Generalized normal ordering

GENERALIZED NORMAL ORDERING, IRREDUCIBLE BRILLOUIN CONDITIONS, AND CONTRACTED SCHRODINGER EQUATIONS... 

Formulating conditions for the energy to be stationary with respect to variations of the wavefunction P in this generalized normal ordering, one is led to the irreducible Brillouin conditions and irreducible contracted Schrodinger equations, which are conditions on the one-particle density matrix and the fe-particle cumulants k, and which differ from their traditional counterparts (even after reconstruction [4]) in being strictly separable (size consistent) and describable in terms of connected diagrams only. 

GENERALIZED NORMAL ORDERING and partial trace relations... 

We introduce the generalized normal ordering in various steps, starting with the traditional particle-hole formalism. 

The results of the last section, which are essentially a reformulation of the traditional particle-hole formalism for excitation operators, were first presented in 1984 [8]. At that time it was not realized that only two very small steps are necessary to generalize this formalism to arbitrary reference states. Only after Mukherjee approached the formulation of a generalized normal ordering on a rather different route [2], did it come to our attention how easy this generalization actually is, when one starts from the results of the last section. 

Then the excitation operators in generalized normal ordering are in a compact form ... 

For the formulation of the generalized Wick theorem corresponding to the generalized normal ordering, we need the matrix element rf, Eq. (65), of the one-hole density matrix and the cumulants kjc, Eqs. (39)-(47), of the fc-particle density matrices. 

Eor products of operators a in the generalized normal order we then get... 

Let us point out, without going into detail, that the generalized normal ordering can, of course, also be formulated in a spin-free form [30]. 

The condition for E to be stationary with respect to unitary one-particle transformations in generalized normal order is... 

The concept of generalized normal ordering is very powerful, but still waiting to become a standard tool in many-body physics. It is the natural generalization of... 

Generalized normal ordering is intimately linked to the cumulants Xk of the k-particle density matrices (for short, density cumulants). The contractions in the sense of the generalized Wick theorem involve the... 

If one formulates the conditions for stationarity of the energy expectation value in terms of generalized normal ordering, one is led to either the irreducible fc-particle Brillouin conditions IBCj or the irreducible A -particle contracted Schrodinger equations (IBC ), which are conditions to be satisfied by y = yj and the k. One gets a hierarchy of k-particle approximations that can be truncated at any desired order, without any need for a reconstruction, as is required for the reducible counterparts. 

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