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In "Lectures on the Infrared Structure of Gravity and Gauge Theories", Strominger considers the so-called asymptotic symmetries. If I got it right, the basic idea is that one chooses a set of falloff boundary conditions near null infinity specifying the asymptotic behavior of the fields and defines an allowed gauge symmetry as one that preserves these conditions, and a trivial gauge symmetry as one that acts trivially on the physical data at infinity. The asymptotic symmetry group is therefore $${\rm ASG}=\dfrac{\text{allowed gauge symmetries}}{\text{trivial gauge symmetries}}.\tag{2.10.1}$$

For electrodynamics one finds out that these asymptotic symmetries are large gauge transformations which transform $A_\mu \mapsto A_\mu + \partial_\mu \varepsilon$ where $$\varepsilon=\varepsilon(z,\bar{z})+{\cal O}\left(\frac{1}{r}\right)\tag{2.10.6}$$

where $(z,\bar{z})$ are holomorphic coordinates on the $S^2$ at null infinity ${\cal I}^\pm \simeq \mathbb{R}\times S^2$.

So this $\varepsilon$ doesn't vanish at infinity, but rather approaches a funcion of the angular coordinates. Still, $\varepsilon$ varies from point to point.

On the other hand, in this PhysicsOverflow post it is mentioned in the answer and comments that:

By the way it is not good to call these "large gauge transformations" because it is really a global symmetry, but with connections as parameters.

Now, I can't see how this transformation is a global transformation. In the most basic approach, a global transformation is one that does not vary from point to point. In fact, take Schwartz "Introduction to QFT and the Standard Model" (page 122):

Symmetries parameterized by a function such as $\alpha(x)$ are called gauge or local symmetries, while if they are only symmetries for constant $\alpha$ they are called global symmetries.

Now, if we were to follow Schwartz terminology, LGT would be local because $\varepsilon$ is obviously not a constant in general. But still in that post people allude to these transformations being global.

My question: how is large gauge transformation a global symmetry if it varies from point to point, i.e., $\varepsilon$ is in fact a function $\varepsilon(x)$? What would it even mean to have a global symmetry with "connections as parameters"? Is the issue here that Schwartz basic characterization of global vs local is not really accurate?

My take on this is that somehow one should add to the definition of local transformation a demand of it being compactly supported, whereas a global one should not be. But this is just a guess and could be totally misguided.

Gold
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1 Answers1

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There's lot of confusing jargon. Let me define the following four terms -

  1. Global symmetry - Continuous symmetry parameterized by a finite number of real numbers (could also be discrete).

  2. Local symmetry - Continuous symmetry parameterized by a function.

  3. Physical symmetry - A true symmetry of the theory. More precisely, such a symmetry implies existence of a conserved charge operator that is non-trivial (i.e. is NOT proportional to the identity operator).

  4. Unphysical symmetry - A symmetry of our description of the system, but not a symmetry of the system itself. This is a fictitious symmetry which is present solely due to the way we choose to describe the system, but is otherwise completely fake. There is no conserved charge operator corresponding to this (see Why do we seek to preserve gauge symmetries after quantization? to understand why such symmetries even exist). This type of symmetry is also sometimes called a gauge symmetry.

Now, the traditional lore is

  • global symmetry = physical symmetry.
  • local symmetry = unphysical symmetry.

In his work, Strominger argues that this is NOT true. There are local symmetries (what he is calling large gauge symmetries) which are indeed physical and satisfies all the relevant properties (including existence of a non-trivial charge operator). He, or others, are perhaps referring to them as global symmetries only in the sense that they are physical.

PS - The name "large gauge transformations" is also confusing because there is a totally unrelated idea (alluding to continuous transformations which are not connected to the identity) which also bears the same name.

Prahar
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