- Group Theory in a Nutshell for Physicists
- Wu-Ki Tung, Group theory in physics
- Why is group theory important?
- Symmetry and group theory throughout physics
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We have therefore identified each point of M with a coset of G. We partition G into cosets by selecting vertex 1 though 2 or 3 would have served equally well. This identification of elements of M with cosets of G provides a useful way of constructing a set M on which G acts transitively. Suppose that H is a subgroup of G. This construction yields the same result no matter which element of the coset we choose. As an example of the construction, we construct a two-element set M on which the group S3 acts transitively. The elements of H carry each point of M into itself; the other three elements of S3 carry [eJ into [ 12 ] and [ 12 J into [el 1.
When G acts on itself by left multiplication, so that a transforms b into ab, then the action is always transitive. There is another natural way for a group to act on itself which leads to less trivial orbits and isotropy subgroups. We say that G acts on itself by conjugation if a acting on b is aba - 1. The orbits of a group under conjugation are called the conjugacy classes of the group. From this equation follow two straightforward consequences: 1 The identity is always a one-element conjugacy class.
It follows that the number of elements in any conjugacy class is always a divisor of the number of elements of the group.
Group Theory in a Nutshell for Physicists
For any group of matrices, conjugate group elements Band C satisfy It follows immediately that matrices Band C have the same eigenvalues, and hence the same determinant and trace. As an example, we list the conjugacy classes of the group S3' One class consists of the identity element e alone. Both these elements represent rotations of an equilateral triangle. The third conjugacy class consists of the remaining three elements, 12 , 13 and 23 , which represent reflections of an equilateral triangle in the perpendicular bisector of a side.
Ordinary table salt, NaCl, if allowed to crystallize under carefully controlled conditions, forms cubic crystals.
The group Oh thus contains 48 elements. NaCl can, however, crystallize in other forms. If a small amount of urea is added, then small equilateral triangles replace the corners of the cube. These triangles are congruent, so the crystal is still invariant under Oh' As more urea is added; the triangles open up into hexagons. Finally, the cube faces disappear altogether and we have an octahedron.
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See Figs. The group preserving the cube is thus the same as the group of the octahedron which is the reason for the notation 0h ' The crystal whose appearance is as in the figure is said to show faces of two different forms; the eight triangular faces constitute one form whereas the six octagonal faces modifications of the original cube faces constitute the other form.
Wu-Ki Tung, Group theory in physics
Crystals that are grown under careful conditions will develop into cubes or octahedra or other regular shapes. But if you go and pick up a crystal in the field, it will most likely have a rather irregular shape. Nevertheless, it does exhibit a symmetry, but in a more subtle sense. The first major discovery, by Nicolas Steno and Christian Huyghens in the field of crystallography is the celebrated 'law of corresponding.
Why is group theory important?
Put another way, suppose that we draw the normals to each of the faces, so as to get a set of points, on the unit sphere. Then up to a rotation of the sphere this set of points is the same for crystals of the same substance, although some of the points may be missing in a given crystal i.
For example, suppose a substance crystallizes on occasion as a cube. Then the set of normals of any crystal of this substance will be invariant under 0h' Steno's 'law' appears in a caption to a figure in the appendix to his book entitled De Solido intra Solidum N aturaliter Contento Florence, The concept of an orbit of a group action allows us to explain in more detail what the crystallographers mean by the concept of the 'form' of a crystal.
As we have already indicated, the observed symmetry of a crystal is best expressed in terms of directions. The law of the constancy of corresponding angles says that it is the collection of normal directions to the faces that is invariant up to an overall rotation. These directions are the same for all crystals, of the same-substance, even though the outward appearance of the crystal might be quite irregular. We can consider these directions as points on the unit sphere.
The group of symmetries of the crystal acts on this set of points. An orbit of.
Symmetry and group theory throughout physics
It is clear that if we pick a generic point x, y, z , with no two coordinates equal in absolute value and no coordinate zero, then the isotropy group G x,y,z consists only of the identity, and thus the orbit of this point contains the full 48 elements. The orbit through x, x, z thus contains 24 elements. Similarly, the orbit through x,x,O contains 12 elements. The orbit through x,x,x contains eight elements and the orbit through 1,0,0 six elements. This last orbit corresponds to the set of faces of a cube - it is called the cubic form of the crystaL The eightelement orbit corresponds to the set of faces of the octahedron - it is called the octahedral form.
The crystallographers use the notation x for - x. The octahedral orbit is the orbit of the point 1,1,1. We can always remember to normalize if we want the points tolie on the unit sphere. It is convenient to have a way of representing points of the sphere on the plane of the paper. A standard convention is to use stereographic projections. Ifwe project from the south pole, as shown in Fig.
We use a solid dot in the interior or boundary of the circle to represent a point coming from the upper hemisphere or equator. Thus, Fig. For points on the lower hemisphere we project from the north pole, and represent the corresponding points which are now also inside the circle by open dots. If points from the upper and lower hemisphere project onto the same point, we use a circle with a center dot.
We can now enumerate the different forms of the group Oh' corresponding to the different possible orbits. For each form it is worthwhile to try to visualize the isotropy group of a point in the orbit see Fig. There are, of course, many different forms corresponding to x, x, z , to x, y, 0 and x, y, z. Representative crystals of the first case have triangular faces, while in the second case the faces are quadrilaterals Figs. These are the 'pure forms' of a crystal of cubic symmetry. The substance may also crystallize in a combination offorms. That is, the set of normals to the faces may constitute more than one orbit.
objectifcoaching.com/components/lewis/quand-rimbaud-rencontre-verlaine.php The relative preponderance of the constituent forms is called the habit of the crystal. If we examine the forms that actually arise for a substance possessing cubic symmetry, we discover a remarkable fact: although, in principle, the values of the unrestricted variables x, y or z in forms of the last four types can be arbitrary, the observed values are, always small integers.
What is observed is that these values are always rational numbers, so, multiplying by a suitable factor, we can arrange that Applications to crystallography - a Fig. This is a special case of the celebrated law of rational indices, discovered by Haiiy in We shall discuss this law later on. We can now pose the following problem: what are the possible observed symmetry groups of crystals, and for each group what are the possible forms? Since a crystal has a finite number offaces, we can clearly tackle the first part of our problem by looking for all possible finite subgroups of the group of Euclidean motions in [R3.
We shall turn to this problem in Section 1. II 'I I,! The topology of SU 2 and SO 3 21 1.
Let m be a point of M and let a be an element of G. We claim that 6. The angles 1. Observe that once we know that any R has such a decomposition, then, in fact, we know that R can be represented by a matrix in SU 2. Indeed; we have already constructed, in Section 1. To prove Euler's theorem, let n be the north pole of the unit sphere Fig. The rotation R is completely determined by a knowledge of the image Rn of n, and of the image of any unit tangent vector to the sphere passing through n.
Thus, D is a rotation about the z axis, i. From this fact, we shall deduce an interesting topological property of the rotation group 80 3. In other words, relative to this parametrization, the group SU 2 looks like a threedimensional unit sphere in the four-dimensional y space.
The identity element e corresponds to the point 1,0,0,0 and - e corresponds to the antipodal point -1,0,0,0. Now the sphere in any positive dimension has the property that any closed curve can be shrunk to a point. By moving y a little we may assume that l' does not pass through the point - 1,0,0,0 for any value of t.