Is the defining property, of the Quantum theory Hilbert space, the Heisenberg algebra or the Poincare algebra? 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.
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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)?
 A: The non-relativistic QM is mathematically based upon the Hilbert space. In physics, we generally have a natural phenomenon that we want to "quantify" using mathematics as a language. For the case of the non-relativistic QM, the mathematics used to describe the phenomena related to QM is the Hilbert space.
A: The Poincaré algebra is necessary in order to encode the laws of special relativity. The entirety of special relativity could be stated as "symmetry of the physics under Poincaré transformations." However, the Heisenberg algebra is necessary in order to encode one of the fundamental quantum effects, that of non-commutative measurements (among other things, I'm sure). In non-relativistic QM, we don't really care about Poincaré symmetry, so why try to encode it in our equations? What we do care about is the Heisenberg algebra, since we want a quantum mechanical theory. Likewise in QFT (from the operator approach) we want the theory to satisfy both the Poincaré and Heisenberg algebras, since it must be relativistic and quantum mechanical. This is why we start in QFT with trying to encode Poincaré symmetry, since we already know we can satisfy the Heisenberg algebra alone, we try to do both, and are led to the concept of Fock space.
