1. Hopf Algebra Structure of a Model Quantum Field Theory Allan Solomon Open University and University of Paris VI Collaborators: Gerard Duchamp, U. Paris 13 Karol Penson, U. Paris VI Pawel Blasiak Inst. of Nuclear Phys., Krakow Andrzej Horzela, Inst. of Nuclear Physics Krakow G26 International Conference on Group Theoretical Methods in Physics New York, June 2006.

2. Abstract Recent elegant work [1] on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. quantum theory of non- commuting non-field-dependent operators. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs [2, 3], analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure[4]. We illustrate these ideas by means of simple solvable models, e.g. the partition function for a free boson gas and for a superfluid bose system. Finally, we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra.

3. References [1] A readable account may be found in Dirk Kreimer's "Knots and Feynman Diagrams", Cambridge Lecture Notes in Physics, CUP (2000). [2] We use the graphical description of Bender, Brody and Meister, J.Math.Phys. 40, 3239 (1999) and arXiv:quant-ph/0604164 [3] This approach is extended in Blasiak, Penson, Solomon, Horzela and Duchamp, J.Math.Phys. 46, 052110 (2005) and arXiv:quant-ph/0405103 [4] A preliminary account of the more mathematical aspects of this work may be found in arXiv: cs.SC/0510041

4. Content of Talk 1. Partition Functions and Generating Functions i: Partition Function Integrand (PFI) ii. Exponential Generating Function iii: Bell numbers and their Graphs vi: Connected Graph Theorem iv: Combinatorial properties of the calculations v: PFI General Case Examples: Free Boson Gas, Superfluid bosons. 3. Relation to other Models (QFT) 2. Model « Feynman » Graphs and the Hopf Algebra of these Graphs

5. QFT and Simplified Model QFT Model Model Analysis Yes Yes No No Graphs (Topology) Yes Yes Algebra Solvable No No Yes Yes

6. Partition Function Free Boson Gas (single mode)

7. Partition Function Free Boson Gas (single mode) Coherent state basis

8. Partition Function Integrand The Partition Function is the integral of a function which has a combinatorial interpretation, as the generating Function of the Bell polynomials in the Free Boson Case. Generally We shall now give a Combinatorial, Graphical interpretation of the Partition Function Integrand in more General cases.

9. Generating Functions In general, for combinatorial numbers c(n), DEFINITION Bell Numbers B(n) : EXAMPLES G(x) is, more precisely, the Exponential Generating Function (egf).

10. Generating Functions For 2-parameter combinatorial numbers c(n,k), DEFINITION EXAMPLES Stirling, Bell

11. Combinatorial numbers Bell numbers B(n) count the number of ways of putting n different objects in n identical containers 1-parameter 2-parameter Stirling numbers (of 2 nd kind) S(n,k) count the number of ways of putting n different objects in k identical (non-empty) containers so ‘Choose’ n C k Stirling 2 nd kind S(n,k)

12. Bell Numbers, Bell polynomials As Bell Numbers are, in a sense which we shall show, generic for the description of Partition Functions, we recap their generating functions.

13. Graphs for Bell numbers B(n) n=1 n=2 1 1 n=3 3 1 Total 1 2 5 B(n) 1 1

14. Connected Graph Theorem* *GW Ford and GE Uhlenbeck,Proc.Nat.Acad.42(1956)122 If C(x) is the generating function of CONNECTED labelled graphs then is the generating function of ALL graphs!

15. Connected Graphs for Bell numbers B(n) n=1n=2 n=3 1 C(n) 1 1 Hence Generating function for ALL graphs is

16. Partition Function Integrand General case w is a ‘word’ in the a,a* operators Limit =1 from normalization of |z> Evaluation depends on finding ‘normal order’ of w n

17. Origin emits single lines only, and vertex receives any number of lines Vertex Strengths V n n=1 n=2 1 1 n=3 31 1 1 General Generating Functions V1V1 V 1 2 +V 2 V 1 3 +3V 1 V 2 +V 3

18. Connected Graphs in General n=1n=2 n=3 V3V3 C(n) V1V1 V2V2 Hence Generating function for ALL graphs is

19. Partition Function Integrand: Expansion for Free Boson Gas n=1 n=2 1 1 n=3 3 1 x=- y=|z| 2 (y+y 2 )x 2 /2! (y+3y 2 +y 3 ) x 3 /3! 1 1 yx V n (y)=y all n

20. Example: Superfluid Boson System This formula generates an integer sequence for c 1, c 2 even integers. Choosing c 1, c 2 and z real to simplify expression, x=-

21. Example: Superfluid Boson System This formula generates an integer sequence for c 1 =2, c 2 =4. Choose c 1 =2, c 2 =4 and y=z 2 real This corresponds to the integer sequence 8,102,1728,35916,...

22. Partition Function Integrand: Expansion for Superfluid Boson Gas n=1 n=2 1 1 n=3 3 1 x=- ,y=z 2 (V 2 (y)+V 1 (y) 2 )x 2 /2! (V 3 +3V 1 V 2 +V 1 3 )x 3 /3! 1 1 V 1 (y)x

23. Model Feynman Graphs These graphs are essentially like the Feynman Diagrams of a zero-dimensional (no space-time dependence, no integration) Model Field Theory. (In the case of a single-mode free boson gas, associated with H = a + a, one can give an alternative more transparent graphical representation). Each graph with m components corresponds to a propagator H = a +m a m We now show how to find the V n associated with general partition functions, thus getting a general graphical expansion for the Partition Function Integrand.

24. Hopf Algebras - the basics Algebra A x,y  A x+y  A xy  A k K kx  A Bialgebra also has Co-product  : A  A  A Co-unit  : A  K Hopf Algebra also has Antipode S : A  A

25. Generic Example Algebra A generated by symbols x,y xxyxyy  A xxy+2 yx  A Unit e e x=x=x e Coproduct  On uniteeXe On GeneratorxxXe + eXx ABA) B algebra map  Antipode S On unitS(e)=e On GeneratorS(x)=-x S(AB)= S(B) S(A) (anti-algebra map) Counit   (e)=1  (x)=0  (AB)=  (A)  (B)

26. Hopf Algebra of Model Diagrams Generators are connected diagrams... Addition + =

27. Hopf Algebra of Model Diagrams (continued) Multiplication = Algebra Identity e (=  ).

28. Hopf Algebra of Model Diagrams (continued) Coproduct  (Examples) on UnitX nGenerator  X = X  =  where A = and B =

29. Hopf Algebra of Model Diagrams (continued) Counit  (Examples) on Unit   otherwise    =  where A = and B =  = 1

30. Hopf Algebra of Model Diagrams (continued) Antipode S (Examples) UnitS Generator S =- S = SSS where A = and B =

31. Relation to other Models Hopf Algebra Morphisms MQSym LDiagFQSym Diag Cocommutative CMM FoissyDecorated Trees Connes-Kreimer

32. Summary We have shown a simple model based on the evaluation of the Partition Function, which leads to a combinatorial description, based on Bell numbers; a graphical description and a related Hopf Algebra. This is a very simple relative of the QFT model, but displays some of the latter’s features.