Summary
From the unitary representations of the Lorentz transformations two four-vector operators,u μ andU μ, are constructed that satisfy the equationsu μ u μ = −1 andU μ U μ=+1 as operator relations, thus behaving like a timelike and a spacelike velocity, respectively. With the help of one of the Casimir operators (K) of the Lorentz group they are represented by difference operators. Eigenvalues and eigenfunctions for the componentsu 4 andU 4 are derived for spin 0 and 1. In contrast to the case of spin 1/2, in which they could be worked out only in approximation, the solutions for the integral spin values can be given in closed form. At the end, a possible use in particle physics of theK-representation is exhibited by the discussion of a linearized infinite-component wave equation.
Riassunto
Dalle rappresentazioni unitarie delle trasformazioni di Lorentz si costruiscono due operatori quadrivettoriali,u μ eU μ, che soddisfano le equazioniu μ u μ=−1 eU μ U μ=+1 come relazioni operatoriali, comportandosi cosí, rispettivamente, come velocità di tipo tempo e di tipo spazio. Con l'aiuto di uno degli operatori di Casimir (K) del gruppo di Lorentz, essi sono rappresentati da operatori di differenza. Si derivano autovalori e autofunzioni per i componentiu 4 eU 4 per spin 0 e 1. Contrariamente al caso con spin 1/2, nel quale possono essere ottenuti solo approssimativamente, si possono dare le soluzioni per valori di spin interi in forma chiusa. Alla fine, si mostra un possibile uso nella fisica delle particelle della rappresentazioneK mediante la discussione di una equazione d'onda linearizzata a infiniti componenti.
Резюме
Из унитарных представлений Лорентц-преобразований конструируются два четырех-векторных оператораu μ иU μ, которые удовлетворяют уравнениямu μ u μ=−1 иU μ U μ=+1, как операторным соотношениям, и которые ведут себя соответственно как времениподобная и пространственноподобная скорости. С помощью одного из операторов Казимира (K) группы Лорентца эти операторы представляются с помощьу разностных операторов. Для спина 0 и 1 выводятся собственные значения и собственные функции для компонентu 4 иU 4. В противоположность случаю спина половина, где можно получить только приближенные решения, для случая целого спина можно вывести решения в замкнутой форме. В заключение, рассматривается возможное использованиеK-представления в физике частиц.
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References
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Postal address: Am Lechblick 3, 8911 Fuchstal, W. Germany.
Traduzione a cura della Redazione.
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Wessel, W. Four-velocities in quantum mechanics and theK-space. Nuov Cim B 44, 337–354 (1978). https://doi.org/10.1007/BF02726797
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DOI: https://doi.org/10.1007/BF02726797