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What you wrote is a lagrangian of a free theory. All free theories have infinite-dimensional symmetry, called higher-spin symmetry. The current you gave is a standard Noether current corresponding to one of the generators of the higher-spin symmetry. Usual space-time symmetry is a subalgebra of the higher-spin algebra. As Maldacena a Zhiboedov showed in 3d, ...

2

The Lagrangian density reads $$\tag{1} {\cal L}~=~\frac{1}{2}\sum_{i=1}^2\left( \partial_{\mu}\phi^i \partial^{\mu}\phi^i -m^2\phi^i\phi^i\right).$$ Consider an infinitesimal transformation $$\tag{2} \delta \phi^i~=~\epsilon Y^i_{\nu_2\ldots \nu_n},$$ with generators $$\tag{3} Y^1_{\nu_2\ldots \nu_n}~:=~ ... 1 The answer is yes; this can be shown by evaluating the variation of the action. The action consists of three terms, we will consider them separately:$$\delta_\epsilon K(\Phi,\bar{\Phi})=\frac{\partial K(\Phi,\bar{\Phi})}{\partial\Phi^i}\delta_\epsilon \Phi^i+\frac{\partial K(\Phi,\bar{\Phi})}{\partial\bar{\Phi}_i}\delta_\epsilon\bar{\Phi}_i=i\epsilon K_i ...

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We can't create noting from something and we can't create something from nothing. Nothing or nonexistence doesn't exist, it is it's definition. The only thing that exist is energy in different forms. So energy exists and cannot become nonexistent, as there is no such thing as nonexistence. Energy can neither come into existence because it has nowhere to come ...

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Comments to the question (v2): It seems that OP assumes that $\alpha$ is independent of $x$, i.e., OP considers a global quasisymmetry $$\delta A_{\mu}=\alpha \dot{A}_{\mu}.$$ The corresponding conserved quantity is energy, cf. Example 1 on the Wikipedia page for Noether's (first) theorem.

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We may approach the problem via differential forms, or ordinary tensor calculus: Differential Forms: The field strength tensor $F$ is a differential form given by the exterior derivative of the 1-form $A$, i.e. $F=\mathrm{d}A$ which in components is $\partial_{[\mu}A_{\nu]}$. To add a total derivative to the form $A$ is equivalent to adding the exterior ...

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