Group Theory and Its Application to Particle Physics

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Abelian Groups

A group is defined as a collection of elements (a set), which are labelled a, b, c, ... and so on, and which are related to one another by the following rules: If a and b are both members of the group G, then their product, c = ab is also a member of the group G. The process is associative, i.e., a(bc) = (ab)c. There must be an element, called the unit element and usually denoted by I, defined so that aI = a, bI = b, and so on for all elements in the group. Each element has an inverse, written as a-1, b-1 and so on, defined so that aa-1 = I and so on.

Figure 01a Parity Operation [view large image]

A group for which ab = ba is an Abelian group. The set of ordinary integer numbers ( ... -3, -2, -1, 0, 1, 2, 3, ...) under "addition" is a simple example of an Abelian group, where "0" is the unit element and the inverse is the same number with opposite sign.

The parity operation (transforming (x,y,z) to (-x,-y,-z)) together with an identity operation (doing nothing) form a very simple discrete group (Figure 01a) that can be used to introduce the concept of representation in group theory.


• Multiplication Table - It is a table to specify the result of multiplying two operations as shown in Figure 01a, where I is the identity operation and P denotes the parity operator. Note that P is its own inverse, i.e., applying P twice is identical to no operation. Thus from the relation P² = 1, P can assume the value of either +1 or -1.



• The number of elements in a finite group G is called the order of G. It is a finite group of order 2 in the example. Note that the elements in this group are not objects, they are instead the actions (operations).



• Representation - It is a mapping of the operators G(g₁, g₂, ...), where the g_i's are the elements in the group G, onto a set of linear operators D(g₁, g₂, ...) with the properties: D(I) = 1 (for the identity operation), and D(g₁)D(g₂) = D(g₁g₂). There are two irreduciable representations for D(P) -- the trivial one D(I) = 1, and the other one D(P) = -1.



• State Vector (Basis) - This is the representation by objects (state vectors) that experience the operations. The state vectors in this example are |I = |g₁ = 1 corresponding to the identity operation and |P = |g₂ corresponding to either the symmetric (no change) or anti-symmetric (change by a minu sign only) state vector after the parity operation. State vectors satisfy the orthonormal conditions such that g_i|g_j = _ij, where _ij = 1 for i = j and _ij = 0 for i j. Note that state vector is different from the spatial point E(x,y,z) acted upon by the partiy operation as shown in Figure 01a, which illustrates the geometrical configuration involved in the operation. Whereas, the state vector is an entity in a mathematical formulation such as the equation in quantum theory.
  
  
• Matrix Representation - This representation is in the form of matrices. They have the special property that both commuting (Abelian) and non-commuting (non-Abelian) representations can be constructed out of them. The matrix elements can be assembled according to the formula: [D(g)]_ij = g_i|D(g)|g_j and the definition D(g₁) |g₂ = |g₁g₂. Thus, the matrix representation in our example is in the form:
  .
  The dimension of a representation is the dimension of the space on which it acts - the above example is 2 dimensional.



• Matrix Operation - The table below lists the changes in state vector and in geometric configuration by various matrix operations. The original state vector is in the form : , while the original configuration is represented by 2 "+" signs (red) along the x-axis and 2 "*" symbols (blue) along the y-axis.
  
                


Note that the combination of some matrix operations produces another operation, for examples, (Rotation 90^o)(Parity) = (Reflection), (Rotation 90^o)(Rotation 90^o) = (Rotation 180^o), and differential scaling is also equivalent to shearing.


• Eigenvalues and Eigenvectors (eigen means special or unique in German) - This is about a special operation on a state vector that not only does not change it but also produces a c-number known as eigenvalue while such state vector is called eigenvector. An example involving D(P) is shown below with the eigenvalue p, and the matrix elements v₁, and v₂ for the eigenvector :
  ,
  which is equivalent to
  p v₁ - v₂ = 0
  v₁ - p v₂ = 0
  The condition for a solution is the vanishing of the determinant of the coefficient of the v's, and this reduces to: p = 1. For p = +1, the normalized eigenvector is |P = which is symmetric; while for p = -1, the normalized eigenvector is |-P = which is anti-symmetric.
  Eigenvalue is very important because it is a measurable quantity in quantum theory. It is the average value of the operator as shown by the example: P|D(P)|P = p, where for example -P| = (1/) ( 1 -1 ) so that -P|D(P)|-P = -1.



• Application to Particle Physics - If the wave function or field (x) and (-x) are the solutions of the equation with the same eigenvalue, then they must be related to each other as: (x) = p(-x) (assuming p to be an unknown now). If we let x -x, then (-x) = p(x), which implies p² = 1 or p = 1 corresponding to even (+) or odd (-) parity.
  
  
  In particle physics, parity is a property of particles called the intrinsic parity.
  
  • For the ferminons with spin-1/2, the particles (electron, neutrino, and quark) have positive parity, while antiparticles (positron, anti-neutrino, and anti-quark) have negative parity.
  • Bosons have same intrinsic parity for both particles and antiparticles. Spin-0 particle with negative parity is called a pseudoscalar such as the and K mesons. Spin-0 with positive parity such as the Higgs boson is called a scalar. A vector boson (photon) has spin-1 and negative parity, while a pseudovector boson has positive parity.
  • Parity is a multiplicative quantum number such that p_total = p_a x p_b x..., which is conserved before and after an electromagnetic or strong interaction; but not in the weak interaction. For example, in the weak interaction process of beta decay: n p + e^- + , p(before) = +1 which is not equal to p(after) = (+1)(+1)(-1) = -1.


• Simultaneous Eigenvectors of Commuting Operators - If two operators commute, i.e., AB - BA = [A, B] = 0 then
  (AB)|a - (BA)|a = A(B|a) - a(B|a) = 0,
  which means that B|a must be a linear combination of the eigenvectors of A, belonging to the same eigenvalue a, i.e., B|a = _kb_k|a, k. In particular, when the irreducible representaion (with the eigenvalue a) appears only once (meaning non-degenerate), i.e., k = 1, then the eigenvector for A is also the eigenvector for B (with its own eigenvalue b). In the parity example, since it is an Abelian group with all elements commuting with each other, the eigenvector |P or |-P for D(P) is also an eigenvector for D(I).
  As a corrollary, if the two operators do not commute, then there is no simultaneous eigenvectors and their eigenvalues cannot be measured simultaneously. This is the case for the momentum operator p and position operator q in quantum theory. Mathematically, the expression for the non-commuting relationship is [p, q] = i, where ~ 10^-27 erg-sec is the Planck's constant. This is the uncertainty principle at the very foundation of quantum theory.



• Inverse of a Matrix - The inverse D^-1 for a matrix D is defined such that DD^-1 = I, where I is the identity matrix. If the elements of D is denoted by a_ij then the elements b_ij for the inverse can be constructed by the formula:
  b_ij = (-1)^i+j[a]_ij / [a],
  where [a] = a_1ia_2ja_3ka_nl - a_2ia_1ja_3ka_nl + is the determinant of the matrix, the plus or minus sign being taken according as the number of permutation of the integers 1, 2, 3, ... is even or odd, and [a]_ij is the minor, which is a determinant of order n - 1 after deleting the ith row and jth column of the determinant. Following this prescription, the inverse D^-1 for the matrix D(P) is just the matrix D itself, i.e., D^-1 = D.



• Hermitean Conjugate Matrices - The Hermitean conjugate matrix is obtained by interchanging rows with columns, and taking the complex conjugate of every element, i.e., (D)_ij = (D^*)_ji. Thus, for the matrix D(P) in the above example, D = D. Matrix satisfies this special relation is called Hermitean matrix, the average value of which is a real number, e.g., -P|D(P)|-P = -1. When D = D^-1 as in the same example, the matrix is called unitary. It has the special property that the length of the state vector is preserved after the operation, e.g., D(P)|-P = -|-P with -P|-P = 1 unchanged. (See further explanation in QM mathematics).



• Similarity Transformation - Group representations are powerful tools because they operate in linear spaces. In such spaces, we are free to choose to represent the states in another way by making a linear transformation. As long as the transformation is invertible, the new states are just as good as the old. The new and old representations are related by: D'(g) = SD(g)S^-1. D' and D are said to be equivalent representations because they differ just by a trivial choice of basis: |g' = S|g. Mathematically, the similarity transformation can be expressed as: SD|g = SDS^-1S|g = D'|g'.



• Reducible Representation - A representation is completely reducible if it can be written into the following form:
  
  where the D_j cannot be contracted into smaller dimension anymore. In the D_j subspace (the irreducible representation), the action of D on any vector in the subspace is still in the subspace. In the parity example, the matrix representation can be transformed into the reducible form:
  
  where D'(p) has been decomposed into two irreducible representations as mentioned earlier. The symmetric and anti-symmetric state vectors are transformed to respectively. Note that each subspace contains its own state vector (an one dimensional vector in this simple example); there is no mixing or crossover.



• Tensor Products - It has just been shown that reducible representations can be taken apart into direct sums of smaller representations. It is also possible to put representations together into larger representations. Suppose that D₁ is an m dimensional representation acting on a space with basis vectors |j for j = 1 to m and D₂ is an n dimensional representation acting on a space with basis vectors |x for x = 1 to n. A (m x n) dimensional space called the tensor product can be constructed by taking basis vectors labeled by both j and x in an ordered pair -- |j, x. On this large space, we can define a new representation call the tensor product representation D₁ D₂ by multiplying the two smaller representations, i.e., .

		Another example is the two-dimensional rotation shown in Figure 01b. It consists of an infinite number of elements in the form of continuously varying parmeter, and is known as continuous group or Lie group. The parameter in this case is the angular displacement c, which is the sum of the rotations a and b. The unit element is a rotation of 0o, and the inverse is a rotation in the opposite direction. For better comprehension of the mathematical formula, Figure 01c shows the coordinate system rotates by an angle a about the origin O instead of rotating the point P. The transformation formula between (x,y) and (x',y') can be written as:
Figure 01b 2-D Rotation [view large image]	Figure 01c Coordinate Rotation [view large image]	x' = [cos()] x + [sin()] y ---------- (1a) y' = -[sin()] x + [cos()] y ---------- (1b) or in matrix notation :

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