Operators In Fock Space Research Articles

Orthogonalization of fermionk-body operators and representability

The reduced k-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of k-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of k-body operators for all k.

Evolution of the squeezing-enhanced vacuum state in the amplitude dissipative channel

We study the evolution of the squeezing-enhanced vacuum state (SEVS) in the amplitude dissipative channel by using the two-mode entangled state in the Fock space and Kraus operator. The explicit formulation of the output state is also given. It is found that the output state does not exhibit sub-Poissonian behavior for the nonnegative value of the Mandel’s Q-parameters in a wide range of values of squeezing parameter and dissipation factor. It is interesting to see that second-order correlation function is independent of the dissipation factor. However, the photon-number distribution of the output quantum state shows remarkable oscillations with respect to the dissipation factor. The shape of Wigner function and the degree of squeezing show that the initial SEVS is dissipated by the amplitude dissipative channel.

4D higher spin black holes with nonlinear scalar fluctuations

We construct an infinite-dimensional space of solutions to Vasiliev’s equations in four dimensions that are asymptotic to AdS spacetime and superpose massless scalar particle modes over static higher spin black holes. Each solution is obtained by a large gauge transformation of an all-order perturbatively defined particular solution given in a simple gauge, in which the spacetime connection vanishes, the twistor space connection is holomorphic, and all local degrees of freedom are encoded into the residual twistor space dependence of the spacetime zero-forms. The latter are expanded over two dual spaces of Fock space operators, corresponding to scalar particle and static black hole modes, equipped with positive definite sesquilinear and bilinear forms, respectively. Switching on an AdS vacuum gauge function, the twistor space connection becomes analytic at generic spacetime points, which makes it possible to reach Vasiliev’s gauge, in which Fronsdal fields arise asymptotically, by another large transformation given here at first order. The particle and black hole modes are related by a twistor space Fourier transform, resulting in a black hole backreaction already at the second order of classical perturbation theory. We speculate on the existence of a fine-tuned branch of moduli space that is free from black hole modes and directly related to the quasi-local deformed Fronsdal theory. Finally, we comment on a possible interpretation of the higher spin black hole solutions as black-hole microstates.

The Vacuum Distributions of the Truncated Virasoro Fields are Products of Gamma Distributions

In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.

Dirac fields in flat FLRW cosmology: Uniqueness of the Fock quantization

We address the issue of the infinite ambiguity that affects the construction of a Fock quantization of a Dirac field propagating in a cosmological spacetime with flat compact sections. In particular, we discuss a physical criterion that restricts to a unique possibility (up to unitary equivalence) the infinite set of available vacua. We prove that this desired uniqueness is guaranteed, for any possible choice of spin structure on the spatial sections, if we impose two conditions. The first one is that the symmetries of the classical system must be implemented quantum mechanically, so that the vacuum is invariant under the symmetry transformations. The second and more important condition is that the constructed theory must have a quantum dynamics that is implementable as a (non-trivial) unitary operator in Fock space. Actually, this unitarity of the quantum dynamics leads us to identify as explicitly time dependent some very specific contributions of the Dirac field. In doing that, we essentially characterize the part of the dynamics governed by the Dirac equation that is unitarily implementable. The uniqueness of the Fock vacuum is attained then once a physically motivated convention for the concepts of particles and antiparticles is fixed.

Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

Squeezing of Dirac's inverse operator in Fock space and the squeezed creation operator's eigenket

By using the singular eigenvector (eigenket) of the creation operator a† we study how Dirac's inverse operators are squeezed and what is the normally ordered expansion of a†coshλ+asinhλ−1. We find the rule for converting annihilator's eigenket to creator's eigenket is: ez→e−∂∂z*δ(z*), where δ(z*) is the Dirac's Delta function in contour integration form. We also derive the squeezed creation operator's eigenket and conclude that a†coshλ+asinhλ−1|0〉 is an excited squeezed state.

Fermion pairing and the scalar boson of the 2D conformal anomaly

We analyze the phenomenon of fermion pairing into an effective boson associated with anomalies and the anomalous commutators of currents, bilinear in the fermion fields. In two spacetime dimensions the chiral bosonization of the Schwinger model is determined by the chiral current anomaly of massless Dirac fermions. A similar bosonized description applies to the 2D conformal trace anomaly of the fermion stress-energy tensor. For both the chiral and conformal anomalies, correlation functions involving anomalous currents, j 5 or T μν of massless fermions exhibit a massless boson 1/k 2 pole, and the associated spectral functions obey a UV finite sum rule, becoming δ-functions in the massless limit. In both cases the corresponding effective action of the anomaly is non-local, but may be expressed in a local form by the introduction of a new bosonic field, which becomes a bona fide propagating quantum field in its own right. In both cases this is expressed in Fock space by the anomalous Schwinger commutators of currents becoming the canonical commutation relations of the corresponding boson. The boson has a Fock space operator realization as a coherent superposition of massless fermion pairs, which saturates the intermediate state sums in quantum correlation functions of fermion currents. The Casimir energy of fermions on a finite spatial interval [0, L] can also be described as a coherent scalar condensation of pairs, and the one-loop correlation function of any number n of fermion stress-energy tensors 〈TT . . . T 〉 may be expressed as a combinatoric sum of n!/2 linear tree diagrams of the scalar boson.

Quantum techniques for studying equilibrium in reaction networks

Anderson, Craciun and Kurtz have proved that a stochastically modelled chemical reaction system with mass-action kinetics admits of a stationary distribution when the deterministic model of the same system with mass-action kinetics admits of an equilibrium solution obeying a certain ‘complex balance’ condition. Here we present a proof of their theorem using tools from the theory of second quantization: Fock space, annihilation and creation operators and coherent states. This is an example of ‘stochastic mechanics’, where we take techniques from quantum mechanics and replace amplitudes by probabilities. We explain how the systems studied here can be described using either of two equivalent formalisms: reaction networks, as in chemistry, and stochastic Petri nets, as in some other disciplines.

Diagonalization of the infinite q-boson system

We present a hierarchy of commuting operators in Fock space containing the q-boson Hamiltonian on Z and show that the operators in question are simultaneously diagonalized by Hall–Littlewood functions and have absolutely continuous spectrum. As an application, the n-particle scattering operator is computed explicitly.

A bijection between rooted trees and fermionic Fock states: grafting and growth operators in Fock space and fermionic operators for rooted trees

We show that fermionic Fock states in the occupation number representation can be indexed uniquely by rooted trees. Our main ingredients in this construction are the Matula numbers, the fundamental theorem of arithmetic, and a relabeling of fermionic quantum states by natural numbers. As a byproduct of the correspondence mentioned above we realize the grafting and the growth operators, comprising important constructions in the context of Hopf algebras, in the fermionic Fock space. Also, we show how to construct fermionic creation and annihilation operators in the context of rooted trees. New representations of the solutions of combinatorial Dyson–Schwinger equations and of the antipode in the Connes–Kreimer Hopf algebra of rooted trees related to the occupation number picture are presented.

Uncertainties of coherent states for a generalized supersymmetric annihilation operator

This study presents supersymmetric coherent states that are eigenstates of a general four-parameter family of annihilation operators. The elements of this family are defined as operators in Fock space that transform a subspace of a definite number of particles into a subspace with one particle removed. The emphasis is on classifying parameter space in various regions according to the uncertainty bounds of the corresponding coherent states. Specifically, the uncertainty in position-momentum is analyzed, with specific focus on characterizing regions of minimum uncertainty states, regions where the uncertainties are bounded from above, and where they grow unbound.

Description of the operator dual to the multiplication operator in fock space

Description of the operator dual to the multiplication operator in fock space

A Vanishing Theorem for Operators in Fock Space

We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space we show that rotation invariance implies the absence of terms which either create or annihilate only a single particle. We outline an application of this result in an operator theoretic renormalization analysis of Hamilton operators, which occur in non-relativistic qed.

Recursive formulation of the multiconfigurational time-dependent Hartree method for fermions, bosons and mixtures thereof in terms of one-body density operators

The multiconfigurational time-dependent Hartree method (MCTDH) [H.-D. Meyer, U. Manthe, L.S. Cederbaum, Chem. Phys. Lett. 165, 73 (1990); U. Manthe, H.-D. Meyer, L.S. Cederbaum, J. Chem. Phys. 97, 3199 (1992)] is celebrating nowadays entering its third decade of tackling numerically-exactly a broad range of correlated multi-dimensional non-equilibrium quantum dynamical systems. Taking in recent years particles’ statistics explicitly into account, within the MCTDH for fermions (MCTDHF) and for bosons (MCTDHB), has opened up further opportunities to treat larger systems of interacting identical particles, primarily in laser-atom and cold-atom physics. With the increase of experimental capabilities to simultaneously trap mixtures of two, three, and possibly even multiple kinds of interacting composite identical particles together, we set up the stage in the present work and specify the MCTDH method for such cases. Explicitly, the MCTDH method for systems with three kinds of identical particles interacting via all combinations of two- and three-body forces is presented, and the resulting equations-of-motion are briefly discussed. All four possible mixtures (Fermi–Fermi–Fermi, Bose–Fermi–Fermi, Bose–Bose–Fermi and Bose–Bose–Bose) are presented in a unified manner. Particular attention is paid to represent the coefficients’ part of the equations-of-motion in a compact recursive form in terms of one-body density operators only. The recursion utilizes the recently proposed Combinadic-based mapping for fermionic and bosonic operators in Fock space [A.I. Streltsov, O.E. Alon, L.S. Cederbaum, Phys. Rev. A 81, 022124 (2010)], successfully applied and implemented within MCTDHB. Our work sheds new light on the representation of the coefficients’ part in MCTDHF and MCTDHB without resorting to the matrix elements of the many-body Hamiltonian with respect to the time-dependent configurations. It suggests a recipe for efficient implementation of the schemes derived here for mixtures which is suitable for parallelization.

Matrices related to some Fock space operators

Matrices of operators with respect to frames are sometimes more natural and easier to compute than the ones related to bases. The present work investigates such opera- tors on the Segal-Bargmann space, known also as the Fock space. We consider in particular some properties of matrices related to Toeplitz and Hankel operators. The underlying frame is provided by normalised reproducing kernel functions at some lattice points.

Implementation problem for the canonical commutation relation in terms of quantum white noise derivatives

The implementation problem for the canonical commutation relation is reduced to a system of differential equations for Fock space operators containing new type of derivatives. We solve these differential equations systematically by means of quantum white noise calculus, and obtain the solution to the implementation problem.

General mapping for bosonic and fermionic operators in Fock space

In this article, we provide an alternative and general way to construct the result of the action of any particle-conserving bosonic or fermionic operator represented in second quantized form on a state vector, without resorting to the matrix representation of operators or even its elements. This approach is based on our proposal to compactly enumerate the configurations (i.e., determinants for fermions, permanents for bosons) that are the elements of the state vector. This extremely simplifies the calculation of the action of an operator on a state vector. The computations of statical properties and the evolution dynamics of a system become much more efficient, and applications to systems made of more particles become feasible. Explicit formulations are given for spin-polarized fermionic systems and spinless bosonic systems, as well as to general (two-component) fermionic systems, two-component bosonic systems, and mixtures thereof.

Massive neutrinos, Lorentz invariance dominated standard model and the phenomenological approach to neutrino oscillations

For the electroweak interactions, the massive neutrino perturbative kinematical procedure is developed in the massive neutrino Fock space. The perturbation expansion parameter is the ratio of neutrino mass to its energy. This procedure, within the Pontecorvo–Maki–Nakagawa–Sakata (PMNS)-modified electroweak Lagrangian, calculates the cross-sections with the new neutrino energy projection operators in the massive neutrino Fock space, resulting in the dominant Lorentz invariant standard model massless flavor neutrino cross-sections. As a consequence of the kinematical relations between the massive and massless neutrinos, some of the neutrino oscillation cross-sections are Lorentz invariance violating. But all these oscillating cross-sections, some of which violate the flavor conservation, being proportional to the squares of neutrino masses are practically unobservable in the laboratory. However, these neutrino oscillating cross-sections are consistent with the original Pontecorvo neutrino oscillating transition probability expression at short time (baseline), as presented by Dvornikov. From these comparisons, by mimicking the time dependence of the original Pontecorvo neutrino oscillating transition probability, one can formulate the dimensionless neutrino intensity-probability I, by phenomenologically extrapolating the time t, or, equivalently the baseline distance L away from the collision point for the oscillating differential cross-section. For the incoming neutrino of 10 MeV in energy and neutrino masses from Fritzsch analysis with the neutrino mixing matrix of Harrison, Perkins and Scott, the baseline distances at the first two maxima of the neutrino intensity are L≃281 and 9279 km. The intensity I at the first maximum conserves the flavor, while at the second maximum, the intensities violate the flavor, respectively, in the final and initial state. At the end some details are given as to how one should be able to verify experimentally these neutrino oscillations away from the collision point. In the material presented here one reinforces the notion that the massless flavor neutrino can be considered as the superposition of three massive neutrinos.

Quantum stochastic integral representations of Fock space operators

An (unbounded) operator Ξ on Boson Fock space over L 2(R +) is called regular if it is an admissible white noise operator such that the conditional expectations give rise to a regular quantum martingale. We prove that an admissible white noise operator is regular if and only if it admits a quantum stochastic integral representation.