Operator fermion annihilation

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

Using Eq. (30), the definition of the antisymmetrized two-electron integral, and the anticommutation relation of fermion annihilation operators, Eq. (33) may be written... 

For fermion creation and fermion annihilation operators and 6, respectively, the relations... 

The Fermion annihilation operator r, which is the adjoint of the creation operator r, can be thought of as annihilating an electron in and is defined to yield zero when operating on the vacuum ket... 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

Normal product of free-field creation and annihilation operators, 606 Normal product operator, 545 operating on Fermion operators, 545 N-particle probability distribution function, 42... 

Particles whose creation and annihilation operators satisfy these relationships are called fermions. It is found that [119] these commutation relations lead to wave functions in space that are antisymmetric. 

Similarly, we define the hermitean conjugate operator, ck, as an annihilation operator, which produces a wave function with a fermion missing from the fcth state, if it was occupied, or zero, if not. Thus, in this case, we have... 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

Since our sets of boson creation and annihilation operators and fermion creation and annihilation operators commute we can write our unperturbed wavefuntion (po) as the product of the fermion state vector /o) and the boson state vector %o), i.e. 

Three distinct sets of linear mappings for the partial 3-positivity matrices in Eqs. (31)-(36) are important (i) the contraction mappings, which relate the lifted metric matrices to the 2-positive matrices in Eqs. (27)-(29) (ii) the linear interconversion mappings from rearranging creation and annihilation operators to interrelate the lifted metric matrices and (iii) antisymmetry (or symmetry) conditions, which enforce the permutation of the creation operators for fermions (or bosons). Note that the correct permutation of the annihilation operators is automatically enforced from the permutation of the creation operators in (iii) by the Hermiticity of the matrices. 

Then one redehnes the annUiilahon operator u, for an occupied spin orbital as the hole creation operator b, and the creation operator a for an occupied spin orbital as the hole annihilation operator bi. The fermion operators for the virtual spin orbitals remain unchanged. 

For a two-electron system in 2m-dimensional spin-space orbital, with and denoting the fermionic annihilation and creation operators of single-particle states and 0) representing the vacuum state, a pure two-electron state ) can be written [57]... 

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

A new possibility of supersymmetry arises when n = nv = n. In this case, using fermionic creation and annihilation operators it is possible to construct the generators of the symplectic group Usp (2n). Consequently a supergroup chain starting with decomposition into the orthosymplectic group U (6/2n) Osp (6/2n) (7)... 

In the superoperator approach, an abstract linear space is introduced [2], The elements of this space are fermion operators generally expressed as linear combinations of products of creation or annihilation operators,... 

We have utilized the fermion creation and annihilation operators denoted a a, and apa, respectively. These operators act on the electron in the pth orbital with the projected spin a. The set 0p(r)) represents the molecular orbitals and the last term, hauc, in Eq- (13-3) is the nuclear repulsion energy. We use the following definitions of the one-electron excitation operator... 

Where s (R, annihilation operators for an electron spin transfer integral is a matrix element connecting one-electron functions ... 

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. 

These relationships can also be expressed in second-quantized form [4-8] by introducing Fermion creation and annihilation operators, which obey the anticommutation relations... 

Here and in the rest of this section 0 shall explicitly represent the quantised fermionic field and also the normal order product AB is written explicitly, which implies that all annihilation operators are to the right of all creation opeartors. All boson operators within an normal ordered product are treated as if they would commute, while all fermion operators are treated as if they would anticommute (see appendix). After applying Wick s theorem a number of terms arise that describe different scattering... 

The corresponding Hamiltonian operator will still be given in terms of proper expansions over bilinear forms of (boson) creation and annihilation operators. (The more complex situations including half-spin particles can be addressed as well by using fermion operators [20].) The general rule is that one introduces a set of (n -I-1) boson operators b, and b (/, y = 1,. . . , n + 1) satisfying the commutation relations... 

Although Eqs. (1.2)-(1.5) contain all of the fundamental properties of the Fermion (electron) creation and annihilation operators, it may be useful to make a few additional remarks about how these operators are used in subsequent applications. In treating perturbative expansions of N-electron wavefunctions or when attempting to optimize the spin-orbitals appearing in such wavefunctions, it is often convenient to refer to Slater determinants that have been obtained from some reference determinant by replacing certain spin-orbitals by other spin orbitals. In terms of second-quantized operators, these spin-orbital replacements will be achieved by using the replacement operator as in Eq. (1.9). 

Having now seen how state vectors that are in one-to-one correspondence with /V-eIcctron Slater determinants can be represented in terms of Fermion creation and annihilation operators, it still remains for us to show how to express one- and two-electron operators in this language. The second-quantized version of any operator is obtained by simply demanding that the operator, when sandwiched between ket vectors of the form [ r vac>, yield exactly the same result as arises in using the first quantized operator between corresponding Slater determinant wavefunctions. For an arbitrary one-electron operator, which in first-quantized language is 5] = i /( i) second quantized equivalent is... 

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