By David Lurie
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Extra resources for Particles And Fields
L) and 0 respectively. ,. ^ 1 t,ru* +,[|tru- 1(200b) 1(200c) VJ lla: t2U* A similar argument shows that ry'o : 0 also for negative energy solutions 1(200d) in the rest frame. Rarita-Schwinger Field 49 S, +io, with eigenvalues +tr, t, - j and -i respectively. The solutions 1(200) are orthogonal, normalized, and satisfy the subsidiary condition 1(198). ,, :0 which are readily seen to be satisfied. The coefficients in 1(200) are just the clebsch-Gordon coefficients for coupling spin 1 and ) to yield spin ].
Indeed, using l(64a), 1(64c) and 1(66), we find : - -CYi, : (YrYtc)' : c'yTy[, : lsTuc : (tuc1' C'Y[ ruC -TrT5c and so on. Accordingly we set t$) : ilty^CA^(x)* t}^,CF^"(x) 1(10e) rvhere A^ and F^n are vector and antisymmetric second-rank tensor fields respectively, and where numerical factors have been introduced for later convenience. We now apply the Bargmann-Wigner equations 1(108) to 1(109). Adding 1(108a) to 1(108b) and using t(64a), we find 0 : iply ", * : y )C A,A t+ ilq r,2 2i 1t2 y A ^C -2pZtnCA + At * ^"C ^u F ^nC u'here we have used the easily derived identity r,2 : 2i6 ^") unT F xn ^C)nF^n 1tZ ^C lT t F 1tD ^+ ^A,+2iy 2i 1t2 y A ^;C t- ^u 2i6 1,tl " Setting the coefficients of yt"and Er, equal tozero, we get the system of coupled equations F^u 0 ^F^, or, in terms of A"(x) by itself, : : A^An- euA^ 1(1 10a) lt2A, 1(110b) ZA"-A"(AtAA) : lr'An 1(111) Equations 1(110a) and 1(110b) or 1(111) are rhe fundamental field equations for a vector field of rest mass,r^r.
If higher order time derivatives of @, appear in L, 2(4) no longer applies, but we shall not need to consider this possibility. functional deriuatiue of Flrp) with respect to the value of E at the point x. From the definition, one easily verifies that the functional derivative satisfies the usual properties associated with differentiation. fua,luu 2(8b) Identifying2(8a) with 2(8b) in the continuum limit we have, since variations at distinct points are independent of one another, 6L(t\ t ALO t) uiilo 6Vi rOiG\ dl,(r) t aL(t) 6Q$,t) 6v;+s \Vt AOtft) 6d$, 2(ea) 2(eb) where x is in the ith cell.