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Legendre Transformation of Lagrangian density ?

  1. May 2, 2015 #1

    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

    bandicam 2015-05-03 11-13-28-477.jpg

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

    and the H density used in textbook is
    bandicam 2015-05-03 11-13-34-326.jpg
    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?
  2. jcsd
  3. May 3, 2015 #2


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  4. May 3, 2015 #3


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    From the lagrangian density [itex]\mathcal{L}[/itex], you can define the canonical stress-energy tensor [itex]\mathcal{T}[/itex] as follows:

    [itex]\mathcal{T}^{\mu \nu} = \dfrac{\partial \mathcal{L}}{\partial \partial_\mu \phi_j} \partial^\nu \phi_j - g^{\mu \nu} \mathcal{L}[/itex]

    This is a conserved current in the first index:

    [itex]\partial_\mu \mathcal{T}^{\mu \nu} = 0[/itex]

    Then you can define a hamiltonian density [itex]\mathcal{H}[/itex] in terms of [itex]\mathcal{T}[/itex]:

    [itex]\mathcal{H} = \mathcal{T}^{tt} =\dfrac{\partial \mathcal{L}}{\partial \partial_t \phi_j} \partial^t \phi_j - g^{tt} \mathcal{L}[/itex]

    This is the same as the expression in the textbook, if you're using the metric where [itex]g^{tt} = +1[/itex] and defining [itex]\dot{\phi_j} = \partial_t \phi_j[/itex]

    The hamiltonian is the integral of the hamiltonian density over all space:

    [itex]H = \int d^3 x \mathcal{H}[/itex]

    It's the hamiltonian, not the hamiltonian density, that is constant:

    [itex]\dfrac{d}{dt} H = 0[/itex]
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