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?

2 comments:

  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

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  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.

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