# Download Communications In Mathematical Physics - Volume 291 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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36) does not give any information about the convergence of the (1) one-particle marginal N ,t , associated with the evolution of the coherent initial state, to the orthogonal projection |ϕt ϕt |. The definition of the marginal density (1) N ,t involves unbounded creation and annihilation operators. 36). 36) requires control of the growth of the expectation of powers of the number of particle operator N only with respect to the limiting dynamics. 3), on the other hand, we need to control the growth of the expectation of N with respect to the N -dependent fluctuation dynamics U N (t; s).

For f ∈ L 2 (R3 ), we define the Weyl-operator W ( f ) = exp a ∗ ( f ) − a( f ) = exp dx ( f (x)ax∗ − f (x)ax ) . 9) Then the coherent state ψ( f ) ∈ F with one-particle wave function f is defined by ψ( f ) = W ( f ) . Notice that ψ( f ) = W ( f ) = e− f (a ∗ ( f ))n n! 10) n! where f ⊗n indicates the Fock-vector {0, . . , 0, f ⊗n , 0, . . }. This follows from exp(a ∗ ( f ) − a( f )) = e− f 2 /2 exp(a ∗ ( f )) exp(−a( f )) which is a consequence of the fact that the commutator [a( f ), a ∗ ( f )] = f 2 commutes with a( f ) and a ∗ ( f ).

Xn ). n≥1 An N particle state with wave function ψ N is described on F by the sequence {ψ (n) }n≥0 where ψ (n) = 0 for all n = N and ψ (N ) = ψ N . The vector {1, 0, 0, . . } ∈ F is called the vacuum, and will be denoted by . On F, we define the number of particles operator N , by (N ψ)(n) = nψ (n) . Eigenvectors of N are vectors of the form {0, . . , 0, ψ (m) , 0, . . } with a fixed number of particles. For f ∈ L 2 (R3 ) we also define the creation operator a ∗ ( f ) and the annihilation operator a( f ) on F by a ∗ ( f )ψ (n) 1 (x1 , .