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In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of Quantum Field Theory (QFT). For instance, considerations extend to formulations pertinent to the free scalar field, chiral bosons, and chiral fermions. Notably, within the realm of Algebraic Quantum Field Theory (AQFT), the principal elements invariably involve smeared operators paired with test functions denoted as $O(f)$, these residing in distinct spaces tailored to specific requirements.

Oftentimes, the mean values of these operators, whether they involve commutators or anticommutators, remarkably intertwine with the inner product within the particle space, delineated as $<f, g>$.

I find myself contemplating whether, given a theory by its Lagrangian, a method exists to determine the corresponding inner product $<f, g>$ inherent to that theory within the particle space encapsulated by the AQFT formalism.

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    $\begingroup$ I think your question needs some clarification. What is the "particle space" that you refer to? A priori, an AQFT need not have a particle interpretation (or have a Lagrangian). The only relevant inner product(s) come from considering representations of its algebra of observables on a Hilbert space. If you start from a Lagrangian, then the Fock space construction provides both the particle interpretation and a Hilbert space of states (at least for free theories, with perturbative interactions not changing the picture that much). That's pretty standard and is described any many QFT textbooks. $\endgroup$
    – Igor Khavkine
    Commented Jan 5 at 4:49

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