# Why don't we formulate quantum theories in real Hilbert spaces? (Part II)

Università degli Studi Trento

38123 Povo (TN)

Tel +39 04 61/281508-1625-1701-3786

dept.math [at] unitn.it

**Luogo:** Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari "-1"**Ore** 15:00

**Relatore: Marco Oppio**(Università di Trento)

**Abstract: **

As shown in the previous seminar, a "naive" attempt to define a relativistic particle on a real Hilbert space by borrowing the well-known approach of Wigner leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real-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 we end up with three possibilities: an equivalent description in terms of Wigner unitary representations on 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 extreme options. Conclusion, the real theory is actually complex

**Referente: **Davide Pastorello