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Perturbative Algebraic Quantum Field Theory pp 3–37Cite as

Algebraic Approach to Quantum Theory

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Abstract

Before entering the realm of the quantum theory of fields, let’s have a look at something simpler and better understood, namely quantum mechanics (QM). To prepare the ground for what follows, we will present an abstract formulation of QM and discuss how it relates to the more standard Dirac–von Neumann axioms (Dirac, The principles of quantum mechanics, 1930, [Dir30], Neumann, Mathematische grundlagen der quantenmechanik, 1932, [vN32]).

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Notes

  1. 1.

    The completion of \(\mathcal {X}\) can be constructed as a set of equivalence classes of Cauchy sequences in \(\mathcal {X}\).

  2. 2.

    A net \(\{T_\alpha \}\) of operators on a Hilbert space \(\mathcal {H}\) converges strongly to an operator T if and only if \(||T_\alpha x-Tx||\rightarrow 0\) for all \(x\in \mathcal {H}\). The definition of a net is at p. 22 in Footnote 5.

  3. 3.

    As sharp localization is physically impossible, operationally we can think of \(A_t\) as the average over some interval \([t-\epsilon ,t+\epsilon ]\) centered at t, for a fixed value of \(\epsilon >0\).

  4. 4.

    An open cover \((V_\beta )_{\beta \in B}\) is a refinement of an open cover \((U_\alpha )_{\alpha \in A}\), if \(\forall \beta \in B\), \(\exists \alpha \) such that \(V_\beta \subseteq U_\alpha \).

  5. 5.

    A net in a topological space \(\mathcal {X}\) is a function from some directed set (nonempty set with a reflexive and transitive binary relation) A to \(\mathcal {X}\).

  6. 6.

    A set A of continuous functions between two topological spaces E and F is equicontinuous at the points \(x_0 \in E\) and \(y_0\in F\) if for any open set \(\mathcal {O}\) around \(y_0\), there are neighborhoods U of \(x_0\) and V of \(y_0\) such that for every \(f \in A\), if the intersection of f(U) and V is nonempty, then \(f(U) \subseteq \mathcal {O}\). One says that A is equicontinuous if it is equicontinuous for all points \(x_0\in E\), \(y_0\in F\). The notion of equicontinuity becomes more intuitive, if we choose E and F to be metric spaces. The family A is equicontinuous at a point \(x_0\) if for every \(\epsilon >0\), there exists a \(\delta > 0\) such that \(d(f(x_0), f(x)) < \epsilon \) for all \(f \in A\) and all x such that \(d(x_0, x) <\delta \). In other words we require all member of the family A to be continuous and to have equal variation over a given neighbourhood.

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    Kasia Rejzner

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Rejzner, K. (2016). Algebraic Approach to Quantum Theory. In: Perturbative Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-25901-7_2

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