Summary
The question of second-quantizing the recently derived most general types of proper Poincaré-invariant Schrödinger wave equations (employing locally covariant wave functions without redundant components and describing arbitrary-spin particles) is considered. As a prelude to carrying out this program the associated relativistically invariant scalar products (with respect to which the Poincaré group is unitary) are determined. Then it is shown that from amongst the infinity of Poincaré-invariantc-number wave equations just two survive the test of second quantization consistent with the microcausality criterion—one being suitable for half-integer spins alone and the other for integer spins only. Our analysis establishes that just the two constraints of proper Poincaré invariance and quantizability lead to the emergence ofseparate T, C, P invariance (in the case of free-field equations) as well as the correct spin-statistics connection.
Riassunto
Si studia la questione della seconda quantizzazione dei tipi più generali, dedotti recentemente, di equazioni d’onda di Schrödinger, invarianti secondo Poincaré, proprie (impiegando funzioni d’onda covarianti localmente senza componenti ridondanti e che descrivono particelle con spin arbitrario). Come preludio all’esecuzione di questo programma si determinano i prodotti scalari associati relativisticamente invarianti (rispetto al quale il gruppo di Poincaré è unitario). Poi si mostra che fra l’infinità di equazioni d’onda di numeroc invarianti secondo Poincaré solo due passano l’esame della seconda quantizzazione consistente con il criterio di microcausalità — una adatta solo per spin seminteri e l’altra per spin interi. La nostra analisi stabilisce che proprio i due vincoli di invarianza propria secondo Poincaré e di quantizabilità portano all’emergenza di invarianzaT, C, P separata (nel caso di equazioni del campo libero) ed anche al corretto collegamento spin-statistica.
Реэюме
Рассматривается вопрос вторичного квантования выведенных недавно наиболее обших типов Пуанкаре-инвариан тных волновых уравнений Щредингера (испольэуюших локально ковариантные волновые функции беэ лищних компонент и описываюших частицы с проиэвольным спином). Сначала определяются релятивистски инвариантные скалярные проиэведения (по отнощению к которым группа Пуанкаре является унитарной). Затем покаэывается, что иэ бесконечного множества Пуанкаре-инвариа нтных е-численных волновых уравнений только два выдерживают проверку вторичного квантования, согласуюшегося с критерием микропричинности, причем, одно иэ них удобно только для полуцелых спинов, а другое только для целых спинов. Нащ аналиэ устанавливает, что именно зти два ограничения собственной Пуанкаре-инвариа нтности и квантуемости приводят к появлениюраэдельной T, C, P инвариантности (в случае уравнений свободных полей), а также правильной свяэи между спином и статистикой.
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Jayaraman, J. Invariant scalar products and quantization of general poincaré-invariant wave equations. Nuov Cim A 14, 343–362 (1973). https://doi.org/10.1007/BF02728958
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DOI: https://doi.org/10.1007/BF02728958