In non relativistic quantum mechanics, we state the Heisenberg algebra $[X, P]=i\hbar$ as one of the postulates. The rotation and translation algebra is discussed later, after we've already defined the Hilbert space. $\renewcommand{\vec}[1]{\boldsymbol{#1}}$ But in Quantum Field Theory, we look for representations of the Poincare algebra and build a Fock space out of it. This is the "particle approach". In the "field approach", we look for representations of the Heisenberg algbera $[\phi(\vec{x}), \pi(\vec{x'})]=i\hbar\ \delta(\vec{x}-\vec{x'})$.
Which algebra is taken as more foundational to quantum theories? Is it that the Heisenberg algebra is more general (as in, a vector space representation of the Heisenberg algebra induces a representation of the Poincare algebra, like in non-relativistic QM)?