Abelian/Non-Abelian, and the Unitary Groups

Abelian Groups
Non-Abelian Groups
Lie Groups
Unitary Groups
Local Gauge Symmetry
Global Gauge Symmetry

Abelian Groups

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 e, defined so that ae = a, be = 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 = e 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.

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. 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.

Eigenvalues and Eigenvectors - These are the c-number (a constant not in matrix form) and the associated state vectors for the matrix representation such as D(P) shown in the above example. The eigenvalue p, and the matrix elements of the eigenvector v₁, and v₂ can be expressed in the following form:
,
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|-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.
- 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.

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 0^o, 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(a)] x - [sin(a)] y ---------- (1a) y' = [sin(a)] x + [cos(a)] y ---------- (1b)

The fact that the laws of physics are unchanged by such a rotation implies conservation of the z component of angular momentum; in general, whenever the physical law is invariant under the operation of a symmetry group, there must be some conserved quantity associated with that operation. This is sometimes referred to as Noether's theorem, and is a useful feature of group theory, which can be used to provide physical insights into the behaviour of interactions and particles such as whether the process is permissible or if an entity is missing without knowing the detailed dynamic. When this rotation group is generalized to an equivalent complex number f = f_R + if_I, and applied to the "internal space" in quantum electro-dynamics, the global operation (same amount of rotation everywhere) leads to the conservation of electric charge; while the local operation (amount of rotation depends on location) associates a gauge boson (the photon) with the electromagnetic interaction. The rotation group in two-dimensional real space is called SO(2); while the one with the complex number is called U(1).

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Non-Abelion Groups

The elements in a non-abelian group do not commute as shown in Figure 01d where the two consecutive 90^o rotations R1 R2 generates a completely different result from R2 R1 with the order reversed. In general, there are three degrees of (rotation) freedom corresponding to rotation about the x, y, and z-axis respectively. This 3-dimensional rotation group in real space is called SO(3). The equivalent 3-dimensional rotation group with two complex numbers and three "phase angles" (parameters) associated to the three non-commuting generators is called SU(2). The precise relation between SO(3) and SU(2) is that each 360^o rotation in three dimensions corresponds to two distinct elements of

Figure 01d Non-Abelian Rotation [view large image]

SU(2). It needed a rotation of 2 x 360^o in the "complex number space" to return to the original form (don't try to visualize such abstract operation).

Mathematically, the SO(3) group is represented by the matrices R_x, R_y, and R_z operating on the vector r:

---------- (2)

For example, the rotations portrayed pictorially in Figure 01d can be expressed as R_x operating on r given r' = R_xr, and then r" = R_zr'. It can be shown that the length of the vector: r² = r'² = r"² = x² + y² + z² is invariant under these operations.

In classical physics, spin is the rotation of rigid body around an axis with certain angular momentum uniquely defined by its magnitude and direction. In quantum theory, the angular momentum is quantized and always comes in multiples of a basic unit, which is equal to

/2. Spin in quantum theory is not well-defined; it is in superposition of different spin states. Figure 01e

	shows the geometrical property of the different kinds of spin. A spin 0 particle looks the same from all directions (A). A spin 1 particle looks the same when it is rotated through a full 360^o (B). A spin 2 particle only needs 180^o to regain its original form (C). Spin 1/2 particles (spinor) must go through two complete rotations before they look the same (D).
Figure 01e Different Kinds of Spin [view large image]

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Lie Groups

Let's consider the phase transformation of a complex function (with real and imaginary parts):

f' = e^ia f ---------- (3)
where a (representing the elements of a continuous group) is either a constant for a global transformation or a function of (x,y,z,t) for local transformation. It is known as the phase angle because if f is a wave represented by f = f_oe^i(kx-t), then a is just the phase shift of the wave. The phase shift interpretation provides another representation other than the rotation in the complex plane as introduced in the followings.

If f and f ' are decomposed into the real and imaginary parts: f = f_R + i f_I ---------- (4)

Then Eq.(3) can be re-written as:

f '_R = [cos(a)] f_R - [sin(a)] f_I ---------- (5)
f '_I = [sin(a)] f_R + [cos(a)] f_I ---------- (6)

which is in the same form as Eqs.(1a,b) for the 2-dimensional rotation, but in the (f_R, f_I) complex plane with the real and imaginary parts of the function as coordinates (Figure 01f). This plane is called an internal space, and the associated symmetry an internal symmetry, if the mathematical formulation is unchanged after the operation. For example, f_R and f_I can be identified to the two components of the scalar field in the Klein-Gordon Equation, e.g., . The internal rotation just presents different aspect of the field components in the complex plane. The Klein-Gordon Equation is invariant (unchanged) under such internal rotation, which is another way to say that the equation is symmetrical with respect to such operation. It is obvious from Figure 01f that length of the arrow has not changed by merely rotating the complex plane. It is something like renaming a person from John to Jack - that

Figure 01f Complex Plane Rotation [view large image]

person remains the same although there may be some legal or financial consequences. The analogy in Figure 01f can be taken with the "arrow" as the person, f and f ' as the different names.

Two successive transformations such as: f

f '

f " ---------- (7)
where f " = e^ib f ', can be written in the form:
f " = e^i(a+b) f = e^ic f ---------- (8)
with c = a + b. This is a transformation to the same form as the original one satifying one of the definitions for a group. The inverse elements are -a, -b, ..., and the identity operator is a

0. Since the phase angle can be infinitesimally small, this kind of group also belongs to the Lie group (continuous group) which is important in studying physical theories such as the Noether's theorem or local gauge transformation.

In general a Lie Group is specified by a finite set of continuous parameters a_i (with i running from 1, 2, ... to N_a) and there are well defined derivatives of the group elements g's with respect to all the parameters. We will simplify the discussion by the example of a group with a single parameter a = a₁. According to the rules for a Lie Group, we can define the generator:
t = -i dg/da|_a=0 ---------- (9)
where the i is included so that if the representation D(a) is unitary, the t will be hermitian operator. The identity representation corresponds to a = 0 is D(0) = 1. Using the Taylor series expansion for small increment

from a = 0, D can be expressed as:
D(

)

1 + i

t
If we apply this operation n times with a = n

, then
D(a) = (1 + iat/n)^ⁿ = e^iat as n

which is very similar to the U(1) group transformation in Eq.(3).

Sophus Lie (1842-1899, Figure 02a) showed how the generators can actually be defined in the abstract group without

		mentioning representations at all. Thus, t = 1 for the U(1) group and the Pauli matrices (see below) are the three generators for the SU(2) group, ... Beside generating the displacement for the a_i, the generators are important for forming a vector space. This means two generators can be added together to obtain a third generator, and the generators can be multiplied by a scalar and still remain generators for the group. A complete vector space can be used as a basis for
Figure 02a Sophus Lie and Symmetry [view large image]	Figure 02b E₈ Lie Group [large image]	representing other vector spaces, hence the generators of a group can be used to represent other vector spaces, e.g., the Pauli matrices can be used to describe any 2x2 matrix.

The E₈ Lie group is the most complicated mathematical structure consisting of 240 vectors in an 8-dimensional space. These vectors are the vertices (corners or dots in Figure 02b) of an 8-dimensional object called the Gosset polytope 4₂₁. The diagram is a 2-dimensional representation of the Gosset polytope 4₂₁. There are 8 concentric circles of 30 vertices each. The lines in the picture connect adjacent vertices in the polytope, with colors chosen according to the length of the 2-dimensional projection. Since the picture is a 2-dimensional projection of an 8-dimensional object, it captures only some of the symmetries of the Gossett polytope.

The Lie algebra is defined by the commutator relation:
[t_a, t_b] = i f_abct_c
where f_abc is the structure constant that can be computed in any representation. It plays a role similar to the multiplication law for the group.

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Unitary Groups

The transformation in Eq.(3) is of unit length, the set of all such transformations forms what mathematicians call a unitary group. In this case U(1) = e^ia means the group of all "unitary" 1-dimensional matrices. A unitary matrix U is one that satisfies
UU = UU = I
where I is the identity (unity matrix) and U is the conjugate of U such that the average value

*|U|

|U*|

; it is called Hermitean if U = U, i.e., the average value is a real number; it is called unitary if U = U^-1. An unitary transformation leaves the normalization of the state vectors unchanged. Specifically, unitary operator preserves the probabilities for quantum transition. It plays a special role in quantum theory. The U(1) group have the simple property that it does not matter in what order they are performed, i.e., a + b = b + a. Thus U(1) = e^ia = e^ia(1) = e^iaI is called an Abelian group in which different transformations commute.

In SU(2) the phase angle a = a

I in U(1) is replaced by a

t = a₁t₁ + a₂t₂ + a₃t₃ - there are now three phase angles
a = (a₁, a₂, a₃), and three 2x2 non-commuting matrices (generators) for t, which are 1/2 of the Pauli matrices:

---------- (10)

where the factor of 1/2 is responsible for the curious property that it takes a 2 x 360^o rotation to return to the original form.

For non-Abelian unitary groups, the number of phase angles (parameters) is determined by the formula N_a = n² - 1, where n is the dimension of the internal space, e.g., N_a = 3 for n = 2 in SU(2). The rank of SU(n) is n - 1. It gives the number of diagonal matrix representation for the generators. Thus, the rank of SU(2) is 1 as shown in Eq.(10).

It can be shown that the absolute square for the determinant of U is 1, i.e., |detU|² = 1 (which implies the trace of the generators equal to zero). Thus |detU| can be either +1 or -1. The "S" in "SU" means the transformations are "Special" satisfying |detU| = +1, which implies continuous roatations without reflections. Therefore SU(2) designates the group of special, unitary 2x2 matrices.

Figure 01e

For the trivial (identity) representation all the generators t_i 0. It can be identified to particle with spin 0 such as the Higgs boson. In this representation, the particle looks the same from every direction.
The fundamental representation of SU(2) is for leptons with spin 1/2. This is the one that has to rotate 2 x 360^o to return to its original form.
There is the representation of SU(2) by the usual 3-dimensional rotations (the SO(3) group) acting on three dimensional vectors. The generators can be produced from the definition in Eq.(9), for example:

t_z = -i (dR_z/d)|_₌₀ =
This is the spin 1 representation and can be used to describe gauge particles such as photon, vector boson, and gluon.

The rotation in internal space has nothing to do with angular momentum. In SU(2), the two complex fields can be identified with the electron and neutrino fields respectively in analogy to the spin up and spin down states. It is called "isospin" or "weak isospin" (for the left-handed electron and neutrino in weak interaction) to distinguish from the ordinary "spin". For example, in terms of the Pauli matrices, the phase angle

, and the complex fields :

a SU(2) transformation of can be expressed as:
' = e^i(/2)₁ = [cos(

/2)

₁ + i sin(

/2)

₁]
It can be shown that the magnitude of the complex fields: u*u + v*v is conserved under such operation.

In SU(3), the three complex fields can be identified to the three quark fields, which can be thought of as the three directions at 120^o to each other. It is called colour isospin or colour spin. In SU(3) N_a = 8, there are eight phase angles and eight 3x3 non-commuting matrices (generators) operating on three complex functions (representing the u, d, s quarks as shown in Eq.(20)):

---------- (11)

The SU(2) and SU(3) transformations are more like the non-commuting 3-dimensional rotations of real space because of the non-commuting generators, e.g., e^{i(at₁+ bt₂)}

e^{i(bt₁+ at₂)}. These groups are called non-Abelian for this reason.

If we define t² = t₁² + t₂² + t₃² + ... ---------- (12)

It can be shown that the phase transformations are characterized by the eigenvalue of t² and t₃ with eigenvector |t,m

:

t² |t,m

= t(t+1) |t,m

---------- (13)
t₃ |t,m

= m |t,m

---------- (14)
where t can be determined by the formula n = 2t +1 (n is the dimension of the generator) and for a given t, the value of m can be -t, -t+1, ..., t-1, t; i.e., there are 2t+1 degenerate states for a given t. For example, in the case of SU(2) n = 2, t = 1/2, and m = -1/2 or +1/2 (an isodoublet). The eigenvalue t is often referred to as isospin to indicate that it is similar to spin but in an abstract space. The eigenvalue for t₃, is sometimes referred to as weak charge (when applied to weak interaction). For SU(3), n = 3, t = 1, and m = +1, 0, -1 forming an isotriplet.

A casimir operator is a nonlinear function of the generators that commutes with all of the generators. The number of casimir operators is equal to the rank of the group. For example, there is only 1 casimir operator in SU(2), e.g., t² = (3/4) I, which obviously commutes with all the t_i. Since the casimir operator is proportional to the identity. This constant of proportionality can be used to classify the representations of the Lie algebra (and hence, also of its Lie group). It is usually related to the mass or (iso)spin. The proportional constant in the example is the "square" of total (iso)spin T² = t (t+1) for t = 1/2.

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Local Gauge Symmetry

In quantum field theory, if we demand the system to be invariant under the local gauge transformation:

where A is the field for the gauge bosons, g_s is the coupling constant, lambda is the generator, and

(x) is now a function of space-time. The second term in Eq.(16) is responsible for the various interactions via the exchange of bosons (see more details in "Quantum Field Theory"). The number of gauage bosons depends on the dimension of the generator. Thus, there is only the photon for the U(1) group applicable to electromagnetic interaction; there are three vector bosons - W⁺, W^-, and Z^o for the SU(2) group applicable to weak interaction; and there are eight gluons for the SU(3) group applicable to strong interaction (see Table 15-02).

Now let us turn to the physical interpretation of the complex function in Eq.(3). For the U(1) group, f_R and f_I can be identified with particles carrying different electric charge. The dynamic of these particles is described by the two-component Klein-Gordon Equation.

Application of the SU(2) group to weak interaction is more complicated because of parity violation. Since there is no right-handed neutrino, the wave functions are split into a left-handed and right-handed parts:

It can be shown that if the U(1) and SU(2) groups are spliced together, i.e., the gauge group becomes U(1)XSU(2), such that

L

L' = e^i(x)/2 e^ia(x)t L,
R

R' = e^i(x) R,

where t is given by the matrices in Eq.(10). Then the electro-weak theory would be invariant under this transformation with corresponding covariant derivative similar to Eq.(16).

The transformation of the quark fields in electro-weak theory is slightly different because the right-handed field involves both the up and down quarks d_R, and u_R:

L

L' = e^-i(x)/3 e^ia(x)t L,
d_R

d_R' = e^-4i(x)/3 d_R ,
u_R

u_R' = e^2i(x)/3 u_R .

The SU(3) transformation for the quark fields in strong interaction theory is similar to the one shown in Eq.(15):

u

u' = e u,

where

is given by the matrices in Eq.(11). In QCD, each flavor of quark is represented by a triplet. For example the u quark is associated with the triplet

where u_r, u_g, u_b are 4-component Dirac spinors, and the subscripts r, g, b label the color states (red, green, blue, which can be considered as the sources of the strong interaction like the electric charge for the electromagnetic interaction, and has nothing to do with color vision). See more in "Standard Model" for strong interaction.

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Global Gauge Symmetry

The global gauge symmetry SU(2) X U(1) can be used to explain the empirical relation of Gell-Mann and Nishijima:

Q = t₃ + (Y/2) ---------- (19)

where Q is the electric charge, t₃ is the third component of the isospin, and Y is the "hypercharge". The hypercharge was invented to explain the state of the electron and neutrino prior to spontaneous symmetry breaking, which endows mass and charge to these particles. Before the occurence of such event these two particles differ only by the weak charge t₃. The hypercharge was introduced to relate the difference in electric charge to the difference in t₃. As shown in Table 01, Y = -1 or 1/3 for the SU(2) isodoublet, and Y = -2 or -2/3 for the U(1) isosinglet. These values of Y reflect the charge state of these particles before symmetry breaking. To satisfy this formula for electro-weak interaction with SU(2) X U(1) symmetry where the SU(2) group splices together with the U(1) group (no mixing), the isodoublet L in Eq.(17) has to be an eigenvector of
SU(2) with t = 1/2 and the isosinglet R an eigenvector of U(1) with t = 0. With the known values of Q and t₃, Y can be computed from Eq.(19). Similar assignment can be given to the u (up), d (down), and s (strange) quarks as listed in Table 01 below:

Particle	Q	t	t₃	Y
	0	1/2	1/2	-1
e_L	-1	1/2	-1/2	-1
e_R	-1	0	0	-2
u	2/3	1/2	1/2	1/3
d	-1/3	1/2	-1/2	1/3
s	-1/3	0	0	-2/3

Table 01 Gell-Mann-Nishijima Relation

In strong interaction with SU(3) symmetry, the hypercharge for the quarks is generated by the

₈/(3)^1/2 generator (see Table 01 and diagrams at the top of Figure 03). The hypercharge for the hadrons, which is the composite particle with quarks, is just the sum of the individual quarks (see Table 02 and 03). The vaule of Y is related to the other quantum number in strong interaction by the formula: Y = S + B, where S is the strangeness number (0 for u and d, -1 for s), and B is the baryon number (+1/3 for quark, -1/3 for anti-quark, +1 for baryon, -1 for anti-baryon, and 0 for other particles). It was used by Gell-Mann to arrange the hadrons into geometrical patterns known as the eightfold way (see Figure 03). It is a plot with Y against t₃. The isosinglet with one member for t = 0 is located at the center. Two isodoublets for t = 1/2 are arranged in the top and bottom row with Y = +1 and -1 respectively. The three members of the SU(3) isotriplet for t = 1 occupy the middle row, the t₃ = 0 member shares the center with the isosinglet member. The particles with same isospin (same multiplet) are supposed to be similar except for the value of t₃. The mass splitting among the isospin members is caused by the electromagnetic interactions as shown in Figure 04.


Figure 03 Eightfold Way [view large image]	Figure 04 Mass Splitting [view large image]

Another application of global gauge symmetry is to build mesons and baryons from quarks in strong interaction, where the wave functions for the quarks u, d, and s constitute the fundamental representation of SU(3):

---------- (20)

Note that the three quarks in Eq.(20) is called "flavor" quark associated with global SU(3). The other three kinds of quark related to the local SU(3) gauge transformation is called "colour" quark participating in strong interaction.

Beginning with the fundamental representation in Eq.(20), all representations of SU(N) can be generated by taking the multiple (tensor) products. For example, the bound states of mesons can be constructed according to the following scheme:

---------- (21)

where the bar "^-" denotes anti-particle.

Table 02 lists the relation between the meson wave functions and the quark pairs. The value within the brackets is the rest mass in Mev.

In these tables J is the total angular momentum quantum number. The mesons in the middle column belongs to J = 0, while those in the right column belongs to J = 1. In terms of the meson wave functions the octet in Eq.(21) can be re-written in the form of Eq.(22). Similarly, the baryon wave functions are related to the quark triplet as shown in Table 03 and Eq.(23).

This is clearly related to the Eightfold Way (Figure 03) originally proposed by Gell-Mann and Ne'eman in 1961.

Note: These elementary particles were referred to as resonances in the 1960s. Now they are considered as the excited states of hadrons with some of their constituent quarks boosted into higher energy levels. Most of them have a very short lifetime about 10^-23 sec.