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Field 
A field is simply a quantity defined at every point throughout some region of space and time. 't Hooft


Form 

Fractal 
Does
neural form follow quantum function? 

Gauge 
As far as gravity is concerned, Einstein’s General Relativity is a beautiful and complete theory. But as Einstein realized it has to be extended to account for other physical forces, the most notable being electromagnetism. It is perhaps no accident that the first and most significant step in this direction was taken by a mathematician – Hermann Weyl. He showed that, by adding a fifth dimension, electromagnetism could also be interpreted as curvature. His idea was that the size of a particle could alter as it passed through an electromagnetic field. In analogy with railways it was called a gauge theory, and this name has stuck through subsequent evolutions of the theory. Unfortunately for Weyl, Einstein immediately objected on physical grounds that this would have meant different atoms of, say hydrogen, would have different sizes depending on their past history, in contradiction with observation. Given this devastating critique, it is remarkable but fortunate that Weyl’s paper was still published, with Einstein’s objection as an appendix. Clearly the beauty of the idea attracted the editor, despite the fatal flaw. In fact, beauty often wins such contests, because with the advent of quantum mechanics, with its complex wave functions, it was pointed out by Kaluza and Klein that Weyl’s gauge theory could be salvaged if one interpreted the variable as a phase rather than a length. A pure phase shift by itself is not physically observable and so Weyl’s theory avoids the Einstein objection.
Atiyah


Gravity 
So the
local gauge symmetry
also requires the introduction of gauge potentials,
which
are responsible for the gauge interactions, to connect
internal
directions at different spacetime points. We also
find
that the role the gauge potentials play in fiberbundle
space
in gauge theory is exactly same as the role
the affine connection plays in curved spacetime in
general
relativity. Cao 

Group 
What we learn from our whole discussion and what indeed has become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure endowed entity Σ try to determine its group of automorphisms, the group of those elementwise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way. After that you may start to investigate symmetric configurations of elements, i.e., configurations which are invariant under a certain subgroup of the group of all automorphisms [...] Weyl
The universe is an enormous direct product of representations of
symmetry groups. ~Weyl The
past 25 years have seen the rise of gauge theories — KaluzaKlein models
of higher dimensions, string theories, and now Mtheory, as physicists
grapple with the challenge of combining all the basic forces of nature
into one all embracing theory. This requires sophisticated mathematics
involving Lie groups, manifolds, differential operators, all of which
are part of Weyl's inheritance. There is no doubt that he would have
been an enthusiastic supporter and admirer of this fusion of
mathematics and physics. No other mathematician could claim to have
initiated more of the theories that are now being explored. His vision
has stood the test of time. Atiyah
1. Closure: If A and B are two elements in G, then the product AB is also in G. 2. Associativity: The defined multiplication is associative, i.e., for all A, B, C ∈ G, (AB)C = A(BC). 3. Identity: There is an identity element I (aka 1, E, or e) such that IA = AI = A for every element A G. 4. Inverse: There must be an inverse (aka reciprocal) of each element. Therefore, for each element A of G, the set contain an element B = A^{1} such that AA^{1} = A^{1}A= I. A group is a monoid each of whose elements is invertible.

