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Translational symmetries of quadratic Lagrangians
International Journal of Modern Physics A ( IF 1.6 ) Pub Date : 2021-05-18 , DOI: 10.1142/s0217751x21500913
Andrea Barducci 1 , Roberto Casalbuoni 1
Affiliation  

In this paper, we show that a quadratic Lagrangian, with no constraints, containing ordinary time derivatives up to the order m of N dynamical variables, has 2mN symmetries consisting in the translation of the variables with solutions of the equations of motion. We construct explicitly the generators of these transformations and prove that they satisfy the Heisenberg algebra. We also analyze other specific cases which are not included in our previous statement: the Klein–Gordon Lagrangian, N Fermi oscillators and the Dirac Lagrangian. In the first case, the system is described by an equation involving partial derivatives, the second case is described by Grassmann variables and the third shows both features. Furthermore, the Fermi oscillator and the Dirac field Lagrangians lead to second class constraints. We prove that also in these last two cases there are translational symmetries and we construct the algebra of the generators. For the Klein–Gordon case we find a continuum version of the Heisenberg algebra, whereas in the other cases, the Grassmann generators satisfy, after quantization, the algebra of the Fermi creation and annihilation operators.

中文翻译:

二次拉格朗日量的平移对称性

在本文中,我们证明了一个没有约束的二次拉格朗日函数,包含高达 1 阶的普通时间导数ñ动态变量,有2ñ对称性在于用运动方程的解来转换变量。我们明确地构造了这些变换的生成器,并证明它们满足海森堡代数。我们还分析了我们之前的陈述中没有包括的其他具体案例:克莱因-戈登拉格朗日,ñ费米振荡器和狄拉克拉格朗日。在第一种情况下,系统由包含偏导数的方程描述,第二种情况由格拉斯曼变量描述,第三种情况显示了这两种特征。此外,费米振子和狄拉克场拉格朗日量导致二类约束。我们证明了在最后两种情况下也存在平移对称性,并且我们构造了生成器的代数。对于 Klein-Gordon 情况,我们找到了海森堡代数的连续版本,而在其他情况下,Grassmann 发生器在量化后满足费米创生和湮灭算子的代数。
更新日期:2021-05-18
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