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MTW Exercise 25.5 b) - killing vectors

  1. Nov 30, 2013 #1
    hi there,
    In this Ex ( see attached snapshot ), point b), the poisson bracket equation is not so straightforward to obtain.
    Please correct my Poisson Bracket expansion here :
    The first one which is provided is simpler :
    [ε,η] = εμδμηρ - ημδμερ = ζη

    and the monster one :

    [pε,pη] = [pμεμ,pβηβ] = (pμεμ)∂μ(pβηβ) -
    (pμημ)∂μ(pβεβ)

    Thanks,
     

    Attached Files:

  2. jcsd
  3. Nov 30, 2013 #2

    WannabeNewton

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    What you have written down doesn't make sense. Thankfully for you it's still quite simple :)

    ##[p_{\mu}\xi^{\mu},p_{\mu}\eta^{\mu}] = \frac{\partial }{\partial x^{\lambda}}(p_{\mu}\xi^{\mu})\frac{\partial }{\partial p_{\lambda}}(p_{\nu}\eta^{\nu})-\frac{\partial }{\partial p_{\lambda}}(p_{\mu}\xi^{\mu})\frac{\partial }{\partial x^{\lambda}}(p_{\nu}\eta^{\nu})## and keep in mind that ##\xi^{\mu}=\xi^{\mu}(x^{\nu})##, ##\eta^{\mu} = \eta^{\mu}(x^{\nu})## as well as ##\frac{\partial p_{\mu}}{\partial p_{\nu}} = \delta^{\nu}_{\mu}##. The rest is a straightforward calculation.

    EDIT: I must admit that this entire exercise is a bit weird. Showing that the commutator of two killing fields is also a killing field is way more straightforward than this.
     
    Last edited: Nov 30, 2013
  4. Nov 30, 2013 #3
    hey Mate,
    Thanks for your reply. Well it seems I got confused here. I had based my second commutation on the first one. In my very first attempt I had indeed written your expression but had forgotten that the other variable was x not either of the killing vectors ε or η....
    Damn that got me wrong all the way !
    At the beginning I had started with the Jacobi identity but did not check out ...
    Thanks very much !
     
  5. Nov 30, 2013 #4
     
  6. Nov 30, 2013 #5

    WannabeNewton

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    Yeah what you wrote down is fine for a Lie bracket of vector fields on a smooth manifold but what you want is a Poisson bracket of scalar fields that are functions of phase space coordinates (cotangent bundle to the smooth manifold).

    Did you get the final result by the way? Feel free to ask if you get stuck elsewhere.
     
  7. Nov 30, 2013 #6
    Ok let us expand out your expression above :

    (pμξμ)[itex]\frac{∂(p_{v}η^{v})}{∂p_{λ}}[/itex]−(pvηv)[itex]\frac{∂(p_{v}ε^{v})}{∂p_{λ}}[/itex] =
    ( pμεμ + pμεμ ) ( ηλ + pv[itex]\frac{∂(η^{v})}{∂p_{λ}}[/itex] ) -
    ( pvηv + pvηv ) ( ελ + pv[itex]\frac{∂(ε^{v})}{∂p_{λ}}[/itex] )
    Now we have : [itex]\frac{∂(ε^{v})}{∂p_{λ}}[/itex] = 0 = [itex]\frac{∂(η^{v})}{∂p_{λ}}[/itex]
    we are left with :
    ( pμεμ + pμεμ ) ( ηλ ) -
    ( pvηv + pvηv ) ( ελ )
    This gives :
    pμεμηλ + pμεμ ηλ - pvηvελ - pvηvελ

    Now the second and the last term would yield the result (i.e. -pζ based on the commutation of ε and η) which means that the first and the third terms would add up to 0. How do you think ?
     
  8. Nov 30, 2013 #7

    WannabeNewton

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    Remember that we are working in phase space coordinates ##\{x^{\mu},p_{\mu}\}## so ##\frac{\partial p_{\mu}}{\partial x^{\nu}} = 0##. This makes the first and third terms vanish.
     
  9. Nov 30, 2013 #8
    Yes there is a much shorter version using Lie derivatives indeed...
     
  10. Nov 30, 2013 #9
     
  11. Nov 30, 2013 #10

    WannabeNewton

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    Haha thanks for the link bud. Take care and have fun!
     
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