a basis) to span that plane the Cartesian coordinate system is just one special case of this general scheme. For instance, one can convert a displacement along a plane into a vector in, or vice versa, once one fixes two linearly independent displacement vectors (i.e.
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Equivalently, one can pick an origin and a unit displacement vector. a point) on a line into a real number, or vice versa, once one fixes a unit of length, an orientation along the line, and an origin on that line. Alternatively, one can replace the unit of length and the orientation by a unit displacement vector along the line. In this case, the coordinate system is specified by the length unit together with the choice of orientation.
#WHAT IS THE BEST BOOST GAUGE HOW TO#
Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis. By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed. But the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations).
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“ Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth.