Quantum mechanics beyond hilbert space

Eduard Prugovečki

Jean-Pierre Antoine

We review the different formalisms that can be used for quantum mechanics, all of them going beyond the standard Hilbert space formulation: Rigged Hilbert Spaces (RHS), partial inner product spaces (PIP spaces), etc. In particular, we consider the spectral analysis of observables and the description of symmetries.

Howard Barnum, Salman Beigi, Sergio Boixo +2 more

We show that, assuming that quantum mechanics holds locally, the finite speed of information is the principle that limits all possible correlations between distant parties to be quantum mechanical as well. Local quantum mechanics means that a Hilbert space is assigned to each party, and then all local positive-operator-valued measurements are (in principle) available; however, the joint system is not necessarily described by a Hilbert space. In particular, we do not assume the tensor product formalism between the joint systems. Our result shows that if any experiment would give nonlocal correlations beyond quantum mechanics, quantum theory would be invalidated even locally.

Eduard Prugovečki, Yuh‐Kang Pan

Jan Myrheim

The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics, keeping the same set of physical states, but admitting more general observables. The standard time reversal operator involves complex conjugation, in this sense it goes beyond the complex theory and may serve as an example to motivate the generalization. Another example is unconventional canonical quantization such that the harmonic oscillator of angular frequency $ω$ has any given finite or infinite set of discrete energy eigenvalues, limited below by $\hbarω/2$.