First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor.

1) The usual process of quantization of a free scalar field on $Q$ is by using Weyl operators $W (f)$ for each $f \in S \subset L^2 (T^*Q)$ (where S is a space of solutions) such that $$W(f)W(g) = \exp (-i \sigma (f, g)) W (f + g)$$ for some (pre-)symplectic form $\sigma$.

This process is explained for the Klein-Gordon equation here The most general procedure for quantization

2) On the other side, in http://arxiv.org/abs/physics/9801019 and http://arxiv.org/abs/math-ph/0408008, given a configuration bundle $$F \twoheadrightarrow X$$ with $X$ a globally hyperbolic Lorentzian manifold of dimension $n + 1$ and a Lagrangian density $$\mathcal{L}: J^1F \rightarrow \Lambda^{n + 1} T^*X$$ such that $\mathcal{L} = Ldx$. Let $$(J^1 (F))^* = \text{Hom} (J^1F, \Lambda^{n + 1}T^*X),$$ $$\mathbb{F} \mathcal{L} : J^1F \rightarrow (J^1F)^*$$ be the Legendre transform, $\omega =- d\theta$ and $\theta$ is the canonical form on $(J^1F)^*$. Then $$\omega_L = -\mathbb{F} \mathcal{L}^* \omega = -d \theta_L$$ for $\theta_L = \mathbb{F} \mathcal{L}^* \theta$ is a a multisymplectic form of degree $n + 2$.

Given Cauchy surface $\Sigma \subset X$, it's possible to produce a (pre-)symplectic form $$\Omega_{\Sigma} = \int_{\Sigma} \omega_L$$ on the off-shell (the space of all sections).

Locally on coordinates $$\theta_L = \frac{\partial L}{\partial v^a_\mu} \wedge du^a \wedge dx_0 \wedge … \wedge \hat{dx_\mu} \wedge … \wedge dx_n + (L - \frac{\partial L}{\partial v^a_\mu}v^a_\mu )dx$$

3) Analogously in http://arxiv.org/1402.1282 and http://ncatlab.org/nlab/show/multisymplectic+geometry, one can consider the first variation $$\delta \mathcal{L} = \sum_a (EL_L)_a \wedge \delta u^a + d \theta_L$$, where $(EL_L)_a$ is the Euler-Lagrange equation and $(x_{\mu}, u^a, v^a_{\mu})$ are the local coordinates of $J^1 F$. In this context $$\omega_L = EL_L + \delta \theta_L $$ is a $n + 2$-form which is (pre-) multisymplectic.

In a lot of references, it's said that the variational principle 3) coincides with 2). It's said too that $\theta_L$ in 3) coincides with the usual boundary condition $$\sum_{a, \mu} \frac{\partial L}{\partial v^a_{\mu}} \delta u^a $$ (which seems impossible). **I would like an answer to these two questions**. These are two minor questions that are probably because of a miscalculation of my part or the authors.

Now my main question is : **How can one relate 1) to 3)?**

**More precisely, are the symplectic forms the same in these two approaches? If not, do canonical quantization of 3) (via Weyl CCR or non-exponeniated CCR) produce an equivalent QFT?**

**EDIT**

The second of my minor question is obviously true, because of the Stokes formula. The boundary condition can be written as $$\int _{\partial V} \sum_{a, \mu} \frac{\partial L}{\partial v^a_{\mu}} \delta u^a vol_{\partial V}= \int_V \sum_{a, \mu} d(\frac{\partial L}{\partial v^a_{\mu}} \delta u^a) dx$$ for some suitable region $V \subset X$.

Anyway the second minor question remain. In other words, is $\theta_L = \frac{\partial L}{\partial v^a_{\mu}} \delta u^a dx$ in 3) equals to $\frac{\partial L}{\partial v^a_\mu} \wedge du^a \wedge dx_0 \wedge … \wedge \hat{dx_\mu} \wedge … \wedge dx_n + (L - \frac{\partial L}{\partial v^a_\mu}v^a_\mu )dx$ ($\theta_L$ in 2))?

All the terms except $Ldx$ seems reasonable to appear in $\theta_L$ of 3).

This post imported from StackExchange Physics at 2016-01-08 10:49 (UTC), posted by SE-user user40276