# Covariant phase space The covariant phase space formalism is a nice formalism for studying and deriving relations for theories with diffeomorphism invariance in a covariant way. They are particularly useful for formulating [[0127 Black hole thermodynamics|black hole thermodynamics]] in a theory-independent way; it can be used to obtain explicit expressions for the conserved charges such as the [[0004 Black hole entropy|black hole entropy]]. ## Notations - solution space $\mathcal{S}$ - a point in $\mathcal{S}$: $\phi$ or $\phi^x$ (a function of $x$) - $\alpha\beta$: the exterior product of a $p$-form and a $q$-form ($\alpha\beta=(-1)^{pq}\beta\alpha$) - $I_V$: anti-derivative (i.e. contraction with vector $V$) - manifold $M$ - a map $Y: M\to M$ - pullback: $Y^*$ - diffeomorphism invariance: if a configuration of tensor fields $\phi$ satisfies EOM then ## Features - Hamiltonian vector fields (HVF) - $£_{\mathbf{X}_f} \mathbf\Omega=0$ - i.e. diffeo. that preserves the symplectic form - 1-to-1 correspondence between HVFs and functions on phase space - definition of HVF: $£_{\mathbf{X}_f} \mathbf\Omega=0$ => (using Cartan formula) - $\mathrm{i}_{\mathbf{X}_f} \mathbf\Omega=\mathbf{d} f$ => $\mathbf{X}_{f}^{A}=\mathbf\Omega^{A B} \partial_{B} f$ (using invertability of symplectic form) - $f$ is called the **charge**, which generates the transformation corresponding to $X_f$ ## Poisson bracket - if $X_f$ and $X_g$ are both HVF, immediately see $[X_f,X_g]$ is also a HVF. Since $X_f$ <-> $f$, and $X_g$ <-> g, we can ask what is one-to-one with $[X_f,X_g]$? - define $\{f,g\}$ <-> $[X_f,X_g]$ - explicit formula: $\{f, g\}=\Omega^{A B} \partial_{A} f \partial_{B} g$ - $= -X_f(g)=X_g(f)=-\Omega(X_f,X_g)$ ## Refs - original - [[1990#Lee, Wald]] - [[1993#Wald]]: black hole entropy for stationary black holes - [[1994#Iyer, Wald]]: black hole entropy for dynamical black holes - extensions - [[2006#Hollands, Marolf]]: [[0332 Supergravity|SUGRA]] - [[2015#Prabhu]]: deals with internal gauge symmetry using the principle bundle ## Related topics - [[0414 Second order formalism]] - [[0360 Poisson bracket]]