Mathematical Prerequisites

Essential Mathematics for QFT

Mathematical foundations required for quantum field theory

Overview

Quantum Field Theory requires a solid foundation in advanced mathematics. This chapter reviews the essential mathematical concepts you'll need throughout this course. For more detailed treatments, see the dedicatedMathematics sectionof this site.

What You Should Know

Linear algebra (vectors, matrices, eigenvalues)
Multivariable calculus and vector calculus
Basic complex analysis
Differential equations (ODEs and PDEs)
Classical mechanics (Lagrangian and Hamiltonian formulations)
Quantum mechanics (at undergraduate level)
Special relativity (4-vectors, Lorentz transformations)

1. Special Relativity & Lorentz Covariance

Minkowski Spacetime

Spacetime is a 4-dimensional manifold with metric signature (+,-,-,-):

$$g_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}$$

4-Vectors

Contravariant 4-vector: x^μ = (t, x, y, z) or (x⁰, x¹, x², x³)

Covariant 4-vector: x_μ = g_μνx^ν = (t, -x, -y, -z)

$$x_\mu = (x_0, x_1, x_2, x_3) = (t, -x, -y, -z)$$

Index Notation

Greek indices (μ, ν, ...) run from 0 to 3 (spacetime)
Latin indices (i, j, k) run from 1 to 3 (space only)
Einstein summation: repeated indices are summed
Raising/lowering: A^μ = g^μνA_ν

Important 4-Vectors

Position:

$$x^\mu = (t, \vec{x})$$

4-Momentum:

$$p^\mu = (E, \vec{p})$$

4-Derivative:

$$\partial_\mu = \frac{\partial}{\partial x^\mu} = \left(\frac{\partial}{\partial t}, \nabla\right)$$

Lorentz Invariants

Quantities unchanged under Lorentz transformations:

$$s^2 = x_\mu x^\mu = t^2 - \vec{x}^2$$

$$m^2 = p_\mu p^\mu = E^2 - \vec{p}^2$$

2. Tensor Analysis

Tensor Basics

A tensor is a multilinear map. In component notation:

Scalar (rank 0): φ
Vector (rank 1): V^μ
Rank 2: T^μν, T_μν, T^μ_ν
Rank n: T^{μ₁...μₙ}

Tensor Operations

Contraction:

$$T^\mu_{\phantom{\mu}\mu} = \sum_{\mu=0}^3 T^\mu_{\phantom{\mu}\mu}$$

Outer Product:

$$(A \otimes B)^{\mu\nu} = A^\mu B^\nu$$

Symmetrization:

$$T^{(\mu\nu)} = \frac{1}{2}(T^{\mu\nu} + T^{\nu\mu})$$

Antisymmetrization:

$$T^{[\mu\nu]} = \frac{1}{2}(T^{\mu\nu} - T^{\nu\mu})$$

Important Tensors in QFT

Metric tensor: g_μν
Field strength: F^μν = ∂^μA^ν - ∂^νA^μ
Energy-momentum tensor: T^μν
Levi-Civita symbol: ε^μνρσ

3. Calculus of Variations

Action Principle

The dynamics of a system is determined by extremizing the action:

$$S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt$$

Euler-Lagrange Equations

For discrete systems (particles):

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0$$

For continuous systems (fields):

$$\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} = 0$$

Functional Derivatives

The functional derivative measures how a functional changes with variations of a function:

$$\frac{\delta F[\phi]}{\delta \phi(x)} = \lim_{\epsilon \to 0} \frac{F[\phi + \epsilon \delta(x-y)] - F[\phi]}{\epsilon}$$

Key property:

$$\frac{\delta \phi(x)}{\delta \phi(y)} = \delta^{(4)}(x-y)$$

4. Complex Analysis

Complex Functions

Complex numbers z = x + iy, with conjugate z* = x - iy

$$|z|^2 = z z^* = x^2 + y^2$$

Contour Integration

Cauchy's residue theorem:

$$\oint_C f(z) \, dz = 2\pi i \sum \text{Residues}$$

Essential for computing Feynman integrals and propagators in QFT.

Analytic Continuation

Used in Wick rotation: t → -iτ to go from Minkowski to Euclidean space.

5. Group Theory & Lie Algebras

Groups

A group (G, ·) satisfies:

Closure: g₁, g₂ ∈ G ⇒ g₁ · g₂ ∈ G
Associativity: (g₁ · g₂) · g₃ = g₁ · (g₂ · g₃)
Identity: ∃ e ∈ G such that e · g = g · e = g
Inverse: ∀g ∈ G, ∃ g⁻¹ such that g · g⁻¹ = e

Important Groups in QFT

U(1): Electromagnetic gauge group

Phase transformations: ψ → e^iαψ

SU(2): Weak isospin

2×2 unitary matrices with det = 1

SU(3): Color gauge group (QCD)

3×3 unitary matrices with det = 1

Lorentz group SO(1,3)

Spacetime symmetries

Poincaré group

Lorentz + translations

Lie Algebras

Infinitesimal generators T^a satisfy commutation relations:

$$[T^a, T^b] = if^{abc}T^c$$

where f^abc are the structure constants.

6. Distributions & Delta Functions

Dirac Delta Function

The Dirac delta δ(x) is a distribution satisfying:

$$\int_{-\infty}^{\infty} f(x) \delta(x-a) \, dx = f(a)$$

Properties

δ(x) = 0 for x ≠ 0
∫ δ(x) dx = 1
δ(-x) = δ(x)
δ(ax) = δ(x)/|a|
xδ(x) = 0

4D Delta Function

$$\delta^{(4)}(x-y) = \delta(x^0-y^0)\delta(x^1-y^1)\delta(x^2-y^2)\delta(x^3-y^3)$$

Fourier Representation

$$\delta^{(4)}(x-y) = \int \frac{d^4p}{(2\pi)^4} e^{-ip \cdot (x-y)}$$

Crucial for defining propagators and Green's functions in QFT.

Mathematical Toolkit Summary

✓ Special Relativity: 4-vectors, Lorentz invariance, Minkowski metric
✓ Tensor Analysis: Index notation, contraction, symmetrization
✓ Variational Calculus: Action principles, Euler-Lagrange equations
✓ Complex Analysis: Contour integration, residues, analytic continuation
✓ Group Theory: U(1), SU(N), Lorentz group, Lie algebras
✓ Distributions: Delta functions, Fourier transforms

For deeper exploration of these topics, visit theMathematics section.

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