# From Metric Spaces to Linear Spaces

1. May 22, 2012

### the_pulp

Hi there, Ive been asking in the Quantum Physics Forum about some foundational stuff and Ive got stuck in why we assume in QM that the states space is linear. I have not found any deeper answer than "because it fits with observation".
So Im now trying the other way around. I believe that the state space should at least be a metric space, so, is there any theorem or something that says something like:

"If we add a Metric Space properties A, B an C then the space is Linear"

I mean, Im looking something like those theorems that state which are the things we have to add to a topological space in order to make it metrizable (that is, to find the only metric that is coherent with its topology). In this line of thought, Id like to find some theorems that state which are the properties that we have to add to a Metric Space in order to make it "Linearizable" (that is, to find the only field, like R or C or Q or Quaternions or Grassmann numbers or whatever, that is coherent with its metric).

Does this exist?

Thanks!

2. May 22, 2012

### chiro

Hey the_pulp.

What kind of metric are you referring to? Also have you checked the Hilbert-space interpretation of Quantum-Mechanics?

http://en.wikipedia.org/wiki/Hilbert_space#Quantum_mechanics

The hilbert-space has an inner-product which means that not only can get norm information (which denotes length information and has the parallelogram property that the metric doesn't), but you can get geometry like angle information using the inner product definition.

3. May 22, 2012

### lavinia

I think in quantum mechanics state spaces are linear because the principle of superposition holds. Superposition implies linearity. There is no need for a metric.

Another way to look at it is that the Shroedinger equation is linear so the solution space will be a vector space.

Topological spaces generally can be metrized in many ways. There is no unique metric.
Generally topological space can not be linearized.

4. May 22, 2012

### the_pulp

Yes, I know. But in fact I was asking in that moment why we assume the Superposition Principle. Because to me, Superposition is equal to linearity and, so, to explain that Linearity must hold because Superposition must hold is like saying that Linearity must hold, because Linearity must hold.

So, yes, I know that we should work with a Hilbert space, but my question was, Why? is there something deeper than "fits with experiments"? And my biggest concern was with the Linearity assumption.

So, now we go again, Im ok with assuming that the State Space is a metric space (because I believe that with the probabilities we can construct some sort of metric), so, is there any theorem that explains what other ingredients we have to add to a metric space to make it a linear space?

Why do I want to know that? because if I find which are those ingredients, perhaps it would be easier to understand why we assume that the state space should have those ingredients?

I think that the following quote of myself resumes what I need to know:

5. May 22, 2012

### lavinia

Superposition fits every experiment ever done. It is not an arbitrary assumption. It is not like saying linearity must hold because linearity must hold. It is a theorem that superposition translates in to linearity of the underlying physical law.

I don't know much about Quantum Mechanics but I think a Hilbert space is assumed because observations correspond to projection operators. Superposition by itself does not require this.

Probabilities do not require a metric. Anyway in Quantum Mechanics probabilities are derived quantities. The fundamental quantities are amplitudes.

There are theorems on when a topological space admits a metric. Certainly it must be a Hausdorff space. But having a metric does not mean it can be a linear space. In my opinion your question is answered by the reverse reasoning. Start with a linear space - because superposition is what you want - and then ask what metrics it can admit.

In Quantum Mechanics though, it is not the metric that is key - it is the inner product. the inner product allows you to project. An arbitrary metric will not allow you to project. An inner product requires that you start with a vector space.

Last edited: May 22, 2012
6. May 22, 2012

### the_pulp

Ok, but, I think that the inner product assumption is too strong. I think that the only assumption that we need is that there should be a function from 2 states to a field (R, C, Grassmann whatever) and then we should put the result of that function in some sort of mechanism that gives back a probability. We dont need the space to be linear in order to have this funtion (or am I wrong?).

So,
1) the State Space should be topological (I sort of thought that the metric assumption should not be debated but Im not sure so from now on I will only say that it is topological) and there should be a function that relates continuously every pair of states to an element of a field. Why then we go on and say that
2) the State Space should be linear and the mentioned function should be the Inner Product?

In order to ask something related to this particular subforum (if you want to continue this QM discusion, I have a thread open in QM forum "on the use of Hilbert Spaces to represent states"), is there a way to add something to 1) that takes me to 2) (like we can add a generic topological space some technical properties and that assures that it is also a Metric Space)???

7. May 22, 2012

### the_pulp

Sorry my ignorance but is there any difference between superposition and linearity?

8. May 22, 2012

### micromass

Staff Emeritus
Since this involves quantum physics, I moved it to that forum.

9. May 22, 2012

### martinbn

How about because Schrodinger's equations is linear. So the set of solutions is a vector space.

10. May 22, 2012

### lavinia

I meant to say that experimentally interference is observed. This means that the amplitudes of the phenomenon superpose which means that at each point they add together. MAahematically, this can be interpreted as a vector space of solution

If the phenomena are solutions to a partial(or ordinary) differential equation then the equation must be linear. So for instance, the wave equation is linear and describes linear waves, waves that superpose. The same is true of the heat equation and the Shoedinger equation.

Last edited: May 22, 2012
11. May 22, 2012

### lavinia

12. May 22, 2012

### Fredrik

Staff Emeritus
the_pulp, you seem to be ignoring an important detail. The pure states aren't members of a vector space; they are the 1-dimensional subspaces of a vector space. So if you're looking for something more general that might include QM as a special case, it seems strange to try to replace the Hilbert space structure with something more general. You should probably be looking specifically at the set of 1-dimensional subspaces instead, since if you drop the vector space structure, the concept of "1-dimensional subspace" will no longer make sense.

It's also reasonable to look at the full set of states instead of just the pure states. One approach is to simply view QM as a generalization of probability theory in the following way: Probability theory is about probability measures defined on σ-algebras. QM is about a kind of generalized probability measures defined on lattices. (It's appropriate to view this as a generalization, since σ-algebras are special kinds of lattices). Now we can look for reasons to consider specific kinds of lattices, like the lattice of Hilbert subspaces of a Hilbert space. This stuff is however really difficult. I don't think there are many people who understand it well. This book looks interesting (but how the bleep did anyone think it would be a good idea to charge almost \$400 for it? I guess they're thinking that a vast majority of the buyers will be university libraries anyway).

Why do you find the "it's simple and it works" answer so unsatisfactory? I hope you realize that the only thing that can explain why it works is a better theory to replace QM, and that if we had one, you'd probably be asking about why that works.

Last edited: May 22, 2012
13. May 22, 2012

### theorem4.5.9

I cannot speak at all about physics, as I have an $\epsilon$ of knowledge there. However I can answer this mathematical question that has thus far seemed to have been ignored.

Frankly, there is not really anything to be gained by approaching your problem this way. The properties you have to add to the metric space is basically an assumption that you began with a vector space. Let me explain this.

A metric space structure only assumes that for every pairing of elements, you have a real number that behaves nicely. You have a notion of comparison, and that's it.

A vector space structure assumes something entirely different, that for every pairing of elements, you have a new element that behaves nicely. You have a notion of addition.

These two properties (at this level) have nothing to do with each other. If you want to try to connect to the ideas, then you assume that your vector space structure also has a norm, for every element you have a real number. Then you have an induced metric, but this metric has nothing to do with your original (or any other) metric your space had.

So summarizing, because a general metric has nothing to do with the addition structure, you cannot induce an addition structure solely from your metric. The opposite is only true because you assume your addition structure has a norm which could act as an intermediary between the two different structures (but the norm fundamentally comes from the addition structure, not the metric structure).

I hope this helps, let me know if I can clarify anything.

14. May 22, 2012

### the_pulp

This is not the answer I was looking, but it looks as the answer, so... thanks!

I was ignoring that important detail. I will rethink it with that in mind and perhaps that solves everything.

I dont know. I want to know why the universe works that way. I know that if I arrive to the basic axioms of something then I wont find anything that explains them. It is just a question of the smell of it. I just smell that, for example, Lorentz covariance is a basic axiom and linearity is not (but I could be totally wrong).