Abstract
In this Chapter we are interested in analysing how, if at all, different quantizations can be represented in Topos Quantum Theory. We already know from the work of [57] that it is indeed possible to define the concept of quantization within a topos. We would like to extend this program to incorporate all possible equivalent quantizations.
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Notes
- 1.
When we think of \(C^{\infty }(S, \mathbb {R})\) as a Lie algebra we will use the notation \(C^{\infty }_{Lie}(S, \mathbb {R})\).
- 2.
There is always an infinite-dimensional transitive group, namely the group of symplectic transformations of S.
- 3.
Note that the use of PG raises the interesting question as to the extent to which the non-commutative structure of the Lie algebra L(G) can be recovered from knowing the poset structure of its abelian Lie subalgebras.
- 4.
Clearly to define an action of G on these poset morphisms we are using a representation of G on \(\mathcal {H}\). By factoring this representation by its Kernel, we can assume that it is indeed faithful.
- 5.
One can think of this as the idea that the elements in each poset L represent labels of physical quantities and these are fixed once and for all.
- 6.
Here \(\Gamma \underline {R}\) denotes the global sections of \( \underline {R}\).
- 7.
In the sense that for each element \(\phi \in (\Lambda \underline {R})_V\), given the open neighbourhood U, p R (U) is open in PG and p R restricted to U ∋ ϕ is a homomorphisms, i.e., \((p_R)|U:U\rightarrow p_{ \underline {R}}(U)\) is a homomorphisms.
- 8.
Recall that \(A:J\rightarrow Sets^{\mathcal {V}(\mathcal {H})}\) is such that A V (j) = A(j)(V ), therefore (J 0(A(j))) ϕ  := A(j) ϕ(V ) = A ϕ(V )(j).
- 9.
Here \(Et(\Lambda \underline {R})\) indicates the etalé bundles over \(\Lambda \underline {R}\).
- 10.
The author in [57] claims that the map \(\tilde {v}\) is faithful, however because of the periodicity of the exponential function it is not clear to us how he justifies his claim.
- 11.
Recall that \(p_Q:\Lambda \underline {Q}\rightarrow \mathbf {C}_o\).
- 12.
In the sense that for each element \(\phi \in (\Lambda \underline {Q})_C\), given the open neighbourhood U, p Q (U) is open in C o and p Q restricted to U ∋ ϕ is a homomorphisms, i.e., p Q | U  : U → p Q (U) is a homomorphisms.
References
C. Flori, Group action in topos quantum physics. J. Math. Phys. 54, 3 (2013). arXiv:1110.1650 [quant-ph]
P.T. Johnstone, Sketches of an Elephant A Topos Theory Compendium I, II (Oxford Science Publications, Oxford, 2002)
S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Springer, London, 1968)
K. Nakayama, Sheaves in quantum topos induced by quantization. arXiv:1109.1192 [math-ph]
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Flori, C. (2018). Quantization in Topos Quantum Theory: An Open Problem. 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_14
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