20th LQP Workshop

Foundations and constructive aspects of QFT

June 29-30, 2007


Leipzig University

Huzihiro Araki (RIMS, Kyoto University)

Dynamics and Potentials

For a system of spins and Fermions (satisfying graded commutation relations) on a lattice, a $C^*$-dynamics can be associated with a potential, which satisfies a natural convergence property and a very convenient standardness property. The existence and uniqueness of the potential with the required properties for any $C^*$-dynamics under consideration, the form of the convergence property and the standardness property are all new results, for which the only assumption for the $C^*$-dynamics is that any strictly localized operator has the time derivative, a condition minimally necessary for the description of a dynamics in terms of a potential.

The standardness property brings about the unique choice of the potential (out of multitudes of equivalent potentials) by distinguishing the genuine $n$-body potential from mingled $k$-body potentials with $k less n$. An energy estimate is shown as a simple and yet non-trivial example of effectiveness of the standardness property, with an aid of non-commutative conditional expectations - an important mathematical tool of this entire analysis.

The key point in the above analysis is a bijective correspondence of a linear space of *-derivations and that of standard derivations, where the *-derivations considered are the restriction of the generators of $C^*$-dynamics to strictly local operators. In the case of supersymmetry, super-derivations replaces *-derivations. By the same analysis as above, a standard (odd) potential can be uniquely associated with a super-derivation.

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Practical Information

Conference Site


Info: Rainer Verch
ITP, Universitaet Leipzig
Vor dem Hospitaltore 1
Phone: +49 341 97 32423