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I've been reading about symmetries and I haven't been able piece this information together. I've the Lorentz transformation
$$x^\mu \mapsto x^{\rho} = \Lambda_\nu^\mu x^\nu$$ First off, arn't we already fixing a representation while doing this? And then, I have the transformation of fields given by $$U(\Lambda)\psi(x)U(\Lambda)^{-1} = \psi(\Lambda^{-1}x)$$ Second, I would like to understand how the above transformation law guarantees or is consistent with the below mentioned rules.

My understanding right now is that when I make a Lorentz transformation on my coordinates, I would want a way to relate the scalar,vector, tensor fields in the new coordinates with the old ones.I know the Lorentz group has various representations and $U(\Lambda)$ is one such representation. But then aren't we a priory assuming a representation $\Lambda_\nu^\mu$.

  • $\Lambda_\nu^\mu$ related to $U(\Lambda)$
  • As per Pg.36 of Peskin & Shroeder, we see that the derivative of a scalar field transforms (by chain rune) as when I do this $x^\mu \mapsto x^{\rho} = \Lambda_\nu^\mu x^\nu$. $$\partial_{\nu}\phi(x) \mapsto (\Lambda^{-1})^\nu_\mu(\partial_{\nu}\phi)(\Lambda^{-1}x)$$ but the Pg.42 spinor transforms as $$\psi(x) \mapsto \Lambda_{\frac{1}{2}}\psi(\Lambda^{-1}x)$$ In this case, didn't I already fixing a representation $\Lambda_\nu^\mu$? How do I have this choice of $\Lambda_\frac{1}{2}$ whereas I didn't in the transformation of $\partial_{\nu}\phi(x)$. There could be some rule by which the underlying representation of $\Lambda$ in $\psi(\Lambda^{-1}x)$ reflects onto $U(\Lambda)$ as in the previous case?

I'm quite certain Wigner's theorem about symmetries being represented as unitary operators has a place amongst all this. I would like to know the precise relevant statement for it.

I've the following questions and its not all linked properly in my head. I know I've mixed up various concepts into a terrible concoction. :( I would greatly appreciate if someone can help me with connecting my understanding.

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    Possible duplicate of Lorentz Algebra Representation and QFT – ACuriousMind Feb 15 '16 at 11:58
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    Not an exact duplicate, but the basic confusion (difference between finite-dimensional non-unitary and infinite-dimensional unitary representation) seems to be the same. – ACuriousMind Feb 15 '16 at 11:59
  • Its not a duplicate. I git a few points from it but the answer wasnt exactly what I was expecting.. –  Feb 19 '16 at 16:47
  • Can you then try to make more clear what is confusing you? I chose the duplicate because I wasn't actually sure what you are trying to ask. – ACuriousMind Feb 19 '16 at 16:51
  • It is taken as axiom how a quantum field transforms under a Lorentz transformation. Without showing how this fact is reflected from the transformation of the underlying Hilbert space. I want to see how tranformation of coordinates implies tranformation of fields in a mathematically consistent manner. While doing so and relating to the above "axioms" that we assumed, I would like to know how they fit in each other. –  Feb 19 '16 at 17:23

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