Abstract
It is shown that Dirac’s “instant form” dynamics provides a theoretical framework in which models of relativistic quantum mechanics can be constructed. The convariant harmonic oscillator formalism discussed in previous papers is shown to be such a model. Dirac’s “point” and “front” forms are shown to generate a space-time geometry convenient for describing Lorentz deformation properties of relativistic extended hadrons.
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References
Yo S. Kim and M. E. Nez, Am. J. Phys. 46. 484 (1978).
R. P. Feynman, M. Kislinger, and F. Ravndal, Phys. Rev. D3, 2706 (1971). The point of this paper is that the inventor of Feynman diagrams stated that it is not practical, if not impossible, to use Feynman diagrams for relativistic bound-state problems. Feynm an et al. suggested that the relativistic harmonic oscillator model, even if it is not totally consistent, can serve useful purposes. The point of Ref. 1 is that the oscillator model does not have to be imperfect, and therefore that it can be made consistent with the known rules of quantum mechanics and special relativity.
The most successful bound-state model in field theory is of course the Bethe-Salpeter equation. However, the Bethe-Salpeter wave function does not yet have proper quantum-mechanical interpretation. See Sec. I of G. C. Wick, Phys. Rev. 96, 1124 (1954). The difficulty in giving a physical interpretation to the relative time-separation variable between two bound-state particles was mentioned earlier by Karplus and Klein. See R. Karplus and A. Klein, Phys. Rev. 87, 848 (1952).
P. A. M. Dirac, Rev. Mod. Phys. 21, 392(1949).
Y. S. Kim, M. E. Noz, and S. H. Oh, Am. J. Phys. 47, 892 (1979); J. Math. Phys. 20, 1341 (1979).
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This exponential form is also derivable from Yukawa’s work. See Eq. (101ofH. Yukawa, Phys. Rev. 91, 416 (19531. For an interpretation of this original paper, see D. Han and Y. S. Kim, Prog. Theor. Phys. 64, 1852 (1980).
The fact that the proton (one of hadrons) is not a point particle and has aspace-time extension was discovered by Hofstadter. See R. Hofstad ter, Rev. Mod. Phys. 28, 214 (1956).
Since Hofstadter’s discovery, there have been many attempts to construct theoretical models for relativistic extended hadrons. See, for instance, V. N. Gribov, B. L. loffe, and I. Va. Pomeranchuk, J. Nuel. Phys. (USSR) 2, 768(1965) or Sov. J. Nucl. Phys. 2, 549 (1966); N. Byers and C. N. Yang. Phys. Rev. 142. 796 (1966); J. D. Bjorken and E. A Paschos, ibid. 185, 1975 (1969); B L Ioffee. Phys. Lett. B 30, 123 (1969):K. Fujimura, T. Kobayashi. and M. Namiki, Prog. Theor. Phys. 43.73 (19701; A. L. Licht and A. Pagnamenta, Phys. Rev. D 2. 1150. 1156 (1970): S.D. DrellandT.M. Yen.Ann. Phys. (NY)60. 578(1971). Y. S. Kim and R. Zaoui. Phys. Rev. D4, 1764 (1971); R.G. Lipes. ibid. 5.2 849 (1972);S. Ishida and J.Otokozawa. Prog. Theor. Phys. 47. 2117 (1972); T. D. Lee, Phys. Rev.D 5.1 738 (1972);G. Feldman, T. Fulton. and J. Town send, ibid. 7, 1814 (1973). Y. S. Kim and M. E. Nez. ibid. 8, 3521 (1973). See also Refs. I, 2, and the references contained therein.
Perhaps one of the curre nt models of extended hadrons is the “MIT bag model,” as is explain ed by K. John son in Sci. Am. 241 (I I, 112 (July J979). One Interestin g question in th is mode l is how bags” wou ld look to moving observers.
The quark confinement problem is regarded as one of the most imporlant current problems in the particl e th eor y front. The ultimate goal of this program is to find a pot ent ial that confines th e qu arks inside hard ons within the field th eoreti c framewor k of QCD (q uantum chro modyn amics). Th e basic quest ion is then th is. What are we going to do with thi s confining pot ent ial? Th e next step is natu rally to construct bound stale wave functions, which eventu all y leads to th e question of their Lorent z tran sformation prope rties. As was noted in QED (quantum electrody namics];’ this does not as yet appe ar to be an easy problem For an introdu cto ry review art icle on QCD, see W. Marcia no and H. Pagels. Phys. Rep. 36 C. 138 (1978).
We have to say th at th e most import ant observation mad e on Lorent zdeformed hadrons was Feynman ‘s part on model. See R. P. Feynm un. In High Energy Collisions, Proceedings of th e 3rd Int ern ati onal Confe rence, Sto ny Brook, New York, edited by C. N. Yang ct al. (Gordo n and Breach, New York, 19691: Photon-Hadron Interactions Benjamin, Reading, MA, 1972).
For an explanat ion of the peculi ariti es in Feynman’s pat to n pict ure, see Y. S. Kim and M. E. Noz, Phys. Rev. 0 15, 335 (1977l. For a graphi cal interp retat ion of the formulas in this paper, see Y. S. Kim and M. E. Noz, Found. Phys. 9. 375 (1979).
For a calcula tion of th e pr oton str uctu re function, see P. E. Hussar. Phys. Rev. D23. 2781 (1981).
For one of the most recent pape rs on this subject, see A. Kihlberg. R. Marnelius, and N. Mukunda, Phy. Rev. D 23, 2201 (1981).
See. for instance, R. Fong and J. Sucher, J. Math. Phys. 5, 456 (1964).
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Han, D., Kim, Y.S. (1988). Dirac’s form of relativistic quantum mechanics. In: Noz, M.E., Kim, Y.S. (eds) Special Relativity and Quantum Theory. Fundamental Theories of Physics, vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3051-3_22
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DOI: https://doi.org/10.1007/978-94-009-3051-3_22
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