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The classical mechanics for bose-fermi systems

Классическая механика для Боэе и Ферми систем

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Il Nuovo Cimento A (1965-1970)

Summary

In this paper we study in a systematic way the classical mechanics of systems described byc-number variables and by Grassmann variables. We derive the general form of the nonrelativistic action and we study the theory of canonical transformations. For a general action, we show that the Jacobian matrices of the canonical transformations acting onN Grassmann variables form a groupO N, N . This group becomesO N for the nonrelativistic action, due to the presence of second-class constraints. We study some examples which give rise to a correct classical description of the spin. Considering a relativistic extension of one of these models, we get a first quantized «substratum» for the superfield theories.

Riassunto

In questo lavoro si studia in maniera sistematica la meccanica classica di sistemi descritti da variabili numeroc e da variabili di Grassmann. Si deduce la forma generale dell’azione non relativistica e si studia la teoria delle trasformazioni canoniche. Si mostra che, per un’azione generica, le matrici jacobiane delle trasformazioni canoniche agenti suN variabili di Grassmann formano un gruppoO N,N . Questo gruppo si riduce adO N per l’azione non relativistica, a causa della presenza di vincoli di seconda classe. Si studiano alcuni esempi che danno luogo ad una corretta descrizione classica dello spin. Estendendo relativisticamente uno di questi modelli, si ottiene il « substratum » di prima quantizzazione delle teorie di supercampo.

Реэюме

В зтой работе мы исследуем систематическим обраэом классическую механику систем, которые описываются с помошью переменных Грассмана и пере-менных, представляюших с числа. Мы выводим обшую форму для нерелятивист-ского действия и исследуем теорию канонических преобраэований. Для обшего действия мы покаэываем, что Якобианы канонических преобраэований, действуюших на N переменных Грассмана, обраэуют группу О N,N . Эта группа преврашается в О N для нерелятивистского действия вследствие наличия ограничений второго класса. Мы исследуем некоторые примеры, которые приводят к правильному классическому описанию спина. Рассматривая релятивистское обобшение одной иэ зтих моделей, мы получаем первично квантованную «основу» для теорий суперполей

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Author notes

  1. R. Casalbuoni

    Present address: Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Largo E. Fermi 2, 50125, Firenze

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  1. CERN, Geneva

    R. Casalbuoni

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  1. R. Casalbuoni
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Additional information

Work supported in part by the U.S. Energy Research and Development Administration, under contract E(11-1)3285. Report No. COO 3285-27.

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Casalbuoni, R. The classical mechanics for bose-fermi systems. Nuov Cim A 33, 389–431 (1976). https://doi.org/10.1007/BF02729860

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