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Lorentz commutator

  1. Jul 24, 2014 #1
    I am trying to derive the algebra and I get a factor of 2 wrong...
    Consider the Lorentz group elements near the identity
    [tex]
    \Lambda_1^\mu\,_\nu = \delta^\mu\,_\nu + \omega_1^\mu\,_\nu, \quad \Lambda_2^\mu\,_\nu = \delta^\mu\,_\nu + \omega_2^\mu\,_\nu
    [/tex]
    and write a representation as
    [tex]
    U(\Lambda)=U(1 +\omega)=1-\frac{1}{2}i\omega^{\mu \nu} M_{\mu \nu}
    [/tex]
    where [itex]M[/itex] is a generator. Now the term [itex]\Lambda \equiv (\Lambda_2^{-1}\Lambda_1^{-1}\Lambda_2 \Lambda_1) [/itex] belongs to the group and up to 2nd order is [itex] 1+[\omega_2,\omega_1] [/itex].
    So its representation is
    [tex]
    U(\Lambda)=U(1+[\omega_2,\omega_1])=1-\frac{1}{2}i[\omega_2,\omega_1]^{\mu \nu} M_{\mu \nu}
    [/tex]
    I know this is wrong and I am supposed to get
    [tex]
    1-i[\omega_2,\omega_1]^{\mu \nu} M_{\mu \nu}
    [/tex]
     
  2. jcsd
  3. Jul 24, 2014 #2

    samalkhaiat

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    Science Advisor

    What algebra? Is it Lorentz algebra?
     
  4. Jul 25, 2014 #3
    yes.
     
  5. Jul 25, 2014 #4
    I am not planning to make the calculation, but did you use the fact that for [tex] \Lambda_{\mu}^{\nu} = \delta_{\mu}^{\nu} + \omega_{\mu}^{\nu}[/tex]
    then its inverse matrix is: [tex] (\Lambda^{-1})_{\mu}^{\nu} = \delta_{\mu}^{\nu} - \omega_{\mu}^{\nu} +O(\omega^2)[/tex]?
     
  6. Jul 25, 2014 #5
    Yes. I used it, and I am pretty sure it leads to [itex]\Lambda = 1+ [\omega_2,\omega_1] [/itex]. I am not sure then how it would lead to
    [tex]
    U(1+[\omega_2,\omega_1])=1-i[\omega_2,\omega_1]^{\mu \nu} M_{\mu \nu}
    [/tex]
    (there is a missing factor 1/2 according to my convention)
     
  7. Jul 25, 2014 #6
    Yes, you are, your calculation is right.

    Do you use a specific reference where this equation is given there?

    I must say that I myself find QFT quite hard as you can follow my posts in the forum in the last month or so.
     
  8. Jul 25, 2014 #7
    Yes. Sorry, I am using the following reference:
    http://www.damtp.cam.ac.uk/user/ho/GNotes.pdf
    and trying to fill in the gaps. This is at the end of page 41, and on page 42. My problem is the top of eq. (4.30) given the definition (4.29).
     
  9. Jul 26, 2014 #8

    samalkhaiat

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    Science Advisor

    No, in fact that is the correct form.

    This is wrong. If you use it, you will have a factor (2) multiplying the RHS of the Lorentz algebra.
     
  10. Jul 26, 2014 #9
    Thank you. you are right, it works out! I stopped when I didn't understand this line, which is a typo, and should have continued in the first place
     
    Last edited: Jul 26, 2014
  11. Jul 27, 2014 #10

    samalkhaiat

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    Science Advisor

    The point of having (1/2) in the [itex]U( \Lambda )[/itex] is to get "mormalized" algebra. There is nothing wrong in choosing [itex]U = 1 + i \omega^{ \mu \nu } M_{ \mu \nu }[/itex], but your algebra, in this case becomes [itex][M , M] = (1/2) ( g M + \cdots )[/itex]. So, when you mormalize [itex]M \rightarrow (1/2) M[/itex] you get the "nice" algebra [itex][M , M] = g M + \cdots [/itex].
    The point to remember is this, once you made a choice, you have to use it for all group elements: [itex] ( \Lambda ), ( \Lambda_{1} \Lambda_{2} ) , \cdots , ( \Lambda_{1} \cdots \Lambda_{n} )[/itex].

    I think it is lazyness not a "typo", because he repeats the same mistake when he "derives" the Poincare' algebra.

    Sam
     
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