aa r X i v : . [ m a t h - ph ] F e b A groupoidification of the fermion algebra
Wei Chen, Bing-Sheng Lin ∗ School of Mathematics, South China University of Technology, Guangzhou 510641, China.
September 4, 2018
Abstract
In this paper, we consider the groupoidification of the fermion algebra. We con-struct a groupoid as the categorical analogues of the fermionic Fock space, and thecreation and annihilation operators correspond to spans of groupoids. The categor-ical fermionic Fock states have some extra structures comparing with the normalforms. We also construct a 2-category of spans of groupoids corresponding to thefermion algebra. The relations of the morphisms in this 2-category are consistentwith those in the graphical category which is represented by string diagrams.
PACS numbers:
Key words: groupoidification, fermion algebra, categorification, 2-category
In recent years, there has been much interest in the studies of categorifications of the-ories in mathematics and theoretical physics [1]-[14]. In general, categorification is aprocess of replacing set-theoretic theorems by category-theoretic analogues. It replacessets by categories, functions by functors, and equations between functions by naturaltransformations of functors [1]. Categorification can be thought of as the process ofenhancing an algebraic object to a more sophisticated one, while decategorification isthe process of reducing the categorified object back to the simpler original object. So auseful categorification should possess a richer structure not seen in the underlying ob-ject. The categorification of physical theories may extend the mathematical structuresof existing physical theories and help us solve the remaining problems in fundamentalphysics, it can also help us better understand the physical essence.Groupoidification is a form of categorification in which vector spaces are replacedby groupoids and linear operators are replaced by spans of groupoids [15]. A groupoidis a special type of category in which every morphism is invertible. It can be seen asa generalization of a group, and a usual group is a groupoid where there is only oneobject. In the framework of groupoidification, the configuration spaces for a physicalsystem are described by some groupoids, and the physical histories are described byspans of groupoids. Furthermore, one can encode the symmetries of the physical systemin arrows in the groupoids.In Ref. [15], the authors considered the groupoidification of the quantum harmonicoscillator system, and the Hilbert space for the quantum harmonic oscillator arises nat-urally from degroupoidifying the groupoid of finite sets and bijections, and these give apurely combinatorial interpretation of creation and annihilation operators. The authorsalso considered the groupoidification of the field operators, their normal-ordered powers ∗ e-mail: [email protected] In the following contents, we will only consider the one-dimensional fermion algebra,and the results can be easily extended to those of higher-dimensional fermion algebras.In normal quantum mechanics, the fermionic creation and annihilation operators ˆ f † , ˆ f satisfy the fermion algebraic relations { ˆ f , ˆ f † } := ˆ f ˆ f † + ˆ f † ˆ f = 1 , { ˆ f , ˆ f } = { ˆ f † , ˆ f † } = 0 . (1)Obviously, we have ˆ f ˆ f = 0 and ˆ f † ˆ f † = 0.The corresponding Hilbert space is spanned only by two states, which can be denotedby | i and | i . These are single mode fermionic Fock states, and satisfy the followingrelations ˆ f | i = 0 , ˆ f † | i = | i , ˆ f | i = | i , ˆ f † | i = 0 . (2)The states | i and | i are orthonormal, h | i = h | i = 1 , h | i = h | i = 0 . (3)We also have the following matrix representationsˆ f = (cid:18) (cid:19) , ˆ f † = (cid:18) (cid:19) , | i = (cid:18) (cid:19) , | i = (cid:18) (cid:19) . (4) In Ref. [17], the authors constructed a diagrammatic categorification of the fermionalgebra, here we will briefly review the main results. Let F be an additive k -linear strictmonoidal category for a commutative ring k , and the set of objects in F is generated byobjects Q + and Q − . An arbitrary object of F is a finite direct sum of tensor products Q ε := Q ε ⊗ · · · ⊗ Q ε n , and ε = ε . . . ε n is a finite sequence of + and − signs. The unitobject is = Q ∅ . The objects Q + and Q − can be regarded as the categorical analoguesof the fermionic creation and annihilation operators ˆ f † , ˆ f .2he morphisms in F are denoted by string diagrams. The diagrams are orientedcompact one-manifolds immersed in the strip R × [0 , { , . . . , m } × { } and { , . . . , k } × { } ,where m and k are the lengths of the sequences ε and ε ′ respectively. The orientationof the one-manifold at the endpoints must agree with the signs in the sequences ε and ε ′ . For example, the diagram (5)is one of the morphisms from Q − + to Q + − . A diagram without endpoints gives anendomorphism of .The space of morphisms Hom F ( Q ε , Q ε ′ ) is the k -module generated by string dia-grams modulo local relations. The local relations for the morphisms in F are as follows.= , = 0 (6)+ = id (7)The relations (6) are equivalent to the following relations= , = 0 (8)The second relation in (6) means that Q ++ ∼ = , where is zero object in the additivecategory F . There is also Q −− ∼ = . These isomorphic relations just correspond to theoperator relations ˆ f † ˆ f † = 0 and ˆ f ˆ f = 0. Furthermore, one may find that all the stringdiagrams with crossings are equal to zero.From the local relations above, one may obtain the following isomorphic relation inthe category F Q − + ⊕ Q + − ∼ = , (9)so in the Grothendieck group K ( F ), we have[ Q − ][ Q + ] + [ Q + ][ Q − ] = 1 , (10)which is the fermion algebraic relation (1). In the framework of groupoidification, groupoids and spans can be degroupoidified tovectors and linear operators, which generally can be represented by matrices. Thegroupoids and spans also have natural physical meanings [15]. For example, let usconsider the following span of sets
M AB g f (11)3 can be considered as a set whose elements are possible initial states for the physicalsystem, and B is the set whose elements are possible final states, then M can be con-sidered as a set of possible events, or histories. For example, let i ∈ A denotes the i thinitial state and j ∈ B denotes the j th final state, then the following subset of MM ji = { m : f ( m ) = i, g ( m ) = j } (12)can be regarded as the set of ways for the physical system to undergo a transition fromits i th initial state to the j th final state. If all the sets M ij are finite, then the ( i, j )-elements of the correspond matrix under degroupoidification are just the cardinalities | M ij | . When A , B and M are groupoids, the objects of A and B can be regarded as thecorresponding physical states. Furthermore, in the groupoids, one can also consider themorphisms between the objects, these morphisms can represent some symmetries of thephysical system.Two spans ( B g ←− M f −→ A ) and ( B g ′ ←− N f ′ −→ A ) are isomorphic if there is anisomorphism h : M → N satisfying the following commuting diagram [18] M AB Ng fg ′ f ′ h (13)In Ref. [16], the authors considered the groupoidification of the Heisenberg algebra.They categorified the bosonic Fock state | n i by the n -element set. This is intuitive andreasonable, since the Fock state | n i means there are n quanta or particles. The bosonicFock space is naturally represented by the groupoid S of finite sets and bijections. Thecreation and annihilation operators ˆ a † and ˆ a are represented by the following spans ofgroupoids S , S SS +1 idˆ a † SS S id +1ˆ a (14)and S +1 −−→ S is the functor taking the disjoint union with the one-element set.For the fermion algebra, the corresponding Fock space is spanned only by two states,namely | i and | i . There are no any nontrivial combinatorial models corresponding tothese states, so we can not construct the categorical analogues of the fermionic Fockstates with the constructions used in Ref. [16].Intuitively, there is some duality between the Fock states | i and | i from the rela-tions (2), so one may consider the categorical analogues of the states | i and | i as someobject A and its dual object A ∗ in some category (e.g., a compact closed category). Forexample, in the category FdVect with finite-dimensional vector spaces as objects andlinear maps as morphisms, A is a finite-dimensional vector space, and A ∗ is its dualspace.Let Ψ be a compact closed category containing an object A and its dual object A ∗ . All the morphisms in Hom Ψ ( A, A ) and Hom Ψ ( A ∗ , A ∗ ) are invertible, namely auto-morphisms, and Hom Ψ ( A, A ∗ ) = Hom Ψ ( A ∗ , A ) = ∅ . In fact, in this construction, themorphisms Hom Ψ ( A, A ) form a group, namely the automorphism group of A , which can4e denoted by Aut Ψ ( A ), or simply Aut( A ). Similarly, there is an automorphism groupAut( A ∗ ) of the object A ∗ . Furthermore, one may naturally assume Aut( A ) ∼ = Aut( A ∗ ). Ψ is also a groupoid, it can be regarded as a groupoidification of the fermionic Fockspace, and the objects A and A ∗ correspond to the states | i and | i , respectively.We may construct the categorical analogues of the fermionic creation and annihi-lation operators ˆ f † , ˆ f with the aid of spans of groupoids. Let us define the spans ofgroupoids as follows, H ΨΨ
T IF † HΨ Ψ
I TF (15)Here H is a full subcategory of Ψ containing only one object A , and Hom H ( A, A ) =Hom Ψ ( A, A ). Obviously, H is also a groupoid, in fact, a group. The inclusion functor I : H → Ψ takes objects and morphisms to themselves, and T is a contravariant functortakes the object A A ∗ and morphisms Hom H ( A, A ) ∋ f f ∗ ∈ Hom Ψ ( A ∗ , A ∗ ). Wedenote these spans by F † and F , respectively. These spans are the categorical analoguesof the fermionic creation and annihilation operators ˆ f † and ˆ f .Similar to Ref. [16], one may construct a 2-category C of spans of groupoids corre-sponding to the graphical category F in the previous section. In C , objects are sometame groupoids, 1-morphisms are isomorphism classes of spans of groupoids, with com-position defined by weak pullback, and 2-morphisms are isomorphism classes of spans ofspans. So Ψ is an object in C , and the spans of groupoids F † and F are just 1-morphismsin C .Now let us consider the composition of 1-morphisms in C , which are just compositionof the corresponding spans. Here we will use the notations used in Ref. [16]. For example,the composition of the spans of groupoids ( B G ←− X H −→ A ) and ( C K ←− Y J −→ B ) is thefollowing span ( J ↓ G ) Y XC B AP Y P X K J G H (16)( J ↓ G ) is a weak pullback groupoid where an object is a triple ( x, y, f ) consisting ofan object x ∈ Ob( X ), an object y ∈ Ob( Y ), and an isomorphism f : G ( x ) → J ( y ) in B . A morphism ( x , y , f ) → ( x , y , f ) in ( J ↓ G ) consists of morphisms x a −→ x and y b −→ y satisfying the following commuting diagram G ( x ) G ( x ) J ( y ) J ( y ) G ( a ) f f J ( b ) (17)5his composite span is just the span ( C K ◦ P Y ←−−−− ( J ↓ G ) H ◦ P X −−−−→ A ).So in the 2-category C , the composition of 1-morphisms F † ◦ F is the followingcomposition of spans, ( I ↓ I ) H HΨ Ψ Ψ π π T I I T (18)The groupoid ( I ↓ I ) has objects which are triples ( A, A, α ), and A α −→ A is an isomor-phism in Ψ . The projection maps π and π act in an obvious way on the objects. Theabove composition can be rewritten as ( I ↓ I ) ΨΨ T ◦ π T ◦ π (19)This is just a 1-morphism Ψ F † ◦ F −−−→ Ψ in the 2-category C .Similarly, the composition of spans F ◦ F † is( T ↓ T ) H HΨ Ψ Ψ π π I T T I (20)The groupoid ( T ↓ T ) has objects ( A, A, β ), and A ∗ β −→ A ∗ is an isomorphism in Ψ . Theabove composition can be rewritten as ( T ↓ T ) ΨΨ I ◦ π I ◦ π (21)This is a 1-morphism Ψ F ◦ F † −−−→ Ψ in the 2-category C .It is easy to see that, there is an isomorphic relation F ◦ F † ⊕ F † ◦ F ∼ = id Ψ , this isjust the relation (9). There are no compositions of 1-morphisms like F ◦ F or F † ◦ F † in the 2-category C . This is consistent with the results of the fermion algebra, since wehave ˆ f ˆ f = 0 and ˆ f † ˆ f † = 0.Now let us consider the 2-morphisms in C , which are isomorphism classes of spansof spans. A span of spans of type ( B G ←− X H −→ A ) → ( B J ←− Y K −→ A ) is a span Y R ←− Z S −→ X equipped with natural isomorphisms G ◦ S µ −→ J ◦ R and H ◦ S ν −→ K ◦ R ,6s indicated by the following diagram [18] XB AYZG HJ KSRµ ν (22)We are considering such diagrams as 2-morphisms only up to isomorphism, namely, theinner span Y R ←− Z S −→ X is only considered up to an isomorphism of spans in the senseof (13).For example, ( I ↓ I ) Ψ ΨΨH T ◦ π T ◦ π id Ψ id Ψ ∆ I T id id (23)where ∆ I is the diagonal functor, it takes objects x ( x, x, id I ( x ) ) and morphisms g ( g, g ). This is a 2-morphism id Ψ η −→ F † ◦ F in C . It can also be represented by thefollowing string diagram η (24)The 2-morphism F ◦ F † ǫ −→ id Ψ corresponds to the following diagram( T ↓ T ) Ψ ΨΨH I ◦ π I ◦ π id Ψ id Ψ ∆ T I id id (25)This 2-morphism can be represented by the following string diagram ǫ (26)7e also have 2-morphisms corresponding to the following diagram η † : F † ◦ F → id Ψ ǫ † : id Ψ → F ◦ F † (27)It is easy to see that, there are no diagrams with crossings, because there are no any2-morphisms like F ◦ F † → F † ◦ F or F † ◦ F → F ◦ F † in C . This is consistent with theresults of string diagrams in the previous section.Using the constructions (23) and (25), one can verify the following relations by somestraightforward calculations,id F ◦ F † = ǫ † ◦ ǫ, id F † ◦ F = η ◦ η † , id id Ψ = ǫ ◦ ǫ † + η † ◦ η. (28)Note that the 2-morphisms in the 2-category C are isomorphism classes of spans of spans,so these spans of spans are equal in the sense of isomorphism. The first relation is equiv-alent to the first diagrammatic relation in (6), the second relation is equivalent to thefirst diagrammatic relation in (8), and the last relation is equivalent to the diagrammaticrelation (7).Using the results in Ref. [16], one may find that there is an ambidextrous adjunctionbetween the spans F and F † , and the 2-morphisms η , ǫ † and ǫ , η † are the correspondingunits and counits. So we have the following adjunction equations( η † ◦ id F † ) ∗ (id F † ◦ ǫ † ) = id F † = (id F † ◦ η ) ∗ ( ǫ ◦ id F † ) , ( ǫ ◦ id F ) ∗ (id F ◦ η ) = id F = (id F ◦ ǫ † ) ∗ ( η † ◦ id F ) , (29)here “ ◦ ” and “ ∗ ” denote the vertical and horizontal composition of 2-morphisms, re-spectively. These are just the zig-zag rules in the string diagrams= = (30)= = (31)This is the isotopy condition of strands in the graphical category. In this paper, based on the methods developed in [15, 16], we studied the groupoidifica-tion of the fermion algebra. We constructed groupoids corresponding to the fermionicFock space in an intuitive way, and the fermionic creation and annihilation operatorscorrespond to some types of spans of groupoids. We find that the construction of spansfor the fermionic creation and annihilation operators are a little different from those inthe bosonic case in Ref. [16]. We also construct a 2-category of spans of groupoids, and8e found that the relations of the 2-morphisms are consistent with those in the graphicalcategory constructed in Ref. [17].Since the fermion algebraic relations and the corresponding Fock space are much sim-pler than those of the boson algebra, we found that the groupoidification of the fermionalgebra is also much simpler than that of the boson algebra. This is consistent with theresults of the diagrammatic categorification with the methods of string diagrams.The fermionic Fock space corresponds to the groupoid Ψ , and the Fock states | i and | i correspond to the object A and its dual object A ∗ in Ψ . If Aut( A ) and Aut( A ∗ )are trivial groups, namely Hom Ψ ( A, A ) = { id A } and Hom Ψ ( A ∗ , A ∗ ) = { id A ∗ } , thenthe groupoid Ψ is just a discrete groupoid, in fact, a set. In this case, the categoricalanalogues of the fermionic Fock states | i and | i have no any extra structures. Thisis a trivial case of categorification. If Aut( A ) and Aut( A ∗ ) are notrivial groups, thenthere are some additional structures on the categorical fermionic Fock states. In thiscase, the categorical fermionic Fock states may have some additional properties, andthese additional properties maybe depend on the concrete physical systems. Differentadditional structures maybe correspond to different fermion systems. So one may usethese categorical fermionic Fock states to describe the fermion systems more finely, andstudy some additional properties of the fermion systems.Our methods can be easily extended to the study of higher-dimensional fermionalgebras. The present work also provides new insight into the fermion algebra and thecorresponding quantum physics. Since the boson and fermion algebras are the simplestand most fundamental algebraic relations in quantum physics, our result is a complementto the study of groupoidification and categorification of physical theories. Acknowledgements
This project is supported by the National Natural Science Foundation of China (Nos.11405060, 11571119).
References [1] J. C. Baez and J. Dolan, “Categorification.”
Contemp. Math. , 1-36 (1998),arXiv:math/9802029.[2] L. Crane and I. B. Frenkel, “Four-dimensional topological quantum field theory,Hopf categories, and the canonical bases.”
J. Math. Phys. , 5136-5154 (1994),arXiv:hep-th/9405183.[3] J. Morton, “Categorified algebra and quantum mechanics.” Theory Appl. Categ. , 785-854 (2006), arXiv:math/0601458.[4] J. Vicary, “A categorical framework for the quantum harmonic oscillator.” Int. J.Theor. Phys. , 3408-3447 (2008), arXiv:0706.0711.[5] C. Heunen, N. P. Landsman and B. Spitters, “A topos for algebraic quantumtheory.” Comm. Math. Phys. , 63-110 (2009), arXiv:0709.4364.[6] S. Abramsky and B. Coecke, “Categorical quantum mechanics.” in:
Hand-book of Quantum Logic and Quantum Structures: Quantum Logic , eds. K. En-gesser, D.M. Gabbay, and D. Lehmann, Elsevier, Amsterdam, pp. 261-323 (2009),arXiv:0808.1023. 97] C. J. Isham, “Topos methods in the foundations of physics.” in
Deep Beauty ,ed. H. Halvorson, Cambridge University Press, Cambridge, pp. 187-206 (2011),arXiv:1004.3564.[8] A. D. Lauda, “A categorification of quantum sl(2).”
Adv. Math. , 3327-3424(2010), arXiv:0803.3652.[9] M. Khovanov, “Heisenberg algebra and a graphical calculus.” (2010),arXiv:1009.3295.[10] S. Cautis and A. Licata, “Heisenberg categorification and Hilbert schemes.”
DukeMath. J. , 2469-2547 (2012), arXiv:1009.5147.[11] A. Licata and A. Savage, “Hecke algebras, finite general linear groups, and Heisen-berg categorification.”
Quantum Topology , 125-185 (2013), arXiv:1101.0420.[12] B. S. Lin and K. Wu, “A categorification of the boson oscillator.” Commun. Theor.Phys. , 34-40 (2012).[13] L. Q. Cai, B. S. Lin and K. Wu, “A diagrammatic categorification of q -boson and q -fermion algebras.” Chin. Phys. B , 020201 (2012).[14] W. Chen, B. S. Lin, “A diagrammatic approach to the categorical coherent state.” J. Math. Phys. , 113506 (2013).[15] J. C. Baez, A. E. Hoffnung and C. D. Walker, “Higher dimensional algebra VII:Groupoidification.” Theory Appl. Categ. , 489-553 (2010), arXiv:0908.4305.[16] J. C. Morton and J. Vicary, “The categorified heisenberg algebra I: a combinatorialrepresentation.” (2012), arXiv:1207.2054.[17] B. S. Lin, Z. X. Wang, K. Wu, and Z. F. Yang, “A diagrammatic categorificationof the fermion algebra.” Chin. Phys. B , 100201 (2013), arXiv:1307.4522.[18] J. C. Morton, “Two-vector spaces and groupoids.” Applied Categorical Structures19