Celebratio Mathematica — Daubechies

Complete Bibliography

Works connected to Diederik Emiel Aerts

Filter the Bibliography List

year	title	people

			clear

1978

D. Aerts and I. Daubechies: “About the structure-preserving maps of a quantum mechanical propositional system,” Helv. Phys. Acta 51 : 5–6 (1978), pp. 637–660. MR 542797 article

Abstract People BibTeX

We study \( c \)-morphisms from one Hilbert space lattice (with dimension at least three) to another one; we show that for a \( c \)-morphism conserving modular pairs, there exists a linear structure underlying such a morphism, which enables us to construct explicitly a family of linear maps generating this morphism. As a special case we prove that a unitary \( c \)-morphism which preserves the atoms (i.e. maps one-dimensional subspaces into one-dimensional subspaces) is necessarily an isomorphism. Counterexamples are given when the Hilbert space has dimension 2.

Diederik Emiel Aerts

@article {key542797m,
	AUTHOR = {Aerts, Dirk and Daubechies, Ingrid},
	TITLE = {About the structure-preserving maps
		 of a quantum mechanical propositional
		 system},
	JOURNAL = {Helv. Phys. Acta},
	FJOURNAL = {Helvetica Physica Acta. Societatis Physicae
		 Helveticae Commentaria Publica},
	VOLUME = {51},
	NUMBER = {5--6},
	YEAR = {1978},
	PAGES = {637--660},
	URL = {https://www.e-periodica.ch/cntmng?pid=hpa-001:1978:51::945},
	NOTE = {MR:542797.},
	ISSN = {0018-0238},
}

Read PDF

1978

D. Aerts and I. Daubechies: “Physical justification for using the tensor product to describe two quantum systems as one joint system,” Helv. Phys. Acta 51 : 5–6 (1978), pp. 661–675. MR 542798 article

Abstract People BibTeX

We require the following three conditions to hold on two systems being described as a joint system:

the structure of the two systems is preserved;
a measurement on one of the systems does not disturb the other one;
maximal information obtained on both systems separately gives maximal information on the joint system.

With these conditions we show, within the framework of the propositional system formalism, that if the systems are classical the joint system is described by the cartesian product of the corresponding phase spaces, and if the systems are quantal the joint system is described by the tensor product of the corresponding Hilbert spaces.

Diederik Emiel Aerts

@article {key542798m,
	AUTHOR = {Aerts, Dirk and Daubechies, Ingrid},
	TITLE = {Physical justification for using the
		 tensor product to describe two quantum
		 systems as one joint system},
	JOURNAL = {Helv. Phys. Acta},
	FJOURNAL = {Helvetica Physica Acta. Societatis Physicae
		 Helveticae Commentaria Publica},
	VOLUME = {51},
	NUMBER = {5--6},
	YEAR = {1978},
	PAGES = {661--675},
	URL = {https://inis.iaea.org/search/search.aspx?orig_q=RN:11511554},
	NOTE = {MR:542798.},
	ISSN = {0018-0238},
}

Read PDF

1979

D. Aerts and I. Daubechies: “A characterization of subsystems in physics,” Lett. Math. Phys. 3 : 1 (1979), pp. 11–17. MR 527173 Zbl 0451.03025 article

Abstract People BibTeX

Working within the framework of the propositional system formalism, we use a previous study [Aerts and Daubechies 1979] of the description of two independent physical systems as one big physical system to derive a characterization of a (non-interacting) physical subsystem. We discuss the classical case and the quantum case.

Diederik Emiel Aerts

@article {key527173m,
	AUTHOR = {Aerts, Dirk and Daubechies, Ingrid},
	TITLE = {A characterization of subsystems in
		 physics},
	JOURNAL = {Lett. Math. Phys.},
	FJOURNAL = {Letters in Mathematical Physics},
	VOLUME = {3},
	NUMBER = {1},
	YEAR = {1979},
	PAGES = {11--17},
	DOI = {10.1007/BF00959533},
	NOTE = {MR:527173. Zbl:0451.03025.},
	ISSN = {0377-9017},
}

Read PDF

1979

D. Aerts and I. Daubechies: “A mathematical condition for a sublattice of a propositional system to represent a physical subsystem, with a physical interpretation,” Lett. Math. Phys. 3 : 1 (1979), pp. 19–27. MR 527174 Zbl 0451.03026 article

Abstract People BibTeX

We display three equivalent conditions for a sublattice, isomorphic to a \( \mathscr{P}(\tilde{\mathscr{H}}) \), of the propositional system \( \mathscr{P}(\mathscr{H}) \) of a quantum system to be the representation of a physical subsystem (see [Aerts and Daubechies 1979]). These conditions are valid for \( \dim\tilde{\mathscr{H}} \) \( \geq 3 \). We prove that one of them is still necessary and sufficient if \( \dim\tilde{\mathscr{H}} \) \( < 3 \). A physical interpretation of this condition is given.

Diederik Emiel Aerts

@article {key527174m,
	AUTHOR = {Aerts, Dirk and Daubechies, Ingrid},
	TITLE = {A mathematical condition for a sublattice
		 of a propositional system to represent
		 a physical subsystem, with a physical
		 interpretation},
	JOURNAL = {Lett. Math. Phys.},
	FJOURNAL = {Letters in Mathematical Physics. A Journal
		 for the Rapid Dissemination of Short
		 Contributions in the Field of Mathematical
		 Physics},
	VOLUME = {3},
	NUMBER = {1},
	YEAR = {1979},
	PAGES = {19--27},
	DOI = {10.1007/BF00959534},
	NOTE = {MR:527174. Zbl:0451.03026.},
	ISSN = {0377-9017},
}

Read PDF

1979

D. Aerts and I. Daubechies: “A connection between propositional systems in Hilbert spaces and von Neumann algebras,” Helv. Phys. Acta 52 : 2 (1979), pp. 184–199. MR 553688 article

Abstract People BibTeX

A theorem of Bade proves that for a complete Boolean sublattice \( \mathscr{L} \) of \( \mathscr{P}(\mathscr{H}) \) the following holds: \[ \mathscr{L} = \{P\in \mathscr{L}^{\prime\prime}: P \textrm{ is orthogonal projection operator}\} \] We prove that this theorem does not hold for the physically interesting class of non-Boolean propositional systems embedded in a \( \mathscr{P}(\mathscr{H}) \); we derive however a necessary and sufficient condition under which the theorem does hold. This condition is automatically satisfied if the propositional system is Boolean.

Diederik Emiel Aerts

@article {key553688m,
	AUTHOR = {Aerts, Dirk and Daubechies, Ingrid},
	TITLE = {A connection between propositional systems
		 in {H}ilbert spaces and von {N}eumann
		 algebras},
	JOURNAL = {Helv. Phys. Acta},
	FJOURNAL = {Helvetica Physica Acta. Societatis Physicae
		 Helveticae Commentaria Publica},
	VOLUME = {52},
	NUMBER = {2},
	YEAR = {1979},
	PAGES = {184--199},
	NOTE = {MR:553688.},
	ISSN = {0018-0238},
}

Read PDF

1983

D. Aerts and I. Daubechies: “Simple proof that the structure preserving maps between quantum mechanical propositional systems conserve the angles,” Helv. Phys. Acta 56 : 6 (1983), pp. 1187–1190. MR 734581 article

Abstract People BibTeX

We show that for any \( c \)-morphism \( \phi \) from the lattice \( \mathscr{P}(\mathscr{H}) \) of closed subspaces of a complex Hilbert space \( \mathscr{H} \) (\( \dim\mathscr{H}\geq 3 \)) to another \( \mathscr{P}(\mathscr{H}^{\prime}) \), a conservation property for the angles holds: \( \forall x,y\in\mathscr{H} \), \( x\neq 0\neq y \): \[ \cos(\mathbb{C}x,\mathbb{C}y) = \cos\bigl(\phi(\mathbb{C}x),\phi(\mathbb{C}y)\bigr) .\] This implies that a technical condition needed in [Aerts and Daubechies 1978] can be dropped: every \( c \)-morphism from \( \mathscr{P}(\mathscr{H}) \) to \( \mathscr{P}(\mathscr{H}^{\prime}) \) is an \( m \)-morphism. Our proof uses Gleason’s theorem.

Diederik Emiel Aerts

@article {key734581m,
	AUTHOR = {Aerts, D. and Daubechies, I.},
	TITLE = {Simple proof that the structure preserving
		 maps between quantum mechanical propositional
		 systems conserve the angles},
	JOURNAL = {Helv. Phys. Acta},
	FJOURNAL = {Helvetica Physica Acta. Physica Theoretica.
		 Societatis Physicae Helveticae Commentaria
		 Publica},
	VOLUME = {56},
	NUMBER = {6},
	YEAR = {1983},
	PAGES = {1187--1190},
	NOTE = {MR:734581.},
	ISSN = {0018-0238},
}

Read PDF