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Prologue.
...Physics is like a cake:
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- First floor: observations, experiments.
- Second floor: differential equations.
- Third floor: geometry - Fourth floor: group theory.
Groups govern geometry, which gives rise to beautiful differential equations.
With differential equations we build things, which are then used to explain or predict what we call physical facts.
...Historically, people began studying and codifying facts, observations, by performing measurements. Then they imagined conservation laws and "physical laws." At the beginning of the century, they started to think that physical laws might be related to geometry.
At the same time, Felix Klein asked: What is a geometry?
Note that he said "a geometry" and not "the geometry" (Erlangen program).
...Klein, Lie, Cartan, and others showed that something lay beneath the surface of geometric appearance. Geometry was not the final floor, the ultimate pinnacle of knowledge in physics. From a group structure, one can construct geometry.
In what follows, we will attempt to show the connection between groups, geometry, and physics.
Along the way, regarding groups—what is it?
...I would tend to say: logic. But logic is a room whose last occupant was Kurt Gödel, a dangerous pyromaniac. With his well-known theorem, he set fire to the furniture, which was completely destroyed. Since that tragedy, the room has been empty.
...That’s why I put a question mark there.
Groups.
...What is a group? In what follows, we limit our study to dynamic groups in physics: a set of square (n,n) matrices obeying defined axioms. These matrices g, elements of a group G, act on each other via standard matrix multiplication (row-column). Among these square matrices, we find the identity matrices.
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...A group obeys the axioms defined by the Norwegian mathematician Sophus Lie. These axioms apply to objects far more general than sets of matrices. But we will restrict our attention to this particular world and use matrix multiplication:
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1 - First axiom of group theory:
The product of two elements g₁ and g₂ of a group G:
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g₃ = g₁ × g₂
obeys:
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Let us give an example of a matrix group depending on a single parameter a. The element is:
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The product of two elements yields:
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or:
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g(a) × g(b) = g(g) = g(a + b) = g(g)
We can write the matrix product:
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which is similar to g₁ and g₂, i.e.:
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Counterexample: Consider the following set of matrices depending on a single parameter a:
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The product of two elements gives:
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which is fundamentally different from (5).
2 - Second axiom of group theory:
In the set of elements, we must find a special one, called the identity element e, which, when combined with any other element, satisfies:
(11) g × e = e × g = g
In groups whose elements are square matrices, this identity element e is always the identity matrix 1.
(12) g × 1 = 1 × g = g Note that we use upright type for scalars and bold type for other objects: square matrices, rows, or columns.
Recall the initial example of a group:
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Observe that:
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