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## Main Question or Discussion Point

I am continuing here the discussion of side issues from another thread, quoting a number of Careful's promotional posts for indefinite spaces, negative probabilities, and other unconventional ideas for a generalized quantum mechanics. (A Nevanlinna space is a vector space equipped with an indefinite sesquilinear form, and perhaps further properties which Careful can surely point out if they are relevant for the discussion. A Krein space

is a vector space equipped with an indefinite sesquilinear form and extra topological structure described, e.g., in http://en.wikipedia.org/wiki/Krein_space)

(i) All of quantum information theory happens in finite-dimensional Hilbert spaces, in which all observables are bounded.

(ii) Most successes of quanrtum mechanics (with thousands of applications) are on the level of nonrelativistic quantum mechanics.

(iii) All free quantum field theories (in any dimension) and all quantum field theories which are known to exist rigorously (in d<4) - in that the solvability of the dynamics can be rigorously shown, were constructed in the Hilbert space framework.

(iv) Nobody knows about the rigorous status of interacting field theories in 4 or higher dimensions, no matter which techniques are employed.

Indefinite state spaces haven't made the slightest impact in cases (i)-(iii). They have had important heuristic successes in case of (iv), where ghost states are an important feature of gauge theories. But even there, the physics happens in the centralizer of the BRST charge, which is a standard Hilbert space.

Thus your language seems to be far too strong, and your assertions far too speculative.

is a vector space equipped with an indefinite sesquilinear form and extra topological structure described, e.g., in http://en.wikipedia.org/wiki/Krein_space)

if you generalize away to Nevanlinna space, unitary operators also become unbounded (on some subspace of zero norm states) so here, the picture of bounded operators completely evaporates. [...] So, I think C^{*} algebra's (as well as Von Neumann algebra's) are (a) not natural and (b) too limited.

So, in my opinion, bounded operators are dead.

I am afraid you confuse negative energy with negative probability. They are different things. It is very easy to define a Lorentz covariant positive energy Hamiltonian with positive and negative norm particles. The interpretation of course happens on a sub-hilbertspace but (a) this one is dynamical and by no means invariant under the Hamiltonian and (b) observer dependent. [...]

Right, there is no substance behind bounded operators. The best proof is that we never use them.

For example, hermitian operators can have a complex spectrum (on the ''ghost'' states) which is totally unbounded. [...]

What would be interesting from the point of view of ''C* algebra's'' is that you try to extend the GNS construction to non-positive states, so that you will get Nevanlinna space representations. This requires of course a change in the C* norm identities in the first place, but it might be good to define such generalized algebra's.

Hilbert space is not only unsuitable because it has only positive norm

Actually on Nevanlinna space, there is no natural algebraic criterion which gives only operators with a real spectrum.

It is possible of course to define bounded operators on Krein space, but it is not the natural class of operators (since their very definition requires a Hilbert space construction!)

All I am pointing out is that from where I stand and how I know quantum gravity to work out, bounded operators have evaporated.

But what I want to do is pull this discussion away from some silly textbook prejudices people have to situations where it really matters. For example to QFT or quantum gravity: that is where these issues really show their teeth, not in standard QM.

And here a few comments:No speculation, operators in Krein space have been rigorously studied as well as spectral decompositions and so on. It is just much less known obviously.

(i) All of quantum information theory happens in finite-dimensional Hilbert spaces, in which all observables are bounded.

(ii) Most successes of quanrtum mechanics (with thousands of applications) are on the level of nonrelativistic quantum mechanics.

(iii) All free quantum field theories (in any dimension) and all quantum field theories which are known to exist rigorously (in d<4) - in that the solvability of the dynamics can be rigorously shown, were constructed in the Hilbert space framework.

(iv) Nobody knows about the rigorous status of interacting field theories in 4 or higher dimensions, no matter which techniques are employed.

Indefinite state spaces haven't made the slightest impact in cases (i)-(iii). They have had important heuristic successes in case of (iv), where ghost states are an important feature of gauge theories. But even there, the physics happens in the centralizer of the BRST charge, which is a standard Hilbert space.

Thus your language seems to be far too strong, and your assertions far too speculative.