Fourier Gauge Theory

[Quetzalcoatl]

Hi,

 

I recently had to think a bit about the property of the Fourier Transform being sort of a rotation in the frequency-time (or more generally, q-dq) domain. That got me thinking of symmetry groups. I also recalled that the Fourier operator is unitary. This fits very nicely into gauge theory.

This reminded me of the Fractional Fourier transform. So it's even a continuous rotation, a Lie group.

Moreover, this transformation is extremely connected to Hilbert space, the underpinning of QM, connecting the position operator with the momentum operator (also later for fields in QFT).

 

So we have under our noses a continuous-unitary-Hilbert rotation operator, which transforms not spacetime, but phase-space.

 

Due to these properties, I imagine there should be an invariant way to describe physics, whether it's in the "time domain" or "frequency domain" OR anywhere in between. This set of transformation of the wave-function, for instance, over phase-space, would define a global symmetry group for the action, and so we can look for a conserved quantity (phase-space's own "angular momentum", for rotations in the q-dq plane). If it sometimes isn't a global symmetry, but only a local one, "a new field can be defined, the quanta of which can be described as particles..."

 

Was this subject ever studied as a proposed gauge theory?

 

Note that an important aspect of this is that we have unitarity. You don't see that every day.

Edited January 21, 2016 by Quetzalcoatl

[ajb]

I have not come across anything exactly like what you are asking about. However, studying symmetries of phase spaces is quite standard.

 

You should look up Marco Cariglia, Hidden Symmetries of Dynamics in Classical and Quantum Physics, which appears in Rev. Mod. Phys. You can get a version from the arXiv.

 

In relation to gauge symmetries on phase spaces, this has been studied, in relation to quantum mechanics anyway. This may not really be what you were looking for by have a look at Siamak Khademi and Sadollah Nasiri, Generalized gauge transformations in phase space picture of quantum mechanics: Kirkwood representation, Journal of Physics: Conference Series 128 (2008) 012016.

 

Maybe their references will also give you more papers to hunt down.

[Quetzalcoatl]

In relation to gauge symmetries on phase spaces, this has been studied, in relation to quantum mechanics anyway.

Thank you!

 

I just realized that a local gauge symmetry over phase-space will probably not quantize to a "particle" per-se, but to some weird analogue of one, as it won't have an energy in the usual sense. That is, I think it won't.

 

I have to admit I don't know enough about phase-space geometry, but remember hearing about things such as, the action behaving like a metric connecting [latex]\dot{q}[/latex] and [latex]p[/latex] like a space and its co-space dual, and about the Poincare form on phase-space.

 

You got me interested though, because it sounds like other symmetries exist, and the oddity of it being in phase-space may add some insight to the Fourier rotation idea. If I have any, I'll post it for others to review.

 

Btw, the thing that forced me to think about Fourier transformations was a Computer Vision fellow student who asked how odd it is that the transform looks very similar to its inverse, as she had no background in the subject and was seeing it for the first time!

 

If anyone else can elaborate on the subject, I'd love to hear what you have to say!

Edited January 21, 2016 by Quetzalcoatl

[ajb]

I have to admit I don't know enough about phase-space geometry, but remember hearing about things such as, the action behaving like a metric connecting [latex]\dot{q}[/latex] and [latex]p[/latex] like a space and its co-space dual,

If the velocity space is just the tangent bundle of some configuration space and the phase space is just the cotangent bundle the a mechanical Hamiltonian without a potential is indeed the 'inverse' of a metric (upon assuming non-degeneracy.)

 

 

...and about the Poincare form on phase-space.

For standard mechanical systems the phase-space is a symplectic manifold, it is just a cotangent bundle. But this need not always be the case.

 

 

You got me interested though, because it sounds like other symmetries exist, and the oddity of it being in phase-space may add some insight to the Fourier rotation idea.

I am not sure how you propose to use Fourier transforms, but classically we do have symmetries of the phase space that are not related to symmetries of the configuration space. We encounter interesting examples of these when dealing with geodesic motion on pesudo-Riemannian manifolds. In particular, we can have constants of motion along the geodesics that are not related to isometries of the metric, ie, not diffeomorphisms of space-time. The generators of such hidden symmetries are known as Killing tensors, which are in fact best examined using symplectic geometry. Maybe we discuss this further another time.

[Quetzalcoatl]

I am not sure how you propose to use Fourier transforms

The key point in the idea is to have q & d/dq on the same footing in the Lagrangian, and that the action is invariant under FT of the q & d/dq operators, meaning if we FT all wave functions and operators, the physics (and action) shouldn't change.

Since there is also a Fractional FT, with a rotation parameter [latex]\alpha[/latex], also under that transformation of all wave functions and operators, the physics (and action) shouldn't change.

 

I don't know how to do the calculations in Lagrangian mechanics, so I tried Hamiltonian mechanics instead. In Hamiltonian mechanics we have q & p coordinates in phase space. In QM we promote them to Q & P operators.

 

There is a relation between Q & P:

[latex]P=\mathfrak{F}^{-1}(\hbar Q)\mathfrak{F}[/latex]

[latex]\hbar Q=\mathfrak{F}P\mathfrak{F}^{-1}[/latex]

 

where [latex]\mathfrak{F}[/latex] is the FT.

 

(taken from http://gregnaber.com/wp-content/uploads/Simple-Harmonic-Oscillator.pdfon page 154)

 

This looks similar for example to the relation between [latex]\vec{e_x}[/latex] & [latex]\vec{e_y}[/latex]:

[latex]\vec{e_y}=R_{\frac{\pi}{2}}[\vec{e_x}][/latex]

 

The idea is that FT is a symmetry of the Hamiltonian operator. For that we demand that the commutator of FT with the Hamiltonian is zero. For example, for a free particle, we have:

[latex]H=\frac{P\cdot P}{2m}[/latex]

But we also demand it be invariant under a 90 degree rotation of the x-y coordinates, R:

[latex]H=\frac{R_{\frac{\pi}{2}}P\cdot R_{\frac{\pi}{2}}P}{2m}[/latex]

 

Similarly, we want to demand invariance under a FT (90 degree rotation of the q-p coordinates):

[latex]H'=\mathfrak{F}H\mathfrak{F}^{-1}=\mathfrak{F}\left(\frac{P\cdot P}{2m}\right)\mathfrak{F}^{-1}=\mathfrak{F}\left(\frac{\mathfrak{F}^{-1}(\hbar Q)\mathfrak{F}\cdot\mathfrak{F}^{-1}(\hbar Q)\mathfrak{F}}{2m}\right)\mathfrak{F}^{-1}=\frac{(\hbar Q)^2}{2m}[/latex]

 

[EDIT: The following is wrong. Fractional FT generator should commute with H...]

The FT operator is a symmetry of the mechanics if it commutes with H:

[latex]\left[\mathfrak{F},H\right]\psi=\mathfrak{F}H\psi-H\mathfrak{F}\psi=\mathfrak{F}\frac{P^2}{2m}\psi-\frac{P^2}{2m}\mathfrak{F}\psi=\frac{1}{2m}\left(\mathfrak{F}P^2-P^2\mathfrak{F}\right)\psi[/latex]

 

So I need to show that:

[latex]\mathfrak{F}P^2-P^2\mathfrak{F}=0[/latex]

 

Calculating both terms:

[latex]\mathfrak{F}P^2\hat{\psi}=\mathfrak{F}\left[p^2\hat{\psi}(p)\right]=+\frac{d^2\psi(-q)}{dq^2}[/latex]

[latex]P^2\mathfrak{F}\hat{\psi}=P^2\left[\psi(-q)\right]=P\left[-i\frac{d\psi(-q)}{dq}\times -1\right]=-\frac{d^2\psi(-q)}{dq^2}[/latex]

 

But then I actually get:

[latex]\left[\mathfrak{F}P^2-P^2\mathfrak{F}\right]\hat{\psi}=2\frac{d^2\psi(-q)}{dq^2}\ne 0[/latex]

 

EDIT:

Looking at this some more, I realized I shouldn't have used the FT in the commutator, which is a discrete rotation of a quarter loop in the q-p domain. I should have used the generator of that transformation. This would be derived somehow from the Fractional FT with [latex]\alpha[/latex] very small. This is getting too complicated. What is the way to do this in Lagrangian mechanics???

 

[ajb]

It is still not very clear what you want to do... the Fourier transformation interchanges the position and momentum representations in quantum mechanics. I do not think we usually see this as a symmetry, but rather changing representations.

 

I have no idea how to look at this in the Lagrangian picture as that is really a classical description and you are clearly using non-relativistic QM, which is better suited to the Hamiltonian picture when using operators.

Edited January 22, 2016 by ajb

[Quetzalcoatl]

I do not think we usually see this as a symmetry, but rather changing representations.

Yes, I realize that, but I'm still going to give it a try. The mathematics of it looks similar to a symmetry, but one in phase space. I'm going to try to reformulate this using a simple relativistic Lagrangian. I'm also going to look at the resources you'd provided above, to see what others did in phase space in perhaps similar situations. No promises of course, but I'll keep posting as this evolves and as I find time to work on this...

 

 

 

As an aside, thinking again about the Hamiltonian commutator, for the Fractional FT generator I might try using the formula:

[latex]genrator=\frac{d}{d\alpha}\mathfrak{F}(\alpha)\mid _{\alpha=0}[/latex]

 

Who knows, it might just end up being the regular FT, the same way as i is the generator of the U(1) group:

[latex]i=\frac{d}{d\alpha}\mathfrak{e^{i\alpha}}\mid _{\alpha=0}[/latex]

Edited January 24, 2016 by Quetzalcoatl