Heisenberg Representation Research Articles

Quantum Gravity in Heisenberg Representation and Self-Consistent Theory of Gravitons in Macroscopic Spacetime

The first mathematically consistent exact equations of quantum gravity in the Heisenberg representation and Hamilton gauge are obtained. It is shown that the path integral over the canonical variables in the Hamilton gauge is mathematically equivalent to the operator equations of quantum theory of gravity with canonical rules of quantization of the gravitational and ghost fields. In its operator formulation, the theory can be used to calculate the graviton S-matrix as well as to describe the quantum evolution of macroscopic system of gravitons in the non-stationary Universe or in the vicinity of relativistic objects. In the S-matrix case, the standard results are obtained. For problems of the second type, the original Heisenberg equations of quantum gravity are converted to a self-consistent system of equations for the metric of the macroscopic spacetime and Heisenberg operators of quantum fields. It is shown that conditions of the compatibility and internal consistency of this system of equations are performed without restrictions on the amplitude and wavelength of gravitons and ghosts. The status of ghost fields in the various formulations of quantum theory of gravity is discussed.

Features of the time evolution of localized quantum states in graphene

In the Heisenberg representation, the phenomena of the interference of quantum states of the conduction and valence bands of graphene, arising for localized electron wave packets are analyzed. The interference leads to temporal oscillations in the values of the average physical quantities.

Theoretical spectroscopy using molecular dynamics: theory and application to CH5+ and its isotopologues

Infrared spectroscopy is a powerful technique to unravel the structure and dynamics of molecular systems of ever increasing complexity. For isolated molecules in the gas phase theoretical approaches that directly rely on solving the Schrödinger equation, either approximately or quasi-exactly, are well established. A distinctly different approach to compute infrared spectra can be based on advanced molecular dynamics, itself being based on classical Newtonian dynamics, in conjunction with concurrent first principles electronic structure calculations. At variance with traditional methods, which are formulated in terms of the Schrödinger representation of quantum mechanics, the molecular dynamics approach stems from Heisenberg's representation and thus relies on computing thermal expectation values of time-correlation functions. Crucial in addition to generating the spectra themselves is their decomposition in terms of modes, which can be assigned to correlated atomic motion. This ab initio molecular dynamics route to compute infrared spectra, and its recent extension to quasiclassical techniques relying on approximate path integral dynamics, is covered in the review part of this Perspective. The usefulness of this unconventional approach, which can be generalized beyond infrared spectroscopy, is demonstrated in detail by applying the full machinery in computing and assigning the infrared spectra of protonated methane and its isotopologues. This particular molecule is often considered to be the most prominent member of the class of floppy or fluxional molecules. CH5(+) has been a longstanding challenge for theoretical infrared spectroscopy because it undergoes intricate large-amplitude motion, which is also reviewed. Molecular dynamics based infrared spectroscopy is general and can be applied to diverse systems such as molecular complexes in the gas phase, chromophores in biomolecular environments, and solute-solvent systems in the liquid phase.

Analysis of Characteristic of Free Particles: Relativistic Concept

A linear Hamiltonian in spatial derivative that satisfies Klein-Gordon equation was used starting from energy momentum relation for free particle was solved in agreement with the matrices and bearing in mind their suitability in terms of anticommutation relations in parallel with the definition of algebraic matrices whose hermicity is fulfilled by i += i and += and in turn linked up to explicit representation of the Dirac matrices. The wave packets of plane Dirac wave obtained as a superposition of plane wave yielding a localized wave function was normalized considering only positive energy of plane wave in which the expectation value with respect to the wave packet resulted from 2p/E>+=gr)> was found to agree with the Ehrenfest theorem in relation to Schrodinger theorem as it relates to true velocity of single particle. A comparison was made between the classical concept with Heisenberg representation from where the combined effect of the positive and negative energy components was considered.

EXTERNAL EDGE CONDITION AND GROUP COHOMOLOGIES ASSOCIATED WITH THE QUANTUM CLEBSCH-GORDAN CONDITION

In this article we determine the structure of a twisted first cohomology group of the first homology of a trivalent graph with a coefficient associated with the quantum Clebsch-Gordan condition. As an application we give a characterization of a combinatorial property, the external edge condition, which is defined by the author in the study of the Heisenberg representation on the TQFT-module.

Stochastic simulation of long-time nonadiabatic dynamics

Using a generalized transition probability, we show how nonadiabatic calculations, within the Wigner–Heisenberg representation of quantum mechanics, can be reliably extended to far longer times than those allowed by a primitive sampling scheme. Tackling the spin-boson model as a paradigmatic example, we provide substantial numerical evidence that the statistical error can be reduced for a wide range of parameter values.

Conditional density matrix: Entangled states in the Heisenberg representation

The application of the conditional density matrix in the Heisenberg representation to the processes caused by entanglement allows elimination of any misunderstanding of these phenomena. We consider quantum entanglement of two- and four-particle systems, quantum correlations different from entanglement, and their causes.

On the uniqueness of unitary representations of the non-commutative Heisenberg–Weyl algebra

In this paper, we discuss the uniqueness of the unitary representations of the noncommutative Heisenberg–Weyl algebra. We show that, apart from a critical line for the noncommutative position and momentum parameters, the Stone–von Neumann theorem still holds, which implies uniqueness of the unitary representation of the Heisenberg–Weyl algebra.

A commutative algebra on degenerate CP1 and Macdonald polynomials

We introduce a unital associative algebra A associated with degenerate CP1. We show that A is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby, we can regard A as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding–Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard–Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t1,t2) of Okounkov–Pandharipande [e-print arXiv:math-ph/0411210].

Asymptotic nonequilibrium steady-state operators

We present a method for the calculation of asymptotic operators for nonequilibrium steady-state quantum systems. The asymptotic steady-state operator is obtained by averaging the corresponding operator in Heisenberg representation over infinitely long time. Several examples are considered to demonstrate the utility of our method. The results obtained within our approach are compared to those obtained within the Schwinger-Keldysh nonequilibrium Green's functions.

Analytic dynamics of the Morse oscillator derived by semiclassical closures

The quantized Hamilton dynamics methodology [O. V. Prezhdo and Y. V. Pereverzev, J. Chem. Phys. 113, 6557 (2000)] is applied to the dynamics of the Morse potential using the SU(2) ladder operators. A number of closed analytic approximations are derived in the Heisenberg representation by performing semiclassical closures and using both exact and approximate correspondence between the ladder and position-momentum variables. In particular, analytic solutions are given for the exact classical dynamics of the Morse potential as well as a second-order semiclassical approximation to the quantum dynamics. The analytic approximations are illustrated with the O-H stretch of water and a Xe-Xe dimer. The results are extended further to coupled Morse oscillators representing a linear triatomic molecule. The reported analytic expressions can be used to accelerate classical molecular dynamics simulations of systems containing Morse interactions and to capture quantum-mechanical effects.

An infinite-temperature limit for a quantum scattering process

We study a quantum dynamical semigroup driven by a Lindblad generator with a deterministic Schrödinger part and a noisy Poisson-timed scattering part. The dynamics describes the evolution of a test particle in ℝn, n = 1, 2, 3, immersed in a gas, and the noisy scattering part is defined by the reduced effect of an individual interaction, where the interaction between the test particle and a single gas particle is via a repulsive point potential. In the limit that the mass ratio λ = m/M tends to zero and the collisions become more frequent as 1/λ, we show that our dynamics Φtλ◊ approaches a limiting dynamics Φt,λ with second order error. Working in the Heisenberg representation, for G ∈ B(L2(ℝn)) n = 1, 3 we bound the difference between Φt,λ(G) and Φt,λ◊(G) in operator norm proportional to λ2.

Continuous Variable Remote State Preparation with a Two-Mode Squeezed Vacuum State

Continuous variable remote state preparation is presented using a two-mode squeezed vacuum state based on the Heisenberg representation. We show that it is possible to remotely prepare this state in principle. Our scheme’s main advantages are: (i) It is practical, because a two-mode squeezed vacuum state is easy achieved in experiment, and our scheme only requires a beam splitter and a feedforward; (ii) Using our scheme decoherence and being cheating can be avoided.

Representations of the Twisted Heisenberg–Virasoro Algebra and the Full Toroidal Lie Algebras

An explicit construction of indecomposable modules for the twisted Heisenberg–Virasoro algebra and representations for the full toroidal Lie algebras are given.

Anti-Hermitian part of the contracted Schrödinger equation for the direct calculation of two-electron reduced density matrices

A recent advance in the theory of the contracted Schroedinger equation (CSE), in which only the anti-Hermitian part of the equation is solved, permits the direct determination of ground-state two-electron reduced density matrices (2-RDM's) that yield 95%-100% of the correlation energy of atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Here we discuss in detail the anti-Hermitian contracted Schroedinger equation (ACSE) and its comparison to the CSE with regard to cumulant reconstruction of RDM's, the role of Nakatsuji's theorem, and the structure of the wave function. The ACSE is also formulated in the Heisenberg representation and related to canonical diagonalization. The solution of the ACSE is illustrated with a variety of molecules including H{sub 2}O, CH{sub 2}, NH{sub 4}{sup +}, HF, and N{sub 2}, and potential energy and dipole-moment surfaces are computed for boron hydride in a polarized double-{zeta} basis set. The computed 2-RDM's very closely satisfy known N-representability conditions.

Quantum dynamical semigroups for finite and infinite Bose systems

A new class of quasifree quantum Markov semigroups on C*-algebras of canonical commutation relations is introduced and discussed. Two applications to decoherence in the Heisenberg representation are given. In the first one the dynamical semigroup which leads to the appearance of decoherence induced superselection rules corresponding to the boundary conditions of a quantum particle in a finite interval is considered. The second example analyzes the possibility of the transition from infinite systems to systems with a finite number of degrees of freedom.

Analyse anaplectique sur la droite réelle

The two irreducible summands of the metaplectic representation in L 2 ( R ) occur for one in the discrete series of the twofold cover of the group G = SL ( 2 , R ) , for the other in the continuation of this series. Starting instead from the complementary series of the universal cover of G, one can associate with every real number ν mod 2 a new ‘analysis’ on the real line. The space A ν that substitutes for L 2 ( R ) consists of functions extending as entire functions of one variable. Once the proper notion of integral on the real line has been introduced, one easily defines a concept of Fourier transformation and a new ‘ ν-anaplectic’ representation: the latter combines in the usual way with the Heisenberg representation, and both representations are pseudo-unitary with respect to some ν-dependent scalar product. The spectrum of the harmonic oscillator in the space A ν is the set ν + 1 2 + Z . To cite this article: A. Unterberger, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Second-order quantized Hamilton dynamics coupled to classical heat bath

Starting with a quantum Langevin equation describing in the Heisenberg representation a quantum system coupled to a quantum bath, the Markov approximation and, further, the closure approximation are applied to derive a semiclassical Langevin equation for the second-order quantized Hamilton dynamics (QHD) coupled to a classical bath. The expectation values of the system operators are decomposed into products of the first and second moments of the position and momentum operators that incorporate zero-point energy and moderate tunneling effects. The random force and friction as well as the system-bath coupling are decomposed to the lowest classical level. The resulting Langevin equation describing QHD-2 coupled to classical bath is analyzed and applied to free particle, harmonic oscillator, and the Morse potential representing the OH stretch of the SPC-flexible water model.

Dynamic Analysis for Bose–Einstein Condensation in Quantum Cavity

For the two-level atoms system interacting withsingle-mode active field in a quantum cavity, the dynamics of theBose–Einstein Condensation (BEC) is analyzed using an ordinary methodsuggested by authors to solve the system of Schrödinger representationin the Heisenberg representation. The wave function of the atoms is given.The stability factor determining the BEC and the selection rules of thequantum transition are solved.

Quantum theory of Manakov solitons

A fully quantum mechanical model of two-component Manakov solitons is developed in both the Heisenberg and Schrodinger representations, followed by an analytical, linearized quantum theory of Manakov solitons in the Heisenberg picture. This theory is used to analyze the vacuum-induced fluctuations of Manakov soliton propagation and collision. The vacuum fluctuations induce phase diffusion and dispersion in Manakov soliton propagation. Calculations of the position, polarization angle, and polarization state fluctuations show an increase in collision-induced noise with a decrease in the relative velocity between the two solitons, as expected because of an increase in the interaction length. Fluctuations in both the polarization angle and state are shown to be independent of propagation distance, opening up possibilities for communications, switching, and logic, exploiting these properties of Manakov solitons. Calculations of the phase noise reveal, surprisingly, that the collision-induced fluctuations can be reduced slightly below the level of fluctuations in the absence of collision, due to cross-correlation effects between the collision-induced phase and amplitude fluctuations of the soliton. The squeezing effect of Manakov solitons is also studied and proven, unexpectedly, to have the same theoretical optimum as scalar solitons.