arXiv:1702.03157

Preserver problems for the logics associated to Hilbert spaces and related Grassmannians

Mark Pankov

Published Feb 10, 2017• 0 citations

Abstract

We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$ and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.

DOI: 10.48550/arxiv.1702.03157

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arXiv:1702.03157

Preserver problems for the logics associated to Hilbert spaces and related Grassmannians

Mark Pankov

Published Feb 10, 2017• 0 citations

arXiv HTML PDF

Abstract

We consider the standard quantum logic

{\mathcal L}(H)

associated to a complex Hilbert space

H

, i.e. the lattice of closed subspaces of

H

together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic

{\mathcal L}(H)

) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of

{\mathcal L}(H)

and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.

DOI: 10.48550/arxiv.1702.03157

Related papers

Orthogonal apartments in Hilbert Grassmannians

Mark Pankov

Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal L}(H)$ be the logic formed by all closed subspaces of $H$ . For every natural $k$ we denote by ${\mathcal G}_{k}(H)$ the Grassmannian consisting of $k$ -dimensional subspaces. An orthogonal apartment of ${\mathcal G}_{k}(H)$ is the set consisting of all $k$ -dimensional subspaces spanned by subsets of a certain orthogonal base of $H$ . Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of ${\mathcal G}_{k}(H)$ . We show that every bijective transformation $f$ of ${\mathcal G}_{k}(H)$ such that $f$ and $f^{-1}$ send orthogonal apartments to orthogonal apartments (in other words, $f$ preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of ${\mathcal L}(H)$ .Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal L}(H)$ be the logic formed by all closed subspaces of $H$. For every natural $k$ we denote by ${\mathcal G}_{k}(H)$ the Grassmannian consisting of $k$-dimensional subspaces. An orthogonal apartment of ${\mathcal G}_{k}(H)$ is the set consisting of all $k$-dimensional subspaces spanned by subsets of a certain orthogonal base of $H$. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of ${\mathcal G}_{k}(H)$. We show that every bijective transformation $f$ of ${\mathcal G}_{k}(H)$ such that $f$ and $f^{-1}$ send orthogonal apartments to orthogonal apartments (in other words, $f$ preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of ${\mathcal L}(H)$.

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According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.

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Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

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arXiv:1604.06973Mathematica SlovacaOct 1, 20183 citations

We consider the standard quantum logic

{\mathcal L}(H)

associated to a complex Hilbert space

H

, i.e. the lattice of closed subspaces of

H

{\mathcal L}(H)

) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of

{\mathcal L}(H)

and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.

Let

H

be an infinite-dimensional complex Hilbert space and let

{\mathcal L}(H)

be the logic formed by all closed subspaces of

H

. For every natural

k

we denote by

{\mathcal G}_{k}(H)

the Grassmannian consisting of

k

-dimensional subspaces. An orthogonal apartment of

{\mathcal G}_{k}(H)

is the set consisting of all

k

-dimensional subspaces spanned by subsets of a certain orthogonal base of

H

. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of

{\mathcal G}_{k}(H)

. We show that every bijective transformation

f

{\mathcal G}_{k}(H)

such that

f

and

f^{-1}

send orthogonal apartments to orthogonal apartments (in other words,

f

preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of

{\mathcal L}(H)

{\mathcal H}

correspond to direct product decompositions H ≃ A × A ⊥

\mathcal{H}\simeq A\times A^\perp

of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.