Abstract
Given a positive-definite operator, such as (m2c4 + P2c2), there is a mathematical theorem that guarantees that there is one, and only one, square root that is also positive definite, denoted by +(m2c4 + P2c2)1/2. Other square roots become possible if we give up positive definiteness. This may appear to spoil the theory by allowing negative energies; but, if the operator is Hermitean, states corresponding to negative energies will be orthogonal to positive-energy states and a sensible physical theory is obtained if we restrict ourselves to the latter. We can, moreover, ensure manifest covariance by looking for an equation not only linear in ∂t but also linear in the space derivatives; that equation, we expect, will describe relativistic spin 1/2 particles, such as the electron1. We then use a multicomponent wave function2, \(\mathop \Psi \limits_ \sim \), and look for an equation linear in the P μ , the Dirac equation,
where the free Dirac Hamiltonian \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{H} _0 \) satisfies
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© 1996 Springer-Verlag Berlin Heidelberg
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Ynduráin, F.J. (1996). Spin 1/2 Particles. In: Relativistic Quantum Mechanics and Introduction to Field Theory. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61057-8_3
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DOI: https://doi.org/10.1007/978-3-642-61057-8_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64674-4
Online ISBN: 978-3-642-61057-8
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