Mathematical Principles of Theoretical Physics

6.5. FIELD THEORY OF MULTI-PARTICLE SYSTEMS 377

then from (6.4.74) we derive that

∆E= 10 −^40 ̄hc/cm= 2 × 10 −^45 eV.

This is a very small value, and it is impossible for experiments to measure.

Remark 6.23.The physical conclusion that all particles have finite and discrete energy dis-
tribution is of very important significance for the quantum field theory. It implies that the
infinity appearing in the field quantization does not exist, and the renormalization theory
needs to be reconsidered.

6.5 Field Theory of Multi-Particle Systems

6.5.1 Introduction

We start with the known model of multi-particle systems. Consider anN-particle system with
particles

(6.5.1) A 1 ,···,AN.

Letxk= (x^1 k,x^2 k,x^3 k)∈R^3 be the coordinate ofAk, and

(6.5.2) ψ=ψ(t,x 1 ,···,xk)

be the wave function describing theN-particle system (6.5.1). Then, the classical theory for
(6.5.1) is provided by the Schr ̈odinger equation

(6.5.3) ih ̄

∂ ψ

∂t

=−

N

∑

k= 1

h ̄^2

2 mk

∆kψ+∑

j 6 =k

V(xj,xk)ψ,

whereV(xj,xk)is the potential energy of interactions betweenAjandAk,mkis the mass of
Ak, and

∆k=

∂^2

(∂x^1 k)^2

+

∂^2

(∂x^2 k)^2

+

∂^2

(∂x^3 k)^2

.

The wave functionψsatisfies the normalization condition

∫

R^3

···

∫

R^3

|ψ|^2 dx 1 ···dxN= 1.

Namely, the physically|ψ(t,x 1 ,···,xN)|^2 represents the probability density ofA 1 ,···,AN
appearing atx 1 ,···,xNat timet.
It is clear that the Schr ̈odinger equation (6.5.3) for anN-particle system is only an ap-
proximate model:

• It is non-relativistic model;


• The model does not involve the vector potentials~Aof the interactions between particles.