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By Andrew Hanson, Tuilio Regge, Claudio Teitelboim

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16) and Eqs. 83)). Hereafter, the explicit dependence of the canonical variables on T will be dropped if no confusion arises. - S4- N O\V \VC cxarnine thc canonical " energy-monlentU111 tensor" in (1", a) space. i P. ' U)L:! == 1IY' == "':r'1J. ' 1/)1) the 2 ' tensor may be taken as [6 c r = }~ u~ b o~ fI! L V • ~__ p ~ IT'l VtJ. Formally, the canonical tensor o. conserved, IS Since [()c](lb ~ ° is obviously first class, ,ve Ina y add linear conlbinations of the first class constraints (4. I 3) to forn1 the density ,vhich is integrated to gi,'c the first class generators of gauge-like transforn-lations in 7 and cr.

23) and the theory is again Poincare-covariant. Applying Eq. 88) are constants of the motion \vith respect to xO == t. The transfornlation of x P' under the Poincare group is altered in the star hrackets. i}* -- { H { P i ,~t. jt* J == _ ,X i 1,* J ~Jj 0 -1 ij k '\ * ~ i;" 1 ,x)- ==0 { 1\. 83) in the ne\v Lorentz fran1e. D. 85). 85) obeying canonical brackets. F'or the top, the appropriate variables correspond to the Pryce-Newton-Wigner variables (Pryce, 1935: N c\:vton and ",Tigner, 1949) supplemented by the I~uler angles.

1M 2 ell.. --- -loi /-Sij \ -SO£) , - L • CiSoi \\Te conclude that if \ve set at son1e point 't'o, \VC (3. 66) ha ve also rO/ == 0 . l'hen from Eq. 63) \ve know that SOi ~ 0 and so SOi vanishes for all T. 65) is therefore an invariant relation provided \VC restrict ourselves to trajectories obeying Eq. 42). \Ve no\v define Since Vi ~ 0 implies -S1)" ~ we must have Pi ~ 0 , - 43- Fro111 Eq. " choose the gauge pO ~ I) == (~ P rL P1L)~, so that or ty,. == o. 'Thus fro111 Eq. 67), \ve find 'T'he equation of motion of pi IS no\v Consistency then forces so that the gauge choice pO ~ P has fixed one of the arbitrary functions In the Han1iltonian as expected.

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Constrained Hamiltonian Systems by Andrew Hanson, Tuilio Regge, Claudio Teitelboim


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