When two global symmetries are distinct? How to differentiate between two separate global symmetries? Any local symmetry is also trivially a global symmetry. Now consider two $U(1)$ symmetries of the Standard Model Lagrangian, the $U(1)_{em}\equiv e^{ieQ\theta(x)}$ (after electroweak symmetry breaking) and $U(1)_B\equiv e^{iB\alpha}$ where $B$ is a global symmetry related to baryon number conservation. 


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*Why can we not regard the $U(1)_B$ symmetry as the global aspect of $U(1)_{em}$ symmetry? Why they should be thought of as distinct?

*When two global symmetries should be thought of as distinct or identical?
Here is how I understand it: Let us consider a special case when $\theta(x)=\theta$. If we also change $B\to B^\prime$ such that $U(1)_B\equiv e^{iB^\prime\theta}$ where $\alpha=\frac{B^\prime}{B}\theta$. Now, $eQ=B^\prime$. If the Baryon numbers of all fields were just a multiple of their electric charge, one can could have regarded $U(1)_B$ to be same as the the global aspect of $U(1)_{em}$. Since this is not the case the $U(1)_B$ is not same as the global part of $U(1)_{em}$.
Is my explanation correct? Is this the correct way to differentiate between two global symmetries (say, $U(1)_L$ and $U(1)_B$)? 
 A: *

*The data for a symmetry is not only given by writing down the group, like $\mathrm{U}(1)$, but also by specifying in which representation every field/dynamical variable of the theory transforms in. The electromagnetic and the baryonic symmetry differ in how they act on the individual fields (or, in a different but equivalent diction, the fields are charged differently under both symmetries).

*Even if two symmetries have exactly the same representations for all fields, they need not be the same: Having a symmetry group $\mathrm{U}(1)\times\mathrm{U}(1)$ where all fields transform in the same representation for the left and the right $\mathrm{U}(1)$ is perfectly possible, just...not very useful.

*You should be much more careful with the expressions you write down. For instance, $\mathrm{U}(1)_\text{em} = \mathrm{e}^{\mathrm{i}eQ\theta(x)}$ does not make any rigorous sense at all - the l.h.s. is a one-dimensional Lie group, the r.h.s. is an element of the group of gauge transformations given by functions $\mathbb{R}^{1,3}\to\mathrm{U}(1),x\mapsto \mathrm{e}^{\mathrm{i}eQ\theta(x)}$ for functions $\theta : \mathbb{R}^{1,3}\to\mathfrak{u}(1)\cong\mathbb{R}$. These things cannot be equal. $\mathrm{U}(1)$ is just the circle group - the group itself does not change, regardless of whether physicists write $\mathrm{U}(1)_\text{em}$ or $\mathrm{U}(1)_\text{B}$ or whatever. Note also that the $e$ and $Q$ appearing there are properties of a particular field, so this is a representation map rather than a map into the abstract $\mathrm{U}(1)$. The subscript denotes "which" of the different $\mathrm{U}(1)$ of your theory it is, i.e. which representations of the fields you have to use, but they aer all isomorphic as groups.
