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Magnetic charge and hyperfine splitting in hydrogen
Volume 63B, number 1
PHYI;ICS LETTERS
5 July 1976
• M A G N E T I C C H A R G E A N D H Y P E R F I N E S P L I T T I N G IN H Y D R O G E N A.O. BARUT Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Received 23 March 1976 The large spatial size of magnetic charges (or dyons) and parity considerations are used to show that in models of the proton based on dyons the hyperfine splitting must be due to the magnetic moment produced by the electric charge. The contributions of the magnetic monopoles cancel, otherwise the proton would have a large electric dipole moment.
In a recent letter, Opat [1 ] reconsiders the problem of hyperfine splitting in the determination of the possible magnetic charge content of the proton [2]. It is correct [2, 3] that clasicaUy the magnetic field at the center of the proton is, if it were produced by a pair of magnetic charges (+g) and ( - g ) , one half in magnitude and the negative of the field produced by a magnetic dipole. Hence, Fermi theory of hyperfine splitting would give AE = - { (1 - ~fi)(pl.Pj)l@(0)l 2 , where 5 is the fraction of the proton magnetic moment due to the magnetic charges, Pl and pj the nucleonic and electronic magnetic moments, respectively. However, there are two important effects, not considered by Opat, which preclude the determination of the magnetic charge content of the proton by this method [I ]. In fact the contribution of the magnetic charges to the magnetic field of the proton must cancel, leaving the contribution of the electric charge alone. Firstly, because spatial size considerations are involved in the above considerations, we must take into account the fact that the spatial size r m that we can attribute to a magnetic charge or dyon must be very large [3, 4], namely r m = g2/m = 1/4c~rn, where m is the mass of the magnetic charge g. The Bohr-radius a m of the bound state of two oppositely charged dyons, on the other hand, is extremely small, a m = 4~/mre d. Consequently the two dyons in the bound state almost completely overlap. In other words, the size of the bound state (e.g. proton) is the same as the classical radius of the magnetic charge. Hence the magnetic fields inside the bound state due to magnetic charges must cancel to a degree am/rm~ ~2 ~ 10-5 (which is of the
order of experimental value o f f in ref. [1 ]). In addition and I think more importantly, in a model of the proton (which is an eigenstate of parity), and of strong interactions (which conserve parity), based on magnetic charges, it is necessary to construct parity eigenstates from the magnetic charges, because a state of magnetic charges, because a state of magnetic charge Ig) is not an eigenstate of parity: Pig) = I-g}. This can be done [5], by using the superposition of states Ig} + I-g}. Indeed, group theoretical construction of proton states which are parity eigenstates lead to such combinations. In these models, the observed dipole form factor of the proton can be accounted only for these parity eigenstates [6]. If this is so, than the magnetic field produced by the magnetic charges must not only to a large degree, but exactly cancel. In fact only quantities quadratic in the magnetic charge g survive. If this cancellation did not occur then the bound state would have a very large electric dipole moment d due to rotating magnetic charges: d ~ er m. We conclude that the magnetic field of the proton, even in the models based on the magnetic charge, must be duetto the rotating electric charge, i.e. to the electric part of the dyon. More detailed theories [7] of the hyperfine splitting do indeed yield a size to the magnetic moment distribution of the proton of the order o f r m ~ 2.5 ( I / M , ) 0.5 fm, about the same size as derived from the form factor of the proton measured in e-p scattering. This size when equated to the size of the dyon given above implies a dyon mass of about 10Mp.
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Volume 63B, number 1
PHYSICS LETTERS
References [1] G.I. Opat, Phys. Letters 60B (1976) 205. [2] This problem was considered a long time ago by W.A. Nirenberg, Annual Review of Nuclear Science, Vol. 7, p. 353 ff. (1957). [3] A.O. Barut, in Topics in Modern Physics, ed. W.E. Brittin and H. Odabasi (Colorado Assoc. Univ. Press, Boulder, 1971) p. 15.
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5 July 1976
[4] C.J. Goebel, in Quanta, eds. P.G.O. Greund, C.J. Goebel, and Y. Nambu (Univ. of Chicago Press, 1970) p. 338. [51 A.O. Barut, Phys. Letters 38B (1972) 97; 46B (1973) 81. [6] A.O. Barut, Phys. Rev. D3 (1971) 1747; Proc. Coral Gables Conference 1970: Fundamental interactions at high energy eds. A. Perlmutter et al. (Gordon & Breach, 1970). [7] W.M. Moellering, A.C. Zemach and F.E. Low, Phys. Rev. 100 (1955) 441.
PHYI;ICS LETTERS
5 July 1976
• M A G N E T I C C H A R G E A N D H Y P E R F I N E S P L I T T I N G IN H Y D R O G E N A.O. BARUT Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Received 23 March 1976 The large spatial size of magnetic charges (or dyons) and parity considerations are used to show that in models of the proton based on dyons the hyperfine splitting must be due to the magnetic moment produced by the electric charge. The contributions of the magnetic monopoles cancel, otherwise the proton would have a large electric dipole moment.
In a recent letter, Opat [1 ] reconsiders the problem of hyperfine splitting in the determination of the possible magnetic charge content of the proton [2]. It is correct [2, 3] that clasicaUy the magnetic field at the center of the proton is, if it were produced by a pair of magnetic charges (+g) and ( - g ) , one half in magnitude and the negative of the field produced by a magnetic dipole. Hence, Fermi theory of hyperfine splitting would give AE = - { (1 - ~fi)(pl.Pj)l@(0)l 2 , where 5 is the fraction of the proton magnetic moment due to the magnetic charges, Pl and pj the nucleonic and electronic magnetic moments, respectively. However, there are two important effects, not considered by Opat, which preclude the determination of the magnetic charge content of the proton by this method [I ]. In fact the contribution of the magnetic charges to the magnetic field of the proton must cancel, leaving the contribution of the electric charge alone. Firstly, because spatial size considerations are involved in the above considerations, we must take into account the fact that the spatial size r m that we can attribute to a magnetic charge or dyon must be very large [3, 4], namely r m = g2/m = 1/4c~rn, where m is the mass of the magnetic charge g. The Bohr-radius a m of the bound state of two oppositely charged dyons, on the other hand, is extremely small, a m = 4~/mre d. Consequently the two dyons in the bound state almost completely overlap. In other words, the size of the bound state (e.g. proton) is the same as the classical radius of the magnetic charge. Hence the magnetic fields inside the bound state due to magnetic charges must cancel to a degree am/rm~ ~2 ~ 10-5 (which is of the
order of experimental value o f f in ref. [1 ]). In addition and I think more importantly, in a model of the proton (which is an eigenstate of parity), and of strong interactions (which conserve parity), based on magnetic charges, it is necessary to construct parity eigenstates from the magnetic charges, because a state of magnetic charges, because a state of magnetic charge Ig) is not an eigenstate of parity: Pig) = I-g}. This can be done [5], by using the superposition of states Ig} + I-g}. Indeed, group theoretical construction of proton states which are parity eigenstates lead to such combinations. In these models, the observed dipole form factor of the proton can be accounted only for these parity eigenstates [6]. If this is so, than the magnetic field produced by the magnetic charges must not only to a large degree, but exactly cancel. In fact only quantities quadratic in the magnetic charge g survive. If this cancellation did not occur then the bound state would have a very large electric dipole moment d due to rotating magnetic charges: d ~ er m. We conclude that the magnetic field of the proton, even in the models based on the magnetic charge, must be duetto the rotating electric charge, i.e. to the electric part of the dyon. More detailed theories [7] of the hyperfine splitting do indeed yield a size to the magnetic moment distribution of the proton of the order o f r m ~ 2.5 ( I / M , ) 0.5 fm, about the same size as derived from the form factor of the proton measured in e-p scattering. This size when equated to the size of the dyon given above implies a dyon mass of about 10Mp.
73
Volume 63B, number 1
PHYSICS LETTERS
References [1] G.I. Opat, Phys. Letters 60B (1976) 205. [2] This problem was considered a long time ago by W.A. Nirenberg, Annual Review of Nuclear Science, Vol. 7, p. 353 ff. (1957). [3] A.O. Barut, in Topics in Modern Physics, ed. W.E. Brittin and H. Odabasi (Colorado Assoc. Univ. Press, Boulder, 1971) p. 15.
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5 July 1976
[4] C.J. Goebel, in Quanta, eds. P.G.O. Greund, C.J. Goebel, and Y. Nambu (Univ. of Chicago Press, 1970) p. 338. [51 A.O. Barut, Phys. Letters 38B (1972) 97; 46B (1973) 81. [6] A.O. Barut, Phys. Rev. D3 (1971) 1747; Proc. Coral Gables Conference 1970: Fundamental interactions at high energy eds. A. Perlmutter et al. (Gordon & Breach, 1970). [7] W.M. Moellering, A.C. Zemach and F.E. Low, Phys. Rev. 100 (1955) 441.