## Popa M.: Feynman Diagrams and Wick products associated with q-Fock space (2003)

Venue: | Proc. Natl. Acad. Sci. USA 100 |

Citations: | 11 - 1 self |

### BibTeX

@INPROCEEDINGS{Effros03popam.:,

author = {Edward G. Effros and Mihai Popa},

title = {Popa M.: Feynman Diagrams and Wick products associated with q-Fock space},

booktitle = {Proc. Natl. Acad. Sci. USA 100},

year = {2003},

pages = {8629--8633}

}

### OpenURL

### Abstract

Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute q-Wick products and normal products in terms of each other. 1.

### Citations

118 | Hopf algebras, renormalization and noncommutative geometry
- Connes, Kreimer
- 1998
(Show Context)
Citation Context ...ecial cases of this material had been considered by Michael Anshelevich in [1]. We will explore q-forms of the Hopf algebraic theory of Kreimer and Connes and Kreimer in a subsequent paper (see [13], =-=[9]-=-). Date: February 27, 2003. E. Effros was partially supported by the National Science Foundation. 12 EDWARD G. EFFROS AND MIHAI POPA 2. q-Fock spaces and Feynman diagrams We begin by recalling the Bo... |

65 |
An example of a generalized Brownian motion
- Bo˙zejko, Speicher
- 1991
(Show Context)
Citation Context ...ent that NC(S) ⊇ SNC(S) ⊇ GF(S). □12 EDWARD G. EFFROS AND MIHAI POPA If S = [n], we will simply write NC(n), etc. If q = 0, we have the commutation relation a− (f)a+(g) =< f,g > I. The convention of =-=[4]-=- is that qc = 0 for c ̸= 0, and q0 = 1. Thus we may drop terms in the 0-Wick theorem for which q is raised to a positive power: E(ξ1ξ2 · · · ξ2n) = ∑ v(γ) γ∈NC(2n) Turning to Wick products, we delete ... |

64 | q-Gaussian processes: noncommutative and classical aspects
- Bo˙zejko, Kümmerer, et al.
- 1997
(Show Context)
Citation Context ...on of the Fock space, which for q = 1, −1, and 0 coincides with the symmetric (Boson), antisymmetric (Fermion), and full (Voiculescu) Fock spaces (some of these ideas had been considered in [10]; see =-=[3]-=-). Bo˙zejko and Speicher’s q-versions of stochastic processes and second quantization have attracted the attention of a large number of researchers. In [3], Bo˙zejko, Kümmerer and Speicher introduced ... |

61 |
On the Analytical Forms Called Trees
- Cayley
(Show Context)
Citation Context ...r. 1. Introduction A recurrent theme in non-commutative analysis is that one may use graphs to efficiently index the terms in complicated sums. One of the first to recognize this principle was Cayley =-=[8]-=-, who introduced rooted trees in order to label differentials (see [7] for additional examples). Currently, the best-known example of graph-theoretic indexing may be found in perturbative quantum fiel... |

44 |
Gaussian Hilbert Spaces. Cambridge Tracts
- Janson
- 1997
(Show Context)
Citation Context ...rturbative quantum field theory. In this context one uses Feynman diagrams to index summands that arise when one evaluates the expectations of products of jointly Gaussian random variables (see [11], =-=[12]-=-). The random variables of quantum field theory correspond to certain selfadjoint operators on symmetric or antisymmetric Fock spaces (see [11],[14]). In 1991, Bo˙zeko and Speicher introduced a remark... |

33 | Runge–Kutta methods and renormalization
- Brouder
- 2000
(Show Context)
Citation Context ...at one may use graphs to efficiently index the terms in complicated sums. One of the first to recognize this principle was Cayley [8], who introduced rooted trees in order to label differentials (see =-=[7]-=- for additional examples). Currently, the best-known example of graph-theoretic indexing may be found in perturbative quantum field theory. In this context one uses Feynman diagrams to index summands ... |

30 | Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces - Bo˙zejko, Speicher - 1994 |

20 |
Some properties of crossings and partitions, Discrete Mathematics 175
- Biane
- 1997
(Show Context)
Citation Context ...n diagrams on S, and we let F(n) = F([n]). We note that more general partitions and their crossings are analyzed by using a succession of semicircles to link the elements of an equivalence class (see =-=[2]-=-). We call the elements in S the vertices of the diagram. We say that a pair (k,l) ∈ γ is a left crossing for (i,j) if k < i < l < j, and we define cl(i,j) to be the number of such left crossings. We ... |

18 |
Partition-dependent stochastic measures and q-deformed cumulants
- ANSHELEVICH
- 2001
(Show Context)
Citation Context ...eynman diagrams to express the Wick products in terms of the normal (operator) products, and vice versa. We note that some special cases of this material had been considered by Michael Anshelevich in =-=[1]-=-. We will explore q-forms of the Hopf algebraic theory of Kreimer and Connes and Kreimer in a subsequent paper (see [13], [9]). Date: February 27, 2003. E. Effros was partially supported by the Nation... |

3 |
On the Hopf algebra of perturbative quantum field theory
- Kreimer
- 1998
(Show Context)
Citation Context ...ome special cases of this material had been considered by Michael Anshelevich in [1]. We will explore q-forms of the Hopf algebraic theory of Kreimer and Connes and Kreimer in a subsequent paper (see =-=[13]-=-, [9]). Date: February 27, 2003. E. Effros was partially supported by the National Science Foundation. 12 EDWARD G. EFFROS AND MIHAI POPA 2. q-Fock spaces and Feynman diagrams We begin by recalling t... |

1 |
The P(φ2) Euclidean Field Theory, Princeton Series in Physics, Princeton Univeristy
- Simon
- 1974
(Show Context)
Citation Context ...cts of jointly Gaussian random variables (see [11], [12]). The random variables of quantum field theory correspond to certain selfadjoint operators on symmetric or antisymmetric Fock spaces (see [11],=-=[14]-=-). In 1991, Bo˙zeko and Speicher introduced a remarkable q-version of the Fock space, which for q = 1, −1, and 0 coincides with the symmetric (Boson), antisymmetric (Fermion), and full (Voiculescu) Fo... |