Abstract
In this chapter, in the framework of Dirac quantization, SU(2) Skyrmion is canonically quantized to yield modified predictions of static properties of baryons. We show that the energy spectrum of this Skyrmion obtained by the Dirac quantization method with a suggestion of generalized momenta is consistent with result of the improved Dirac Hamiltonian formalism [42]. We next apply the improved Dirac Hamiltonian method to the SU(2) Skyrmion and directly obtain the first class Hamiltonian. We also find that Poisson brackets of first class physical fields in extended phase space have the same structure as the well-known Dirac brackets. Furthermore, in this improved Dirac Hamiltonian scheme, effects of Weyl ordering correction on a baryon energy spectrum are shown to modify static properties of baryons. On the other hand, following BFV formalism [23, 26, 79–82] we derive a BRST invariant gauge fixed Lagrangian as well as an effective action corresponding to the first class Hamiltonian.
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- 1.
Here, one can easily check that the Skyrmion Lagrangian can be rewritten as \(L = -E + 2\mathcal{I}\vec{\alpha }^{2}\) by defining the new variables \(\alpha ^{k} = a^{0}\dot{a}^{k} -\dot{ a}^{0}a^{k} +\epsilon ^{\mathit{kpq}}a^{p}\dot{a}^{q}\).
- 2.
In Ref. [137], the authors did not include the last term so that one cannot clarify the relations between the improved Dirac scheme and the Dirac bracket one. Also one can easily see that \(\Pi ^{\mu }\) is not the canonical momenta conjugate to the collective coordinates a μ any more since \(\Pi ^{\mu }\) depend on a μ, as expected.
- 3.
Here, the first three terms are nothing but the three-sphere Laplacian [135] given in terms of the collective coordinates and their derivatives to yield the eigenvalues l(l + 2).
- 4.
- 5.
For the delta magnetic moments, we use the experimental data of Nefkens et al.[165].
- 6.
Here, one notes that the Poisson brackets of \(\tilde{\mathcal{F}}\)’s have the same structure as that of the corresponding Dirac brackets [75].
- 7.
Here, we have the modified predictions c = 0. 22 and \(\bar{c} = 0.34\) of the standard rigid rotator without the pion mass since the numerical evaluation for the inertia parameters should be fixed with the values \(\mathcal{I}_{1}^{-1} = 196\,\mathrm{MeV}\), \(\mathcal{I}_{2}^{-1} = 528\,\mathrm{MeV}\), \(\Gamma _{0}^{-1} = 182\,\mathrm{MeV}\) and E = 866 MeV, instead of \(\mathcal{I}_{1}^{-1} = 211\,\mathrm{MeV}\), \(\mathcal{I}_{2}^{-1} = 552\,\mathrm{MeV}\), \(\Gamma _{0}^{-1} = 202\,\mathrm{MeV}\) and E = 862 MeV which yields c = 0. 28 and \(\bar{c} = 0.35\) [126], to be consistent with the parameter fit \(f_{\pi } = 64.5\,\mathrm{MeV},\ e = 5.45\) used in the massless standard SU(2) Skyrmion [158]. One notes also that the bound state approach does not include the quartic terms in the kaon field.
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Hong, ST. (2015). Hamiltonian Quantization and BRST Symmetry of Skyrmion Models. In: BRST Symmetry and de Rham Cohomology. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9750-4_6
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DOI: https://doi.org/10.1007/978-94-017-9750-4_6
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