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A Evaluation of functional determinants

  1. Nov 4, 2016 #1
    Consider the evaluation of the following functional determinant:

    $$\text{log}\ \text{det}\ (\partial^{2}+m^{2})$$

    $$=\text{Tr}\ \text{log}\ (\partial^{2}+m^{2})$$

    $$= \sum\limits_{k} \text{log}\ (-k^{2}+m^{2})$$

    $$= VT \int\frac{d^{4}k}{(2\pi)^{4}}\ \text{log}\ (-k^{2}+m^{2})$$

    $$= iVT \int\frac{d^{4}k_{E}}{(2\pi)^{4}}\ \text{log}\ (k^{2}_{E}+m^{2})$$

    $$=-iVT\frac{\partial}{\partial\alpha}\int \frac{d^{4}k_{E}}{(2\pi)^{4}}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}\Bigg|_{\alpha=0}$$

    $$=-iVT\frac{\partial}{\partial\alpha}\left(\frac{1}{(4\pi)^{d/2}} \frac{\Gamma\left(\alpha-\frac{d}{2}\right)}{\Gamma(\alpha)}\frac{1}{(m^{2})^{\alpha-d/2}}\right)\bigg|_{\alpha=0}$$

    $$=-iVT\frac{\Gamma(-d/2)}{(4\pi)^{d/2}}\frac{1}{(m^{2})^{-d/2}}$$

    ----

    1. In the first line to the second line, a common identity is used to write the logarithm of the determinant of an operator as the trace of the determinant of the operator.
    2. In the second line to the third line, the trace sums over the eigenvalues of the operators of the determinant.
    3. In the third line to the fourth line, the summation over ##k## is converted to an integral over ##k##.
    4. In the fourth line to the fifth line, the integral is analytically continued into the complex plane via Wick rotation.
    5. In the fifth line to the sixth line, dimensional regularisation is used with the regulator ##\alpha##.
    6. In the sixth line to the seventh line, the integral over ##k_{E}## is evaluated.
    7. In the seventh line to the eighth line, the derivative is taken with respect to ##\alpha## and then ##\alpha## is set to ##0##. This step uses the fact ##\Gamma(\alpha)\rightarrow 1/\alpha## as ##\alpha\rightarrow 0##.

    ---

    1. Why is the analytic continuation via Wick rotation justified - after all, the the integral is not defined on the complex plane, and the definition of the integral in the complex plane via Wick rotation appears arbitrary - why not use some other analytic continuation?
    2. In the sixth line to the seventh line, how is the integral evaluated? How is the parameter ##d## introduced?
    3. In the seventh line to the eighth line, how are the gamma functions differentiated? How is ##\Gamma(\alpha)\rightarrow 1/\alpha## as ##\alpha\rightarrow 0## used to obtain the final result?
     
  2. jcsd
  3. Nov 9, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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