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Suppose we wish to obtain the energy density of the free complex scalar field $\varphi$ as a Legendre transform of the corresponding action. From Wikipedia, writing the action of a free complex scalar field in Minkowski space as

$$S(\varphi)=\int d^4x \,\mathcal L(\partial_0\varphi,\partial_i\varphi,\,\varphi)$$ the energy density should be the Legendre transform of the Lagrangian density, where the Lagrangian density is defined as $$\mathcal L(\partial_0\varphi,\partial_i\varphi,\,\varphi)=\langle \varphi, \,(\Delta+m^2)\varphi\rangle.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$

Using the definition of Legendre transform from the Wikipedia page on the Legendre transformation (https://en.wikipedia.org/wiki/Legendre_transformation#Definition), $$\mathcal E(\partial_0\varphi^*,\partial_i\varphi,\,\varphi):=\max_{\partial_0\varphi}\,\,\langle\partial_0\varphi^*,\partial_0\varphi\rangle-\mathcal L(\partial_0\varphi,\partial_i\varphi,\,\varphi)$$ From which it may be obtained that (using the maximum condition to obtain a derivative formula for the conjugate variable $\,\partial_0\varphi^*$): $$\mathcal E(\partial_0\varphi^*,\partial_i\varphi,\,\varphi)=\langle\partial_0\varphi,\partial_0\varphi\rangle-\mathcal L(\partial_0\varphi,\partial_i\varphi,\,\varphi)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\langle\varphi,\,(\Delta_3+m^2)\varphi\rangle\,\,\,\,\,\,$$ where $\Delta_3$ is the induced Laplacian on a Cauchy surface inside the Lorentzian spacetime which includes the point at which the energy density is being evaluated.

Unfortunately, the formula obtained is not the energy density that matches anything I can find online. This is significant, because this means that the energy density is not the Legendre transform of the Lagrangian density for the case of complex variables. Is this a valid conclusion? Is there a natural redefinition of the energy density in the complex setting?

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    $\begingroup$ Hint: One should insert complex conjugation in appropriate places to ensure that the Lagrangian density is real and the energy density is non-negative. $\endgroup$
    – Qmechanic ♦
    Nov 14, 2017 at 11:31
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    $\begingroup$ Another consideration: A complex scalar field is equivalent to two real scalar fields, namely its real and imaginary parts. Writing everything in terms of the real scalars and doing the Legendre transform, do you obtain the same as with your method? If not, you know you did something wrong. $\endgroup$
    – ACuriousMind ♦
    Nov 14, 2017 at 11:39
  • $\begingroup$ Hey Qmechanic, I'm already assuming that all inner-products are hermitian in the calculation, so the action is definitely real-valued $\endgroup$
    – David Roberts
    Nov 14, 2017 at 19:23
  • $\begingroup$ Hey @David Roberts. How does your result deviate from what you could find online? $\endgroup$
    – Qmechanic ♦
    Nov 17, 2017 at 18:04

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