Applications of Field Theory to Statistical Mechanics by L. Garrido

By L. Garrido

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55) that p2 = 0. This is a secondary constraint. 53), as expected. 55) to the tertiary constraint p3 = 0. 53). Hence in this example, the primary, secondary and tertiary constraints yield the full set of Lagrange equations of motion, and the Lagrange multiplier v in the Hamilton equations of motion remains undetermined. 6 We emphasize that, although the secondary constraints are hidden in the Hamilton equations of motion, which include in principle only the primary constraints, it will be important to make all the constraints manifest before embarking on the quantization of the theory.

In fact φ(1,1) + φ(0,1) ≡ 0 . This leads to a new gauge identity, (0,1) u(1,1) · E (1) + u1 (1) (1) (0) · E (0) = E1 + E4 + E2 ≡0. 35) by arbitrary functions α(t) and β(t), respectively, (1) (0) (1) (0) d and noting that E1 = E1 and E4 = dt E2 one is then led to the identity (0) (0) (0) βE1 + (β − β˙ + α)E2 + αE3 + d (0) (βE2 ) ≡ 0 . 23), we conclude that the following transformations leave the action invariant: δx1 = β, δx2 = β − β˙ + α, δx3 = α . 36) Generator of gauge transformations and Noether identities In the previous section we have related local symmetries of a Lagrangian to gauge (or Noether) identities.

22) For later purposes we recall some fundamental properties of Poisson brackets: i) Antisymmetry {f, g} = −{g, f } ii) Linearity {f1 + f2 , g} = {f1 , g} + {f2 , g} iii) Associative product law {f1 f2 , g} = f1 {f2 , g} + {f1 , g}f2 iv) Jacobi-identity (bosonic) {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 . 23) From these properties it follows that for any function of the phase space variables F (q, p), F˙ ≈ {F, HT } . 2 Alternative derivation of the Hamilton equations Before proceeding with our Hamiltonian analysis of singular systems, it is instructive to rederive the above equations of motion by starting from a nonsingular Lagrangian depending on one or more parameters, for which the Hamilton equations of motion are the standard ones, and which reduces to the singular Lagrangian of interest by taking an appropriate limit [Rothe 2003a].

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