Quantum Numbers as Representations of Gauge Groups

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As emphasized by Klein and Weyl, a group is a collection of operations leaving a certain “object” unchanged. This amounts to classifying the symmetries of the object. When the “object” in question is the laws of Physics in a space-time with negligible gravitation-induced curvature, the symmetries can be classified as follows: (i) No point in four- dimensional space-time is privileged, hence one can shift or translate the origin of space-time arbitrarily in four directions. Noether’s theorem then implies there are four associated conserved quantities, namely the three components of space momentum and the energy. These four quantities naturally constitute the components of a 4-vector Pμ, μ = 0, 1, 2, 3. (ii) No direction is special in space; leading to three conserved quantities Ji, i = 1, 2, 3. (iii) There is no special inertial frame; the same laws of Physics hold in inertial frames moving with constant speed in any one of the three independent directions. What is generally known as Noether’s Theorem states that if the Lagrangian function for a physical system is not affected by a continuous change (transformation) in the coordinate system used to describe it, then there will be a corresponding conservation law; i.e. there is a quantity that is constant. For example, if the Lagrangian is independent of the location of the origin then the system will preserve (or conserve) linear momentum. If it is independent of the base time then energy is conserved. If it is independent of the angle of measurement then angular momentum is conserved.

As we suggested above, it is possible to get a non-mathematical insight into Noether’s theorem relating symmetries to conserved quantities. Consider a single particle moving in a completely homogeneous space. It cannot come to a stop or change its velocity because this would have to happen at some particular point, but all points being equal, it is impossible to choose one. Hence the particle has no choice but to move at constant velocity or, in other words, to conserve its linear momentum, which was anciently called “impetus”. It is easy to extend the argument to a rotating object in an isotropic space and conclude that it cannot come to a stop at any particular angle since there is no special angle; hence its angular momentum is conserved.

(ii) and (iii) amount to covariance of the laws of physics under rotations in a four-dimensional space with a metric that is not positive-definite. The squared length of a 4-vector defined via this metric must then be an important invariant independent of the orientation or the velocity of the frame. Indeed, for the 4-vector Pμ this is the squared mass m2 of the particle, and it is one of the two invariant labels used in specifying the representation. The other label is the squared length of another 4-vector called the Pauli-Lubanski vector. It then follows from the algebra of the group that this squared length takes on values s(s + 1) and that in contrast to m2, which assumes continuous values, s can only be zero, or a positive integer, or half a positive odd integer. The unitary representation of the Poincaré group for a particle of mass m and spin s provides its relativistic quantum mechanical wave function. The equation of motion the wave function must obey also comes with the representation; it is the Bargmann-Wigner equation for that spin and mass. In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles of arbitrary spin j, an integer for bosons (j = 1, 2, 3 …) or half-integer for fermions (j = 123252 …). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. The procedure of second quantization then naturally promotes the wave functions to quantized field operators, and in a sense demotes the particles to quanta created or destroyed by these operators. Pauli’s spin-statistics theorem, based on a set of very general requirements such as the existence of a lowest energy vacuum state, the positivity of energy and probability, microcausality, and the invariance of the laws of Physics under the Poincaré group, leads to the result that the only acceptable quantum conditions for field operators of integer-spin particles are commutation relations, while those corresponding to half-integer spin must obey anticommutation relations. The standard terms for the two families of particles are bosons and fermions, respectively. The Pauli’s Exclusion Principle, or the impossibility of putting two electrons into the same state, is now seen to be the result of the anticommutation relation between electron creation operators: to place two fermions in the same state, the same creation operator has to be applied twice. The result must vanish, since the operator anticommutes with itself.

The symmetries of space-time are reflected in the fields which are representations of the symmetry groups; a quantum mechanical recipe called quantization then turns these fields into operators capable of creating and destroying quanta (or particles, in more common parlance) at all space-time points. Actually, the framework we have described only suffices to describe “Free fields” which do not interact with each other. In order to incorporate interactions, one has to resort to another kind of symmetry called gauge symmetry, which operates in an “internal” space attached to each point of space-time. While the identity of masses and spins of, say, electrons can be attributed to space-time symmetries, the identities of additional quantum numbers such as charge, isospin and “color” can only be explained in terms of the representations of these gauge groups.

Cyclotomic Fields

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A cyclotomic extension of a field F is a splitting field E for the polynomial

f(X) = Xn − 1

over F. The roots of f are called nth roots of unity. The nth roots of unity form a multiplicative subgroup of the group E× of non-zero elements of E, and thus must be cyclic. A primitive nth root of unity is an nth root of unity whose order in E× is n. It is denoted ζn.

From now on, we will assume that we work in a characteristic char(F) such that char(F) does not divide n. (Otherwise, we have n = mchar(F) and 0 = ζnn − 1 = (ζm − 1)char(F) and the order of ζn is less than n.)

The nth cyclotomic polynomial is defined by

Ψn(X) = ∏(i,n) (X−ζni)

where the product is taken over all primitive nth roots of unity in C.

The degree of Ψn(X) is thus deg(Ψn) = φ(n)

We have

Xn −1 = ∏d|n Ψd(X)

In particular, if n = p a prime, then d is either 1 or p and

Xp − 1 = Ψ1(X)Ψp(X) = (X − 1)Ψp(X)

Ψp(X) = (Xp −1)/(X−1) = Xp−1 +Xp−2 +···+ X + 1

Proof:

Equality is proved by comparing the roots of both monic polynomials. If ζ is a nth root of unity, then by definition ζnn = 0 and its order d divides n. Thus ζ is actually a primitive dth root of unity, and a root of Ψd(X). Conversely, if d|n, then any root of Ψd(X) is a dth root hence a nth root of unity.

The nth cyclotomic polynomial Ψn(X) satisfies

Ψn(X) ∈ Z[X].

Proof:

We proceed by induction on n. It is true for n = 1 since X − 1 ∈ Z[X]. Let us suppose it is true for Ψk(X) where k is up to n−1, and prove it is also true for n. Using the above proposition, we know that

Xn −1 = ∏d|n Ψd(X)

= Ψn(X) ∏d|n,d<n Ψd(X).
The aim is to prove that
Ψn(X) ∈ Z[X] :
Ψn(X) = (Xn −1)/∏d|n,d<n Ψd(X)
First note that Ψn(X) has to be monic (by definition), and both Xn − 1 and Ψd(X) (by induction hypothesis) are in Z[X]. We can thus conclude invoking the division algorithm for polynomials in Z[X].

Finite Fields

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A finite field is a field with a finite field order (i.e., number of elements), also called a Galois field. The order of a finite field is always a prime or a power of a prime. For each prime power, there exists exactly one (with the usual caveat that “exactly one” means “exactly one up to an isomorphism“) finite field GF(p^n), often written as Fpn in current usage.

Theorem:

Let E be a finite field of characteristic p.

1. The cardinality of E is

|E| = pn, for some n ≥ 1. It is denoted E = Fpn

Furthermore, E is the splitting field for the separable polynomial f(X) = Xpn − X

over Fp, so that any finite field with pn elements is isomorphic to E. In fact, E coincides with the set of roots of f.

Proof:

1. Let Fp be the finite field with p elements, given by the integers modulo p. Since E has characteristic p, it contains a copy of Fp. Thus E is a field extension of Fp, and we may see E as a vector space over Fp. If the dimension is n, then let α1,…,αn be a basis. Every x in E can be written as

x = x1α1 +···+ xnαn

and there are p choices for each xi, thus a total of pn different elements in E.

2. Let E× be the multiplicative group of non-zero elements of E. If α ∈ E×, then

αpn−1 = 1

by Lagrange’s Theorem, so that

αpn

∀ α in E (including α = 0). Thus each element of E is a root off, and f is separable.

Now f has at most pn distinct roots, and we have already identified the pn elements of E as roots of f.

Corollary: If E is a finite field of characteristic p, then E/Fp is a Galois extension, with cyclic Galois group, generated by the Frobenius automorphism

σ : x → σ(x) = xp, x ∈ E

Proof:

By the above proposition, we know that E is a splitting field for a separable polynomial over Fp, thus E/Fp is Galois.

Since xp = x ∀ x in Fp, we have that

Fp ⊂ F(⟨σ⟩)

that is Fp is contained in the fixed field of the cyclic subgroup generated by the Frobenius automorphism σ. But conversely, each element fixed by σ is a root of Xp − X so F(⟨σ⟩) has at most p elements. Consequently

Fp = F(⟨σ⟩)

and

Gal(E/Fp) = ⟨σ⟩

This can be generalized when the base field is larger than Fp.

Corollary: Let E/F be a finite field extension with |E| = pn and |F| = pm. Then E/F is a Galois extension and m|n. Furthermore, the Galois group is cyclic, generated by the automorphism

τ : x → τ(x) = xpm, x ∈ E

Proof:

If the degree [E : F] = d, then every x in E can be written as

x = x1α1 +···+ xdαd and there are pm choices for each xi, thus a total of

(pm)d = pn different elements in E, so that

d = m/n

and

m|n

The same proof as for the above corollary holds for the rest.

Thus a way to construct a finite field E is, given p and n, to construct E = Fpn as a splitting field for Xpn − X over Fp

Theorem:

If G is a finite subgroup of the multiplicative group of an arbitrary field, then G is cyclic. Thus in particular, the multiplicative group E× of a finite field E is cyclic.

Proof:

The proof relies on the following fact: if G is a finite abelian group, it contains an element g whose order r is the exponent of G, that is, the least common multiple of the orders of all elements of G.

Assuming this fact, we proceed as follows: if x ∈ G, then its order divides r and thus

xr = 1

Therefore each element of G is a root of Xr − 1 and

|G| ≤ r

Conversely, |G| is a multiple of the order of every element, so |G| is at least as big as their least common multiple, that is

and

|G| ≥ r |G| = r

Since the order of |G| is r, and it coincides with the order of the element g whose order is the exponent, we have that G is generated by g, that is G = ⟨g⟩ is cyclic. Since E× is cyclic, it is generated by a single element, say α : E = Fp(α) and α is called a primitive element of E. The minimal polynomial of α is called a primitive polynomial.