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Analyticity in quantum field theory

Аналитичность в квантовой теории поля. - II: Причинность и дисперсионные соотнощения

II. Causality and dispersion relations

  • Published:
Il Nuovo Cimento A (1971-1996)

Summary

The consequences from causality in high-energy physics are studied, with particular reference to the well-known claim that a strong connection exists between causality and global analyticity of the scattering amplitudes. We demonstrate that this relation is not as tight as has been conventionally taken for granted, but depends on additional assumptions imposed on the scattering amplitudes in the unphysical as well as in the physical region; at present no experimental test of these extra assumptions can be given. Therefore we try to discard them and explore a new concept of causality, which assures the usual (covariant) cause-event structure only over macroscopic distances and allows for violations in microscopic regions. Quantum field theory with « shadow states » is examined and found to be consistent with macroscopic causality, though not necessarily leading to the so-called normal analytic scattering amplitudes.

Riassunto

Si studiano le conseguenze della causalità nella fisica delle alte energie, con particolare riferimento alla ben nota asserzione che esiste una forte connessione fra causalità ed analiticità delle ampiezze di scattering. Si dimostra che questa relazione non è così stretta come si è convenzionalmente dato per certo, ma dipende da ipotesi aggiuntive imposte alle ampiezze di scattering sia nella regione non fisica che in quella fisica; attualmente non si può dare alcuna riprova sperimentale di queste ipotesi aggiuntive. Si cerca quindi di scartarle e di esplorare un nuovo concetto di causalità, che assicuri la usuale struttura causa-evento (covariante) solo su distanze macroscopiche e consenta violazioni nelle regioni microscopiche. Si esamina la teoria quantistica dei campi con « stati ombra » e si trova che essa è consistente con la causalità macroscopica, senza portare necessariamente alle cosiddette ampiezze di scattering analitiche normali.

Реэюме

Исследуются следствия причинности в фиэике высоких знергий. Особое внимание уделяется хорощо иэвестному требованию, что сушествует сильная свяэь между причинностью и глобальной аналитичностью амплитуд рассеяния. Мы покаэываем, что зто соотнощение не такое жесткое, как обычно принимается, а эависит от дополнительных предположений, накладываемых на амплитуды рас-сеяния в нефиэической и в фиэической областях. В настояшее время невоэможно провести зкспериментальную проверку зтих дополнительных предположений. Поз-тому мы пытаемся откаэаться от них и испольэовать новую концепцию причинности, которая обеспечивает обычную (ковариантную) структуру " причина-событие " лищь на макроскопических расстояниях и допускает нарущения в микроскопических об-ластях. Исследуется квантовая теория поля с "теневыми состояниями" и полу-чается, что зта теория не противоречит макроскопической причинности, хотя не обяэательно приводит к так наэываемым нормальным аналитическим амплитудам рассеяния.

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References

  1. A. Sommerfeld:Phys. Zeits.,8, 841 (1907).

    Google Scholar 

  2. See for more detailsA. Sommerfeld:Ann. der Phys.,44, 177 (1914).

    Article  ADS  Google Scholar 

  3. H. A. Kramers:Nature,117, 775 (1925);Atti del Congresso Internazionale dei Fisici, Vol.2 (Como, 1927), p. 545;R. de L. Kronig:Journ. Opt. Soc. Am.,12, 547 (1926).

    Google Scholar 

  4. H. Rechenberg:Optical dispersion relations with finite wave packets, Austin preprint (AEC 91-CPT 144) (June, 1972).

  5. H. Rechenberg andE. C. G. Sudarshan:Nuovo Cimento,12 A, 541 (1972), quoted as I below.

    Article  ADS  Google Scholar 

  6. We refer,e.g., toD. Iagolnitzer andH. P. Stapp:Comm. Math. Phys.,14, 15 (1969).

    Article  ADS  MathSciNet  Google Scholar 

  7. E. C. G. Sudarshan:Fields and Quanta,2, 175 (1971);H. Rechenberg:The triangle graph in quantum field theory with shadow states: causality, unitarity and interpretation, Internal report, CPT Austin, August 1971 (unpublished);H. Rechenberg andE. C. G. Sudarshan:Analyticity in quantum field theory. - III:University, in preparation.

    Google Scholar 

  8. As far as we know,Dirac was the first to discuss generally the use of complex « observables » inProc. Roy. Soc., A160, 48 (1937).

  9. By « normal analyticity » one means in particular the following property: each scattering amplitude is an analytic function at a pointP, given by a set of complex variabless, t, …, unlessP lies in the union of positive Landau surfaces of the process in question. For further details see,e.g.,Iagolnitzer andStapp, quoted in footnote (6) p. 21.

    Article  ADS  MathSciNet  Google Scholar 

  10. See,e.g.,R. Karplus,C. M. Sommerfield andE. H. Wichmann:Phys. Rev.,114, 376 (1959).

    Article  ADS  MathSciNet  Google Scholar 

  11. SeeR. Karplus, C. M. Sommerfield andE. H. Wichmann:Phys. Rev.,114, 380 (1959), who obtain in the first case yB24=+0.999, and in the second case y-B24=0.899. In the latter example we find agains a belows phys.

    Article  ADS  Google Scholar 

  12. It has been noted already in the late twenties that,e.g., the position of a photon is not an observable in the strict sense of quantum theory; it can only be obtained in the limit of geometrical optics. We refer toW. Pauli:Die allgemeinen Prinzipien der Wellenmechanik, inHandbuch der Physik, Vol.24/1, edited byH. Geiger andK. Scheel (Berlin, 1933), p. 260; for somewhat related technical remarks seeR. Acharya andE. C. G. Sudarshan:Journ. Math. Phys.,1, 532 (1960).

  13. K. Symanzik:Phys. Rev.,105, 743 (1957).

    Article  ADS  Google Scholar 

  14. See the paper quoted in footnote (15) andN. N. Bogoliubov, B. V. Medvedev andM. K. Polivanov:Fortschr. Phys.,6, 169 (1958);H. J. Bremermann, R. Oehme andJ. G. Taylor:Phys. Rev.,109, 2178 (1958).

    Article  ADS  Google Scholar 

  15. H. Lehmann:Suppl. Nuovo Cimento,14, 153 (1959).

    Article  ADS  Google Scholar 

  16. J. G. Taylor:Ann. of Phys.,5, 391 (1958). In the case of the two-point function the variable is clearly (x−y)2, and the special value is zero; the corresponding momentum variable is the energy of the incoming particle.

    Article  ADS  Google Scholar 

  17. See,e.g.,K. Hepp:Helv. Phys. Acta,37, 639 (1964).

    MathSciNet  Google Scholar 

  18. W. Pauli:Die allgemeinen Prinzipien der Wellenmechanik, inHandbuch der Physik, Vol.24/1, edited byH. Geiger andK. Scheel (Berlin, 1933), p. 272.

  19. G. Wanders:Nuovo Cimento,14, 168 (1959).

    Article  MathSciNet  Google Scholar 

  20. G. Wanders:Nuovo Cimento,14, 175 (1959).

    Article  MathSciNet  Google Scholar 

  21. N. G. Van Kampen:Phys. Rev.,91, 1267 (1953).

    Article  ADS  Google Scholar 

  22. J. Sucher:Nucl. Phys.,14, 263 (1959/60). He found that in systems with the responseg(t) satisfying\(\int\limits_{ - \infty }^{ + \infty } {K\left( {t - t'} \right)} f\left( {t'} \right)dt' = 0\) fort−t′<T, i.e. which have « finitely acausal » kernelsK(τ)=0 forτ<T, dispersion relations can still be derived.

    Article  MathSciNet  Google Scholar 

  23. SeeD. Iagolnitzer andH. P. Stapp:Comm. Math. Phys.,14, 15 (1969);C. Chandler andH. P. Stapp:Journ. Math. Phys.,10, 826 (1969).

    Article  ADS  MathSciNet  Google Scholar 

  24. These wave packets have been introduced byR. Omnès:Phys. Rev.,146, 1123 (1966); they spread out in time with the constantγ.

    Article  ADS  Google Scholar 

  25. D. Iagolnitzer andH. P. Stapp:Comm. Math. Phys.,14, 24 (1969).

    Article  ADS  Google Scholar 

  26. D. Iagolnitzer andH. P. Stapp:Comm. Math. Phys.,14, 29 (1969).

    Article  Google Scholar 

  27. A. March:Zeits. Phys.,104, 93, 161 (1936).

    Article  ADS  Google Scholar 

  28. SeeE. C. G. Sudarshan:Fields and Quanta,2, 175 (1971);Proceedings of the Symposium on Basic Questions in Elementary Particle Physics, Dedicated to Werner Heisenberg (Munich, 1971), p. 96.

    Google Scholar 

  29. C. Nelson andE. C. G. Sudarshan:Phys. Rev. D,6, 3651, 3658 (1972);R. J. Moore:Shadow channels in potential theory, Austin preprint CPT 98-AEC 48 (March 1971).

    Article  ADS  Google Scholar 

  30. A. M. Gleeson, R. J. Moore, H. Rechenberg andE. C. G. Sudarshan:Phys. Rev. D,4, 2242 (1971).

    Article  ADS  Google Scholar 

  31. SeeH. Rechenberg:The triangle graph in quantum field theory with shadow states: causality, unitarity and interpretation, Internal report, CPT Austin (August 1971) (unpublished);H. Rechenberg andE. C. G. Sudarshan:Analyticity in quantum field theory. - III:Unitarity, in preparation;J. L. Richard:Formal scattering theory with shadow states, Austin preprint CPT 149-ORO 3992/89.

  32. The first indication of that idea might be found in papers byH. T. Flint:Proc. Roy. Soc., A117, 630 (1927); and later byV. Ambarzumian andD. Iwanenko:Zeits. Phys.,64, 563 (1930);W. Heisenberg:Zeits. Phys.,65, 4 (1930).

    ADS  Google Scholar 

  33. In the literature we find the statement that the present experimental data on pion-nucleon dispersion relations have proved that the usual geometry still holds down to distances of the order 10−17 cm. (See,e.g.,S. J. Lindenbaum:Fields and Quanta,1, 255 (1971).)

    Google Scholar 

  34. A. Einstein:Phys. Zeits.,10, 185 (1909);W. Ritz:Phys. Zeits.,10, 224 (1909);W. Ritz andA. Einstein:Phys. Zeits.,10, 323 (1909).

    Google Scholar 

  35. The first action-at-a-distance theory in modern science is ascribed toI. Newton when he introduced the law of gravitation (1686). A new aspect was brought into the theory byC. F. Gauss who, in a letter toW. Weber, dated March 19, 1845, talked about the propagation of effects with finite velocity. In this century contributions have been made in particular byK. Schwarzschild:Nachr. Kgl. Ges. Wiss. Gött. (1903), p. 128, 132, 245;H. Tetrode:Zeits. Phys.,10, 317 (1922);G. N. Lewis:Proc. Nat. Acad. Sci.,12, 22 (1926);A. D. Fokker:Zeits. Phys.,58, 386 (1929);Physica,9, 33 (1929);12, 145 (1932);R. P. Feynman andJ. A. Wheeler:Rev. Mod. Phys.,17, 157 (1945);21, 425 (1949);A. Schild:Phys. Rev.,131, 2762 (1963);A. Schild andJ. A. Schlosser:Journ. Math. Phys.,6, 1299 (1965);H. Van Dam andE. Wigner:Phys. Rev.,138, B 1576 (1965);E. C. G. Sudarshan:Fields and Quanta,2, 175 (1971).

  36. See the papers ofTetrode, Feynman andWheeler, andSudarshan, quoted above.

  37. From the « essential » nonlocality of shadow state theory we predict an explicit violation of dispersion relations for scattering amplitudes. Different consequences were derived in the literature some time ago for nonlocal scattering byN. G. Van Kampen:Phys. Rev.,89, 1072, 1267 (1953); and byR. Oehme:Phys. Rev.,100, 1503 (1955). The problem was asked in the following way: given a nonlocal potential of finite spherical extensiona, thus that an incident physical particle (signal) can arrive at the center of this potential at a timea/c later than the scattered wave left the region of the potential, what properties does the scattering amplitude have? The answer was that dispersion relations in the energy variableω did not exist for the amplitude itself, but only for exp [iωa]F(ω). Hence the actual scattering amplitudeF(ω) should beanalytic in the finite upper ω-plane. The difference in the nonlocality assumptions is discussed in the paper quoted in footnote (4). The main point is that the nonlocality in shadow state theory leads to a finite minimal « uncertainty of the position » of macroscopic objects, and then analyticity of the scattering amplitudes can only be derived for energies smaller than the inverse of that nonlocality.

    Article  ADS  Google Scholar 

  38. M. Gundzik andE. C. G. Sudarshan:Phys. Rev. D,6, 798 (1972).

    Article  ADS  Google Scholar 

  39. D. I. Blokhintsev andG. I. Kolerov:Nuovo Cimento,34, 163 (1965).

    Article  Google Scholar 

  40. H. Yukawa:Progr. Theor. Phys.,3, 453 (1948);6, 133 (1951);Phys. Rev.,77, 219 (1950);Suppl. Progr. Theor. Phys.,37/38, 512 (1966).

    ADS  Google Scholar 

  41. M. Sekine:Progr. Theor. Phys.,39, 1038 (1968);41, 843 (1969);45, 322 (1971).

    Article  ADS  Google Scholar 

  42. It is well known that the classical result will hold then. (See,e.g.,H. A. Kramers:Quantentheorie des Elektrons und der Strahlung, inHand- und Jahrbuch der chemischen Physik, edited byA. Eucken andK. G. Wolf, Vol.1 (Leipzig, 1938), p. 439.)

  43. T. D. Lee:A relativistic complex pole model with indefinite metric, inQuanta—Essays in Theoretical Physics Dedicated to Gregor Wentzel, edited byP. G. O. Freund, C. J. Goebel andY. Nambu (Chicago, 1970), p. 290. See alsoK. Baumann:Stückweise Analytizität von S-Matrixelementen und Kausalität, Diplomarbeit, München, 1971;K. Baumann andH. P. Dürr:Nuovo Cimento,12 A, 969 (1972).

  44. H. P. Dürr, W. Heisenberg, H. Mitter, S. Schlieder andK. Yamazaki:Zeits. Naturf.,14 a, 441 (1959);H. P. Dürr, W. Heisenberg, H. Yamamoto andK. Yamazaki:Nuovo Cimento,38, 1220 (1965).

    ADS  Google Scholar 

  45. See,e.g.,A. Salam:Fields and Quanta,1, 1 (1970); for speculations in a slightly different direction see alsoH. Rechenberg:Phys. Blätter,28, 353 (1972).

    ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Max-Planck-Institut für Physik und Astrophysik, München

    H. Rechenberg

  2. Center for Particle Theory, The University of Texas at Austin, Austin, Tex.

    H. Rechenberg

  3. Center for Particle Theory, The University of Texas at Austin, Austin, Tex.

    E. C. G. Sudarshan

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  1. H. Rechenberg
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Work supported in part by AEC under Contract No. AT(40-1)3992.

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Rechenberg, H., Sudarshan, E.C.G. Analyticity in quantum field theory. Nuov Cim A 14, 299–322 (1973). https://doi.org/10.1007/BF02728955

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