Operators creation and annihilation

D. V. Averim and K. K. Likharev developed a theory for describing the behavior of small tunneling junctions based on electron interactions. They had started from previous work on Josephson junctions (Likharev and Zorin 1985, Ben-Jacob 1985, Averin and Likharev 1986b) and established the fundamental features of the single-charging phenomena. Their work is based on a quantization theory and handles the tunneling phenomenon as a perturbation, described by annihilation and creation operators of a Hamiltonian. 

The functions fk and are the counterparts of the so-called destruction (annihilation) and creation operators in the Heisenberg-Dirac picture. It is noted in anticipation that these operators occur as the solutions a,k(t) = lulkt of the Hamiltonian equation... 

Defining the time- and temperature-dependent annihilation and creation operators through the Bogoliubov transformation... 

The main idea of TFD is the following (Santana, 2004) for a given Hamiltonian which is written in terms of annihilation and creation operators, one applies a doubling procedure which implies extending the Fock space, formally written as Ht = H H. The physical variables are described by the non-tilde operators. In a second step, a Bogolyubov transformation is applied which introduces a rotation of the tilde and non-tilde variables and transforms the non-thermal variables into temperature-dependent form. This formalism can be applied to quite a large class of systems whose Hamiltonian operators can be represented in terms of annihilation and creation operators. 

As all quantities discussed in this publication are understood within the no-pair approximation, we will omit the index np in the following for brevity). In Eqs. (2.21, 2.22) bk and b are the annihilation and creation operators for positive energy KS states, which allow to write the electronic ground state as... 

By relaxing the condition that a von Neumann density be positive semidefinite, a graded family of approximations can be constructed. Since an operator can be represented as a polynomial in the annihilation and creation operators, it can be... 

Definition 15 A -body operator is a Hermitian operator that can be represented as a polynomial of degree 2 A in the annihilation and creation operators, and is of even degree in these operators. In addition, a A -body operator must be orthogonal to all k — l)-body operators, all k — 2)-body operators,. .., and all scalar operators, with respect to the trace scalar product. 

Let us make clear now the correspondence between our treatment here and Erdahl s 1978 treatment [4, Sec. 8]. Erdahl works in general Fock space and his operators conserve only the parity of the number of nuclei. He exhibits two families of operators that are polynomials in the annihilation and creation operators containing a three-body and a one-body term. Generic instances of these operators are denoted y and w. The coefficients are real, and Erdahl stresses that this is essential for his treatment. The one-body term is otherwise unrestricted, but the three-body term must satisfy conditions to guarantee that y+y or H +w does not contain a six-body term. For the first family the conditions amount to the three-body term being even under taking the adjoint, and for... 

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]... 

Here are the annihilation and creation operators of mode A, respectively, and is the coupling amplimde to mode A. 

Exercise. The annihilation and creation operators obey the commutation rules... 

Most formulations of MCSCF theory are based on the second quantization formalism. We therefore review briefly in this section the basic definitions of the annihilation and creation operators, and the expansion of quantum mechanical operators in products of them. 

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 ... 

These operator relations allow manipulating the operators independently of the function they are operating on. In general we will work with products of the operators. These can then often be simplified by the use of (3 5) or relations derived from them. Important operator products are those that preserve the number of particles. They always contain equally many annihilation and creation operators. A basic operator of this kind is the single excitation operator, which excites an electron from orbital i to orbital j ... 

The corresponding transformation of the spin-orbitals is obtained by multiplying (3 26) with an a or P spin function. When we make the transformation from one set of spin-orbitals to the other, the annihilation and creation operators will change. The following relations are easily established, by operating with the creation operator in the primed space on the vacuum state ... 

Alternatively it is possible to write the transformed annihilation and creation operators in the following form ... 

We have thus shown the relations (3 28) for the transformation of the annihilation and creation operators to a new spin-orbital basis. We can use these relations to express an arbitrary Slater determinant in the new basis in terms of the determinants in the original basis. In order to do so, we generate the Slater determinant by applying a sequence of creation operators on the vacuum state ... 

Express the spin dgeantbS in terms of annihilation and creation operators. Then show that the excitation operators Ey commute with these spin operators. 

For some one-determinant state ai,...,ajy) we can completely change over from particle description to hole description if, instead of electron creation and annihilation operators, we introduce, respectively, annihilation and creation operators for holes... 

Both A(3> and B(3> are longitudinally directed and are nonzero in the vacuum. Both A(3> and B(3> are phaseless, but propagate with the radiation [47-62] and with their (1) and (2) counterparts. The radiated vector potential A<3 does not give rise to a photon on the low-energy scale, because it has no phase with which to construct annihilation and creation operators. On the high-energy scale, there is a superheavy photon [44] present from electroweak theory with an SU(2)x SU(2) symmetry. The existence of such a superheavy photon has been inferred empirically [44], However, the radiated vector potential A<3) is not zero in 0(3) electrodynamics from first principles, which, as shown in this section, are supported empirically with precision. 

We then have a description of an infinite number of harmonic oscillators with every possible mode at every point in space. The electromagnetic field is quantized in a cavity with a volume V by defining annihilation and creation operators by redefining these raising and lower operators as... 

A very useful and convenient method that fulfils this postulate is the method of second quantization. That means, we describe all quantities (relevant for our system under consideration) in terms of so-called annihilation and creation operators a(b) and a+(b), respectively. 

Equation (29) is particularly useful for the quantum theory because we know how to represent the Coulomb gauge vector potential operator in terms of photon annihilation and creation operators since the Green s function g(x,x ) remains a c-numbcr. (29) gives an operator representation of the vector potential in an arbitrary gauge. 

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