I am reading Andy's lectures.

On page 22, one has the symplectic form for free electrodynamics as $$\Omega_{\Sigma}=-\frac{1}{e^2}\int_{\Sigma}\delta(*F)\wedge\delta A,\tag{1}$$ where $\Sigma$ is any Cauchy hypersurface (3-dimensional) and $\delta$ is the variation in the phase space. Subsequently, (1) gives the equation with indices as $$\Omega_{\Sigma}=-\frac{1}{e^2}\int_{\Sigma}d\Sigma^{\mu}\delta F_{\mu\nu}\wedge\delta A^{\nu},\tag{2}$$ where $d\Sigma^{\mu}$ is the induced mearsure times the unit norm vector to $\Sigma$.

Anybody know how to derive Eq.(2) from Eq.(1)? Further, what does it mean by "induced measure" here? Does it mean $\sqrt{|h|}dx^1dx^2dx^3$, where $h_{\mu\nu}$ is the induced metric? If it is indeed the case, then when the surface is null hypersurce $\mathcal{I}^{\pm}$, we have $h=0$.

  • 2
    $\begingroup$ Possible duplicate of Induced metric on a null hypersurface $\endgroup$
    – JamalS
    Oct 9 '17 at 6:53
  • 1
    $\begingroup$ Note that, in a Q&A site, every post is going to be a question, so it makes little sense to give this post the title you have. Please consider a title that actually describes the problem you have, rather than the statement that you have a problem. $\endgroup$
    – Kyle Kanos
    Oct 9 '17 at 11:43
  • $\begingroup$ Expanding on Kyle's comments, here's a proposed title at a much more informative level. $\endgroup$ Oct 9 '17 at 12:22

Let us here only address the last conceptional part of OP's question (v4).

Let there be given a 4-dimensional orientable Lorentzian manifold $(M,\mathbb{g})$ with canonical pseudovolume$^1$ form $$\Omega~:=~\sqrt{|g|}\mathrm{d}x^{0}\wedge \mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}~\in~\Gamma(\bigwedge \! {}^4(T^{\ast}M)), \qquad g~:=~\det(g_{\mu\nu}). \tag{A}$$ [Note that this Omega has nothing to do with the symplectic form in eqs. (1) & (2)] Hamiltonian quantization in field theory typically rely on a choice of a vector field $X\in \Gamma(TM)$ that represents the flow of an evolution parameter, cf. e.g. this Phys.SE post. For a time-like vector field we can always normalize it, if we so wish; but if it is light-like, there is no canonical choice of $X$.

We next pick a Cauchy surface $\Sigma\subset M$ of co-dimension 1, i.e. a hypersurface, such that $$T^{\ast}_p\Sigma~=~{\rm Ker}(X_p), \qquad p~\in~\Sigma.\tag{B}$$ Here we have identified the vector $$X_p: T^{\ast}_pM~\to~ \mathbb{R}\tag{C}$$ with a functional on the cotangent space $T^{\ast}_pM$. We can then define a pseudovolume form $\omega \in \Gamma(\bigwedge\! {}^3(T^{\ast}\Sigma))$ on the Cauchy surface $\Sigma$ via a contraction $$\omega_p~:=~i_{X_p}\Omega_p, \qquad p~\in~\Sigma.\tag{D}$$


$^1$ A pseudovolume form transforms as $$ \Omega^{\prime} ~=~{\rm sgn }(J)~\Omega, \qquad J~:=~\det(\frac{\partial x^{\prime \nu}}{\partial x^{\mu}}), \tag{E} $$ under general coordinate transformations $x^{\mu} \to x^{\prime \nu}=f^{\nu}(x)$.


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