Please pardon me if my words are not rigorous mathematically, but I hope you understand what i mean.

In the fibre bundle description of gauge theories, what is actually the difference between dimension of spacetime and dimensionality (or in other words, degrees of freedom) of the bundle?

I mean, if spacetime is represented by $\mathbb{R^4}$ and the gauge fields are described by the fiber bundles (with $SU(2)$ or $SU(3)$ as fibers) with spacetime being the base space. I don't understand the idea of clubbing the spacetime which is something "physical" with the gauge groups which are "mathematical" or non-physical

  • $\begingroup$ What does "physical" mean? $\endgroup$
    – WillO
    Mar 12 at 2:31
  • 1
    $\begingroup$ The dimension of the bundle usually refers to the dimension of the total space, as a manifold. For vector bundles, this will be the dimension of the base space plus the rank of the fiber. $\endgroup$
    – WillO
    Mar 12 at 3:35
  • $\begingroup$ Although, I can't precisely define what is physical and what is not, you can think of a space is physical if you can attach real geometric objects to it, for example the shape of a ball is spherical, where the noun ball is attached to the real world objects. OTOH if i say the Lagrangian of a physical system is spherically symmetrical, i can't immediately attach anything physical, in the sense that we don't have a physical object attached to it. $\endgroup$
    – Eden Zane
    Mar 12 at 4:23
  • $\begingroup$ If you do not think a fiber bundle is a real geometric object, you probably haven't spent much time thinking about them. $\endgroup$
    – WillO
    Mar 12 at 5:17

1 Answer 1


Wind is a physical thing right? Wind speed can be represented as a 3-vector. So I can define a fibre bundle to represent the wind speed over spacetime. In this case the wind speed would be three dimensional.

In the case of gauge theories, the symmetry groups live in spaces each with their own dimensionality. Whether you want to consider gauge symmetries as physical things or not is up to you, but the theory as a whole works in that it is successful at making measurable predictions. In that sense, I consider them to be physical.

  • $\begingroup$ Thanks a lot. THis make sense to me. SO you are saying [gauge field- spacetime] is analogous to the [wind-space] combination? In that case i have another question, It is agreeable that the wind is physical as well, but what about the dimensionality of both, can the wind vector have higher dimensionality than the space? in other words can we have a 4-vecctor wind in 3d space? $\endgroup$
    – Eden Zane
    Mar 12 at 6:00
  • $\begingroup$ Although I cannot offhand think of an example, I cannot see any reason why you cannot have a situation where the dimension of the fibre bundle is larger than the space over which it is defined. (QCD would be such an example, but I guess you would argue that it is not "physical.") $\endgroup$ Mar 13 at 5:29

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