Fermion-like operator

Eq. (58) represents the starting point for all approximate propagator methods. Even though in the derivation we only discussed the linear response functions or polarization propagators, a similar equation holds for the electron propagator. The equation for this propagator has the same form but there are differences in the choice of h and in the definition of the binary product (Eq. (52)), which for non-number-conserving, fermion-like operators should be... 

Such a treatment can, with advantage, be expressed in terms of the superoperators introduced in Eq. (4.19) and in terms of a basis of field operators. The basis of fermion-like operators Xj = a, aj[aja, ,a aja, a ap, - is chosen, such that the electron field operators correspond to the SCF spin orbitals. The field operator space supports a scalar product (XjlXj) = ([A , X,]+) = Tr /9[Xl,Xj]+, where p is the density operator defined in Eq. (4.33). The superoperator identity and the superoperator hamiltonian operate on this space of fermion-like field operators and, in particular, Xi HXj) = [x/, [H,Xj - J. ) = Tt p[xI[H,X ] U. ... 

Thus the current operator indeed transforms like a vector. This must be the case in order that the equation Qdu(x) = ju(x) transform properly, assuming the transformation property (11-267) for Au(x). We now inquire briefly into tike question of the uniqueness of the U(ia) operator, in particular into the question of the phase associated with the fermion field operator. Note that the phase of the photon field operator is uniquely determined (Eq. (11-267)) by the fact that An is a hermitian field which commutes with the total charge operator Q. The negaton-positon field operator on the other hand does not commute with the total charge operator, in fact... 

Restricting f to only some simple fermion-like products of electron field operators will generate only certain types of diagrams that can then be summed to all orders. For instance, self-energy diagrams in third order of the ring and ladder types, which can easily be generalized in any order. It is notable that between consecutive interaction lines, there only occur one hole line and two particle lines or vice versa [see Fig. 9.1]. 

All expectation values of concern can be expressed in the form X a-r ), with X a fermion-like annihilation operator. Eq.(11.64) and the property (11.87) are applied to obtain... 

All electrons, protons and neutrons, the elementary constituents of atoms, are fermions and therefore intrinsically endowed with an amount h/2 of angular momentum, known as spin. Like mass and charge, the other properties of fermions, the nature of spin is poorly understood. In quantum theory spin is treated purely mathematically in terms of operators and spinors, without physical connotation. 

The purpose of these notes is to show how some strongly correlated electron models like the one-band Hubbard model with infinite electron repulsion on rectangular and triangular lattices can be described in terms of spinless fermions and the operators of cyclic spin permutations. We will consider in detail the... 

Having established that creation and annihilation operators are rank 1 covariant and contravariant tensors, respectively, with respect to the operator ( )L,S, we can define an rath-rank boson operator as consisting of a like number of fermion creation and annihilation operators. Then the normal product of an rath-rank boson operator is a natural definition for the irreducible tensor. 

Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141 134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented. 

---