Abstract
In this Chapter we will explain an alternative way of describing group actions in topos quantum theory. The definition of group and group action in topos quantum theory was first introduced in [27]. Later, an alternative definition was put forward in [13]. In the following chapter we will explain this new definition which rests on the idea of flows in the spectral presheaf .
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Notes
- 1.
\(\mathcal {B}(\mathcal {H})\) indicates the algebra of bounded operators on the Hilbert space.
- 2.
Note that since ϕ is a *-homomorphisms, then the restriction ϕ| C is norm-closed and hence ϕ| C (C) is a C ∗-algebra.
- 3.
Note that from [50, 4.2.7] we know that when the base map Ï• is surjective on objects then the induced essential geometric morphisms is surjective. Moreover, from [50, 4.2.12] we know that when the base map Ï• is full and faithful, then the induced essential geometric morphisms is an inclusion. It then follows that in our case the essential geometric morphism induced by the isomorphism Ï• is itself an isomorphism.
- 4.
For this reason van den Berg and Heunen introduced their definition as partial C*-algebras, but the term was subsequently changed to piecewise C*-algebra [41].
- 5.
We recall that a forgetful functor ‘forgets’ or drops some or all of the input’s structure or properties ‘before’ mapping to the output.
- 6.
As stated in [13], this quotient is unproblematic when considering single systems and their unitary evolution.
- 7.
Note that in this context the Jordan algebras \(\tilde {\mathcal {N}}\)and \(\tilde {\mathcal {M}}\) associated to the von Neumann algebras \(\mathcal {N}\) and \(\mathcal {M}\) respectively, are JBW-algebras, hence, their ground field is \(\mathbb {C}\).
- 8.
Here \( Aut(\tilde {\mathcal {N}})\) represents the group of Jordan *-automorphisms associated to \(\mathcal {N}\).
- 9.
Similarly as done in [13, 14], from now on, we will define the one-parameter group by \((\hat {U}_{-t})_{t\in \mathbb {R}}\) rather than by \((\hat {U}_{t})_{t\in \mathbb {R}}\). This is in accordance with the fact that, as seen above, in canonical quantum theory time evolution is given by conjugating by \(\hat {U}^*_t=\hat {U}_{-t}\) instead of \(\hat {U}_t\).
- 10.
Recall that the definition of outer daseinisation is given by \(\delta ^o(\hat {P})=\bigwedge \{\hat {Q}\in \mathcal {P}(V)|\hat {Q}\geq \hat {P}\}\).
- 11.
Recall that, for each V ′⊆ V , the morphisms \( \underline {\Sigma }^{\mathcal {N}}(i_{V'V})\) are continuous and hence measurable. This implies that the inverse \( \underline {\Sigma }^{\mathcal {N}}(i_{V'V})^{-1}\) maps measurable sets to measurable sets.
References
A. Doering, Flows on generalised Gelfand spectra of non-abelian unital C*-algebras and time evolution of quantum systems (2012). arXiv:1212.4882 [math.OA]
A. Doering, Generalised Gelfand spectra of nonabelian unital C*-algebras (2012). arXiv:1212.2613 [math.OA]
H.A. Dye, On the geometry of projections in certain operator algebras. Ann. Math. 61(1), 73–89 (1955)
C. Flori, A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol. 868 (Springer, Heidelberg, 2013)
C. Flori, Group action in topos quantum physics. J. Math. Phys. 54, 3 (2013). arXiv:1110.1650 [quant-ph]
J. Hamhalter, Quantum Measure Theory. Fundamental Theories of Physics, vol. 134 (Kluwer, Dordrecht, 2003)
C. Heunen, M.L. Reyes, Active lattices determine AW∗-algebras. J. Math. Anal. Appl. 416(1), 289–313 (2014)
P.T. Johnstone, Sketches of an Elephant A Topos Theory Compendium I, II (Oxford Science Publications, Oxford, 2002)
S. Kochen, E. Specker, The problem of hidden variables in quantum mechanics. J. Math. Mech. 17(1), 59–87 (1967)
S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Springer, London, 1968)
nLab, Stuff, structure, property (2014). http://ncatlab.org/nlab/revision/stuff,+structure,+property/38
B. van den Berg, C. Heunen, Noncommutativity as a colimit. Appl. Categ. Struct. 20(4), 393–414 (2012)
S. Vickers, Topology via Logic (Cambridge University Press, New York, 1989)
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Flori, C. (2018). Alternative Group Action in Topos Quantum Theory. In: A Second Course in Topos Quantum Theory. Lecture Notes in Physics, vol 944. Springer, Cham. https://doi.org/10.1007/978-3-319-71108-9_3
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