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Energy-Momentum Conservation Implies Translation Invariance: Some Didactic Remarks

J. Math. Phys. 8, 1807 (1967); doi:10.1063/1.1705424

Issue Date: September 1967

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David N. Williams
Service de Physique Théorique, Centre d'Etudes Nucléaires de Saclay, BP n° 2,91, Gif-sur-Yvette, France
We discuss from a rigorous viewpoint two more-or-less familiar cases where energy-momentum conservation implies invariance under space-time translations. First, if a closed linear operator on a Hilbert space has a domain that is invariant under spectral projections belonging to the four-momentum operators, and if it ``conserves energy-momentum,'' it necessarily commutes with the appropriate representation of the translations. (Bounded operators, such as the S matrix, are a special case.) At least for separable spaces, the domain restriction characterizes the closed operators for which the theorem is true. Second, if a bounded bilinear form between momentum states of m and n particles in a Fock space (or more generally, a bounded multilinear form) conserves energy momentum, the corresponding tempered distribution has a conservation delta function at points where the mass shell is a C[infinity] manifold; but no derivatives of delta functions can occur. In this connection, we are led to a result that seems to be new: the cluster parameters (``connected amplitudes'') of a family of bounded bilinear forms, labeled by (m, n), are also bounded bilinear forms. The two systems, of course, mutually conserve energy momentum. ©1968 The American Institute of Physics
History: Received 10 October 1966
Permalink: http://link.aip.org/link/?JMAPAQ/8/1807/1
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0022-2488 (print)   1089-7658 (online)
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REFERENCES (34)
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  1. E. Noether, Nachr. Akad. Wiss. Goettingen, II. Math. Physik. Kl., 235 (1918).
  2. A. Messiah, Mécanique Quantique (Dunod Cie., Paris, 1960), Vol. II, Chap. XV.
  3. Although we should not be surprised to learn that the argument in question is known, we have not succeeded in finding it in the literature. For the case of the S matrix, H. P. Stapp [Phys. Rev. 125, 2139 (1962)] mentions without proof that translation invariance and energy-momentum conservation are equivalent. For bounded operators, the exercise is indeed not only simple but trivial, given the standard results of the spectral theory.
  4. E.g., G. F. Chew, Sci. Progr. (G.B.), 51, 529 (1963);
    5. H. P. Stapp in Ref. 3;
    E. Lubkin, Nuovo Cimento 32, 171 (1964).
  5. This strikes us as a radical view because we are not able to imagine all possible theories by means of which the concept could acquire an operational meaning. We do not intend by that a value judgement on the plausibility of theories motivated by such a view.
  6. There have been several interesting attempts in this direction, based on the S matrix. Among them we mention M. L. Goldberger and K. M. Watson, Phys. Rev. 127, 2284 (1962);
    7. M. Froissart, M. L. Goldberger, and K. M. Watson, ibid. 131, 2820 (1963);
    H. P. Stapp, ibid. 139, B257 (1965);
    A. Peres, Ann. Phys. (N.Y.) 37, 179 (1966). The last of these contains a more complete list of references.
  7. This is, for example, the attitude of Stapp. See Ref. 3.
  8. Very readable summaries on the “SNAG” theorem are given by R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (W. A. Benjamin, Inc., New York, 1964), pp. 91–93;
    9. R. Jost, The General Theory of Quantized Fields (American Mathematical Society, Providence, Rhode Island, 1965), pp. 16–17.
  9. Recall that the Borel sets of Rn are the smallest family of sets that contains all denumerable unions, intersections, and complements of open sets.
  10. F. Riesz and B. Sz.-Nagy, Functional Analysis (Frederick Ungar Publishing Company, New York, 1955), Chap. VIII, Secs. 114–116.
  11. L. Schwartz, Théorie des Distributions (Hermann et Cie., Paris, 1959), Vol. II, Chap. VII.
  12. It will generally be obvious how to take into account the case m or n = 0, corresponding to the vacuum with zero energy momentum, so we most often do not mention it explicitly.
  13. I am indebted to D. Iagolnitzer for drawing my attention to this point, as well as to the fact that translation invariance of the probabilities does not imply translation invariance of the amplitudes (although Poincare invariance does).
  14. This simple proof occurred to several other people before it occurred to the author. In the literature, the question is raised indirectly by E. H. Wichmann and J. H. Crichton [Phys. Rev. 132, 2788 (1963)],
    15. who give a lucid discussion of the cluster decomposition property which assumes that the S-matrix amplitudes are tempered distributions. K. Hepp [Helv. Phys. Acta 37, 659 (1964)]
    states it as a fact, without giving the proof. We have found references to the proof in J. R. Taylor, Phys. Rev. 142, 1236 (1966),
    and D. Iagolnitzer, “S-Matrix Theory and Phenomenological Space-Time Description,” Saclay preprint (to be published).
  15. L. Gårding and J. L. Lions, Nuovo Cimento, Suppl. 14, 9 (1959).
  16. The domain specified is translation invariant, but not invariant under spectral projections. It follows from Theorem A, with Theorem B in Sec. IV, that if the operators conserve energy momentum, they are not closed on this domain.
  17. K. Hepp, Commun. Math. Phys. 1, 7 (1965).
  18. L. Auslander and R. E. MacKenzie, Introduction to Differentiable Manifolds (McGraw-Hill Book Company, Inc., New York, 1963), Chap. II.
    19. Our excuse for sketching the proof of these well-known facts about the mass shell is that it takes only a few words, and hopefully makes things clearer. For more details, see K. Hepp, Helv. Phys. Acta 36, 355 (1963);
    and 37, 55 (1964);
    H. P. Stapp, “Studies in the Foundations of S-Matrix Theory,” University of California, Lawrence Radiation Laboratory Report, UCRL 10843.
  19. L. Schwartz, Ref. 11, Vol. I, pp. 100–103, applied to Ex. 2, p. 114, and using the temperedness of Bmn.
  20. See Ref. 14.
  21. By analogy with the definition of truncated Wightman functions due to R. Haag, Phys. Rev. 112, 669 (1958). Any other consistent choice of momentum-dependent, but measurable phases in this definition would be harmless for our purpose.
  22. F. Riesz and Sz.-Nagy, Ref. 10, pp. 305–306.
  23. N. Dunford and J. T. Schwartz, Linear Operators (Interscience Publishers, Inc., New York, 1958), Part II, Chap. XI, Sec. 6.
  24. The reader who treats the following argument as a recipe for pencil and paper will find it straightforward.
  25. Note that T(b)A[subset, equals]AT(b) for all translations implies by definition that T(b)D(A)[subset or is implied by]D(A), and hence from the group property that T(b)A = AT(b).
  26. F. Riesz and Sz.-Nagy, Ref. 10, p. 383.
  27. Ref. 10, p. 302.
  28. Ref. 10, Sec. 126.
  29. M. A. Naimark, Normed Rings (P. Noordhoff Ltd., Groningen, The Netherlands, 1964), Appendix III.
  30. P. R. Halmos, Measure Theory (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1950), p. 86.
  31. Ref. 30, Ex. 2, p. 86.
  32. Ref. 29, p. 28.
  33. Ref. 29, p. 33.
  34. Ref. 29, p. 129.

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