Many-particle operator Hilbert space

A possible way to construct a bctsis for the Hilbert spaw e Y may also be the definition of operator sets A and B, such that the set of states A,B) A A,B B span the Hilbert space Y. In the context of traditional one- and two-particle propagators of many-body theory, such complete operator manifolds have been found [23] and used for deriving approximation schemes (see e. g. [24] and references cited therein). In the present context, however, such a construction seems difficult because of the complicated nature of the extended states A,B). It is certainly an interesting open question whether a convenient construction can be found. 

Quantum mechanics taJces many different guises. For instance, one can use a Hilbert space realination in terms of time-dependent wavefunctions ( 1, 2,- of an 7V-particle system, where = (r, C) is a compound configuration space and spin variable for a particle. Operators acting on this Hilbert space are obtained by making the common identifications p —ihV for the momentum and f— fioi position vector of a particle, which can be referred to as first quantization, producing quantum mechanical operators out of classical expressions. One can equally well use time-dependent field operators (so-called second quantization) tp, t) and their adjoints build Hilbert spaces (or rather... 

This is a very useful expression for considering it associated with the mono-density operators when the many-fermionic systems are treated, although similar procedure applies for mixed (sample) states as well. There is immediate to see that for Af formally independent partitions the Hilbert space corresponding to the iV-mono-particle densities on pure states, we individually have, see Eqs. (4.176), (4.181), (4.182) and (4.184),... 

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