# What are the symmetries associated to the conservation of quantum numbers?

According to the beautiful Noether’s theorem, a conservation law is associated to a specific symmetry such as for example energy $\leftrightarrow$ translation in time, momentum $\leftrightarrow$ translation in space, etc... In the context of the Standard Model a lot of new conservation laws come in play, such as charge conservation, lepton number conservation, baryon number conservation TCP, etc... Is there a symmetry associated with the conservation of those quantum numbers? It seems like this would shed some light on the nature of those quantum numbers, which are otherwise so abstract.

I also noticed that the symmetries associated to the (classical) conservation laws by Noether are the same associations as in the uncertainty principle. Can the two be related in some way, or deduce one from the other?

Thank you very much in advance.

Julien

## 2 Answers

According to the beautiful Noether’s theorem, a conservation law is associated to a specific symmetry [...] Is there a symmetry associated with the conservation of those quantum numbers?

Yes, there are. Judging by your question I think a very brief and basic comment on how symmetries work in QM will be helpful. Let $\mathcal H$ be the space of states of your system, $G$ the symmetry group of your system. Elements $g$ of the group $G$ act on the states $|\psi> \in \mathcal H$ through
$$|\psi> \rightarrow U(g) |\psi>$$
and on observables through $$A \rightarrow U(g)\,A\,U(g)^{\dagger}$$ Where $U(g)$ are unitary operators that form a representation of G. Usually $G$ is either a Lie Group or a discrete group. If it's a Lie group, under some hypoteses one can write $U(g)= e^{c_a T_a}$, where $c_a T_a$ is a linear combination of (representatives of) the generators of the group's algebra. The number of generators is equal to the dimension of the group, but for the present discussion let's just consider a single generator and let's write just $e^{c T}$. The $T$'s can be taken to be self-adjoint. Requiring that the Hamiltonian $H$ is invariant under the action of the group means $$H = U(g)\, H \,U(g)^{\dagger} = e^{c T} H e^{-cT} \qquad \forall g(\iff \forall c)$$ Taking the derivative of this line with respect to $c$ you get: $$0= TH -HT = [T,H] \quad \Rightarrow \quad \dot T=-i[T,H]=0 \qquad \qquad (1)$$ And this means that the generator $T$ of the symmetry is conserved (I'm of course talking in Heisenberg picture). Notice that from $(1)$ also the reverse is true: if you have a conserved observable $A$, you can generate a symmetry acting with $e^{cA}$. This is the link between symmetries and conservation in QM. There are many details I'm glossing over, and complications may arise, but this should give you the idea.

...charge conservation, lepton number conservation, baryon number conservation TCP...

Charge, lepton and baryon conservation arise from this exact mechanism. The symmetries are the global gauge symmetries of the Lagrangian, and groups involved for example are $U(1)$ for the electric charge in QED, $SU(2)$ for the leptonic number in electroweak theory; their generators (and combinations thereof) give the operators associated with those quantum numbers.

I also noticed that the symmetries associated to the (classical) conservation laws by Noether are the same associations as in the uncertainty principle. Can the two be related in some way, or deduce one from the other?

I think the link you mention is accidental. The uncertainty principle (which is actually a theorem) holds for every couple of observables and states that for every two observables $A,B$ $$\sigma_A \sigma_B \geq \langle \frac{[A,B]}{2i} \rangle$$

Where $\sigma$ is the square root of the variance, and all means are computed on the same state.

• Thank you for your very complete answer, I guess that settles it regarding the symmetries. About the uncertainty principle, I realize now that the reason why I found that so troubling is that the pairs are canonically conjugate momenta usually, but as you stated HUP can be applied to any pair of observables. Do you know if pairs of conjugate momenta are always non-commuting though? Edit: according to Wiki it is always $i \hbar$. – Jxx Feb 16 '18 at 20:02
• In the "straighforward" cases conjugate variable are non-commuting "by construction" I would say. This has to do with the procedure of quantization. To quantize a system in the operatorial formalism, you start from your classical hamiltonian $H(q,p)$ with $q,p$ conjugate (classical) variable. They have poisson brackets $\{q^i,p_j\}=\delta^i_j$; to quantize the system you make them into operator imposing the canonical commutation relations $[q^i,p_j]=i \delta^i_j$ (basically you substitute the poisson bracket with $\frac{[\, , \,]}{i}$. So they are automatically non-commuting – tbt Feb 16 '18 at 20:13

charge conservation, lepton number conservation, baryon number conservation TCP, etc...

You must notice that these conservation laws are on integers. There is no continuum. They have given us the clue to group symmetries, as seen in the eightfold way, i.e. the particles belong to discrete levels in group representations. The meson octet. Particles along the same horizontal line share the same strangeness, s, while those on the same diagonals share the same charge, q.

You state:

same associations as in the uncertainty principle.

Could you expand on this? dx*dp for example, momentum goes into a conservation law, but space?

• Thank you first of all for your answer. Good point regarding the integers. And about the symmetry you mean that conservation of strangeness $\leftrightarrow$ translation on the horizontal line for example? Is there more to it though? Because your example states also conservation of charge $\leftrightarrow$ translation on the diagonal, but conservation of charge also means on a broader sense invariance in gauge transformations as far as I know. – Jxx Feb 16 '18 at 18:18
• Regarding the uncertainty principle I meant that $\Delta E \Delta t$ relates energy change to translation in time and that $\Delta p \Delta x$ relates momentum change to translation in space. Those associations seem to me at first glance similar to the Noether symmetry-conservation law associations, but maybe that's just my fantasy. – Jxx Feb 16 '18 at 18:19
• I meant the structures, they are stable because of the conservation of the quantum numbers, and the underlying level of quarks is responsible. That the symmetry displayed by the integers is due to the conservation but not in the sense of Noether. On the heisenberg un. energy is conserved but not time . I do not see a connection of H to N – anna v Feb 16 '18 at 18:44