- #1
blankvin
- 15
- 1
I am attempting to read my first book in QFT, and got stuck.
A Lorentz transformation that preserves the Minkowski metric [itex]\eta_{\mu \nu}[/itex] is given by [itex]x^{\mu} \rightarrow {x'}^{\mu} = {\Lambda}^\mu_\nu x^\nu [/itex]. This means [itex] \eta_{\mu \nu} x^\mu x^\nu = \eta_{\mu \nu}x'^\mu x'^\nu [/itex] for all [itex]x[/itex], which implies that [itex] \eta_{\mu \nu} = \eta_{\sigma \tau} {\Lambda}^\sigma_\mu {\Lambda}^\tau_\nu [/itex].
I am wondering if this is the right direction so as to arrive at the implication:
[itex] x^\sigma \rightarrow x'^\sigma = \Lambda^\sigma_\mu x^\mu, x^\tau \rightarrow x'^\tau = \Lambda^\tau_\nu x^\nu [/itex]
Now, I am not sure how to go beyond this point. And I assume that [itex] \eta_{\mu \nu} = \eta_{\sigma \tau} {\Lambda}^\sigma_\mu {\Lambda}^\tau_\nu [/itex] is an operator, so it does not matter if the [itex] x^i [/itex] are included or not.
Thanks.
A Lorentz transformation that preserves the Minkowski metric [itex]\eta_{\mu \nu}[/itex] is given by [itex]x^{\mu} \rightarrow {x'}^{\mu} = {\Lambda}^\mu_\nu x^\nu [/itex]. This means [itex] \eta_{\mu \nu} x^\mu x^\nu = \eta_{\mu \nu}x'^\mu x'^\nu [/itex] for all [itex]x[/itex], which implies that [itex] \eta_{\mu \nu} = \eta_{\sigma \tau} {\Lambda}^\sigma_\mu {\Lambda}^\tau_\nu [/itex].
I am wondering if this is the right direction so as to arrive at the implication:
[itex] x^\sigma \rightarrow x'^\sigma = \Lambda^\sigma_\mu x^\mu, x^\tau \rightarrow x'^\tau = \Lambda^\tau_\nu x^\nu [/itex]
Now, I am not sure how to go beyond this point. And I assume that [itex] \eta_{\mu \nu} = \eta_{\sigma \tau} {\Lambda}^\sigma_\mu {\Lambda}^\tau_\nu [/itex] is an operator, so it does not matter if the [itex] x^i [/itex] are included or not.
Thanks.