Evaluation of functional determinants

[spaghetti3451]

Consider the evaluation of the following functional determinant:

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

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

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

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

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

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

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

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

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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 $\alpha\rightarrow 0$.

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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?

 

[jvicens]

1. The analytic continuation via Wick rotation is justified because it allows for the evaluation of the integral in a simpler and more manageable way. By rotating the integration contour into the complex plane, the integral can be evaluated using methods from complex analysis, such as the residue theorem. This is a common technique in quantum field theory and has been shown to give consistent and reliable results. Other analytic continuations may also be possible, but the Wick rotation has proven to be the most effective in this case.
2. The integral is evaluated using the method of dimensional regularisation. This involves introducing a regulator $\alpha$ into the integral and then taking the limit as $\alpha\rightarrow 0$. This allows for the evaluation of the integral in a way that is independent of the specific dimension of the space. The parameter $d$ is introduced as the number of dimensions in the space, which is a free parameter that can be varied to obtain different results.
3. The gamma functions are differentiated using standard rules of calculus. The fact that $\Gamma(\alpha)\rightarrow 1/\alpha$ as $\alpha\rightarrow 0$ is used to simplify the expression and remove the singularity at $\alpha=0$. This allows for the final result to be obtained in a form that is more easily interpretable and usable.