From Special Relativity to Feynman Diagrams.pdf

Let us apply the same argument of invariance to the

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conservation laws for energy, linear momentum and angular momentum. Let us apply the same argument of invariance to the Hamiltonian of a dynamic system: A canonical transformation is an invariance of the system if it leaves the Hamilton equations of motion invariant in form, and this is the case if the functional dependence of the Hamiltonian on the old and new canonical variables is the same 13 H ( p , q , t ) = H ( p , q , t ). (8.99) If we consider infinitesimal canonical transformations ( 8.89 ), ( 8.90 ), ( 8.91 ), p , q differ from p , q by infinitesimals δ p , δ q , so that ( 8.99 ) amounts to requiring: δ H = H ( p , q ) H ( p , q ) = − δθ r { H , G r } = − δθ r G r t , (8.100) where we have used ( 8.94 ) and ( 8.95 ). From Eqs. ( 8.69 ) and ( 8.100 ), being δθ r arbitrary, we conclude that dG r dt = G r t + { G r , H } = 0 , namely that the G r are constants of motion. In particular we see that the infinitesi- mal generators G r of the canonical transformations in the Hamiltonian formalism correspond to the Noether charges Q r of the Lagrangian formalism. As an example, we want to retrieve once again the three conservation laws of linear momentum, angular momentum and energy, in the Hamiltonian formalism. Let us start implementing the condition of invariance of a system of n particles under space translations. The (Cartesian) coordinates and momenta of the particles are denoted, as usual, by x ( k ) and p ( k ) , ( k = 1 , . . . , n ) , respectively. Let us perform the infinitesimal translations x ( k ) x ( k ) = x ( k ) ; | | 1 p ( k ) p ( k ) = p ( k ) , which are supposed to leave the action invariant. To compute the infinitesimal gen- erator of a translation on the k th particle we use ( 8.92 ) and ( 8.93 ): δ x i ( k ) = − j G j p i ( k ) = − i ; δ p i ( k ) = j G j x i ( k ) = 0 . From the above equations it follows that: G j x i ( k ) = 0 ; G j p i ( k ) = δ i j G j ( x ( k ) , p ( k ) ) = k p j ( k ) = P j ( tot ) , 13 In this subsection we use the notations p i , q i instead of P i , Q i .
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8.4 Canonical Transformations and Conserved Quantities 231 where P j ( tot ) is the j th component of the total linear momentum. If the Hamiltonian is invariant under space-translations then the total linear momentum P tot has vanishing Poisson bracket with the Hamiltonian, which in turn implies that it is conserved: d P ( tot ) dt = 0 . By the same token we deduce the conservation of the total angular momentum. Indeed under an infinitesimal rotation we have: x ( k ) x ( k ) = x ( k ) δ θ × x ( k ) , p ( k ) p ( k ) = p ( k ) δ θ × p ( k ) , from which it follows that δ x i ( k ) = − δθ r G r p i ( k ) = − δθ r ir j x j ( k ) = − ( δ θ × x ( k ) ) i , (8.101) δ p i ( k ) = δθ r G r x i ( k ) = − ir j δθ r p j ( k ) = − ( δ θ × p ( k ) ) i . (8.102) From ( 8.101 ) and ( 8.101 ) we obtain: G i = k i jr x j ( k ) p r ( k ) = M i ( tot ) , where M i ( tot ) is the i th component of the total angular momentum. Therefore, if the system is invariant under rotations, the total angular momentum M ( tot ) commutes with the Hamiltonian, implying that it is conserved: d dt M ( tot ) = 0 .
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