By Paul Roman
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Extra info for Advanced Quantum Theory
E. different phases in a region may describe the same physical situation. 11. They also point out that the phase factor for a curve C arises naturally within the ﬁber bundle formulation—it is just the holonomy of C. Let me explain. The electromagnetic potential is represented by the connection on a principal bundle with structure group U(1)—the multiplicative group of complex numbers of modulus 1. A connection on a principal ﬁber bundle determines a corresponding set of holonomies as follows. Consider a horizontal lift of a closed curve C beginning and ending at m.
This is simply related to the Lie-algebra-valued one-form ω on P (it is the covariant derivative of ω—see appendix B). A bundle section also maps onto a Lie-algebra-valued two-form iF on (a subset of) M, where F is a tensor ﬁeld that (for the purely magnetic case in which A = A) may be identiﬁed with the magnetic ﬁeld B or (for the full electromagnetic case A = Aμ ) the electromagnetic ﬁeld strength Fμν . F is 6 γ ˜ is called a horizontal lift of γ since γ is the image of γ˜ under π and the tangent vector to γ˜ always lies in the horizontal subspace H of the tangent space at each point along γ˜ .
All the gauge theories considered in this book admit of a ﬁber bundle formulation, and this offers a perspective which can deepen one’s understanding of them. But the beauty of the mathematics can also prove a confusing distraction from attempts to resolve important interpretative issues. Worse still, it can mislead one into thinking that these are either rendered unimportant by, or admit of a straightforward technical resolution in, this new framework. This is true in particular of the ﬁber bundle formulation of classical electromagnetism, or so I shall argue later in this chapter.