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An Axiomatic Basis for Quantum Mechanics

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Abstract

In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of quantum mechanics. The key results in this derivation are the co-ordinatization of generalized geometries and a theorem of Solér which characterizes Hilbert spaces among the orthomodular spaces. A generalized Wigner theorem is applied to reduce some of the assumptions of Solér’s theorem to the theory of symmetry in quantum mechanics. Since this reduction is only partial we also point out the remaining open questions.

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Notes

  1. Clearly, this assumption could also be posed under Axiom 1 but we refrain of doing it.

  2. The paper of Mielnik [45] contains an extensive analysis of possible state changes, including some nonlinear processes.

  3. This property has independently been introduced in [65] and [33] and it is known to be equivalent to the fact that each \(\alpha \in \mathbf S\) has a (unique) support in \(\mathbf L\) [3, Theorem11.4.3].

  4. Our proof is an adaption of the corresponding results in [3]. Another source leading to this conclusion is given by the results of Sects. 2.5.2 of [55].

  5. For a detailed discussion of this theorem, see, e.g. [63]

  6. If \(K=\mathbb R\) then \({}^*\) is the identity. For \(K=\mathbb C\) the map \({}^*\) cannot be the identity and if it is continuous then it is the complex conjugation. For \(K=\mathbb H\) the map is the quaternionic conjugation.

  7. Various definitions of the notion of symmetry in quantum mechanics are studied e.g. in [9, 48].

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Acknowledgments

We are grateful to Drs. Paul Busch and Maciej Ma̧czynski for their valuable comments in earlier versions of this manuscript.

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Authors and Affiliations

  1. Department of Physics, University of Genova, and INFN Sezione di Genova, Genoa, Italy

    Gianni Cassinelli

  2. Department of Physics and Astronomy, Turku Centre for Quantum Physics, University of Turku, Turku, Finland

    Pekka Lahti

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  1. Gianni Cassinelli
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Correspondence to Pekka Lahti.

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Cassinelli, G., Lahti, P. An Axiomatic Basis for Quantum Mechanics. Found Phys 46, 1341–1373 (2016). https://doi.org/10.1007/s10701-016-0022-y

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