Lagrangian treatment

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The Lagrangian density for this field is a relativistic invariant, as is the Klein Gordon equation. For the non-interacting real field, the density is:

\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2.

Since the field is described by a scalar, it is also invariant under Lorentz transformations. It therefore describes particles with spin 0.

[edit] Canonical Variables

For the complex field, we can treat the real and imaginary components as two independent fields, letting \phi = \frac{1}{\sqrt{2}}\left(\phi_R + i \phi_I\right). The Lagrangian then contains identical contributions from both components, and we can treat either \phi_R\, and \phi_I\, or \phi\, and \phi^*\, as our dynamical variables:

\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi -m^2 \phi^*\phi, or
\mathcal{L} = \frac{1}{2} \partial_\mu \phi_R \partial^\mu \phi_R - \frac{1}{2} m^2 \phi_R^2 + \frac{1}{2} \partial_\mu \phi_I \partial^\mu \phi_I - \frac{1}{2} m^2 \phi_I^2.

The conjugate momentum to \phi_R\, is

\pi^\mu_R\, = \frac{\partial \mathcal{L} } { \partial( \partial_\mu \phi_R ) },
= \frac{\partial } { \partial( \partial_\mu \phi_R ) } \left(\frac{1}{2} g^{\alpha\beta} \partial_\alpha \phi_R \partial_\beta \phi_R \right),
= \frac{1}{2} g^{\mu\beta} \partial_\beta \phi_R + \frac{1}{2} g^{\alpha\mu} \partial_\alpha \phi_R (no sum over \mu\,),
= \partial^\mu \phi_R\,,

and similarly for \phi_I\,. Meanwhile, the conjugate momentum to \phi^*\, is

\pi^{*\mu} = \frac{\partial \mathcal{L} } { \partial( \partial_\mu \phi^* ) } = \partial^\mu \phi\,,

while

\pi^{\mu} = \partial^\mu \phi^*\,.

The canonically conjugate momentum to \phi^*\, is defined to be

\pi^* \equiv \pi^{*t} = \frac{\partial \mathcal{L} } { \partial( \partial_t \phi^* ) } = \partial^t \phi = \frac{1}{\sqrt{2}}\left(\partial_t \phi_R + i \partial_t \phi_I \right) = \frac{1}{\sqrt{2}}\left(\pi_R + i \pi_I \right),

while that of \phi\, is

\pi \equiv \pi^t = \partial_t \phi^* = \frac{1}{\sqrt{2}}\left(\partial_t \phi_R - i \partial_t \phi_I \right) = \frac{1}{\sqrt{2}}\left(\pi_R - i \pi_I \right).

(Because of our choice of metric, \partial_t = \partial^t\,)

In the canonical quantization of the field, the canonically conjugate variables \phi\, and \pi\, will obey a canonical commutation relation.

[edit] Equations of Motion

When we apply the Euler-Lagrange equations, we can vary the field with respect to either \phi\, or \phi^*\,:

\partial_\mu \frac{\partial\mathcal{L}}{\partial(\partial_\mu \phi^*)} - \frac{\partial\mathcal{L}}{\partial\phi^*} = 0 \rightarrow \left(\partial_\mu \partial^\mu + m^2\right)\phi = 0,

and

\partial_\mu \frac{\partial\mathcal{L}}{\partial(\partial_\mu \phi)} - \frac{\partial\mathcal{L}}{\partial\phi} = 0 \rightarrow \left(\partial_\mu \partial^\mu + m^2\right)\phi^* = 0.

We therefore find that both the field and its complex conjugate must satisfy the Klein Gordon equation. This is of course by construction.

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