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EDIT: The main question of this post is Why do we apply the Legendre transformation with a partial derivative $\partial_\mu$ by foliating spacetime rather than with the covariant derivative $\nabla_\mu$ without foliating?

I was thinking there could be some limitations from the definition of the Legendre transformation that made it necessary and obligatory, but I couldn't find any. Unless I understood it wrong, there shouldn't be any problem taking the time covariant derivative $\nabla_0$ rather than $\partial_0$ after foliation.

I thought it might be obligatory to work on (flat) tangent bundles for the Hamiltonian mechanism to work but I weren't able to find any source that confirms or discards this.

But there is a problem with this approach: De Donder-Weyl theory. They use the full covariant derivative rather than only $\nabla_0$ or $\partial_0$:

$$p^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial\left(\nabla_\mu A_\nu\right)},\tag{1}$$

$$p^{\mu} = \frac{\partial \mathcal{L}}{\partial\left(\nabla_\mu \phi\right)},\tag{2}$$

So here we don't we foliate anything, right? That's the idea of De Donder-Weyl theory, so there's no point on that. If I did it correctly, we keep using covariant derivative here with polymomenta, right? Then (1) there shouldn't be a mathematical restriction itself on $\nabla_\mu$ that makes it unfeasible to Legendre transform, or (2) we cannot Legendre transform De Donder-Weyl theory on curved spacetime.

And if that's the case and there is no problem applying the Legendre transformation using covariant derivatives as the operator, why foliate anyway? Is it just because otherwise we have $$\nabla_\sigma g_{\mu\nu} = 0\tag{3}$$ identically and thus a singular Hamiltonian in quantum gravity?

Qmechanic
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Antoniou
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1 Answers1

2

  1. In $n$ spacetime dimensions, instead of a De Donder-Weyl (DDW) Legendre transformation from polyvelocities $$v_{\mu}~=~\partial_{\mu}q, \qquad \mu~\in~\{0,1,\ldots, n\!-\!1\}, \tag{A}$$ to polymomenta $p^{\mu}$, OP effectively suggests a Legendre transformation$^1$ $${\cal H}(q, p)~:=~\sup_w \left\{\sum_{\mu=0}^{n-1}p^{\mu}w_{\mu} -{\cal L}(q, w)\right\}.\tag{B}$$ from polyfunctions$^2$ $$w_{\mu}~=~f_{\mu}(q,\partial q), \qquad \mu~\in~\{0,1,\ldots, n\!-\!1\},\tag{C}$$ to polymomenta $p^{\mu}$. This in principle works for a regular 1st-order$^3$ Lagrangian density ${\cal L}(q, w)$, even in a curved spacetime, cf. OP's question.

  2. The corresponding Euler-Lagrange equations $$ \begin{align}\frac{\partial {\cal L}(q, w)}{\partial q}~+~&\sum_{\mu=0}^{n-1}\frac{\partial {\cal L}(q, w)}{\partial w_{\mu}}\frac{\partial f_{\mu}(q,v)}{\partial q}\cr ~=~&\sum_{\mu,\nu=0}^{n-1}\frac{d}{dx^{\nu}}\left\{\frac{\partial {\cal L}(q, w)}{\partial w_{\mu}}\frac{\partial f_{\mu}(q,v)}{\partial v_{\nu}}\right\}\end{align}\tag{D} $$ turn into the Hamilton's equations $$\begin{align} f_{\mu}(q,v)~=~&\frac{\partial {\cal H}(q, p)}{\partial p^{\mu}}, \qquad \mu~\in~\{0,1,\ldots, n\!-\!1\}, \cr \sum_{\mu=0}^{n-1}p_{\mu}\frac{\partial f_{\mu}(q,v)}{\partial q} ~-~&\sum_{\mu,\nu=0}^{n-1}\frac{d}{dx^{\nu}}\left\{ p_{\mu}\frac{\partial f_{\mu}(q,v)}{\partial v_{\nu}}\right\}~=~\frac{\partial {\cal H}(q, p)}{\partial q}.\end{align}\tag{E} $$

  3. The foliation into Cauchy hypersurfaces is not needed for the (modified) DDW Legendre transformation itself, but to fix a canonical Poisson structure. The latter is needed in the quantization process. The introduction of polyfunctions (C) may make the pertinent canonical Poisson structure less transparent.

$^1$ Explicit $x$-spacetime dependence and component indices on the field $q$ are suppressed in our notation. One might have to use infimum rather than supremum, depending on signs. Concerning DDW, see also e.g. my related Phys.SE answer here.

$^2$ The polyfunctions could be covariant derivatives $$f_{\mu}(q,v)~=~\nabla_{\mu}q\tag{F}, \qquad \mu~\in~\{0,1,\ldots, n\!-\!1\},$$ as OP suggests. [Note that covariant derivatives of an $r$-form field is the same as a partial derivatives for torsionfree connections, cf. OP's eqs. (1) & (2). Also note that the vanishing Levi-Civita covariant derivative of the metric tensor (3) is obviously not useful.] The covariance further simplifies the form of the EOMs (D) & (E).

$^3$ It's believed to also work for singular Legendre transformations of 1st-order Lagrangian gauge theories. In this answer we stop short of investigating higher-order Lagrangian theories.

Qmechanic
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