Abstract
The quantum logic formalism is another interesting, albeit more abstract, way to formulate quantum physics. The bonus of this approach is that one does not need to start from the assumption that the set of observables of a physical system is embodied with the algebraic structure of an associative unital algebra. As discussed in the previous section, this assumption that one can “add” and “multiply” observables is already a highly non trivial one. This algebraic structure is natural in classical physics since the observables form a commutative Poisson algebra, addition and multiplication of observables reflect the action of adding and multiplying results of different measurements (it is the Poisson bracket structure that is non trivial).
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Notes
- 1.
An eminent mathematician, not to be confused with his father, the famous G.D. Birkhoff of the ergodic theorem.
- 2.
In a very loose sense, I am not discussing mathematical logic theory.
- 3.
with the usual reservation on the lecturer’s qualifications.
- 4.
The point discussed here is a priori not connected to the classical versus intuitionist logics debate. Remember that we are not discussing a logical system.
- 5.
One should be careful for infinite dimensional Hilbert spaces and general operator algebras. Projectors correspond in general to orthogonal projections on closed subspaces.
- 6.
In French: treillis orthomodulaire, in German: Orthomodulare Verband.
- 7.
A module is the analog of a vector space, but on a ring instead of a (commutative) field.
- 8.
A division ring is the analog of a field, but without commutativity.
Bibliography
E. Artin, Geometric Algebra, vol. 3 (Interscience Publishers, New York, 1957)
R. Baer, Linear Algebra and Projective Geometry (Dover, Mineola, 2005)
E.G. Beltrametti, G. Cassinelli, The Logic of Quantum Mechanics. Encyclopedia of Mathematics and Its Applications, vol. 15 (Addison-Wesley, Reading, 1981)
G. Birkhoff, J. von Neumann, The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)
G. Gasinelli, P. Lahti, A theorem of Solér, the theory of symmetry and quantum mechanics. Int. J. Geom. Methods Mod. Phys. 9(2), 1260005 (2012)
A.M. Gleason, Measures on the closed subspaces of a Hilbert space. Indiana Univ. Math. J. 6, 885–893 (1957)
S.S. Holland, Orthomodularity in infinite dimensions; a theorem of M. Solèr. Bull. Am. Math. Soc. (N.S.) 32, 205–234 (1995)
J.M. Jauch, Foundations of Quantum Mechanics (Addison-Wesley, Reading, 1968)
G.W. Mackey, Mathematical Foundation of Quantum Mechanics (Benjamin, New York, 1963)
F. Maeda, S. Maeda, Theory of Symmetric Lattices. Grundlehren der mathematischen Wissenschaften (Springer, Berlin, 1971)
C. Piron, “Axiomatique” quantique. Helv. Phys. Acta 37, 439–468 (1964)
C. Piron, Foundations of Quantum Physics (Benjamin, New York, 1976)
M.P. Solèr, Characterization of Hilbert spaces with orthomodular spaces. Commun. Algebra 23, 219–243 (1995)
V.S. Varadarajan, The Geometry of Quantum Mechanics (Springer, New York, 1985)
J. von Neumann, Mathematische Grundlagen der Quantenmechanik. Grundlehren der mathematischen Wissenschaften, volume Bd. 38. (Springer, Berlin, 1932)
J. von Neumann, Mathematical Foundations of Quantum Mechanics. Investigations in Physics, vol. 2. (Princeton University Press, Princeton, 1955)
J. von Neumann, Continuous Geometry. Princeton Mathematical Series, vol. 25. (Princeton University Press, Princeton, 1960)
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David, F. (2015). The Quantum Logic Formalism. In: The Formalisms of Quantum Mechanics. Lecture Notes in Physics, vol 893. Springer, Cham. https://doi.org/10.1007/978-3-319-10539-0_4
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