Anti-commutator

The opposite of a creation operator is an annihilation operator a which removes orbital i from the wave function it is acting on. The a-a product of operators removes orbital j and creates orbital i, i.e. replaces the occupied orbital j with an unoccupied orbital i. The antisymmetry of the wave function is built into the operators as they obey the following anti-commutation relationships. 

Proof The magnetic moment of a particle is given by the expectation value of the operator Jd3x r x j(x). Hence since Ue is unitary, and anti commutes with j... 

These conditions define a matrix algebra which requires at least four anticommuting, traceless (hence even-dimensioned) matrices. The smallest even dimension, n = 2, can only accommodate three anti-commuting matrices, the Pauli matrices ... 

Each pair has an anti-commutator, cr, cr = OjOk + OkOj = 2 jk- Based on this the required representation of the Bi matrices follows immediately as... 

It may be shown that the two operators are the adjoint of each other. The annihilation and creation operators fulfill the following anti-commutator relations ... 

The Dirac operator Ho is interpreted as the operator corresponding to the energy of a free particle, which fits well to its role as the generator of the time evolution. As a consequence of the anti-commutation relations (5), the square of the Dirac Hamiltonian is a diagonal matrix. It is simply given by... 

As the matrix product is an elementary example of a noncommutative product, it is very natural to look for n x n matrices in order to satisy the anticommutation relations (5). Here we show that matrices a and 0 obeying the anti-commutation relations have to be at least four-dimensional. 

Now the operators (a ), (x) may be called the operators of the quantized electron-positron field. These operators are defined in the Fock space and act on the state vector ). The creation and annihilation operators satisfy the anti-commutation relations ... 

En effet, si A et B sont deux objets de (k, A B est encore une algebra anti-commutative graduee ([9] chap. Ill 4 n° 9 Proposition 14) de plus, la compatibilite de 1 isomorphisms de commutativite i et du produit tensoriel resultant de la relation... 

If the operators A and B anti-commute (commute) the extended state I A, B) is anti-symmetric (symmetric) under permutation of A and B ... 

The square bracket [.,. ] denotes the usual anti-commutator (commutator). Again, the upper (lower) sign applies in the fermionic (bosonic)... 

This expression simplifies further in the case when the anti-commutators (commutators) of any pair of the operators A, B, C, and D simply are complex numbers ... 

Example Let ot (or) denote the creation (destruction) operator for a single fermion and let the canonical anti-commutation relations be fulfilled ([at,Oj]+ = 5rs, etc.). We then find that the extended states aJ,aj) are normalised and orthogonal with respect to the //-product ... 

Normal ordering of the operators is introduced so that annihilation operators are placed to the right of creation operators allowing all anti-commutators to vanish in the process. A normally-ordered product of operators is denoted, for example, by , where the normal ordering is indicated by colons. The... 

Cautious readers may feel uneasy about the apparently surreptitious disappearance of the negative-energy states from this theory. The anti-commutation relations between the creation and annihilation operators of electrons and positrons are sufficient to derive how the the negative-energy states enter in electron correlation problems. Labzowsky [15] and Sapirstein [16] have derived... 

To write the Hamiltonian in a compact from we have used a, so-called, second quantization notation,dj and dj, respectively, create and annihilate an electron in orbital i and the requirements of the Pauli exclusion principle are satisfied by imposing the (anti)commutation relation... 

Back to commutators and anti-commutators, there is useful quoting their main properties... 

He was so self-assured, that he said eq. (3.48) has to be satisfied at any price by finding suitable unknowns ai (independent of coordinates and time). The a s have to satisfy the following relations (anti-commutation relations)... 

Indeed, using the anti-commutation relations one recovers the Klein-Gordon equation ... 

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. 

The vector or cross product determines a so-called axial vector with orthogonal orientation from two spatial vectors. The vector product is anti-commutative ... 

Hint You may want to make use of the commutator and anti-commutator relations of the Pauli spin matrices... 

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