A question on the Radion field

[robousy]

I'm trying to understand how a field is also a dimension.

For example, consider the following from a brane paper "...the distance between branes is a massless degree of freedom, the radion field."

Can anyone help me understand this? The field - dimensional radius relationship.

 

[Haelfix]

Yea, its a bit subtle, b/c how you introduce the radion field is somewhat model dependant.

So you will have a background metric in general (say RS) and you will have a parameter in front of the coordinate describing the compactified small dimensions (call it Y). So 0<y<2pi * R. Where R is really the radius of the orbifold. Convenient variable changes will lead to something like R^2 dphi^2

The radius R in the first version of RS is not determined by the dynamics so the radion field is said to be a massless degree of freedom. In other models it can be different.

The super naive way of going from the constant R to a scalar field (The Radion) is simply to promote it to r(X) (X is the other coordinates of the metric). Intuitively it sort of governs the interspacing between branes. But this naive way is not quite right, but its almost right. Anyway I leave the details in any number of arxiv preprints on the subject (I don't have one handy atm)

 

[BenTheMan]

See if this paper helps: http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3844v1.pdf.

The second author (Jay Hubisz) was at OSU last winter.

You might also check out the Goldberger-Wise paper, who were the first people to address this problem: http://arxiv.org/abs/hep-ph/9907447.

 

[robousy]

Thanks for the references, I'm taking a look at the GW paper now.

And thanks for the insight Haelfix.

 

[javierR]

The scalar (radion) field is not a dimension; it gives a measure of distance in a dimension. More specifically, the idea comes from general relativity in which a spacetime has no metric *a priori*, so no physical spatial and temporal measurements can be made. You can have coordinate frames, which can be arbitrarily chosen, but measurements using these frames are not physically meaningful. This is the purpose of the metric: to ascribe geometry to a (topological) space. Now, in the example of theories in 5 dimensions with a circular 5th dimension, the 5-dim metric field splits into a 4-dim metric for the 4-dim (non-compact) spacetime + a scalar "metric" for the 5th dimension + something else...The background value (vacuum expectation value in quantum field theory) of this scalar field is the size of the circle dimension *in units of the coordinate frame you choose*. (Hence the name "radion"). Another simple example is to slice 5-dim (non-compact) space with 4-dim walls; the distance between each wall corresponds to an independent scalar field as above. The idea of a scalar field corresponding to a length is more general than these examples, though.

 

[BenTheMan]

The scalar (radion) field is not a dimension; it gives a measure of distance in a dimension. More specifically, the idea comes from general relativity in which a spacetime has no metric *a priori*, so no physical spatial and temporal measurements can be made. You can have coordinate frames, which can be arbitrarily chosen, but measurements using these frames are not physically meaningful. This is the purpose of the metric: to ascribe geometry to a (topological) space. Now, in the example of theories in 5 dimensions with a circular 5th dimension, the 5-dim metric field splits into a 4-dim metric for the 4-dim (non-compact) spacetime + a scalar "metric" for the 5th dimension + something else...The background value (vacuum expectation value in quantum field theory) of this scalar field is the size of the circle dimension *in units of the coordinate frame you choose*. (Hence the name "radion"). Another simple example is to slice 5-dim (non-compact) space with 4-dim walls; the distance between each wall corresponds to an independent scalar field as above. The idea of a scalar field corresponding to a length is more general than these examples, though.

So the radion is basically the G_55 component of the five dimensional metric G? This makes sense. It's a scalar field in the same sense that g_{\mu\nu} is a graviton.