New method of applying conformal group to quantum fields

  • Most of previous work on applying the conformal group to quantum fields has emphasized its invariant aspects, whereas in this paper we find that the conformal group can give us running quantum fields, with some constants, vertex and Green functions running, compatible with the scaling properties of renormalization group method (RGM). We start with the renormalization group equation (RGE), in which the differential operator happens to be a generator of the conformal group, named dilatation operator. In addition we link the operator/spatial representation and unitary/spinor representation of the conformal group by inquiring a conformal-invariant interaction vertex mimicking the similar process of Lorentz transformation applied to Dirac equation. By this kind of application, we find out that quite a few interaction vertices are separately invariant under certain transformations (generators) of the conformal group. The significance of these transformations and vertices is explained. Using a particular generator of the conformal group, we suggest a new equation analogous to RGE which may lead a system to evolve from asymptotic regime to nonperturbative regime, in contrast to the effect of the conventional RGE from nonperturbative regime to asymptotic regime.
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  • [1] Budini P. Czechoslovak Journal of Physics B, 1979, 29: 6[2] LIU Yu-Fen, MA Zhong-Qi, HOU Bo-Yuan. Commun. Theor. Phys., 1999, 31: 481[3] Bateman H. Proc. London Math. Soc., 1910, 8: 223[4] Cunningham E. Proc. London Math. Soc., 1910, 8: 77[5] Polchinski J. Nucl. Phys. B, 1988, 303: 226[6] Nakayama Y. arXiv:1302.0884 [hep-th][7] Gross D J, Wess J. Phys. Rev. D, 1970, 2: 753[8] Kastrup H A. Annalen der Physik, 2008,17: 631; arXiv:0808.2730 [physics.hist-ph][9] Dirac P A M. Annals of Mathematics, 1935, 37: 429[10] Lüscher M, Mack G, Commun. Math. Phys., 1975, 41: 203; Mack G. Commun. math. Phys., 1977, 55: 1[11] Mack G, Salam A. Ann. Phys., 1969, 53: 174[12] Ryder L H. J. Phys. A: Math. Nucl. Gen., 1974, 7: 1817[13] Wheeler J A, Feynman R P. Rev. Mod. Phys., 1945, 17: 157; 1949, 21: 425[14] SU Jun-Chen, YI Xue-Xi, CAO Ying-Hui. J. Phys. G: Nucl. Part. Phys., 1999, 25: 2325[15] SU Jun-Chen, WANG Hai-Jun. Phys. Rev. C, 2004, 70: 044003[16] Curtis G. Callan, Jr. Phys. Rev. D, 1970, 2: 1541[17] Symanzik K. Comm. Math. Phys., 1970, 18: 227[18] Budini P, Furlan P, Raczka R. IL Nuovo Cimento A 52, 191 (21 Luglio 1979).[19] Cartan E. The Theory of Spinors, Dover Publications, Inc. 1981. This is a republication of the first version published by Hermann, Paris. 1966[20] Esteve A, Sona P G. IL Nuovo Cimento XXXII, 1964, 473[21] WANG Hai-Jun. J. Math. Phys., 2008, 49: 053508[22] Feynman R P. Quantum Electrodynamics-A Lecture Note and Preprint Volume, W. A. Benjamin, Inc. 1962. 44, please refer to the Tenth Lecture on Equivalence Transformation[23] Mandl F, Shaw G. Quantum Field Theory, 2nd Edition (Jhon Wiley and Sons Ltd., 2010). page 93, Section 2.4 and Appendix A.7[24] Francesco P, Mathieu P, Senechal D. Conformal Field Theory, Springer-Verlag New York, Inc. 1997[25] Jack I, Osborn H. Nuclear Physics B, 2014, 883: 425C500[26] Kusafuka Y, Terao H. Phys. Rev. D, 2011, 84: 125006[27] Baume F, Keren-Zur B, Rattazzi R et al. JHEP, 2014, 08: 152[28] Osborn H. Nuclear Physics B, 1991, 363: 486
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HAN Lei and WANG Hai-Jun. New method of applying conformal group to quantum fields[J]. Chinese Physics C, 2015, 39(9): 093102. doi: 10.1088/1674-1137/39/9/093102
HAN Lei and WANG Hai-Jun. New method of applying conformal group to quantum fields[J]. Chinese Physics C, 2015, 39(9): 093102.  doi: 10.1088/1674-1137/39/9/093102 shu
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Received: 2014-12-04
Revised: 2015-04-22
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New method of applying conformal group to quantum fields

Abstract: Most of previous work on applying the conformal group to quantum fields has emphasized its invariant aspects, whereas in this paper we find that the conformal group can give us running quantum fields, with some constants, vertex and Green functions running, compatible with the scaling properties of renormalization group method (RGM). We start with the renormalization group equation (RGE), in which the differential operator happens to be a generator of the conformal group, named dilatation operator. In addition we link the operator/spatial representation and unitary/spinor representation of the conformal group by inquiring a conformal-invariant interaction vertex mimicking the similar process of Lorentz transformation applied to Dirac equation. By this kind of application, we find out that quite a few interaction vertices are separately invariant under certain transformations (generators) of the conformal group. The significance of these transformations and vertices is explained. Using a particular generator of the conformal group, we suggest a new equation analogous to RGE which may lead a system to evolve from asymptotic regime to nonperturbative regime, in contrast to the effect of the conventional RGE from nonperturbative regime to asymptotic regime.

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