Causality and Analyticity in Formal Scattering Theory

Gerald Minerbo

doi:10.1103/PhysRevD.3.928

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Causality and Analyticity in Formal Scattering Theory

Gerald Minerbo

Adelphi University, Garden City, New York 11530

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Phys. Rev. D 3, 928 – Published 15 February, 1971

DOI: https://doi.org/10.1103/PhysRevD.3.928

Abstract

The connection between causality and analyticity in scattering theory is formulated in terms of Hilbertspace concepts. The usual rules of nonrelativistic quantum mechanics are assumed to hold for the "in" and "out" states of the scattering system. We show that there is a (physically verifiable) causality condition which implies that each diagonal $S$ -matrix element $S_{α α} (E)$ must be the limit of an analytic function of the energy $E$ , regular in $Im E > 0$ . The implications for partial-wave amplitudes and for the forward scattering amplitude in elastic two-body collisions are discussed.

References (30)

N. G. van Kampen, Phys. Rev. 91, 1267 (1953)
R. J. Eden and P. V. Landshoff, Ann. Phys. (N. Y.) 31, 370 (1965)
A. Peres, Ann. Phys. (N. Y.) 37, 179 (1966)
G. R. Screaton, Phys. Rev. 165, 1610 (1968) ibid.182, 1415 (1969)
D. Iagolnitzer and H. P. Stapp, Commun. Math. Phys. 14, 15 (1969)
H. M. Nussenzveig, Phys. Rev. 177, 1848 (1969)
[3,5,6]
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (Ungar, New York, 1961), Sec. 61
N. Dunford and J. T. Schwartz, Linear Operators (Interscience, New York, 1963), Sec. XII.3
[9], p. 516
([12])
N. N. Khuri, Phys. Rev. 107, 1148 (1957)
[8], Vol. 1, pp. 63, 72, and 85
[8], Secs. 61, 66, and 68
M. H. Stone, Linear Transformations in Hilbert Space (American Mathematical Society, New York, 1932), Chap. 6
[9]
Omitted endnote
[8], Sec. 71
[15]
E. J. Beltrami and M. R. Wohlers, Distributions and the Boundary Values of Analytic Functions (Academic, New York, 1966), Sec. 1.1
Omitted endnote
[20]
[8], Sec. 74
M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964), Chap. 5
Omitted endnote
([24], Chap. 6
I. Saavedra, Nucl. Phys. 29, 137 (1962)
R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 6
[8], Sec. 65
Omitted endnote

[c1] N. G. van Kampen, Phys. Rev. 91, 1267 (1953)

[c2] R. J. Eden and P. V. Landshoff, Ann. Phys. (N. Y.) 31, 370 (1965)

[c3] A. Peres, Ann. Phys. (N. Y.) 37, 179 (1966)

[c4] G. R. Screaton, Phys. Rev. 165, 1610 (1968) ibid.182, 1415 (1969)

[c5] D. Iagolnitzer and H. P. Stapp, Commun. Math. Phys. 14, 15 (1969)

[c6] H. M. Nussenzveig, Phys. Rev. 177, 1848 (1969)

[c7] [3,5,6]

[c8] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (Ungar, New York, 1961), Sec. 61

[c9] N. Dunford and J. T. Schwartz, Linear Operators (Interscience, New York, 1963), Sec. XII.3

[c10] [9], p. 516

[c11] ([12])

[c12] N. N. Khuri, Phys. Rev. 107, 1148 (1957)

[c13] [8], Vol. 1, pp. 63, 72, and 85

[c14] [8], Secs. 61, 66, and 68

[c15] M. H. Stone, Linear Transformations in Hilbert Space (American Mathematical Society, New York, 1932), Chap. 6

[c16] [9]

[c17] Omitted endnote

[c18] [8], Sec. 71

[c19] [15]

[c20] E. J. Beltrami and M. R. Wohlers, Distributions and the Boundary Values of Analytic Functions (Academic, New York, 1966), Sec. 1.1

[c21] Omitted endnote

[c22] [20]

[c23] [8], Sec. 74

[c24] M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964), Chap. 5

[c25] Omitted endnote

[c26] ([24], Chap. 6

[c27] I. Saavedra, Nucl. Phys. 29, 137 (1962)

[c28] R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 6

[c29] [8], Sec. 65

[c30] Omitted endnote

Physical Review D

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Causality and Analyticity in Formal Scattering Theory

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