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Dynamics and potential


Speaker: Huzihiro Araki, RIMS, Kyoto University
Location: 1147 MSB
Start time: Tue, Oct 24 2006, 4:10PM

An infinite quantum lattice system is considered.

Traditional (physics) approach is as follows. The interaction of the system is described in terms of an operator (more precisely an element of a C*-algebra) for each finite subset I of lattice points, which represent the interaction energy among physical objects (such as spins and/or Fermions in our present consideration) on lattice points in the set I and is called the (many body) potential (for the set). Then the time derivative (at time 0) of any element of the C*-algebra is given as i (imaginary unit) times the sum of the commutator of the element with the many body potentials, summed over all finite subsets I of the lattice. In order to guarantee the convergence of the sum, the element for which the time derivative is so defined is limited to those with strictly local support (with respect to the local structure of the algebra relative to the lattice) and some convergence condition (chosen simply for the sake of the convergence purpose) is assumed for the potentials. By integrating this time derivative (if the existence and uniqueness of the integration holds), the time translation automorphism of the algebra is obtained and one has a C*-dynamical system.)

Our approach is in the opposite direction. Given any C*-dynamical system satisfying the sole condition that the time derivative of strictly local operators exist, we prove the existence and uniqueness of the associated potential, which describes its time derivative as above. Furthermore this potential satisfies a natural convergence condition and in addition very convenient standardness condition. The existence and uniqueness is for potentials satisfying these two conditions (in addition to usual simple conditions). We demonstrate the effectiveness of the standardness condition for the potential by showing a simple proof of the so-called energy estimate.