Why are spatial derivatives of a field not used to define the conjugate momentum?

Question

In mechanics, the conjugate momentum is defined to be $\pi=\dfrac{\partial L}{\partial \left(\frac{dx}{dt}\right)}.$ This can be regarded as a partial derivative with respect to a derivative of a dependent variable ($x$ depends on $t$, the latter being the independent variable). In (scalar) field theory, the number of independent variables increases: $\phi$ depends on $\{t,x,\cdots\}$. One may hence define multiple quantities: $$\dfrac{\partial L}{\partial \left(\frac{\partial\phi}{\partial t}\right)},\dfrac{\partial L}{\partial \left(\frac{\partial\phi}{\partial x}\right)},\cdots.$$ And yet, only the first quantity is regarded as the conjugate momentum and in canonical quantization scheme, is used to obtain the "equal time" commutation relations. Why don't we use the remaining quantities? Is it because $\{x,\cdots\}$ are still regarded as dependent variables (not on the same footing with $t$)? What is going on from a purely mathematical point of view?

Answer 1

The Hamiltonian formulation explicitily breaks Lorentz by choosing a particular direction in spacetime, namely $t$. More generally, one can foliate spacetime with spacetime surfaces $\Sigma_t$, where $t$ denotes a coordinate along a congruence, which can be thought as a time coordinate. This construction is used in Hamiltonian formulation of GR, which is known as ADM Formalism https://en.wikipedia.org/wiki/ADM_formalism.

There is some research of defining this construction without breaking general covariance, you may find this paper by D.Harlow useful https://arxiv.org/abs/1906.08616.