Quantum Field Theory III: Gauge Theory

Frontmatter

0. Prologue

Abstract

This prologue should help the reader to understand the sophisticated historical development of gauge theory in mathematics and physics. We will not follow a strict logical route. This will be done later on. At this point, we are going to emphasize the basic ideas. It is our goal to show the reader how the methods of modern differential geometry work in the case of Einstein’s theory of general relativity, which describes the gravitational force in nature.

Eberhard Zeidler

1. The Euclidean Space E 3 (Hilbert Space and Lie Algebra Structure)

Abstract

One has to distinguish between

• the Euclidean space E ₃ (a set of vectors), and
• the Euclidean manifold \(\mathbb{E}^{3}\) (a set of points).

The Euclidean space E ₃ is a real 3-dimensional Hilbert space equipped with the inner product

$$\langle \mathbf{x}|\mathbf{y}\rangle:= \mathbf{x}\mathbf{y}$$

of vectors x,y. Additionally, the Euclidean space E ₃ is a Lie algebra equipped with the vector product

$$[\mathbf{x}, \mathbf{y}]:= \mathbf{x}\times \mathbf{y}.$$

Eberhard Zeidler

2. Algebras and Duality (Tensor Algebra, Grassmann Algebra, Clifford Algebra, Lie Algebra)

Abstract

Operator algebras play a fundamental role in algebraic quantum field theory. In order to understand this, one has first to understand the crucial algebraic structures of the Euclidean space. The point is that relevant products possess an invariant meaning, that is, they are independent of the choice of a basis of the Euclidean space.

Eberhard Zeidler

3. Representations of Symmetries in Mathematics and Physics, and Elementary Particles

Abstract

The representation of symmetry groups plays a crucial role in physics. In this chapter we discuss the elements of the representation theory of Lie groups and Lie algebras. In particular, we apply representations of the Lie group SU(3) and the Lie algebra su(3) to the quark model in strong interaction.

Eberhard Zeidler

4. The Euclidean Manifold $\mathbb{E}^{3}$

Abstract

Let us use the notation introduced at the beginning of Sect. 1.2 on page 71. Consider the motion

$$P=P(t), \qquad t \in \mathbb{R}$$

of a particle (Fig. 4.1). Equivalently, we write

$$\mathbf{x}= \mathbf{x}(t), \qquad t\in \mathbb{R}.$$

Here, x(t) denotes the position vector starting at the origin O at time t with the terminal point P(t)=O+x(t). Let E ₃(P) denote the space of all the position vectors starting at the point P. This is a real 3-dimensional Hilbert space equipped with the inner product 〈u|w〉 _P :=uw and the norm \(|\mathbf{u}|_{P}:= \sqrt {\langle \mathbf{u}|\mathbf{u}\rangle_{P}}\) for all u,w∈E ₃(P).

Eberhard Zeidler

5. The Lie Group U(1) as a Paradigm in Harmonic Analysis and Geometry

Abstract

The theory of Lie groups and Lie algebras is nothing else than a far-reaching generalization of Euler’s exponential function. The simplest case is the Lie group U(1) defined by

$$U(1): = \{z \in \mathbb{C}: \;|z|=1\}$$

equipped with the usual multiplication of complex numbers. Equivalently,

$$U(1)= \{\textrm{e}^{\textrm{i} \varphi}: \; \varphi \in \mathbb{R}\}.$$

The set U(1) is a real one-dimensional manifold, namely, the unit circle. This manifold is called the group manifold of the Lie group U(1). In particular, a Lie group \(\mathcal{G}\) is called compact iff \(\mathcal{G}\) is a compact manifold. For example, the Lie group U(1) is compact. In fact, the unit circle is a compact manifold.

Eberhard Zeidler

6. Infinitesimal Rotations and Constraints in Physics

Abstract

The operator A:E ₃→E ₃ is called unitary iff it is linear and it respects the inner product, that is,

$$\langle A\mathbf{x}|A\mathbf{y}\rangle = \langle \mathbf{x}|\mathbf{y}\rangle \qquad \mbox{for all}\quad \mathbf{x}, \mathbf{y}\in E_3.$$

The symbol U(E ₃) denotes the set of all unitary operators A:E ₃→E ₃. We have

$$A \in U(E_3)\qquad \mbox{iff}\qquad A^\dagger A =I.$$

In fact, it follows from (6.1) that

$$\langle \mathbf{x}|A^\dagger A\mathbf{y}\rangle = \langle A\mathbf{x}|A\mathbf{y}\rangle =\langle \mathbf{x}|\mathbf{y}\rangle \qquad \mbox{for all}\quad \mathbf{x}, \mathbf{y}\in E_3.$$

Hence A ^† A=I. Conversely, A ^† A=I implies (6.1).

If A∈U(E ₃), then det (A)=±1.

In fact, I=A ^† A implies 1=det I=det A ^†det A=(det A)^†det A=|det A|².

Eberhard Zeidler

7. Rotations, Quaternions, the Universal Covering Group, and the Electron Spin

Abstract

Euler’s rotation formula (6.6) can be elegantly written by using Hamilton’s quaternions. This was discovered independently by Hamilton and Cayley in 1844, one year after Hamilton’s discovery of quaternions. In the language of quaternions, Euler’s rotation formula (6.6) reads elegantly as

$$\fbox{$\mathbf{x}' = q\cdot \mathbf{x}\cdot q^{\dagger}, \qquad \mathbf{x}\in E_{3}.$}$$

Here, the given quaternion

$$q: = \cos \tfrac{\varphi}{2} + \sin \tfrac{\varphi}{2} \; \mathbf{n} $$

contains the information about the rotation angle ϕ and the rotation axis vector n of length one. In particular, for the norm of the quaternion q we get

$$|q| = \sqrt{ \cos^2 \tfrac{\varphi}{2} +\mathbf{n}^2\sin^2\tfrac{\varphi}{2}}=1.$$

Hence q∈U(1,ℍ).

Eberhard Zeidler

8. Changing Observers – A Glance at Invariant Theory Based on the Principle of the Correct Index Picture

Abstract

Invariant theory plays a crucial role in all branches of mathematics and in modern physics. Invariant theory has its roots in celestial mechanics (Lagrange’s contributions to the three-body problem), the motion of rigid bodies (Euler’s spinning top), Cauchy’s theory of elasticity, number theory, projective geometry, and differential geometry. In his fundamental work Disquisitiones arithmeticae on number theory from 1801, Gauss (1777–1855) studied invariants of quadratic forms under unimodular linear substitutions with integral coefficients. Later on, more general results on quadratic forms were obtained by Jacobi (1804–1851), Sylvester (1814–1897), and Hermite (1822–1901).

Eberhard Zeidler

9. Applications of Invariant Theory to the Rotation Group

Abstract

We want to use the method of orthonormal frames in order to define

• the gradient grad Θ of a smooth temperature field Θ, and
• both the divergence, div v, and the curl, curl v, of a smooth velocity vector field v on the Euclidean manifold \(\mathbb{E}^{3}\).

The physical meaning of grad Θ, div v, and curl v will be discussed in Sect. 9.1.4.

Eberhard Zeidler

10. Temperature Fields on the Euclidean Manifold $\mathbb{E}^{3}$

Abstract

In mathematics and physics, differentiation describes the linearization of analytic objects like physical fields. In this chapter, let us study the directional derivative of a temperature field Θ on the Euclidean manifold \(\mathbb{E}^{3}\). To this end, let

$$\varTheta: \mathbb{E}^3\to \mathbb{R}$$

be a smooth function. In terms of physics, we regard Θ(P) as the temperature at the point P on \(\mathbb{E}^{3}\). We are given the smooth curve

$$C: P=P(t), \qquad t\in \mathbb{R}$$

on \(\mathbb{E}^{3}\) with P ₀:=P(0). In terms of position vectors at the origin, we describe the curve C by the smooth vector function x=x(t),t∈ℝ. The derivative

$$\fbox{$d_{\mathbf{h}}\varTheta (P_{0}):= \frac{d\varTheta (\mathbf{x}(t))}{dt}_{|t=0}$}$$

is called the directional derivative of the temperature field Θ along the trajectory C at the point P ₀.

Eberhard Zeidler

11. Velocity Vector Fields on the Euclidean Manifold $\mathbb{E}^{3}$

Abstract

We want to study vector fields

$$\mathbf{w}=\mathbf{w}(P), \quad P \in \mathbb{E}^3$$

on the 3-dimensional Euclidean manifold \(\mathbb{E}^{3}\). For example, this concerns velocity vector fields or force fields like

• Newton’s gravitational field w=F _grav,
• Maxwell’s electric field w=E, or
• Maxwell’s magnetic field w=B.

We will frequently use the intuitive picture of the velocity vector field of a fluid. For such vector fields w on \(\mathbb{E}^{3}\), one has to distinguish between

• the covariant directional derivative D _v w, and
• the Lie derivative \(\mathcal{L}_{\mathbf{v}}\mathbf{w}=D_{\mathbf{v}}\mathbf{w}-D_{\mathbf{w}}\mathbf{v}\). Here, v is the velocity field of the flow of fluid particles on \(\mathbb{E}^{3}\).

Eberhard Zeidler

12. Covector Fields and Cartan’s Exterior Differential – the Beauty of Differential Forms

Abstract

The calculus of differential forms was introduced by Élie Cartan (1869–1951) in 1899. It was Cartan’s goal to study Pfaff systems

$$\sum _{k=1}^n a_{jk}(x ^1, \ldots, x^n) dx^k=0, \qquad j= 1, \ldots, m$$

by using a symbolic method. It turns out that:

Cartan’s calculus is the proper language of generalizing the classical calculus due to Newton (1643–1727) and Leibniz (1646–1716) to real and complex functions with n variables.

The key idea is to combine the notion of the Leibniz differential df with the alternating product a∧b due to Grassmann (1809–1877). Cartan’s calculus has its roots in physics. It emerged in the study of point mechanics, elasticity, fluid mechanics, heat conduction, and electromagnetism. It turns out that Cartan’s differential calculus is the most important analytic tool in modern differential geometry and differential topology, and hence Cartan’s calculus plays a crucial role in modern physics (gauge theory, theory of general relativity, the Standard Model in particle physics). In particular, as we will show in Chap. 19, the language of differential forms shows that Maxwell’s theory of electromagnetism fits Einstein’s theory of special relativity, whereas the language of classical vector calculus conceals the relativistic invariance of the Maxwell equations.

Eberhard Zeidler

13. The Commutative Weyl U(1)-Gauge Theory and the Electromagnetic Field

Abstract

In what follows, we will consider the following two transformations:

(i)

transformation of the space and time coordinates, and

(ii)

gauge transformations of the physical field (local symmetry transformations).

Our final goal is to establish a mathematical formalism which is invariant under both transformations.

Eberhard Zeidler

14. Symmetry Breaking

Abstract

We want to study the typical behavior of physical fields near a ground state (also called vacuum). It happens frequently that the ground state of a many-particle system is not unique. In this case, the system can oscillate near different ground states which, as a rule, corresponds to different physical behavior. Therefore, the choice of the ground state plays a crucial role. Historically, Pauli criticized the formulation of gauge field theories by Yang and Mills in 1954; Pauli emphasized that the corresponding interacting gauge particles are massless, in contrast to physical experiments. This defect of gauge theories could be cured in the 1960s by using the so-called Higgs mechanism which equips the gauge bosons with mass. This way, the W ^±-bosons and the Z ⁰-boson obtain their mass in the Standard Model of particle physics. Physicists speak of symmetry breaking (or loss of symmetry) for the following reason.

• The original theory possesses a family of ground states which can be transformed into each other by using the symmetry group \(\mathcal{G}\) of the theory.
• In nature, physical systems oscillate frequently near a distinguished ground state. These realistic states are not anymore symmetric under the original symmetry group \(\mathcal{G}\). In this sense, the symmetry group \(\mathcal{G}\) is broken.

Eberhard Zeidler

15. The Noncommutative Yang–Mills SU(N)-Gauge Theory

Abstract

Fix N=1,2,… Let \(\mathcal{G}\) be a closed subgroup of the Lie matrix group GL(N,ℂ) of invertible complex (N×N)-matrices. Then, \(\mathcal{G}\) is a Lie group. As a prototype, the reader should have the special case in mind where N=2 and

$$\mathcal{G} = SU(2).$$

This gauge group was used by Yang and Mills in 1954. Recall that the Lie group SU(2) consists of all the unitary (2×2)-matrices U with det (U)=1. The corresponding Lie algebra su(2) consists of all the complex (2×2)-matrices A with A ^†=−A and tr (A)=0. Further examples for the Lie group \(\mathcal{G}\) are the Lie groups U(N),SU(N),GL(N,ℂ),SL(N,ℂ), and SO(3) with N=3. We will show how the U(1)-gauge theory from Chap. 13 has to be modified in the case of a noncommutative gauge group \(\mathcal{G}\) (e.g., \(\mathcal{G}= SU(N)\) with N=2,3,…).

Eberhard Zeidler

16. Cocycles and Observers

Abstract

There exist two approaches to the theory of bundles, namely,

(i)

the observer approach based on cocycles, and

(ii)

the axiomatic geometric approach.

In (i), roughly speaking, we will use a cocycle in order to glue together product bundles. This will be considered in the present chapter. The geometric approach will be considered in the next chapter; this is the most elegant approach based on a few geometric axioms. For didactic reasons, we start with (i). The idea is to use bundle charts (product bundles) and to describe the change of bundle coordinates by a cocycle. In fact, the two approaches (i) and (ii) are equivalent to each other. In (ii), the cocycle corresponds to the transition maps between the bundle charts.

Eberhard Zeidler

17. The Axiomatic Geometric Approach to Bundles

Abstract

Our strategy is to define the notion of vector bundles and principal bundles in an invariant way by only using geometric properties of manifolds. In order to prove further geometric properties of these manifolds (e.g., curvature or parallel transport), we use the fact that, by definition, these properties do not depend on the choice of local bundle coordinates. Therefore, we can pass to special bundle coordinates. This is the situation of product bundles considered in Sects. 15.1 through 15.3. This way, the general results are immediate consequences of our special results about product bundles.

In this chapter, we tacitly assume that all the objects are smooth, that is, they are described by smooth functions with respect to local coordinates.

Eberhard Zeidler

18. Inertial Systems and Einstein’s Principle of Special Relativity

Abstract

The Einstein convention. In what follows, we will sum over equal upper and lower Greek (resp. Latin) indices from 0 to 3 (resp. 1 to 3). For example,

$$a^\mu b_\mu = a^0b_0+a^1b_1+a^2b_2+a^3b_3 =a^0b_0+a^jb_j.$$

Eberhard Zeidler

19. The Relativistic Invariance of the Maxwell Equations

Abstract

Consider a fixed inertial system. We have to distinguish between

• the Maxwell equations in a vacuum, and
• the Maxwell equations in materials.

The basic quantities are

• the electric vector field E,
• the magnetic vector field B,
• the electric charge density ϱ and
• the electric current density vector J.

Eberhard Zeidler

20. The Relativistic Invariance of the Dirac Equation and the Electron Spin

Abstract

In this chapter, we restrict ourselves on discussing the basic ideas. A detailed investigation of the Dirac equation together with the Seiberg–Witten equation and the relations to spinor calculus, Clifford algebras, and spin geometry can be found in Vol. IV on quantum mathematics.

Eberhard Zeidler

21. The Language of Exact Sequences

Abstract

We will show that exact sequences are a basic tool for the following topics:

• systems of linear equations (linear operator equations),
• homology and cohomology in topology,
• the general potential equation in gauge theory; here, Betti numbers are counting the linearly independent constraints for the existence of physical fields on manifolds: Betti numbers are also counting the gauge degrees of freedom of the potentials (modulo cohomology).

Eberhard Zeidler

22. Electrical Circuits as a Paradigm in Homology and Cohomology

Abstract

We want to show that:

(i)

Electric currents J are 1-cycles: ∂ J=0.

(ii)

Voltages V are 1-coboundaries: V=−dU (U is the electrostatic potential).

(iii)

There exists a duality relation between electric currents and voltages: 〈V|J〉=0 (orthogonality).

(iv)

If the electrical circuit is connected, then we get β ₀=1 for the zeroth Betti number. In the general case, β ₀ is equal to the number of connectivity components of the electrical circuit.

(v)

If the electrical circuit has s ₀ nodes and s ₁ connections, then the Euler characteristic is given by χ=s ₀−s ₁.

(vi)

This yields the first Betti number β ₁=β ₀−χ.

(vii)

The space of electric currents is a linear space of dimension β ₁.

Modern computers are based on huge electrical circuits. In this section, we would like to study the theory of electrical circuits as a paradigm for important generalizations in modern physics and mathematics.

Eberhard Zeidler

23. The Electromagnetic Field and the de Rham Cohomology

Abstract

Gauge theory is based on potentials. The de Rham cohomology of a manifold M relates the existence of potentials to the topology of the manifold M. To begin with, let us explain this for the real line and the unit circle \(\mathbb{S}^{1}\).

Eberhard Zeidler

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