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Proceedings of the Physical Society

Second Quantization of the General Many Fermion Operator

Edward R. Pike

Published Mar 1, 1963• 4 citations
Publisher

Abstract

The second-quantized equivalent of the general particle-conserving operator in the configuration space of Slater determinants is found by a direct calculation. The usual one- and two-particle operators are found as special cases and their equivalence with the corresponding configuration-space operators is thus shown without a lengthy comparison of either signs or matrix elements.

DOI
10.1088/0370-1328/81/3/304

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The second-quantized equivalent of the general particle-conserving operator in the configuration space of Slater determinants is found by a direct calculation. The usual one- and two-particle operators are found as special cases and their equivalence with the corresponding configuration-space operators is thus shown without a lengthy comparison of either signs or matrix elements.
In this article, we provide an alternative and general way to construct the result of the action of any particle-conserving bosonic or fermionic operator represented in second quantized form on a state vector, without resorting to the matrix representation of operators or even its elements. This approach is based on our proposal to compactly enumerate the configurations (i.e., determinants for fermions, permanents for bosons) that are the elements of the state vector. This extremely simplifies the calculation of the action of an operator on a state vector. The computations of statical properties and the evolution dynamics of a system become much more efficient, and applications to systems made of more particles become feasible. Explicit formulations are given for spin-polarized fermionic systems and spinless bosonic systems, as well as to general (two-component) fermionic systems, two-component bosonic systems, and mixtures thereof.
Second-quantized operators are derived algebraically from first-quantized counterparts, thus providing an alternative to all existing methods, most of which rely upon demonstrating the equivalence of matrix elements of the two operators. This derivation establishes an operator identity between a second-quantized operator and a first-quantized operator whose domain of definition is extended explicitly from an N-particle Hilbert space to Fock space. The existence of such an identity eliminates the need to address the issue of the representations used for the two operators, which often obscures the underlying simple relationship between them.
We demonstrated in the previous chapter (Chapter 1) that the evaluation of matrix elements between states represented by the Slater determinant can be evaluated unambiguously using Slater’s rule [1, 28, 40]. While the determinantal representation and the evaluation matrix elements between the states are quite straightforward, this procedure can eventually be ineffective and cumbersome with increasing size of the system. The occupation number formalism or second quantization technique introduced by Dirac in quantum field theory, in this context has been found to be quite formidable and elegant. This formalism not only allows us to represent the states and the operators in a compact and convenient form but also enables us to deal with systems containing variable numbers of particles without explicitly specifying the number of particles. It implicitly assumes the existence of an unspecified number of functions in a single-particle basis. These one-particle functions, which constitute the Hilbert space for N particles, are generally chosen to be spinorbital because of their orthogonality. In the case of fermions, the many-particle basis is the anti-symmetric product of N one-particle basis functions (Slater determinant). The vector space consisting of all possible anti-symmetric Hilbert spaces is referred to as the Fock space, in which the operators and functions are represented without specifying the number of particles.
In Section 14 we considered many nucleon states as the antisymmetrized products of single nucleon wave functions. Due to the antisymmetrization, the wave function is completely determined by the single nucleon states which are occupied. This suggests that a direct way to deal with many nucleon states is by using a representation in which occupied single nucleon states, rather than nucleon coordinates, are specified. To do this we have to define a fixed set of single nucleon states. The results ( 14.6 ), ( 14.7 ) indicate that we should provide matrix elements of relevant single nucleon operators, calculated in the basis of these single nucleon wave functions. The results ( 14.11 ), ( 14.12 ) and ( 14.13 ) show that we should also calculate matrix elements of two-body operators in the basis of single nucleon wave functions. These calculations should be performed in the way shown in preceding sections. They are carried out in configuration space by using wave functions of space coordinates as well as spin (and isospin) variables. Equipped with these matrix elements we can introduce the representation in terms of occupied states by using the formalism of second quantization. This formalism will now be briefly described without entering into serious discussions and rigorous proofs.
The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.