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I am currently reading the book "A First Course in Loop Quantum Gravity" by Gambini and Pullin. On page 55, they claim that the vanishing of the Poisson bracket between the smeared Gauss law, which is a constraint, and the Hamiltonian means that the "orbits" generated by this constraint leave the theory invariant. I am not sure what these "orbits" are or what the theory is invariant under. By my understanding, reading from other sources, their claim means that there is a curve $(E(\lambda),A(\lambda))$ in phase space given by

$$\frac{dE^a}{d\lambda}=\{G(\lambda),E^a\}$$

$$\frac{dA_a}{d\lambda}=\{G(\lambda),A_a(x)\}$$

such that anywhere along this curve the theory should have the same formulation and give the same physical predictions. This curve would be one of the "orbits" they talk about and there would be many orbits since the solutions to the differential equations above may not be unique.

Is my interpretation correct? And if not, what is the right one?

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    $\begingroup$ I agree with what you said: The Poisson brackets as Lie-brackets should generate curves (orbits) on your manifold. And since H commutes with the constrain, H stays invariant in this curve $\endgroup$
    – Confuse-ray30
    Commented Jun 11 at 9:27

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