[qft] Srednicki 2.3 Lorentz group generator commutator

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1. The problem statement, all variables and given/known data
Verify that (2.16) follows from (2.14). Here [tex]\Lambda[/tex] is a Lorentz transformation matrix, [tex]U[/tex] is a unitary operator, [tex]M[/tex] is a generator of the Lorentz group.

2. Relevant equations
2.8: [tex]\delta\omega_{\rho\sigma}=-\delta\omega_{\sigma\rho}[/tex]

[tex]M^{\mu\nu}=-M^{\nu\mu}[/tex]

2.14: [tex]U(\Lambda}^{-1})M^{\mu\nu}U(\Lambda})=\Lambda^\mu_\rho\Lambda^\nu_\sigma M^{\rho\sigma}[/tex]

2.16: [tex][M^{\mu\nu},M^{\rho\sigma}]=i\hbar (g^{\mu\rho}M^{\nu\sigma}-
g^{\nu\rho}M^{\mu\sigma}+g^{\nu\sigma}M^{\mu\rho}-g^{\mu\sigma}M^{\nu\rho})
[/tex]


2.12: [tex]U(1+\delta\omega)=1+{i \over 2\hbar}\delta\omega_{\mu\nu}M^{\mu\nu}[/tex]

3. The attempt at a solution
I assume that [tex]\Lambda[/tex] is a small transformation, as hinted by Srednicki: [tex]\Lambda=1+\delta\omega[/tex] and rewrite the 2.14:

[tex](1-{i \over 2\hbar}\delta\omega_{\rho\sigma}M^{\rho\sigma})M^{\mu\nu}(1+{i \over 2\hbar}\delta\omega_{\rho\sigma}M^{\rho\sigma})=
(\delta^\mu_\rho+\delta\omega^\mu_\rho)(\delta^\nu_\sigma+\delta\omega^\nu_\sigma)M^{\rho\sigma}[/tex].

Then I cross out the [tex]M^{\mu\nu}[/tex] that come on the both sides, throw out the double-omega pieces and rewrite the omegas in the following manner

[tex]\delta\omega^\nu_\sigma=g^{\rho\nu}\delta\omega_{\rho\sigma}[/tex]

I come to

[tex]
[M^{\mu\nu},M^{\rho\sigma}]=2i\hbar (g^{\mu\sigma}M^{\rho\nu}-g^{\rho\nu}M^{\mu\sigma})
[/tex]

which seems to be incorrect. Where do I make the mistakes, if I do? How to derive the (2.16) without involvement of certain expression of the Lorentz generator?

Thanks.
 
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you shouldn't just throw away the omega. Take into accout the fact that omega_ab and omega_ba are not independent but related by the relation omega_ab=-omega_ba. i.e. when you consider omega_ab, you shoud also consider omega_ba.
 
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you shouldn't just throw away the omega. Take into accout the fact that omega_ab and omega_ba are not independent but related by the relation omega_ab=-omega_ba. i.e. when you consider omega_ab, you shoud also consider omega_ba.
I throw away the double omega's, that is [tex]\delta\omega^\mu_\rho\delta\omega^\nu_\sigma[/tex]. I think this is reasonable, because this kind of term is second-order (very small).

I have also come to having [tex]\delta\omega_{\rho\sigma}[/tex] on both sides, before cancelling them out (that is, we are only dealing with asymmetric pieces).

Or have I misunderstood your comment?
 
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I guess the expressioni "throw away" in my comment was misleading.
I'll explain it again.

What you actually got in the calculation must have been someting like this.
[tex]\delta\omega_{\rho\sigma}[M^{\mu\nu},M^{\rho\sigma}]=2i\hbar \delta\omega_{\rho\sigma} (g^{\mu\sigma}M^{\rho\nu}-g^{\rho\nu}M^{\mu\sigma})[/tex]

Now, you want to take away [tex]\delta\omega_{\rho\sigma}[/tex] from the above expression, as the above equation holds for arbitrary [tex]\delta\omega[/tex]

If each component [tex]\delta\omega_{\rho\sigma}[/tex] is truely arbitrary, you can do that. Yet, in your case, [tex]\delta\omega[/tex] is antisymmetric so that some of its components are related. You should take this fact into account.
 
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Yes, I have got the equation

[tex]\delta\omega_{\rho\sigma}[M^{\mu\nu},M^{\rho\sigma}]=2i\hbar \delta\omega_{\rho\sigma} (g^{\mu\sigma}M^{\rho\nu}-g^{\rho\nu}M^{\mu\sigma})[/tex]

but [tex]M^{\rho\sigma}[/tex] is antisymmetric, too, and I thought this gives us the right to cancel the [tex]\delta\omega_{\rho\sigma}[/tex] 's.

Thank you for pointing out the mistake, however, I still do not know how I take into account that some of the components are related. Should I try to write a new equation with interchanged indices [tex]\rho\leftrightarrow\sigma[/tex] and try to combine the two? Is there some resource in the net that would give me this kind of elementary understanding about equations with antisymmetric coefficients?
 
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In the above equation, [tex]\rho [/tex] and [tex]\sigma [/tex] are summed over. Lets expand this summation and look into the 12 and 21 components, as an example.

[tex]\delta\omega_{\rho\sigma}[M^{\mu\nu},M^{\rho\sigma}]=2i\hbar \delta\omega_{\rho\sigma} (g^{\mu\sigma}M^{\rho\nu}-g^{\rho\nu}M^{\mu\sigma}) [/tex]

->

[tex]... + \delta\omega_{12}[M^{\mu\nu},M^{12}] + \delta\omega_{21}[M^{\mu\nu},M^{21}] +...= ... + 2i\hbar \delta\omega_{12} (g^{\mu 2}M^{1\nu}-g^{1\nu}M^{\mu 2}) + 2i\hbar \delta\omega_{21} (g^{\mu 1}M^{2\nu}-g^{2\nu}M^{\mu 1})+...[/tex]

First, lets assume every component of [tex]\delta\omega[/tex] is independent. Then, for the above equation to hold for an arbitrary [tex]\delta\omega[/tex], the following equations should be satisfied.

...
[tex][M^{\mu\nu},M^{12}] = 2i\hbar (g^{\mu 2}M^{1\nu}-g^{1\nu}M^{\mu 2})[/tex]
[tex][M^{\mu\nu},M^{21}] = 2i\hbar (g^{\mu 1}M^{2\nu}-g^{2\nu}M^{\mu 1})[/tex]
...

What if there is a constraint that [tex]\delta\omega_{12} = -\delta\omega_{21} [/tex]?
 
Last edited:
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[tex]\delta\omega_{12}[M^{\mu\nu}, M^{12}]-\delta\omega_{12}[M^{\mu\nu}, M^{21}]=2i\hbar\delta\omega_{12}(g^{\mu2}M^{1\nu}-g^{1\nu}M^{\mu2}-g^{\mu 1}M^{2\nu}+g^{2\nu}M^{\mu 1}[/tex]

that, by crossing out [tex]2\delta\omega_{12}[/tex] and writing rho instead of 1 and sigma instead of 2 is

[tex]
[M^{\mu\nu},M^{\rho\sigma}]=i\hbar (g^{\mu\sigma}M^{\rho\nu}-g^{\rho\nu}M^{\mu\sigma}-g^{\mu\rho}M^{\sigma\nu}+g^{\sigma\nu}M^{\mu\rho})
[/tex]

and this is the same as (2.16). Thank you very much, weejee, that was a really great help :)
 

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