Cluster Decomposition Properties of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>Matrix

Wichmann, Eyvind H.; Crichton, James H.

doi:10.1103/physrev.132.2788

Cited by 82 publications

(46 citation statements)

References 11 publications

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“…Cluster separability [24,25] means that a Green's function describing the propagation of clusters of particles, when there are no interactions between the clusters, consists of a product of independent factors, one for each cluster. Each cluster's factor is the same as if the other clusters were not present.…”

Section: The Three-dimensional Three-body Green's Functionmentioning

confidence: 99%

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Relativistic three-body bound states and the reduction from four to three dimensions

Dulany

Wallace

1997

Phys. Rev. C

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Beginning with an effective field theory based upon meson exchange, the Bethe-Salpeter equation for the three-particle propagator (six-point function) is derived. Using the one-boson-exchange form of the kernel, this equation is then analyzed using time-ordered perturbation theory, and a threedimensional equation for the propagator is developed. The propagator consists of a pre-factor in which the relative energies are fixed by the initial state of the particles, an intermediate part in which only global propagation of the particles occurs, and a post-factor in which relative energies are fixed by the final state of the particles. The pre-and post-factors are necessary in order to account for the transition from states where particles are off their mass shell to states described by the global propagator with all of the particle energies on shell. The pole structure of the intermediate part of the propagator is used to determine the equation for the three-body bound state: a Schrödinger-like relativistic equation with a single, global Green's function. The role of the pre-and post-factors in the relativistic dynamics is to incorporate the poles of the breakup channels in the initial and final states. The derivation of this equation by integrating over the relative times rather than via a constraint on relative momenta allows the inclusion of retardation and dynamical boost corrections without introducing unphysical singularities. 21.30.Fe, 21.45.+v, 13.75.Cs

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Section: The Three-dimensional Three-body Green's Functionmentioning

confidence: 99%

“…Cluster separability (CS) implies that if we describe particles propagating using a Green's function, and if one cluster of particles does not interact with another cluster of particles, then we can perform a separation of variables (between these two clusters) on the Green's function [24,25].…”

Section: Appendix C: Cluster Separability and Toptmentioning

confidence: 99%

Relativistic three-body bound states and the reduction from four to three dimensions

Dulany

Wallace

1997

Phys. Rev. C

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“…It first appeared in quantum statistical physics where it was used for instance to establish the proportionality of extensive thermodynamic quantities to the volume in a quantum framework [7,8]. It occurred also in S-matrix theory [9] where it is often expressed as showing why an experiment in Geneva is insensitive to another experiment in Brookhaven. In the present case, it would mean that an atom far enough from the alpha track feels very little influence from this particle, at least during some time or far enough away.…”

Section: A Amentioning

confidence: 99%

Local Properties of Entanglement and Application to Collapse

Omnès

2013

Found Phys

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When a quantum system is macroscopic and becomes entangled with a microscopic one, this entanglement is not immediately total, but gradual and local. A study of this locality is the starting point of the present work and shows unexpected and detailed properties in the generation and propagation of entanglement between a measuring apparatus and a microscopic measured system. Of special importance is the propagation of entanglement in nonlinear waves with a finite velocity. When applied to the entanglement between a macroscopic system and its environment, this study yields also new results about the resulting disordered state. Finally, a mechanism of wave function collapse is proposed as an effect of perturbation in the growth of local entanglement between a measuring system and the measured one by waves of entanglement with the environment.Keywords: entanglement, kinetic physics, decoherence, wave function collapse _________ This is a revised version of arxiv.org.quant.phys.13093472, partly rewritten for clarity and with two previous key assumptions reconsidered, page 20: II as still convenient but unnecessary and III as being now proved.It became clear early in the theory of quantum measurements -particularly in Schrödinger's works [1, 2]-that entanglement is the stumbling block forbidding emergence of a unique datum in a measurement. More recently many experiments confirmed this viewpoint by realizing conditions that were similar to a measurement but involved only a few atoms or photons [3]: Von Neumann's standard description of a measurement as a creation of entanglement was found perfectly adequate [4], even when there was decoherence, and there was as predicted no glimpse of collapse in these experiments.An obvious consequence is that collapse, or the emergence of reality, is essentially a macroscopic phenomenon. This conclusion was drawn particularly in the theory of spontaneous collapse at a small but macroscopic scale by Ghirardi, Rimini and Weber [5], but in a form requiring modification in the quantum principles: The existence of a collapse effect with universal range and universal rate was added to these principles, the essential role of this effect being to break down the deadly obstruction from entanglement.But one may also wonder whether entanglement is so total and rigid that a new principle is needed to break it. Is it impossible to look at it more ordinarily as a physical phenomenon needing time for its growth and space for its expansion, rather than remaining an absolute mathematical property of wave functions? This question can stand in some sense as the starting point of the present work.When describing the nature of entanglement, Schrödinger considered an example where two quantum systems A and B, initially independent, begin to interact at some time zero and separate again after some more time [2]. Both systems are initially in a pure state but, although this is still true of the compound system AB after their interaction, it is not

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“…In the case of three space dimensions the difficulty signaled above makes it unlikely that one can define pair potentials for which the S-operator has the customary cluster properties [43], known to hold in nonrelativistic quantum mechanics [22,441 and relativistic quantum field theory [45]. We shall come back to this in Section 5.2.…”

Section: N Equal-mass Particles In One Dimensionmentioning

confidence: 99%

A positive energy dynamics and scattering theory for directly interacting relativistic particles

1980

Annals of Physics

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Starting from the tensor product of N irreducible positive energy representations of the PoincarC group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ((I k I2 + 1)1/2, k) and a free "reduced Hamiltonian"Hro, which is a positive operator acting only on internal variables, and then replace HrO by an interacting reduced Hamiltonian H, = H> + V, where V commutes with the Lorentz group and is such that H, is a positive operator. The resulting product form is shown to imply that the wave operators intertwine the free and interacting representations so that the s-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrodinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the spin-& case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the spin-$ case, coupling of a quantized radiation field. In view of eventual applications to "completely integrable" one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics.

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Cluster Decomposition Properties of theSMatrix

Cited by 82 publications

References 11 publications

Relativistic three-body bound states and the reduction from four to three dimensions

Relativistic three-body bound states and the reduction from four to three dimensions

Local Properties of Entanglement and Application to Collapse

A positive energy dynamics and scattering theory for directly interacting relativistic particles

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