# Dimensional Reduction for scalar fields

The main motivation for this question is the paper "Supersymmetric Yang-Mills Theories" by Brink, Schwarz and Scherk where they use dimensional reduction to go from Yang-Mills in $D=4$ to $D=2$. But let's consider a a simpler case.

Consider the action $S[\Phi]$ for a massless scalar field in $D=5$ dimensions. I have coordinates $x^M=(x^\mu,y)$ on a spacetime $\mathbb{R}^{1,3}\times S^1$, hence$$S[\Phi]=\int d^4x dy \partial_M\Phi(x^\mu,y)\partial^M\Phi(x^\mu,y)$$

$y$ is the coordinate on the circle with radius $R$, hence $\Phi(x^\mu,y+R)=\Phi(x^\mu,y)$

The field is periodic in $y$, hence we can expand in a fourier series $$\Phi(x^\mu,y)=\sum_{n=-\infty}^\infty \phi^n(x)e^{2\pi iny/R}$$

Now, in most sources on Kaluza-Klein reduction they focus on the equation of motion for $\Phi$, which is $$\partial_M\partial^M\phi=0$$

from which we obtain $$(\partial_\mu\partial^\mu-m_n^2)\phi^n(x)$$ where $m_n=\left(2\pi n/R\right)^2$.

Now, for each of them, since they are non-interacting, I could go and build an individual action for each of them to produce these equations of motion, eg $$S[\phi^n]=\int d^4x \partial_\mu \phi^n\partial^\mu\phi^n+m_n^2\phi^n\phi^n$$ Hence I can say I have constructed an infinite number of physically valid model of scalars on $D=4$ via dimensional reduction (neverminding the fact that the non-zero modes are tachyonic).

But this is not what Brink, Schwarz, and Scherk do. They expand in the periodic coordinate as I have done and obtain an infinite tower of fields and say "by letting $R$ approach zero and rescaling the fields we keep only the zeroth coefficient in the fourier expansion and are left with a finite number of fields".

So they obtain a $D=4$ action with a finite number of fields.

No sources I have seen go into further details about what is going on here. So, my concrete questions are

1. How does this rescaling work and why does it leave us with a finite number of fields, or are they just throwing away the non-zero modes? I think this is okay since they are not interacting and we can get a perfectly valid model only considering one mode, but wouldn't explain their rescaling.

2. What happens to the integration over $y$? In my method I can bypass it by finding the e.o.m. and finding an action to produce this e.o.m. but I do not know if this is what they do.

Any sources on this would also be very welcome.

EDIT: Original referece. Here My questions concern the bottom of page 8 onwards.

• By the way, if you take $R\to 0$, you will get that the massive states have a very large mass. Forgetting about masses higher than the energies of your interest, you can effectively have a finite number of states. That answers your first point. But I don't understand the second, since I don't know the original reference your are using. Finally, a good reference about this KK compactification is Nastase AdS/CFT book (not the note on the arxiv, the book is more complete.) – CGH Jun 7 '16 at 19:36
• I've reconciled some of what I have with the paper. The main issue is that in the paper the fermions are complex, yet I'm considering a real scalar. For complex fields you can do a nice integration over $y$. Also one can change their fourier series conventions to make things nicer too. I will update with a full answer soon. – Okazaki Jun 8 '16 at 16:47