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I was looking at the lagrangian and conserved currents for the free complex scalar field and it looks like it has a striking similarity to the conserved current for probability:

[tex]

\frac{\partial \rho}{\partial t}=\nabla\cdot \vec{j}

[/tex]

where [itex]j_i =-i(\psi^{\ast}\partial_i \psi - \psi\partial_i \psi^{\ast})[/itex] and [itex]\rho[/itex] is the probability density. Then with the action

[tex]

\mathcal{L}=\partial_\alpha \psi^{\ast}\partial^\alpha \psi

[/tex]

the conserved current is

[tex]

j^{\alpha}=-i(\psi^\ast \partial^\alpha \psi - \psi \partial^\alpha \psi^\ast )

[/tex]

Then I had the thought that with the conservation of probability current, the above lagrangian appears to be a lagrangian for a free field of...probability. Now I'm aware that the complex scalar field is used to describe various spin-0 particles, but has anyone heard of any other possible thoughts on this lagrangian, maybe back when it was first put forward, or when anyone was just looking at relativistic quantum mechanics about 100 years ago?

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# Complex Scalar Field and Probability Field

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