标题： 关于规范化对辛结构的影响、与CP^1模型耦合的Hopf项以及分数自旋 显示英文标题

标题： On effects of gauging on symplectic structure, the Hopf term coupled to CP^1 model, and fractional spin

摘要： 我们将霍普夫项耦合到相对论性的$CP^1$模型，并在经典水平上进行哈密顿分析。 发现由相空间变量的狄拉克括号集合所确定的模型的辛结构与纯$CP^1$模型的辛结构相同。 该辛结构被证明是从全局SU(2)不变的$S^3$模型继承而来，在规范U(1)子群时没有发生修改，除了出现一个额外的一类约束生成U(1)规范变换外。 然后我们研究了霍普夫项在经典水平上传递的分数自旋问题。 为此，我们通过对称能量-动量张量以及通过诺特定理构造角动量表达式。 对于该模型，两种表达式一致，表明此术语在经典水平上未传递分数自旋——这一结果与文献中的说法不符。 我们提供了一个解释这种差异的论点，并通过考虑固定辐射规范的霍普夫项与$CP^1$模型耦合来证实我们的论点，在该模型中得到了所需的分数自旋，并以孤子数表示。 最后，通过添加 Chern-Simons 项使$CP^1$模型的规范场成为动力学场，该模型不再是一个$CP^1$模型，这一点与其非相对论对应物的情况相同。 还显示此模型揭示了“异常”自旋的存在。 然而，这被表述为系统的总电荷，而不是任何孤子数。 显示英文摘要

摘要： We couple the Hopf term to the relativistic $CP^1$ model and carry out the Hamiltonian analysis at the classical level. The symplectic structure of the model given by the set of Dirac Brackets among the phase space variables is found to be the same as that of the pure $CP^1$ model. This symplectic structure is shown to be inherited from the global SU(2) invariant $S^3$ model, and undergoes no modification upon gauging the U(1) subgroup, except the appearance of an additional first class constraint generating U(1) gauge transformation. We then address the question of fractional spin as imparted by the Hopf term at the classical level. For that we construct the expression of angular momentum through both symmetric energy-momentum tensor as well as through Noether's prescription. Both the expressions agree for the model indicating no fractional spin is imparted by this term at the classical level-a result which is at variance with what has been claimed in the literature. We provide an argument to explain the discrepancy and corroborate our argument by considering a radiation gauge fixed Hopf term coupled to $CP^1$ model, where the desired fractional spin is reproduced and is given in terms of the soliton number. Finally, by making the gauge field of the $CP^1$ model dynamical by adding the Chern-Simons term, the model ceases to be a $CP^1$ model, as is the case with its nonrelativistic counterpart. This model is also shown to reveals the existence of `anomalous' spin. This is however given in terms of the total charge of the system, rather than any soliton number.