I've learned the quantization of Klein-Gordon field using Fourier expansion. I understand that this process is kind of exchanging complex fourier coefficients to operator and makes it satisfying the canonical relation.

The result of usual process is given as $$ \Phi(x) = \int (d^3k)a(k)e^{-ikx} + a(k)^\dagger e^{ikx}\bigr|_{k^0 = \omega_k} $$

I wonder if I can do a quantization with other complete orthonormal systems, rather than $e^{ikx}$'s. (For example, radial fourier expansion or legendre polynomials)

If not, what makes it difficult? If we can, can you suggests the result with detail, the proper choice of commutation relation and Hamiltonian? Are they also determined by the quantization of classical Hamiltonian equation of motion as usual result be?

  • $\begingroup$ The exponentials are chosen as the complete orthogonal basis because they satisfy the classical equations of motion. $\endgroup$ – Sounak Sinha Oct 7 at 7:04
  • $\begingroup$ You also put $k^0=\omega_k$. That imples the definition of $\Phi$ is not a simple Fourier transform. $\endgroup$ – Sounak Sinha Oct 7 at 7:24
  • $\begingroup$ @SounakSinha I think that is not critical reason because that form is just a result of elliminating dirac delta function($\delta(k^2 - m^2)$) from original form. $\endgroup$ – ChoMedit Oct 7 at 7:28
  • $\begingroup$ @SounakSinha Do you mean that there is no other complete orthogonal basis which satisfies canonical quantization condition? $\endgroup$ – ChoMedit Oct 7 at 7:31
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    $\begingroup$ You can find more information about this by searching for the keywords Bogoliubov transformation. $\endgroup$ – Chiral Anomaly Oct 9 at 0:20

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