Under the infinitesimal translation ##x^{\nu} \rightarrow x^{\nu}-\epsilon^{\nu}##,(adsbygoogle = window.adsbygoogle || []).push({});

the field ##\phi(x)## transforms as ##\phi_{a}(x) \rightarrow \phi_{a}(x) + \epsilon^{\nu}\partial_{\nu}\phi_{a}(x)##.

I don't understand why the field transforms as above. Let me try to do the math.

The Taylor expansion of ##f(x+\delta x)##, where the argument ##x+\delta x## is a ##4##-vector and ##f## is a scalar, is ##f(x+\delta x)=f(x)+\frac{\partial f}{\partial x^{\nu}}(\delta x)^{\nu} + \dots##

So, ##\phi_{a}(x) \rightarrow \phi_{a}(x-\epsilon) = \phi_{a}(x) + (-\epsilon^{\nu})\partial_{\nu}\phi_{a}(x)##.

Now, where did I go wrong?

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# A Field transformation under an active transformation

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