Abstract
We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\)-algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities between the roots of a polynomial \(p(x)\in K[x]\) result from the limitations of the field \(K\). We discuss the relation between this epistemic interpretation of the Galois–Grothendieck duality and Leibniz’s principle of the identity of indiscernibles. We then use the conceptual framework provided by Klein’s Erlangen program to propose an alternative ontologic interpretation of this duality. The Galoisian symmetries are now interpreted in terms of the automorphisms of the symmetric geometric figures that can be placed in a background Klein geometry. According to this interpretation, the Galois–Grothendieck duality encodes the compatibility condition between geometric figures endowed with groups of automorphisms and the ‘observables’ that can be consistently evaluated at such figures. In this conceptual framework, the Galoisian symmetries do not encode the epistemic indiscernibility between individuals, but rather the intrinsic indeterminacy in the pointwise localization of the figures with respect to the background Klein geometry.
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Notes
It is also worth mentioning that Da Costa, Rodriguez and Bueno developed, in the wake of the works of Jose Sebastiao e Silva and Mark Krasner, a “generalized or abstract Galois theory” (see Da Costa and Rodrigues 2007; Da Costa 2011 and references therein). This theory relates the invariance under symmetry transformations on the one hand and the notion of definability with respect to a given language on the other.
It is also worth noting that the formal analogy between Grothendieck–Galois theory and the Tannaka–Krein reconstruction theorem for compact topological groups points towards the possible relevance of the former in algebraic quantum field theory (see Chargois 2005; Joyal and Ross 1991; Szyld 2011). Indeed, the Tannaka-Krein formalism is at the heart of the analysis of superselection sectors in algebraic quantum field proposed by Doplicher, Haag, and Roberts (DHR) (see Halvorson and Mueger 2006 and references therein).
In turn, this connection might open the possibility of relating the Galois–Grothendieck theory and Cartan’s generalization of Klein’s program (Sharpe 1997). It is worth noting that the resulting Cartan geometries are the geometric underpinning of some current attempts to reformulate gravitational theories as gauge theories Wise (2010).
A \(K\)-algebra \(A\) is a ring endowed with a scalar multiplication by elements of the field \(K\) such that \(k(aa')=(ka)a'\) with \(k\in K\) and \(a,a'\in A\). Equivalently, a \(K\)-algebra is a \(K\)-vector space endowed with a multiplication. A typical example of a \(K\)-algebra is the power \(K^{n}\) where the addition and the multiplications are defined componentwise. The \(K\)-algebra \(K^{n}\) for \(n>1\) is not a field. For instance, the element \((1,0)\in K^{2}\) does not have a multiplicative inverse (where the unit of \(K^{2}\) is \((1,1)\)).
It can be shown that a splitting field always exist and that two such fields are necessarily isomorphic (see Lang 2005, p. 236).
Let’s note that by writing the set \(X\) as \(X=\{\alpha _1, \alpha _2,\ldots , \alpha _n\}\), we have ordered the set, i.e. we have chosen a bijection \(Ord: \{1,2,\ldots ,n\} \rightarrow X\). The following discussion does not depend on this election.
One can prove that this definition of \(K\)-indiscernibility is equivalent to that of Model Theory: for every first-order \(K\)-formula \(\phi \) (not only the atomic ones) and every permutation \(\sigma \), \(K(\overrightarrow{\alpha })\models (\phi (\overrightarrow{\alpha })\Leftrightarrow \sigma \cdot \phi (\overrightarrow{\alpha }))\). See Marker (2002, p. 178) for details.
Indeed, \(id_X \in S_K(X)\). If \(\sigma _1,\sigma _2\in S_K(X)\), then \(\forall q\), \(q(\overrightarrow{\alpha })=0 \Rightarrow \sigma _1 \cdot q(\overrightarrow{\alpha }) = 0\) . Since \(\sigma _{1}\cdot q\) is an algebraic relation in the \(\alpha _{i}\)’s, \(\sigma _{2}\cdot (\sigma _{1}\cdot q)=(\sigma _{2}\cdot \sigma _{1}) \cdot q\) is also an algebraic relation. Finally, every \(\sigma \) in \(S_{X}\) is of finite order. Therefore, \(\sigma ^{-1}=\sigma ^{p}\) for some \(p\in \mathbb {N}\).
With respect to the relations between the notions of individuality and discernibility (or distinguishability), we report the reader to French and Krause (2006). It is worth noting that in French and Krause (2006), hypothetical elements that are absolutely indiscernible, i.e. elements that can be distinguished solo numero in any possible domain of rationality, are called non-individuals. Now, the notion of indiscernibility formalized by Galois theory is a weaker notion, since it is relative to the different possible domains of rationality. Therefore, we have decided to call the corresponding elements \(K\) -indiscernible individuals.
Note that this definition of \(Gal_K(p)\) depends on the choice of a splitting field \(K^p\). Hence, the Galois group of \(p(x)\) is actually defined up to isomorphisms.
The kernel of a ring homomorphism \(\sigma : K\rightarrow L\) is an ideal of \(K\). But a field \(K\) has only one proper ideal, namely \(\{0\}\). Hence, a field homomorphism \(\sigma : K\rightarrow L\) is necessarily injective.
It is worth stressing that the minimal polynomial \(p_{\alpha , K}\) depends of the base field \(K\). For instance, the minimal polynomial of \(\root 4 \of {3}\) is \(X^{4}-3\) in \({\mathbb {Q}}\) and \(X^{2}-\sqrt{3}\) in \({\mathbb {Q}}(\sqrt{3})\).
From the equation \(p_{\alpha , K}(\alpha )=0\), we can express \(\alpha ^{n}\) as a linear combination of \(\left\{ 1,\alpha ,\ldots ,\alpha ^{n-1}\right\} \) with coefficients in \(K\). We can then express every \(g(\alpha )\) in \(K(\alpha )\) as a linear expression in \(\left\{ 1,\alpha ,\ldots ,\alpha ^{n-1}\right\} \). The minimality of \(p_{\alpha , K}\) implies that these elements are linearly independent.
Indeed, \(q(\vec {\alpha })=0\) implies that \(\sigma \cdot q(\vec {\alpha })=q(\tilde{\sigma }^{-1}(\vec {\alpha }))=\tilde{\sigma }^{-1}(q(\vec {\alpha })) =\tilde{\sigma }^{-1}(0)=0\).
This definition does not depend on the choice of \(q\). Indeed, \(y=q(\vec {\alpha })=q'(\vec {\alpha })\Rightarrow (q-q')(\vec {\alpha })=0 \Rightarrow \sigma \cdot (q-q')(\vec {\alpha })=0\) since \(\sigma \in Gal_K(p)\) and finally \(\sigma \cdot q(\vec {\alpha })=\sigma \cdot q'(\vec {\alpha })\).
Consider for instance the polynomial \(p(x)=x^{3}-2\in {\mathbb {Q}}\). The unique \({\mathbb {R}}\)-root of \(p\) is \(\alpha =\root 3 \of {2}\). If \(\sigma \in Aut({\mathbb {Q}}(\alpha ):{\mathbb {Q}})\), then \(\sigma (\alpha )\) must also be a root of \(p\) in \({\mathbb {Q}}(\alpha )\). Since the other two roots of \(p\) are the complex numbers \(j\alpha \) and \(j^{2}{\alpha }\) with \(j=e^{2i\pi /3}\) (which are not in \({\mathbb {Q}}(\alpha )\)), necessarily \(\sigma (\alpha )=\alpha \). Hence, \(G =Aut({\mathbb {Q}}(\alpha ):{\mathbb {Q}})=\{id_{{\mathbb {Q}}(\alpha )}\}\) and \({\mathbb {Q}}(\alpha )^G={\mathbb {Q}}(\alpha )\ne {\mathbb {Q}}\). Since not all the roots of \(p(x)\) are in \({\mathbb {Q}}(\alpha )\), the transformation that permutes \(\alpha \) and \(j\alpha \) is not in \(G\), but rather in \(Aut({\mathbb {Q}}(j,\alpha ):{\mathbb {Q}})\). One can prove that actually \(({\mathbb {Q}}(j,\alpha ):{\mathbb {Q}})\) is a Galois extension and that \(Gal({\mathbb {Q}}(j,\alpha ):{\mathbb {Q}})\cong S_3\).
The French philosopher Albert Lautman described this succession of extensions from \(K\) to \(K^p\) as an “ascent towards the absolute” by means of which the “the imperfection of a base field \(K\) with respect to a given polynomial” progressively decreases (see Lautman 2011, pp. 126–128).
For instance, the diagonal \(K\)-algebra \(K^{m}\) is not an integral domain, since \((1,0,\ldots ,0)(0,1,\ldots ,0)=(0,0,\ldots ,0)\). Now, the roots in \(K^{m}\) of a polynomial \(p(x)=(x-\alpha _{1})\ldots (x-\alpha _{n})\) with \(\alpha _{i}\in K\), are vectors \(\overrightarrow{\alpha }\) whose components are given by the \(K\)-roots of \(p(x)\). Each \(K\)-root \(\alpha _{i}\) can be identified with the \(K^{m}\)-root \((\alpha _{i},\ldots ,\alpha _{i})\) of \(m\) identical components. However, in addition to the \(K^{m}\)-roots induced by the \(n\) \(K\)-roots, one has all the \(K^{m}\)-roots given by vectors in \(K^{m}\) whose components are not all equal. These extra \(K^{m}\)-roots of \(p(x)\) do not cancel any of the \(n\) linear factors of \(p(x)\). Hence, the reducible minimal polynomial of each extra \(K^{m}\)-roots \(\overrightarrow{\alpha }\) is the product of the factors \((x-\alpha _{i})\) where the \(\alpha _i\)’s are the components of \(\overrightarrow{\alpha }\).
In the context of infinitary Galois theory, the corresponding dual spaces are indeed profinite topological \(G\)-spaces (see Borceux and Janelidze 2001, p. 62). In what follows, we shall only consider the case of finite \(K\)-algebras and finite dual spaces. In this restricted framework, the topology of the dual spaces is the discrete topology. Hence, the profinite topological \(G\)-spaces of the general theory will be simply replaced by finite \(G\)-sets.
A \(K\)-homomorphism \(\varepsilon \in Hom_{K}(\frac{K[x]}{\langle p(x)\rangle },A)\) is completely specified by fixing the element \(\alpha \in A\) to which \(\overline{x}\) (i.e. the class of the polynomial \(x\in K[x]\)) is sent. In other terms, \(\varepsilon (\bar{f}(x))=f(\alpha )\), where \(\bar{f}(x)\) denotes the class of \(f(x)\) in \(K[x]/\langle p(x)\rangle \). Now, in order to guarantee that such a morphism is a homomorphism of \(K\)-algebras, the zero in the quotient algebra \(\frac{K[x]}{\langle p(x)\rangle }\) has to be sent to the zero in \(A\). In particular, the polynomial \(\bar{p}(x)\), being zero in \(\frac{K[x]}{\langle p(x)\rangle }\), has to be sent to the zero in \(A\). This means that \(\varepsilon (\bar{p}(x))=p(\alpha )=0\). Hence, \(\overline{x}\) has to be sent to a root \(\alpha \) of \(p(x)\) in \(A\). In this way, a morphism in \(Hom_{K}(\frac{K[x]}{\langle p(x)\rangle },A)\) is completely specified by choosing a root \(\alpha \) of \(p(x)\) in \(A\) (Borceux and Janelidze 2001, p. 22).
Since we only consider the finite case, where \(V_{p}(A)\) is discrete, the algebra \({\mathcal {C}}(V_{p}(A),A)\) of continuous functions coincides with the algebra \(A^{V_{p}(A)}\) of all functions.
The spectrum \(Spec({\mathcal {U}})\) of a \(C^{*}\)-algebra \({\mathcal {U}}\) is the (locally compact) space of its characters. A character is a non-null morphism of \({\mathbb {C}}\)-algebras \(\chi : {\mathcal {U}} \rightarrow {\mathbb {C}}.\) If \({\mathcal {U}}\) is unital, \(Spec({\mathcal {U}})\) is a compact space. In functional analysis and modern mathematical-physics, a state of a \(C^{*}\)-algebra \({\mathcal {U}}\) is defined as a morphism of \({\mathbb {C}}\)-vector spaces \(\phi : {\mathcal {U}} \rightarrow {\mathbb {C}}\) satisfying certain conditions (see for instance Landsman 1998). This explains why we have decided to call the points of \(V_p(A)\) (i.e. the \(K\)-algebra morphisms \(\alpha : \frac{K[x]}{\langle p(x)\rangle } \rightarrow A\)) \(A\) -states.
By representation, we understand here a \(K\)-algebra homomorphism. Equivalently, we can say that \(\frac{K[x]}{\langle p(x)\rangle }\) is represented in \({\mathcal {C}}(V_{p}(A),A)\) by means of multiplicative operators acting in \(A\).
For the characterization of points as representations of algebraic structures, see Cartier (2001, Sect. 5).
The fact that the functor of points is representable allows us to generalize the previous construction to any \(K\)-algebra, i.e. to \(K\)-algebras that are not necessarily of the form \(\frac{K[x]}{\langle p(x)\rangle }\). Given any \(K\)-algebra \(B\), we can define its functor of points as \(V_{B}(-)=Hom_{K}(B,-)\). The set \(V_{B}(A)\) can be considered the set of \(A\)-points of \(B\). We can now define a kind of Gelfand transform as \(Gel_{A}: B \rightarrow A^{V_{B}(A)}\) given by \(f \mapsto \hat{f}\) where \(\hat{f}(\alpha )=\alpha (f)\in A\). Given a morphism \(\phi : A\rightarrow C\) of \(K\)-algebras, the composition of morphisms of sets guarantees that \(\phi \) induces a map \(Hom_{K}(B, A) \rightarrow Hom_{K}(B, C)\) given by \(f \mapsto \phi \circ f\). This means that the representable functor \(V_{B}(-)\) is a covariant functor.
This way of interpreting \(\widehat{f}(\alpha )\) is justified by considering an analogous situation in the framework of the GNS (Gelfand-Naimark-Segal) construction for \(C^{*}\)-algebras applied to the commutative case. In this context, each \({\mathbb {C}}\)-valued state \(\chi :\mathfrak {U}\rightarrow {\mathbb {C}}\) on an abstract commutative \(C^{*}\)-algebra \(\mathfrak {U}\) defines a different representation \(\pi _{\chi }:\mathfrak {U}\rightarrow \mathcal {B}({\mathcal {H}}_{\chi })\) of \(\mathfrak {U}\) in a Hilbert space \({\mathcal {H}}_{\chi }\) defined by \(\chi \). The Hilbert space \({\mathcal {H}}_{\chi }\) is the closure of \(\mathfrak {U}/{\mathcal {N}}_{\chi }\), where \({\mathcal {N}}_{\chi }\) is the ideal \({\mathcal {N}}_{\chi } = \{f \in \mathfrak {U} / \chi (f^{*}f)=0\}\). The representation \(\pi _{\chi }\) of \(\mathfrak {U}\) in \({\mathcal {H}}_{\chi }\) is given by \(\pi _{\chi }(f)(\bar{g}) \doteq \overline{fg}\), where the bar denotes the equivalence classes in \(\mathfrak {U}/{\mathcal {N}}_{\chi }\) (see Landsman 1998, p. 53). The Gelfand transform \(Gel: \mathfrak {U} \rightarrow {\mathcal {C}}_0(Spec(\mathfrak {U}),{\mathbb {C}})\) associates to each abstract algebra element \(f \in \mathfrak {U}\) an observable \(\hat{f}\) on the pure state space \(\mathcal {P}(\mathfrak {U}) = Spec(\mathfrak {U})\). It is easy to see that the representation \(\pi _{\chi }\) (which is irreducible if \(\chi \in Spec(\mathfrak {U})\)) of \(f\) acting on \(\bar{g}\) amounts to multiply \(\bar{g}\) by the complex number \(\hat{f}(\chi )\in {\mathbb {C}}\), i.e. that \(\pi _{\chi }(f)(\bar{g})\doteq \overline{fg} = \hat{f}(\chi )\bar{g}\). Indeed, \(\overline{fg} - \hat{f}(\chi )\bar{g}=0\) in \({\mathcal {H}}_{\chi }\) iff \(\overline{fg} - \hat{f}(\chi )\bar{g}\in {\mathcal {N}}_{\chi }\). Now, \(h\in {\mathcal {N}}_{\chi }\) iff \(\hat{h}(\chi )=0\) (indeed, \(h \in {\mathcal {N}}_{\chi }\) iff \(\chi (h^{*}h) = 0\) iff \(\widehat{h^{*}h}(\chi ) = \hat{h}^{*}(\chi )\hat{h}(\chi ) = |\hat{h}(\chi )|^{2} = 0\) iff \(\hat{h}(\chi ) = 0\)). In our case, \(\widehat{(fg - \hat{f}(\chi )g)}(\chi )=\hat{f}(\chi )\hat{g}(\chi ) - \hat{f}(\chi )\hat{g}(\chi )=0\). It is also worth noting that the set of numerical values \(\left\{ \hat{f}(\chi )=\chi (f)\right\} _{\chi \in Spec(\mathfrak {U})}\) that the observable \(\hat{f}\) takes coincides with the spectrum of \(f\in \mathfrak {U}\) (i.e. with the set of elements \(\lambda \in {\mathbb {C}}\) such that \(f-\lambda 1_{\mathfrak {U}}\) has no inverse in \(\mathfrak {U}\)). Indeed \(\hat{f}- \lambda \cdot \hat{1}\) is not invertible iff there is some \(\chi \in Spec(\mathfrak {U})\) such that \(\hat{f}(\chi ) - \lambda = 0\).
By duality we mean here that there exists a functor \(Hom_{K}(-, A)\) that permits to canonically associate the space \(V_p(A)=\) \(Hom_{K}(\frac{K[x]}{\langle p(x)\rangle }, A)\) to the \(K\)-algebra \(\frac{K[x]}{\langle p(x)\rangle }\). This duality is for the moment weak in the sense that the functor does not necessarily define an equivalence of categories.
For instance, the set of \({\mathbb {R}}\)-points of the polynomial \(x^{2}+1\) is the empty set. Hence, the algebra \(\frac{{\mathbb {R}}[x]}{\langle x^{2}+1\rangle }\simeq {\mathbb {C}}\) cannot be faithfully represented as an observable algebra on such a space.
Let’s consider an element \(f\in K[x]\) such that \(\widehat{f}(\alpha )=0\) for every \(\alpha \in V_{p}(K^{p})\). Then, since \(p\) is assumed to be separable, the polynomial \(p\) divides \(f\). Therefore, \(\overline{f}=0\) in \(\frac{K[x]}{\langle p(x)\rangle }\). We have thus shown that a null function \(\widehat{f}\) on \(V_{p}(K^{p})\) necessarily comes from the null element in \(\frac{K[x]}{\langle p(x)\rangle }\). Hence, \(Gel_{K^p}\) is injective.
Take for instance \(K={\mathbb {R}}\) and \(p(x)=x^2+1\). Then \(K^p \cong \frac{K[x]}{\langle p(x)\rangle } \cong {\mathbb {C}}\) and \(V_p({\mathbb {C}}) \cong 2\). But by no means \( Gel_{{\mathbb {C}}}: {\mathbb {C}}\rightarrow {\mathbb {C}}^2\) can be surjective. In the general case, it may happen that a \(K\)-algebra \(B\) does not induce, via the corresponding Gelfand transform, all the \(A\)-valued observables on the \(A\)-dual space \(V_{B}(A)\). For instance, if we take \(B=K\) and \(A=K^{p}\), the \(A\)-dual space is composed of a single point \(V_{K}(K^{p})=Hom_{K}(K,K^{p})=\left\{ id_{K}\right\} \) corresponding to the inclusion \(K\hookrightarrow K^{p}\). Now, the image of the \(K\)-algebra \(K\) by the Gelfand transform is the algebra of \(K\)-valued functions on a one-point set, which coincides with \(K\) itself. Hence, it is evident that this observable algebra is not isomorphic to the algebra of \(K^{p}\)-valued observables on \(V_{K}(K^{p})\), which is \(K^{p}\).
A \(K\)-algebra \(A\) is split by an extension \(L\) of \(K\) if 1) \(A\) is algebraic over \(K\) and 2) the minimal polynomial over \(K\) of every element \(\alpha \in A\) factorizes in \(L[x]\) into linear polynomials with distinct roots. It can be shown that \(A\) is split by \(L\) if and only if there is some \(n\) such that \(A \otimes L \cong L^n\) as \(L\)-algebras (Borceux and Janelidze 2001, pp. 23–24). It can also be shown that a \(K\)-algebra \(B\) split by \(L\) is a product \(\prod _{i}B_{i}\) of intermediate fields \(K\subseteq B_{i}\subseteq L\). It is worth noting that the fact that a \(K\)-algebra \(A\) is split by a field \(L\) does not mean that \(L\) contains all the roots of the minimal polynomials of every \(\alpha \in A\). Since \(L\) is a field, it cannot contain all the roots belonging to \(K\)-algebras that are not fields.
By the Primitive Element Theorem (see Lang 2005, p. 243), \(L\) is of the form \(L = K(a) \cong K[x]/\langle q(x) \rangle \), where the irreducible and separable polynomial \(q(x)\in K[x]\) has all its roots in \(L\). This means that \(L\) is a splitting field \(K^q\) of \(q(x)\). Hence the Gelfand transform is injective (see footnote 30).
It is worth noting that the possibility of making products in the algebraic category is a consequence of Grothendieck’s generalization from fields to \(K\)-algebras. Indeed, while the product of \(K\)-algebras is a \(K\)-algebra, the product of two fields is not a field. This has as a consequence that the geometric category, far from only including homogeneous \(G\)-sets, includes general \(G\)-sets composed of many \(G\)-orbits.
If \(B_1\) and \(B_2\) are two \(K\)-algebras, it is well-known that \(B_1 \otimes _{K} B_2\) is their coproduct in the category of commutative \(K\)-algebras. Now if \(B_1\) and \(B_2\) are split by \(L\), we have to check that \(B_1 \otimes _{K} B_2\) is also split by \(L\). As \(K\)-algebras, \(B_1 \otimes _{K} L \cong L^n\) and \(B_2 \otimes _{K} L \cong L^m\) imply that \(B_1 \otimes _{K} B_2 \otimes _{K} L \cong B_1 \otimes _{K} L^m \cong (B_1 \otimes _{K} L)^m \cong L^{nm}\), where we have used that \(B \otimes _{K} L^m \cong (B \otimes _{K} L)^m\) (see Lang 2005, p. 608).
Indeed, it can be shown that the application \(\mu : K^X \otimes K^Y \rightarrow K^{X \times Y}\) defined by \(\mu (f\otimes g)(x,y) \doteq f(x) g(y)\) is an isomorphism if \(X\) and \(Y\) are finite sets.
Indeed, let’s consider the restriction map \(\varphi _{res}: \frac{G}{H} \rightarrow Hom_K(Fix(H),L)\) defined by \([g] \mapsto g|_{Fix(H)}\). This map is well-defined in the sense that it does not depend on the choice of \(g\) in the class \([g]\). If we take another \(g' \in [g]\), we have \(g'=g \cdot h\) for some \(h\in H=Gal(L:Fix(H))\) and \(g'|_{Fix(H)}=g|_{Fix(H)}\) since \(h|_{Fix(H)}=id_{Fix(H)}\). The map \(\varphi _{res}\) is surjective because every \(f \in Hom_K(Fix(H),L)\) can be extended to a \(g \in Hom_K(L,L)=G\). Indeed, if \(\overline{K}\) is an algebraic closure of \(K\) that contains \(L\), \(f\) can be extended to a \(K\)-morphism \(\overline{f}: L \rightarrow \overline{K}\) (see Theorem 2.8 in Lang 2005, p. 233). We can then take \(g = \overline{f}|_{Fix(H)}\). The normality of \((L:K)\) implies that \(Im(g) \subseteq L\). Indeed, every \(a\) in \(Fix(H)\) is a root of \(p_{K, a}\) that belongs to \(L\). Then \(g(a)\) is also a root of \(p_{K, a} \in K[x]\). By normality \(g(a) \in L\) and thus \(g\in G\). Finally, \(\varphi _{res}\) is injective since \(\varphi _{res}(g)=\varphi _{res}(g')\Rightarrow (g'\cdot g^{-1})|_{Fix(H)}=id_{Fix(H)} \Rightarrow g'\cdot g^{-1} \in H \Leftrightarrow \overline{g'}=\overline{g}\). The map \(\varphi _{res}\) is obviously a \(G\)-morphism.
We shall show that this is the case for a homogeneous \(G\) space \({\mathcal {O}}\simeq G/H\). Given a \(G\)-morphism \(\phi :\frac{G}{H} \rightarrow L\), we want to define an element \(a\in Fix(H)\). Let’s take \(a=\phi (\overline{e})\). We have to verify that \(a\) is in \(Fix(H)\): for all \(h\in H\), \(h(a)=h(\phi ([e]))= \phi (h\cdot [e])=\phi ([h\cdot e]) = \phi ([h])=\phi ([e])=a\). Reciprocally, given \(a\in Fix(H)\) we want to define a morphism \(\phi _{a}:\frac{G}{H}\rightarrow L\) and show that it is indeed a \(G\)-morphism. Let’s define \(\phi _{a}\) by \(\phi _{a}([g])=g(a)\). This definition does not depend on the choice of the representative \(g\) in the class \([g]\). For example, if we take another \(g' \in [g]\), we have \(g'=g\cdot h\) for some \(h\in H=Gal(L:Fix(H))\). Hence \(h(a)=a\) and \(g'(a)=g\cdot h(a)=g(a).\) Moreover, \(\phi _{a}(g'\cdot [g])=\phi _{a}([g'\cdot g])= (g'\cdot g)(a)=g'\cdot g(a) = g'\cdot \phi _{a}([g])\), which means that \(\phi _{a}\) is indeed a \(G\)-morphism. It is easy to verify that the two constructions are inverses of each other.
As we have argued in Sect. 5, it is the \(L\)-algebra \(B\otimes _{K}L\) that is isomorphic to the whole algebra of \(L\)-valued observables on \(Spec_L(B)\).
According to the Galois correspondence, \(H=Gal(L:Fix(H))\). For every \(f \in Fix(H)\), \(\widehat{f}([g_1])= \widehat{f}([g_2]))\) if and only if \(\widehat{f}(g_1|_{Fix(H)})= \widehat{f}(g_2|_{Fix(H)})\), where we have used the isomorphism (5). Now, this will be the case if and only if \(g_1(f)=g_2(f)\), that is if and only if \(g_2^{-1}g_1 \in H\). This implies that \([g_1] = [g_2]\).
Indeed the L-valued observable algebra induced by \(Fix(H'')\) is \({\mathcal {C}}(G/H'',L)\), which can be identified with the algebra of functions \(f: G \rightarrow L\) that are constant on every \(H''\)-orbit.
The relationship between Galois theory and the notion of individuals was also addressed in Vuillemin (1962), notably in Sect. 32, pp. 286–289.
Since the subgroup \(gH_0g^{-1}\) is canonically isomorphic to \(H_0\) for every \(g \in G\), we can simply say that \(H_0\) is the group of automorphisms of every \(x\in M\). If \(H_0\) is a normal subgroup of \(G\), then \(H_0\) and \(gH_0g^{-1}\) coincide.
It is worth noting that the Galois–Grothendieck theory provides just a toy model for the analysis of Klein’s ideas given that the ‘Klein geometries’ appearing in the Galois–Grothendieck duality are just discrete (\(0\)-dimensional) \(G\)-spaces.
More generally, we could have started with a Klein geometry isomorphic to \(G/H_0\) with \(H_0\) a strict subgroup of \(G\). Since \(G/H_0\) is not necessarily a group, the elements in \(G/H_0\) do not form a group algebra, but just a \(L\)-vector space. In this case, the figures of the Klein geometry are the elements of this vector space.
The main difference between the figures as subsets of \(G\) and the figures as elements in \(L[G]\) is the following. The figure obtained by “superposing” two figures-as-point-sets of \(G\) is simply the set-theoretic union of the corresponding subsets. On the contrary, the “superposition” of two figures in \(L[G]\) is given by the addition of the corresponding group algebra elements. Let’s consider for instance the “‘superposition” of a circle and a point in the circle. If we consider the circle and the point as subsets of the geometry, their superposition just gives back the circle. On the contrary, if we consider them as elements in \(L[G]\), their superposition (i.e. their addition as group algebra elements) yields a pointed circle.
The characterization of the transformations between figures of the same type in terms of motions is borrowed from Cartan (1974).
We have used here the terminology used by Cartan (1974, p. 15): “If indeed one tries to clarify the notion of equality, which is introduced right at the beginning of Geometry, one is led to say that two figures are equal when one can go from one to the other by a specific geometric operation, called a motion [déplacement].” (quoted and translated by Marquis 2009, p. 19).
In Page and Catren (2014), we argue that this ontologic interpretation of the Galois–Grothendieck duality points towards the possibility of a Galoisian interpretation of Heisenberg indeterminacy principle in quantum mechanics. In a restricted formal framework defined by a Galois extension satisfying certain conditions, we argue that Heisenberg indeterminacy principle (at least for the case of squeezed coherent states) can be understood as a consequence of the correspondence between states endowed with non-trivial groups of automorphisms and the compatible observables that can be evaluated at these states.
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Acknowledgments
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013 Grant Agreement No. 263523). We also thank Daniel Bennequin and Mathieu Anel for helpful discussions concerning the relationship between Galois theory and physics. The reading of Bennequin’s article Questions de physique galoisienne was one of the original motivations for writing this work.
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Catren, G., Page, J. On the notions of indiscernibility and indeterminacy in the light of the Galois–Grothendieck theory. Synthese 191, 4377–4408 (2014). https://doi.org/10.1007/s11229-014-0528-9
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DOI: https://doi.org/10.1007/s11229-014-0528-9