By Andrew Hanson, Tuilio Regge, Claudio Teitelboim
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Additional info for Constrained Hamiltonian Systems
T. \. RELATIVISTIC SPINNING PARTICLE REVIE\V OF LAGRANGIAN ApPROACH TO Top We next consider the Lagrangian approach to classical relati vistic spinning particles developed by Hanson and Regge (1974). The treatment gi \'en here will be slightly n10re general than that in the original paper, \vhich dealt only with spherical tops. In order to ensure Poincare-inyariance, unphysical degrees of freedon1 are introduced into the Lagrangian. \\lhen \ve in1posc constraints to eliminate the un\'lanted yariahles, \ve must use the Dirac formalism to find a consistent I-Iamiltonian systen1 and a consistent quantum system corresponding to the original Lagrangian.
A2 Q'i' (0) Q'Y (0')). • - 64- Here we ha ve defined the functions p (8) === ~_ ~ 11. 72 a) E: eillO I n=:pO 7l 8 (8) === p (8) 2 U1t) \vith the periodic properties + 2 1t) == P (8) E: (8 + 2 1t) == E: (8) + z ~ (8 + Z 1t) == 0 (8) . 72 b) One may verify that all of the brackets (4. 7 1 ), (4. 64). 71) are also con1patible \vith the constraints (4. 6 9). ') E: (8 - t G) , Eq. 69). Examining Eqs. 2 == 0 It" === for the D-2 independent degrees of freedoll1. 0 \Ve are free to choose the gauge in such a \vay that the theory is as simple as possible.
We make no atten1pt 2 here to resolve ordering problen1s \vhich n1ay arise \:vhen 1\1 (J) is expressed as an operator). 109) is non10ca1; a local equation, \vhich cculd be the starting point for a quantu111 field theory of relativistic tops, I11ight take the form (3. 110) We ccnclude \vith the relnark that [)irac's pr·ocedures can also be carried out for the electrically charged rclati vistic top. I-Io\vever, since this systen1 becolnes exceedingly c0I11plicated and is still ilnperfectly understood, \:ve \vil1 not atten1pt to treat it here (see Hanson and l~egge, 1974).