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Weyl gauge theory

Hermann Klaus Hugo Weyl, ForMemRS (German: [vaɪl]; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.
His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him.

Newton and the Theory of Gravity

Newton’s three basic laws of motion outlined in Principia helped him arrive at his theory of gravity. Newton’s law of universal gravitation states that two objects attract each other with a force of gravitational attraction that’s proportional to their masses and inversely proportional to the square of the distance between their centers. These laws helped explain not only elliptical planetary orbits but nearly every other motion in the universe: how the planets are kept in orbit by the pull of the sun’s gravity; how the moon revolves around Earth and the moons of Jupiter revolve around it; and how comets revolve in elliptical orbits around the sun. They also allowed him to calculate the mass of each planet, calculate the flattening of the Earth at the poles and the bulge at the equator, and how the gravitational pull of the sun and moon create the Earth’s tides. In Newton's account, gravity kept the universe balanced, made it work, and brought heaven and Earth together in one great equation.
In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this framework, the Hamiltonian formalism in mechanics is generalized to field theory in the way that a field is represented as a system that varies both in space and in time. This generalization is different from the canonical Hamiltonian formalism in field theory which treats space and time variables differently and describes classical fields as infinite-dimensional systems evolving in time.

Weyl's equation

Hermann Weyl developed a relativistic wave equation for massless particles of spin 1/2. These particles are called Weyl fermions.
weyl equation
In wikipedia this equation can be found instead of the above equation :

weyl equation
From this equation we can define Weyl spinor as a two component wave function. Here Weyl spinor is a solution to Weyl equation. But it have a different meaning in the abstract algebra of spinors which is a single point is spacetime. Spinors are geometric objects of study in abstract algebra.
The plane wave solution of Weyl equation is the spinor in the context of relatvistic wave mechanics.

Weyl spinor
In the diagram below a table of four fundamental forces is given :
grand unification
The spinor of Weyl equation bring us to the twistor theory which Roger Penrose devloped.

Twistor theory

Twistor theory is the study of twistor spaces. There are two types of twistor spaces: one is non-projective twistor space and other is projective twistor space. Non-projective twistor space T is the four-dimensional complex vector space with the coordinates denoted by

projective space
Where w[A] and π[A`] are the two constant Weyl spinors.
There is a specific transformation rule which takes a point in minkowski space to a point in twistor space.
Projective twistor space PT is a complex three-dimensional manifold - complex three-space CP3. Here a point x < M (complex non-projective space) determines a line in PT. It is the duality principle of projective geometry.

Weyl's gauge theory continued

The theory to be considered in this chapter is , from a geometrical point of view , a ntural generalization of artbitrariness of coordinates of Einstein's general theory of relativity; from a physical point of view , it fits electromagnetism into the deductive system, which Einstein's theory does not. The theory is due to Hermann Weyl and will be found in space, time and matter(1922).
The puzzles about measurement considered at another chapter naturally suggest the point of view from which Weyl starts. As he says : " The same certainty that charactrizes the relativity of motion accompanies the principle of relativity of magnitude:" Measurement is a comparism of lengths , and weyl suggests that , when lengths in different places are to be compared , the results may depend upon the route pursued in passing from the one place to the other. Lengths at the same place (i.e having one end identical) , if small , he regards as directly comparable ; also he assumes continuity in the changes accompanying transportation. This is not the sum-total of his assumptions nor the most general way of stating them; but before we can state them adequately certain explanation are necessary.
Reduced to its simplest terms, the conception used by Weyl may be expressed as follow. Given a vector at a point , what are to mean by the statement that a vector at another point is equal to it? There must be some element of convention in out definition; let us therefore , as our first step, set up a unit of length in each place, and see what limitation it is desirable to impose on our initial arbitrariness.
There is, to begin with , an assumption which is made almost tacitly,and that is, that we can recognize something in one place as the same vector as something at another place. We may perhaps take this sameness as being merely analytical: the two are the same function of co-ordinates at their repective places. I do not think this is all that is meant, since a vector is supposed to have some physical significance; but if more is meant, it is not clear how it is to be defined. We will therefore assume that given a function of the co-ordinates which is a vector , we shall regard the same function of other values of the co-ordinates as the same vector at another place.
We next have to define "parallel displacement" This may be defined in various ways. Perhaps the most graphic description is to say it is displacement along a geodesic. Another definition is that it is a displacement such that the "co-variant derivative" vanishes, the covariant derivative of a vector A(u) with respect to v being defined as A(uv) where :
A(uv) = dA(u)/dx(v) + ∑{uv, α}A(α) [α = 1,2,3,4]
For the definition of {uv,α} , it is known as the Christoffel symbols. In the tensor calculus , covariant differentiation takes the place of ordinary differentiation for many purposes, since the covariant derivative of a tensor is a tensor, wheras the ordinary derivative is in general not a tensor. We assume that our units of length in different places are so choosen that , when a small displacement is moved to a neighbouring place by parallel displacement , the change in the measure of its length is small and is propotional to its length. We assume , in short, that the ratio of the increase of length to the initial length for a change of coordinates( dx1, dx2, dx3, dx4) is :
k1dx1 + k2dx2 + k3dx3 + k4dx4
So that (k1, k2,k3, k4) form a vector k(u). Now it is possible to express Maxwell's equations in terms of a vector which may be identified with the above vector k(u). Hence it is possible to regrad electromagnetic phenomena as explained by the variation of what is taken as the unit of length as we pass from point to point. I shall not attempt to explain the theory , as it in any case be necessary to read a full account in order to grasp its significance.
Here perhaps even more than elsewhere in relativity theory, it is diifficult to disentangle the conventional elements from those having physical significance. On the face of it, it might seem as though we ere attempting to account for actual physical phenomena by means of a mere convention as to choice of units. But this, of course, is not what is meant. The way the unit is assigned in different places is called by Eddington the "gauge-system": this is only partially arbitrary and is in part the representation of the physical state of the world. This has to do with the fact that vectors are not purely analytical expressions, but also corresponds to physical facts. It would seem, however, that the theory has not yet been expressed with the logical purity that is to be desired, chiefly because it is not prefaced by any clear account of what is to be understood by "measurement" - or, what comes to much the same thing from the standpoint of theory, what we are to mean when we talk of "moving" a vector whether by parallel displacement or in any other way. To "move" something , we must be able to recognize some identity between things in different places. Perhaps all this is quite clear in the minds of competent exponents of the theory, but if so they have not succeded in conveying their thoughts without loss of clarity to readers who have not their back-ground. When Eddington says : "Take a displacement at P and transfer it by parallel displacement to an infinitely near point P". I fing myselfwondering how exactly, the displacement is to preserve its indentity throughout the transfer and the only answer suggested by the accompanying formulae is that the identity is that of an algebraic expression in terms of the co-ordinates. This , however, is clearly insufficient.
Professor Eddington , after expounding Weyl's theory, proceeds to generalize it and some of his accompanying elucidations are relevant to our present difficulties. Thus he says :
" In Weyl's theory , a gauge-system is partly physical and partly conventional; lengths in different directions but at the same point are supposed to compared by experimental (optical) methods; but lengths at different points are not supposed to be comparable by physical methods ( transfer of rods and clocks) and the unit of length at each point is laid down by convention. I think this hybrid definition of length is undesirable, and that length should be treated as a purely conventional or else a purely physical conception."
He proceeds to a generalised theory in which , at first, length is purely conventional, for comparisons at a point as well as for comparisons between different points. This generalized theory does not seem to involve the same kind of difficulties as those which have been troubling us. the following passage, for example , states the matter with great clearness:
" The relations of displacement between point events and the relation of "equivalence" between displacements form part of one idea, which are only separated for convenience of mathematical manipulation. That the relation of displacement between A and B amounts to such-and-such a quantity conveys no absolute meaning; but that the relation of displacement between A and B is equivalent to the relation of dispacement between C and D is ( or at any rate may be) an absolute assertion. Thus four points is minimum number for which an assertion of absolute structural relation can be made. The ultimate elements of structure are thus four-point elements. By adopting the condition of affine geometry, (geometry where the origin is removed ) , I have limited the possible assetion with regard to a four-point element to the statement that the four points do, or do not, form a parallelogram. The defence of affine geometry thus rests on the not unplausible view that four-point elements are recognized to be differentiated from one another by a single character-viz, that they are or are not of a particular kind which is conventionally named parallelogramical. Then the analysis if the parallelogram property into a double equivalence of AB to CD and AC to BD, is merely a definition of what is meant by the equivalence of displacements."
Here we have a logically satisfactory theoretical basis for a metric. We may suppose that , as a matter of fact, there are important properties of groups of four points which are "parallelogramical" and that the actual physical measurement is an approximate method of discovering which groups have this property. We shall find certain laws approximately fulfilled by rough-and-ready measurements, and fulfilled with increasing accuracy as we introduce refinements into the process of measurement. Consider, for example, Euclid's first axiom, Things which are equal to the same thing are equal to one another. Presumably Euclid regarded this as a logically necessary proposition, and so do people who are engaged in the practice of measurement. If two lengths each equal to a metre are found to be not equal to each other, the plain man assumes that there must be a mistake somewhere. We are therefore continually refining the actual operations of measurement with a view to verifying Euclid's first axiom as nearly as possible. But wih the above qouted definiton of equality of length the first axiom becomes a substantial proposition, namely: if ABCD is a parallelogram and likewise DCEF, then ABEF is a parallelogram. If this proposition is true, then it is theoretically possible to define measurement in such a way that two lengths each equal to a metre shall always be equal to each other. What is called "accuracy" is , speaking generally , an attempt to obtain a result comfortable with some ideal standard supposed to be logical but in fact physical. What do you mean by saying that a length has been "wrongly" measured? whatever result we obtain from measuring a given length , the result represents a fact in the world. But in what we call a wrong measurement , the fact assertained is complex and of small universality. If the observer has simply misread a scale , the fact ascertained involves reference to his psychology. If he has neglected a physical correction-e.g for the temperature of his measure- the fact refers only to a measurement carried out with that particular apparatus on that particular occasion. In relativity theory we have another set of what might be called "inaccurate" measurement-e.g measurement of mass of α-particle or β-particle emitted from radio-active bodies must be corrected for their motion relative to the observer before they acquire any general significance. It is always the search for simple relations which enter into general laws that govern successive refinements. But the existence of such relations ( where they do exist) is an empirical fact., so that much that seems prima facie to be logically necessary is really contingent. On the other hand, the number of premises in a deductive system which has to agree with an emirical science can, by logical skill, be diminished to an extent which may be astonishing. Of, this theory of general relativity is a remarkable example. The theory is a combination of two diverse elements: on the one hand, new experimental data; on the other, a new logical method. It must be regarded as a happy accident that the two appeared together; if the right kind of theoretical genius had not happened to be forthcoming, we might have had to be content for a long time with patched-up hypotheses such as the FitzGerald contraction. As it is, the combination of experiment and theory has produced one of the supreme triumphs of human genius.

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Herman weyl first combined electromagnetism and General theory of relativity. In the context of General theory of relativity a vector when transported around a closed loop does not coincide with itself. The vector's direction is changed. This apparent change of vector's direction define Riemann curvature tensor. What if both magnitude and direction are changed when vectors are thus transported? Weyl incorporated this idea and thought of a more generalized theory of gravity. He thought of units of length as railway gauge which can be different in different place.
Suppose a vector is defined as 4-tuples x1, x2, x3, x4. The length of the vector is changed in proportion with the incremental distance dx so that the ration between the length of the new vector and original are k1(dx1), k2(dx2), k3(dx3) and k4(dx4). These four numbers k1, k2, k3 and k4 can be interpreted as the electromagnetic field in Maxwell's theory. Thus Weyl was able to explain electromagnetic by generalizing general theory of relativity. Mathematically it can be interpreted as a conformal change of metric tensor.

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Gravitation potential

In Newtonian potential gravitational potential denotes the amount of work required to bring a mass m from inifinity to a point at at distant R from another massive object. Thus the potential difference between two points which are at a distance a and b respective is :
gravitational potential

Reference materials:

Classical electromagnetic theory
Analysis of matter by Russell
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf

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