A What exactly does 'Locality' in Gauge Theory mean?

[The Tortoise-Man]

What means exactly the principle of 'locality' in context of gauge theory? Motivation: David Tong wrote in his notes on Gauge Theory (p 115): "their paper (the 'original' paper by Yang & Mills introducing their theory) suggests that global symmetries of quantum f ield theory– specifically SU(2) isospin– are not consistent with locality."

Well, the question is what they meant by locality? And why this statement seems 'plausible' even if Tong pointed out that it is not the modern view.

 

[vanhees71]

The idea is that you have a symmetry like isospin. Take as an example a world of two quarks, up and down. These we can put in an "isospin doublet", i.e., we have two Dirac fields, $u$ and $d$ and construct a 2D "iso-spinor" $\psi=\binom{u}{d}$ and write down the free Lagrangian as
$$\mathcal{L}=\bar{\psi} (\mathrm{i} \gamma^{\mu} \partial_{\mu}-m) \psi.$$
This is obviously symmetric under global SU(2) transformations of $\psi$, i.e.,
$$\psi'(x)=\hat{U}(\vec{\theta}) \psi(x)=\exp(g \mathrm{i} \vec{\theta} \hat{\vec{\sigma}}/2) \psi(x).$$
Now Yang's and Mill's idea was to make the "symmetry local" in the sense that $\vec{\Theta}$ may be chosen as an arbitrary $x$-dependent field, $\vec{\Theta}(x)$. Now the Lagrangian is no longer invariant, because auf the derivative. The way out is to introduce a corresponding gauge field and define the gauge-covariant derivatives
$$\mathrm{D}_{\mu} = \partial_{\mu} + \mathrm{i} g \vec{A}_{\mu} \cdot \hat{\vec{\sigma}}/2.$$
This makes the Lagrangian invariant, if you transform the quark fields and the gauge-boson field as
$$\psi'(x)=\hat{U}[\vec{\theta}(x)] \psi(x), \quad \vec{A}_{\mu}' \dot{\vec{\sigma}}{2} = \hat{U} \vec{A}_{\mu}' \dot{\vec{\sigma}}{2} \hat{U}^{\dagger} -\frac{\mathrm{i}}{g} \hat{U} \partial_{\mu} \hat{U}^{\dagger}.$$
This makes the "symmetry local", i.e., substituting for all $\partial_{\mu}$'s in the Lagrangian $\mathrm{D}_{\mu}$'s, the Lagrangian is invariant under the "local gauge transformations", where the $\vec{\theta}=\vec{\theta}(x)$ are arbitrary space-time-dependent group parameters:
$$\mathcal{L}=\bar{\psi} (\mathrm{i} \gamma^{\mu} \mathrm{D}_{\mu}-m) \psi.$$
Now the gauge fields should also become dynamical, i.e., one needs some "kinetic term" for them, and we still want to keep the Lagrangian gauge invariant. For this we need something with first derivatives of the $\vec{A}_{\mu}$, and the obvious candidate is the "curvature", i.e., the commutator of two gauge-covariant derivatives,
$$F_{\mu \nu}=\frac{1}{\mathrm{i} g} [\mathrm{D}_{\mu},\mathrm{D}_{\nu}].$$
Now this "field-strenght tensor" transforms under local (!) gauge transformations as
$$F_{\mu \nu}'=\hat{U} F_{\mu \nu} \hat{U}^{\dagger}.$$
And thus you can build the gauge-field Lagrangian as
$$\mathcal{L}_{\text{gauge}}=-\frac{1}{2} \mathrm{Tr} (F_{\mu \nu} F^{\mu \nu}),$$
which includes kinetic terms for the three gauge fields (note that $F_{\mu \nu}$ is a Hermitean traceless $2 \times 2$ matrix, i.e., it's su(2) valued, where su(2) is the Lie algebra of SU(2)), which look like for photons in electrodynamics but also interactions with three and four gauge bosons. This is the important addition when going from an Abelian local gauge theory (as is electrodynamics/QED) to a non-Abelian local gauge theory.

 

[The Tortoise-Man]

Yes, this procedure replacing global by local action is what Yang & Mills did to 'repair' this 'defect' which they reconized as 'global symmetries being not consistent with locality'. So roughly how I understand it is that their idea was: their hypothesis was that global symmetries are incompatible with 'locality'. How we remedy it? We make global symmetries local.

But my question is essentially 'what Yang & Mills understood as 'locality', (with which global symmetries are inconsistent according to Yang & Mills finally motivating them to introduce this construction you explaned above)

 

[vanhees71]

One should be aware that this argument his entirely heuristic and not a rigorous first-principle argument. The point is that if we define a certain field in a theory with such a "flavor symmetry" to represent a certain particle, that's arbitrary, because of the symmetry. E.g., take the three pions. They are isospin triplets, i.e., they belong to a representation of the isospin SU(2) with $I=1$. Usually we use the usual convention that we write everything in terms of eigenstates of $\hat{t}_3$ with eigenvalues $(1,0,-1)$, and these represent the positive, neutral and negative pion. Now you can argue that we physicists on Earth do this by just agreeing on this convention.

Now at some distant other place in the universe, maybe the physicists on another planet use the convention to represent the isospin state as the eigenstates of $\hat{t}_1$, of course with the same eigenvalues and then they associate the three pions with these isospin eigenstates. Their physics of pions is of course precisely the same as that on Earth, i.e., it doesn't depend on this choice of conventions. Also all experiments physicists can do are in principle pretty "local", i.e., what we observe with our experiments is influenced by the circumstances not too far away. So it's plausible to demand that the symmetries should also be local, i.e., the choice of which iso-spin component I use to define the charge states of the pion within this theory may vary locally. This was the heuristic argument by Yang and Mills to introduce the idea of local gauge symmetries, but of course they are useful for completely different reasons, but that's another story.

 

[The Tortoise-Man]

yes exactly, my concern was really focused on gaining a bit 'deeper' intuition on this heuristic principle of locality. May I try to pose a very vague thought experiment which came into my mind reading your last very insightful explanation:

essentially the "choice" of working with $\hat{t}_1$ or $\hat{t}_3$ in this context is matter of choice of concrete vector space basis, right? Can this 'principle of locality' phrased semiformally as follows: say we have our field $\psi$ on space manifold $M$. At that point a remark (which some time ag confused me a bit)

RMK: indeed if the manifold is complex enough then a field is not a function$\psi: M \to \mathbb{C}$, but a section of certain line bundle, but locally for appropriate localization $U \subset M$ the field $\psi$ can be regarded as a ' honest function' $\psi: U \to \mathbb{C}$

Now the funny part. Assume we are sitting at point $m_0\in M$ and want to performs an observation involving field $\psi$. Due to fundamental principle of QM this act involves always an active interaction with field which in turn changes it. Mathematically that is modeled by an appropriate trasformation : $\psi(x) \to e^{i\theta(x)} \psi(x)$. Theta is some appropriate element in Lie algebra which is exponentiated to Lie group, don't want to specify it here, it's really about the conceptional issues. (to draw connection to above with the example of SU(2)-action and the eigenvalues of $\hat{t}_1$ or $\hat{t}_3$, one can interpret this transformation as continuous from point to point changing choices of concrete basis of the vector vector space we are working with)

Now the question: can the locality principle in sense of Yang & Mills be vague understood as this kind of interaction with field cause by us as 'observers' sitting in $m_0\in M$ be understood as that the transformation of $\psi$ we cause has the property that $e^{i\theta(x)} \neq \mathbb{1}$ only in some neighborhood of $m_0$, but 'far away' from us always MUST BE given as identity? Ie in the sense that our interaction literally only change it locally?

Is this consideration - at least at vague level - resonable?

 

[The Tortoise-Man]

Recently I tried once moreto reflect your thought experiment but I'm not pretty sure if I completely got how it explains the locality philosophy by Yang and Mills, especially how does it interplay with concept of "gauge" they introduced. Could you check if I now understand your explanation correctly? (I think in my last comment I confused a lot of things)

So, say as before we model our universe as certain manifold $M$ and wonder how "our physics practiced on Earth" (say arround spatial point $m_E \in M$ on the manifold) of eg pions differs from physics of pions "practiced" by physicists on "another planet" around point $m_P \in M$ far away from $m_E$.

As you said assume we on Earth choose the convention to write everything in terms of eigenstates of $\hat{t}_3$ and on another planet the choose the convention to work with eigenstates of $\hat{t}_1$.

One should expect that "the physics on both places" is the same. To keep it precise: Do you mean "the physics should be same" phrase in the naive sense that the Lagrangian describing the physics on Earth $\mathcal{L}_E$ should differ from the Lagrangian $\mathcal{L}_P$ with which they work on the other planet only by certain symmetry transformation $\psi(x) \to \widetilde{\psi}:=e^{i \theta(x)} \phi$, right so far? Then, so far, nothing new.But then you write

Also all experiments physicists can do are in principle pretty "local", i.e., what we observe with our experiments is influenced by the circumstances not too far away. So it's plausible to demand that the symmetries should also be local, i.e., the choice of which iso-spin component I use to define the charge states of the pion within this theory may vary locally.

and I'm not completely sure what you mean there. Do you mean it in the sense that you regard within this thought experiment the procedure of taking "an explicit choice of which iso-component our model locally use" as a symmetry transformation itself? Right?

If yes, then, what do you the precisely mean by

the choice of which iso-spin component I use to define the charge states of the pion within this theory may vary locally.

in mathematical terms? Do you mean by this that the message of locality philosophy by Yang and Mills is that every such "family of choices varying from point to point" $(\hat{t}_{s(m)})_{m \in M}$ aka symmetry transformation on the underlying manifold which iso-spin component is used by physicists at point $m \in M$ to model the pion physics vary "continuously"? In other words, that gauge transformations must be "continuous" on underlying space as mathematical function, so no "jumps" etc?

And then in order to legitimize the naive locality, we continue the reasoning by, say on the Earth - ie at point $m_E$ - we change with time our choice of which iso-component our pion model use, ie $s(m_E)=s(m_E)(T)$ is time dependent. Then the physicists on the other planet at $m_P$ percive our permanent procedure to vary with time the iso-component we are working with only after "some time has past", ie as only after some time has passed the $t_s(m_P)$ can be affected by "our wheeling and dealing on Earth".

Is that what they mean by "locality"?

Or do I mising the pun you intended to emphasise in your thought experiment?