In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as quantum triviality.
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The Kondo model (sometimes referred to as the s-d model) is a model for a single localized quantum impurity coupled to a large reservoir of delocalized and noninteracting electrons. The quantum impurity is represented by a spin-1/2 particle, and is coupled to a continuous band of noninteracting electrons by an antiferromagnetic exchange coupling J. The Kondo model is used as a model for metals containing magnetic impurities, as well as quantum dot systems.
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Figure 3. An asymmetric dendrimer with a ferrocene core No applications have been identified for ferrocene-containing dendrimers. They exhibit multielectron redox indicating that the ferrocenyl moieties are essentially noninteracting redox centers.
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We note that an analogous collision- induced light scattering (CILS) or Raman process also exists, which is well studied and is in many ways completely analogous to CIA and CIE. CILS arises from interaction-induced polarizability increments of molecular complexes; the excess polarizability of a complex, relative the sum of polarizabilities of the noninteracting molecules.
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In a conformal field theory, the only truly massless particles are noninteracting singletons (see singleton field). The other "particles"/bound states have a continuous mass spectrum which can take on any arbitrarily small nonzero mass. So, we can have spin-3/2 and spin-2 bound states with arbitrarily small masses but still not violate the theorem. In other words, they are infraparticles.
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In quantum field theory wave function renormalization is a rescaling (or renormalization) of quantum fields to take into account the effects of interactions. For a noninteracting or free field the field operator creates or annihilates a single particle with probability 1. Once interactions are included, however, this probability is modified in general to Z eq 1\. This appears when one calculates the propagator beyond leading order; e.g.
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Lattice gauge theory is also important for the study of quantum triviality by the real-space renormalization group. The most important information in the RG flow are what's called the fixed points. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be trivial or noninteracting.
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A Gaussian fixed point is a fixed point of the renormalization group flow which is noninteracting in the sense that it is described by a free field theory. The word Gaussian comes from the fact that the probability distribution is Gaussian at the Gaussian fixed point. This means that Gaussian fixed points are exactly solvable (trivially solvable in fact). Slight deviations from the Gaussian fixed point can be described by perturbation theory.
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J is half the difference in energy between the two possibilities. Magnetic interactions seek to align spins relative to one another. Spins become randomized when thermal energy is greater than the strength of the interaction. For each pair, if ::J_{ij} > 0 the interaction is called ferromagnetic ::J_{ij} < 0 the interaction is called antiferromagnetic ::J_{ij} = 0 the spins are noninteracting A ferromagnetic interaction tends to align spins, and an antiferromagnetic tends to antialign them.
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This is called demagnetization remanence or DC demagnetization remanence and is denoted by symbols like Md(H), where H is the magnitude of the field. Yet another kind of remanence can be obtained by demagnetizing the saturation remanence in an ac field. This is called AC demagnetization remanence or alternating field demagnetization remanence and is denoted by symbols like Maf(H). If the particles are noninteracting single- domain particles with uniaxial anisotropy, there are simple linear relations between the remanences.
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The classical band structure of solid state physics predicts the Fermi level to lie in a band gap for insulators and in the conduction band for metals, which means metallic behavior is seen for compounds with partially filled bands. However, some compounds have been found which show insulating behavior even for partially filled bands. This is due to the electron-electron correlation, since electrons cannot be seen as noninteracting. Mott considers a lattice model with just one electron per site.
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New York, NY, John Wiley, 1996 He also introduced to MRI the first report on parallel acquisition of MRI data from multiple non-interacting surface coils.Hyde, J. S., Jesmanowicz, A., Froncisz, W., Kneeland, J. B., Grist, T. M., and Campagna, N. F.: Parallel Image Acquisition from Noninteracting Local Coils. J. Magn. Reson. 70:512-517, 1986 Gradient coils are required in order to make an image, and it was natural to consider use of a local gradient coil that was tailored to the anatomy of interest.
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Non-interacting systems are relatively easy to solve, as the wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic energy functional of such a system is known exactly. The exchange–correlation part of the total energy functional remains unknown and must be approximated. Another approach, less popular than KS DFT but arguably more closely related to the spirit of the original H–K theorems, is orbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the noninteracting system.
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The physics of the zero- temperature phase behavior of jellium is driven by competition between the kinetic energy of the electrons and the electron-electron interaction energy. The kinetic-energy operator in the Hamiltonian scales as 1/r_s^2, where r_s is the Wigner–Seitz radius, whereas the interaction energy operator scales as 1/r_s. Hence the kinetic energy dominates at high density (small r_s), while the interaction energy dominates at low density (large r_s). The limit of high density is where jellium most resembles a noninteracting free electron gas.
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Intuitively, it consists of elementary particles or bound states that are sufficiently well separated that their interactions with each other are ignored. The idea is that whatever physical process one is trying to study may be modeled as a scattering process of these well separated bound states. This process is described by the full Hamiltonian , but once it's over, all of the new elementary particles and new bound states separate again and one finds a new noninteracting state called the out state. The S-matrix is more symmetric under relativity than the Hamiltonian, because it does not require a choice of time slices to define.
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Of special interest is the limit of infinite strength repulsion, in which case the Lieb–Liniger model becomes the Tonks–Girardeau gas (also called the hard- core Bose gas, or impenetrable Bose gas). In this limit, the bosons may, by a change of variables that is a continuum generalization of the Jordan–Wigner transformation, be transformed to a system one-dimensional noninteracting spinless fermions. The nonlinear Schrödinger equation is a simplified 1+1-dimensional form of the Ginzburg–Landau equation introduced in 1950 in their work on superconductivity, and was written down explicitly by in their study of optical beams. Multi-dimensional version replaces the second spatial derivative by the Laplacian.
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This theorem has since been extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states. The second H–K theorem defines an energy functional for the system and proves that the correct ground-state electron density minimizes this energy functional. In work that later won them the Nobel prize in chemistry, the H–K theorem was further developed by Walter Kohn and Lu Jeu Sham to produce Kohn–Sham DFT (KS DFT). Within this framework, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting electrons moving in an effective potential.
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DOPE, or Discrete Optimized Protein Energy, is a statistical potential used to assess homology models in protein structure prediction. DOPE is based on an improved reference state that corresponds to noninteracting atoms in a homogeneous sphere with the radius dependent on a sample native structure; it thus accounts for the finite and spherical shape of the native structures. It is implemented in the popular homology modeling program MODELLER and used to assess the energy of the protein model generated through many iterations by MODELLER, which produces homology models by the satisfaction of spatial restraints. The models returning the minimum molpdfs can be chosen as best probable structures and can be further used for evaluating with the DOPE score.
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In classical 1,4-didehydrobenzene experiments, heating to 300 °C, [1,6-D2]-A readily equilibrates with [3,2-D2]-B, but does not equilibrate with C or D. The simultaneous migration of deuterium atoms to form B, and the fact that none of C or D is formed can only be explained by a presence of a cyclic and symmetrical intermediate–1,4-didehydrobenzene. center Two states were proposed for 1,4-didehydrobenzene: singlet and triplet, with the singlet state lower in energy. Triplet state represents two noninteracting radical centers, and hence should abstract hydrogens at the same rate as phenyl radical. However, singlet state is more stabilized than the triplet, and therefore some of the stabilizing energy will be lost in order to form the transition state for hydrogen cleavage, leading to slower hydrogen abstraction.
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He leaves out the treatment of relativistic wave mechanics altogether in his otherwise complete introduction to modern applications of quantum mechanics, explaining: "It seems to me that the way this is usually presented in books on quantum mechanics is profoundly misleading." (From the preface in Lectures on Quantum Mechanics, referring to treatments of the Dirac equation in its original flavor.) Others, like Walter Greiner does in his series on theoretical physics, give a full account of the historical development and view of relativistic quantum mechanics before they get to the modern interpretation, with the rationale that it is highly desirable or even necessary from a pedagogical point of view to take the long route. In quantum field theory, the solutions of the free (noninteracting) versions of the original equations still play a role. They are needed to build the Hilbert space (Fock space) and to express quantum field by using complete sets (spanning sets of Hilbert space) of wave functions.
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