(adsbygoogle = window.adsbygoogle || []).push({}); This is not correct. Notions such as strangerep said: ↑The issue is that I've heard various purist advocates of the algebraic approach suggesting that Hilbert space is not essential to quantum physics,completeness(by a norm) andcontinuity(i.e., boundedness) of any element of an operator algebra need to be defined with respect to some vector spacetopology. Hermitianadjointcan only be defined on a vector space withscalar product. Moreover, every (abstract) non-commutative [itex]C^{*}[/itex]-algebra can be realized as (i.e.,isomorphicto) a norm-closed , *-closed subalgebra of [itex]\mathcal{L}(\mathcal{H})[/itex], the algebra of bounded operators on some Hilbert space [itex]\mathcal{H}[/itex]. Precisely speaking, for every abstract [itex]C^{*}[/itex]-algebra [itex]\mathcal{A}[/itex], there exists a Hilbert space [itex]\mathcal{H}[/itex] and injective *-homomorphism [itex]\rho : \mathcal{A} \to \mathcal{L}(\mathcal{H})[/itex]. That is [itex]\mathcal{A} \cong \rho (\mathcal{A}) \subset \mathcal{L}(\mathcal{H})[/itex], as every *-homomorphism is continuous (i.e., norm-decreasing).

In general, one can say the following aboutquantization: Given a locally compact group [itex]G[/itex], its (projective) unitary representation on some Hilbert space [itex]\mbox{(p)Urep}_{\mathcal{H}}(G)[/itex] and the group (Banach) *-algebra [itex]\mathcal{A}(G)[/itex], then you have the followingbijectivecorrespondence [tex]\mbox{(p)URep}_{\mathcal{H}}(G) \leftrightarrow \mbox{Rep}_{\mathcal{H}}\left(\mathcal{A}(G)\right) \ , \ \ \ \ (1)[/tex] where [itex] \mbox{Rep}_{\mathcal{H}}\left(\mathcal{A}(G)\right)[/itex] is the representation of the (Banach) *-algebra [itex]\mathcal{A}(G)[/itex] on the same Hilbert space [itex]\mathcal{H}[/itex], i.e., *-homomorphism from [itex]\mathcal{A}(G)[/itex] into the algebra of bounded operators [itex]\mathcal{L}(\mathcal{H})[/itex] on [itex]\mathcal{H}[/itex]. Similar bijective correspondence exists when [itex]\mathcal{A}[/itex] is a C*-algebra. Andboth endsof the correspondence lead to quantization. When [itex]G = \mathbb{R}^{2n}[/itex] is the Abelian group of translations on the phase-space [itex]S = T^{*}\left(\mathbb{R}^{n}\right) \cong \mathbb{R}^{2n}[/itex] (or its central extension [itex]H^{(2n+1)}[/itex], the Weyl-Heisenberg group) then (a) the left-hand-side of the correspondence leads (via the Stone-von Neumann theorem) to the so-called Schrodinger representation on [itex]\mathcal{H} = L^{2}(\mathbb{R}^{n})[/itex] [Side remark: of course Weyl did all the work, but mathematicians decided (unjustly) to associate Heisenberg’s name with the group [itex]H^{2n+1}[/itex]], while (b) the right-hand-side of the correspondence leads to the Weyl quantization which one can interpret as deformation quantization (in effect, Weyl quantization induces a non-commutative product (star product) on the classical observable algebra, thus deforming the commutative associative algebra of functions [itex]C^{\infty}(\mathbb{R}^{2n})[/itex]).

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# Insights Mathematical Quantum Field Theory - Fields - Comments

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