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Given the following lagrangian for the non-linear sigma model:

$$ \mathcal{L}=\frac{1}{2}\sum_{a,b}\partial_\mu\phi^a\partial^\mu\phi^b f_{ab}(\phi) $$

where $f_{ab}(\phi)$ is a matrix function.

My question is how can I work with those matrix elements in order to solve the Euler-Lagrange equation and then finally start with the canonical quantization of the theory?

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    $\begingroup$ In my opinion, it is much straightforward to quantise non-linear sigma model using path integrals as the interaction term of NLSM is contained in the constraint imposed on the fields: $\vec{\phi}^2 = 1$ which is not contained within the lagrangian. However, if you still wish to see canonical quantisation, this paper seem to go over it: arxiv.org/abs/1704.00735. $\endgroup$
    – emir sezik
    Jun 25, 2023 at 17:23
  • $\begingroup$ @emirsezik I'm just looking at canonical quantization for now... But that papers seems a good help, thanks. Although it seems I might not need to solve the EL equation, right? $\endgroup$
    – Syn1110
    Jun 25, 2023 at 18:03

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If you had not resolved the constraint, e.g., of a hyperspherical O(N) model, you'd use the standard Dirack bracket procedure, not needed here.

Here, you only have Goldstone scalars, and no "σ", so you quantize it like a standard interacting scalar theory, where the metric $f_{ab}$ provides the interaction. The canonical procedure is illustrated in section 13.3 for d=2 in the standard text of Peskin & Schroeder, and in many QFT textbooks such as that of Itzykson & Zuber.

P&S derive and interpret, directly, the asymptotic freedom of the hyperspherical O(N) model. Recall, in canonical quantization you quantize the free theory and address the interactions perturbatively.

This is not rocket science. All of the above (& some other ) formulations should/do provide the same answers, unless you believe, and argue, that you have stumbled on an unlikely paradoxical mismatch.

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