37 Multi-particle Systems and Field Quantization

The multi-particle formalism developed in chapter 36 is based on the idea of taking as dual phase space the space of solutions to the free particle Schr¨odinger equation, then quantizing using the Bargmann-Fock method. Continuous basis elements $A(p),\overline{A}(p)$ are momentum operator eigenstates which after quantization become creation and annihilation operators $a^{†}(p),a(p)$ . These act on the multi-particle state space by adding or subtracting a free particle of momentum $p.$

Instead of the solutions $A(p)$ , which are momentum space delta-functions localized at $p$ at $t=0$ , one could use solutions $\Psi (x)$ that are position space delta-functions localized at x at $t=0$ . Quantization using these basis elements of ${\mathcal{H}}_1$ (and conjugate basis elements of ${\overline{\mathcal{H}}}_1)$ will give operators ${\hat{\Psi }}^{†}(x)$ and $\hat{\Psi }(x)$ 2 which are the quantum field operators. These are related to the operators $a^{†}(p),a(p)$ by the Fourier transform.

The operators $\hat{\Psi }(x)$ and ${\hat{\Psi }}^{†}(x)$ can be given a physical interpretation as acting on states by subtracting or adding a particle localized at x. Such states with localized position have the same problem of non-normalizability as momentum eigenstates. In addition, unlike states with fixed momentum, they are not stable energy eigenstates so will immediately evolve into something non-localized (just as in the single-particle case discussed in section 12.5). Quantum fields are however very useful in the study of theories of interacting particles, since the interactions in such theories are typically local, taking place at a point $x$ and describable by adding terms to the Hamiltonian operator involving multiplying field operators at the same point $x$. The dificulties involved in properly defining products of these operators and calculating their dynamical efects will keep us from starting the study of such interacting theories.

中文目录

Chapter contents

• 37.1 Quantum field operators
• 37.2 Quadratic operators and dynamics
• 37.3 The propagator in non-relativistic quantum field theory
• 37.4 Interacting quantum fields
• 37.5 Fermion fields
• 37.6 For further reading

Previous chapter: 36 Multi-particle Systems: Momentum Space Description

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 394、395、396、397、398、399、400、401、402、403

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：6347d8c41762e939255697161b57ae546648e7995817037af878b299b5717877

OCR 产物 SHA-256：6347d8c41762e939255697161b57ae546648e7995817037af878b299b5717877