Cylindrical Group and Massless Particles
It is shown that the representation of the E(2)-like little group for photons can be reduced to the coordinate transformation matrix of the cylindrical group, which describes movement of a point on a cylindrical surface. The cylindrical group is isomorphic to the two-dimensional Euclidean group. As in the case of E(2), the cylindrical group can be regarded as a contraction of the three-dimensional rotation group. It is pointed out that the E(2)-like little group is the Lorentz-boosted O(3)-like little group for massive particles in the infinite-momentum/zeromass limit. This limiting process is shown to be identical to that of the contraction of O (3) to the cylindrical group. Gauge transformations for free massless particles can thus be regarded as Lorentz-boosted rotations.
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- 1.E. Inonu and E. P. Wigner, Proc. Natl. Acad. Sci. USA 39, 510 (1953); J. D. Talman, Special Functions, A Group Theoretical Approach Based on Lectures by E. P. Wigner (Benjamin, New York, 1968 ). See also R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications in Physics ( Wiley, New York, 1974 ).Google Scholar
- 2.E. P. Wigner, Ann. Math. 40, 149 (1939); V. Bargmann and E. P. Wigner, Proc. Natl. Acad. Sci. USA 34, 211 (1946); E. P. Wigner, Z. Phys. 124, 665 (1948); A. S. Wightman, in Dispersion Relations and Elementary Particles, edited by C. De Witt and R. Omnes (Hermann, Paris, 1960); M. Hamer-mesh, Group Theory (Addison-Wesley, Reading, MA, 1962); E. P. Wigner, in Theoretical Physics, edited by A. Salam (IAEA, Vienna, 1962); A. Janner and T. Jenssen, Physics 53, 1 (1971); 60, 292 (1972); J. L. Richard, Nuovo Cimento A 8, 485 (1972); H. P. W. Gottlieb, Proc. R. Soc. London Ser. A 368, 429 (1979); H. van Dam, Y. J. Ng, and L. C. Biedenharn, Phys. Lett. B 158, 227 (1985). For a recent textbook on this subject, see Y. S. Kim and M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, Holland, 1986 ).Google Scholar
- 3.E. P. Wigner, Rev. Mod. Phys. 29, 255 (1957). See also D. W. Robinson, HeIv. Phys. Acta 35, 98 (1962); D. Korff, J. Math. Phys. 5, 869 (1964); S. Weinberg, in Lectures on Particles and Field Theory, Brandeis 1964,edited by S. Deser and K. W. Ford (Prentice-Hall, Englewood Cliffs, NJ, 1965) Vol. 2; S. P. Misra and J. Maharana, Phys. Rev. D 14, 133 (1976); D. Han, Y. S. Kim, and D. Son, J. Math. Phys. 27, 2228 (1986).Google Scholar
- 4.S. Weinberg, Phys. Rev. B 134, 882 (1964); B 135, 1049 (1964); J. Kuperzstych, Nuovo Cimento B 31, 1 (1976); D. Han and Y. S. Kim, Am. J. Phys. 49, 348 (1981); J. J. van der Bìj, H. van Dam, and Y. J. Ng, Physica A 116, 307 (1982); D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 31, 328 (1985).Google Scholar
- 5.D. Han, Y. S. Kim, and D. Son, Phys. Rev. D 26, 3717 (1982). For an earlier effort to study the E(2)-like little group in terms of the cylindrical group, see L. J. Boya and J. A. de Azcarraga, An. R. Soc. Esp. Fis. Quim. A 63, 143 (1967). We are grateful to Professor Azcarraga for bringing this paper to our attention.Google Scholar
- 6.P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949); L. P. Parker and G. M. Schmieg, Am. J. Phys. 38, 218, 1298 (1970); Y. S. Kim and M. E. Noz, J. Math. Phys. 22, 2289 (1981).Google Scholar
- 7.D. Han, Y. S. Kim, and D. Son, Phys. Lett. B 131, 327 (1983); D. Han, Y. S. Kim, M. E. Noz, and D. Son, Am. J. Phys. 52, 1037 (1984). These authors studied the correspondence between the contraction of 0(3) to E(2) and the Lorentz boost of the 0(31-like little group.Google Scholar