Introduction Flashcards
Introduction
What are symmetries and what can group theory do with them?
Invariance under suitable transformations.
Examples: bilateral (mirror) symmetry (human body), five-fold rotational symmetry (startfish), etc.
- symmetry trafos: can be composed
- groups: describing algebra of symmetries
So group theory can convert qualitative information into quantitative info.
Introduction
What’s a point group?
A group of matrices that describe crystalline structure characterized by its symmetries.
- helps determine the amount of independent elastic moduli in any dimension
Fundamental concepts
What’s a group? What is the order of a group?
A group is a set of G elements together with an associative and unital binary operation, such that an inverse and the identity exists.
- in summary: 1. associativity, 2. inverse, 3. identity
- the inverse is unique to each value
- order: the cardinality of its set of elements (number of elements)
Fundamental concepts
What’s a binary operation?
A rule that assigns to two elements of a set a well-defined third element of that same set.
- multiplicative infix notation: the notation used for the product of the binary operation
- they can be: associative (x(yz) = (xy)z), commutative (xy = yx), unital (existence of identity)
Fundamental concepts
When is a group Abelian?
When its product is commutative.
- completely different behaviour from non-Abelian groups
- subvariety of the algebraic variety of groups
Fundamental concepts
When are groups isomorphic?
When there exists a bijective map that preserves products.
- in other words: applying the mapping to the product is the same as applying the mapping separately and then making a product
- properties of isomorphism: reflexive, symmetric, transitive
- the orders of isomorphic groups are the same
Fundamental concepts
What’s the isomorphy principle?
Isomorphic groups cannot be distinguished from each other by algebraic means
Meaning they have the same algebraic structure.
Fundamental concepts
What’s an automorphism?
An isomorphism of a group with itself (self-isomorphism).
- automorphism group (symmetry group): the collection of all automorphisms of a group is itself a group
Fundamental concepts
What’s a subgroup?
A subset H of elements of a group G is a subgroup, denoted H < G, if
the inverse and the product of any of its elements also belongs to H.
- these conditions ensure that the identity element of G is contained in its subgroups
- the relation of being a subgroup is an ordering
- every subgroup is a group (this ordering is a binary operation that satisfies thes group axioms)
- every subgroup of a subgroup is itself a subgroup (the subgroups of a groups form a partially ordered set, not every subgroup can be composed)
Fundamental concepts
How can the concept of groups be generalized?
We can relax each of the conditions that are required for a set of elements to be group.
- monoids: relaxing the existence of inverses (pl.: renormalization)
- quasi-groups: relaxing the associativity of the product results (pl: combinatorial applications)
- groupoids: partially defined product (pl.: topology, description of quasi crystals)
Partially defined product: to some pairs of elements we associate a product which is a group elements, but not to every pair.
Examples of groups
What are the properties of additive groups?
They are the sets of all integer/rational/real/complex numbers with the binary operation of addition: (Z, +), (Q, +), (R, +), (C, +).
- binary operation of addition: associative, commutative (hence the groups are abelian), unital
- existance of additive inverse = negative
- integer multiples of n also forms a group (nZ, +)
Z is also an infinite cyclic group (the others are not?).
Examples of groups
What’s the modulus operation?
It creates the remainder of an integer upon division by another nonzero integer.
- the remainder upon division by b of a1 + a2 only depends on the remainders of a1 and a2 separately
Examples of groups
What are additive groups of mod n?
What is a residue class mod n?
Additive group of mod n: finite abelian group of order n with the operation of addition mod n.
- addition mod n: assigns to x,y from Z the remainder upon division by n of their sum
Residue class mod n: the collection of all integers that have remainder k upon division by n
Examples of groups
What’s a matrix and what are its properties?
It’s a rectangular array of numbers, with the matrix element A(ij) denoting the number found at the intersection of the ith row and jth
column.
- square matrix: n rows, n columns
- addition of matrices of the same shape element-wise
- matrix product that’s only defined is the # of columns of the first matrix equals the # of rows of the second matrix
- identity matrix: diagonal matrix elements equal to 1, others 0
- determinant rule: the determinant of a product is the product of the determinants
- invertibility: true for square matrices (there exists a second matrix for which AB = BA = 1 is true if its determinant had a multiplicative inverse)
Examples of groups
What’s the general linear group?
For any number ring R and positive integer n the collection of all invertible n-by-n matrices with entries from R with the operation of the matrix product.
- the group is infinite if R is
- the group is only abelian for n = 1
Examples of groups
What are important examples for subgroups of the GL(R) of order n?
- monomial matrices: each row and column contains exactly one nonzero entry
- permutation matrices: subgroup of the monomial matrices, the nonzero elements are all 1
- diagonal matrices: nonzero elements = diagonal elements, abelian subgroup
- orthogonal matrices: inverse = transpose (more generally: metric tensor of flat space)
- symplectic matrices: where J(n) is the block-diagonal matrix made up of n copies of the Pauli-matrix
- special linear and special orthogonal groups: subsets of all the above groups for which detA = 1
Every monomial matrix is the product of a diagonal and a permutation matrix.
Examples of groups
What are permutations? What’s the symmetric group?
Permutation: a bijective self-map of a set onto itself (‘reshuffing’)
- product: composition of corresponding maps
- inverse: inverse map
Symmetric group: the collection of all permutations of a (finite) set X forms a group Sym(X)
- not commutative if |X| > 2
- |Sym(X)| = |X|!*
- symmetric groups are isomorphic only if the orders of the original groups are equal, so it’s enought to consider only symmetric groups of order n
Examples of groups
What’s a transposition? What is the alternating group?
Transposition: interchange of two elements
- any permutation can be decomposed (in many ways) into a product of transpositions
- odd and even permutations can be distinguished
Alternating group: the group of even permutations, Alt(x)
Examples of groups
What is a cycle? What is its connection to permutations?
A cycle (orbit) of a permutation is a set of points that are taken into each other by successive applications of the permutation (a fixed point
is a cycle of length 1).
- the cycles of a permutation π ∈ Sym(X) partition the set X
- a permutation is called cyclic if it has only one cycle of length greater than one, and the length of this cycle is its order
- every permutation can be decomposed into a product of cyclic ones
- cycle notation… exists
Examples of groups
How to describe the symmetry of a geometric figure?
The symmetry of a geometric figure is rigid motion mapping the figure (as a
set of points) onto itself.
- rigid motion: mapping of Euclidean space onto itself that preserves the distance of points
- types of rigid motion: translations, rotations, reflections and different composites of these
Examples of groups
What are regular polygons? What’s the dihedral group?
Regular polygon: convex plane figure all of whose sides are congruent and angles between neighboring sides are equal
- for each integer n > 2 there is exactly one regular n-gon
Dihedral group of degree n: the symmetries of a regular n-gon, composed of rotations around the center (by multiples of 2π/n) and reflections across lines passing through the center and some vertex
- |D(n)| = 2n*, since there are n different refelction axes and n different rotations
Examples of groups
How to make a Cayley table?
It describes the group structure for finite groups. We put all the elements of the group into a column and also a row and make a table with their products.
Examples of groups
What’s the connection between symmetries of polygons and permutations?
Since symmetries map the polygon onto itself, any set of distinguished subfigures (like vertices, edges, medians, etc.) is also mapped onto itself by a symmetry transformation. As a consequence, each symmetry transformation induces a permutation of any chosen set of distinguished subfigures.
Examples of groups
What are Platonic solids? What are their symmetry groups?
Convex spatial figures all of whose bounding facets are congruent regular polygons.
There are five types: tetrahedron, octahedron, icosahedron, cube, dodecahedron.
Their symmetry groups:
- tetrahedral group: tetrahedron (isom.: A4)
- octahedral group: octahedron, cube (isom.: S4)
- icosahedral group: icosahedron, dodecahedron (isom.: A5)
Examples of groups
What’s a point group for molecules? What point groups are there in 3D?
A finite group of geometric transformations under which charge density in molecules is invariant. (This leads to restrictions on various aspects of molecules.)
3D point groups:
- polyhedral groups: chiral tetrahedral, tetrahedral, pyritohedral, chiral octahedral, octahedral, chiral icosahedral, icosahedral
- infinite families of axial groups: cyclic, pyramidal, Z(n) x Z(2), dehidral, anti-prismatic, prismatic, S(2n) ~ Z(2n)
This is a complete classification of molecular symmetry which gives a lot of info without having to solve the Schrödinger equation.
Examples of groups
How does anisotropy affect crystals? From where does it originate?
Some homogeneous substances exhibit anisotropic (direction dependent) behavior on a macroscopic scale which comes from ordered microscopic inhomogeneity/anisotropy.
- examples: ferro- and ferrimagnetic materials
- macroscopic order can come from discrete translational symmetries
Examples of groups
What’s a crystalline substance? What groups describe its symmetries?
In crystalline substances microscopic components (atoms/molecules/ions) are distributed periodically in space and are localized (in the absence of defects) around the lattice points of a 3D periodic lattice (the crystal lattice).
- quasi-crystal: aperiodic structure exhibiting long range order
The groups describing crytalline symmetries:
- space group: full symmetry group of the crystal, taking into account the symmetries of its microscopic constituents (translations, rotations, reflections, inversions and combinations of these)
- translation group: group of translations taking the lattice point of the crystal into itself
- point group: a finite group describing the rotational and reflection symmetries of the crystal
Examples of groups
How are crystals classified? What’s a consequence?
Crystal structures are grouped into crystal classes, families and systems according to their point groups and translation subgroups.
- crystallographic restriction: the number of integers coprime to the order of any element of the point group cannot exceed the dimension of space
- consequence: there are only finitely many different crystal structures in
any space dimension
Examples of groups
What’s an inertial frame? What is it useful for?
It’s a type of reference frame (system of bodies with known relative motions) in which the inertial motion of isolated bodies is uniform translation.
- these frames move at constant speed relative to each other
Examples of groups
What are the universal symmetries of natural laws?
Galileo’s relativity principle: not only the law of inertial motion, but all mechanical laws look the same in every inertial frame.
In inertial frames
- both space and time are homogeneous,
- space is isotropic,
- any reference frame obtained via a boost from an inertial one is itself inertial.
boost: constant speed uniform translation
Examples of groups
What is the Galilei group and what are its elements? What are the corresponding conserved quantities?
The group formed by the symmetries of classical mechanics.
Its constituents: 10 continuous parameters + some discrete ones
- space translations (3) – (linear) momentum
- time translations (1) – energy
- spatial rotations (3) – angular momentum
- (Galilean) boosts (3) – center of mass
- discrete reflection symmetries
Though it turns out that this is not the symmetry of nature.
Examples of groups
What’s Noether’s theorem?
To each one-parameter group of continuous
symmetries of a physical system corresponds a conserved quantity.
- Galilean symmetries correspond to universal first integrals
- the symmetries of the Poincaré group correspond to the relativistic first integrals
Examples of groups
What’s the Poincaré group? Elements and corresponding conserved quantities?
The isometry group of 4D Minkowski space (not completely Euclidean) which is also the symmetry group of Maxwell’s equations.
- the Galilei group can be obtained from this one by making the c —» ∞ limit
- this is the actual symmetry group of physical laws
Elements:
- space-time translations (4) – four-momentum
- space-time rotations including the spatial rotations and Lorentz boosts (6) – angular momentum
In general relatvity the flat Minkowski space-time is instead a curved manifold and the Poincaré group is replaced by the de Sitter group.
