1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Real Scalar Field, Hamiltonian, Conjugate Momentum

  1. Oct 29, 2016 #1
    ## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density for a real scalar field in 4-d, ##u=0,1,2,3 = t,x,y,z##, below ##i = 1,2,3 =x,y,z##

    In order to compute the Hamiltonian I first of all need to compute the conjugate momentum:

    ## \Pi (t, x) = \frac{\partial L}{\partial \dot{\phi (x)}} ##

    I can see it's coming from the second term : ##- \partial_{u} \phi(x) \partial^{u} \phi (x)) = - \partial_{0} \phi(x) \partial^{0} \phi (x)) - \partial_{i} \phi(x) \partial^{i} \phi (x)) ##, where I'm only interested in ##\partial_{0} \phi(x) \partial^{0} \phi (x)) ##,

    But I'm unsure how to deal with the one upper and one lower index.

    The Hamiltonian is ## H = \int d^{3} x \dot{\phi} \Pi - L(t, {x}) = \int d^{3} x (1/2m^{2}\phi^{2} + 1/2( \partial_{i} \phi )^{2} + 1/2\dot{\phi^{2}}) ##

    I can see I clearly need to lower an index. So if do ##g_{\alpha u} \partial ^{\alpha} \phi \partial_{u} \phi = (\partial _{u} \phi )^{2} = (\partial_{0} \phi(x))^{2} - (\partial_{i} \phi(x))^{2} ## , I get the correct answer that ##\Pi = \dot{\phi} ##

    however then surely I have found ##g_{\alpha u} \Pi ## and subbed in ## g_{\alpha u} L ## in H, as a pose to ##L##

    Many thanks in advance.
     
    Last edited: Oct 29, 2016
  2. jcsd
  3. Oct 30, 2016 #2
    anyone?

    I believe the following is related:

    Showing that ##\partial^{u}\alpha(\phi^*\partial_{u}\phi)=\partial_{u}\alpha(\phi^*\partial^{u}\phi) ##

    I don't really know how to approach this, since I need to raise one index and lower the other, but they are the same index so I can't use the obvious like ##g_{ab}x^{b}=x_{a}## etc.
     
  4. Nov 2, 2016 #3
    it's okay got it :thumbup:, thanks for the help guys
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted