Error in Modern Theory of Critical Phenomena by Shang-Keng Ma

[matematikuvol]

In book Modern theory of critical phenomena author Shang - Keng Ma in page 17.

$$\sigma_{\vec{k}}=V^{-\frac{1}{2}}\int d^3\vec{x}e^{-i\vec{k}\cdot\vec{x}}\sigma(\vec{x})$$

$$\sigma(\vec{x})=V^{-\frac{1}{2}}\sum_{\vec{k}}e^{i\vec{k}\cdot \vec{x}}\sigma_{\vec{k}}$$

Is this correct? How can inversion of continual FT be discrete FT? Thanks for your answer.

 

[DrDu]

The sum is to be understood as an integral over a comb of very narrow functions which approach delta functions in the limit V to infinity, i.e. think of the FT of a product of a periodic function with a rectangle of width V.
The integral over the delta functions is then equivalent to a sum over their locations.

 

[torquil]

In book Modern theory of critical phenomena author Shang - Keng Ma in page 17.

$$\sigma_{\vec{k}}=V^{-\frac{1}{2}}\int d^3\vec{x}e^{-i\vec{k}\cdot\vec{x}}\sigma(\vec{x})$$

$$\sigma(\vec{x})=V^{-\frac{1}{2}}\sum_{\vec{k}}e^{i\vec{k}\cdot \vec{x}}\sigma_{\vec{k}}$$

Is this correct? How can inversion of continual FT be discrete FT? Thanks for your answer.

If the x-domain is bounded, the k-domain will be discrete. This is what happens for an ordinary Fourier series for functions on a bounded domain, or periodic:

http://en.wikipedia.org/wiki/Fourier_series