# Non-Euclidean spaces in Quantum Mechanics

In quantum mechanics, I have been going through basics of the subject. In general the space of quantum states is Hilbert space (which is Euclidean - I presume). Being just curious, are there any instances where Non-Euclidean spaces are used in quantum mechanics ? I have heard that there is something called Topological QFT, is it associated to my question ?

I know my question is a little ridiculous, but it just out of sheer curiosity.

• What do you mean by "Euclidean" in this context? – Valter Moretti Feb 4 '14 at 6:34
• Perhaps the OP are referring to non positively defined scalar products. Krein-Hilbert spaces. Gupta-Bleuler formalism to deal with quantized EM field has something to do with them. But I think it is too technical to discuss as an example for the OP. – Valter Moretti Feb 4 '14 at 8:26
• @V.Moretti : I am sorry, my knowledge is quite limited. The only kind of Non-Euclidean space that I learnt so far is the Riemannian metric space whose metric is not a kronecker delta. – user38249 Feb 5 '14 at 3:31

The Hilbert space of physical states of any physical system is a positively definite complex vector space i.e. the squared proper length may be computed as $$ds^2 = |da_1|^2 + |da_2|^2 + \dots$$ We may also split the complex coordinates ("amplitudes") $a_i$ to the real parts and imaginary parts which turns the $N$-dimensional complex space to a $2N$-dimensional real Euclidean space. Under this transition, we may also define the angles between two vectors, via $$\cos(\alpha) = \frac{|\langle u| v \rangle|}{|u|\cdot |v|}$$ There are several basic generalizations of this "Euclidean" space to a non-Euclidean one.
First, some of the terms $|da_i|^2$ in the formula for the $ds^2$ could be given negative coefficients; more generally, the bilinear form could be indefinite. If it is negatively definite, we obtain an isomorphic Hilbert space and we should just flip the overall sign of $ds^2$ to achieve the usual, positively definite convention. But if $ds^2$ is really indefinite, i.e. allowing both signs, we have a problem with the interpretation of quantum mechanics because $ds^2$ is interpreted as a probability by quantum mechanics and probabilities just can't be negative (can't have both signs).
So the indefinite Hilbert spaces are not possible for a theory that may be physically interpreted. However, indefinite Hilbert spaces actually often appear in modern theories as an intermediate step – in theories with gauge symmetries, bad ghosts, and good ghosts (BRST quantization). For example, it is natural to have a Hilbert space with 4 polarizations of a photon for each allowed vector $k^\mu$; the signature of this 4-complex-dimensional space is $3+1$, just like for the spacetime. But the gauge symmetry and the related Gauss' law render $1+1$ polarizations unphysical, leaving just 2 physical polarizations (say $x,y$) with a positively definite norm. This is a master example for all analogous situations.
One could also try to discuss noncommutative generalizations of the Hilbert space. But due to the irrelevance of the overall scaling of a vector, $|\psi\rangle \to a |\psi\rangle$, there should be no parameter analogous to the "noncommutativity parameter" on the Hilbert space, either. Noncommutative spaces are useful in quantum mechanics but they generalize classical phase spaces (because $p,x$ don't commute in quantum mechanics), not Hilbert spaces.