Scattering and reflection positivity in relativistic Euclidean quantum mechanics

Polyzou, W. N.

doi:10.1103/physrevd.89.076008

Cited by 5 publications

(2 citation statements)

References 11 publications

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“…The advantage of the Hilbert space representation are that the input involves solutions of relatively well-behaved Euclidean Green functions and there are explicit expressions for the Hamiltonian and the other nine self-adjoint Poincaré generators on this space. Because the Hilbert space representation is the physical representation, direct calculations of scattering observables [90] [91] [92] can be preformed without analytic continuation. The challenges are to ensure that the kernel is reflection positive.…”

Section: Dynamicsmentioning

confidence: 99%

Representations of relativistic particles of arbitrary spin in Poincaré, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

Polyzou

2019

Phys. Rev. C

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Background: Relativistic treatments of quantum mechanical systems are important for understanding hadronic structure and dynamics at sub-nucleon scales. Relativistic invariance of a quantum system means that there is an underlying unitary representation of the Poincaré group. This is equivalent to the requirement that the quantum observables (probabilities, expectation values and ensemble averages) for equivalent measurements performed in different inertial reference frames are identical. Different representations are used in practice, including Poincaré covariant forms of dynamics, representations based on Lorentz covariant wave functions, Euclidean covariant representations and representations generated by Lorentz covariant fields.Purpose: The purpose of this work is to illustrate the relation between the different equivalent representations of states in relativistic quantum mechanics.Method: The starting point is a description of a particle of mass m and spin j using irreducible representations of the Poincaré group. Since any unitary representation of the Poincaré group can be decomposed into a direct integral of irreducible representations, these are the basic building blocks of any relativistically invariant quantum theory. The equivalence is established by constructing equivalent Lorentz covariant irreducible representations from Poincaré covariant irreducible representations and constructing equivalent Euclidean covariant irreducible representations from Lorentz covariant irreducible representations.Results: Equivalent descriptions for positive mass representations of arbitrary spin are presented in each of these frameworks. Dynamical realizations of the different representations are briefly discussed.Conclusion: Poincaré covariant, Lorentz covariant and Euclidean covariant realizations of relativistic dynamics are shown to be equivalent by explicitly relating the positive-mass positive-energy irreducible representations of the Poincaré group that appear in the direct integral.

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Section: Dynamicsmentioning

confidence: 99%

Representations of relativistic particles of arbitrary spin in Poincaré, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

Polyzou

2019

Phys. Rev. C

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“…We emphasize that the disconnected terms in following arguments in [18], W. N. Polyzou has shown the same −3/2 behavior for the integrand in (5.16) that one gets non-relativistically (see [27]). This is sufficient to satisfy the Cook condition.…”

Section: Injection Operators and One-body Solutionsmentioning

confidence: 57%

An application of the theory of moments to Euclidean relativistic quantum mechanical scattering

Aiello¹

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To my brother, Alan. I admire you for becoming the man you are today and I truly look up to you for having made the climb. Your thoughtfulness, generosity, and devotion impact my life on a daily basis for which I am immeasurably grateful. I am proud to be your brother. To my father, James. Through your example I have learned the importance of values, integrity, dedication, and measuring twice to cut once. That's what makes you that guy. I am proud to be your son. Go Lions! To my love and inspiration, Brittany. Thank you for continually believing in me and being there every step of the way. This road would have been entirely impassable if not for the light you've given to my life. Finally, to my mother, Mary. There are no words. Thank you for always believing in me. Thank you for teaching me to value the beauty of dreams. Thank you for showing me how to sustain grace, conviction, and courage under fire. Thank you for teaching me the difference one person can make. Thank you for showing me what real heroes look like. I am proud to be your son. This one's definitely for you, Ma!

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Zero energy scattering calculation in Euclidean space

Carbonell

Karmanov

2016

Physics Letters B

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We show that the Bethe-Salpeter equation for the scattering amplitude in the limit of zero incident energy can be transformed into a purely Euclidean form, as it is the case for the bound states. The decoupling between Euclidean and Minkowski amplitudes is only possible for zero energy scattering observables and allows determining the scattering length from the Euclidean Bethe-Salpeter amplitude. Such a possibility strongly simplifies the numerical solution of the Bethe-Salpeter equation and suggests an alternative way to compute the scattering length in Lattice Euclidean calculations without using the Luscher formalism. The derivations contained in this work were performed for scalar particles and one-boson exchange kernel. They can be generalized to the fermion case and more involved interactions.Comment: 11 pages, 3 figures, to be published in Phys. Lett.

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Scattering and reflection positivity in relativistic Euclidean quantum mechanics

Cited by 5 publications

References 11 publications

Representations of relativistic particles of arbitrary spin in Poincaré, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

Representations of relativistic particles of arbitrary spin in Poincaré, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

An application of the theory of moments to Euclidean relativistic quantum mechanical scattering

Zero energy scattering calculation in Euclidean space

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