- #1
geoduck
- 258
- 2
If you Lorentz transform a scalar:
[tex]U^{-1}(\Lambda)\phi(x)U(\Lambda)=\phi(\Lambda^{-1}x) [/tex]
If you now perform another Lorentz transform, would it it look like this:
[tex]U^{-1}(\Lambda')U^{-1}(\Lambda)\phi(x)U(\Lambda)U(\Lambda')=\phi(\Lambda'^{-1}\Lambda^{-1}x) [/tex] ?
But isn't this wrong, because this expression is equal to:
[tex]U^{-1}(\Lambda\Lambda')\phi(x)U(\Lambda\Lambda')=\phi([\Lambda\Lambda']^{-1}x) [/tex]
and not:
[tex]U^{-1}(\Lambda'\Lambda)\phi(x)U(\Lambda'\Lambda)=\phi([\Lambda'\Lambda]^{-1}x) [/tex]
[tex]U^{-1}(\Lambda)\phi(x)U(\Lambda)=\phi(\Lambda^{-1}x) [/tex]
If you now perform another Lorentz transform, would it it look like this:
[tex]U^{-1}(\Lambda')U^{-1}(\Lambda)\phi(x)U(\Lambda)U(\Lambda')=\phi(\Lambda'^{-1}\Lambda^{-1}x) [/tex] ?
But isn't this wrong, because this expression is equal to:
[tex]U^{-1}(\Lambda\Lambda')\phi(x)U(\Lambda\Lambda')=\phi([\Lambda\Lambda']^{-1}x) [/tex]
and not:
[tex]U^{-1}(\Lambda'\Lambda)\phi(x)U(\Lambda'\Lambda)=\phi([\Lambda'\Lambda]^{-1}x) [/tex]