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Gauge Theory in Physics

Introduction

Physical systems are described in terms of a coordinate system for a manifold. The behavior of the physical sytem should be independent of transformations of that coordinate system. This independence with respect to transformations of the coordinate system implies certain quantities are invariant over time, the conservation laws. For exmample, invariance to translations of the coordinate system implies the conservation of linear momentum. Likewise invariance to rotation of the coordinate system implies the conservation of angular momentum. This link between invariance to trnasformations of the coordinate system and conservation laws is the embodiment of Noether's Theorem.

In electromagnetic theory it was found that the electric and magnetic field could be expressed in terms of a scalar and vector potential. But the vector potential for an electromagnetic field is not uniquely determined. The vector potential function can be varied in certain ways without altering the corresponding electric and magnetic fields. This phenomenon is much like the invariance to transformations of the coordinate system for locations. This suggested that the complete description of a physical system requires the specification of another dimension besides the locational dimensions. In electromagnetic phenomena, for example, this other dimension could be the phase angle of the wave vector.

Modern physics has incorporated the mathematical field of fiber bundle topology to describe this theory. At each location in physical space, or manifold to use the mathematical term, there is a fiber that represents another manifold associated with that point.

Gauge theory involves transformations of the description of a physical system in terms of the additional manifold located at each point. An advanced field of particle physics, called String Theory, emphasizes hidden dimensions to the physical world. While the physical world appears to be three dimensional, String Theory holds there are hidden dimensions that make the world ten dimensional. For an instance of hidden dimension, consider a very thin tube like a soda straw. This appears to be one dimensional but in reality is two dimensional. Location on this tube could be described by a longitudinal position and an angle identifying position on the circular cross-section. One interesting aspect of this example is that the longitudinal dimension might be unconstrained but the "fiber" is a closed circle.

This field is relevant to low-dimensional semiconductors because the constrained dimensions could be considered to constitute a fiber associated with each position in the unconstrained dimensions.

The Theory of Electromagnetic Fields

(To be continued.)


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