Summary
Describing the spin degree of freedom in terms of Grassmann variables, we study the quantization problem of nonrelativistic and relativistic spinning particles interacting with crossed electric and magnetic fields by using the path integral method. The problem presents some difficulties due to the odd-dimensional nature of the phase space for the spin variables. We show how to deal with these difficulties by bringing back all the problems to the study of the quantum mechanics of a single Grassmann variable. Then it is straightforward to perform the path integrals in all the cases considered here, simply by solving the classical equations of motion.
Riassunto
Descrivendo i gradi di libertà di spin in termini di variabili di Grassmann, si studia il problema della quantizzazione per particelle con spin (non relativistiche e relativistiche) interagenti con campi elettromagnetici usando il metodo degli integrali di percorso. Il problema presenta qualche difficoltà poiché lo spazio delle fasi per le variabili di spin è di dimensione dispari. Si mostra come superare queste difficoltà riportando il problema allo studio della meccanica quantistica di una singola variabile di Grassmann. Si può allora effettuare l’integrazione funzionale in tutti i casi considerati semplicemente risolvendo le equazioni classiche di moto.
Резюме
Описывая спиновую степень свободы в терминах переменных Грассмана, мы исследуем проблему квантования нерелятивистских и релятивистских спиновых частиц, взаимодействующих со скрещенными электрическим и магнитным полями, используя метод интегралов по траекториям. Рассматриваемая проблема обладает рядом трудностей, обусловленных нечетным числом измерений фазового пространства для спиновых переменных. Мы показываем, как можно обойти эти трудности, если свести рассмотрение всех проблем к исследованию квантовой механика для одной переменной Грассмана. Затем непосредственно проводится интегрирование по траекториям во всех рассмотренных здесь случаях, просто решая классические уравнения движения.
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Barducci, A., Bordi, F. & Casalbuoni, R. Path integral quantization of spinning particles interacting with crossed external electromagnetic fields. Nuov Cim B 64, 287–315 (1981). https://doi.org/10.1007/BF02903290
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DOI: https://doi.org/10.1007/BF02903290