One can represent a quantum system by the Weyl algebra (which is a C*algebra). For instance, a 1 degree of freedom system can be represented by the algebra generated by $e^{\imath t Q}, e^{\imath s P}$. My guess for modeling a quantum field would be to look at C*algebras with an uncountably infinite number of generators, but in AQFT this is done thanks to a net of C*algebras. Is there a link between the approach I think of as being intuitive and the AQFT one?
The Weyl algebra construction can be done abstractly for any real vector space (even infinitedimensional) endowed with an antisymmetric bilinear form, thanks to B. Blackadar's universal C*algebra construction using generators and relations ("Shape theory for C∗algebras", Math. Scand. 56 (1985) 249–275). However, since you asked for a concrete correspondence, here is a more explicit description. In the case of (free) real scalar fields $\phi$ in a globally hyperbolic spacetime $(M,g)$ subject to the KleinGordon equation $$ \Box_g\phi+m^2\phi=0\ ,$$ the Weyl unitaries are concretely given by the functionals $$ \mathscr{C}^\infty(M,\mathbb{R})\ni\phi\mapsto W_f(\phi)=e^{i\int_M f\phi\ d\mu_g}\ ,\quad f\in\mathscr{C}^\infty_c(M,\mathbb{R})\ ,$$ where $d\mu_g$ is the volume form associated to the Lorentzian metric $g$. One can understand the test functions $f$ as "component indices", just like we do in the case of the $2n$dimensional symplectic (phase) space generated by $n$ positions $x_1,\ldots,x_n$ and $n$ momenta $p_1,\ldots,p_n$. Given any nonvoid open subset $O\subset M$ of the spacetime manifold $M$, let $\tilde{\mathfrak{A}}(O)$ be the unital $*$algebra generated by the Weyl unitaries $W_f$ as $f$ runs over the realvalued smooth functions compactly supported in $O$ (so we say that such $W_f$'s are localized in $O$), once we endow such functionals with the following operations:
The above operations are then extended to general elements in the usual way, and they entail that the Weyl unitaries are worthy of their name, since we clearly have $W^*_f W_f=\mathbb{1}$. This $*$algebra has nontrivial $*$representations (namely, Fock representations associated to quasifree states), hence it admits a minimal nontrivial C$*$norm $\\cdot\$. The Weyl algebra $\mathfrak{A}(O)$ associated to $O$ is the C$*$completion of $\tilde{\mathfrak{A}}(O)$ with respect to this C$*$norm, and the correspondence $O\mapsto\mathfrak{A}(O)$ is obviously an isotonous net of C$*$algebras. Moreover, due to the causal support of $\Delta_{m,g}$ entailed by the hyperbolicity of the KleinGordon operator, this net is also causal (i.e. elements localized in causally disjoint regions commute). Finally, uniqueness of retarded and advanced fundamental solutions for this operator guarantees that the isometry group of $(M,g)$ acts on this net as it should  namely, for any isometry $\psi$ of $(M,g)$, the action $$ \alpha_\psi(W_f)(\phi)=W_f(\psi^*\phi)=W_{\psi_*f}(\phi) $$ uniquely extends to unitpreserving *isomorphisms satisfying $$\alpha_\psi\circ\alpha_{\psi'}=\alpha_{\psi\circ\psi'}\ ,\quad\alpha_\psi(\mathfrak{A}(O))=\mathfrak{A}(\psi(O))\ .$$ In the above formula, $\psi^*\phi=\phi\circ\psi$ is the pullback of the field configuration $\phi$, and $\psi_*f=f\circ\psi^{1}$ is the pushforward of the test function $f$. To summarize, we have obtained a HaagKastler net of C$*$algebras. The above argument is a somewhat modernized version of a construction due to J. Dimock ("Algebras of local observables on a manifold", Comm. Math. Phys. 77 (1980), no. 3, 219–228). See also C. Bär, N. Ginoux and F. Pfäffle, "Wave equations on Lorentzian manifolds and quantization", ESI Lectures in Mathematics and Physics, European Mathematical Society (2007), arXiv:0806.1036 [math]. One can perform a similar construction for Dirac fields (see for instance C. Dappiaggi, T.P. Hack and N. Pinamonti, "The extended algebra of observables for Dirac fields and the trace anomaly of their stressenergy tensor", Rev. Math. Phys. 21 (2009) 12411312, arXiv:0904.0612 [mathph]), with the simplification that for the CAR algebra it's not necessary to exponentiate the smeared fields to get C*algebra elements. For interacting fields, if one wants to keep close to what physicists do and steer clear of constructive methods a la GlimmJaffe (which are of course a better choice from the viewpoint of rigor but severely limit the models one may study in their present state of the art), one has to recourse to formal perturbation theory, which means one has to work with formal power series in the coupling constant and Planck's constant. This also means abandoning C$*$algebras and working with more general *algebras. Once one accepts this, perturbative renormalization can be dealt with in a rigorous way, using a language close to the one adopted above. See for instance R. Brunetti, M. Dütsch and K. Fredenhagen, "Perturbative algebraic quantum field theory and the renormalization groups", Adv. Theor. Math. Phys. 13 (2009) 1541–1599, arXiv:0901.2038 [mathph]. 


At a crude intuitive level, yes, something like that  you want to replace each classical degree of freedom with a onedimensional quantum particle. Quantum fields have infinitely many degrees of freedom, okay (but only countably many, sorry). The point about nets of C*algebras is that for each spacetime region you quantize the portion of a field that lies in that region, and the resulting C*algebras cohere in the manner described by the net. This really becomes important when you take relativity into account. If you're quantizing a nonrelativistic field I don't think you need to worry about nets. 

