## Positive representations of general commutation relations allowing wick ordering (1995)

Venue: | FUNCT ANAL |

Citations: | 30 - 7 self |

### BibTeX

@ARTICLE{Jørgensen95positiverepresentations,

author = {P. E. T. Jørgensen and L. M. Schmitt and R. F. Werner},

title = {Positive representations of general commutation relations allowing wick ordering},

journal = {FUNCT ANAL},

year = {1995},

pages = {33--99}

}

### OpenURL

### Abstract

We consider the problem of representing in Hilbert space commutation relations of the form aia ∗ j = δij1 + ∑ kℓ T kℓ ij a ∗ ℓ ak, where the T kℓ ij are essentially arbitrary scalar coefficients. Examples comprise the q-canonical commutation relations introduced by Greenberg, Bozejko, and Speicher, and the twisted canonical (anti-)commutation relations studied by Pusz and Woronowicz, as well as the quantum group SνU(2). Using these relations, any polynomial in the generators ai and their adjoints can uniquely be written in “Wick ordered form ” in which all starred generators are to the left of all unstarred ones. In this general framework we define the Fock representation, as well as coherent representations. We develop criteria for the natural scalar product in the associated representation spaces to be positive definite, and for the relations to have representations by bounded operators in a Hilbert space. We characterize the relations between the generators ai (not involving a ∗ i) which are compatible with the basic relations. The relations may also be interpreted as defining a non-commutative differential calculus. For generic coefficients T kℓ ij, however, all differential forms of degree 2 and higher vanish. We exhibit conditions for this not to be the case, and relate them to the ideal structure of the Wick algebra, and conditions of positivity. We show that the differential calculus is compatible with the involution iff the coefficients T define a representation of the braid group. This condition is also shown to imply improved bounds for the positivity of the Fock representation. Finally, we study the KMS states of the group of gauge transformations defined by aj ↦ → exp(it)aj.

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Citation Context ...ntations of the twisted canonical anti-commutation relations not assuming relations between the ai. The “untwisted” case is the theory of Clifford algebras, and was treated from this point of view in =-=[JW]-=-. Finally, we briefly consider the KMS states associated with the one parameter automorphism group α acting on the generators as αt(aj) = e −it aj. In Fock space this automorphism is generated by the ... |

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Citation Context ...en we explicitly define γ α ∗n Ω = ˆγ ν n α ∗n Ω . One readily verifies that this is the coherent representation, and, comparing with the known irreducible representations of the C*-algebra of SνU(2) =-=[Mei]-=- we find the result. □ Example 1.2.5: Braid relations. Examples 1.2.1 and 1.2.3 have a common feature, namely the validity of the identity ∑ ∑ = efh T fc be T he ak T mf hℓ deh T eb ad T mc eh T ℓh dk... |

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Citation Context ...t the uniform bound ‖π(i)‖ ≤ βi holds in every bounded representation. If only the first condition holds, one can still find a universal representation in the class of “inverse limits” of C*-algebras =-=[Phi]-=-, or LMC*-algebras [Smü]. Since in all the examples we study below either both conditions hold or none of them, we will not elaborate on this structure. 1.2. Examples. In this section we describe the ... |

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1 |
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Citation Context ...nterplay between the structure of Wick ideals and the positivity of representations, we consider three structures defined by Woronowicz and Pusz, namely their twisted canonical commutation relations (=-=[PW]-=-, see (1.2.3)), the twisted canonical anti-commutation relations ([Pus], see (1.2.4)), and the quantum group SνU(2) ([Wo1], see (1.2.5)). In each case the original definition uses relations of the for... |

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Citation Context ...akes only values ±1, but the indices cannot be grouped into “bosonic” and “fermionic” (which would be equivalent to qij = (1 + qi + qj − qiqj)/2, qi = ±1) one speaks of anomalous statistics. Speicher =-=[Sp2]-=- has used the known Fock positivity in this case to show it also for general real qij with |qij| ≤ 1. Below in Theorem 2.6.2 we will extend this Fock positivity result to complex qij with |qij| ≤ 1. A... |

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