Direct sum of subspaces

The notation concerns are easily overcome by the following simple construct bearing the name of second quantization formalism.21 Let us consider the space of wave functions of all possible numbers of electrons and complement it by a wave function of no electrons and call the latter the vacuum state vac). This is obviously the direct sum of subspaces each corresponding to a specific number of electrons. It is called the Fock space. The Slater determinants eq. (1.137) entering the expansion eq. (1.138) of the exact wave function are uniquely characterized by subsets of spin-orbitals K = k,, k2,..., fc/v which are occupied (filled) in each given Slater determinant. The states in the list are the vectors in the carrier space of spin-orbitals (linear combinations of the functions of the (pk (x) = ma (r, s) basis. We can think about the linear combinations of all Slater determinants, may be of different numbers of electrons, as elements of the Fock space spanned by the basis states including the vacuum one. 

The real Hilbert space is always partitioned into a direct sum of subspaces, each representing a different energy eigenvalue of the spectrum of the hamiltonian operator ... 

The ON vectors in (1.1.2) constitute an orthonormal basis in the 2"-dimensional Fock space F M). This Fock space may be decomposed as a direct sum of subspaces F(M, N)... 

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See also in sourсe #XX -- [ ]

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