# Global vs. Local (gauge) Symmetry

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• paralleltransport
In summary, gauge symmetry, also known as gauge redundancy, refers to the mapping of multiple representations to the same physical state. There are conflicting definitions of "large" and "smaller" gauge transformations in the literature. A global symmetry should not be called a gauge symmetry as it can lead to confusion. Local gauge symmetries cannot be spontaneously broken and are better described as the "Higgs mechanism". A paper on superconductivity and electromagnetic gauge symmetry provides a clear explanation of these concepts.
paralleltransport
TL;DR Summary
Let's pick a good definition of gauge "symmetry"
Gauge symmetry is highly confusing, partly because many definitions differ in the literature. Strictly speaking gauge symmetry should be called gauge redundancy since you are mapping multiple representations to the same physical state.

What is your favourite definition of what "large" gauge vs. "smaller" gauge transformations are?
What subtle points do you know about the distinction between a global vs. a local gauge transformation is (any examples)?

I'm polling because I have seen conflicting definitions in the literature.

vanhees71
I'd never call a global symmetry gauge symmetry, because that's indeed confusing. A gauge "symmetry" is indeed exactly defined as you write in your first paragraph, and to call it "gauge redundancy" would be a much more accurate choice of terminology (it implies also that local gauge symmetries cannot be spontaneously broken, which is known as Elitzur's theorem; it's rather the "Higgs mechanism" than spontaneous symmetry breaking).

A very nice and pedagogical paper on these issues the following in connection with superconductivity and electromagnetic gauge symmetry:

https://arxiv.org/abs/cond-mat/0503400
https://doi.org/10.1016/j.aop.2005.03.008

ohwilleke and paralleltransport

## 1. What is global symmetry and how does it differ from local (gauge) symmetry?

Global symmetry refers to a symmetry that is present throughout the entire system, meaning that the system looks the same no matter where you are in it. Local (gauge) symmetry, on the other hand, refers to a symmetry that varies from point to point within the system. This means that the system may look different at different points, but there is still an underlying symmetry that holds true.

## 2. What is the significance of global vs. local symmetry in physics?

The presence of global vs. local symmetry in a physical system can have important implications for the laws of physics that govern that system. For example, global symmetry can lead to conservation laws, while local symmetry can give rise to gauge theories, which are fundamental in understanding the behavior of particles and forces in the universe.

## 3. Can global symmetry be broken?

Yes, global symmetry can be broken in certain physical systems. This occurs when the symmetry is no longer present in the system due to external influences or interactions. In contrast, local symmetry cannot be broken, but it can be spontaneously "hidden" or "unmasked" in certain situations.

## 4. How does the Higgs field relate to global vs. local symmetry?

The Higgs field is a fundamental component of the Standard Model of particle physics and is responsible for giving particles mass. It is also intimately connected to global vs. local symmetry. In particular, the Higgs field breaks the global symmetry of the system, while preserving the local symmetry, which is essential for the consistency of the Standard Model.

## 5. Are there any real-world examples of global vs. local symmetry?

Yes, there are many real-world examples of global vs. local symmetry. One of the most well-known is the symmetry of a sphere. When viewed from any angle, a sphere appears the same, making it an example of global symmetry. On the other hand, a honeycomb structure exhibits local symmetry, as each individual cell may look different but has the same underlying structure.

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