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I began to study the basics of QED.

Now I am studying Lagrangian and Hamiltonian densities of Dirac Equation.

I'll call them L density and H density for convenience :)

Anyway, the derivation of the H density from L density using Legendre transformation confuses me :(

I thought because parameters of them are space-time components, it should be

But I found that this is related to the De Doner - Weyl Theory,

and the H density used in textbook is

where the dot represents time derivative.

So, my question is,

why we consider Legendre transformation on only time derivative of phi ?

Is it just 'defined' to consider energy of the system?

Then what does the covariant H density defined in the De Donder - Weyl theory mean?

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# Legendre Transformation of Lagrangian density ?

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