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.

  1. 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?

  2. 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 $U(1)_B$)?


1 Answer 1

  1. 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).

  2. 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.

  3. 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.


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