By C.C. Wang, C. Truesdell

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Unitary Representation of the Time-Evolution If the algebra changes globally as time passes, then a representation may change at any moment into an inequivalent representation, and it is not 42 1 Systems with Many Particles possible to represent the time-evolution with a group of unitary transformations within the representation. Yet if the representation is based on a time-invariant state, then the other vectors of the representation differ only locally, and thus do not change in time, from the global point of view.

Lieb's Theorem. Let a and b be non-negative, a, b, C E PJJ(Jf'), and o ~ oc ~ 1. Then the functions a --+ Tr exp(c + In a) and (a, b) --+ Tr a~cbl-~c* are concave. Proof 1. By the spectral theorem and lensen's inequality, for any unit vector Ii), (ilk(a)li) ~ k«ilali», and therefore (ilk(a)li) ~ Lik«ilali». 1 The Ordering of the States Equality holds if the Ii) are eigenvectors of a. It suffices to take the supremum over finite sets {I i)}. 2. )b.

5) for finite systems does not hold any more. Thus it would be desirable to find a point of view that organizes them somehow. 4), as an algebra with a trivial center, Z = {Co( ·1}. On a finite-dimensional space it amounts to a direct sum of equivalent irreducible representations. The first step in any decomposition is to collect the equivalent irreducible representations together in factors and then write the whole representation as a sum of various factors. In the finite-dimensional case this appears as shown in Figure 2.