Scaling Transformation for Nonlocal Interactions
Author(s)
Hai-Jun Wang*
Affiliation(s)
Center for Theoretical Physics and School of Physics, Jilin University, Changchun, China.
Center for Theoretical Physics and School of Physics, Jilin University, Changchun, China.
ABSTRACT
In the light of their relationships with renormalization, in this paper we associate the scaling transformation with nonlocal interactions. On one hand, the association leads us to interpret the nonlocality with locally symmetric method. On the other hand, we find that the nonlocal interaction between hadrons could be test ground for scaling transformation if ascribing the running effects in renormalization to scaling transformation. The nonlocal interaction Lagrangian turns out to vary under scaling transformation, analogous to running cases in renormalization. And the total Lagrangian becomes scale invariant only under some extreme conditions. The conservation law of this extreme Lagrangian is discussed and a contribution named scalum appears to the spin angular momentum. Finally a mechanism is designed to test the scaling effect on nonlocal interaction.
In the light of their relationships with renormalization, in this paper we associate the scaling transformation with nonlocal interactions. On one hand, the association leads us to interpret the nonlocality with locally symmetric method. On the other hand, we find that the nonlocal interaction between hadrons could be test ground for scaling transformation if ascribing the running effects in renormalization to scaling transformation. The nonlocal interaction Lagrangian turns out to vary under scaling transformation, analogous to running cases in renormalization. And the total Lagrangian becomes scale invariant only under some extreme conditions. The conservation law of this extreme Lagrangian is discussed and a contribution named scalum appears to the spin angular momentum. Finally a mechanism is designed to test the scaling effect on nonlocal interaction.
KEYWORDS
Scaling Transformation, Renormalization, Conservation Law, Spin Crisis, Nonlocal Scattering
Scaling Transformation, Renormalization, Conservation Law, Spin Crisis, Nonlocal Scattering
Cite this paper
Wang, H. (2015) Scaling Transformation for Nonlocal Interactions. Journal of Modern Physics, 6, 670-690. doi: 10.4236/jmp.2015.65073.
Wang, H. (2015) Scaling Transformation for Nonlocal Interactions. Journal of Modern Physics, 6, 670-690. doi: 10.4236/jmp.2015.65073.
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http://dx.doi.org/10.1103/RevModPhys.84.449 [4] Timofeyuk, N.K. and Johnson, R.C. (2013) Physical Review Letters, 110, Article ID: 112501.
http://dx.doi.org/10.1103/PhysRevLett.110.112501 [5] Eliezer, D.A. and Woodard, R.P. (1989) Nuclear Physics B, 325, 389-469.
http://dx.doi.org/10.1016/0550-3213(89)90461-6 [6] Moffat, J.W. (1989) Physical Review D, 39, 3654-3671.
http://dx.doi.org/10.1103/PhysRevD.39.3654 [7] Evens, D., Moffat, J.W., Kleppe, G. and Woodard, R.P. (1991) Physical Review D, 43, 499-519.
http://dx.doi.org/10.1103/PhysRevD.43.499 [8] Kleppe, G. and Woodard, R.P. (1992) Nuclear Physics B, 388, 81-112.
http://dx.doi.org/10.1016/0550-3213(92)90546-N [9] Moffat, J.W. (1990) Physical Review D, 41, 1177-1184.
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http://dx.doi.org/10.1103/PhysRevLett.105.211601 [11] Campo, D. (2010) arXiv:1004.5324.
http://arxiv.org/abs/1004.5324 [12] Bruuning, E. and Nagamachi, S. (2004) Journal of Mathematical Physics, 45, 2199.
http://dx.doi.org/10.1063/1.1737055 [13] Haque, A. and Joglekar, S.D. (2009) Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 065401.
http://dx.doi.org/10.1088/1751-8113/42/6/065401 [14] Dowker, J.S. (2009) arXiv:0902.4180.
http://arxiv.org/abs/0902.4180 [15] Barnaby, N. (2011) Nuclear Physics B, 845, 1-29.
http://dx.doi.org/10.1016/j.nuclphysb.2010.11.016 [16] Calcagni, G., Montobbio, M. and Nardelli, G. (2008) Physics Letters B, 662, 285-289.
http://dx.doi.org/10.1016/j.physletb.2008.03.024 [17] Hofman, D.M. and Strominger, A. (2011) Physical Review Letters, 107, Article ID: 161601.
http://dx.doi.org/10.1103/PhysRevLett.107.161601 [18] Moffat, J.W. (2011) arXiv:1104.5706.
http://arxiv.org/abs/1104.5706 [19] Zhou, L.-J., Wu, Q. and Ma, W.-X. (2008) Communications in Theoretical Physics, 50, 161-174.
http://dx.doi.org/10.1088/0253-6102/50/1/33 [20] Pimikov, A.V., Bakulev, A.P. and Stefanis, N.G. (2010) Nuclear Physics B—Proceedings Supplements, 198, 182-185.
Also at arXiv:0909.4624v1 [hep-ph]. [21] Wataghin, G. (1934) Zeitschrift für Physik, 88, 92-98.
http://dx.doi.org/10.1007/BF01352311 [22] Yukawa, H. (1950) Physical Review, 80, 1047-1052. (ibid, 77, 849 (1950); ibid 76, 300 (1949)) [23] Yennie, D.R. (1950) Physical Review, 80, 1053-1061.
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http://dx.doi.org/10.1007/BF02815288 [27] Hayashi, C. (1953) Progress of Theoretical Physics, 10, 533-548. ibid, 11, 226 (1954). [28] Lee, T.D. and Yang, C.N. (1957) Physical Review, 108, 1611-1614.
http://dx.doi.org/10.1103/PhysRev.108.1611 [29] Dirac, P.A.M. (1962) Proceedings of the Royal Society A, 268, 57-67.
http://dx.doi.org/10.1098/rspa.1962.0124 [30] Kirzhnits, D.A. (1967) Soviet Physics Uspekhi, 9, 692-700. Translated from Uspekhi Fizicheskikh Nauk, 90, 129-142. (Sep. 1966). [31] Efimov, G.V. (1967) Communications in Mathematical Physics, 5, 42-56.
http://dx.doi.org/10.1007/BF01646357 [32] Efimov, G.V. (1972) Annals of Physics, 71, 466-485.
http://dx.doi.org/10.1016/0003-4916(72)90127-3 [33] Alebastrov, V.A. and Efimov, G.V. (1973) Communications in Mathematical Physics, 31, 1-24.
http://dx.doi.org/10.1007/BF01645588 [34] Alebastrov, V.A., Efimov, G.V. and Seltzer, S.Z. (1973) Annals of Physics, 76, 251-280.
http://dx.doi.org/10.1016/0003-4916(73)90450-8 [35] Bloch, C. and Dan, K. (1950) Vidensk. Selsk. Mat.-Fys. Medd, 26. [36] Marnelius, R. (1974) Physical Review D, 10, 3411. (Equation (2.3) and Equation (2.8); ibid, 8, 2472 (1973)) [37] Yang, C.N. and Feldman, D. (1950) Physical Review, 79, 972-978.
http://dx.doi.org/10.1103/PhysRev.79.972 [38] Kleppe, G. and Woodard, R.P. (1992) Nuclear Physics B, 388, 81-112.
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