Operators In Fock Space Research Articles

Inverse operators in Fock space studied via a coherent-state approach.

The Fock representation of the inverse of harmonic-oscillator creation and annihilation operators [see, e.g., C. L. Mehta, Anil K. Roy, and G. M. Saxena, Phys. Rev. A 46, 1565 (1992)] are naturally derived via a coherent-state approach. The completeness relation of Fock space is reexpressed by the inverse operators. Applications of the inverse operators in studying the phase operators and in normally ordering some operators are given.

Covariance and uniqueness of the canonical fermion coherent state.

A unitary Fock-space operator that leaves the canonical quantization rule for fermions covariant is introduced. The defining equation of a canonical fermion coherent state retains the symmetry under this unitary transformation. This fact mandates that the coherent state must accompany a partner state, and that the form of a state in the pair is related to the partner state by the symmetry. A careful definition of the phase in the extended fermion Fock space is crucial for the uniqueness of the normalized fermion coherent state. Also the proper choice of the phase renders the canonical fermion coherent state equal to a displaced-vacuum state

Fock space formulation of Chern-Simons theories

A Fock space operator formulation of Chern-Simons-Witten theories is presented, and its connection with the usual holomorphic quantization is established. The conformal blocks are obtained as projections of suitable states on a Fock vacuum. We study in detail the abelian case and sketch its extension to non-abelian groups where the powerfulness of the formalism to compute the corresponding WZW conformal blocks on arbitrary Riemann surfaces is emphasized.

Infinite abelian subalgebra of W(sl( n))

A representation theoretical construction of the conservation laws of affine Toda type systems is described. The construction employs the completely degenerate representations of the extended conformal algebras W(sl( n)). The conserved charges are shown to generate an infinite-dimensional abelian subalgebra of W(sl( n)). Different characterizations of this subalgebra are obtained: As space of physical Fock space operators with dihedral symmetry, as constants of commuting flows of quantum KdV-type equations and as subalgebra of the sl( n) singlets in affine sl ( n) level-1 modules. The existence of the subalgebras is established for low-rank cases by means of an algorithmic Fock space procedure.

The General Form of non-Fock Coherent Boson States

Specifying their (normally ordered) characteristic functions we determine all states of the boson C *-Weyl algebra which satisfy Glauber's coherence condition and are not realizable as density operators in Fock space. The pure ones are shown to be just the eigenstates of the annihilation operators in their GNS-representations (in contrast to the Fock case) and are characterized in many equivalent manners. The central decomposition of an arbitrary coherent state has the macroscopic phase variable as parameter and is supported by the pure coherent states, which is in fact the only way for a maximal decomposition. The set of all coherent states with the same absolute factorizing function is proven to be a Bauer simplex. The appearence of a classical coherent field part is studied in detail in the GNS-representations and shown to correspond to an enlargement of the set of one boson states by just one additional mode.

Coupled-cluster method in Fock space. III. On similarity transformation of operators in Fock space.

Coupled-cluster method in Fock space. III. On similarity transformation of operators in Fock space.

Connected-diagram expansions of effective Hamiltonians in incomplete model spaces. I. Quasicomplete and isolated incomplete model spaces

In this and the following paper, we formulate a Fock space theory for incomplete model spaces (IMS) that applies both to coupled-cluster expansions and to perturbation theory. We stress in this paper that the concept of the ‘‘connected’’ nature of extensive quantities like an effective Hamiltonian Heff is more fundamental than the ‘‘linkedness’’ that is conventionally used in many-body perturbation theory. The ‘‘connectedness’’ of Heff follows when the wave operator W is multiplicatively separable into noninteracting subsystems. This is ensured by writing W as an exponential Fock space operator with the exponent connected. It is demonstrated in particular that the connectedness of the exponent in W requires that the normalization condition of W be separable as well. Unlike the situation in a complete model space, the definition of ‘‘diagonal’’ or ‘‘nondiagonal’’ operators depends generally on the particular m-valence IMS. There are, however, special categories of IMS, the ‘‘quasicomplete’’ and the ‘‘isolated’’ model spaces, for which these definitions are possible without reference to the particular IMS. The formal properties of these IMS are discussed. It is shown that for the quasicomplete model space, the intermediate normalization is not separable, while it is so for the isolated model space.

Quantum nonlinear Schrödinger equation and its classical counterpart

The quantum nonlinear Schrödinger equation (QNLS) has attracted much attention recently as a simplest exactly soluble nonlinear model of the quantum field theory in 1 + 1 space-time. There are two approaches in the literature to solution of QNLS. One is a quantization prescription for the solution of classical nonlinear Schrödinger equation by the inverse scattering method. It is called the quantum inverse method (QIM). The other is an elaboration of the Bethe Ansatz technique. It is called the method of intertwining operators (MIO). The two approaches produce formally different expansions for the QNLS field and for certain Fock space operators associated with it. In this work we give a comparative exposition of both methods. We then show that the QIM expansions and the MIO expansions define the same operators on Fock space.

Fock space perturbation theory

Diagonal and non-diagonal operators in Fock space are defined. With a universal Fock space wave operator W the Fock space hamiltonian H can be transformed to a diagonal operator L containing all relevant information about eigenvalues of H for arbitrary particle number in a simply coded form. W and L are constructed by perturbation theory, even in a spinfree form, and illustrated diagrammatically.

Method for construction of operators in Fock space

It is shown, by providing a general method for the construction that any Fock space linear operator defined on the dense linear manifold spanned by the particle number representation basis can be represented in terms of the annihilation and creation operators. The normal form of the representation is unique.

On positivity preservation for free quantum fields

Let A be a real Bose or Fermi one-particle operator with ∥ A ∥ ⩽ I. Using Kaplansky's density theorem, a simple proof is given of the fact that Γ( A), the operator in Fock space induced by A, is positivity preserving in the relevant L 2 -space.

Creation and annihilation operators in Fock space

A discussion is presented of creation and annihilation operators in Fock space, without reference to occupation-number space. This leeds to anticommutation relations for fermion operators and commutation relations for boson operators in a more transparent way.

On certain non-relativistic quantized fields

We consider a Euclidean invariant interaction Hamiltonian which is a polynomial in smeared Fermion field operators (the smearing function providing an ultraviolet cut-off). By considering Guenin's perturbation series for the time-development of the theory, we show that time-displacements define a one-parameter group of automorphisms of the field algebra att=0, which acts continuously in the time-parameter. Results are obtained for any dimension of space and for both relativistic and nonrelativistic forms for the free Hamiltonian. In special cases the total Hamiltonian is a positive self-adjoint operator in Fock space, thus defining a concrete non-relativistic quantized field with non-trivial particle production.

On the nonlinear properties of retarded products

We start from a set of functions t( p 1, … p n ) that satisfy the unitarity equations appropriate for time-ordered functions of n current operators, n = 4, 5, …. In terms of functions t, we define r α ( p 1, … p n ) of each order n which correspond to the generalized retarded functions of axiomatic local quantum field theory. Using the technique of generating functionals, the corresponding “products of operators” T and R are defined. The t( p 1, … p n ) and r α ( p 1, … p n ) are well defined as distributions in momentum space, possibly not tempered; and the time-ordered and retarded operators are also distributions in momentum space, and as operators in Fock space have reasonable domains of definition. We find that the commutator [ R α , R β ] between any two retarded operators can be expressed as sums and differences between retarded operators of order n + m, where n is the order of R α and m that of R β . The various r α -functions coincide in some region of momentum space, and any given r α coincides with the t-function in the region of momentum space appropriate to the function r α . The R α -operators trivially satisfy the Steinmann relations. Thus many of the “linear” properties of the retarded operators that follow from causality and Lorentz invariance also follow from unitarity, when no locality is assumed.