Theoretical and Mathematical Physics

The following sections are included:

• Preface
• Contents
• Notation

Arithmetic series, geometric series and harmonic series play a central role in problems in theoretical and mathematical physics.

An arithmetic series is the sum of a sequence {s_k}, k = 0, 1, … in which each term is calculated from the previous one by adding (or subtracting) a constant c. This means s_k = s_k−1 + c = s_k−2 + 2c = … = s₀ + c · k, where k = 1, 2, …. In particular one has (n ≥ 1)

Let X, Y be two non-empty sets. Then a map of X into Y is a function f with X as domain and Y as codomain, i.e. f maps X into Y . We write f : X → Y . With x ∊ X and f(x) = y ∊ Y y is called the value of the function f corresponding to the argument x. If f is a map of X into Y we say that f is surjective (onto) when f(X) = Y ; injective (one-one) when for all x, x′ ∊ X, f(x) = f(x′) implies x = x′; bijective (one-one, onto) when it is both injective and surjective…

A complex polynomial is a complex function of the form

Let 𝔽 be a field, for example the set of real numbers ℝ or the set of complex numbers ℂ. Let m, n ≥ 1 be two integers. An array A of numbers in 𝔽

Let A be an n × n matrix over ℂ. The eigenvalue equation is given by

A complex number z can be written as

Let f : [a, b] ↦ ℝ and let g be a function such that dg/dx = f(x) for all x ∊ [a, b]. Then (Newton’s integral)

In a normed vector space V, for example ℂⁿ, the triangular inequality is

Let M be a smooth manifold and f be a real valued function of class C⁽¹⁾ on some open set containing M. We consider the problem of finding the extrema of the function f|M. This is called a problem of constrained extrema…

Consider the system of first order differential equations

Let (x₁, …, x_m) ∊ ℝ^m (m ≥ 2) be the independent variables and (u₁, …, u_n) be the dependent variables. So one can consider the system of partial differential equations (i = 1, …, m; j = 1, …, n; k = 1, …, p)…

A Hilbert space is a set, ℋ of elements, or vectors, (f, g, h, …) which satisfies the following conditions (1) to (5).

(1) If f and g belong to ℋ, then there is a unique element of ℋ, denoted by f + g, the operation of addition (+) being invertible, commutative and associative.

(2) If c is a complex number, then for any f in ℋ, there is an element cf of ℋ; and the multiplication of vectors by complex numbers thereby defined satisfies the distributive conditions

(3) Hilbert spaces ℋ possess a zero element, 0, characterized by the property that 0 + f = f for all vectors f in ℋ.

(4) For each pair of vectors f, g in ℋ, there is a complex number 〈f|g〉, termed the inner product or scalar product of f with g, such that

(5) If {f_n} is a sequence in ℋ satisfying the Cauchy condition that

Legendre, Hermite, Laguerre and Chebyshev polynomials play an important role in mathematical physics, for example as an orthonormal basis in the Hilbert space of square integrable functions. Jacobi elliptic functions and Weierstrass elliptic functions play a role in the solution of nonlinear differential equations. The Legendre polynomials are given by the Rodrigue’s formula

The vector space D(ℝⁿ) denotes all infinitely-differentiable functions in ℝ with compact support. The functions ϕ ∊ D(ℝⁿ) are called test functions. Each linear continuous functional over the vector space of test functions D(ℝⁿ) is called a generalized function…

A group G is a set of objects { a, b, c, … } (not necessarily countable) together with a binary operation which associates with any ordered pair of elements a, b in G a third element ab in G (closure). The binary operation (called group multiplication) is subject to the following requirements:

• There exists an element e in G called the identity element such that eg = ge = g for all g ∊ G.
• For every g ∊ G there exists an inverse element g⁻¹ in G such that gg⁻¹ = g⁻¹g = e.
• Associative law. The identity (ab)c = a(bc) is satisfied for all a, b, c ∊ G.

If ab = ba for all a, b ∊ G we call the group commutative. If G has a finite number of elements it has finite order n(G), where n(G) is the number of elements. Otherwise, G has infinite order…

In combinatorics we study the enumeration, combination, and permutations of sets of elements. Furthermore finite or countable discrete structures (for example finite sets, graphs) are studied. Enumerative combinatorics focuses on counting the number of certain combinatorial objects. A bijection of a finite set S onto itself is called a permutation of S. If S consist of n elements then there are n! possible permutations. The fundamental principle of counting tells us that if one event has j ∊ ℕ possible outcomes and a second independent event has k ∊ ℕ possible outcomes, then the number of possible outcomes for the combined events is given by j · k. Let S be a finite set with n elements. Then the number of subsets is 2ⁿ which includes the set S itself and the empty set. Let n ≥ 1, S be a set containing n elements and n₁, …, n_r be positive integers with n₁+ n₂+ … + n_r = n. Then there exist

In number theory we are dealing with positive integers ℕ, non-negative integers ℕ₀, integers ℤ, rational numbers ℚ, real numbers ℝ and complex numbers ℂ, where ℕ, ℕ₀, ℤ, ℚ are countable. We have

Let A be an m × n matrix and let B be a p × q matrix. The Kronecker product of A and B is defined as

Let W₁ be some topological vector space and f a (nonlinear) mapping f : W → W₁. We call f Gateaux differentiable in u ∊ W if there exists a mapping θ ∊ L(W, W₁) such that for all v ∊ W

Soliton equations are special partial differential equations. They have a number of interesting properties such as a Lax representation, solvable by the inverse scattering method, infinite many conservation laws. They also admit Bäcklund transformations and Lie-Bäcklund symmetries. Most of them also pass the Painlevé test. Lax representations can also be found for some system of ordinary differential equations…

The Hirota technique plays a central role in finding solutions to soliton equations. Let f, g be smooth functions. The Hirota bilinear operators D_x and D_t are defined as

A Lie algebra is defined as follows: A vector space L over a field F, with an operation L × L → L denoted by (x, y) → [x, y] and called the commutator of x and y, is called a Lie algebra over F if the following axioms are satisfied:

• (L1) The bracket operation is bilinear:
• (L2) [x, x] = 0 for all x ∊ L
• (L3) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (x, y, z ∊ L).

Axiom (L3) is called the Jacobi identity…

We define differential p-forms of class C^∞ on an open set Ω of ℝⁿ to be the expressions

Let G be the gravitational constant and c be the speed of light in vacuum. The Einstein equation are given by

Let M be an n-dimensional C^∞ differentiable manifold with local coordinates x_j, j = 1, …, n. Real valued C^∞ vector fields and real valued C^∞ differential forms on M can be considered. The components of the vector field V are denoted by V_j∂/∂x_j . This means

Let S₁, S₂, S₃ be the spin matrices for spin

Elementary particles can be classified in two categories according to their value of the spin s. Half-integer spin particles (such as the electron and positron) obey Fermi-Dirac statistics and are called Fermions whereas integerspin particles (such as the photon and graviton) obey Bose-Einstein statistics and are known as Bosons…

Gauge theories study invariance of a field theory with Lagrange function or Lagrange density under internal global symmetry transformations of a Lie symmetry group G which are extended to transformations which are extended to transformations which can be different at different space-time points. So global Lie symmetry groups are independent of the space time point. Local or gauge transformations depend on (x₀, x₁, x₂, x₃) (x₀ = ct). If a set of physical laws is invariant under a global Lie symmetry transformation, the requirement of invariance under gauge Lie symmetries demands the introduction of new fields, the so-called gauge fields. The simplest case is the Lie group U(1). In most cases the starting point is the Lagrange function or Lagrange density…

The following sections are included:

• Bibliography
• Index