Example of a factor space in physics

1. Nov 12, 2015

Geofleur

Suppose $V$ is a vector space and $W$ is a subspace of $V$. Then for $|a\rangle, |b\rangle \in V$, say that $|a\rangle$ and $|b\rangle$ are equivalent (in the sense of an equivalence relation), if $|a\rangle - |b\rangle \in W$. Denote the equivalence class of $|a\rangle$ by $[\![ a ]\!]$. The factor space (also called the quotient space) is defined via the underlying set $\{ [\![ a ]\!] \; | \; | a\rangle \in V \}$ together with the addition rule $\alpha [\![ a ]\!] + \beta [\![ b ]\!] = [\![ \alpha a + \beta b ]\!]$ for combining elements in this set, with $\alpha$ and $\beta$ scalars of the original space $V$.

For an intuitive idea of the meaning of a factor space, let $V = \mathbb{R}^2$, $W$ be a line passing through the origin, and $| v \rangle$ be a position vector originating from the origin. Then $[\![ v ]\!]$ is the line obtained by shifting every point in $W$ (regarded as a position vector) by $|v\rangle$. The factor space is the set of all lines that are parallel to $W$.

Does anyone know of a good example, from physics, of a factor space?

2. Nov 12, 2015

Staff: Mentor

Do Lorentz groups count?

3. Nov 12, 2015

andrewkirk

The Fock Space is an important example of such a space, that occurs in quantum mechanics. It is obtained by taking quotients of tensor spaces. But tensor spaces are also vector spaces. The quotient is used to model the indistinguishability of particles.

4. Nov 12, 2015

Staff: Mentor

But where is the quotient there? Bosonic it's a tensor algebra and fermionic a Graßmann algebra. Graßmann algebras can be viewed as factor algebras. Is the Pauli principle basically the equivalence relation here? Can it be viewed this way?

Last edited: Nov 12, 2015
5. Nov 12, 2015

andrewkirk

As you note, there's a quotient taken in constructing the Grassman Algebra, which can model collections of fermions. I can't remember how it works for bosons. I thought there was a quotient taken in there as well but my memory is dim.

I wrote some good notes on this topic a few years ago but I went looking for them a couple of months ago and couldn't find them anywhere. It's most disconsoling. It all came out of a thread on here where I was ranting about the sloppiness with which most QM texts approach the issue of indistinguishable particles, whereby they just start breaking all the rules they've laid down in the previous chapters, without justification. Most posters said don't worry about it, it's just pedantry. But one helpful person pointed me in the direction of Fock spaces which, IIRC, led me on a most satisfying journey through Wedge Products and Symmetric and Antisymmetric Algebras. I think I go to the point where I decided the universe is - GR and string theory aside - just one enormous Fock Space.

I'll see if I can dig up the thread (and, if possible, the notes). Anything further I say without consulting my notes or recreating them from scratch is likely to be wrong.

6. Nov 12, 2015

Staff: Mentor

@andrew: Thank you. Your remark on the universe remembers me on the SUSYs. First their gauge groups got larger and larger, next they graded, i.e. doubled their playground. As long as they don't come up with an s-particle I can't avoid to think: Yes, if you blow up your dimensions long enough you can embed and model everything within.

But my last question was meant seriously. The fermionic Fock space is a Graßmann algebra and therewith its quotient is $span \{ v_1 ⊗ ... ⊗ v_n | v_i = v_j$ for a $i ≠ j \}$. That looks pretty much like the Pauli principle.
Is this interpretation correct?

Last edited: Nov 12, 2015
7. Nov 12, 2015

andrewkirk

I found the old thread. Here it is. There are some links in there to uni notes that I think may answer some of these questions. Maybe if I follow them I'll remember where I put my own notes.

I think you're probably right about the Pauli principle @fresh_42. I'm not seeing right now the equivalence between your span definition above and the quotient definition $\Lambda(V)\equiv T(V)\ /\ I$ where $I$ is the two-sided ideal generated by $\{x\otimes x\ |\ x\in V\}$. But once I reflect on it and refresh my understanding of these objects I daresay it will become clearer to me.

8. Nov 12, 2015

Staff: Mentor

Thank you. Something learned today. Thought it were easier to find an example for the OP. I was on de Sitter spaces ...(I don't see the equivalence of the 2 definitions either at once. But I remember they are. It's the usual construction of universal (sic!) objects of this kind.)

9. Nov 13, 2015

Geofleur

Thanks guys!

10. Nov 24, 2015

andrewkirk

@fresh_42 In case you're still interested, I managed to eventually find the notes I wrote a while ago on this.
It's saved as PDF here.
I can't be absolutely sure it has everything I wrote in it, but it seems to contain most of the things I remember poking into.
When I get a nice chunk of time, I'll go over it myself and try to convert it to latex so it's more amenable to putting on the web, and I can refresh my knowledge, correct errors and fill in gaps along the way.

11. Nov 24, 2015

Staff: Mentor

I am, thank you. I am dealing with an invariant of Lie Algebras which has some interesting properties. I'm asking myself whether it's just a nice mathematical fact or if there's anything "real" about it. It's a leftover of a thesis I once started a long time ago. And regarding the whole thing as a quotient of a tensor algebra is one possible view on it.