International Journal of Theoretical Physics

, Volume 56, Issue 7, pp 2071–2080 | Cite as

A Groupoidification of the Fermion Algebra

  • Wei Chen
  • Bingsheng Lin


In this paper, we consider the groupoidification of the fermion algebra. We construct a groupoid as the categorical analogues of the fermionic Fock space, and the creation and annihilation operators correspond to spans of groupoids. The categorical fermionic Fock states have some extra structures comparing with the normal forms. We also construct a 2-category of spans of groupoids corresponding to the fermion algebra. The relations of the morphisms in this 2-category are consistent with those in the graphical category which is represented by string diagrams. One may use these formalisms to describe the fermion systems more finely, and study some additional properties of the fermion systems.


Groupoidification Fermion algebra Categorification 2-category 



This work is supported by the National Natural Science Foundation of China (Grant Nos. 11405060, 11571119).


  1. 1.
    Baez, J.C., Dolan, J.: Categorification. Contemp. Math. 230, 1–36 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Crane, L., Frenkel, I.B.: Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. J. Math. Phys. 35, 5136–5154 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Morton, J.: Categorified algebra and quantum mechanics. Theory Appl. Categ. 16, 785–854 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Vicary, J.: A categorical framework for the quantum harmonic oscillator. Int. J. Theor. Phys. 47, 3408–3447 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Heunen, C., Landsman, N.P., Spitters, B.: A topos for algebraic quantum theory. Comm. Math. Phys. 291, 63–110 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Abramsky, S., Coecke, B.: Categorical quantum mechanics. In: Engesser, K., Gabbay, D.M., Lehmann, D (eds.) Handbook of Quantum Logic and Quantum Structures: Quantum Logic, pp. 261–323. Elsevier, Amsterdam (2009)CrossRefGoogle Scholar
  7. 7.
    Isham, C.J.: Topos methods in the foundations of physics. In: Halvorson, H (ed.) Deep Beauty, pp. 187–206. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  8. 8.
    Lauda, A.D.: A categorification of quantum sl(2). Adv. Math. 225, 3327–3424 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Khovanov, M.: Heisenberg algebra and a graphical calculus. arXiv:1009.3295 (2010)
  10. 10.
    Cautis, S., Licata, A.: Heisenberg categorification and Hilbert schemes. Duke Math. J. 161, 2469–2547 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Licata, A., Savage, A.: Hecke algebras, finite general linear groups, and Heisenberg categorification. Quantum Topol. 4, 125–185 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lin, B.S., Wu, K.: A categorification of the boson oscillator. Commun. Theor. Phys. 57, 34–40 (2012)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Cai, L.Q., Lin, B.S., Wu, K.: A diagrammatic categorification of q-boson and q-fermion algebras. Chin. Phys. B 21, 020201 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Chen, W., Lin, B.S.: A diagrammatic approach to the categorical coherent state. J. Math. Phys. 54, 113506 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Baez, J.C., Hoffnung, A.E., Walker, C.D.: Higher dimensional algebra VII: groupoidification. Theory Appl. Categ. 24, 489–553 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Morton, J.C., Vicary, J.: The categorified heisenberg algebra I: a combinatorial representation. arXiv:1207.2054 (2012)
  17. 17.
    Lin, B.S., Wang, Z.X., Wu, K., Yang, Z.F.: A diagrammatic categorification of the fermion algebra. Chin. Phys. B 22, 100201 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    Morton, J.C.: Two-vector spaces and groupoids. Appl. Categ. Struct. 19, 659–707 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of MathematicsSouth China University of TechnologyGuangzhouChina

Personalised recommendations