A Course in Mathematical Physics 1 and 2

The foundations of the part of mechanics that deals with the motion of point-particles were laid by Newton in 1687 in his Philosophiae Naturalis Principia Mathematica. This classic work does not consist of a carefully thought-out system of axioms in the modern sense, but rather of a number of statements of various significance and generality, depending on the state of knowledge at the time. We shall take his second law as our starting point: “Force equals mass times acceleration.” Letting x_i(t) be the Cartesian coordinates of the i-th particle as a function of time, this means where F_i, denotes the force on the i-th particle.

The intuitive picture of a smooth surface becomes analytic with the concept of a manifold. On the small scale a manifold looks like a Euclidean space, so that infinitesimal operations like differentiation may be defined on it.

A function f from an open subset U of ℝⁿ into ℝ^m is differentiable at a point x ε U if it may be approximated there with a linear mapping Df: ℝⁿ→ℝ^m We can make this notion more precise by requiring that for all ε > 0 there exists a neighborhood U of x such that.

A 2-form is canonically defined on the cotangent bundle of a manifold. Diffeomorphisms leaving this 2-form invariant are called canonical transformations.

The Lagrangian (2.3.23) defines a bijection T(M) T*(M): q→p, and the corresponding local flow on T*(M) satisfies Hamilton’s equations. The flow has the special property of preserving the symplectic structure of T*(M), which is determined by the canonical 2-form.

The study of free particles is the foundation of kinematics, and can be used as a basis of comparison for realistic systems. The canonical flow for free particles is linear.

In this section we apply the mathematical methods that we have developed to the problems posed in (1.1.1) and (1.1.2). We begin with the trivial case of a free particle, in order to illustrate the various concepts. In other words, let M = ℝ³ and T*(M) = ℝ⁶, and choose a chart (ℝ⁶,1), calling the coordinates x_i and p_i. By (2.3.26) H has the simple form .

The theory of special relativity replaces the Galilean group with the Poincaré group. This makes the equations of motion of a particle in an external field only slightly more complicated. However, physics at high velocities looks quite different from its nonrelativistic limit.

Newton’s equations, as we know, are only an approximation, and have to be generalized to (1.1.4) or (1.1.6) when the speed of a particle approaches the speed of light. In order to solve these equations in some physically interesting cases, we first put (1.1.4) into Hamiltonian form. This means that the motion takes place in extended configuration space, which is a particular subset of ℝ⁴. We shall just concern ourselves with one-body problems, since even when there are only two bodies only special solutions are known if the interaction is relativistic—and therefore not instantaneous (cf. [12]).

In physics, space and time are defined by the way yardsticks and clocks behave, which in turn is determined by the equations of motion. It is this reasoning that gives a concrete significance to the mathematical structure of our formalism.

The first step towards a theory of relativity was the recognition that space and time are homogeneous. Homogeneity is expressed in the invariance of physical laws under spatial and temporal displacements, and implies that no point of the manifold is special. However, it is compatible with a structure in which certain directions are favored over others.

Electric and magnetic fields are dynamically interconnected in such a way that an electromagnetic disturbance propagates with a universal velocity in empty space. By studying this phenomenon we gain a qualitative understanding of field radiation and are led to expect analogous gravitational behavior.

The unification of the theories of electric and magnetic phenomena was one of the great scientific events of the nineteenth century. Whereas stationary electric fields E have sources at the positions of the charges but are irrotational (∇ × E = 0), changing magnetic fields produce circulating electromotive forces. In contrast, magnetic fields B are always sourceless and circulate around currents and places where there is a time-dependent electric field. The dynamical interrelation of the two fields is described by Maxwell’s equations: If we consider empty space (no sources or currents), then in units where c = 1 they require that .

If the field equations originate from a stationary-action principle, then a conserved current can be constructed for each parameter of an invariance group.

Field theory may be regarded as a generalization of the mechanics of point particles, in which the dynamical variables q_i(i) are replaced with fields Φ(x, t), such as E(x, t) and B(x, t). The discrete index i goes over to the continuous variable x, and, accordingly, the sum Σ_i is replaced with an integral ʃ d³x. A direct transcription of the formalism of I, §3, leads to infinite-dimensional manifolds, which we would prefer to avoid. Instead, we merely generalize the stationary-action principle (I:2.3.20) in order to find the analogues of the constants arising from the invariance properties. It is clear that in field theory the action ʃ dt L(q, q) involves an integral over a four- dimensional submanifold N4, and thus requires a 4-form, which allows the construction of a chart-independent integral.

The superconductor is a simple model of a coupled system of equations for charged matter and an electromagnetic field. As a perfect conductor and diamagnet it excludes all electric and magnetic fields from its interior.

Realistic situations do not very closely resemble the idealization discussed in the preceding chapter, where the charge distribution is prescribed. The field in turn influences the motion of the charges, so it would be more correct to analyze the coupled system. For a point-particle the analysis is subject to the difficulties encountered in §2.4. Moreover, the charge-carriers in matter, electrons and atomic nuclei, are governed by the laws of quantum mechanics, and their motion is a very complicated many-body problem. Every phenomenological description of matter is of necessity either highly idealized or else so general as to contain little information. Notwithstanding that objection, in order to formulate the ideas of this chapter mathematically, we shall single out one of the many models for a superconductor, which can be cast in a simple mathematical form. It is good enough for our purposes, as we shall always consider an extreme case in the examples, for which the charge-carriers in matter are numerous and move about freely. By responding instantaneously to any applied field, they cause the net field within the material to disappear entirely. Later, when we treat the gravitational interaction, this model will serve as our prototype of charged matter.

The covariant derivative defines the rate of change of a tensor field in the direction of a vector. Covariant derivatives in two different directions do not in general commute; their commutator determines the curvature of space.

In field theory one has to deal with derivatives of vector fields and in modern theories there appear quantities which are vectors not in space-time but in an internal space. In both cases one deals with vector bundles where vectors at different points are not canonically oriented towards each other. A chart independent notion of a derivative requires an additional structure, the so-called connection. It will be the subject of this chapter. As one hopes that eventually space, time and internal space will turn out to be only different directions in a unifying entity we start with some definitions which allow us to treat both cases in the same way.