TITLE: Complex scalar field
QUESTION [1 upvotes]: In his book on Quantum Field Theory, Ryder mentioned in p. 91 under the title Complex Scalar Fields and Electromagnetism, the following:
He said that under a global phase transformation $$\phi \rightarrow e^{-i\Lambda}\phi        $$
and $$ \phi^* \rightarrow e^{-i\Lambda}\phi ^*       $$ the Lagrangian density
$$ {\cal L}=(\partial _\mu \phi)(\partial ^\mu \phi^*) -m^2\phi^* \phi$$ [with Minkowski signature $(+,-,-,-)$] is invariant. 
He then said, that this gauge transformation of the first kind doesn't involve spacetime (it is purely 'internal'). 
My question is can we make it involve spacetime? And does the fact that it doesn't involve spacetime make it a less important symmetry if I shall put it this way?

REPLY [1 votes]: It is unclear what you mean by transformation involving space-time, well, at least, I found two possibilities.
The mentioned transformations are not gauge, since $\Lambda$ does not depend on space-time. So one way to make it involve space-time is to set $\Lambda \rightarrow \Lambda(x)$. The given Lagrangian is not invariant under these gauge transformations: you have to add an additional term called the connection and it will become something like
$$ L = (D_{\mu} \phi)(D^{\mu} \phi^{*}) - m^2 \phi^{*} \phi; \qquad D_{\mu}\equiv \partial_{\mu} + i e A_{\mu}. $$
This describes how charged scalars interact with electromagnetic potential $A_{\mu}$, which is the whole point of gauge theory: the requirement of gauge-invariance reduces to the need for an additional gauge field $A$.
Nevertheless, you can still say that gauge transformations do not involve space-time in sense that the reason for their existense is the specifics of the structure of the local Lagrangian. It does not make sense (at least I don't know of any attempts) to extend it further. As an example of gauge transformations which actually involve space-time, you can check out GR with its diffeomorphism-invariance.
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