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The connection as a choice of horizontal subspace? 
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#1
Nov2912, 03:36 PM

P: 12

Hi,
I'm trying to understand the fiber bundle formulation of gauge theory at the moment, and I'm stuck on the connection. Every reference I've found introduces the idea of a connection on a principle bundle as a kind of partitioning of the tangent space at all points in the total space into a "vertical space" and a "horizontal space". The vertical space V_{p} consists of vectors in T_{p}P which are also tangent to the fiber at p, and the horizontal space H_{p} is a set of vectors such that V_{p}+H_{p}=T_{p}P. What I don't understand is why finding V_{p} doesn't uniquely specify H_{p}. It should be possible to construct T_{p}P without defining a connection, right? If so, wouldn't H_{p} just be every element of T_{p}P that is not also in V_{p}? I don't see how we are free to make this partition ourselves. Where am I going wrong? Thanks for reading! 


#2
Nov2912, 06:48 PM

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HW Helper
PF Gold
P: 4,771

Take Rē for instance, and for simplicity, assume that V = {(0,y)  y in R}. Then you're saying "take H:= Rē  V". But that's not a subspace! (Perhaps you overlooked the fact that H is supposed to be a vector subspace?)
On the other hand, H:={(x,0)  x in R} is a natural candidate... but there are (infinitely many) other choice as H:={ (x,ax)  x in R} for any a in R would do just as well. 


#3
Nov2912, 09:07 PM

P: 12




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