Functional Methods in Quantum Field Theory and Statistical Physics

A.N. Vasiliev

• Publisher: Routledge
• Year: 2019
• eBook ISBN: 9781351446808
• Topic: Physical Sciences
• Subtopic: Mechanics

Providing a systematic introduction to the techniques which are fundamental to quantum field theory, this book pays special attention to the use of these techniques in a wide variety of areas, including ordinary quantum mechanics, quantum mechanics in the second-quantized formulation, relativistic quantum field theory, Euclidean field theory, quant

Chapter 1

THE BASIC FORMALISM OF FIELD THEORY

1.1 Fields and Products

1.1.1 Canonical quantization

In classical physics the dynamics of a system with a finite number of degrees of freedom are determined by the Lagrangian ℒ, which is given as a function of the generalized coordinates q = {q₁…q_n} and velocities ${\dot{q}}_i\equiv {\partial }_t{q}_i$ . The classical equations of motion are obtained by requiring that the action functional S = ∫ dtℒ(t) be stationary.

The classical system is usually quantized using the canonical formalism, i.e., in the language of coordinates and momenta. The momenta p_i conjugate to the coordinates q_i are given by $p_i=\partial ℒ/\partial {\dot{q}}_i$ . If these relations can be solved uniquely for the velocities, it is said that the Lagrangian is nondegenerate. In this case the classical Hamiltonian $\mathscr{H}=\text{Σ}{p}_i{\dot{q}}_{.i}-ℒ$ can be viewed as a known function of the coordinates and momenta.

In going to the quantum theory the coordinates and momenta are realized as linear Hermitian operators ${\widehat{q}}_i$ and ${\widehat{p}}_i$ acting in some Hilbert space and satisfying the canonical commutation relations ${\widehat{q}}_s{\widehat{p}}_m-{\widehat{p}}_m{\widehat{q}}_s=i{\text{δ}}_{sm}$ . The quantum analog of a classical observable F(p, q) is the operator $F\left(\widehat{p},\widehat{q}\right)$ ; in particular, the quantum energy operator, the Hamiltonian H, is by definition $\mathscr{H}\left(\widehat{p},\widehat{q}\right)$ . These rules for going from the classical theory to the quantum theory are the well known recipe for canonical quantization.

A function of noncommuting operators $\mathscr{H}\left(\widehat{p},\widehat{q}\right)$ is not defined uniquely even when the natural additional requirement of Hermiticity is imposed. First of all, there is an arbitrariness in the ordering of the factors $\widehat{p}$ and $\widehat{q}$ , i.e., an arbitrariness in the choice of how the classical Hamiltonian ℋ(p, q) is written. For example, the identical classical functions p² + q² and (p + iq)(p − iq) correspond to different operators when quantized. This implies that, in general, a given classical...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Foreword
• Preface
• Chapter 1 THE BASIC FORMALISM OF FIELD THEORY
• Chapter 2 SPECIFIC SYSTEMS
• Chapter 3 THE MASSLESS YANG-MILLS FIELD
• Chapter 4 EUCLIDEAN FIELD THEORY
• Chapter 5 STATISTICAL PHYSICS
• Chapter 6 VARIATIONAL METHODS AND FUNCTIONAL LEGENDRE TRANSFORMS
• Appendix 1 NONSTATIONARY PERTURBATION THEORY FOR A DISCRETE LEVEL
• Appendix 2 GRAPHS AND SYMMETRY COEFFICIENTS
• REFERENCES
• SUBJECT INDEX