aa r X i v : . [ qu a n t - ph ] J un A categorical framework for thequantum harmonic oscillator
Jamie VicaryImperial College London [email protected]
June 5, 2007
Abstract
This paper describes how the structure of the state space of the quantum harmonicoscillator can be described by an adjunction of categories, that encodes the raisingand lowering operators into a commutative comonoid. The formulation is an entirelygeneral one in which Hilbert spaces play no special role.Generalised coherent states arise through the hom-set isomorphisms defining the ad-junction, and we prove that they are eigenstates of the lowering operators. Surprisingly,generalised exponentials also emerge naturally in this setting, and we demonstrate thatcoherent states are produced by the exponential of a raising morphism acting on thezero-particle state. Finally, we examine all of these constructions in a suitable cate-gory of Hilbert spaces, and find that they reproduce the conventional mathematicalstructures.
With [1], a research programme was begun to describe many of the familiar aspects ofquantum mechanics — such as Hilbert spaces, unitary operators, states, inner products,superposition and entanglement — in terms of the structure of a category. This programmesuggests that we do quantum mechanics in an entirely new way: rather than explicitlyworking with the mathematics of Hilbert spaces, we should construct the category of Hilbertspaces and identify those parts of the categorical structure which are needed for the quantummechanics that we want to do. Our quantum mechanics can then be done abstractly, in termsof this categorical structure. Such an approach leads to deeper insights into the mathematicalstructures which are necessary for quantum mechanics, as well as providing new avenues forits generalisation.In this paper, we describe a way to extend the categorical description of quantum me-chanics to encompass the quantum harmonic oscillator, one of the most basic and importantquantum systems. We accomplish this with the theory of adjunctions, itself one of the mostbasic and important tools of category theory. The approach is based on the well-knownfact, first discussed in [3], that the symmetric Fock space over a given single-particle Hilbert1pace is given by the canonical free commutative monoid object over that Hilbert space. Infact, we shall see that much more can be obtained from the free commutative monoid con-struction; every part of the adjunction structure corresponds naturally to some aspect of thetraditional mathematical treatment of the quantum harmonic oscillator. A key observationis that the natural isomorphisms F ( A ⊕ B ) ≃ F ( A ) ⊗ F ( B )should be unitary, where F ( A ) represents the Fock space over a Hilbert space A , and ⊕ and ⊗ represent direct sum and tensor product respectively. The standard free commutativemonoid functor will not, however, give rise to a unitary natural isomorphism in general.We include introductions to the necessary category theory, and to the necessary physics,in the form of self-contained sections which can be read or skipped as desired. In this section we introduce the category theory that is useful for describing finite-dimensionalquantum mechanics, which we will be using throughout this paper to work with the quantumharmonic oscillator. The structure that we will define by the end of this section is that of † -compact-closed categories with † -biproducts.All of the structures in this section can be applied to the category FdHilb , which hasfinite-dimensional Hilbert spaces as objects and bounded linear operators as morphisms. Aswe introduce each element of the structure, we will show how it can be formulated in thiscategory, and describe how it captures some aspect of quantum mechanics as discussed in[1].
A symmetric monoidal category C has a functor ⊗ : C × C - C called the tensor product, and a monoidal unit object I , such that for all objects A , B and C in C , the following isomorphisms exist: ρ A : A ⊗ I ≃ Aλ A : I ⊗ A ≃ Aα A,B,C : A ⊗ ( B ⊗ C ) ≃ ( A ⊗ B ) ⊗ C swap ⊗ A,B : A ⊗ B ≃ B ⊗ A We also require that the swap ⊗ isomorphisms satisfy the equationswap ⊗ B,A ◦ swap ⊗ A,B = id A ⊗ id B A and B . We require that all of these isomorphisms are natural; in otherwords, that they can be expressed as the stages of natural transformations. They must alsobe compatible with each other, so that any diagram built solely from these isomorphismsand their inverses is commutative. These isomorphisms imply that, up to isomorphism, ⊗ is a commutative, associative monoidal operation on our category, with unit object I . Forreadability we shall often exploit the coherence theorem [7], which proves that any monoidalcategory is equivalent to one for which all structural isomorphisms are identities; this allowsus to neglect these isomorphisms (except for the swap ⊗ isomorphism) when it is convenient.A good general reference for the theory of symmetric monoidal categories is [7].The category FdHilb is a symmetric monoidal category, with the operation ⊗ given bythe tensor product of Hilbert spaces. The tensor unit I is the one-dimensional Hilbert space C , the complex numbers. Observing that we can access the multiplicative monoid structureof the complex numbers by the endomorphisms of C , we will refer to the endomorphismmonoid Hom( I, I ) in any monoidal category as the monoid of scalars . It can be proved thatthe scalars in any monoidal category are commutative [5].Also, we note that the vectors in a finite-dimensional Hilbert space A are in correspon-dence with bounded linear operators C - A in FdHilb . Since a state of a Hilbert spaceis given by a nonzero element of that Hilbert space, we can generalise the notion of statedirectly: given an arbitrary object B in a symmetric monoidal category C , the states of B are the nonzero morphisms Hom C ( I, B ). (We will encounter an abstract formulation of zeromorphisms in section 2.4.)
Monoidal categories have a useful graphical representation. We represent objects of thecategory as vertical lines, and the tensor product operation as lines placed side-by-side.Morphisms are junction boxes, with lines coming in and lines going out. For example, givenobjects A , B , and C , and morphisms f : A ⊗ B - B ⊗ A and g : C ⊗ A - I , we representthe composition (id B ⊗ g ) ◦ (id B ⊗ swap ⊗ A,C ) ◦ ( f ⊗ id C ) : A ⊗ B ⊗ C - B using the following diagram: f gAAB B C The diagram is read from bottom to top. We represent the junction boxes using a wedgeshape as we will later be rotating and reflecting them, and we want to break the symmetryso that their orientation can be identified. The tensor unit I is represented by a blank space;3n other words, it is not represented at all. The swap ⊗ isomorphism is represented by acrossing of lines, and because we are working in a symmetric monoidal category, it makesno difference which line goes over which. This graphical representation makes the structuralisomorphisms ρ , λ , α and swap ⊗ intuitive, and so is often a useful way to understand complexconstructions in monoidal categories. In some categories, products and coproducts become unified in a particularly elegant way.
Definition 2.1.
A category has a zero object , written 0, when it has isomorphic initial andterminal objects. Each hom-set Hom(
A, B ) then gains a zero morphism , written 0
A,B , whichis the unique morphism in the hom-set that factors through the zero object.
Definition 2.2.
A category has biproducts if and only if, for all objects A and B , the uniquearrow w A,B : A + B - A × B that satisfies the equations π A ◦ w A,B ◦ i A = id A π B ◦ w A,B ◦ i A = 0 A,B π A ◦ w A,B ◦ i B = 0 B,A π B ◦ w A,B ◦ i B = id B is an isomorphism, where i A and i B are the coproduct injections and π A and π B are theproduct projections.In a category with biproducts, we represent the isomorphic product and coproduct by thesymbol ⊕ , and call it a biproduct.A category with biproducts is enriched over commutative monoids ; in other words, anyhom-set Hom( A, B ) carries the structure of a commutative monoid, with unit 0
A,B . Weinterpret the monoid action as being addition, and we define it in the following way:
A f + g - BA ⊕ A ∆ A ? f ⊕ g - B ⊕ B ∇ B (1)Here, ∆ A is the diagonal for the product, and ∇ A is the codiagonal for the coproduct.In a category with biproducts, we can always choose the canonical injections and pro-jections to satisfy some useful properties, and when we talk about ‘the’ injections and projec-tions, it will be these well-behaved ones which are meant. For all objects B := A ⊕ A ⊕ . . . ⊕ A N ,we can choose projection morphisms π n : B - A n and injection morphisms i n : A n - B ,such that the following properties hold:1. π n ◦ i n = id A n for all n ;2. π m ◦ i n = 0 A n ,A m when n = m ; 4. P Nn =1 ( i n ◦ π n ) = id B , where this sum is as defined in diagram (1).In fact, in a category which is enriched over commutative monoids, this can be taken as thedefinition of a biproduct structure.Naturality of the diagonal and codiagonal morphisms implies that in a category withbiproducts, composition is linear. In other words, for all f, f ′ : A - B and g, g ′ : B - C ,we have g ◦ ( f + f ′ ) = ( g ◦ f ) + ( g ◦ f ′ ) ( g + g ′ ) ◦ f = ( g ◦ f ) + ( g ′ ◦ f )The scalars in the category interact nicely with a biproduct structure. Not only can weadd elements of hom-sets, but we can also multiply them by scalars in a well-defined way. Definition 2.3.
For any morphism f : A - B , for any objects A and B , and for any scalar s : I - I , the scalar multiple s · f is defined in the following way : A s · f - BI ⊗ Aλ − A ? s ⊗ f - I ⊗ Bλ B (2)In fact, the scalars interact with the biproduct structure in such a way that each hom-setgains the structure of a commutative semimodule , a weakening of the notion of a vectorspace. In particular, for all scalars s and p and all morphisms f , we have( s ◦ p ) · f = s · ( p · f ) . Scalar multiplication is also well-behaved with respect to composition, meaning that for any g with g ◦ f well-defined, we have g ◦ ( s · f ) = ( s · g ) ◦ f = s · ( g ◦ f ) . The category
FdHilb has biproducts, given by the direct sum of Hilbert spaces. Inthis case, the i n are injections of subspaces, and the π n are projections of subspaces. In anarbitrary symmetric monoidal category with biproducts, the commutative monoid structureinduced on the hom-sets gives us a way to add states: given arbitrary φ, ψ : I - A , thesuperposition is given by φ + ψ : I - A . In FdHilb , this agrees with the usual notion ofaddition of vectors. † -categories Definition 2.4.
A category C is a † -category if it is endowed with a contravariant endo-functor † : C - C , which is the identity on objects, and which satisfies † ◦ † = id C . Of course, there are other equivalent definitions. † -category then we must have a particular one in mind, which we will refer to as † . Wealso note that any † -category must be isomorphic to its opposite.We can use this structure to adapt some useful terminology from the mathematics ofHilbert spaces. Definition 2.5.
For any morphism f , we call f † its adjoint . Definition 2.6.
A morphism f : A - A is self-adjoint if it satisfies f † = f . Definition 2.7.
A morphism f : A - B is an isometry if it satisfies f † ◦ f = id A ; in otherwords, if its adjoint is its retraction. Definition 2.8.
A morphism f : A - B is unitary if it satisfies f † ◦ f = id A and f ◦ f † = id B ;in other words, if both f and f † are isometries. In this case, A and B are of course isomorphic.If a † -category has additional structure, we will often require that the additional structurebe compatible with the † functor. Definition 2.9. A † -category has † -biproducts iff it has biproducts, such that the canonicalprojections and injections are related by the † functor. Definition 2.10. A symmetric monoidal † -category is defined in the obvious way, but withthe extra constraints that the canonical isomorphisms associated to the symmetric monoidalstructure be unitary. Definition 2.11 (Notation due to Selinger [10]) . An equaliser d : D - A is a † -equaliser if d is an isometry (note that it must automatically be monic, by the properties of the equaliser.)Finally, we note that natural transformations defined between functors connecting † -categories may themselves admit a notion of adjoint. Definition 2.12.
For † -categories C and D , a functor J : C - D is compatible with the † -structures iff for all morphisms f in C , we have J ( f † ) = ( J ( f )) † , where we employ the † on C and D respectively. Lemma 2.13.
Given a natural transformation n : J ˙ - K for functors J, K : C - D compatible with † -structures on C and D , then n has an adjoint natural transformation n † : K ˙ - J , defined at each stage A of C by ( n † ) A := ( n A ) † . † -compact closure A symmetric monoidal category has duals , or equivalently is compact-closed , if for everyobject A there exists a second object A ∗ and morphisms ζ A : I - A ⊗ A ∗ θ A : A ∗ ⊗ A - I A ζ A ⊗ id A - A ⊗ A ∗ ⊗ A id A ⊗ θ A - AA id A - A (3) A ∗ id A ∗ ⊗ ζ A - A ∗ ⊗ A ⊗ A ∗ θ A ⊗ id A ∗ - A ∗ A ∗ id A ∗ - A ∗ (4)These equations imply that A ∗ is unique up to isomorphism, and that ( A ∗ ) ∗ ≃ A . However,it is useful to talk about A ∗ as if it were unique, and to use ( A ∗ ) ∗ = A as if it held as anequation, knowing that what we do will only hold up to isomorphism. We refer to A ∗ as thedual of A .We can use the duals to define a natural isomorphism of hom-sets S A,B,C : Hom( A ⊗ B, C ) ≃ Hom(
B, C ⊗ A ∗ ) (5)for all objects A , B and C , as shown in the following diagrams for any f : A ⊗ B - C and f ′ : B - C ⊗ A ∗ related by the isomorphism: B f ′ = S A,B,C ( f ) - C ⊗ A ∗ B ⊗ A ⊗ A ∗ id B ⊗ ζ A ? swap ⊗ B,A ⊗ id A ∗ - A ⊗ B ⊗ A ∗ f ⊗ id A ∗ (6) A ⊗ B f = S − A,B,C ( f ′ ) - CB ⊗ A swap ⊗ A,B ? f ′ ⊗ id A - C ⊗ A ∗ ⊗ A id C ⊗ θ A (7)Here, the morphisms of the form swap ⊗ A,B are the symmetry isomorphisms that make up partof the symmetric monoidal structure. Equations (3) and (4) ensure that the diagrams areinverse to each other in the correct way.The graphical representation of the dual structure is especially powerful. If an object A is represented by a line with arrow oriented up the page, then its dual A ∗ has an arrowpointing down the page, and vice-versa. The duality morphisms ζ A and θ A then take theform of lines which loop back on themselves: A A ∗ AA ∗ ζ A : I - A ⊗ A ∗ θ A : A ∗ ⊗ A - I A AA ∗ = A AA ∗ A ∗ = A ∗ This is very intuitive; when working in monoidal categories with duals, it is often mucheasier to work with expressions graphically rather than symbolically, as the eye can easilyspot simplifications.The isomorphism (5) also has a straightforward interpretation in the graphical represen-tation. To obtain S A,B,C ( f ) : B - C ⊗ A ∗ from f : A ⊗ B - C , one simply ‘bends around’the line representing the object A : A BCf A ∗ BCff : A ⊗ B - C θ
A,B,C ( f ) : B - C ⊗ A ∗ The inverse to the isomorphism bends the line around again, and by the ‘straightening-out’rule this gives back the original morphism f .There are many things we can achieve by bending lines. For example, for any morphism h : A - B , we could bend around both the A line and the B line. This gives a morphism h ∗ : B ∗ - A ∗ , which we call the transpose or dual of h . This operation is involutive, so( h ∗ ) ∗ = h . It is also functorial, in the sense that it defines a contravariant functor( − ) ∗ : C op - C satisfying (( − ) ∗ ) ∗ = id C . We call this the duality functor . In our graphical representation,there is a simple way to represent the duality: A ∗ B ∗ h = A ∗ B ∗ h ∗ We ‘straighten out the lines’, rotating the junction box for h by 180 ◦ as we do so.8iven h : A - B , we could also choose to bend around just the A line, giving a morphism p h q : I - B ⊗ A ∗ . We call this the name of h . Given a second morphism k : B - C ,we can perform the composition k ◦ h in terms of the names p h q and p k q by applying theequation (id C ⊗ θ B ⊗ id A ∗ ) ◦ ( p k q ⊗ p h q ) = p k ◦ h q , which is simple to prove using the graphicalrepresentation: C ⊗ A ∗ C ⊗ B ∗ ⊗ B ⊗ A ∗ id C ⊗ θ B ⊗ id A ∗ I p k q ⊗ p h q = k h A ∗ C BB ∗ = k ◦ h A ∗ C = C ⊗ A ∗ I p k ◦ h q A compact-closed category is † -compact-closed , or is a † -category with duals , if there alsoexists a conjugation functor ( − ) ∗ : C - C (8)satisfying (( − ) ∗ ) ∗ = id C , which is compatible with the duality functor, in the sense that(( − ) ∗ ) ∗ = (( − ) ∗ ) ∗ . This composite is then called the adjoint functor or † -functor , as de-scribed in section 2.5, and is denoted in the following way:( − ) † := (( − ) ∗ ) ∗ . (9)We extend our graphical notation in a standard way to represent ( − ) ∗ as flipping over avertical axis, and changing the direction of arrows. A vertical-axis flip commutes with a180 ◦ rotation to produce a horizontal-axis flip, and so we use a horizontal-axis flip alongwith a change of arrow direction to represent ( − ) † .All of these structures appear in FdHilb . The dual of a Hilbert space A given by is itsdual in the usual sense, the Hilbert space of bounded linear operators A - C . Given anybasis v n of A , we obtain a basis of A ∗ from the linear operators v ∗ n := ( − , v n ) A , where ( − , − ) A is the inner product on A . We can then define the unit and counit as ζ A = P n v n ⊗ v ∗ n and θ A ◦ ( v ∗ n ⊗ v m ) = δ nm respectively. These are bounded linear operators, and so are validmorphisms in the category . The transpose of a morphism is the matrix transpose in theusual sense, and the conjugation functor ( − ) ∗ is complex conjugation. This then producesthe adjoint functor ( − ) † as the familiar matrix conjugate-transpose operation.Given the existence of an isomorphism A ≃ A ∗ , we can interpret the unit ζ A : I - A ⊗ A ∗ as preparation of a particular Bell entangled state, and the counit θ A : A ∗ ⊗ A - I as per-forming a Bell measurement. As discussed in [1], this allows use of the graphical calculus toaid design of quantum algorithms, such as the quantum teleportation protocol. A conventional monoid is built from a set S of elements, along with a multiplication map g : S × S - S , where × is the cartesian product, and a unit map u : 1 - S , where 1 is the Unfortunately, they are only bounded because the underlying Hilbert spaces are finite-dimensional. Theinfinite-dimensional case raises significant difficulties, which we discuss in section 6. g and u must satisfy.However, this can be generalised: given any monoidal category C , we can replace thecartesian product × in this definition with the monoidal product ⊗ , and the one-element set 1with the monoidal unit object I . In a symmetric monoidal category, an internal commutativemonoid ( A, g, u ) + consists of an object A , a multiplication morphism g : A ⊗ A - A and aunit morphism u : I - A , which satisfy associativity, unit and commutativity diagrams. Ifthe context is clear, we shall often just refer to them as monoids. We will use the followinggraphical representation for the multiplication and unit morphisms for a monoid: g : A ⊗ A - A u : I - A Here, the vertical lines are all instances of the object A . In terms of this graphical represen-tation, the associativity, unit and commutativity laws are as follows: (10)(11)(12)=Associativity law (10) = =Unit laws (11) =Commutativity law (12)The dual notion is an internal comonoid . An internal comonoid ( A, h, v ) × consists of anobject A , a comultiplication morphism h : A - A ⊗ A and a counit morphism v : A - I ,such that these morphisms satisfy the coassociativity, counit and cocommutativity laws,which are just the associativity, unit and commutativity laws with the arrows reversed. Thesubscript × for a comonoid and + for a monoid is inspired by the behavior of products andcoproducts in category theory: an object together with its diagonal and terminal morphismsforms a comonoid, and an object together with its codiagonal and initial morphisms formsa monoid.We require morphisms of internal monoids to preserve the multiplication and the unit,just as for homomorphisms of conventional monoids. Given a symmetric monoidal category C , and two internal commutative monoids ( A, g, u ) + and ( B, h, v ) + , a morphism m : A - B is a morphism of comonoids if and only if the following diagram commutes: A ⊗ A m ⊗ m - B ⊗ BAg ? m - Bh ? Iu =============== I v (13)10he dual definition, of a morphism of comonoids, is obtained by reversing all the arrowsin this diagram. We use the notion of morphism of monoids to define C + , the category ofinternal commutative monoids in the symmetric monoidal category C , which has internalcommutative monoids in C as objects and morphisms of monoids as arrows. We can similarlyconstruct C × , the category of internal cocommutative comonoids in C .We now look at this in the context of all the structure that we have developed, andconsider a category C with the structure of a † -category with biproducts. We can makethe following observations about internal monoids and comonoids in C , all of which havereasonably simple proofs based on the contents of this section.Firstly, using the † -structure, it is clear that every monoid ( A, g, u ) + gives rise to acomonoid ( A, g † , u † ) × . If m : ( A, g, u ) + - ( B, h, v ) + is a morphism of monoids, then theadjoint morphism m † : ( B, h † , v † ) × - ( A, g † , u † ) × is a morphism of comonoids. It followsthat the categories C × and C + are opposite to each other.We now consider the symmetric monoidal structure. We can use the structural isomor-phisms to define a commutative comonoid on the monoidal unit: I × := ( I, λ − I , id I ) × . (14)For any comonoid ( A, g, u ) × , the only morphism of comonoids ( A, g, u ) × - I × is u itself; inother words, I × is the terminal object in C × . In fact, the category C × has finite products,with binary product and projections defined as follows:( A, g, u ) × × ( B, h, v ) × ≃ (cid:16) A ⊗ B, (id A ⊗ swap ⊗ A,B ⊗ id B ) ◦ ( g ⊗ h ) , u ⊗ v (cid:17) × p ( A,g,u ) × = id A ⊗ vp ( B,h,v ) × = u ⊗ id B The biproduct structure of C gives rise to an initial object in C × . We can define a comonoidon the zero object in the following way:0 × := (0 , , ⊗ , ,I ) × . (15)For any comonoid ( A, g, u ) × , there is only one morphism of comonoids 0 × - ( A, g, u ) × ,given by the zero morphism 0 ,A , and so 0 × is the initial object in C × . Also, there isno morphism ( A, g, u ) × - × unless ( A, g, u ) × is isomorphic to 0 × . If the tensor productdistributes naturally over the biproduct in C , then C × will in fact have coproducts, but wewill not define these here as we do not need them for this paper.The category C × therefore bears a clear resemblance to the structure of many categoriesof spaces, such as the category of sets: it is a category with distinct products and coproducts,such that the only object with a morphism to the initial object is the initial object itself.This gives us the motivation to think of C × as a category of spaces, and for all comonoids( A, g, u ) × , to think of morphisms I × - ( A, g, u ) × in C × as representing its points . In thespecial case of commutative comonoids which are dual to commutative C*-algebras, this is awell-used construction: the points of the comonoid are precisely the elements of the spectrumof the C*-algebra . We note that internal monoids in
FdHilb will not, in general, give rise to C*-algebras: elements of themonoid lack a canonical involution, and the Banach algebra condition || a b || ≤ || a || || b || will not necessarilybe satisfied. In fact, the Banach algebra condition seems quite unnatural in this context.
11f we can find both a comonoid structure (
A, g, u ) × and a monoid structure ( A, h, v ) + onthe same object A , then we can consider the compatibility of the comonoid morphisms g and u with the monoid morphisms h and v . If g and u are both morphisms of monoids, then wesay that the combined structure ( A, g, u, h, v ) × + is a bialgebra . This requires the definition ofa monoid on A ⊗ A ; we choose this monoid to have unit given by v ⊗ v , and multiplication givenby ( h ⊗ h ) ◦ (id A ⊗ swap ⊗ A ⊗ id A ). (This is equivalent to a similar specification of a comonoidon A ⊗ A , and requiring that h and v be morphisms of comonoids.) Diagrammatically, thecompatibility conditions that arise are as follows: A ⊗ A ⊗ A ⊗ A id A ⊗ swap ⊗ A ⊗ id A - A ⊗ A ⊗ A ⊗ AA ⊗ Ag ⊗ g h - A g - A ⊗ Ah ⊗ h ? A u - IIv = = = = = = = = = A g - A ⊗ AIv ≃ - I ⊗ Iv ⊗ v A (cid:27) h A ⊗ AIu ? ≃ - I ⊗ Iu ⊗ u ? The quantum harmonic oscillator is one of the simplest quantum systems that can be studied.It is also one of the most important; in particular, quantum field theory can be interpretedas a perturbation on top of an infinite-dimensional harmonic oscillator. The state space ofthe conventional quantum harmonic oscillator is known as Fock space, a Hilbert space ofcountably-infinite dimension. We will examine it in detail in section 6, to see how it arises asa special case of the more general categorical description. In this section we will give a briefintroduction to the necessary physics, focusing on giving an intuitive description of the statespace of the quantum harmonic oscillator and the tools that physicists use to work with it.The quantum harmonic oscillator is the name given to the mathematical model describinga quantum particle trapped in a quadratic potential. Energy levels of this system are notcontinuous, as they would be for the classical harmonic oscillator; they are discrete, givenby E n := hf ( n + ) for all natural numbers n ≥
0, where h is Planck’s constant and f is thecharacteristic frequency of the system. In particular, we note that the n = 0 state has non-zero energy, which is surprising from the point of view of classical mechanics. Specifyinga state of the quantum harmonic oscillator amounts to specifying an amplitude for thesystem to be found in any of these energy levels. The state space is clearly countably-infinitedimensional, since there is a countable number of energy levels for the system.Inspired by quantum field theory, we will refer to the different energy levels of eachharmonic oscillator as counting numbers of particles ; for example, a harmonic oscillatorwith energy E is in a three-particle state. The lowest energy state is the zero-particle state,or the vacuum. This is a useful language, but perhaps one that should not be taken too12eriously without better motivation. We will explore the particle interpretation further insection 4.4.We shall now describe symmetric Fock space . This is the state space of a compound sys-tem formed from a countable number of harmonic oscillators, but with the extra requirementthat permutation of particles is a symmetry of the state space; our particles are symmetri-cally distinguishable . So, given the symmetric Fock space over two harmonic oscillators, apossible state is one in which there are two particles on the first oscillator; that is, the firstoscillator has energy E and the second oscillator has energy E . But there is no state in thespace for which a first particle is on the first oscillator and a second particle is on the secondoscillator, since such a state is not invariant under interchange of the particles. There is astate with two particles for which each oscillator has energy E , but no information can begathered as to which particle is where.In general, a state of symmetric Fock space is fully described by specifying, separately foreach possible total number of particles, a complex amplitude for each way that this numberof particles could be distributed between all of the available oscillators. Note that sucha specification contains no information about which particle is where, and so respects thesymmetric indistinguishability criterion. If we write A for the single-particle subspace of thisstate space, sufficient to describe the possible ways that a single particle could be distributedamong the oscillators, then we see that the total symmetric Fock space, written F ( A ), hasthe following structure: F ( A ) = C ⊕ A ⊕ ( A ⊗ s A ) ⊕ ( A ⊗ s A ⊗ s A ) ⊕ · · · . The symbol ⊕ represents disjoint union (or direct sum) of state spaces, and the symbol ⊗ s represents symmetric tensor product. We refer to each n -fold symmetric tensor productof A as the n -particle subspace of F ( A ). The zero-particle subspace C consists of a singlecomplex number, since if there are no particles, there is no information to give about howthese particles are arranged; one need only give the amplitude that this is the case.To work with symmetric Fock space, physicists have developed a number of tools. Themost important are the raising and lowering linear operators, a † φ : F ( A ) - F ( A ) and a φ : F ( A ) - F ( A ) respectively, sometimes known as the creation and annihilation operators.Here, φ is a state of A , the single-particle subspace. Applying the raising operator a † φ to an n -particle state of F ( A ) — that is, a state which is zero except in the n -particle subspace —creates an ( n + 1)-particle state of F ( A ), with the same particle content as before, except forthe addition of a new particle described by the state φ . The lowering operator a φ performsthe adjoint to this process, turning an ( n + 1)-particle state into an n -particle state byremoving a particle in the state φ . If there was no amplitude to find a particle in the state φ in the first place, then applying a φ will annihilate the state, giving zero. Also, applyingthe lowering operator to the zero-particle state will give zero.These raising and lowering operators satisfy various commutation relations, describingthe different effects created by applying them in different orders. If we add two particleswith the raising operators a † φ and a † ψ , it should not matter in which order we choose toapply them, and similarly for the case of two different lowering operators. So we expect the The alternative, which we do not consider here, is that the particles be antisymmetrically indistinguish-able, which implies a change of sign under every permutation. φ and ψ : h a ψ , a φ i ⊂ h a † ψ , a † φ i ⊂ a φ and a † ψ , it is not clear that the order should be unimportant. Infact, the physics of the harmonic oscillator tells us that the commutation relation should beas follows: h a φ , a † ψ i ⊂ ( φ, ψ ) idHere, ( φ, ψ ) is the inner product of the two single-particle states, linear in ψ and antilinear in ψ , and id is the identity on the Hilbert space. The raising and lowering operators are obtainedas a combination of the position and momentum operators, and this commutation relationarises directly from Heisenberg’s uncertainty relation, which states that measurements ofposition and momentum do not commute. An intriguingly direct argument is presented in[9], where this commutation relation is related to the fact that there is one more way to firstput a ball into a box and then take a ball out, then there is to first take a ball out of a boxand then put one in.Finally, we describe a family states of symmetric Fock space known as the coherent states.These are parameterised by the single-particle states; for each state φ of A , we write Coh( φ )for the corresponding coherent state of F ( A ). We can construct Coh( φ ) explicitly in thefollowing way: Coh( φ ) := 1 ⊕ φ ⊕ √ φ ⊗ s φ ! ⊕ √ φ ⊗ s φ ⊗ s φ ! ⊕ · · · These states always have finite norm; we have || Coh( φ ) || = e || φ || . Physically, coherent statescontain an indeterminate number of particles, all of which are in the same state. There areother states with these properties that are not of this form, but the defining feature of thecoherent states are that they satisfy the following equation, for all single-particle states φ and ψ : a φ Coh( ψ ) = ( φ, ψ ) Coh( ψ )The coherent states are eigenstates for the lowering operators; intuitively, removing a particlefrom a coherent state only modifies the state by a factor. Two more interesting propertiesenjoyed by the coherent states is that they can be copied and deleted: there exists linearmaps d : F ( A ) - F ( A ) ⊗ F ( A ) and e : F ( A ) - C such that d Coh( φ ) = Coh( φ ) ⊗ Coh( φ ) e Coh( φ ) = 1for all single-particle states φ . For these reasons, the coherent states are often thought of ashaving classical properties. Finally, we note that the coherent states in the form given canbe constructed using the following identity, where we define v , the ‘vacuum state’, to be the14tate in which there is an amplitude of 1 to find zero particles, and no amplitude to find anygreater number: exp( a † φ ) v = Coh( φ )This provides an interesting connection between the zero-particle state, the raising operators,the coherent states and the exponential function.This completes our tour of the classical treatment of the quantum harmonic oscillator. Itwould seem that the most crucial part is the structure of symmetric Fock space; the rest of thestructure could be regarded merely as tools developed by physicists to aid its study. However,we shall see that the categorical approach efficiently reproduces (and indeed generalises) allof the structures and techniques described in this section: not only symmetric Fock space,but also the zero- and single-particle subspaces, the raising and lowering operators andtheir commutation relations, the coherent states and their copying and deleting maps, theexponential function, and all the equations relating them which we have described. We begin with our categorical description of the quantum harmonic oscillator.
Definition 4.1.
Given a symmetric monoidal † -category C with finite † -biproducts, an harmonic oscillator adjunction hh Q, η, ǫ ii is a right adjoint Q : C - C × for the forgetfulfunctor R : C × - C , with unit η : id C × ˙ - Q ◦ R and counit ǫ : R ◦ Q ˙ - id C , that has thefollowing properties:1. The functor Q preserves finite products unitarily;2. The natural transformation ǫ † is an isometry at every stage; that is, ǫ ◦ ǫ † = id id C ;3. The endofunctors R ◦ Q : C - C and † : C - C commute.For each object A in C , we interpret Q ( A ) in C × as the harmonic oscillator constructedover A . Defining the endofunctor F : C - C as F := R ◦ Q, (16)we interpret F ( A ) as the state space of the comonoid Q ( A ), obtained by ‘forgetting’ thecomultiplication and counit morphisms. States of Q ( A ) are therefore given by non-zeromorphisms φ : I - F ( A ) in C , following the general framework of categorical quantummechanics as discussed in section 2.2. F can be thought of a generalised Fock space functor,or a generalised ‘second quantisation’ functor. We also see that the adjunction R ⊣ Q gives F the structure of a comonad . This allows us to update the old adage: “First quantization is a mystery, but second quantization is acomonad.”
15 related approach, developed in parallel to this work by another author [4], is to considerthe comonad (
F, ǫ, RηQ ) as primary rather than the adjunction. This is a more generalframework, but one in which the counit morphisms Rη ( A,g,u ) × will not be available for allcommutative comonoids ( A, g, u ) × . We will make substantial use of these morphisms laterin the paper, and so the current construction is more convenient for our purposes.To work with the comultiplication and counit of each categorical harmonic oscillator Q ( A ) more easily, we make the following definition for the remainder of the paper: (cid:16) F ( A ) , d A , e A (cid:17) × := Q ( A ) . (17)We will explore in the next few sections of this paper how this structure, along with theunit and counit natural transformations, endow Q ( A ) with many of the properties of aconventional quantum harmonic oscillator.Before we go any further, we must make clear what it means in definition 4.1 for Q topreserve products unitarily. Products in C are given by the † -biproduct structure, and in C × they are given by the underlying tensor product in C , as discussed in section 2:( A, g, u ) × × ( B, h, v ) × ≃ (cid:16) A ⊗ B, (id A ⊗ swap ⊗ A,B ⊗ id A ) ◦ ( g ⊗ h ) , u ⊗ v (cid:17) × . As Q is a right adjoint it must preserve finite products up to unique natural isomorphism,and therefore for all A and B in C there exist in C × unique natural isomorphisms k A,B : Q ( A ⊕ B ) - Q ( A ) × Q ( B ) (18) k : Q (0) - I × (19)which make the following diagrams commute: Q ( A ) (cid:27) Q ( π A ) Q ( A ⊕ B ) Q ( π B ) - Q ( B ) Q ( A ) (cid:27) id Q ( A ) × e B Q ( A ) × Q ( B ) k − A,B k A,B ? e A × id Q ( B ) - Q ( B ) Q (0) I × k − k ? We include the trivial right-hand diagram for completeness. For Q to preserve productsunitarily means that the natural isomorphisms k A,B and k are unitary, when viewed asmorphisms in C using the forgetful functor R : Rk − A,B = ( Rk A,B ) † Rk − = ( Rk ) † We can use the natural isomorphisms k A,B and k to obtain explicit expressions for thecomultiplication d A and counit e A associated to each categorical harmonic oscillator Q ( A ). In terms of the comonad rather than the adjunction, morphisms corresponding to the Rη ( A,g,u ) × arise as coalgebras for the comonad. If there exists a coalgebra for every comonoid, then the category of coalgebras(the co-Eilenberg-Moore category) will be equivalent to the category of comonoids, and the comonad mustarise from an adjunction in precisely the sense of definition 4.1. emma 4.2. For all objects A in C , we have e A = Rk ◦ F (0 A, ) (20) d A = Rk A,A ◦ F (∆ A ) (21) where we view e A and d A as morphisms in C rather than C × .Proof. Since terminal morphisms are unique, and terminal objects are preserved by Q , equa-tion (20) must hold automatically. The unit law for Q ( A ) can be reinterpreted as the expres-sion that d A is the diagonal for the object Q ( A ) in C × ; in other words, d A = h id Q ( A ) , id Q ( A ) i in C × . It must therefore be related by c A,A to the image under Q of the diagonal ∆ A := h id A , id A i in C , implying equation (21).In fact, the commutative comonoids in the image of the functor Q have more structurethan is immediately apparent; as discussed in [4], they are bialgebras, for which the multi-plication and unit morphisms are given by the adjoint of the comultiplication and counit in C . Lemma 4.3 (Fiore) . For each object A in C , (cid:16) F ( A ) , d A , e A , d † A , e † A (cid:17) × + is a bialgebra.Proof. In a category with biproducts, the diagonal and codiagonal along with terminal andinitial morphisms form a bialgebra with respect to the monoidal structure of the biproduct, ascan be checked by working through all of the necessary diagrams. This bialgebraic structureis inherited by d A , e A and their adjoints, since they are formed ‘naturally’ from the biproductstructure in C , the unitary isomorphisms k A,B and k translating between the biproduct andtensor product structures.We can also prove that the comultiplication d A is additive , in the following sense. Lemma 4.4.
For all morphisms f , g : A - B in C , F ( f + g ) = d † B ◦ ( F ( f ) ⊗ F ( g )) ◦ d A . (22) where the sum f + g is defined by the † -biproduct structure.Proof. Using using naturality of c A,A and compatibility of F with ( − ) † , we obtain: F ( A ) F (∆ A ) - F ( A ⊕ A ) Rk A,A - F ( A ) ⊗ F ( A ) F ( f ) ⊗ F ( g ) - F ( B ) ⊗ F ( B ) ( Rk B,B ) † - F ( B ⊕ B ) F (∆ B ) † - F ( B ) F ( A ⊕ A ) K ( f ⊕ g ) - F ( B ⊕ B ) Rk B,B - F ( B ) ⊗ F ( B ) F ( B ⊕ B ) F ( ∇ B ) - F ( B ) F ( B ⊕ B ) ========================= F ( B ⊕ B ) F ( A ) F ( f + g ) - F ( B ) Although the natural transformations e and ǫ are not directly related by any equationsin the construction of the harmonic oscillator adjunction, they nevertheless automaticallysatisfy some compatibility conditions. 17 emma 4.5. The natural transformations ǫ and e are normalised and orthogonal; that is,at every stage A , ǫ A ◦ ǫ † A = id A (23) e A ◦ e † A = id I (24) e A ◦ ǫ † A = 0 A,I (25)
Proof.
Equation (23) holds by construction, since it was a requirement in definition 4.1 ofa harmonic oscillator adjunction. To tackle equation (24), we use lemma 4.2 to write e A interms of k ; we then obtain e A ◦ e † A = Rk ◦ F (0 A, ) ◦ F (0 ,A ) ◦ ( Rk ) † = Rk ◦ F (0 , ) ◦ ( Rk ) † = Rk ◦ id F (0) ◦ ( Rk ) † = id I , where we have employed unitarity of Rk . For equation (25), we have e A ◦ ǫ † A = Rk ◦ F (0 A, ) ◦ ǫ † A = Rk ◦ ǫ † ◦ A, = 0 A,I , where we use naturality of ǫ † and the fact that only zero morphisms can factor through zeromorphisms.In fact, as is well-known in other contexts [3], we can always write k A,B directly in terms of e and ǫ in the following way. Lemma 4.6.
Under the canonical hom-set isomorphism H Q ( A ) × Q ( B ) ,A ⊕ B : Hom C (cid:16) F ( A ) ⊗ F ( B ) , A ⊕ B (cid:17) ≃ Hom C × (cid:16) Q ( A ) × Q ( B ) , Q ( A ⊕ B ) (cid:17) induced by the adjunction, the family of morphisms in C given by r A,B := i A ǫ A ⊗ e B + e A ⊗ i B ǫ B , where i A and i B are canonical injections into the biproduct A ⊕ B , produce the morphisms k − A,B in C × ; that is, for all A and B in C , k − A,B = H Q ( A ) × Q ( B ) ,A ⊕ B ( r A,B ) . (26) Proof.
The morphisms H Q ( A ) × Q ( B ) ,A ⊕ B ( r A,B ) defined in this manner are clearly well-definedmorphisms of comonoids. We must show that it mediates between the product structuresin C × ; it suffices to show that for any A and B , Q ( i † A ) ◦ k − A,B = id Q ( A ) × e B . These are18orphisms in Hom C × ( Q ( A ) × Q ( B ) , Q ( A )), and we apply the hom-set isomorphism onceagain to view them as morphisms in Hom C ( F ( A ) ⊗ F ( B ) , A ). We obtain ǫ A ◦ F ( i † A ) ◦ R ( H Q ( A ) × Q ( B ) ,A ⊕ B ( r A,B ))= ǫ A ◦ F ( i † A ) ◦ F ( i A ǫ A ⊗ e B + e A ⊗ i B ǫ B ) ◦ Rη Q ( A ) × Q ( B ) = ǫ A ◦ F ( ǫ A ⊗ e B ) ◦ Rη Q ( A ) × Q ( B ) = ( ǫ A ⊗ e B ) ◦ ǫ F ( F ( A ) ⊗ F ( B )) ◦ Rη Q ( A ) × Q ( B ) = ǫ A ⊗ e B . But this is equal to ǫ A ◦ R (id Q ( A ) × e B ), and so the product-preservation equation holds.We can use this to prove another very useful result. Lemma 4.7.
At any stage A , we can write ǫ A ◦ d † A : F ( A ) ⊗ F ( A ) - A as ǫ A ◦ d † A = ǫ A ⊗ e A + e A ⊗ ǫ A . (27) Proof.
Using lemmas 4.2 and 4.6, along with naturality of ǫ and one of the adjunctionequations, we obtain ǫ A ◦ d † A = ǫ A ◦ F ( ∇ A ) ◦ F ( i A ǫ A ⊗ e A + e A ⊗ i A ǫ A ) ◦ Rη Q ( A ) × Q ( A ) = ǫ A ◦ F ( ǫ A ⊗ e A + e A ⊗ ǫ A ) ◦ Rη Q ( A ) × Q ( A ) = ( ǫ A ⊗ e A + e A ⊗ ǫ A ) ◦ ǫ F ( F ( A ) ⊗ F ( A )) ◦ Rη Q ( A ) × Q ( A ) = ǫ A ⊗ e A + e A ⊗ ǫ A . We will develop a graphical representation for the extra structure associated with a categor-ical harmonic oscillator, which will help us to prove theorems more easily. The most basicstructure is the functor F : C - C . We represent it as a pair of dashed lines, one oneach side of its argument, for its action on both objects and morphisms. Functoriality of F then means that graphical components within the dashed lines can be manipulated as if thedashed lines were not there; F acts as an ‘inert container’. This principle is illustrated inthe following diagram, which holds for all f : A - B and g : B - C in C : F ( A ) F ( C ) F ( B ) fg = F ( A ) F ( C ) g ◦ f
19o represent the comultiplication d A and counit e A graphically for each comonoid Q ( A ),we extend the graphical representation for comonoids which we developed in section 2. Thegraphical components that we will use are as follows, defined for all objects A in C : (28)(29) d A : F ( A ) - F ( A ) ⊗ F ( A ) e A : F ( A ) - I As we have done here, we will often not annotate our diagrams whenever this does notintroduce ambiguities. We will also frequently work with the adjoints d † A and e † A withoutremark; following the conventions laid out in section 2, these are given by flipping thediagrams along a horizontal axis, and then reversing the orientation of the arrows. Ofcourse, these graphical components obey the dual versions of the associativity, unit andcommutativity laws (10), (11) and (12).Finally, we introduce the representations for the unit and counit natural transformations.For each stage η ( A,g,u ) × : ( A, g, u ) × - Q ( A ) ,ǫ A : F ( A ) - A, we employ the following diagrams: Rη ( A,g,u ) × Rη ( A,g,u ) × : A - F ( A ) ǫ A : F ( A ) - A The graphical representation of ǫ A features an unblocked ‘mouth’ for the dashed lines. Thisis an intuitive way to represent of the naturality of ǫ , which can be written algebraically as ǫ B ◦ F ( f ) = f ◦ ǫ A for any f : A - B . In other words, graphical components can be freelymoved across e A , into and out of the functor F , traversing the ‘mouth’ of the dashed lines: f = f For Rη ( A,g,u ) × , however, we mark the end of the functor F by a double line; this is ‘harder’ forgraphical components to cross, as naturality of η implies that only morphisms of comonoidsmay pass. 20inally, there are compatibility equations satisfied by the natural transformations η and ǫ which define the adjunction. We summarise these here, along with their graphical repre-sentations. Q ηQ - Q ◦ R ◦ QQQǫ ? i d Q - Rη Q ( A ) = (30) R Rη - R ◦ Q ◦ RRǫR ? i d R - Rη ( A,g,u ) × = (31)These are the graphical representations for arbitrary stages A and ( A, g, u ) × respectively.These adjunction equations are of course entirely category-theoretical, but they will proveessential to the physics, allowing us to demonstrate in theorem 4.13 that coherent statesare eigenstates of the lowering operators, and in theorem 5.9 that the coherent state can bewritten as the exponential of the raising operator. In this section, we will construct generalised raising and lowering operators associated to thecategorical harmonic oscillators.For all morphisms f : A - B in C , Q ( f ) : Q ( A ) - Q ( B ) is a morphism of comonoidsby the definition of Q , and so must satisfy the following diagram drawn in C : F ( A ) ⊗ F ( A ) F ( f ) ⊗ F ( f ) - F ( B ) ⊗ F ( B ) F ( A ) d A F ( f ) - F ( B ) d B Ie A ? ======================= I e B ? (32)However, this is exactly the condition for d A and e A to be the stages of natural transforma-tions d : F - F ⊗ Fe : F - C I C I : C - C is the functor that sends all objects in C to the monoidal identity object I , and all morphisms to id I . We therefore have four basic natural transformations, arisingfrom the comonoid structure and from the adjunction, which we can summarise using thefollowing diagram: F ⊗ F ComonoidC I (cid:27) e Fd RηQ - F F
Adjunction id C ǫ ? (33)We will define some new natural transformations, and use them to define the raising andlowering morphisms. Definition 4.8.
The lowering natural transformation a : F ˙ - F ⊗ id C is defined as thefollowing composite natural transformation: F a - F ⊗ id C F d - F ⊗ F id F ⊗ ǫ - F ⊗ id C (34)The raising natural transformation a † : F ⊗ id C ˙ - F is the adjoint to this. Definition 4.9.
The lowering morphism a φ : F ( A ) - F ( A ) associated to the state φ : I - F ( A ) is defined as follows: F ( A ) a φ - F ( A ) F ( A ) a A - F ( A ) ⊗ A id F ( A ) ⊗ φ † - F ( A ) (35)The raising morphism a † φ : F ( A ) - F ( A ) associated to φ is defined similarly: F ( A ) a † φ - F ( A ) F ( A ) id F ( A ) ⊗ φ - F ( A ) ⊗ A a † A - F ( A ) (36)The lowering morphism a φ has a state as its subscript, and the stage a A of the loweringnatural transformation has an object as its subscript, so they can be differentiated. Ofcourse, the raising and lowering morphisms are related directly by the † functor: a † φ ≡ ( a φ ) † . φ φ † Raising morphism Lowering morphism a † φ : F ( A ) - F ( A ) a φ : F ( A ) - F ( A )We apply the raising morphism a † φ to F ( A ) by creating a new state φ of A , embedding itinto F ( A ), then multiplying with our original instance of F ( A ). To lower with respect to φ ,we comultiply, then extract the single-particle state from one of the legs and take the innerproduct with φ .A cornerstone of the conventional analysis of the quantum harmonic oscillator is the setof canonical commutation relations, or CCRs, as described in section 3: h a φ , a † ψ i ⊂ ( φ, ψ ) id h a † φ , a † ψ i ⊂ h a φ , a ψ i ⊂ Theorem 4.10.
For all objects A in C , and all states φ, ψ : I - A , the following commu-tation relations hold: a φ ◦ a † ψ = a † ψ ◦ a φ + ( φ † ◦ ψ ) · id F ( A ) (37) a † φ ◦ a † ψ = a † ψ ◦ a † φ (38) a φ ◦ a ψ = a ψ ◦ a φ (39)We first note that the domain issue in the conventional case is not relevant here; indeed, itis not even expressible. Secondly, we have rearranged the equations to some extent, as wewill not in general have a way to express negatives in our category. Proof.
Given the definitions of the categorical raising and lowering morphisms, equations (38)and (39) follow straightforwardly from cocommutativity and coassociativity of the comonoid Q ( A ). We demonstrate this using the graphical representation for the a † commutation23elation; the proof for a is analogous. F ( A ) F ( A ) a † φ F ( A ) a † ψ = φ ψ = φ ψ = ψ φ = ψ φ = F ( A ) F ( A ) a † ψ F ( A ) a † φ A in C . a A ◦ a † A = == ++ += + 0 + 0 += a † A ◦ a A + id A ⊗ id F ( A ) . .4 Physical interpretation and path-counting For each object A in C , F ( A ) is the generalised Fock space over A . We interpret the state e † A : I - F ( A ) as containing no ‘particles’; we think of it as the ‘vacuum state’. The mor-phism ǫ † A : A - F ( A ) injects the ‘single-particle space’ into F ( A ); if a state of F ( A ) factorsthrough this arrow, then we say that it is a ‘single-particle state’. We make this interpreta-tion not only because it mirrors the particle interpretation for Fock space in quantum fieldtheory; in fact, we will argue that it fits naturally with the extra structure that we haveavailable.The multiplication d † A : F ( A ) ⊗ F ( A ) - F ( A ) is interpreted as combining the statesof two identical physical systems, to produce a state of a single physical system. Applyingthe adjoint, we obtain d A : F ( A ) - F ( A ) ⊗ F ( A ), which we interpret as constructing thesuperposition of every possible way of splitting the state of the system between two systems;or alternatively, of splitting the particle content of the system between two systems. Theidentity d A ◦ ǫ † A = ǫ † A ⊗ e † A + e † A ⊗ ǫ † A now becomes transparent: splitting a single-particle statebetween two systems can be done in one of two ways, placing the single particle in eitherthe first or second system, the other system being left in the vacuum state.An even more concrete physical analogy is to think of d A : F ( A ) - F ( A ) ⊗ F ( A ) as a beamsplitter , a half-silvered mirror which splits an obliquely incident stream of photons intotwo separate outgoing beams, one which passes through the mirror and one which reflectsfrom it. In general, these outgoing beams will be entangled with one another, in state whichis a superposition of all of the possible ways that the photon content of the incident beamcould be distributed between the two outgoing beams.We interpret the counit morphism e A : F ( A ) - I as a measurement of the vacuumstate. Essentially, it asserts that there are no particles present, and in doing so contributesa scalar factor representing the amplitude that this is the case.We will see that this ‘particle interpretation’ of the components of the categorical struc-ture gives rise to a useful algorithm for evaluating a large class of morphism compositions.The following diagrams must hold in our framework, the first by additivity of the comulti-plication and the second by functoriality of F : (40)(41) n m = n + m nm = m ◦ n (40) (41)The symbols n and m are scalars I - I in C , and n + m is their sum as defined by thebiproduct structure. The scalars are understood to be evaluated inside the functor F , inthe sense F ( n · id A ). We can evaluate these diagrams by imagining a single, classical particlepassing along them, in the direction of the arrows. For each possible route that the particlecould take, we calculate a total amplitude for that route by composing the scalars m or26 through which the particle passes. The total amplitude w for the process is the sum ofthe amplitudes for each of the different routes, and the value for the diagram is F ( w · id A ).Diagram (40) has two paths, contributing amplitudes n and m respectively; diagram (41)has one path, contributing an amplitude n ◦ m .This can be considered an application of the path-integral approach for quantum me-chanics: the amplitude for a quantum process is a given by a sum of the amplitudes for eachof the classical ways that the process could be completed. The novel aspect here is that thebranching point at which the classical particle must make an exclusive choice is an explicitpart of the theory, represented by the comultiplication morphisms d A .With this evaluation algorithm in mind, we would expect the following diagrams toevaluate to the same morphism, since in each case, there is only one path from each ‘in’ legto each ‘out’ leg: = (42)But this is precisely one of the axioms of a bialgebra, which we have already established inlemma 4.3.We can now appreciate the difficulties that would be introduced here if our category hadduals. In that case, we would be able to construct closed loops, and there could be an infinitenumber of paths by which to navigate a finite diagram in our category. However, lackinginfinite scalars in general, we would not be able to make sense of this. The harmonic oscillator adjunction R ⊣ Q implies particular isomorphisms of hom-sets, bythe basic definition of an adjunction. Specifically, for all ( A, g, u ) × in C × and all B in C wehave a natural isomorphism of sets: H ( A,g,u ) × ,B : Hom C ( A, B ) ≃ Hom C × (( A, g, u ) × , Q ( B )) . (43)We recall the explicit definition for this isomorphism in terms of the unit and counit of theadjunction. If morphisms f : A - B in C and f ′ : ( A, g, u ) × - Q ( B ) in C × are relatedby the isomorphism, then we must have the following: H ( A,g,u ) × ,B ( f ) := Q ( f ) ◦ η ( A,g,u ) × = f ′ (44) H − A,g,u ) × ,B ( f ′ ) := ǫ B ◦ R ( f ′ ) = f. (45)Generally, objects A in C do not have a natural cocommutative comonoid structure, sogiven only an f : A - B there will be no canonical choice of an ( A, g, u ) × with which27o implement equation (44), producing a morphism of comonoids in C × . But there areexceptions: in particular, the monoidal unit object I in C has a natural cocommutativecomonoid structure I × := ( I, λ † I , id I ), which is the terminal object in C × as discussed insection 2.Making this natural choice of I × for the comonoid on I , we can establish the followingisomorphism for all objects A in C : H I × ,A : Hom C ( I, A ) ≃ Hom C × ( I × , Q ( A )) . If we interpret elements of Hom C ( I, A ) as states of A , then this tells us that there is anisomorphism between states of A in C and points of the comonoid Q ( A ) in C × . We cancalculate the morphisms of comonoids to which these points correspond: given a state φ : I - A in C , we use equation (44) to obtain a composite morphism H I × ,A ( φ ) = I × η I × - ( F ( I ) , d I , e I ) Q ( φ ) - ( F ( A ) , d A , e A ) . We interpret this as a generalised coherent state.
Definition 4.11.
The coherent state
Coh( φ ) : I - F ( A ) associated to the single-particlestate φ : I - A is given by Coh( φ ) := F ( φ ) ◦ Rη I × . (46)We make this interpretation because states of this form have the properties that we expectfrom coherent states, as described in section 3: they can be copied, and they are eigenstatesof the lowering morphisms. Theorem 4.12.
Coherent states are copied and deleted by the comultiplication and counitmorphisms defined by the harmonic oscillator adjunction, as described by the following equa-tions@ d A ◦ Coh( φ ) = Coh( φ ) ⊗ Coh( φ ) (47) e A ◦ Coh( φ ) = id I . (48) Proof.
To prove equation (47), we use the fact that F ( φ ) and Rη I × are morphisms ofcomonoids. I Coh( φ ) - F ( A ) d A - F ( A ) ⊗ F ( A ) I Rη I × - F ( I ) F ( φ ) - F ( A ) F ( I ) d I - F ( I ) ⊗ F ( I ) F ( φ ) ⊗ F ( φ ) - F ( A ) ⊗ F ( A ) I d = λ † I - I ⊗ I Rη I × ⊗ Rη I × - F ( I ) ⊗ F ( I ) I Coh( φ ) ⊗ Coh( φ ) - F ( A ) ⊗ F ( A ) We note that this construction can still be made in the case of a model of linear logic on a category withbiproducts, such as that described in [4], where in place of Rη I × we have δ : F (0) ≃ I - F F (0) ≃ F ( I ),where δ : F ˙ - F F is the comultiplication for the comonad. e A ◦ Coh( φ ) = e A ◦ F ( φ ) ◦ Rη I × is in the image of the hom-setHom C × ( I × , I × ) under the functor R : C × - C . But I × is terminal, and functoriality of R then implies e A ◦ Coh( φ ) = id I . Theorem 4.13.
Coherent states are eigenstates of the lowering morphisms, satisfying theequation a ψ ◦ Coh( φ ) = ( ψ † ◦ φ ) · Coh( φ ) . (49) Proof.
The theorem is proved by the following commuting diagram. In order, we employ that F ( φ ) is a morphism of comonoids, that η I × is a morphism of comonoids, the construction ofthe H I × ,A isomorphism, and the properties of the symmetric monoidal structure. I Coh( φ ) - F ( A ) a ψ - F ( A ) I η I × - F ( I ) F ( φ ) - F ( A ) d A - F ( A ) id F ( A ) ⊗ ǫ A - F ( A ) ⊗ A id A ⊗ ψ † - F ( A ) F ( I ) d I - F ( I ) ⊗ F ( φ ) ⊗ - F ( A ) I λ − I - I ⊗ I Rη ⊗ I × - F ( I ) ⊗ I ⊗ I F ( φ ) ◦ Rη I × (cid:17) ⊗ φ - F ( A ) ⊗ A I ( ψ † ◦ φ ) · Coh( φ ) - F ( A ) We also give the same proof using the graphical calculus. In this form, it is much morereadable, although it has exactly the same content as the previous symbolic proof. Here, thedouble horizontal parallel lines all represent the morphism Rη I × : I - F ( I ). The explicituse of the adjunction equation (31) is very clear here. φ ψ † = φ φ ψ † = φ ψ † φ Let C be a symmetric monoidal † -category with † -biproducts. We will see how a harmonicoscillator adjunction hh Q, η, ǫ ii can be used to create exponentials of elements of commutative29onoids in C ; that is, we will use commutative monoids ( A, g, u ) + to convert states φ : I - A in C into exponential states exp ( A,g,u ) + ( φ ) : I - A in C .Since we will often find ourselves needing to work with with commutative monoids aswell as cocommutative comonoids, we first introduce some new notation. Definition 5.1.
Given a free cocommutative comonoid functor Q : C - C × , we define theassociated free commutative monoid functor to be Q + : C - C + . It is therefore naturalto refer to Q itself as Q × . We also define the associated forgetful functors R × : C × - C and R + : C + - C . The adjunction R × ⊣ Q × with unit and counit η × and ǫ × induces anadjunction Q + ⊣ R + with unit and counit ǫ + and η + , where for all monoids ( A, g, u ) + and C -objects A , R + η +( A,g,u ) + = ( R × η × ( A,g † ,u † ) × ) † and ǫ + A = ( ǫ × A ) † . Definition 5.2.
Given a commutative monoid (
A, g, u ) + in C , and a state φ : I - A in C ,then the exponential exp ( A,g,u ) + ( φ ) : I - A in C is defined in the following way, where wegive both the diagrammatic and graphical representations: I exp ( A,g,u ) + ( φ ) - AI R × η × I × - F ( I ) F ( φ ) - F ( A ) R + η +( A,g,u ) + - A φ R + η +( A,g,u ) + R × η × I × (50)The state φ is evaluated ‘inside a box’ built from the categorical structure. Intuitively, wecan think of exp ( A,g,u ) + ( φ ) as being given by the infinite sumexp ( A,g,u ) + ( φ ) = (cid:18) · u (cid:19) + (cid:18) · φ (cid:19) + (cid:18) · g ◦ ( φ ⊗ φ ) (cid:19) + (cid:18) · g ◦ ( g ⊗ id A ) ◦ ( φ ⊗ φ ⊗ φ ) (cid:19) + . . . , (51)as will in fact be the case in a suitable category of Hilbert spaces. The correspondence withthe conventional notion of exponential is very clear.This construction has several nice properties. First, we demonstrate that these expo-nentials behave well under composition, and that the unit for the monoid is given by theexponential of the zero morphism. Lemma 5.3.
Exponentials of elements of commutative monoids compose additively; that is,for morphisms φ, ψ : I - A and ( A, g, u ) + , g † ◦ (cid:16) exp( φ ) ⊗ exp( ψ ) (cid:17) = exp( φ + ψ ) (52) where the exponentials are defined with respect to the monoid ( A, g, u ) + , and φ + ψ is definedby the biproduct structure in the underlying category. roof. We can prove the lemma in a straightforward way using the graphical representation.We employ that R × η × I × is a morphism of comonoids, that R + η +( A,g,u ) + is a morphism ofmonoids, and the additivity of the comultiplication for comonoids in the image of Q . φ ψR × η × I × R + η +( A,g,u ) + R × η × I × R + η +( A,g,u ) + = φ ψR × η × I × R + η +( A,g,u ) + = φ + ψ R × η × I × R + η +( A,g,u ) + Lemma 5.4.
For all monoids ( A, g, u ) + , we have exp ( A,g,u ) + (0 I,A ) = u. (53) Proof.
We demonstrate the lemma with the following diagram. We introduce the identityin the form Rk − ◦ Rk , and employ the fact that I × is terminal in C × , the definition of e † A in terms of F (0 I, ), and the fact that R + η +( A,g,u ) + is a morphism of monoids. I R × η × I × - F ( I ) F (0 I,A ) - F ( A ) R + η +( A,g,u ) + - AF ( I ) F (0 I, ) - F (0) F (0 ,A ) - F ( A ) F (0) Rk - = = = = = = = I Rk − - F (0) ======= I ====================== I e † A - F ( A ) I u - A Lemma 5.5.
For any pair of commutative monoids ( A, g, u ) + and ( B, h, v ) + , and a mor-phism of monoids m : ( A, g, u ) + - ( B, h, v ) + , the following naturality condition holds forall φ : I - A : exp ( B,h,v ) + ( m ◦ φ ) = m ◦ exp ( A,g,u ) + ( φ ) . (54) Proof.
Straightforward, from naturality of η .Intuitively, this implies that the exponential construction is only sensitive to the smallestsubmonoid containing the element to be exponentiated.Given an element α N of a noncommutative monoid ( N, k, s ) n+ , where the subscript ‘n’stands for noncommutative, we cannot apply this exponential construction directly, as thecounit η + associated to the harmonic oscillator adjunction is only defined for commutativemonoids. However, inspired by lemma 5.5, if we have some commutative monoid ( A, g, u ) + and a morphism of monoids m : ( A, g, u ) + - ( N, k, s ) n + such that for some α A : I - A we have m ◦ α A = α N , then we may defineexp ( M,k,s ) n + ( α M ) := m ◦ exp ( A,g,u ) + ( α A ) . (55)We calculate the exponential using the commutative monoid, and embed the result into N by using m . A common application of exponentials in functional analysis is to construct the exponentialof an operator on a Hilbert space, defined using the familiar power series expansion. Theanalogue in our setting is to construct the exponential of an endomorphism f : A - A .To apply the generalised exponential construction which we have just developed, it seemsthat we would require a monoid which ‘knows’ about arrow composition. Such a monoid iscanonically present if our category has duals , as defined in section 2.6. Definition 5.6.
In a monoidal category with duals, the endomorphism monoid (cid:13) A n+ on anobject A is defined in the following way: (cid:13) A n+ := (cid:16) A ⊗ A ∗ , id A ⊗ θ A ⊗ id A , ζ A (cid:17) n+ . Intuitively, elements of the monoid are names of endomorphisms on A , multiplication is en-domorphism composition, and the unit for the monoid is the name of the identity morphism.We can use this monoid to define endomorphism exponentials. This is no great surprise: a monoidal category has duals if, seen as a 2-category in a particular canonicalway, each 1-morphism has a categorical adjoint. But each such adjunction then induces a monad in thefamiliar way, and this monad is precisely the monoid that we define here. efinition 5.7. In a symmetric monoidal † -category with duals and † -biproducts, and aharmonic oscillator adjunction, the name of the exponential of an arbitrary endomorphism f : A - A is given by p exp ( f ) q := exp (cid:13) A n+ ( p f q ) , (56)where the exponential over a noncommutative monoid is defined by equation (55).To construct the exponential of a particular operator, we would therefore need to find acommutative monoid, which embeds as a monoid into (cid:13) A n+ , and through which the nameof our operator factors. Although this is not straightforward, it intuitively seems likely thateach endomorphism should generate a commutative monoid, and that this generated monoidshould embed into the full endomorphism monoid. While it may not be straightforward inpractice to construct such a commutative monoid, there does exist a canonical embeddinginto the endomorphism monoid. Lemma 5.8.
In a monoidal category with duals, any monoid ( N, k, s ) n + has a canonicalmonic embedding m ( N,k,s ) n+ : ( N, k, s ) n+ - (cid:13) N n+ into the endomorphism monoid on N ,given by m ( N,k,s ) n+ := ( k ⊗ id N ∗ ) ◦ (id N ⊗ ζ N ) . (57)Note that, in particular, commutative monoids will also have such an embedding. Proof.
The embedding has the following graphical representation: N ⊗ N ∗ Nm ( N,k,s ) n+ =Intuitively, each element of the monoid ( N, k, s ) n+ is taken to the name of the endomor-phism of A which performs multiplication by that element. It is simple to demonstrate that m ( N,k,s ) n+ is monic, since it has a retraction: (id N ⊗ s ∗ ) ◦ m ( N,k,s ) n + = id N , although notethat this retraction will not be a morphism of monoids in general.We first show that m ( N,k,s ) n + preserves multiplication. We must demonstrate that follow-ing diagram commutes: N ⊗ N m ( N,k,s ) n+ ⊗ m ( N,k,s ) n+ - N ⊗ N ∗ ⊗ N ⊗ N ∗ id N ⊗ θ N ⊗ id N ∗ - N ⊗ N ∗ N ⊗ N k - N m ( N,k,s ) n+ - N ⊗ N ∗ (58)Using the graphical representation, it can be seen that this equation holds, using the dual33quations and associativity of the multiplication: N ⊗ N ∗ N ⊗ N ∗ ⊗ N ⊗ N ∗ id N ⊗ θ N ⊗ id N N ⊗ N m ⊗ N,k,s ) n+ = = = N ⊗ N ∗ N m ( N,k,s ) n+ N ⊗ N k Secondly, we show that m ( N,k,s ) n+ preserves units, where we employ the unit law for themonoid m ( N,k,s ) n+ : N ⊗ N ∗ Nm ( N,k,s ) n+ Is = = = N ⊗ N ∗ Iζ N So m ( N,k,s ) n+ is a well-defined morphism of monoids, for all monoids ( N, k, s ) n+ .We now describe a quantum-mechanical application of this construction. From the theoryof the conventional quantum harmonic oscillator, we expect that for all objects A and for all φ : I - A in C , Coh( φ ) = exp( a † φ ) ◦ e † A , (59)where exp( a † φ ) is the endomorphism exponential of a † φ .The name p a † φ q of the raising morphism a † φ : F ( A ) - F ( A ) is an element of the endo-morphism monoid (cid:13) F ( A )n+ . Noting that p a † φ q = m Q + ( A ) ◦ ǫ † A ◦ φ , we apply equation (55) toobtain exp (cid:13) F ( A )n+ (cid:16) p a † φ q (cid:17) = exp (cid:13) F ( A )n+ (cid:16) m Q + ( A ) ◦ ǫ † A ◦ φ (cid:17) = m Q + ( A ) ◦ exp Q + ( A ) (cid:16) ǫ † A ◦ φ (cid:17) This simplifies further, using the following lemma.
Lemma 5.9.
The coherent state is expressed as an exponential in the following way:
Coh( φ ) = exp Q + ( A ) (cid:16) ǫ † A ◦ φ (cid:17) . (60) Associativity of (
N, k, s ) + is clearly necessary for performing this embedding. For this reason, is doesnot seem useful to consider the commutative Jordan ‘algebra’ generated by the monoid, as this will fail tobe associative in general. roof. Writing this out with the graphical calculus, the result follows straightforwardly fromthe adjunction equation (30).Writing the endomorphism exponential exp( a † φ ) in terms of its name, we obtainexp( a † φ ) ≡ (id A ⊗ θ A ) ◦ (cid:16) p exp( a † φ ) q ⊗ id A (cid:17) = (id A ⊗ θ A ) ◦ (cid:16) exp (cid:13) F ( A )n+ (cid:16) p a † φ q (cid:17) ⊗ id A (cid:17) = (id A ⊗ θ A ) ◦ (cid:16)(cid:16) m Q + ( A ) ◦ Coh( φ ) (cid:17) ⊗ id A (cid:17) . Writing graphically and simplifying, we obtain F ( A ) F ( A )exp( a † φ ) = φ R × η × I × (61) Theorem 5.10.
The exponential of the raising morphism applied to the zero-particle stateproduces the coherent state:
Coh( φ ) = exp( a † φ ) ◦ e † A . ( ) Proof.
The proof is immediate from the graphical representation (61) and the unit law forthe monoid Q + ( A ). In this section, we will construct a categorical harmonic oscillator from a category of Hilbertspaces. The structures that we obtain will be those used to study the harmonic oscillator inconventional quantum mechanics. However, constructing the category in which we will workwill not be straightforward. We will unavoidably need to work with infinite-dimensionalseparable Hilbert spaces, and with unbounded linear maps, but the category of separableHilbert spaces and all unbounded linear maps is not well-defined, as two unbounded maps f : A - B and g : B - C may not have a well-defined composite g ◦ f . We overcomethis problem by working instead with inner product spaces which are not necessarily com-plete; this will allow us to include a restricted class of unbounded maps in our category.A second problem is that the trace operation for an infinite-dimensional Hilbert space is A separable Hilbert space is one of finite or countably-infinite dimension.
Definition 6.1.
The category
Inner has objects given by countable-dimensional complexinner-product spaces. Morphisms are given by everywhere-defined linear maps f : A - B ,such that f † : B - A is also everywhere-defined.Note that this category will contain both complete and non-complete inner-product spaces. Lemma 6.2.
The category
Inner is a symmetric monoidal † -category with † -biproducts.Proof. The symmetric monoidal structure is given by the tensor product of inner-productspaces, the monoidal unit object being given by the one-dimensional space C . The † -structureis given by the adjoint, which by construction is well-defined. The † -biproduct structure isgiven by the direct sum of inner-product spaces, along with the canonical normalised andorthogonal projections.It may seem that we have abandoned Hilbert spaces, but we have done so only in a verymild way: every inner-product space A has a canonical embedding into its completion ¯ A ,which is a Hilbert space. Moreover, all bounded linear operators A - B can be extendedto bounded linear operators ¯ A - B , and further to bounded linear operators ¯ A - ¯ B . Infact, this new definition makes no difference at all in the finite-dimensional case, since everyfinite-dimensional inner product space is complete, and therefore a Hilbert space; as a result, FdHilb is a full subcategory of
Inner . However, in the infinite-dimensional case it makes abig difference: our unbounded operators can now be everywhere-defined, which ensures thatthey will always compose well with other morphisms.Unfortunately, we will not be able to find duals in this category. The trace on an infinite-dimensional inner-product space A , considered as a linear map tr A : A ∗ ⊗ A - C whichtakes the name of an operator to its trace, is not bounded, as the trace of a normalisedoperator name can be arbitrarily large. This is not a problem; the set of operator nameshaving finite trace (the trace-class operators) do form an inner-product space. But theadjoint to this map has empty domain: tr † A (1) = p id A q , but tr A ( p id A q ) is infinite, and sotr † A (1) is not an element of the space. This causes a problem for the construction of duals,for which we require a trace operator θ A : A ∗ ⊗ A - C and its transpose, the name ofthe identity operator p id A q ≡ ζ A : C - A ⊗ A ∗ , which we have shown cannot exist. Tosummarise: although the trace is a well-defined linear operator, it must be left out of ourcategory since we lack an infinite scalar to represent the trace of the identity. Duals do notplay a big part in our construction, but this does mean that we will be unable to implementthe endomorphism exponential construction. Although we are primarily interested in the category
Inner at this point, the constructionof the adjunction that we are going to make can be carried out in a wide class of categories.We work with a symmetric monoidal † -category C , with countably-infinite † -biproducts,sufficient countably-infinite sums, and symmetric † -subspaces (which we define below.) Note36hat, although we require the existence of a countably-infinite extension to the † -biproductstructure, we do not require the existence of countably-infinite diagonals and codiagonals.Rather, we require for each countable set A i of objects in C the existence of a † -biproductobject L i A i , along with canonical projection morphisms π j : L i A i - A j for each elementof the biproduct. Definition 6.3.
The n-fold tensor product functor T n : C - C is defined on all objects A and all morphisms f as T n ( A ) := O n A =: A ⊗ n T n ( f ) := O n f =: f ⊗ n A ⊗ is defined to be I , the tensor unit, and f ⊗ is id I . Definition 6.4.
Given a tensor product object A ⊗ n , we write (swap ⊗ A ) i,j : A ⊗ n - A ⊗ n forthe symmetry isomorphism which exchanges the i th and j th factors of the tensor product. Definition 6.5.
We define the n -fold symmetric † -subspace A ⊗ s n of an object A as thefollowing † -coequaliser, A ⊗ n id A ⊗ n - (swap ⊗ A ) , - ... (swap ⊗ A ) ( n − ,n - A ⊗ n s nA - A ⊗ s n We are taking the coequaliser of all permutations of the tensor factors of A ⊗ n , or at least ofa generating set for the permutation group. A ⊗ s is defined to be I , the tensor unit.The † -coequaliser property implies that for all A and n , we have s nA ◦ s nA † = id A ⊗ sn , asdiscussed in section 2.5. Lemma 6.6.
The projections onto the symmetric subspace s nA † ◦ s nA : A ⊗ n - A ⊗ n arenatural.Proof. This follows from the fact that, for all arrows f : A - B , f ⊗ n ◦ s nA † ◦ s nA is a cone forthe equaliser diagram defining s nB † . Definition 6.7.
The n-fold symmetric tensor product functor S n : C - C is defined in thefollowing way, for all objects A and all morphisms f : A - B in C : S n ( A ) := A ⊗ s n S n ( f ) := s nB ◦ f ⊗ n ◦ s nA † Functoriality of composition follows from lemma 6.6.
Definition 6.8.
The symmetric subspace natural projection s n : T n ˙ - S n is defined at eachstage A as ( s n ) A := s nA . It can be shown to be natural using lemma 6.6.37 efinition 6.9. The commutative ladder functor L C : C - C on a symmetric monoidal † -category C with † -biproducts and symmetric † -subspaces is defined as L := ∞ M n =0 S n (62)where the † -biproduct is understood as being applied componentwise for each stage, so forall objects A in C , G ( A ) = L ∞ n =0 S n ( A ) = S ( A ) ⊕ S ( A ) ⊕ · · · .. Definition 6.10.
The ladder functor L has natural projections onto its components, whichare derived from the † -biproduct structure. The projectors form the ladder projection naturaltransformations , for all natural numbers n : p n : L ˙ - S n . (63)Note that this is not an exponential notation; n is just an index. Definition 6.11.
Any cocommutative comonoid (
A, g, u ) × in C × has a series of n -foldcomultiplication maps g n − : A - A ⊗ n for all n ≥
0. The exponential notation is onlysuggestive; for the comultiplication g : A - A ⊗ A , of course, we cannot construct g ◦ g .However, we can construct ( g ⊗ id A ) ◦ g : A - A ⊗ A ⊗ A . The associativity axiom meansthat this morphism is the same as (id A ⊗ g ) ◦ g , so there is no need to make a choice here.For n ≥
2, we define g n − := (cid:16) g ⊗ id ( n − A (cid:17) ◦ · · · ◦ ( g ⊗ id A ) ◦ g . (64)For the special case of n = 0, we define g − := u : A - I ; and for n = 1, we define g := id A . Also notice that g ≡ g , which gives the notation some intuitive consistency. Ifthe notion of an n -fold comultiplication seems mysterious, that is only because we are notused to working with comonoids: it is no different to using the associative multiplication ofa monoid to perform n -fold multiplication. Definition 6.12.
An adjunction h R, P, κ, λ i , where R : C × - C is the forgetful functor,is ladder-style if the right adjoint P : C - C × , the unit and the counit are of the followingform, for all objects A and B and all morphisms f : A - B in C : P ( A ) = (cid:16) L C ( A ) , b A , c A (cid:17) × P ( f ) = L C ( f ) b A = ∞ X m,n =0 B m,n · (cid:16) p mA † ⊗ p nA † (cid:17) ◦ (cid:16) s mA ⊗ s nA (cid:17) ◦ (cid:16) s m + nA (cid:17) † ◦ p m + nA c A = C · p A κ ( A,g,u ) × = ∞ X n =0 K n · p nA † ◦ s nA ◦ g n − λ A = L · p A The objects B m,n , C , K n and L are all scalars in Hom C ( I, I ), and are functions of the naturalnumbers m and n where appropriate. 38hese scalars cannot be freely chosen, as the counit, coassociativity and cocommutativeequations, the adjunction equations, and the requirement that κ ( A,g,u ) × is a morphism ofcomonoids will all give constraints on their values. There is also the need to ensure that allmorphisms actually exist in the category C , which will not necessarily be the case for allchoices of scalars, given that all countable sums will not necessarily be defined. We will notenter into a complete analysis of the possible values that the scalars can take, but it is worthmentioning that setting all scalars equal to id I does satisfy all of the constraints, and so willprovide an adjunction between C and C × if all the required sums exist. This gives rise tothe ‘canonical’ free cocommutative comonoid construction. Lemma 6.13.
If a ladder-style adjunction h R, P, κ, λ i is a harmonic oscillator adjunction,then the extra constraints imposed on the scalar coefficients, in addition to those arising fromthe condition that the adjunction be well-defined, are as follows:1. Both C and L are unitary.2. B n,m ◦ B † n,m = ( n + m )! n ! m ! . We note that ( n + m )! /n ! m ! is always a natural number, and so exists as a scalar in ourcategory as a repeated sum of the scalar identity id I . Lemma 6.14.
If a ladder-style adjunction h R, P, κ, λ i is a harmonic oscillator adjunction,then the following equations must hold for the coefficients K n :1. K = C † .2. K = L † .3. ( n + 1) K n +1 K † n +1 = K n † K n . It is straightforward to see from the equations given that, in a category with inverses to allscalars, we must have K n K † n = 1 n ! . (65)It is no surprise that these are the coefficients of the power series of the exponential function,given the role that the natural transformation κ plays in defining the morphism exponentials.We now investigate this structure in the category Inner . Definition 6.15.
In the † -category Inner , we define the countably-infinite † -biproduct L n A n as the inner-product space consisting of sequences a = ( a , a , . . . ) of vectors, with a n ∈ A n , such that P ∞ n =1 c n ( a n , a n ) A n is finite for all complex numbers c , and ( − , − ) A n isthe inner product of A n . The inner product on L n A n is given by ( a, b ) = P ∞ n =1 ( a n , b n ) A n ,which will always be finite (just choose c = 1.) There is a fourth equation arising from the adjunction equation
P λ ◦ κP = id P , but this is not so easyto express in components.
39e choose the rather strong requirement that P ∞ n =1 c n ( a n , a n ) A n be finite for all complexnumbers c to allow us to define a large class of linear maps on L n A n . If we used the‘standard’ definition, where we only require the sum to converge for c = 1, then the onlyeverywhere-defined linear maps with source object L n A n would be the bounded linear maps.This would be too restrictive for our purposes; we will see that unbounded linear maps forma crucial part of the structure. Definition 6.16.
The harmonic oscillator adjunction hh Q, η, ǫ ii on Inner is a ladder-styleadjunction, defined by coefficients B n,m = s ( n + m )! n ! m ! , K n = 1 √ n ! ,C = 1 , L = 1 . There exist other harmonic oscillator adjunctions, but we choose this one as it is moststraightforward.We conjecture that this harmonic oscillator adjunction is well-defined. The difficulty liesin establishing that the natural transformation η : id Inner × ˙ - Q ◦ R is defined at everystage; more work is needed here.Assuming the model is valid, it is simple to demonstrate that we recover the conventionalraising and lowering operators from hh Q, η, ǫ ii : a † φ = d † A ◦ (cid:16) ǫ † A ⊗ id F ( A ) (cid:17) ◦ (cid:16) φ ⊗ id F ( A ) (cid:17) = ∞ X m,n =0 s ( m + n )! m ! n ! · (cid:16) p mA † ⊗ p nA † (cid:17) ◦ (cid:16) s mA ⊗ s nA (cid:17) ◦ (cid:16) s m + nA (cid:17) † ◦ p m + n = ∞ X m,n =0 s ( m + n )! m ! n ! · p m + nA † ◦ s m + nA ◦ (cid:16) s mA † ⊗ s nA † (cid:17) ◦ (cid:16) p mA ⊗ p nA (cid:17) ◦ (cid:16)(cid:16) p A † ◦ φ (cid:17) ⊗ id F ( A ) (cid:17) = ∞ X n =0 √ n + 1 · p n +1 A † ◦ s n +1 A ◦ (cid:16) id A ⊗ s nA † (cid:17) ◦ (cid:16) φ ⊗ p nA (cid:17) We also obtain the conventional coherent states:Coh( φ ) = F H ( φ ) ◦ R H η I × = " ∞ X n =0 p nA † ◦ s nA ◦ φ ⊗ n ◦ s nI † ◦ p nI ◦ " ∞ X m =0 √ m ! · p mI † ◦ s mI ◦ (cid:16) λ † I (cid:17) m − = ∞ X n =0 √ m ! · p nA † ◦ s nA ◦ φ ⊗ n ◦ (cid:16) λ † I (cid:17) n − = id I , φ, √ φ ⊗ s φ, √ φ ⊗ s φ ⊗ s φ, . . . , √ n ! φ ⊗ s n , . . . ! We now turn to the morphism exponentials. Given a commutative monoid (
A, g, u ) + and40n element φ : I - A , we can construct the exponential of φ :exp ( A,g,u ) + ( φ ) = R + η +( A,g,u ) + ◦ F ( φ ) ◦ R × η × I × = (cid:16) Rη ( A,g † ,u † ) × (cid:17) † ◦ F ( φ ) ◦ Rη I × = ∞ X m,n,p =0 √ m ! p ! · g m − ◦ s mA † ◦ p mA ◦ p nA † ◦ s nA ◦ φ ⊗ n ◦ s nI † ◦ p nI ◦ p pI † ◦ s pI ◦ (cid:16) λ † I (cid:17) p − = ∞ X m =0 m ! · g m − ◦ s mA † ◦ s mA ◦ φ ⊗ m ◦ s mI † ◦ s mI ◦ (cid:16) λ † I (cid:17) p − = ∞ X m =0 m ! · g m − ◦ φ ⊗ m . (suppressing λ † I isomorphisms)This is clearly a direct generalisation of the conventional notion of exponential. We do nothave duals in the category, so we cannot demonstrate endomorphism exponentials; however,if one allows the duality morphisms to ‘formally’ exist in the category, it is clear the usualoperator exponential construction is obtained. In this paper, we presented a categorical description of the quantum harmonic oscillator inany symmetric monoidal † -category with finite † -biproducts, giving categorical definitions ofthe zero-particle state, single-particle state and raising and lowering morphisms, and demon-strating the canonical commutators. We described how coherent states could be defined asthe points of certain comonoids in C × , and proved that they can be copied and deleted,and that they are eigenstates of the lowering morphisms. A formalism was developed forconstructing exponentials of elements of commutative monoids, and we demonstrated thatthese exponentials are additive in a familiar way, and that the exponential of the appropriatezero element gives the unit for the monoid.We then added duals to our category and demonstrated that endomorphism exponentials,generalising operator exponentials in conventional Hilbert space theory, could be defined ingeneral, and that they interacted well with the existing categorical structures, coherent statesbeing defined by the endomorphism exponentials of the raising morphisms.Finally, we explored these structures in the context of a suitably-defined category ofinfinite-dimensional inner-product spaces, demonstrating that the conventional mathemati-cal structures are produced by the categorical construction in this case. We conjectured thatthe construction was well-defined, and described the remaining open problem.However, there are many issues which are still unclear. Philosophically, perhaps thebiggest problem with the existing framework for categorical quantum mechanics is the lackof any nontrivial categorical description of dynamics. For this reason, it is questionablewhether the system under study in this paper deserves to be called the harmonic oscillatorat all: without a description of dynamics, all that has really been defined is the state space,but many different systems have isomorphic state spaces.41onventionally, a quantum system is described by a Hilbert space A and a self-adjointoperator H : A - A . This operator describes the energy of different states of the system,and generates time evolution via Schr¨odinger’s first-order differential equation. Work hasbeen done on differential structure in categories, as we discuss below, but an elegant cat-egorical description of the other ingredients of the Schr¨odinger equation, such as Planck’sconstant h and the imaginary unit i , is far from apparent. The author is hopeful that theendomorphism exponential construction presented here might somehow prove relevant tothis problem, given Stone’s theorem in the classical case, which describes how the opera-tor exponential induces an isomorphism between self-adjoint operators and certain unitaryone-parameter groups.We also note that the operator exponential construction, by way of the unit naturaltransformation η , calculates infinite sums, which, given that each term of the sum can begiven a graphical representation, are strongly reminiscent of the infinite sums of Feynmandiagrams which are used to defined an interacting field theory perturbatively. It wouldbe interesting to explore whether a correspondence can be made here, and the formalismdescribed in this paper extended to give a categorical framework for perturbative quantumfield theory.However, lurking behind the theory of the endomorphism exponential is the simple factthat interesting models of it seem impossible to come by. We recall from section 6 that ifa category of Hilbert spaces contains infinite-dimensional objects, then it cannot be madecompact closed, since the trace of the identity on an infinite-dimensional space is infinite.There is no hope of adding an infinite scalar to the category: it is demonstrated in [6] that asemiring can only be given countably-infinite sums iff it admits a partial order, compatiblewith the semiring multiplication and addition, but the complex numbers admit no suchordering.It seems that there is a fundamental incompatibility between duals, the existence of aright adjoint to the forgetful functor R : C × - C , and non-trivial (by which we meannon-partially-orderable) scalars. The reason is this: if there exists a right adjoint Q to R ,then it seems that objects in the image of Q must be infinite-dimensional. Duals allow usto take the trace of the identities on these infinite objects, but that will involve an infinitesum of scalars, which cannot be defined if the scalars are non-trivial. Dropping any oneof these three properties, models are easy to come by: the category FdHilb lacks a rightadjoint to R , the category Inner lacks duals, and the category
Rel of sets and relationslacks non-trivial scalars, but they all have the other two of the three properties.A completely different interpretation of the categorical structures described in this paperis as a model for the differential calculus of polynomials in one or more variables (see [2],and also [4] developed in parallel to this work.) In this viewpoint, the raising operatorperforms multiplication by a homogeneous polynomial of degree 1, and the lowering operatorperforms differentiation. It is meaningless to discuss whether the mathematics presentedhere is more ‘honestly’ a model of calculus or of the quantum harmonic oscillator, butthere are certainly differences between the viewpoints: it is not conventional to considermultiplication and differentiation as being literally adjoint to each other, for example, as isnatural for the quantum raising and lowering operators. The quantum point of view alsosuggests different avenues for generalisation: there are many other quantum systems, andit would be fascinating to explore the extent to which they too have natural categorical42escriptions. Questions concerning the categorical descriptions of the dynamics also seemdifficult to ask from the differential calculus point of view.In fact, rather then being alternatives, the two viewpoints may have an interesting andnontrivial connection. As mentioned in section 3, we expect the position and momentumoperators — that is, the multiplication and differentiation operators — to arise as linear com-binations of the raising and lowering operators, as x φ = ( a † φ + a φ ) / √ p φ = i ( a † φ − a φ ) / √ x φ , where φ ranges over a basisfor the single-particle space, represents physical space for the case of a harmonic oscillatorwhich is free to oscillate in all directions.It is also striking that the framework presented in this paper has so much in commonwith the model theory for linear logic [4, 8]. However, some aspects which are key to thedescription of the quantum harmonic oscillator — such as the raising and lowering operators,and their commutation relations — seem syntactically absent from linear logic, and aretherefore quite mysterious from that point of view. In light of this, it would be interestingto explore whether linear logic might itself admit modification that would render it moresuitable for performing deductions about quantum systems, such as the quantum harmonicoscillator considered here. I would like to thank Bob Coecke, Marcelo Fiore, Robin Houston, Andrea Schalk and espe-cially Chris Isham for useful discussions.
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