Whether group theory can have similar successes in fields like linguistics or anthropology seems a bit doubtful to me.
Jim Farmelant
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My apologies for taking so long. I had to look up Noether's theorem* and that didn't do much good. Here is what it looks like:
http://www.cc.emory.edu/PHYSICS/Faculty/Benson/361/notes/29/29.html
The formalism is dense and the underlying machinery is subtle and difficult to explain. On the other hand, the philosophical implications are wide ranging and comprehensive.
To make the background and implications of Noether's Theorem as understandable as possible I had to retrace a fair amount of ground. I came to the ideas in physics through an abstract interest in space. The picture I created in my mind for these concepts relies almost entirely on geometric ideas. These serve very well, but they obscure what is known as the energy picture. Noether's theorem is a key part of an energy picture. In mechanics, a spatial view results from considering motion as a path through space that inscribes that space. In this view an electromagnetic or gravitational field has a shape which is the trace of a particle as it moves in the field. In an energy picture, motion is considered a state that can change with respect to time. That is, the energy picture is the time evolution of a system.
The energy picture is often called the Hamiltonian, after Hamilton's principle that says the motion of a system is governed by the principle of least action. If there are a family of paths between two positions in time, the principle of least action means that a system will follow the shortest path. This idea expresses the conservation of energy. In mathematical terms the condition of least action means that of the possible paths between two positions, the path that has the smallest integral is the path taken.
Imagine a closed path that circumscribes a region. The area of region will not change in value if the length of the perimeter remains the constant, despite changes in its shape. The changes in perimeter shape are reflected in a spatial view, but the constant value of its length and of that of the inscribed area is not immediately obvious. The constant or conserved quantity of area or perimeter length, despite arbitrary changes in spatial deformations is the concept in Noether's theorem.
To paraphrase the theorem is relatively simple. It says that charge (or unit of energy measure) is conserved as a system progresses from one temporal position to another in a field.
In elementary mechanics the path of a particle is a curve with end-points and the area under the curve represents the energy, say momentum. Such a path is closed by virtue of dotted lines at the end points that intersect the coordinate axis and therefore define the area. Integrating the equation for the path, yields a value for the area, and therefore the energy. More advanced methods compute the line integral itself. In the vector picture, integrating with respect to the orthonormal tangent to the path achieves a similar result.
Only a closed path circumscribes an area. So in order to find the energy associated with a moving particle, its path must be closed. The path must be continuous and circumscribe a simply connected region in order to insure the shape of the path is independent of the integral for its area. There are some subtle difficulties involved here that are covered in the theorems of Gauss, Stokes and Green for line and surface integrals.
The closed path requirement poses a problem in the generalization because obviously most paths are not closed, unless coordinate frames, together with initial and ending positions are specified. In addition, changes in orientation and coordinate frames create spurious results. So, for the general case, it is necessary to find the invariants of such a path and identify those with the fundamental properties that must be preserved during various motions or transformations.
In this case, some relation between closed paths and their areas should remain invariant in order that the energy of a system remains constant. This reflects the idea that if there have been no inputs of energy to change its state, then the particle (or system) should maintain its state, whatever that is. In other words, linear and angular momenta, and energy are be conserved.
So, the question is how to close every path, even if no path can be closed? The answer is that all paths are turned into closed paths by transforming them into circles. Or equivalently, transforming Euclidean space into a space such that the only paths that can be taken in this space are closed.
Transforming lines into circles is accomplished by what is called a conformal transformation, or an inversive transformation of the plane into a circle. The simplist version of this transformation is the map of whole numbers into their fractions: n => 1/n.
This map has the following geometric representation. Evenly space the whole numbers along the x-axis with a unit circle at origin. Construct two lines from each number point, above and below the x-axis and tangent to the circle. Draw a cord between the tangent points on the circle and through the x-axis, marking 1/n on the x-axis, in the interior of the unit circle.
Notice that the reflected points 1/n in the circle never reach the center at 0. This is because the center lays on a diameter with tangents parallel to the x-axis. This is a map of inf => 0. The point at infinity is mapped to the center of the circle.
A model for a 2D map of lines to circles is accomplished by constructing a polar to a point with respect to a circle. There are two conjugate versions of this model. The first assumes the point is outside the circle. Make the same construction as before, but choose any point on or off the x-y axis (coordinate independent). Draw a line between the circle center and the exterior point. Draw tangents from the point to the circle. The cord connecting the tangent points in this case is the polar to the external point.
For the second version, choose a point inside the circle. Draw two intersecting cords through the point and construct a pair of tangents from each cord. Draw a line between the pair of intersect points of the two tangents. This exterior line is the polar to the point interior to the circle.
Choose an arbitrary line exterior to the circle, select points along the line and find their image point on the interior of the circle. After choosing more points along the exterior line and connecting their interior image points, you will discover these form a small interior circle, with one point as the center of the circle. Lines far away from the circle will have small circle images. Lines very near the circle will have large circle images inside the greater circle.
In this construction all the points on the exterior of the circle can be mapped to points on the interior of the circle. Exterior lines become circles on the interior of the circle with one point at the center, which represents the two infinitely distant ends of the exterior line. All points infinitely distant from the circle are mapped to the center point. Thus, all lines are closed paths, if the center point is added, which is reciprocal to the points at infinity. Notice that all lines share this center point and therefore it is one of the invariants in this construction.
The polar, point, circle construction is very concept rich, but it takes awhile to see that it is a complete generalization of the map n => 1/n. In other words whole coordinate systems can be transformed under this map. One can either define the space with these features, or one can define operators that reproduce equivalent algebraic results.
Technically what I have been describing is a plane geometry model for complex analysis. To see a color animation go here:
http://www.math.psu.edu/dna/complex-j.html
In mechanics, the transformation from Euclidean space to a conformal space requires that the conservation laws are invariant. Paths are mapped to paths, areas to areas, orientation is reflected. The Euclidean metric or absolute lengths and angles are not preserved. What is preserved under conformal transformations are ratios, n:1/n, and the center or points at infinity. With regard to the ratio of areas, it is possible to take the path integral and form a quotient with its transformed image. What is preserved is the limiting value (ratio) as the quotient integral is taken. This invariant is called the Jacobian or Jacobian determinant. In crude terms what the above means is divide one path from another so many times until one vanishes. The count of the number of times is the ratio of magnification.
The most general form of linear transformations are called projective transformations or the general linear group GL(). Under the projective transformations, the length invariants are cross ratios. If ABCD are four points on a straight line, then AB/AC:CD/BC remains invariant.
Draw any triangle ABC and medians AL, BM, CN concurrent at center P. The product of the ratios in which the sides AB, BC, CA are divided at L, N, M is +1. The ratios l,m,n uniquely define point P and P uniquely defines the ratios: l:m:n. These numbers (l,m,n) form the homogeneous coordinates of P. In such a space, arbitrary deformations of shape have no quantitative effect, since all such transformation on their homogeneous coordinates are equivalent.
As far as I understand it, the machinery behind Noether's Theorem defines how to construct such invariants for the continuous version of conformal and projective transformations (Lie groups) in their respective spaces (conformal and Hilbert space). Selected invariants compose the conservation laws (specifically the conservation of charge for the Hamiltonian). See:
(http://theory1.physics.wisc.edu/~ldurand/711html/courseinfo/NoethersThm.html)
With all that I have completely forgotten what the point was. Oh, yeah, was group theory artificial? What are the philosophical implications of these ideas? See another post under the same subject later.
Chuck Grimes
*In case people don't know, Emmy Noether is considered the greatest woman mathematician of all time. She taught (under sexist duress as an equivalent to temporary adjunct) at Gottingen with David Hilbert as chair. But she was with about every luminary of early 20thC math. Noether's speciality was the algebra of rings and ideals. I think that Noether's theorem was named after her because it uses her contributions to the theory of invariants on differentiable manifolds (thesis topic) for its formulation. This is like the name of Hilbert for Hilbert space. Hilbert created the machinery to build such a space, but didn't in fact do it himself. When the nazis took over, most of the fabulous Gottingen math faculty were fired or quit almost immediately since many were Jewish. Noether came the US and taught at Princeton for a year or so, before she died in 1935.