What is meant by winding state of a field?

[arroy_0205]

Can anybody explain what is meant by winding modes of excitation? How do such states contribute to total energy? For example, in the case of a scalar field in 2 dimensional toroidal space, how to calculate these?

 

[alphanzo]

the winding number just counts how many times a field or object raps around a compactified dimension (a.k.a a circle). You might have a field or object that goes something like $$\phi (x,y)=\phi (x) e^{i2\pi n y/R}$$

Now let's say that y is defined modulo R, y ~ y + R (that is: the compactified dimension has length R). Then you can see that the field above will wind up n times by the time y goes from 0 to R. So n is the winding number. Assuming that a scalar field on a 2D torus is analogous to a closed string then the contribution goes like $$\Delta E = n^{2}R^{2}$$

 

[arroy_0205]

Thanks for your response, but I don't fully understand. Why should the energy contribution go like
$$\Delta E = n^{2}R^{2}$$
Even dimensionally how is that possible? Can you please elaborate a bit?

 

[humanino]

The winding number is not only defined in situation where you have such a thing as a "compactified dimension". This is a very specific term that people use in string theory, and using it here is confusing, especially if you refer to a circle, which is not what people use in string theory.

Winding numbers appear for instance in QCD (or any Yang-Mills in general). Instantons for instance have a non vanishing Pontryagin index, and are essential in many models of the vacuum and/or bound states. It is a topological number indexing equivalent vacua.

Winding numbers appear in situation as simple as a closed loop in a plane. Suppose such a loop encloses the origin. How do you know ? Draw a line from the origin all the way outside the loop. Choose an orientation of the loop. Count positive every time the crossing is (say) right, negative otherwise. If you get a non-zero number, the origin is on the other side of the outside, that is what you define as inside. If the loop winds 3 times, you'll get 3 (or -3). This is also called the Brouwer degree in that case. No matter how complicated you deform the curve as long as you don't cut it : topological property are (what is) stable against continuous deformation.

 

[blechman]

Thanks for your response, but I don't fully understand. Why should the energy contribution go like
$$\Delta E = n^{2}R^{2}$$
Even dimensionally how is that possible? Can you please elaborate a bit?

You have to be careful about what units you're working with (what are you setting "equal to 1") - I always thought of winding number along the same lines as a "particle in a box" from ordinary, nonrelativistic quantum mechanics. There, you have:

mE ~ n^2/R^2, n = integer

(setting hbar = 1 and ignoring pi's and 2's and whatnot). Generalizing this to a relativistic situation, mE -> E^2, and we have the formula for winding number states.

In the case of a "compactified extra dimension" you can think of this extra dimension as a "box", and the above logic holds, with n now physically representing the quantum of momentum in the extra dimension (this follows directly from solving the Schrödinger - or klein gordan if relativistic - equation).

Hope that helps.