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Transformations to a moving coordinate system) this does not hold. 116): st ~ exp ( - ~ "'&01) ~ exp ( + ~ a l) dt", SCI. e. 124) S-1=yoSt yo . This formula will be used repeatedly. 125) but it was not shown that it is really a four-vector. 125) transforms under Lorentz transformations as PIl(X') = clfI,t(X')yOylllfl'(X') = Clflt(X)St yOyIlSIfI(X) = clflt(X)yOS-lyIlSIfI(X) = ca~ IfI t (x) yO Y v IfI(X) = a~r(x). 65). 127) 52 2. The Wave Equation for Spin-l12 Particles This proves in particular that the probability density jO(x) = c g(x) = c IjIt ljI(x) is indeed the time component of a four vector.

From non-relativistic quantum mechanics it is known that the Pauli equation describes particles with spin 112. Hence we can conclude that also the Dirac equation - as the relativistic generalization of the Pauli equation - has to describe spin-II2 particles. This is especially so, since the spin, as an intrinsic property, should be present independently whether the particle moves with relativistic or non-relativistic velocities. e. invariant in form when changing from one inertial frame to another.

10) These anticommutation relations define an algebra for the four matrices. In order to fulfil condition (b) these matrices have to Hermitian: ait = ai, /Jt = /J . 11) Hence the eigenvalues of the matrices ai and /J are real, and since aT = 11 and /J2 = 11 we conclude that the eigenvalues can only be ± 1. This is most easily seen in the eigenrepresentation of the matrices, where the matrices are diagonal. For example, has then the form a; Ai [ ai= o: o 0... A~ 0 0 0 0 ... OA}y J where A ~ (v = 1,2 ...

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an introduction to ringand modules

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