San José State University |
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Symmetries and Algebraic Structures |
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The theory of mathematical groups has proved useful in several fields of quantum theory, particularly in particle physics. This material makes the point that the link between quantum mechanics and algebraic structures such as groups is through the matter of symmetries.
Consider an object that is square in shape and let t denote turning it 90 degrees. Furthermore assume there is an arrow on the object. When the object is turned 90, 180, 270 or 360 degrees it matches the original object but the arrow is turned. Turning the object 180 degrees can be represented as t². Turning it 270 degrees is then t³. Turning it four times is the same as not turning it at all. Let not turning it at all be denoted as e.
So there is an algebraic structure having four elements; i.e., e, t, t² and t³. The successive combinations of these four elements can be represented as a multiplication table.
e | t | t² | t³ |
t | t² | t³ | e |
t² | t³ | e | t |
t³ | e | t | t² |
This is an operation, a binary function, which has an identity element, namely e. Furthermore for each element a there is a b such that ab=e; i.e., each element has an inverse. Thus this set of elements and the above operation constitutes a mathematical group. It is also such that for any two elements ab=ba. It is therefore called an abelian group.
The above group is derived from a square object, an object with fourfold symmetry. An object with n-fold symmetry likewise generates an abelian group.
Now consider a cuble situated at the center of a set of xyz axes. As in the above assume there is an arrow in the cube. Let t_{x}, t_{y} and t_{z} represent 90 degree turns about the x. y and z axes respectively.
(To be continued.)
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