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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