Dimensional reduction

In field theory

In statistical or quantum field theory, dimensional reduction is the limit of a compactified theory where the size of the compact dimension(s) goes to zero. For a system with p compact dimensions, the excitations that survive after taking this limit are zero modes that depend on the remaining d = D - p non-compact coordinates.

For example, consider a periodic compact dimension with period L. Let x be the coordinate along this dimension. Any field ${\displaystyle \varphi }$ can be described as a sum of the following terms:

${\displaystyle {\varphi }_n(x)=A_n\cos \left(\frac{2\pi nx}{L}\right)}$

with ${\displaystyle A_n}$ a constant. According to quantum mechanics, such a term has momentum ${\displaystyle nh/L}$ along x, where h is the Planck constant. Therefore, as L goes to zero, the momentum goes to infinity, and so does the energy, unless ${\displaystyle n=0}$:

${\displaystyle E_n\propto \frac{n^2}{L^2}.}$

At finite energy E, we can therefore only excite the ${\displaystyle n=0}$ mode. So in the limit where L goes to zero, ${\displaystyle \varphi }$ must be a multiple of ${\displaystyle {\varphi }_0}$, which doesn't depend on x.

This argument generalizes. The compact dimension imposes specific boundary conditions on all fields, for example periodic boundary conditions in the case of a periodic dimension, and typically Neumann or Dirichlet boundary conditions in other cases. Now suppose the size of the compact dimension is L; then the possible eigenvalues under gradient along this dimension are integer or half-integer multiples of 1/L (depending on the precise boundary conditions). In quantum mechanics this eigenvalue is the momentum of the field, and is therefore related to its energy. As ${\displaystyle L\to 0}$ all eigenvalues except zero go to infinity, and so does the energy. Therefore, at this limit, with finite energy, zero is the only possible eigenvalue under gradient along the compact dimension, meaning that nothing depends on this dimension.

This idea applies to thermal quantum field theory, since a thermal system at high temperatures can be thought off as having a compactified circle of radius ${\displaystyle 1/T\to 0}$.