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I am trying to prove the Lorentz invariance of the (left-handed) Weyl Lagrangian: $$\mathcal L=i\psi^\dagger\bar\sigma^\mu\partial_\mu\psi$$

A Lorentz transformation is realized as $\psi\to M\psi$, where $M=\exp[-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}]$, and $S$ are the generators of the representation, i.e., $S^{0i}=-\frac{i}{2}\sigma^i$ and $S^{ij}=\frac{1}{2}\epsilon^{ijk}\sigma^k$. Therefore, the Lagrangian transforms as $$\mathcal L=i\psi^\dagger\bar\sigma^\mu\partial_\mu\psi\to i\psi^\dagger M^\dagger\bar\sigma^\mu M\partial_\mu \psi$$ so that if it actually is invariant, we must have $M^\dagger\bar\sigma^\mu M=\bar\sigma^\mu$ (and, in particular, $M^\dagger M=\mathbb I$). I think this is a nice relation, which expresses the invariance of the symbol $\bar\sigma^\mu$.

Now the thing is, I think this relation is not true. Take, for example, a boost in the $z$-direction with rapidity $\omega$. This means $$M=\exp[-i\omega S^{03}]=\exp\left[\frac{\omega}{2}\begin{pmatrix}-1& 0\\0&1\end{pmatrix}\right]=\begin{pmatrix}e^{-\omega/2}&0\\0&e^{\omega/2}\end{pmatrix}$$

In this case, it is easly seen that $M^\dagger M\neq\mathbb I$. My question is: what went wrong in my argument?

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  • $\begingroup$ Possible duplicate by OP: physics.stackexchange.com/q/195460/2451 $\endgroup$ – Qmechanic Jul 23 '15 at 20:17
  • $\begingroup$ Certaily a duplicate. The question is more clearly stated in this post, which also has the solution. Should I delete the older one? $\endgroup$ – AccidentalFourierTransform Jul 23 '15 at 21:03
  • $\begingroup$ Yes. In the future, please edit rather than repost. $\endgroup$ – Qmechanic Jul 23 '15 at 21:09
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Lorentz invariance refers to the action $S=\int\mathcal{L}(x)\,\mathrm{d}x$, not to the Lagrangian. To determine the condition on the Lagrangian which we must have, we make the coordinate change $x\to \Lambda x=:x'$ (a Lorentz transformation) and use the general fact that the Jacobian of a Lorentz transformation is unity, so $$\int\mathcal{L}(x)\,\mathrm{d}x\to\int\mathcal{L}(x')\,\mathrm{d}x'=\int\mathcal{L}(\Lambda x)\,\mathrm{d}x$$ We must thus show $\mathcal{L}(x)\to\mathcal{L}(\Lambda x)$, where $\mathcal{L}(x)=\mathrm{i}\xi^\dagger(x)\bar\sigma^\mu\partial_\mu\xi(x)$.

The Weyl spinors transform as $\xi(x)\to D(\Lambda^{-1})\xi(\Lambda x)$, so the first step is$$\mathrm{i}\xi^\dagger(x)\bar\sigma^\mu\partial_\mu\xi(x)\to\mathrm{i}\xi^\dagger(\Lambda x)D^\dagger(\Lambda^{-1})\bar\sigma^\mu\partial_\mu D(\Lambda^{-1})\xi(\Lambda x)$$ The Pauli 4-vector $\bar\sigma^\mu =(1,\vec\sigma)$ transforms as $\bar\sigma^\mu\to D^\dagger(\Lambda)\bar\sigma^\mu D(\Lambda)=\Lambda^\mu{}_\nu\bar\sigma^\nu$, so the transformed Lagrangian is $$\mathrm{i}\xi^\dagger(\Lambda x)\bar\sigma^\nu(\Lambda^{-1})^\mu{}_\nu\partial_\mu \xi(\Lambda x)$$ Finally, the partial derivative transformation rule is $\partial_\mu=\Lambda^\nu{}_\mu\partial_\nu'$, and the transformed Lagrangian is $$\mathrm{i}\xi^\dagger(\Lambda x)\bar\sigma^\nu(\Lambda^{-1})^\mu{}_\nu\Lambda^\nu{}_\mu\partial_\nu' \xi(\Lambda x)=\mathrm{i}\xi^\dagger(\Lambda x)\bar\sigma^\nu\partial_\nu' \xi(\Lambda x)=\mathcal{L}(\Lambda x)$$ This shows the Lagrangian $\mathrm{i}\xi^\dagger\bar\sigma^\mu\partial_\mu\xi$ leads to a Lorentz-invariant action, as desired.

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  • $\begingroup$ I've been thinking about this for some time, and I think this is the right answer. Nevertheless, I still think that there should be a way of proving the relation $D^\dagger \bar \sigma^\mu D=\Lambda^\mu_\nu \hat \sigma^\nu$, which expresses the transformation rule of the Pauli vector (instead of just stating it). Do you know how can this be proven, or any source where I can find the proof? $\endgroup$ – AccidentalFourierTransform Jul 23 '15 at 19:15
  • $\begingroup$ @qftishard I imagine you'd be able to find it in S. Weinberg, The Quantum Theory of Fields, Vol. I (1995). Chapter 5 would be a likely location for it. $\endgroup$ – Ryan Unger Jul 23 '15 at 19:27
  • $\begingroup$ @qftishard K. Cahill, Physical Mathematics (2013) has a nearly complete partial proof on page 390, with some minor details left to the reader (a Taylor expansion and arithmetic). $\endgroup$ – Ryan Unger Jul 23 '15 at 19:47
  • $\begingroup$ @qftishard The second reference answers your other question, but since the answer requires multiple pages of matrix algebra and group theory, I am hesitant to post an answer on that question. $\endgroup$ – Ryan Unger Jul 23 '15 at 19:48
  • $\begingroup$ Well that's really helpful. Thank you very much. I'll check them out. $\endgroup$ – AccidentalFourierTransform Jul 23 '15 at 21:04

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