Fermionic creation operator

The reference state of A-electron theory becomes a reference vacuum state 4>) in the field theory. A complete orthonormal set of spin-indexed orbital functions fip(x) is defined by eigenfunctions of a one-electron Hamiltonian Ti, with eigenvalues ep. The reference vacuum state corresponds to the ground state of a noninteracting A-electron system determined by this Hamiltonian. N occupied orbital functions (el < pi) are characterized by fermion creation operators a such that a] ) =0. Here pt is the chemical potential or Fermi level. A complementary orthogonal set of unoccupied orbital functions are characterized by destruction operators aa such that aa < >) = 0 for ea > p and a > N. A fermion quantum field is defined in this orbital basis by... 

Let us now define a creation operator, c , such that, if nk = 0, its operation will yield a wave function with nk = 1 and, if nk = 1 already, its operation will give zero, since a fermion can not be created in a state that is occupied. Hence, we have... 

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]... 

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)... 

The fermion creation and destruction operators are defined such that apa +a ap = Spq. In analogy to relativistic theory, and more appropriate to the linear response theory to be considered here, the elementary fermion operators ap can be treated as algebraic objects fixed in time, while the orbital functions are solutions of a time-dependent Schrodinger equation... 

Away from the charged stripes, creation operators of approximate fermion basis states of coupled holon-spinon and excession-spinon pairs are constructed [2], Together with the small-U states they form, within the auxiliary space, a basis to quasi-electron (QE) states, created by (/ (k). The bare QE energies ef (k) form quasi-continuous ranges of bands within the BZ. 

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 ... 

Second quantization for composite particles, in the context of quantum chemistry, was elaborated by several authors, e.g. by Girardeau [ 101 ], Kvasnicka [102], Fukutome [103, 92], and Valdemoro [104, 105], to name a few. The present author used creation operators composed of two fermion operators to describe geminals in orthogonal [106, 107, 108, 98] and non-orthogonal [109, 110, 111, 112, 113] basis sets. Second quantization for geminals will be reviewed in Sect. 3.2. 

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... 

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... 

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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