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Physical Review

Feynman Diagrams Without Field Theory

Asher Peres

Published Feb 8, 1965• 2 citations
Publisher

Abstract

The purpose of this note is to present a "generalized quantum mechanics" in which the number of particles is not given a priori in each problem, and, in particular, is not a constant of the motion. The quantum-mechanical Hilbert space thus contains states with 0, 1, 2,... particles. The interaction picture and the $S$ matrix are introduced as usual. It is shown (assuming relativistic invariance) that the calculation of $S$-matrix elements can be performed by means of graphical rules identical to those of Feynman.

DOI
10.1103/physrev.137.b696

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The purpose of this note is to present a "generalized quantum mechanics" in which the number of particles is not given a priori in each problem, and, in particular, is not a constant of the motion. The quantum-mechanical Hilbert space thus contains states with 0, 1, 2,... particles. The interaction picture and the SSS matrix are introduced as usual. It is shown (assuming relativistic invariance) that the calculation of SSS-matrix elements can be performed by means of graphical rules identical to those of Feynman.
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Scattering processes are a standard topic covered in introductory courses on quantum mechanics and particle physics. Unfortunately, a full mathematical treatment tends to be overwhelming for undergraduate students. This article introduces some toy models that are easy to comprehend but still contain the essential features of quantum theory. We define a Hilbert space with state vectors and use creation/annihilation operators to construct transition matrices and S-matrices. We show how perturbation theory gives rise to Feynman diagrams and Feynman rules. We also discuss how we can use symmetry and group theory to restrict what interactions are possible.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
Basically, there are two approaches to the formulation of quantum mechanics, i.e., the operator and the Feynman path integral approach. The main object in quantum mechanics is referred to as the transition amplitude between those states, matrix elements of physical operators and, in particular, the S- matrix in the problem of quantum scattering. On the basis of quantum mechanics, the operator method is closed, well formulated mathematically, and so is an appropriate universal language describing quantum field theory, the quantum physics of condensed states and quantum relativistic statistical physics. The differential equation defining the path of the system in the configuration space is called the equation of motion of the system. The aim of classical as well as quantum mechanics is to predict the time evolution of the system, starting from knowledge of the state of the system at a given point in time.