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dennis
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I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a SU(2) gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(\cdot)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation equalsis the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a SU(2) gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(\cdot)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation equals the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a SU(2) gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(\cdot)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation is the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

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dennis
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  • 10

I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a $SU(2)$SU(2) gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(r)$$\omega(\cdot)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation equals the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a $SU(2)$ gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(r)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation equals the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a SU(2) gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(\cdot)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation equals the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

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Qmechanic
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I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a SU(2)$SU(2)$ gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(r)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdfhttp://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation equals the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a SU(2) gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(r)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation equals the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a $SU(2)$ gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(r)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation equals the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

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