Abstract
We generalize the theory of Lorentz-covariant distributions to broader classes of functionals including ultradistributions, hyperfunctions, and analytic functionals with a tempered growth. We prove that Lorentz-covariant functionals with essential singularities can be decomposed into polynomial covariants and establish the possibility of the invariant decomposition of their carrier cones. We describe the properties of odd highly singular generalized functions. These results are used to investigate the vacuum expectation values of nonlocal quantum fields with an arbitrary high-energy behavior and to extend the spin–statistics theorem to nonlocal field theory.
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REFERENCES
R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin, New York (1964).
N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, and I. T. Todorov, General Principles of Quantum Field Theory [in Russian], Nauka, Moscow (1987); English transl., Kluwer, Dordrecht (1990).
N. N. Meiman, JETP, 20, 1320 (1965).
A. Jaffe, Phys. Rev., 158, 1454 (1967).
M. Z. Iofa and V. Ya. Fainberg, Teor. Mat. Fiz., 1, 187 (1969).
M. Z. Iofa and V. Ya. Fainberg, JETP, 29, 880 (1969).
V. Ya. Fainberg and A. V. Marshakov, Phys. Lett. B, 211, 82 (1988).
M. Z. Iofa and V. Ya. Fainberg, Nuovo Cimento A, 5, 273 (1971).
V. Ya. Fainberg, “On quantum theories with a nonpolynomial growth of matrix elements [in Russian],” in: Problems in Theoretical Physics (V. I. Ritus, ed.), Nauka, Moscow (1972), p. 119.
V. Ya. Fainberg and M. A. Soloviev, Ann. Phys., 113, 421 (1978).
S. B. Giddings, Phys. Rev. D, 61, 106008 (2000).
G. V. Efimov, Nonlocal Interactions of Quantum Fields [in Russian], Nauka, Moscow (1977).
G. V. Efimov, Problems in the Quantum Theory of Nonlocal Interactions [in Russian], Nauka, Moscow (1985).
J. W. Moffat, “Quantum field theory solution to the gauge hierarchy and cosmological constant problems,” hep-ph/0003171 (2000).
M. A. Solov'ev, Theor. Math. Phys., 7, 458 (1971).
S. Nagamachi and N. Mugibayashi, Commun. Math. Phys., 46, 119 (1976).
S. Nagamachi and N. Mugibayashi, Commun. Math. Phys., 49, 257 (1976).
M. A. Solov'ev, Theor. Math. Phys., 15, 317 (1973).
U. Moschella and F. Strocchi, Lett. Math. Phys., 24, 103 (1992).
M. A. Soloviev, Lett. Math. Phys., 41, 265 (1997).
A. G. Smirnov and M. A. Solov'ev, Theor. Math. Phys., 123, 709 (2000).
M. A. Solov'ev, Trudy Fiz. Inst. Lebedev., 209, 121 (1993).
A. I. Oksak and I. T. Todorov, Commun. Math. Phys., 14, 271 (1969).
M. A. Soloviev, Theor. Math. Phys., 121, 1377 (1999).
I. M. Gelfand and G. E. Shilov, Functions and Generalized Function Spaces, Vol. 2 of Generalized Functions [In Russian], Fizmat, Moscow (1958); English transl., Acad. Press, New York (1968).
V. P. Palamodov, Russ. Math. Surv., 26, 1 (1971).
H. H. Schaefer, Topological Vector Spaces, MacMillan, New York (1966).
I. M. Gelfand and N. Ya. Vilenkin, Applications of Harmonic Analysis, Vol. 4 of Generalized Functions [in Russian] by I. M. Gelfand and G. E. Shilov, Fizmatgiz, Moscow (1961); English transl., Acad. Press, New York (1964).
L. H¨ormander, Distribution Theory and Fourier Analysis, Vol. 1 of The Analysis of Linear Partial Differential Operators, Springer, Berlin (1983).
A. Lambert, Ann. Inst. Fourier, 29, 57 (1979).
P. Schapira, Th´eorie des hyperfonctions (Lect. Notes. Math., Vol. 126), Springer (1970).
T. Kawai, J. Fac. Sci. Univ. Tokyo. Sect. 1A. Math., 17, 467 (1970).
D. A. Raikov, Sib. Math. J., 7, 287 (1966).
A. Grothendieck, Mem. Amer. Math. Soc., 16, 1 (1955).
V. S. Retakh, Sov. Math. Dokl., 11, 1384 (1970).
V. Ya. Fainberg and M. A. Soloviev, Theor. Math. Phys., 93, 1438 (1992).
M. A. Soloviev, Lett. Math. Phys., 33, 49 (1995).
D. P. Zhelobenko, Compact Lie Groups and Their Representations [in Russian], Nauka, Moscow (1970).
A. S. Wightman, Adv. Math. Suppl. Stud., 7B, 769 (1981).
M. A. Soloviev, Commun. Math. Phys., 184, 579 (1997).
V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables [in Russian], Nauka, Moscow (1964); English transl., MIT, Cambridge, Mass. (1966).
G. E. Shilov, Mathematical Analysis: Second Special Course [in Russian], Nauka, Moscow (1965).
N. Bourbaki, Espaces vectoriels topologiques, Vol. 5 of Les structures foundamentales de l'analyse, Hermann, Paris (1955).
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Soloviev, M.A. Lorentz-Covariant Ultradistributions, Hyperfunctions, and Analytic Functionals. Theoretical and Mathematical Physics 128, 1252–1270 (2001). https://doi.org/10.1023/A:1012368004774
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DOI: https://doi.org/10.1023/A:1012368004774