Notiziario Scientifico

Notiziario dei seminari di carattere matematico
a cura del Dipartimento 'G. Castelnuovo'
Sapienza Università di Roma

Settimana dal 24 al 30 luglio 2017

Mercoledì 26 luglio 2017
Ore 16:00, aula Dal Passo, dipartimento di Matematica, Università di Roma Tor Vergata
Marco Oppio (Università di Trento)
Quantum theory in real or quaternionic Hilbert space: How the complex Hilbert space structure emerges from Poincaré
(Joint work with Valter Moretti) In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Soler's theorem (1995) closing a long standing problem that can be traced back to von Neumann's mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems (in Wigner's approach) cannot be described in real/quaternionic Hilbert spaces as a consequence of some peculiarity of continuous unitary projective representations of SL(2,C) related with the theory of polar decomposition of operators. Indeed such a 'naive' attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the 'naive' result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. In conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while there exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is completely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators with quaternionic coefficients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc Hilbert spaces. Clearly, this property turns out to be crucial. [Based on the paper V. Moretti and M. Oppio: 'Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry', Rev. Math. Phys. 29 (2017) 1750021 and V. Moretti and M. Oppio: 'Quantum theory in quaternionic Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry', in preparation]

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