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By Jakob Schwichtenberg
This is a textbook that derives the basic theories of physics from symmetry.
It starts off by means of introducing, in a totally self-contained means, all mathematical instruments had to use symmetry rules in physics. Thereafter, those instruments are positioned into motion and by utilizing symmetry constraints, the basic equations of Quantum Mechanics, Quantum box idea, Electromagnetism, and Classical Mechanics are derived.
As a outcome, the reader is ready to comprehend the elemental assumptions in the back of, and the connections among the fashionable theories of physics. The e-book concludes with first purposes of the formerly derived equations.
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Extra resources for Physics from Symmetry
Example text
A special type of Group Theory is Lie Theory, which deals with continuous symmetries, as we encounter them often in nature. Symmetry is defined as invariance under a set of of transformations and therefore, one defines a group as a collection of transformations. Let us get started with two easy examples to get a feel for what we want to do: 1. A square is mathematically a set of points (for example, the four corner points are part of this set) and a symmetry of the square is a transformation that maps this set of points into itself.
In the language of linear algebra this means that the determinant of our matrices must be +1. 5. 1 Rotations with Unit Complex Numbers There is a quite different way to describe rotations in two dimensions that makes use of complex numbers: Rotations about the origin by angle θ can be described by multiplication with a unit complex number ( z = a + ib which fulfils the condition15 |z|2 = z z = 1). The unit complex numbers form a group, called16 U (1) under ordinary complex number multiplication, as you can check by looking at the group axioms.
In this description, the rotation operator ei90 acts on complex numbers instead of on vectors. To describe a rotation in two dimensions, one parameter is necessary: the angle of rotation φ. A complex number has two degrees of freedom and with the constraint to unit complex numbers |z| = 1, one degree of freedom is left as needed. We can make the connection to the previous description by representing complex numbers by real 2 × 2 matrices. We define 1= 1 0 0 1 , i= 0 1 −1 . 15) Fig. 5: The unit complex numbers lie on the unit circle in the complex plane.