I am getting rather confused by the dimensions of the Hilbert space in which a state $\psi$ lives, and with regards to the distinction between the Hilbert space and projective Hilbert Space.

Consider a two-state system like spin. This describes a two dimensional Hilbert space over the field of complex numbers. Now my first point of confusion is regarding this. Surely, for each dimension we have two independent degrees of freedom: the magnitude of the complex number and its phase. However we still speak of dimensionality two of the state space.

I see that when we consider the physical interpretation and corresponding demands on the normalisability of psi, and what is experimentally verifyable, we could argue that a) the normalisability of psi eliminates one degree of freedom and b) only the relative phase of the coefficients for the two basis vectors matters, eliminating another degree of freedom. This is the understanding I get from the top answer in this post corresponding to the motivation behind the Bloch Sphere respresentation; but it is, to my mind, unsatisfactoraly realist about psi. I don't think that a vector space should care at all about the physical demands on the state and what gives rise to physically distinct states, so normalisability requirements and the phsyical interpretation of relative phases should, I believe, not have any bearing on the space-and its dimensionality- in which the state lives. It can, however, motivate representations of the state, which is what I think the Bloch sphere is aiming at. And besides, we still refer to psi being in a two dimensional Hilbert space over the Complex numbers, and in this space we do allow linear combinations of basis states that are not normalised. So trying to explain the difference in representaing the state in $\mathbb{C}^n$ rather than $\mathbb{R}^{2n}$ considering the physical significance of psi is conceptually ugly and unsatisfactory.

Secondly, the dimensionality doesn't even match up right when considering higher dimensional vector spaces. Normalisation always eliminates one degree of freedom, and the physical significance of relative phases only also eliminates one degree of freedom (set one of the phases arbitrarily to 0, and all other phases are then important and measurable). So this reasoning does not eliminate the additional degrees of freedom of considering $\mathbb{C}^n$ compared with $\mathbb{R}^{2n}$

A final unanswered question in my mind pertains to the projective Hilbert Space representation, with the state space of psi being a hypersphere in an $n+1$ dimensional space. The number of degrees of freedom on moving along the hypersphere must be the degrees of freedom the state can have. But then what is the basis for this higher dimensional vector space? It has one extra dimension!

EDIT: I have just seen this questions which is essentially a concentrated form of the part of my question above relating to the Bloch sphere, however I do not feel the response actually answers any of the questions, only clarifies the relation between where the degrees of freedom go for a two state system, and does not generalise to higher dimensions.

  • 2
    $\begingroup$ Can you edit the question to add a succinct summary of your question(s)? You've written a lot here, but it's not entirely clear what you're asking. $\endgroup$ – Michael Seifert Jun 26 '18 at 13:07
  • 1
    $\begingroup$ Also, related (and possibly duplicate): Hilbert space vs. Projective Hilbert space $\endgroup$ – Michael Seifert Jun 26 '18 at 13:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.