Why does one not represent states with equivalence classes in the common formalism?

Question

Some context:

Usually, one describes states formally through elements of a Hilbert space $\mathcal{H}$ (e.g. the n-dimensional vector space of complex numbers with the standard basis and standard scalar product). This way the representation is not unique - two representations $|\psi_1\rangle$ and $|\psi_2\rangle=e^{i\gamma}|\psi_1\rangle$ represent the same state.

A unique way of representation is achieved by the equivalence relation $|\psi_1\rangle\sim|\psi_2\rangle:\Leftrightarrow \exists\gamma\in\mathbb{R}:|\psi_1\rangle=e^{i\gamma}|\psi_2\rangle$. Marinescu (978-0-12-383874-2) defines states this way in the first place.

Question:

Why does one usually still not calculate with equivalence classes but with elements of $\mathcal{H}$? Also Marinescu abandons the idea of "rays" (meaning equivalence classes) right after introducing them and goes on by using "the state $|\psi\rangle\in\mathcal{H}$".

Problematic scenario: Using equivalence classes would come in handy e.g. in the following scenario. Usually in physics, people do identify a mathematical representation with the label of something. In this case, a state is called / labeled $|\psi\rangle$ using the mathematical representation $|\psi\rangle=(1,0)^T$ for example. Since two representations $|\psi_1\rangle$ and $|\psi_2\rangle=e^{i\gamma}|\psi_1\rangle$ represent the same state (since measurement statistics don't differ), one would have two different terms/labels for the same, which is not welcomed. If one used equivalence classes, this problem wouldn't exist.

Edit: I quickly want to touch on the fact that I said "states are represented by elements of a Hilbert space" and not "states are elements of a Hilbert space". In my opinion, this doesn't matter for the question, since "identifying a mathematical representation with the label of something" as explained in my "problematic scenario" is exactly what this is. Although I see many books don't see a more general term "state" above the mathematical entity - as a sidenote: is there a reason for this?

Answer 1

It's not clear how doing calculations with "equivalence class of quantum states" instead of with psi function like it is usually done, would look like - if you know, maybe add some examples into the question. Density matrix induced by single psi function $\rho(\mathbf r,\mathbf r') = \psi^*(\mathbf r)\psi(\mathbf r')$ has diagonal terms that do not depend on the phase and thus on the representation, but it still has off-diagonal terms that are not unique -they can change if we change to $\psi'(\mathbf r) = e^{i\gamma(\mathbf r)} \psi(\mathbf r)$ with coordinate-dependent $\gamma$.

However, I am not sure about the motivation. What would be the value? Calculations in physics very often use non-unique representations of all things involved - coordinates, physical quantities, even numbers. These are not unique, but refer to some frame of reference, convention, agreed upon definition, or numeral system. Uniqueness of representation of these things is not very important in physics.

Answer 2

I guess part of the confusion is that there are multiple levels of modeling involved. At the very top, you have the actual physical phenomena, which may be complicated and have aspects beyond our knowledge.

Then you narrow that down to only the aspects that are relevant to quantum mechanics: your conceptual model of a quantum state - the physical state. This is the thing that ultimately determines the behavior observed in quantum phenomena and measurement outcomes.

It is an abstraction, and, to quote Edsger W. Dijkstra: "The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise." An abstraction defines conceptually what aspects are relevant (and therefore also what can be ignored or assumed away). So, this abstract state does not necessarily completely describe every aspect of the physical system - but it does completely describe it within the confines of the theoretical model.

Then you have the formal mathematical model of the physical state, a ray in projective Hilbert space (the "ray" terminology comes from the concept of projective spaces). But rays are kind of inconvenient to work with, so you represent states with elements of a Hilbert space. Someone more knowledgeable than me will have to explain why exactly that's the case, but I have a feeling that it has to do with the apparatus of linear algebra and calculus being available for use on Hilbert spaces, and also with the fact that this formalism is more readily utilized by computers.

So, now instead of having rays as first class elements of your model, you have equivalence classes "in the background", but you work with the elements of a Hilbert space - and you keep track of the fact that some formally different elements are going to represent the same conceptual physical state (those lying on the same ray / in the same equivalence class). Because of this, you have to be careful about how you manipulate them mathematically (e.g. how you add them, you're careful to normalize things, etc).

Then keeping all that in mind, you use the term "state" somewhat loosely (e.g. you say "state $|\psi\rangle$" instead of "state represented by $|\psi\rangle$", or some such thing), but ultimately, you're really concerned with the physical states.