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While going through a book named Mirror Symmetry, I came across a path integral,

$Z(\beta) = \int\limits_{X(t+\beta) = X(t)} DX(t) \exp\left(-\int\frac{1}{2}( \dot{X}^2 + X^2)dt\right)dt $

where $X(t)$ has a periodicity of $\beta$.

Using orthonormal eigenfunctions of the operator $\Theta = -\frac{d^2}{dt^2}+1 $, the exponential term is given as $\exp(-\frac{1}{2}\sum_n \lambda_n c^2_n) $, where the $\lambda_n$ are the eigenvalues. To perform the above integral, the measure is transformed as, $ DX(t) = \prod_n \frac{dc_n}{\sqrt{2\pi}}$ I didn't get how is it derived, especially the $\frac{1}{\sqrt{2\pi}}$ factor in front. How is the jacobian transformation is done here ?

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    @Fabian : As far as I know, the definition is bit sloppy, but we physicists live with it.2012-09-27

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The constant $\sqrt{2\pi}$ is just there for convenience. It has been arbitrarily introduced as a normalization factor without changing anything both for mathematics or physics. The advantage to have it there is that the integral $ \int_{-\infty}^\infty\frac{dc_n}{\sqrt{2\pi}}{e^{-\frac{1}{2}\lambda_nc_n^2}}=\frac{1}{\lambda_n} $ and you will get at the end for the partition function exactly the inverse of the determinant of the operator $\Theta$.