Sine-Gordon form factors in finite volume

We compare form factors in sine-Gordon theory, obtained via the bootstrap, to finite volume matrix elements computed using the truncated conformal space approach. For breather form factors, this is essentially a straightforward application of a previously developed formalism that describes the volume dependence of operator matrix elements up to corrections exponentially decaying with the volume. In the case of solitons, it is necessary to generalize the formalism to include effects of non-diagonal scattering. In some cases it is also necessary to take into account some of the exponential corrections (so-called μ-terms) to get agreement with the numerical data. For almost all matrix elements the comparison is a success, with the puzzling exception of some breather matrix elements that contain disconnected pieces. We also give a short discussion of the implications of the observed behavior of μ-terms on the determination of operator matrix elements from finite volume data, as occurs e.g. in the context of lattice field theory.     

Exact form factors in integrable quantum field theories: the sine-Gordon model

We provide detailed arguments on how to derive properties of generalized form factors, originally proposed by one of the authors (M.K.) and Weisz twenty years ago, solely based on the assumption of ‘maximal analyticity” and the validity of the LSZ reduction formalism. These properties constitute consistency equations which allow the explicit evaluation of the n-particle form factors once the scattering matrix is known. The equations give rise to a matrix Riemann-Hilbert problem. Exploiting the “off-shell” Bethe ansatz we propose a general formula for form factors for an odd number of particles. For the sine-Gordon model alias the massive Thirring model we exemplify the general solution for several operators. In particular we calculate the three-particle form factor of the soliton field, carry out a consistency check against the Thirring model perturbation theory and thus confirm the general formalism.     

Exact form factors in (1 + 1)-dimensional field theoretic models with soliton behaviour

A system of equations is derived which must be satisfied by multiparticle matrix elements of any local operator in field theories with soliton behaviour. Form factors of various operators of interest are calculated exactly by means of the known exact S-matrices in the sine-Gordon, massive Thirring, non-linear σ^?, and Gross-Neveu models. The finite sine-Gordon wave function renormalization constant is determined exactly.     

The lattice quantum Sine-Gordon model

The integrable statistical physics model on the rectangular two-dimensional lattice which we call the L-model is constructed. This model generates the integrable quantum sine-Gordon model on the one-dimensional lattice in the same way as the ice model generates the XXZ model.     

Exact form factors in integrable quantum field theories: the sine-Gordon model (II)

A general model independent approach using the ‘off-shell Bethe Ansatz’ is presented to obtain an integral representation of generalized form factors. The general techniques are applied to the quantum sine-Gordon model alias the massive Thirring model. Exact expressions of all matrix elements are obtained for several local operators. In particular soliton form factors of charge-less operators as for example all higher currents are investigated. It turns out that the various local operators correspond to specific scalar functions called p-functions. The identification of the local operators is performed. In particular the exact results are checked with Feynman graph expansion and full agreement is found. Furthermore all eigenvalues of the infinitely many conserved charges are calculated and the results agree with what is expected from the classical case. Within the frame work of integrable quantum field theories a general model independent ‘crossing’ formula is derived. Furthermore the ‘bound state intertwiners’ are introduced and the bound state form factors are investigated. The general results are again applied to the sine-Gordon model. The integrations are performed and in particular for the lowest breathers a simple formula for generalized form factors is obtained.     

XIII. Exact S-matrices and form factors in 1 + 1 dimensional field theoretic models with soliton behaviour

A review is given of the derivation of exact S-matrices in field theoretic models with soliton behaviour, that means models obeying infinitely many conservation laws which imply the factorization of the S-matrix. Form factors of various operators are calculated exactly by means of Watson's theorem. The exact value of the finite Sine-Gordon wave function renormalization constant is determined.     

Exact form factors in integrable quantum field theories: the scaling Z(N)-Ising model

A general form factor formula for the scaling $Z(N)$-Ising model is constructed. Exact expressions of all matrix elements are obtained for several local operators. In addition, the commutation rules for order, disorder parameters and para-Fermi fields are derived. Because of the unusual statistics of the fields, the quantum field theory seems not to be related to any classical Lagrangian or field equation.     

Electro-magnetic nucleon form factors and their spectral functions in soliton models

It is demonstrated that in simple soliton models essential features of the electro-magnetic nucleon form factors observed over three orders of magnitude in momentum transfer t are naturally reproduced. The analysis shows that three basic ingredients are required: an extended object, partial coupling to vector mesons, and relativistic recoil corrections. We use for the extended object the standard skyrmion, one vector meson propagator for both isospin channels, and the relativistic boost to the Breit frame. Continuation to time-like t leads to quite stable results for the spectral functions in the regime from the 2- or 3-pion threshold to about two rho masses. Especially the onset of the continuous part of the spectral functions at threshold can be reliably determined and there are strong analogies to the results imposed on dispersion theoretic approaches by the unitarity constraint.     

Electro-magnetic nucleon form factors and their spectral functions in soliton models

It is demonstrated that in simple soliton models essential features of the electro-magnetic nucleon form factors observed over three orders of magnitude in momentum transfert are naturally reproduced. The analysis shows that three basic ingredients are required: an extended object, partial coupling to vector mesons, and relativistic recoil corrections. We use for the extended object the standard skyrmion, one vector meson propagator for both isospin channels, and the relativistic boost to the Breit frame. Continuation to time-liket leads to quite stable results for the spectral functions in the regime from the 2- or 3-pion threshold to about two rho masses. Especially the onset of the continuous part of the spectral functions at threshold can be reliably determined and there are strong analogies to the results imposed on dispersion theoretic approaches by the unitarity constraint.     

Quantum fluctuations in Sine-Gordon like magnetic chains: Corrections to soliton energies

The effect of quantum fluctuations on solitons in the easy-plane ferromagnetic chain is considered within the semiclassical approximation. In accordance with the low temperature ideal gas picture we treat the solitons as a Boltzmann gas and impose quantisation on the spin wave spectrum. We present a method which allows to calculate quantum corrections in a systematic perturbation expansion in 1/S, whereS is the spin length. We use this method to obtain the soliton energy to second order at zero temperature. Our results indicate that the semiclassical approach reasonably describes quantum effects on soliton properties.     

Exact soliton solutions of spinor Bose-Einstein condensates

We study matter-wave solitons in Bose-Einstein condensates of ultracold gaseous atoms with spin degrees of freedom and present a class of exact solutions based on the inverse scattering method. The one-soliton solutions are classified with respect to the spin states. We analyze collisional effects between solitons in the same or different spin state(s), which reveals a very interesting possibility: we can manipulate the spin dynamics by controlling the parameters of colliding solitons.     

Nucleon form factors in a projected chiral soliton model with dynamical confinement

We calculate nucleon form factors in the framework of a chiral chromodielectric model. The model state describing the nucleon is an angular momentum and isospin eigenstate with obtained by means of Peierls-Yoccoz projection from the hedgehog. We present results for the electromagnetic form factors and also for the axial form factors of the nucleon. There is a fairly good agreement with the data for small momentum transfers. For high momentum transfers (i.e. q² > 0.1 GeV² the agreement becomes poorer. As a general rule the calculated form factors fall too rapidly.     

Point-form quantum field theory and meson form factors

We comment on canonical quantization of relativistic field theories on a Lorentz-invariant surface of the form x ² = τ². By this choice of the quantization surface all components of the four-momentum operator become interaction dependent, whereas the generators of Lorentz transformations stay free of interactions – a feature characteristic for Dirac’s “point-form” of relativistic dynamics. In the sequel we demonstrate how field theoretical concepts may enter the framework of relativistic quantum mechanics. To this aim we employ a Poincaré-invariant approximation scheme, which allows to reduce a field theoretical many-body problem to a multichannel problem for a Bakamjian-Thomas-type mass operator. As an application of this multichannel formalism we will discuss the scattering of an electron by a (confined) quark-antiquark pair. It will be sketched how an electromagnetic meson form factor can be extracted from the one-photon exchange optical potential.     

Exact integral operator form of the Wigner distribution-function equation in many-body quantum transport theory

A formal derivation of a generalized equation of a Wigner distribution function including all many-body effects and all scattering mechanisms is given. The result is given in integral operator form suitable for application to the numerical modeling of quantum tunneling and quantum interference solid state devices. In the absence of scattering and many-body effects, the result reduces to the noninteracting-particle Wigner distribution function equation, often used to simulate resonant tunneling devices. The derivation uses a Weyl transform technique which can easily incorporate Bloch electrons. Weyl transforms of self-energies are derived. Various simplifications of a general quantum transport equation for semiconductor device analysis and self-consistent numerical simulation of a quantum distribution function in the phase-space/frequency-time domain are discussed. Recent attempts to include collisions in the Wigner distribution-function approach to the numerical simulation of tunneling devices are clearly shown to be non-self-consistent and inaccurate; more accurate numerical simulation is needed for a deeper understanding of the effects of collision and scattering.     

Exact breathing soliton solutions in combined time-dependent harmonic-lattice potential

We investigate the explicit novel localized nonlinear matter waves of the cubic-quintic nonlinear Schr6dinger equation with spafiotemporal modulation of the nonlinearities and the harmonic-lattice potential using a modified similarity trans- formation. We also find that when the modulus of the Jacobian elliptic function in the limit closes to 1, the shapes of the breathing solitons may exhibit some interesting features, i.e., one breathing soliton dividing into two in the ground state. The stability of the exact solutions is investigated numerically such that some stable breathing soliton solutions are found.     

Ion-acoustic dressed soliton in electron-ion quantum plasma

Ion acoustic dressed soliton in an unmagnetized two-species electron-ion quantum plasma is studied. Using reductive perturbation technique, a higher order inhomogeneous (KdV-type) differential equation is derived for the second order correction. The nonsecular solution is obtained by using renormalization procedure. A new technique is used to obtain the particular solution of the higher order inhomogeneous equation which is found to be simpler compared to the technique used by previous investigators.     

The quantum lifetime of the Davydov soliton

Considering an infinite spine in the Alpha-helix, stationary states should be eigenstates of a translational operator. These states should be nonlocalized in contradiction to a localized soliton. The difference in energy between localized and nonlocalized (Bloch) states is due to zero point motion and gives information about the quantum stability of the Davydov soliton. We develop a theory of statinary states and show that only for the limiting case of a classical lattice the product ansatz by Davydov is exact. Finally, we calculate the width of the soliton band to get information on the lifetime of the localized soliton.     

Exact two-qubit universal quantum circuit

We provide an analytic way to implement any arbitrary two-qubit unitary operation, given an entangling two-qubit gate together with local gates. This is shown to provide explicit construction of a universal quantum circuit that exactly simulates arbitrary two-qubit operations in SU(4). Each block in this circuit is given in a closed form solution. We also provide a uniform upper bound of the applications of the given entangling gates, and find that exactly half of all the controlled-unitary gates satisfy the same upper bound as the CNOT gate. These results allow for the efficient implementation of operations in SU(4) required for both quantum computation and quantum simulation.     

Exact uncertainty approach in quantum mechanics and quantum gravity

The assumption that an ensemble of classical particles is subject to nonclassical momentum fluctuations, with the fluctuation uncertainty fully determined by the position uncertainty, has been shown to lead from the classical equations of motion to the Schrödinger equation. This ‘exact uncertainty’ approach may be generalised to ensembles of gravitational fields, where nonclassical fluctuations are added to the field momentum densities, of a magnitude determined by the uncertainty in the metric tensor components. In this way one obtains the Wheeler-DeWitt equation of quantum gravity, with the added bonus of a uniquely specified operator ordering. No a priori assumptions are required concerning the existence of wave functions, Hilbert spaces, Planck's constant, linear operators, etc. Thus this approach has greater transparency than the usual canonical approach, particularly in regard to the connections between quantum and classical ensembles. Conceptual foundations and advantages are emphasised.