the a themselves that enter into the linear

i i

combinations of states This interplay b etween the complex co ecients and the interpretation of

their squares as probabilities is what distinguishes the probabilistic asp ects of quantum mechanics

from statistical mechanics which also has a probabilistic asp ect but where one directly manipulates

probabilities

The dynamics of systems is describ ed by a rstorder dierential equation called Schro edingers

equation Thus the evolution of states in quantum mechanics is determinate just as in classical

mechanics The indeterminacy usually asso ciated with quantum mechanics app ears in the fact that

the state of a system may not b e an eigenstate of the observable b eing measured so the outcome

of the measurement may b e indeterminate

Quantum mechanics is designed to handle systems in which the numb er of interacting entities

usually called particles is xed On the other hand exp eriment tells us that at suciently high

energies particles may b e created or destroyed Quantum eld theory was invented to account for

such pro cesses The original formulations of this theory due to Dirac Heisenb erg Fo ck Jordan

Pauli Wigner and many others was quite heuristic Now a reasonably rigourous theory is available

see the b o ok by Baez Segal and Zhou BSZ for a recent exp osition of quantum eld theory

The rst need in a manyparticle theory is a space of states which can describ e variable numb ers

of particles this is what Fo ck space is Ge The second ingredient is the availability of op erators

that can describ e the creation and annihilation of particles Of course there is much more that

needs to b e said in order to see how all this formalism translates into calculations of realistic

physical pro cesses but that would require a very thick b o ok which in any case has b een written

many times over

Given a Hilb ert space H in quantum mechanics representing the states of a single particle one

can construct a manyparticle Hilb ert space as F H Supp ose that H one interprets the

element of H H as a twoparticle state with one particle in the state and the other

s s

in the state Similarly for the other summands of F H The reason for the symmetrization

is that one is dealing with indistinguishabl e particles so that the nparticle states have to carry

representations of the p ermutation group Thus one could have particle states that were symmetric

or antisymmetric under interchange leading to the b osonic or fermionic Fo ck spaces resp ectively

It is a remarkable fact that b oth typ es of particles are observed in nature Notice that is

identically zero hence one cannot have manyparticle states in the antisymmetric Fo ck space in

which b oth particles are in the same oneparticle state This is observed in nature as the exclusion

principle Fo ck space is the space of states for quantum eld theory and is constructed from the

space of states for quantum mechanics

The presentation of Fo ck space ab ove emphasized the concept of manyparticle states Math

ematically however F H is just a Hilb ert space and can b e presented dierently As we have

shown in the last section it can b e presented as the space of holomorphic functions of a Hilb ert

space the details are somewhat dierent from the Banach space case but the ideas are essentially

the same The space of holomorphic functions has as its inner pro duct

Z

z z

hg f i f z g dz d z e z

i

See IZ page for example What do the creation and annihilation op erators lo ok like

from this p ersp ective For simplicity let us lo ok at p ower series in a single variable z This

amounts to only lo oking at the manyparticle states of the form tensored with itself The creation

op erator is just z while the annihilation op erator is just ddz One can easily check that

AC CAf dz f dz z df dz f in other words the basic algebraic relation holds

Furthermore one can ask what the eigenstates of A and C lo ok like Clearly the eigenstate of C is

z

just the zero vector The eigenstate of A is the state represented by the holomorphic function e

These states actually exist in nature and are called coherent states they o ccur for example in

lasers The key p oint ab out coherent states is that they lo ok classical one can remove a particle

without changing the state As such they b ear a resemblance to the role of formulas in linear

logic

Acknowledgements

We wish to thank several collegues for helpful conversations Samson Abramsky John Baez Mike

Barr Radha Jagadeesan Steve Vickers and esp ecially Abhay Ashtekar

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App endixContractibility of the Exp onential Isomorphism

It is straightforward to verify that the isomorphism

A B A B

constructed on vector spaces lifts to an isomorphism on Banach spaces The only issue is whether

the maps so constructed are contractive

We now show that the two maps in this isomorphism are contractive We need some preliminary

facts ab out norms on tensor pro ducts of Banach spaces Supp ose that A and B are two Banach

spaces Let a b e in A and b in B The norm of a b is k a kk b k the socalled crossnorm Not

all elements of A B are of the form a b for some a in A and some b in B in general an element

of A B will b e a sum of such terms Furthermore there is no unique representation as such a

sum The norm is then dened as follows

k u k inf f k a kk b k a b u ia A b B g

i i i i i i i

AB

Note also that the norm is a continuous but not linear function Thus one cannot argue in terms

of basis elements

We consider F A F B F A B rst Supp ose that u and v are pure tensors in

F A and F B resp ectively then clearly k u v k k u kk v k since just takes the tensor

of u and v and we have the ab ove remark ab out the norm of such elements in the tensor pro duct

space Now let y F A and z F B b e arbitrary We consider the action of on y z this

is still not the most general situation As usual we write y for the ith comp onent of y in the

i

standard basis of Fo ck space thus y is a pure tensor Similarly for z

i j

k y z k k y z k

ij i j

k y z k k y z k k y kk z k k y kk z k k y z k

ij i j ij i j ij i j

The equalities are obvious the inequality is the triangle inequality

Now supp ose that we have an element u F A F B and that u has the decomp osition

i i

y z as well of course as other such decomp ositions Now we have using the linearity of

i

the triangle inequality and the argument just ab ove

i i

k u k k y z k

i

However we do not know that the right hand side is less than k u k in fact using the triangle

inequality we would get the opp osite inequality But since the norm of u is dened as the inmum

of the ab ove sum of norms across all such decomp ositions Now if the inmum is actually realized

by such a decomp osition then we still have the ab ove inequality but now we know that the right

hand side is indeed k u k If the inmum is not realized there is a sequence of decomp ositions

with the right hand sides as ab ove converging to the inmum and since k u k is less than all

such sums it must b e less than the inmum Thus in all cases k u k k u k and hence is

contractive

To show that is contractive we need a fact ab out how symmetrization aects norms Supp ose

that we have u v B where B is any banach space Now u v u v v u We claim

s

that

k u v k k u v k

s

Clearly one decomp osition of u v is u v v u if this were the one realizing the

s

inmum in the denition of the norm we would b e done Now supp ose that there were another

decomp osition p q for u v such that k u v k k p q k k u v k Consider the

i i i s s i i i

expression p q v u u v Computing norms and using the triangle inequality we get

i i i

k u v k k p q v u k k p q k k u kk v k

i i i i i i

In other words we have by simple arithmetic

k u v k k u kk v k k p q k

i i i

This contradicts the assumption ab ove Thus symmetrization preserves norms It is clear that the

ab ove argument could have b een carried out for symmetrization over more than two elements

The map just undo es symmetrization thus on pure tensors is norm preserving Now

consider an arbitrary element x of F A B We have

s

k x k k h x i k

n n

k h x i k

n n

k h x i k

n n

k x k k x k

n n

Thus is contractive as well It immediately follows that the morphism

A B A B

is a morphism in B AN C O N and satises the necessary prop erties