Wednesday, July 9, 2014

units of spectra

can you please show that the unit of the fluctuation spectrum $P_\phi$ of a field $\phi(\vec{x})$ is given by the unit of the variable $\vec{x}$ on which the field depends to the power of the dimension of the space on which the field is defined times the squared unit of the field amplitude?


  1. Can be seen immediately from the definition of the power spectrum, the square of the Fourier transform: P_phi = V |1/V int dV phi(x) exp(ikx)|^2

  2. the spectrum is defined by the variance of Fourier modes, $\langle\phi(\vec{k})\phi(\vec{k}^\prime)\rangle\propto P_\phi(k)\delta_D(\vec{k}-\vec{k}^\prime)$. the Fourier-transformed field acquires additionally to the unit of $\phi$ itself the unit of a volume, $\phi(\vec{k})=\int\mathrm{d}^n k\:\phi(\vec{x}\exp(-\mathrm{i}\vec{k}\vec{x})$. likewise, the Dirac-function has the unit of a volume, which can be seen through the relation $\int\mathrm{d}^n k\:\delta_D(k)=1$. putting everything together yields for the unit of $P_\phi(k)$ the square of the amplitude of $\phi$ times a volume.