TITLE: What will happen if one used $[\phi (x), \frac{\partial L}{\partial (\partial _x\phi(x))}]=i\hbar$ to get a Quantum Field Theory?
QUESTION [4 upvotes]: Classical field theory does not discriminate between space and time, but canonical quantisation does. We use the relation $$[\phi (x), \frac{\partial L}{\partial (\partial _t\phi(x))}]=i\hbar$$ to get the quantum theory. What if we instead used $$[\phi (x), \frac{\partial L}{\partial (\partial _x\phi(x))}]=i\hbar~?$$
In this, the canonical momentum has a $\partial_ x$ instead of $\partial _t$. I know this would give us a bad a theory which would not agree with experiments, but is there a mathematical motivation to disregard this? Why does nature prefer to implement probabilities in the time direction? Could this also be the reason consciousness perceives the universe in the time direction?
EDIT One thing I found is that we would need to deal with inverted harmonic oscillators if we tried this weird quantisation. This is because the new mode expansion of the field is:
$$\sum_{p_0,p_2,p_3} ae^{-i (\omega x+p_0t+p_2y+p_3z)} + a^{\dagger}e^{i (\omega x+p_0t+p_2y+p_3z)},$$
where $\omega = \sqrt {p_0^2-p_2^2-p_3^2}$. So we need to constrain the value of $(p_0,p_2,p_3)$ to not end up with imaginary $\omega$ values. If we don't constrain them, then we get imaginary frequencies which, when quantised, will result in Quantum Inverted Harmonic Oscillators.
From what I've found about Inverted Quantum Oscillators, they have difficulties with a probabilistic interpretation. Is this enough of a mathematical reason to discard this quantisation? Did nature choose the time direction to implement probabilities because this is the only direction where probabilities can be implemented? The time direction differs from the other directions in the metric sign.
In the case of Euclidean spacetime, we end up with $\omega =\sqrt {-p_1^2-p_2^2-p_3^2}$. Now there is no way to avoid imaginary $\omega$ even with constraints. Could this be used to mathematically motivate why nature chose Minkowski spacetime?

REPLY [2 votes]: Conventionally, we assume there exists an initial/Cauchy surface $\Sigma$ of co-dimension$^1$ 1 in spacetime, and that the fields satisfy a hyperbolic PDE. The CCR arises from the symplectic structure on $\Sigma$, cf. my Phys.SE answers here and here.

OP's title qustion is ruled out because the corresponding Cauchy surface would have timelike directions.

Similarly, spacetimes with Euclidean signature are ruled out because they corresponds to an elliptic PDE.


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$^1$This is related to that spacetime is a Lorentzian manifold. For theories with multiple temporal dimensions, see e.g. this Phys.SE post.
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