Look up!
Field to Group

Main menu

vector field sphere

A field is simply a quantity defined at every point throughout some region of space and time.

't Hooft

When a beam of light falls on the human eye, certain sensations are produced, from which the possessor of that organ judges of the color and luminance of the light. Now, though everyone experiences these sensations and though they are the foundation of all the phenomena of sight, yet, on account of their absolute simplicity, they are incapable of analysis...


A speck in the visual field, though it need not be red must have some color; it is, so to speak, surrounded by color-space.


It is just like the mathematics of the addition of vectors, where (a, b, c) are the components of one vector, and (a', b', c') are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians. In fact, Schrödinger wrote a wonderful paper on color vision in which he developed this theory of vector analysis as applied to the mixing of colors.


There is nothing else except these fields: the whole of the material universe is built of them. 


This is the characteristic mathematical property of a classical field: it is an undefined something which exists throughout a volume of space and which is described by sets of numbers, each set denoting the field strength and direction at a single point in the space.



Main menu

EM 2-form

Neural form follows quantum function

Does neural form follow quantum function?


Main menu

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 electro-magnetism. 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 electro-magnetic 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.



Main menu

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 space-time points. We also find that the role the gauge potentials play in fiber-bundle space in gauge theory is exactly same as the role the affine connection plays in curved space-time in general relativity.



Main menu

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 element-wise 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 [...]

group theory

The universe is an enormous direct product
of representations of symmetry groups. ~Weyl


The past 25 years have seen the rise of gauge theoriesKaluza-Klein models of high dimensions, string theories, and now M-theory, 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.


A group
G is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. The operation with respect to which a group is defined is often called the "group operation," and a set is said to be a group "under" this operation. Elements A, B, C, ... with binary operation between A and B denoted AB form a group if

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 element of 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-1A= I.

A group is a monoid each of whose elements is invertible.