By R. Gobel
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Les ? ‰l? ©ments de math? ©matique de Nicolas Bourbaki ont pour objet une pr? ©sentation rigoureuse, syst? ©matique et sans pr? ©requis des math? ©matiques depuis leurs fondements. Ce troisi? ?me quantity du Livre sur les Groupes et alg? ?bres de Lie, neuvi? ?me Livre du trait? ©, poursuit l ? ©tude des alg?
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The Schwinger model is solved by several methods. The bosonization is one of them and we will discuss it below. Also, the Schwinger model has been solved by the Bogoliubov transformation method. In principle, the Bogoliubov transformation method is an approximate scheme for the four fermion Schwinger Boson in Two Dimensional QED 51 interaction models. However, the correct mass of the Schwinger boson is obtained by the Bogoliubov transformation method. Until now, it is not clarified why the Bogoliubov transformation method can give an exact mass for the Schwinger model.
However, this is extremely difficult to do it properly, and indeed the RPA eigenvalue equation is not Hermite, and thus it is not clear whether the effect is taken into account in a better way or worse. The examination and the validity of the RPA equation will be given else where. 25) can be easily obtained, and the boson mass spectrum for the NJL model is shown in Fig. 1. Note that the boson mass is measured in units of the cutoff momentum Λ. As can be seen from the figure, there is a massive boson for some regions of the values of the coupling constant.
320 The energy Evtrue of the true vacuum state can be written as Evtrue = −Λ N0 + 1 + 2(N0 + 1) −1 g tan π 2 . 13) From the distributions of the negative energy particles, one sees that this solution breaks the chiral symmetry. This situation can be easily seen from the analytical solutions since the absolute value of the momentum of the negative g Λ . Therefore, we can define the efenergy particles is higher than tan−1 π 2 fective fermion mass MN by MN = Λ −1 g tan . 14) In Table 1, we show the calculated results of the new vacuum as well as the symmetric vacuum energies as the function of the particle number N.
Abelian Group Theory by R. Gobel