Behavior of Commutator Matrix Elements at Small Distances. II. Equal-Time Limits of Charge Moments and Time Derivatives

A. H. Völkel

doi:10.1103/PhysRevD.3.917

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Behavior of Commutator Matrix Elements at Small Distances. II. Equal-Time Limits of Charge Moments and Time Derivatives

A. H. Völkel^*,†

Institut für Theoretische Physik, Freie Universität Berlin, Berlin, Germany
Instituut voor Theoretische Fysica, Universiteit Nijmegen, Nijmegen, Netherlands

^*Present address.
^†Supported in part by the Stichting voor Fundamenteel Onderzoek der Materie (FOM).

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Phys. Rev. D 3, 917 – Published 15 February, 1971

DOI: https://doi.org/10.1103/PhysRevD.3.917

Abstract

From general principles of quantum field theory (especially locality and Poincaré invariance, but without use of the spectrum condition), it is shown that the equal-time limits of current-density commutators exist if the limits for the commutators between one current density and one generalized charge exist. If the equal-time limits of the current-density commutators containing at least one zeroth component exist, then the limits of the space-space components exist also. If the equal-time limits between one current density and the $n$ th time derivative of the generalized charges exist, then also the limits of all time derivatives up to order $n$ of the density commutators exist. Explicit expressions for the first time derivative of currentdensity commutators in terms of the Gell-Mann $Σ$ and meson commutators are derived.

References (29)

B. Renner, Current Algebras and their Applications (Pergamon, New York, 1968)
S. L. Adler and R. F. Dashen, Current Algebras and Applications to Particle Physics (Benjamin, New York, 1968)
B. Schroer and P. Stichel, Commun. Math. Phys. 3, 258 (1966)
A. H. Völkel, Phys. Rev. D 1, 3377 (1970)
Omitted endnote
F. Treves, Topological Vector Spaces, Distributions and Kernels (Academic, New York, 1967), Sec. 28
([3])
N. Kroll, T. D. Lee, and B. Zumino, Phys. Rev. 157, 1376 (1967)
C. A. Orzalesi, University of Maryland Technical Report No. 833, 1968 (unpublished)
([11])
I. M. Gelfand and G. E. Schilow, Verallgemeinerte Funktionen (Distributionen) (Deutscher Verlag der Wissenschaften, Berlin, 1962), Vol. II
J. D. Bjorken, Phys. Rev. 148, 1467 (1966)
R. A. Brandt and J. Sucher, Phys. Rev. 177, 2218 (1969)
R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That (Benjamin, New York, 1964)
L. Gårding and A. S. Wightman, Arkiv Fysik 28, 129 (1964)
R. Jost, The General Theory of Quantized Fields (American Mathematical Society, Providence, 1965)
K. Hepp, in Axiomatic Field Theory and Particle Symmetries (Gordon and Breach, New York, 1965), Vol. 1
Omitted endnote
S. Łojasiewicz, Studia Mathematica T. XVII, 1 (1958), Sections 4.3-4.5
A. H. Völkel, Commun. Math. Phys. 5, 57 (1967)
Uta Völkel and A. H. Völkel, Commun. Math. Phys. 7, 261 (1968)
Uta Völkel and A. H. Völkel, Nuovo Cimento 63A, 203 (1969)
L. Schwartz, Théorie des distributions I/II (Hermann, Paris, 1957/59)
Uta Völkel, B. Schroer, and A. H. Völkel, Commun. Math. Phys. 10, 69 (1968)
M. Gell-Mann and M. Lévy, Nuovo Cimento 16, 705 (1960)
Omitted endnote
T. K. Kuo and M. Sugawara, Phys. Rev. 163, 1716 (1967)
Omitted endnote
Omitted endnote

[c1] B. Renner, Current Algebras and their Applications (Pergamon, New York, 1968)

[c2] S. L. Adler and R. F. Dashen, Current Algebras and Applications to Particle Physics (Benjamin, New York, 1968)

[c3] B. Schroer and P. Stichel, Commun. Math. Phys. 3, 258 (1966)

[c4] A. H. Völkel, Phys. Rev. D 1, 3377 (1970)

[c5] Omitted endnote

[c6] F. Treves, Topological Vector Spaces, Distributions and Kernels (Academic, New York, 1967), Sec. 28

[c7] ([3])

[c8] N. Kroll, T. D. Lee, and B. Zumino, Phys. Rev. 157, 1376 (1967)

[c9] C. A. Orzalesi, University of Maryland Technical Report No. 833, 1968 (unpublished)

[c10] ([11])

[c11] I. M. Gelfand and G. E. Schilow, Verallgemeinerte Funktionen (Distributionen) (Deutscher Verlag der Wissenschaften, Berlin, 1962), Vol. II

[c12] J. D. Bjorken, Phys. Rev. 148, 1467 (1966)

[c13] R. A. Brandt and J. Sucher, Phys. Rev. 177, 2218 (1969)

[c14] R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That (Benjamin, New York, 1964)

[c15] L. Gårding and A. S. Wightman, Arkiv Fysik 28, 129 (1964)

[c16] R. Jost, The General Theory of Quantized Fields (American Mathematical Society, Providence, 1965)

[c17] K. Hepp, in Axiomatic Field Theory and Particle Symmetries (Gordon and Breach, New York, 1965), Vol. 1

[c18] Omitted endnote

[c19] S. Łojasiewicz, Studia Mathematica T. XVII, 1 (1958), Sections 4.3-4.5

[c20] A. H. Völkel, Commun. Math. Phys. 5, 57 (1967)

[c21] Uta Völkel and A. H. Völkel, Commun. Math. Phys. 7, 261 (1968)

[c22] Uta Völkel and A. H. Völkel, Nuovo Cimento 63A, 203 (1969)

[c23] L. Schwartz, Théorie des distributions I/II (Hermann, Paris, 1957/59)

[c24] Uta Völkel, B. Schroer, and A. H. Völkel, Commun. Math. Phys. 10, 69 (1968)

[c25] M. Gell-Mann and M. Lévy, Nuovo Cimento 16, 705 (1960)

[c26] Omitted endnote

[c27] T. K. Kuo and M. Sugawara, Phys. Rev. 163, 1716 (1967)

[c28] Omitted endnote

[c29] Omitted endnote

Physical Review D

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Behavior of Commutator Matrix Elements at Small Distances. II. Equal-Time Limits of Charge Moments and Time Derivatives

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Behavior of Commutator Matrix Elements at Small Distances. I. Existence and Structure of Equal-Time Limits

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