Quasisecular scattering in the <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>φ</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></math></span> model

B. B. Varga; S. Ø. Aks

doi:10.1103/PhysRevD.10.614

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Quasisecular scattering in the $φ^{4}$ model

B. B. Varga and S. Ø. Aks

University of Illinois, Chicago, Illinois 60680

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Phys. Rev. D 10, 614 – Published 15 July, 1974

DOI: https://doi.org/10.1103/PhysRevD.10.614

Abstract

The first-order renormalized Hamiltonian, calculated by the quasisecular perturbation method applied to the weakly nonlinear $φ^{4}$ model of quantum field theory, is used to discuss the scattering of two particles. The scattering amplitude is found by an approximation which is accurate for weak coupling. For one space dimension a resonance in the $s$ -wave scattering occurs at threshold for both signs of the $φ^{4}$ coupling. For two space dimensions a resonance occurs only for the sign of coupling which also gives a bound state as found in previous work, and the resonance lies above threshold by the bound-state energy. For three dimensions the scattering amplitude shows very little structure. Comparison of the scattering amplitudes is made with the Bethe-Salpeter equation.

References (13)

L. Van Hove, Physica 21, 901 (1955) ibid.22, 343 (1956)
S. Aks, J. Sienicki, and B. Varga, Phys. Rev. D 6, 520 (1972)
B. Varga and S. Aks, Phys. Rev. D 6, 529 (1972)
S. Aks and B. Varga, Phys. Rev. D 6, 2773 (1972)
B. Varga and S. Aks, J. Math Phys. 15, 149 (1974)
E. Frieman, J. Math. Phys. 4, 410 (1963) G. Sandri, Ann. Phys. (N.Y.) 24, 332 (1963)
H. Fröhlich, Proc. R. Soc. A215, 291 (1952)
G. Barton, Introduction to Advanced Field Theory (Interscience, New York, 1963), p. 126
B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms (Princeton Univ. Press, Princeton, N. J., 1971), p. 90
T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966)
L. D. Faddeev, Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory (Israel Program for Scientific Translations, Jerusalem, 1965), p. 10
Eq. (5.1) of [4]
J. Combes, Commun. Math. Phys. 12, 283 (1969)

[c1] L. Van Hove, Physica 21, 901 (1955) ibid.22, 343 (1956)

[c2] S. Aks, J. Sienicki, and B. Varga, Phys. Rev. D 6, 520 (1972)

[c3] B. Varga and S. Aks, Phys. Rev. D 6, 529 (1972)

[c4] S. Aks and B. Varga, Phys. Rev. D 6, 2773 (1972)

[c5] B. Varga and S. Aks, J. Math Phys. 15, 149 (1974)

[c6] E. Frieman, J. Math. Phys. 4, 410 (1963) G. Sandri, Ann. Phys. (N.Y.) 24, 332 (1963)

[c7] H. Fröhlich, Proc. R. Soc. A215, 291 (1952)

[c8] G. Barton, Introduction to Advanced Field Theory (Interscience, New York, 1963), p. 126

[c9] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms (Princeton Univ. Press, Princeton, N. J., 1971), p. 90

[c10] T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966)

[c11] L. D. Faddeev, Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory (Israel Program for Scientific Translations, Jerusalem, 1965), p. 10

[c12] Eq. (5.1) of [4]

[c13] J. Combes, Commun. Math. Phys. 12, 283 (1969)

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Quasisecular scattering in the $φ^{4}$ model

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Quasisecular scattering in the φ4 model

Abstract

References (13)

Quasisecular scattering in the $φ^{4}$ model