Fock space operator

Although the theory has been formulated in terms of excitation operators only, the extension to arbitrary Fock-space operators is straightforward. 

Other studies of Fock space h formulations by Kutzelnigg and Koch consider two different definitions of the diagonal part of a Fock space operator. Fock space transformations yield different h s from those produced by the analogous Hilbert space transformations with their second definition [54], but apparently not with their first one [53, 119]. Their numerous VF transformation variants yield h s which can be classified using the same method utilized above with the four transformations that Kutzelnigg et al. find to yield connected h s. Table IV presents this classification. 

The colons stand for normal order of the Fock space operators, and... 

The abstract formalism introduced in this chapter builds the fundament of the theory of extended two-particle Green s functions. Our approach is very general in order to allow for a unified treatment of the different species of extended Green s functions discussed in the main part of this paper. Since the discussed propagators can be applied to a wide variety of physical situations, the emphasis of this chapter lies on the unifying mathematical structure. The formalism is developed simultaneously for (projectile) particles of fermionic and bosonic character. We will define the general extended states which serve to define the primary or model space of the extended Green s functions. We also define the /.j-product under which the previously defined extended states fulfil peculiar orthonormality conditions. Finally we introduce a canonical extension of common Fock-space operators and super-operators to the space of the extended states. 

Note that there is no canonic way to define the application of a general Fock-space operator to an element of the space Y. We will see later how to construct so-called extended operators that operate in the space Y from Fock-space operators. 

The specification super-operator is common in quantum chemical emd physical literature for linear mappings of Fock-space operators. It is very helpful to transfer this concept to the extended states A, B) and define the application of super-operators by the action on the operators A and B. We will see later how this definition helps for a compeict notation of iterated equations of motion and perturbation expansions. In certain cases, however, the action of a super-operator is fully equivalent to the action of an operator in the Hilbert space Y. The alternative concept of Y-space operators allows to introduce approximations by finite basis set representations of operators in a well-defined and lucid way. 

Definition Given a Fock-space operator U, we define U by the Unear mapping of an operator A onto the commutator of U with A ... 

The symbols and are understood as operations that map a Fock-space operator onto the corresponding super-operator and Y-space operator,... 

The operations and are linccir in the sense that if the operator / is a linear combination of Fock-space operators V and W... 

So far we have used the picture of operators like H and p, acting on states like l rs) and Q/) in the Hilbert space Y. For developing perturbation theoretic expansions, however, it is useful to use the complementary concept of super-operators acting on Fock-space operators as defined in Sec. IIC. Using the super-operator H, the definition for the extended particle-hole Green s function of Eq. (23) can be written as... 

Definition Given a set of primary states V = ] A,B) A A,B B defined by sets of Fock-space operators A tind B, we define the space Y as the smallest closed linear space containing P as a subset V CV that is closed under action of the extended operator H, i. e. 

The expression to the right of the dot is the model state P f >Rei = f o >Rei, which implies that the expression to the left is the relativistically covariant wave operator (also a Fock-space operator)... 

It remains only to express in terms of the a s the Fock-space operator A equivalent to the Schrodinger operator A . We consider first the one-electron operator (e.g. a term h(i) in the Hamiltonian)... 

In later chapters we shall sometimes formulate general arguments using the algebraic properties of the Fock-space operators sometimes we shall employ the more traditional methods and in some chapters both approaches will be used side by side. 

What combination of Fock-space operators will produce, when acting on a closed-shell state vector o)> a resultant vector equivalent to the wavefunction used in Problem 8.1 Use the second-quantization form of the Hamiltonian (p. 82) and the anticommutation properties of the creation and annihilation operators (p. 81) to give an alternative derivation of the energy expression found in Problem 8.1. Hint Use operators a , etc. to destroy or create up-spin or down-spin electrons in orbital ipr- It is convenient to use indices i, j,... for orbitals in 0o and m, n,... for the virtual set used in the O.]... 

The connection with the Schrodinger formalism is now immediate, since we know (Section 3.10) that a Fock-space operator such as ajaj will have a Schrodinger equivalent E (i). where E (i) works on functions of Xi (which can be expanded in terms of spin-orbitals iprixi)) and turns any component c,0, into CsE fps CsM>r- Formally, E may be represented as the ket-bra product 0r)(0f or as an integral operator (cf. p. 177) with kernel 0r(- i)0 ( /). Similarly, for a spatial function expanded in terms of orbitals 0r(/ ) we may introduce the operator = I0r)(0j. the analogue of (10.4.9), such that... 

Fock Space Representation of Operators.—Let F be some operator that neither creates nor destroys particles, and is a known function in configuration space for N particles. In symbols such an operator must by definition have the following matrix elements in Fock space ... 

We now define the projection operator in Fock space as an obvious generalization of Eq. (8-164) ... 

For any linear operator 22 defined in Fock space, we can similarly prove, by following an argument like that leading to Eq. (8-189), that the trace in Fock space of WB is the grand-ensemble-average of 22 ... 

Flow, control of, 265 Flow function on network, 258 Flow, optimal, method for, 261 Fock amplitude for one-particle system, 511 Fock space, 454 amplitudes, 570 description of photons, 569 representation of operators in, 455 Schrodinger equation in, 459 vectors in, 454 Focus, 326 weak, 328... 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

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. 

Equation (4.1) is sometimes referred to as a state vector in Fock space and its use requires that the Hamiltonian be expressed in terms of operators that can act on such vectors. 

The definition of the pair part or, equivalently the no-pair part of Hmat is not unique. The precise meaning of no-pair implicitly depends on the choice of external potential, so that the operator Hm t depends implicitly on the external potential, whereas the sum Hmat Hm t + Hm t is independent of the choice of external potential. Since the no-pair part conserves the number of particles (electrons, positrons and photons) we can look for eigenstates of Hm j in the sector of Fock space with N fermions and no photons or positrons. Following Sucher [18,26,28], the resulting no-pair Hamiltonian in configuration space can be written as... 

The operators W, A, occurring above, should be taken in the second-quantization form, free of explicit dependence on particle number, and Tr means the trace in Fock space (see e.g. [10] for details). Problems of existence and functional differentiability of generalized functionals F [n] and r [n] are discussed in [28] the functional F [n] is denoted there as Fi,[n] or Ffrac[n] or FfraoM (depending on the scope of 3), similarly for F [n]. Note that DMs can be viewed as the coordinate representation of the density operators. 

A Hermitian operator p is a von Neumann density if it is nonnegative and has unit trace. In more concrete terms, if is the finite-dimensional Fock space for a quantum model where electrons are distributed over a finite number of states, then p is a von Neumann density if (i) v,pv) > 0 for all operators v on and (ii) l)g = 1. By the formula v,pv) we mean the trace scalar product of the operators v and pv, that is, v,pv) = traceg(u pu) since (p, l)g = tracegp = 1 we have used this scalar product to express the trace condition. More generally. 

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... 

The Fock-space Hamiltonian H is equivalent to the configuration-space Hamiltonian H insofar as both have the same matrix elements between n-electron Slater determinants. The main difference is that H has eigenstates of arbitrary particle number n it is, in a way, the direct sum of aU // . Another difference, of course, is that H is defined independently of a basis and hence does not depend on the dimension of the latter. One can also define a basis-independent Fock-space Hamiltonian H, in terms of field operators [11], but this is not very convenient for our purposes. 

In the method based on the unitary transformation, we start by writing the exact wavefunction th in terms of the reference function and a unitary transformation operator in Fock space ... 

W. Kutzelnigg, Quantum chemistry in Fock space. I. The universal wave and energy operator. J. Chem. Phys. 77, 3081 (1982). 

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