Towards Quantum Field Theory in Categorical Quantum Mechanics
BBob Coecke and Aleks Kissinger (Eds.):14th International Conference on Quantum Physics and Logic (QPL)EPTCS 266, 2018, pp. 349–366, doi:10.4204/EPTCS.266.22 c (cid:13)
S. Gogioso & F. GenoveseThis work is licensed under the Creative CommonsAttribution-Noncommercial-Share Alike License.
Towards Quantum Field Theoryin Categorical Quantum Mechanics
Stefano GogiosoUniversity of Oxford [email protected]
Fabrizio GenoveseUniversity of Oxford [email protected]
In this work, we use tools from non-standard analysis to introduce infinite-dimensional quantumsystems and quantum fields within the framework of Categorical Quantum Mechanics. We define adagger compact category ∗ Hilb suitable for the algebraic manipulation of unbounded operators, Diracdeltas and plane-waves. We cover in detail the construction of quantum systems for particles in boxeswith periodic boundary conditions, particles on cubic lattices, and particles in real space. Not quitesatisfied with this, we show how certain non-separable Hilbert spaces can also be modelled in ournon-standard framework, and we explicitly treat the cases of quantum fields on cubic lattices andquantum fields in real space.
The rigorous diagrammatic methods of Categorical Quantum Mechanics [1, 3, 5, 6, 7, 8] have found widelysuccessful application to quantum information, quantum computation and the foundations of quantumtheory. Until very recently, however, a major limitation of the framework was its lack of applicabilityto infinite-dimensional quantum systems, which include many iconic examples from textbook quantummechanics and quantum field theory. Previous work by the authors [11] partially overcame this limitation,using non-standard analysis `a la Robinson [23] to define a dagger compact category (cid:63)
Hilb of infinite-dimensional separable Hilbert spaces. A non-standard approach was chosen because it enables a consistentmathematical treatment of infinitesimal and infinite quantities, such as those involved in the manipulationof unbounded operators, Dirac deltas, plane waves, and many other gadgets and structures featuring intraditional approaches to quantum mechanics.The debate about the physical interpretation of infinitesimals and infinities is as old as calculusitself [16], and their use always attracts a healthy dose of scepticism. In time, this has lead to an interestingdichotomy, where infinitesimals are used as a quick way to convince oneself of the validity of a statement,but limits are then required for formal justification. Non-standard analysis simply provides a frameworkto completely replace limits with a consistent algebraic treatment of infinitesimals and infinities. As longas one is willing to assign physical meaning to non-convergent limit constructions—and mainstreamquantum mechanics certainly seems to be—there should be little or no problem of physical interpretation.The original definition of (cid:63)
Hilb from Ref. [11] featured unital †-Frobenius algebras on all objects,the main ingredient of Categorical Quantum Mechanics (CQM) which was missing from the categoryHilb of infinite-dimensional Hilbert spaces and bounded operators [2]. It enabled some first, successfulapplications of CQM methods to infinite-dimensional quantum systems, but was otherwise somewhatlimited: most notably, it could not be applied to the case of unbounded quantum particles on real spaces(the single most important textbook example), nor could it tackle the non-separable Hilbert spaces requiredfor the treatment of quantum fields as will be understood as part of this work.50
Towards Quantum Field Theory in Categorical Quantum Mechanics
The limitations of the original definition were self-imposed, aimed at keeping the framework simple andmore easily relatable, and did not play any relevant role in most constructions we presented. The startingpoint of this work, in Section 2, is a re-definition of the category (cid:63)
Hilb, addressing those unnecessarylimitations. Specifically, we remove the requirement that the underlying Hilbert spaces for the objects of (cid:63)
Hilb be standard and separable, and we provide a basis-independent formulation of the objects themselves:aside from making the formalism significantly more powerful, this choice has the categorically pleasingeffect of identifying (cid:63)
Hilb as a full subcategory of the Karoubi envelope for the category of non-standardHilbert spaces and (cid:63) C -linear maps.The rest of this work is dedicated to the explicit construction of quantum systems of interest ina number of traditional applications of quantum mechanics. In Section 3 we construct the space ofwavefunctions in an n -dimensional box with periodic boundary conditions T n , while in Section 4 weconstruct the space of wavefunctions on an n -dimensional lattice Z n ; both these examples were sketchedin the original Ref. [11], and are here reproduced in additional detail. In Section 5 we construct the spaceof unbounded wavefunctions in an n -dimensional real space R n , which we approximate using an infinitelattice of infinitesimal mesh (a well-tested trick in non-standard analysis [23]). For each of these threeconstructions, we provide a strongly complementary pair corresponding to the position and momentumobservables. In Sections 6 and 7, we proceed to treat two cases of non-separable standard Hilbert spaces,exploiting a somewhat surprising fact about exponentials of infinite non-standard integers: in Section 6we construct the space of quantum fields on an n -dimensional lattice Z n ; in Section 7, we once again usean infinite lattice of infinitesimal mesh to construct the space of quantum fields in real space R n .Before moving on, we should remark that the originality of this work and the work of Ref. [11]does not lie in the application of non-standard methods to conventional quantum theory, for which arich literature already exists [10, 19, 20, 21, 22, 26]. Rather, it lies in the application of non-standardmethods to solve a set of issues—lack of Frobenius algebras, compact closed structure and stronglycomplementary pairs, to mention just a few—which prevented the algebraic/diagrammatic methods ofCategorical Quantum Mechanics from being applied to the infinite-dimensional setting. (cid:63) Hilb
We define the symmetric monoidal category (cid:63)
Hilb (read:
Star Hilb ) to have objects in the form of pairs H : = ( | H | , P H ) , where | H | is a non-standard Hilbert space (the underlying Hilbert space ) and P H : | H | → | H | is an internal non-standard (cid:63) C -linear map which satisfies the following requirements. • The map P H is a self-adjoint idempotent (we refer to it as the truncating projector ). • There is some family | e n (cid:105) Dn = of orthonormal vectors in | H | , for some D ∈ (cid:63) N , such that: P H = D ∑ n = | e n (cid:105)(cid:104) e n | (2.1)The existence of such families is guaranteed by Transfer Theorem. Again by Transfer Theorem, D is the same for all such choices of orthonormal families, and we can consistently define the dimension of H to be dim H : = D ∈ (cid:63) N .The morphisms of our re-defined (cid:63) Hilb take the same form as those given in the original definition (theyare internal non-standard (cid:63) C -linear maps, with the truncating projectors acting as identities):Hom (cid:63) Hilb [ H , G ] : = { P G ◦ F ◦ P H | F : | H | → | G | internal linear map } . (2.2) Note that we dropped the requirement that | H | be separable, or even that | H | = (cid:63) V for some standard Hilbert space V . Note that we dropped the requirement that | e n (cid:105) Dn = be (a non-standard extension of) a standard orthonormal basis. . Gogioso & F. Genovese (cid:63) Hilb is now a genuine full subcategory of the Karoubi envelope for the category ofnon-standard Hilbert spaces and non-standard (cid:63) C -linear maps. The rest of the construction proceedsexactly as it did in the original definition, and (cid:63) Hilb is a dagger symmetric monoidal category (†-SMC,for short). Morphisms can still be represented as matrices (although no longer in a canonical way), bychoosing orthonormal sets which diagonalise the relevant truncating projectors:¯ F : = P G ◦ F ◦ P H = dim G ∑ m = H ∑ n = | f m (cid:105) (cid:16) (cid:104) f m | F | e n (cid:105) (cid:17) (cid:104) e n | . (2.3)The tensor, symmetric braiding and dagger can be defined as usual by looking at the matrix decomposition,and by Transfer Theorem they are invariant under different choices of diagonalising orthonormal sets.Similarly, unital special commutative †-Frobenius algebras can be constructed for all orthonormal bases ofan object H (i.e. for all orthonormal families diagonalising the truncating projector P H ). The interestedreader is referred to Ref. [11] for the details of the original constructions, which are unchanged.Contrary to the dagger symmetric monoidal structure, the compact closed structure in the originaldefinition was given in terms of the chosen orthonormal basis for each object, and needs to be adaptedto our new basis-invariant definition. Consider an object H of (cid:63) Hilb, together with a diagonalisation P H = ∑ dim H n = | e n (cid:105)(cid:104) e n | of its truncating projector. Also consider the dual | H | ∗ to the underlying Hilbertspace | H | of H (which exists by Transfer Theorem), together with the orthonormal set | ξ n (cid:105) dim H n = ofvectors in | H | ∗ specified by the adjoints of the states in the orthonormal family | e n (cid:105) dim H n = . We define the dual object H ∗ to be given by the pair H ∗ : = ( | H | ∗ , P H ∗ ) , where the truncating projector is defined by P H ∗ : = ∑ dim H n = | ξ n (cid:105)(cid:104) ξ n | (it is a standard check that this definition is basis-invariant). The compact closedstructure is defined as follows (once again, it is a standard check that this definition is basis-invariant): dim H ∑ n = | ξ n (cid:105) ⊗ | e n (cid:105) : = : = dim H ∑ n = (cid:104) e n | ⊗ (cid:104) ξ n | (2.4)The dual object H ∗ has the same dimension as H , and the non-standard natural number dim H coincideswith the scalar given by the definition of dimension in dagger compact categories: dim H ∑ n = (cid:104) ξ n | ξ n (cid:105) ⊗ (cid:104) e n | e n (cid:105) = = dim H (2.5)Let (cid:63) sHilb be the full subcategory of (cid:63) Hilb given by those objects H such that | H | = (cid:63) V forsome separable standard Hilbert space V , and such that the truncating projector spans all near-standardvectors. Let (cid:63) sHilb ( std ) be the sub-†-SMC of (cid:63) sHilb given by near-standard morphisms. In particular,the original (cid:63) Hilb from Ref. [11] is a full subcategory of the newly defined (cid:63) sHilb, and the original (cid:63)
Hilb ( std ) featuring in Theorem 3.4 of Ref. [11] is a full subcategory of the newly defined (cid:63) sHilb ( std ) .We can define a standard part functor st : (cid:63) sHilb ( std ) → sHilb, which acts as H (cid:55)→ | H | on objectsand as F (cid:55)→ st ( F ) on morphisms. The standard part functor is C -linear, and identifies two near-standardmaps F , G : H → K if and only if F − G has infinitesimal operator norm; this defines an equivalencerelation on morphisms in (cid:63) sHilb ( std ) , which we denote by ∼ and refer to as infinitesimal equivalence .The equivalence relation ∼ respects composition, tensor product and dagger, and endows (cid:63) sHilb ( std ) withthe structure of a †-symmetric monoidal 2-category. We can also define a weak truncation functor I.e. | ξ n (cid:105) is the vector in | H | ∗ specified by the linear operator (cid:104) e n | on | H | . By weak we mean that composition and tensor product are respected only up to infinitesimal equivalence, i.e. that we havelift ω [ f ◦ g ] ∼ lift ω [ f ] ◦ lift ω [ g ] and lift ω [ f ⊗ g ] ∼ lift ω [ f ] ⊗ lift ω [ g ] Towards Quantum Field Theory in Categorical Quantum Mechanics lift ω : sHilb → (cid:63) sHilb ( std ) , which acts as V (cid:55)→ ( V , P ( V ) ) on objects and sends the standard morphism f : V → W to the non-standard morphism P ( W ) ◦ (cid:63) f ◦ P ( V ) (here (cid:63) f is the non-standard extension of f ).The following result relates sHilb and (cid:63) sHilb ( std ) ω , the full subcategory of (cid:63) sHilb ( std ) spanned by thoseobjects H having dimension dim H ∈ (cid:63) N which is either a finite natural or the infinite natural ω . Theorem 2.1 ( Updated version of Theorem 3.4 from Ref. [11] ) . The standard part and truncation functors determine a weak equivalence between sHilb and (cid:63) sHilb ( std ) ω :(i) st is a full dagger monoidal functor, which is surjective on objects;(ii) lift ω is a weak dagger monoidal faithful functor from a † -SMC to a † -symmetric monoidal 2-category,which is injective and essentially surjective on objects (its restriction to fHilb is strict);(iii) st is a left inverse to lift ω ;(iv) for all objects H of (cid:63) sHilb ( std ) ω there is a canonical unitary isomorphism ¯ u H : H → lift ω [ st ( H )] ,the unique one which satisfies st ( ¯ u H ) = id st ( H ) ;(v) for all morphisms F : H → G in (cid:63) sHilb ( std ) ω we have ¯ u † G ◦ lift ω [ st ( F )] ◦ ¯ u H ∼ F.Proof.
Essentially the same of Theorem 3.4 from Ref. [11], using the fact that the subspace defined bythe truncating projector P H contains at least all near-standard vectors in H . Remark 2.2.
Unfortunately, the updated version of Theorem 3.4 only covers standard separable Hilbertspaces, while we will see that our re-defined (cid:63)
Hilb allows us to treat some standard non-separable spacesas well. We believe Theorem 3.4 to be likely to extend to some subcategory of standard non-separablespaces, but an exact determination of the necessary and sufficient conditions is left to future work.
The essence of Theorem 2.1 is that sHilb is equivalent to the subcategory (cid:63) sHilb ( std ) ω of (cid:63) Hilb given bynear-standard morphisms, as long as we take care to equate morphisms which are infinitesimally close.The equivalence allows one to prove results about sHilb by working in (cid:63)
Hilb and taking advantage of thefull CQM machinery, according to the following general recipe:(i) start from a morphism in sHilb;(ii) lift to (cid:63) sHilb ( std ) ω via the lifting functor; (iii) work in (cid:63) Hilb, obtain a result in (cid:63) sHilb ( std ) ω ;(iv) descend to sHilb via the standard part functor.This procedure is conceptually akin to using the two directions of the transfer theorem to prove results ofstandard analysis using non-standard methods. When proving equalities of morphisms in sHilb, it is infact sufficient to lift both sides via lift ω , and prove the equality in (cid:63) Hilb without further constraints (this isbecause both sides will necessarily be lifted to (cid:63) sHilb ( std ) ω ).The arbitrary choice of infinite natural ω in the lifting part of the recipe might seem unnatural atfirst, as the objects lift ω [ V ] and lift ω (cid:48) [ V ] are not isomorphic in (cid:63) Hilb for different infinite naturals ω (cid:54) = ω (cid:48) .However, this is not actually an issue: from the perspective of sHilb, the two spaces are equivalentfor all intents and purposes, and any proof that can be performed in one space can also be performedin the other. This could be made precise by saying that setting Φ ω , ω (cid:48) : = ( lift ω (cid:48) ◦ st ) defines a weakequivalence Φ ω , ω (cid:48) : (cid:63) Hilb ( std ) ω → (cid:63) Hilb ( std ) ω (cid:48) such that st ( Φ ω , ω (cid:48) ( ¯ F )) = st ( ¯ F ) holds for all morphisms ¯ F in (cid:63) Hilb ( std ) ω . From this point of view, the same limiting objects (e.g. Diract deltas, plane-waves andunbounded operators) corresponding to different choices of ω can be interpreted as incarnations of thesame conceptual objects seen at different values for the “infinite cut-off parameter” ω . The truncating projectors P ( V ) are defined by appropriately choosing an orthonormal basis | e ( V ) n (cid:105) n for each separable V , andletting P ( V ) : = ∑ dim Vn = | e ( V ) n (cid:105)(cid:104) e ( V ) n | (where we set dim V : = ω for infinite-dimensional standard Hilbert spaces V ). . Gogioso & F. Genovese Consider a quantum particle in a n -dimensional box with periodic boundary conditions: this can equiva-lently be seen as a quantum particle in an n -dimensional torus , and hence it corresponds to the Hilbertspace L [ T n ] (the n -dimensional torus is acted freely and transitively upon by its translation group T n ).The momentum eigenstates for the particle form a countable complete orthonormal basis for L [ T n ] , withthe eigenstate of momentum k ¯ h ( k ∈ Z n ) given by the following square-integrable function T n → C : | χ k (cid:105) : = x → e − i π k · x (3.1)We take a non-standard extension | χ k (cid:105) + ω k ,..., k n = − ω of the standard orthonormal basis of momentum eigen-states, where ω ∈ (cid:63) N is some infinite non-standard natural, and we consider the (cid:0) ( ω + ) n (cid:1) -dimensionalobject H T n : = ( (cid:63) L [ T n ] , P H T n ) defined by the following truncating projector: P H T n : = + ω ∑ k = − ω ... + ω ∑ k n = − ω | χ k (cid:105)(cid:104) χ k | (3.2)We take this object to be our model in (cid:63) Hilb of a quantum particle in an n -dimensional box with periodicboundary conditions, and we refer to the family | χ k (cid:105) + ω k ,..., k n = − ω as the momentum eigenstates for H T n .Pretty much by definition, this object comes with a unital special commutative †-Frobenius algebracorresponding to the momentum observable for the particle:: = + ω ∑ k = − ω ... + ω ∑ k n = − ω (cid:104) χ k | : = + ω ∑ k = − ω ... + ω ∑ k n = − ω | χ k (cid:105) ⊗ | χ k (cid:105) ⊗ (cid:104) χ k | (3.3)The labels of the momentum eigenstates can be endowed with the infinite abelian group structure of ( (cid:63) Z n ω + , ⊕ , ) , which is defined in full detail in the Appendix, and we can consider the linear extensionof the group multiplication and unit:: = | χ (cid:105) : = + ω ∑ k , h = − ω ... + ω ∑ k n , h n = − ω | χ k ⊕ h (cid:105) ⊗ (cid:104) χ k | ⊗ (cid:104) χ h | (3.4)It is immediate to check that these two maps form, together with their adjoints, a unital quasi-specialcommutative †-Frobenius algebra, with normalisation factor ( ω + ) n . Furthermore, a result of Ref. [12]guarantees that ( , ) is a strongly complementary pair of unital †-Frobenius algebras . The following position eigenstates | δ x (cid:105) , indexed by x ∈ ω + (cid:63) Z n ω + (a lattice in the non-standard torus (cid:63) T n ) are the-classical states (they are orthogonal and have square norm ( ω + ) n ; see the Appendix for a proof): | δ x (cid:105) : = + ω ∑ k = − ω ... + ω ∑ k n = − ω χ k ( x ) ∗ | χ k (cid:105) (3.5)For all x ∈ ω + (cid:63) Z n ω + and all standard smooth f ∈ L [ T n ] , the position eigenstates satisfy the identityst ( (cid:104) δ x | f (cid:105) ) = f ( st ( x )) by Transfer Theorem: hence the position eigenstates (as we defined them) behaveexactly as expected from Dirac delta functions, and we can legitimately refer to as the positionobservable for the particle. Note the duality between the large-scale cut-off on momenta in (cid:63) Z n (whichare bounded in magnitude by √ n ω ¯ h ) and the small-scale cut-off on positions in (cid:63) T n (which are discretisedonto a lattice of infinitesimal mesh ω + ): in our non-standard framework, this well-known phenomenonarises in a purely algebraic way from the copy condition for -classical states (see the Appendix). The relevant result is Theorem 3.4 of Ref. [12], which requires to have enough classical states (the momentum eigenstates), ( , ) to endow the -classical states with the structure of a group (the group (cid:63) Z n ω + ), and to be -classical. They are orthogonal because they are copied by a quasi-special commutative †-FA in a SMC where scalars form a field [9]. Towards Quantum Field Theory in Categorical Quantum Mechanics
Consider a particle on an n -dimensional cubic lattice, corresponding to the Hilbert space L [ Z n ] . Theposition eigenstates for the particle form a countable complete orthonormal basis for L [ Z n ] , with theeigenstate of position k ∈ Z n given by the following square-integrable function Z n → C : | δ k (cid:105) : = h (cid:55)→ (cid:40) k = h | δ k (cid:105) ω k ,..., k n = − ω of the standard orthonormal basis of position eigenstates,where ω ∈ (cid:63) N is some infinite non-standard natural, and we consider the (cid:0) ( ω + ) n (cid:1) -dimensional object H Z n : = ( (cid:63) L [ Z n ] , P H Z n ) defined by the following truncating projector: P H Z n : = + ω ∑ k = − ω ... + ω ∑ k n = − ω | δ k (cid:105)(cid:104) δ k | (4.2)We take this object to be our model in (cid:63) Hilb of a quantum particle on an n -dimensional lattice, and we referto the family | δ k (cid:105) + ω k ,..., k n = − ω as the position eigenstates for H Z n . Pretty much by definition, this objectcomes with a unital special commutative †-Frobenius algebra corresponding to the position observable for the particle: : = + ω ∑ k = − ω ... + ω ∑ k n = − ω (cid:104) δ k | : = + ω ∑ k = − ω ... + ω ∑ k n = − ω | δ k (cid:105) ⊗ | δ k (cid:105) ⊗ (cid:104) δ k | (4.3)The labels of the position eigenstates can be endowed with the infinite abelian group structure of ( (cid:63) Z n ω + , ⊕ , ) , and we can consider the linear extension of the group multiplication and unit:: = | δ (cid:105) : = + ω ∑ k , h = − ω ... + ω ∑ k n , h n = − ω | δ k ⊕ h (cid:105) ⊗ (cid:104) δ k | ⊗ (cid:104) δ h | (4.4)It is immediate to check that these two maps form, together with their adjoints, a unital quasi-specialcommutative †-Frobenius algebra, with normalisation factor ( ω + ) n . Furthermore, a result of Ref. [12]guarantees that ( , ) is a strongly complementary pair of unital †-Frobenius algebras.The infinite abelian group ( (cid:63) Z ω + , ⊕ , ) is defined in full detail in the Appendix, and has the intervalof non-standard integers {− ω , ..., + ω } as its underlying set. Remarkably, it contains the standard integers Z as a subgroup: as a consequence, the group ( (cid:63) Z n ω + , ⊕ , ) is a legitimate non-standard extension of thetranslation group Z n for the n -dimensional lattice. In a sense, we are seeing standard infinite lattices areactually being periodic, but circling around “beyond standard infinity” rather than at some finite point.A proof on the same lines of the one mentioned in the previous section shows that the following momentum eigenstates | χ x (cid:105) , indexed by x ∈ ω + (cid:63) Z n ω + , are the -classical states (they are orthogonaland have square norm ( ω + ) n ): | χ x (cid:105) : = + ω ∑ k = − ω ... + ω ∑ k n = − ω e − i π k · x | δ k (cid:105) (4.5)From their formulation, it is immediately clear that these are exactly the plane waves, and as a consequencewe can legitimately refer to as the momentum observable for the particle on the lattice. Similarly tothe previous Section, we have a nice duality between the large-scale cut-off on positions in (cid:63) Z n (whichare bounded in magnitude by √ n ω ) and the small-scale cut-off on momenta in (cid:63) T n (which are discretisedonto a lattice of infinitesimal mesh ¯ h ω + ). . Gogioso & F. Genovese Consider a quantum particle in n -dimensional real space, corresponding to the Hilbert space L [ R n ] .This is a separable space, but for obvious reasons it does not come with a choice of complete countableorthonormal basis of standard states which is invariant under the translation group R n of the underlyingspace. However, we can try and approximate one using a subdivision of R n with infinitesimal mesh,as done in Ref. [23] to model Riemann integration (as well as differentiation and a number of otherfundamental constructions of calculus).We fix two infinite natural numbers: an infrared (IR) infinity ω ir , which will govern the largescale limit of our approximation, and a ultraviolet (UV) infinity ω uv , which will govern the small scalelimit. We require both to be odd, and we set 2 ω + = ω uv ω ir . Consider the following internal lattice—isomorphic to (cid:63) Z n ω + , of which it inherits the group structure—in n -dim non-standard real space (cid:63) R n :1 ω uv (cid:63) Z n ω + : = (cid:26) p ∈ (cid:63) R n (cid:12)(cid:12)(cid:12)(cid:12) p = k ω uv , k ∈ (cid:63) Z n ω + (cid:27) ⊂ (cid:63) R n (5.1)Note that the group R n is not a subgroup of ω uv (cid:63) Z n ω + , as the elements of the latter are non-standardrational numbers. However the former can be approximated by the latter in the following rigorous sense:there is a subgroup (cid:0) ω uv (cid:63) Z n ω + (cid:1) (cid:47) ω uv (cid:63) Z n ω + given by taking the near standard elements, and taking aquotient of this subgroup through the standard part function yields R n as the quotient group (see Appendix).We can interpret the group ω uv (cid:63) Z n ω + as a lattice with infinitesimally fine mesh (specified by ω uv ) whichapproximates R n , which covers it entirely, and which circles around “beyond standard infinity” (at apoint specified by ω ir ). Having seen this, we consider the following orthonormal family of non-standardfunctions (cid:63) R n → (cid:63) C , indexed by the points p ∈ ω uv (cid:63) Z n ω + of the lattice which we introduced above:: | χ p (cid:105) : = x (cid:55)→ √ ω uv e − i π ( p · x ) (5.2)Just like the family of momentum eigenstates defined in Section 3, this family is orthonormal, andhence we can take it to define a (cid:0) ( ω + ) n (cid:1) -dimensional object H R n : = ( (cid:63) L [ R n ] , P H R n ) of (cid:63) Hilb, byconsidering the following truncating projector: P H R n : = + ω ir ∑ p = − ω ir ... + ω ir ∑ p n = − ω ir | χ p (cid:105)(cid:104) χ p | : ≡ + ω ∑ k = − ω ... + ω ∑ k n = − ω | χ k / ω uv (cid:105)(cid:104) χ k / ω uv | (5.3)The shorthand notation ∑ ω ir p = − ω ir f ( p ) : ≡ ∑ ω k = − ω f ( k / ω uv ) adopted above for summation over elements of ω uv (cid:63) Z n ω + will be used through the rest of this work. We take this object to be our model in (cid:63) Hilb of anunbounded quantum particle in an n -dimensional real space, and we refer to the family | χ p (cid:105) + ω ir p ,..., p n = − ω ir as the momentum eigenstates for H R n . Pretty much by definition, this object comes with a unitalspecial commutative †-Frobenius algebra corresponding to the momentum observable for the unboundedparticle: : = + ω ir ∑ p = − ω ir ... + ω ir ∑ p n = − ω ir (cid:104) χ p | : = + ω ir ∑ p = − ω ir ... + ω ir ∑ p n = − ω ir | χ p (cid:105) ⊗ | χ p (cid:105) ⊗ (cid:104) χ p | (5.4)The labels of the momentum eigenstates come endowed with the infinite abelian group structure of ( ω uv (cid:63) Z n ω + , ⊕ , ) , and we can consider the linear extension of the group multiplication and unit:: = | χ (cid:105) : = + ω ir ∑ p , q = − ω ir ... + ω ir ∑ p n , q n = − ω ir | χ p ⊕ q (cid:105) ⊗ (cid:104) χ p | ⊗ (cid:104) χ q | (5.5)56 Towards Quantum Field Theory in Categorical Quantum Mechanics
It is immediate to check that these two maps form, together with their adjoints, a unital quasi-specialcommutative †-Frobenius algebra, with normalisation factor ( ω + ) n . Furthermore, a result of Ref. [12]guarantees that ( , ) is a strongly complementary pair of unital †-Frobenius algebras.The same proof from Section 3 can be adapted to show that the following position eigenstates ,indexed by x ∈ ω ir (cid:63) Z n ω + , are the -classical states (orthogonal and with square norm ( ω + ) n ): | δ x (cid:105) : = + ω ir ∑ p = − ω ir ... + ω ir ∑ p n = − ω ir χ p ( x ) ∗ | χ p (cid:105) (5.6)Note once again the nice duality between the cut-offs of the momenta (which have magnitude boundedabove by √ n ω ir ¯ h and are discretised onto a lattice of infinitesimal mesh ¯ h ω uv ) and the cut-offs of the posi-tions (which have magnitude bounded above by √ n ω uv and are discretised onto a lattice of infinitesimalmesh ω ir ). For all x ∈ ω ir (cid:63) Z n ω + and all standard smooth f ∈ L [ R n ] , the position eigenstates can easilybe shown to satisfy the identity st ( (cid:104) δ x | f (cid:105) ) = f ( st ( x )) : hence, the position eigenstates (as we defined them)behave exactly as expected from Dirac delta functions, and we can legitimately refer to as the positionobservable for the unbounded particle. Furthermore, standard smooth functions span the entirety ofL [ R n ] , and hence the delta functions defined above show that the subspace defined by the truncatingprojector P H R n spans the near-standard vectors in (cid:63) L [ R n ] . Weyl CCRs using diagrams in (cid:63)
Hilb . As a sample application of our non-standard framework, weprovide a diagrammatic proof of the Weyl Canonical Commutation Relations, using the (cid:63)
Hilb object H R n which we have just constructed. For all x ∈ R n , let U x be the unitary on L [ R n ] corresponding tospace-translation of wavefunctions by x . For all p ∈ R n , let V p be the unitary corresponding to momentum-boost by p ¯ h . Let x (cid:48) ∈ ω ir (cid:63) Z n ω + and p (cid:48) ∈ ω uv (cid:63) Z n ω + be such that st ( x (cid:48) ) = x and st ( p (cid:48) ) = p . The unitaries U x and V p can be expressed diagrammatically as follows (because acts as space-translation on deltafunctions, and acts as momentum-boost on plane-waves): U x V p δ x (cid:48) st = st = χ p (cid:48) (5.7)We can then deduce the Weyl Canonical Commutation relations for the position and momentum observ-ables of a quantum particle in n -dimensional real space: V p U x e i π p · x U x V p χ p (cid:48) χ p (cid:48) δ x (cid:48) δ x (cid:48) = χ † − p (cid:48) δ x (cid:48) = st st = (5.8)The proof of the central equality can be carried out fully diagrammatically in the non-standard framework,using strong complementarity for ( , ) together with the fact that delta functions are -classical statesand that plane-waves are -classical states: δ x (cid:48) = χ † − p (cid:48) δ x (cid:48) = δ x (cid:48) χ † − p (cid:48) χ p (cid:48) (5.9) = δ x (cid:48) = χ † − p (cid:48) δ x (cid:48) δ x (cid:48) χ † − p (cid:48) δ x (cid:48) χ † − p (cid:48) χ p (cid:48) χ † − p (cid:48) δ x (cid:48) (5.10) . Gogioso & F. Genovese In Section 5, we have seen that there are so many non-standard rational numbers that the standard realscan be approximated with ease by considering an infinite lattice with infinitesimal mesh. In this Section,we will see that there also are so many non-standard integers that quantum fields living on lattices,traditionally forming a non-separable Hilbert space, can be modelled in our framework.If V is a standard finite-dimensional Hilbert space with dimension greater than one, and X is somecountably infinite set, then it is well known that the space V ⊗ X of V -valued quantum fields on X is anon-separable standard Hilbert space: any orthonormal basis would have cardinality ( dim V ) X , which isstrictly larger than ℵ whenever dim V > X is infinite (we use the infinite direct product V ⊗ X ofvon Neumann [17]). Our model begins with the following observation: if D , m ∈ N then we have D m ∈ N ,and hence by Transfer Theorem if D , µ ∈ (cid:63) N then we must have D µ ∈ (cid:63) N . Fix D , m ∈ N + , and observethat the natural numbers between 1 and D m can be constructively interpreted as strings of length m withcharacters chosen in { , ..., D } . Denote the corresponding decoding/encoding functions as follows:dec D , m : { , ..., D m } → { , ..., D } m enc D , m : { , ..., D } m → { , ..., D m } (6.1)By Transfer Theorem, we obtain a pair of corresponding decoding/encoding functions dec D , µ and enc D , µ for each pair of positive non-standard naturals D , µ ∈ (cid:63) N + ; we will use underlined letters s to denotestrings seen as functions s ∈ { , ..., D } µ , and undecorated letters s to denote the corresponding encodingsof strings as numbers s ∈ { , ..., D µ } .Take some orthonormal family | e s (cid:105) ∈ | H | of vectors in some non-standard Hilbert space | H | , anda family ψ s ∈ (cid:63) C of non-standard complex numbers, both indexed by the internal set of all strings s ∈ { , ..., D } µ for some positive non-standard naturals D , µ ∈ (cid:63) N + . By using the decoding function, wecan always construct a vector | ψ (cid:105) of | H | as follows: | ψ (cid:105) : = D µ ∑ s = ψ dec D , µ ( s ) | e dec D , µ ( s ) (cid:105) (6.2)Now consider an object H : = ( | H | , P H ) of (cid:63) Hilb, with P H : = ∑ dim H d = | e d (cid:105)(cid:104) e d | for some orthonormalfamily | e d (cid:105) dim H d = : we wish to construct an object H ⊗ (cid:63) Z n ω + corresponding to a H -valued quantum fieldliving on the lattice (cid:63) Z n ω + . Define the shorthands D : = dim H and µ : = ( ω + ) n , and consider thefollowing orthonormal family | e s (cid:105) of non-standard states in | H | ⊗ µ (a non-standard Hilbert space whichexists by Transfer Theorem) indexed by strings s ∈ { , ..., D } µ : | e s (cid:105) : = + ω (cid:79) k = − ω ... + ω (cid:79) k n = − ω | e s ( k ) (cid:105) (6.3)We introduced the following shorthand to access the characters of the indexing strings: s ( k ) : ≡ s (cid:16) enc ω + , n (cid:0) k + ω + , ..., k n + ω + (cid:1)(cid:17) ∈ { , ..., D } (6.4)We have chosen to introduce the shorthand above because s is technically s : { , ..., µ } → { , ..., D } , but itis more physically significant to treat it as s : {− ω , ..., + ω } n → { , ..., D } , since we are working in thecontext of n -dimensional lattices.58 Towards Quantum Field Theory in Categorical Quantum Mechanics
Traditionally, this procedure would result in an uncountable family, which cannot be adequatelysummed in the standard framework. In our non-standard framework, on the other hand, we can use theencoding/decoding trick to sum it and define a legitimate truncating projector: P H ⊗ (cid:63) Z n ω + : = D µ ∑ s = | e dec D , µ ( s ) (cid:105)(cid:104) e dec D , µ ( s ) | (6.5)We take the corresponding object H ⊗ (cid:63) Z n ω + : = ( | H | ⊗ µ , P H ⊗ (cid:63) Z n ω + ) of (cid:63) Hilb to model H -valued quan-tum fields on the lattice (cid:63) Z n ω + within our framework. As a matter of convenience, we will adopt a moreslender notation for sums over strings, leaving the decoding step s (cid:55)→ s : = dec D , µ ( s ) implicit: P H ⊗ (cid:63) Z n ω + = ∑ s | e s (cid:105)(cid:104) e s | (6.6)Now consider a decomposition P H : = ∑ Dd = | φ ( k ) d (cid:105)(cid:104) φ ( k ) d | of the truncating projector P H for the quantumsystem H in terms of orthonormal families specified at each point k of the lattice, and let (cid:0) ( k ) (cid:1) k ∈ (cid:63) Z n ω + bethe associated non-degenerate observable on H . Then the truncating projector P H ⊗ (cid:63) Z n ω + for the quantumfield is correspondingly decomposed as P H ⊗ (cid:63) Z n ω + : = ∑ s | φ s (cid:105)(cid:104) φ s | , where | φ s (cid:105) : = (cid:78) + ω k = − ω ... (cid:78) + ω k n = − ω | φ ( k ) s ( k ) (cid:105) .The associated non-degenerate observable for the quantum field is given by the following unital specialcommutative †-Frobenius algebra:: = ∑ s (cid:104) φ s | : = ∑ s | φ s (cid:105) ⊗ | φ s (cid:105) ⊗ (cid:104) φ s | (6.7)In Sections 3 and 4, we constructed objects H T n and H Z n having (cid:63) L [ T n ] and (cid:63) L [ Z n ] as theirunderlying Hilbert spaces, and we considered truncating projectors obtained from the extension of astandard orthonormal basis. In Section 5, we constructed an object H R n having (cid:63) L [ R n ] as its underlyingHilbert space, but we constructed the truncating projector using a genuinely non-standard orthonormalbasis. In all three cases, the underlying Hilbert space is the non-standard extension (cid:63) V of a separa-ble standard Hilbert space V , and the connection to the traditional quantum mechanical formalism isguaranteed by Theorem 2.1. Unfortunately, Theorem 2.1 is not applicable in this Section: in order toestablish a connection between H ⊗ (cid:63) Z n ω + and the traditional model for quantum fields on lattices, we willformulate a suitable universal property for H ⊗ (cid:63) Z n ω + . In the remainder of this Section, we will assumethat | H | = (cid:63) V for some standard separable Hilbert space V .Consider the infinite direct sum of standard Hilbert spaces ∏ k ∈ Z n V . If W is another standardHilbert space, we say that a function ˜ f : (cid:0) ∏ k ∈ Z n V (cid:1) → W is multilinear if for each k ∈ Z n and each u : Z n \{ k } → V the function ˜ f (cid:12)(cid:12) k , u : V → W defined by ˜ f (cid:12)(cid:12) k , u ( v ) : = ˜ f ( u ∪ { k (cid:55)→ v } ) is linear. Nowconsider the non-standard extension ∏ k ∈ (cid:63) Z n (cid:63) V of the direct sum, and let ∏ k ∈ (cid:63) Z n ω + H be the objectof (cid:63) Hilb obtained by restricting non-standard internal maps (cid:63) ϕ : (cid:63) Z n → (cid:63) V to non-standard internalmaps (cid:63) ϕ : (cid:63) Z n ω + → (cid:63) V , extending the projector P H to the direct sum ∏ k ∈ (cid:63) Z n ω + (cid:63) V by pointwise action.Multilinear maps ˜ F : (cid:0) ∏ k ∈ (cid:63) Z n ω + H (cid:1) → K are defined analogously to the standard case. Theorem 6.1 ( Universal property for the infinite tensor product H ⊗ (cid:63) Z n ω + ) . Let | H | = (cid:63) V , and define θ : (cid:0) ∏ k ∈ (cid:63) Z n ω + H (cid:1) → H ⊗ (cid:63) Z n ω + to be the following multilinear map: θ (cid:0) k (cid:55)→ D ∑ d = v d ( k ) | e d (cid:105) (cid:1) : = ∑ s (cid:16) + ω (cid:79) k = − ω ... + ω (cid:79) k n = − ω v s ( k ) ( k ) | e s ( k ) (cid:105) (cid:17) (6.8) . Gogioso & F. Genovese Then for every object K with | K | = (cid:63) W and every multilinear map ˜ F : (cid:0) ∏ k ∈ (cid:63) Z n ω + H (cid:1) → K there is aunique linear map F : H ⊗ (cid:63) Z n ω + → K such that the following diagram commutes: ∏ k ∈ (cid:63) Z n ω + H H ⊗ (cid:63) Z n ω + K θ F ˜ F (6.9) Proof.
The map F can be defined on the standard orthonormal basis of H ⊗ (cid:63) Z n ω + as follows: F (cid:16) + ω (cid:79) k = − ω ... + ω (cid:79) k n = − ω | e s ( k ) (cid:105) (cid:17) : = ˜ F (cid:16) k (cid:55)→ | e s ( k ) (cid:105) (cid:17) (6.10)Commutativity of Diagram 6.9 and uniqueness of F are both consequences of the following equation for˜ F , together with the observation that θ (cid:0) k (cid:55)→ | e s ( k ) (cid:105) (cid:1) = (cid:78) + ω k = − ω ... (cid:78) + ω k n = − ω | e s ( k ) (cid:105) :˜ F (cid:16) k (cid:55)→ D ∑ d = v d ( k ) | e d (cid:105) (cid:17) = ∑ s (cid:16) ˜ F (cid:0) k (cid:55)→ | e s ( k ) (cid:105) (cid:1) ∏ k ∈ (cid:63) Z n ω + v s ( k ) ( k ) (cid:17) (6.11)Equation 6.11 itself is a consequence of Transfer Theorem, because the corresponding standard statementis valid for all standard multilinear ˜ f : (cid:0) ∏ k ∈{− m ,..., + m } n V (cid:1) → W , for any choice of m ∈ N . In Section 5, we have seen that there are so many non-standard rational numbers that the standard realscan be approximated with ease by considering an infinite lattice with infinitesimal mesh. In Section 6, wehave seen that there are so many non-standard integers that quantum fields on lattices can be modelled inour framework. In this Section, we will put both tricks together to show that, in fact, quantum fields onunbounded real space can also be modelled in our framework (a similar argument applies to tori).Consider again an object H : = ( | H | , P H ) of (cid:63) Hilb, with P H : = ∑ Dd = | e d (cid:105)(cid:104) e d | for some orthonormalfamily | e d (cid:105) Dd = . We fix two odd infinite natural numbers ω uv , ω ir ∈ (cid:63) N , and let 2 ω + = ω uv ω ir . We wishto construct the system H ⊗ ω uv (cid:63) Z n ω + corresponding to a H -valued quantum field living on the lattice ω uv (cid:63) Z n ω + in (cid:63) R n , which we have seen in Section 5 to approximate the real space R n to infinitesimalmesh ω uv (and all the way up to some infinity ω ir , where the lattice “circles around”).As objects of (cid:63) Hilb, the space H ⊗ (cid:63) Z n ω + we constructed in the previous Section and the space H ⊗ ω uv (cid:63) Z n ω + we wish to construct in this Section are isomorphic: to see this, it suffices to consider theisomorphism of abelian groups sending k ∈ (cid:63) Z n ω + to p : = ω uv k ∈ ω uv (cid:63) Z n ω + . As a consequence, theonly formal distinction is that we will write our summations over p rather than k , i.e. we will use the sameshorthand ∑ + ω ir p = − ω ir f ( p ) : ≡ ∑ + ω k = − ω f ( k / ω uv ) that we introduced in Section 5.When it comes to quantum field theory, however, the two objects H ⊗ (cid:63) Z n ω + and H ⊗ ω uv (cid:63) Z n ω + havevery different interpretations, corresponding to the different ways in which the underlying lattice isimmersed into n -dimensional non-standard real space (cid:63) R n (and successively related to standard realspace R n ). In the previous Section, the underlying lattice was embedded as the lattice (cid:63) Z n ω + of standard60 Towards Quantum Field Theory in Categorical Quantum Mechanics mesh 1 in (cid:63) R n , while in this section the underlying lattice is embedded as the lattice ω uv (cid:63) Z n ω + ofinfinitesimal mesh ω uv . From the perspective of non-standard reals, the two lattices are equivalent(rescaling by a factor of ω uv ∈ (cid:63) R ), but from the perspective of standard reals they are extremely different:restricting to near-standard reals and quotienting by infinitesimal equivalence sends (cid:63) Z n ω + to the lattice Z n , while ω uv (cid:63) Z n ω + covers the entirety of R n . We now provide justification for the interpretation of H ⊗ ω uv (cid:63) Z n ω + as a space of quantum fields on real space.Just as we did in the previous Section, we will assume that | H | = (cid:63) V for separable standard V .Consider the direct integral of Hilbert spaces (cid:82) ⊕ R n V d p (in the sense of von Neumann [18]), i.e. the spaceof square-integrable functions ϕ : R n → V , together with the inner product (cid:104) ψ | ϕ (cid:105) : = (cid:82) R n (cid:104) ψ ( p ) | ϕ ( p ) (cid:105) d p .Consider its non-standard extension, and let ∏ p ∈ ω uv (cid:63) Z n ω + H be the object obtained by restricting non-standard internal maps (cid:63) ϕ : (cid:63) R n → V to non-standard internal maps (cid:63) ϕ : ω uv (cid:63) Z n ω + → V , and appropri-ately extending the projector P H to the direct sum ∏ p ∈ ω uv (cid:63) Z n ω + (cid:63) V .The universal property for the infinite tensor product H ⊗ ω uv (cid:63) Z n ω + takes a form similar to that ofTheorem 6.1, and has a similar proof: one only needs to use ∏ p ∈ ω uv (cid:63) Z n ω + H in place of the original ∏ k ∈ (cid:63) Z n ω + H . The non-trivial part is the connection between ∏ p ∈ ω uv (cid:63) Z n ω + H and the direct integral (cid:82) ⊕ R n V d p . In the previous Section, the connection between ∏ k ∈ (cid:63) Z n ω + H and the direct sum ∏ k ∈ Z n V was immediate: a function ϕ : Z n → V extends to a function (cid:63) ϕ : (cid:63) Z n → (cid:63) V , which then restricts to (cid:63) ϕ : (cid:63) Z n ω + → (cid:63) V and then back to ϕ : Z n → V . In this Section, we note that (cid:82) ⊕ R n V d p is the L completionof the space of continuous square-integrable functions ϕ : R n → V , so it is enough to show that thesecan be reconstructed from the corresponding (cid:63) ϕ : ω uv (cid:63) Z n ω + → (cid:63) V . But this is indeed the case: if ϕ iscontinuous then we have that st ( x ) = st ( y ) implies st ( (cid:63) ϕ ( x )) = st ( (cid:63) ϕ ( y )) , and in particular we can obtain ϕ back from (cid:63) ϕ in a unique way by setting ϕ ( z ) : = st ( (cid:63) ϕ ( x )) for any x ∈ ω uv (cid:63) Z n ω + such that st ( x ) = z . A legitimate question to ask at this point is: How does Quantum Field Theory fit into the framework wedescribed? Why are we talking about “quantum fields” in the context of certain infinite tensor products?As part of canonical quantisation, classical fields from the Lagrangian formalism are translated intocertain operator-valued distributions, also known as field operators , acting upon quantum states livingin a Fock space. Using the field operators, the classical Lagrangian can be translated into the dynamicsand interactions of the quantum field theory, so it is no surprise that they occupy the vast majority of theliterature dedicated to the subject.It is worth noting, however, that the field operators play a very different role from the classical fieldsthat they originally quantised: classical fields are states of a classical system, while field operators actupon states of a quantum system (e.g. the vacuum). In this sense, the closest correspondents in quantumfield theory to the fields of classical field theory or the wavefunctions of quantum mechanics are, in fact,the quantum states in the Fock space. Just as C is the space of quantum states for a qubit, so the Fockspace is the space of quantum states for a quantum field. And just as we freely refer to the object C asa qubit, so we take the liberty to refer to the Fock space as a quantum field . We will use the term fieldoperator when talking about the operator-valued distributions obtained by canonical quantisation.Let’s consider the textbook example of the real scalar field, a relativistic classical field φ ( x , t ) satisfyingthe Klein-Gordon equation : ∂ µ ∂ µ φ + m φ = . Gogioso & F. Genovese φ ( p , t ) , the Klein-Gordon equation becomes: (cid:16) ∂ ∂ t + ( | p | + m ) (cid:17) φ ( p , t ) = φ ( p , t ) to the Klein-Gordon equation can be thought of as a field ofsimple harmonic oscillators, each oscillator vibrating with its own amplitude and at a frequency given by ν p : = (cid:113) | p | + m for each point p ∈ R of momentum space. In order to quantise the real scalar field φ ,we simply need to quantise the simple harmonic oscillators. We do so in our non-standard framework.Consider the object H of (cid:63) Hilb defined as follows, where τ is some infinite non-standard natural and | n (cid:105) n ∈ (cid:63) N is the standard orthonormal basis for (cid:63) L [ N ] : H : = (cid:16) (cid:63) L [ N ] , τ ∑ n = | n (cid:105)(cid:104) n | (cid:17) (8.3)We will think of H as the non-standard counterpart for a quantum harmonic oscillator: the states | n (cid:105) correspond to energy eigenstates for the oscillator, and we extended our range of energy values all theway up to some infinite natural τ . We define the ladder operators a and a † on H as follows: a | n (cid:105) = (cid:40) n = √ n | n − (cid:105) otherwise a † | n (cid:105) = (cid:40) n = τ √ n + | n + (cid:105) otherwise (8.4)It is easy to check that these operators satisfy the usual canonical commutation relations, up to a correctionfactor accounting for the truncation of energy above the infinite τ : [ a , a † ] = id H − ( τ + ) | τ (cid:105)(cid:104) τ | (8.5)When restricting ourselves to finite energy states, these operators are exactly the ladder operators for thequantum harmonic oscillator. We then proceed to define the number operator N : = a † a , and we obtainthe usual property and commutators for it (no correction this time): N | n (cid:105) = n | n (cid:105) [ N , a † ] = a † [ N , a ] = − a (8.6)The number operator is associated to a †-SCFA on H , the number observable , with | n (cid:105) as classicalstates. For a quantum harmonic oscillator of frequency ν , the Hamiltonian can finally be defined as: H : = ¯ h ν N (8.7)Aside perhaps for the correction term in the canonical commutation relation, this is exactly what wewould expect the non-standard version of the quantum harmonic oscillator to look like, and the traditionalquantum harmonic oscillator is recovered exactly by restricting to states of finite energy.We saw before that a solution to the Klein-Gordon can be interpreted to describe a field of simpleharmonic oscillators at each point p ∈ R of momentum space, vibrating independently with frequenciesgiven by the expression ν p = (cid:113) | p | + m . The natural quantisation of such a scenario involves consideringindependent quantum harmonic oscillators at each point p ∈ R of momentum space, i.e. an infinite directproduct of separable Hilbert spaces over the 3-dimensional continuum. Because such a space would bemathematically unwieldy, and because only finite energy states are deemed to be physically interesting,62 Towards Quantum Field Theory in Categorical Quantum Mechanics the infinite direct product of quantum harmonic oscillators is never constructed, and the Fock space isconsidered instead. The Fock space is the Hilbert space of joint states for the quantum harmonic oscillatorswhich is spanned by those separable states involving only finitely many oscillators not in their groundstate: the state | n (cid:105) for the oscillator at point p ∈ R is considered to count the number of quantum particleswith definite momentum p , and the Fock space is spanned by all states containing finitely many particles.Within our non-standard framework, we don’t have to worry about infinite tensor products, and wedon’t have to restrict ourselves to finite energy states or finite number of particles: as a consequence,we quantise the real scalar field φ by constructing the field of quantum harmonic oscillators in allits glory. This can be done by considering the space H ⊗ ω uv (cid:63) Z ω + defined in Section 7 above: wediscretise momentum space to an infinite lattice ω uv (cid:63) Z ω + of infinitesimal mesh 1 / ω uv , and we place anindependent quantum harmonic oscillator H at each point of the lattice (with varying frequency ν p ).For each p ∈ ω uv (cid:63) Z ω + , we write a p and a † p for the ladder operators acting on the quantum Harmonicoscillator at p (tensored with the identity on all other oscillators), and | n , p (cid:105) τ n = for the orthonormal basisof the oscillator at p . We define the rescaled versions a ( p ) : = (cid:112) ω uv a p and a † ( p ) : = (cid:112) ω uv a † p , whichsatisfy the commutation relations [ a ( p ) , a ( q )] = [ a † ( p ) , a † ( q )] = [ a ( p ) , a † ( q )] = (cid:40) ω uv (cid:0) id − ( ν + ) | ν , p (cid:105)(cid:104) ν , p | (cid:1) if p = q π ( x ) and φ ( x ) can be defined from a ( p ) and a † ( p ) through the followingdiscretised integral, for all points x ∈ ω ir (cid:63) Z ω + in space: φ ( x ) : = ∑ p ω uv (cid:112) ν p (cid:104) a ( p ) e i π p · x + a † ( p ) e − i π p · x (cid:105) π ( x ) : = ∑ p ω uv ( − i ) (cid:112) ν p (cid:104) a ( p ) e i π p · x − a † ( p ) e − i π p · x (cid:105) (8.9)The field operators satisfy commutation relations similar to the ones of the rescaled ladder operators,as would be expected. For every n : ω uv (cid:63) Z ω + → { , ..., τ } , we can define the state | n (cid:105) : = ⊗ p | n ( p ) , p (cid:105) .Then the discretised integral of the (rescaled) number observables for all quantum harmonic oscillators atall points p ∈ ω uv (cid:63) Z ω + of momentum space gives rise to the number operator N on H ⊗ ω uv (cid:63) Z ω + : N : = ∑ p ω uv a † ( p ) a ( p ) = ∑ p a † p a p = ∑ n (cid:16) ∑ p n ( p ) (cid:17) | n (cid:105)(cid:104) n | (8.10)The Hamiltonian for the quantum field is similarly obtained as a discretised integral: H : = ∑ p ω uv ¯ h ν p a † ( p ) a ( p ) = ∑ p ¯ h ν p a † p a p = ∑ n (cid:16) ∑ p ¯ h ν p n ( p ) (cid:17) | n (cid:105)(cid:104) n | (8.11)The traditional Fock space is recovered by considering the states | n (cid:105) with finite energy (cid:104) n | H | n (cid:105) (i.e. thosewith a finite number of particles, all having finite momenta). The corresponding number of particles at astandard point q ∈ R of standard momentum space, which we will denote by st ( n )( q ) , is then given bythe following expression: st ( n )( q ) : = ∑ p ∈ ω uv (cid:63) Z ω + such that st ( p )= q n ( p ) (8.12)The further development of traditional QFT machinery within our framework is left to future work. . Gogioso & F. Genovese In the first section of this work, we have presented a more mature formulation of the category (cid:63)
Hilb,refining and expanding the original definition from Ref. [11] in a number of ways. Firstly, the newdefinition is no longer restricted to standard Hilbert spaces and non-standard extensions of standardorthonormal bases, but instead allows all kinds of non-standard Hilbert spaces and orthonormal families.Secondly, the new definition is basis-independent and has a neater categorical presentation as a fullsub-category of the Karoubi envelope for the category of non-standard Hilbert spaces. Thirdly, objects areno longer self-dual, and compact closure is now formulated in a basis-independent way, analogous to theone used in fHilb. Backward compatibility is guaranteed by the fact that the category (cid:63)
Hilb originallydefined in Ref. [11] is equivalent to a full-subcategory of the category (cid:63)
Hilb redefined in this work.Our new definition allowed us to push the framework beyond its original limitations, and the bulkof this work was dedicated to the explicit constructions of five families of infinite-dimensional quantumsystems that are of interest to the practising quantum theorist. In Sections 3 and 4 we have presentedthe quantum systems for particles in boxes with periodic boundary conditions and particles on lattices:both constructions were already within reach of the original definition of (cid:63)
Hilb, and the special case of aparticle in a one-dimensional box with periodic boundary conditions was already explored in Ref. [11].The first real application of our extended (cid:63)
Hilb category has come in Section 5, where it was put to workin presenting the quantum system for particles in R n . Key to this construction have been the use of a trulynon-standard orthonormal basis (i.e. not the extension of a standard one), together with an approximationof R n achieved by using a non-standard lattice of infinitesimal mesh in (cid:63) R n .We have also seen that our extended definition allows for the treatment of certain cases of interest inquantum field theory. Thanks to a key observation about exponentials of infinite natural numbers—whichare themselves infinite natural numbers by Transfer Theorem—and exploiting the freedom to work withnon-separable spaces, we have constructed in Section 6 a quantum system suitable for the treatment ofquantum fields on a cubic lattice Z n . Finally, in Section 7 we have combined the ideas of Sections 5 and 6to construct a quantum system suitable for the treatment of quantum fields in R n , and in Section 8 we haveprovided a first direct link to the traditional quantum field theoretic framework.This work is a significant development of the original Ref. [11], and provides a solid basis forthe application of algebraic and diagrammatic methods from CQM to infinite-dimensional quantummechanics and quantum field theory. From here, we foresee a number of interesting further developmentsand applications, some of which are briefly detailed below. Future work.
To begin with, an extension of Theorem 2.1 to the entirety of (cid:63)
Hilb should be a priorityin future developments, as it would establish a uniformly tight relationship between our frameworkand more traditional approaches to quantum mechanics and quantum field theory. In the same spirit ofrelating to mainstream works, we endeavour to explicitly construct more quantum systems of widespreadinterest—such as wavefunctions/fields over locally compact groups—and to explore more sophisticatedapplications to quantum field theory and quantum gravity (e.g. constructing analogues of algebraicquantum field theory and introducing Feynman diagrams).On a different note, we believe that it would be extremely interesting to analyse the natural infinite-dimensional extension of a number of quantum protocols already formalised in CQM. Examples mightinclude simple protocols—such as quantum teleportation and quantum key distribution—or more elabo-rate protocols—such as the generalised Mermin-type non-locality arguments of Ref. [13], and infinite-dimensional extensions of the work on tight reference-frame-independent quantum teleportation ofRef. [25]. To kick-start this line of research, an application of (cid:63)
Hilb to the Hidden Subgroup Problem forthe infinite group Z n has already appeared in Ref. [12], based on the original (cid:63) Hilb from Ref. [11].64
Towards Quantum Field Theory in Categorical Quantum Mechanics
Acknowledgements.
The authors would like to thank Samson Abramsky, Bob Coecke, DavidReutter and Masanao Ozawa for comments, suggestions and useful discussions, as well as SukritaChatterji and Nicol`o Chiappori for their support. SG gratefully acknowledges funding from EPSRCand the Williams Scholarship offered by Trinity College. FG gratefully acknowledges funding from theAFSOR grant “Algorithmic and logical aspects when composing meaning”.
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A The non-standard cyclic group (cid:63) Z ω + (cid:63) Z ω + is defined to be the internal set of non-standard integers {− ω , ..., + ω } endowedwith 0 as unit and with the following binary operation ⊕ as group multiplication: k ⊕ h : = k + h if − ω ≤ k + h ≤ + ω k + h − ( ω + ) if + ω < k + hk + h + ( ω + ) if k + h < − ω (A.1) The group (cid:63) Z ω + has the integers as a subgroup: if k , h ∈ Z are standard integers, then certainly − ω ≤ k + h ≤ + ω , and hence k ⊕ h = k + h . More in general, the group ( (cid:63) Z n ω + , ⊕ , ) contains Z n asa subgroup, and as a consequence it is a legitimate non-standard extension of the translation group Z n of an n -dimensional lattice. Furthermore, the group of automorphisms of (cid:63) Z n ω + contains the group ofautomorphisms of Z n as a subgroup (rotations and reflections about the origin are the same, but there aremore translations of (cid:63) Z n ω + than there are of Z n ).It is not hard to show that (cid:63) Z ω + ∼ = Z × C for some dense abelian group C . As the elements of C , wetake exactly one representative from each full copy of Z in {− ω , + ω } , plus a single element representingboth the final segment + ω − N and the initial segment − ω + N . For each full copy of Z , we take therepresentative to be the zero element of that copy, and in particular we let 0 C : = + ω − N and the initial segment − ω + N asglued together to form a single copy of Z , and without loss of generality we pick the representative to be − ω (so that − ω is the zero element for that that virtual copy of the integers). Given these considerations,we can always decompose k ∈ (cid:63) Z ω + uniquely as ( k (cid:48) , θ k ) in terms of a standard integer component k (cid:48) ∈ Z and a representative θ k ∈ C : k (cid:48) = k − θ k if θ k (cid:54) = − ω k + ω if θ k = − ω and k ∈ − ω + N k − ( ω + ) if θ k = − ω and k ∈ + ω − N (A.2) The standard integer components can be added independently of the representatives in C (with somecare taken for the boundary case of θ k = − ω ), so that this defines a group isomorphism (cid:63) Z ω + ∼ = Z × C .Also, recall that the infinite positive and negative integers form two dense, uncountable sets, each havingno maximum or minimum, and with the finite integers Z in between [23]: as a consequence, the group C is dense and uncountable (but, unlike the non-standard integers, it is not totally ordered).66 Towards Quantum Field Theory in Categorical Quantum Mechanics
There are three different embeddings of the periodic non-standard cubic lattice (cid:63) Z n ω + that are of interestin this work, reflecting distinct applications to the modelling of cubic lattices Z n , the approximation ofreal space R n , and the approximation of real toroidal space T n :(i) the embedding as the lattice (cid:63) Z n ω + in (cid:63) R n , where we send k ∈ (cid:63) Z n ω + to k ∈ (cid:63) R n (not a subgroup);(ii) the embedding as the lattice ω uv (cid:63) Z n ω + in (cid:63) R n , where ω : = ω uv ω ir for some infinite ω uv , ω ir ∈ (cid:63) N + and we send k ∈ (cid:63) Z n ω + to p : = k / ω uv ∈ (cid:63) R n (also not a subgroup);(iii) the embedding as the subgroup ω + (cid:63) Z n ω + in (cid:63) T n , where we send k ∈ (cid:63) Z n ω + to ω + k ∈ (cid:63) T n .The first embedding uses (cid:63) Z n ω + to approximate Z n , under the observation that the latter is a subgroup ofthe former. The second embedding instead uses (cid:63) Z n ω + to approximate R n : this is a bit more complicated,as R n cannot be seen as a subgroup of (cid:63) Z n ω + (the latter is discrete, while the former is dense). However,we can consider the subgroup of (cid:63) Z n ω + given by those k such that p : = k / ω uv is a near-standard vectorin (cid:63) R n , and we can quotient it by infinitesimal equivalence of vectors to obtain the group R n . Hence thesecond embedding can be seen to approximate R n by using a non-standard lattice of infinitesimal mesh, andworking up to infinitesimal equivalence. The third embedding is used similarly to the second embedding,but to approximate T n instead of R n (with a quotient group homomorphism ω + (cid:63) Z n ω + (cid:16) T n ).It should be noted that ω uv (cid:63) Z n ω + is a lattice, and as such it does not enjoy the same symmetries of thecontinuum R n : finite translations can be approximated up to infinitesimals, but rotations cannot. This is incontrast to the (cid:63) Z n ω + case, the automorphisms of which contain the automorphisms of Z n as a subgroup.When working with quantum systems, however, we are not really interested in the symmetries Φ of R n ,but rather in the unitary automorphisms U Φ of L [ R n ] that they induce: because the subspace defined bythe truncating projector P H R n spans the near-standard vectors, all these unitaries lift from L [ R n ] to H R n .From the point of view of the non-standard quantum system H R n , it is “as if” ω uv (cid:63) Z n ω + really possessedall the symmetries of R n . B Delta functions, plane waves and position/momentum cut-offs
We present here the proof that the position eigenstates | δ x (cid:105) defined in Section 3 are the classical states forthe group algebra of (cid:63) Z n ω + , exactly when x ∈ ω + (cid:63) Z n ω + : ◦ (cid:0) | δ x (cid:105) (cid:1) = ∑ n ∑ k | χ k (cid:105) ⊗ | χ n (cid:9) k (cid:105) (cid:104) χ n | δ x (cid:105) = ∑ n ∑ k | χ k (cid:105) ⊗ | χ n (cid:9) k (cid:105) χ n ( x ) ∗ = ∑ n ∑ k | χ k (cid:105) ⊗ | χ n (cid:9) k (cid:105) χ k ( x ) ∗ χ n (cid:9) k ( x ) ∗ e i π ( ω + ) s · x = (cid:104) ∑ n (cid:48) χ n (cid:48) ( x ) ∗ | χ n (cid:48) (cid:105) (cid:105) ⊗ (cid:104) ∑ k χ k ( x ) ∗ | χ k (cid:105) (cid:105) = | δ x (cid:105) ⊗ | δ x (cid:105) . (B.1)In the third line, the extra phase e i π ( ω + ) s · x appears because χ k is a character of Z n , not of (cid:63) Z n ω + : thevalue of s ∈ {− , , + } n keeps track of whether some modular reductions were necessary to go from k ⊕ ( n (cid:9) k ) to n . It is cancelled out if and only if we require x to be in the form x = j ω + , for some j ∈ (cid:63) Z n ω + . Hence the duality between the large-scale cut-off for momenta and the small-scale cut-offfor positions, a well-understood phenomenon in quantum mechanics, arises as a consequence of a purelyalgebraic requirement in our non-standard framework. Similar phases appear for the setups of Section4 and Section 5, with similar large/small-scale dualities following from the algebraic requirement ofcopiability for classical states.8