Rovelli's relativistic phase space

Physics Asked by Syr on May 9, 2021

I’m trying to understand the definition of relativistic phase space given by Rovelli in his book Quantum Gravity. At chapter 3 he states those following definitions

  • The relativistic phase space $Gamma$ is the space of relativistic states – p.107;
  • The relativistic phase space $Gamma$ is the space of orbits of $dtheta$ in $Sigma$ – p.102;
  • The relativistic phase space $Gamma$ is the space of physical motions (i.e: the space of all solutions of the equations of motion) -p.102;

relativistic states definition: he define using the pendulum example: we have the pendulum physical motion given by the evolution equation $f(alpha,t)=0$ where $alpha$ is elongation and $t$ is time. So, $(alpha,t)$ is just a point in relativistic configuration space $mathcal{C}$. But, the evolution equation $f$ changes when we disturb the pendulum: so we have two parameters $(A,phi)$ such that $f(alpha,t;A,phi)=alpha-Asin{(omega t+phi)}$ and we define $(A,phi)$ as a state.

orbits definition: we define a degenerated 2-form on $mathbb{R}times T^*mathcal{C}$ such that $Sigma$ is the hypersurface which $(dtheta)(X)=0$ and the orbits of $dtheta$ is the integral curves of such $X$ (called the null vector space).

My question: Actually there isn’t difficult link the last two definitions. My problem is link the first (where $Gamma$ is space of states) with the others (like where $Gamma$ is the space of all physical motions).

Also, I’m looking for more about it, but I can’t find those definitions in any other book. Therefore, I would also be grateful for new references ! Thanks in advance.

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