# Srednicki 2.8 / Inverse Lorentz Transformation

## Homework Statement

I need to demonstrate the relation $$[\varphi(x),M^{\mu\nu}]=\matchal{L}^{\mu\nu}\varphi(x)$$

where
$$\mathcal{L}^{\mu\nu}\equiv \frac{\hbar}{i}(x^\mu\partial^\nu-x^\nu\partial^\mu)$$.

## Homework Equations

$$U(\Lambda)^{-1}\varphi(x)U(\Lambda) = \varphi(\Lambda^{-1}x)$$

$$\Lambda = 1+\delta\omega$$

$$\Lambda^\mu_\nu = \delta^\mu_\nu + \delta\omega^\mu_\nu$$

$$U(1+\delta\omega) = 1 + \frac{i}{2\hbar}\delta\omega_{\mu\nu}M^{\mu\nu}$$

## The Attempt at a Solution

I can pretty easily evaluate the LHS of the expression to give

$$\varphi(x) + \frac{i}{2\hbar}\delta\omega_{\mu\nu} [\varphi(x),M^{\mu\nu}]$$,

but I am having a ton of trouble with deciphering what is meant by $$\varphi(\Lambda^{-1}x)$$, particularly how I am supposed to get derivatives of the type $$\partial^\mu$$ when I had thought $$x\equiv x^\mu$$ here, implying derivatives of the kind $$\partial_\mu$$.

I tried, for instance, defining an inverse transformation so

$$\bar{x}^\mu = {\Lambda_\nu}^\mu x^\nu = (\delta^\mu_\nu - \delta\omega^\mu_\nu)x^\nu$$ and finding a Taylor series about x, but I don't really know how to do this in terms of indices. I get

$$\varphi(\Lambda^{-1}x) = \varphi(x^\mu-\delta\omega^\mu_\nu x^\nu)$$ so that

$$\sim \varphi(x) - \delta\omega^\mu_\nu x^\nu\bar{\partial_\mu}\varphi(\Lambda^{-1}x)$$, which is sort of close, but not right.

If I magically change the inverse transformation to $$\Lambda_{\mu\nu}x^\nu = x_\mu - \delta\omega_{\mu\nu}x^\nu$$ it works, but I can't justify this.

I think the problem is with my inverse transformation and taking the derivative. Help would be greatly appreciated.

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