Mathematical Principles of Theoretical Physics

6.5. FIELD THEORY OF MULTI-PARTICLE SYSTEMS 391

Thus, we have two kinds of classifications (6.5.55) and (6.5.56) of sub-systems for atomic
orbital electrons, which lead to two different sets of field equations.

A. FIELD EQUATION OF SYSTEMSn. If we ignore the orbit-orbit interactions, then we
take (6.5.55) as anN-particle system. LetSnhaveKnelectrons:

(6.5.57) Sn:Ψn= (ψn^1 ,···,ψnKn), Kn≤Nn, 1 ≤n≤N,

whereNnis as in (6.5.54). Hence, the model of (6.5.57) is reduced to theSU(K 1 )× ··· ×
SU(KN)gauge fields of fermions. Referring to the single fermion system (6.5.15)-(6.5.21),
the action of (6.5.57) is

(6.5.58) L=

∫ N

∑

n= 1

(LSU(Kn)+LDn)dx,

where

(6.5.59)

LSU(Kn)=−

1

4 hc ̄

gμ αgν βAaμ νnAaα βn 1 ≤an≤Kn,

LDn=Ψn(iγμDμ−

mec

h ̄

)Ψn 1 ≤n≤N,

Aaμ νn =∂μAaνn−∂νAaμn−

e

̄hc

λbanncnAbμnAcνn,

DμΨn= (∂μ−

ie

hc ̄

A^0 μ−

ie

hc ̄

Aaμnτan)Ψn,

whereAaμnare theSU(Kn)gauge fields representing the electromagnetic(EM)potential of

the electrons inSn,λbanncnare the structure constants ofSU(Kn)such thatGanbn=^12 tr(τanτb†n) =

δanbn,A^0 μis theEMpotential generated by the nuclear,g=−e(e> 0 )is the charge of an
electron, andmeis the electron mass.
The PID gradient operators forSU(K 1 )× ··· ×SU(KN)in (6.5.19) are given by

(6.5.60) Dnμ=

1

̄hc

[

∂μ+

e

hc ̄ k∑ 6 =n

A(μk)

]

for 1≤n≤N,

whereA
(k)
μ =α

Kk

aKA

ak
μ is the total EM potential ofSkshell as defined in (6.5.10).
Then by (6.5.58)-(6.5.60), the field equations of (6.5.57) can be written in the following
form

∂νAaν μn+

e

hc ̄

(6.5.61) λbanncngα βAbα μnAcβn+eΨγμτanΨn

=

[

∂μ+

e

hc ̄ k∑ 6 =n

A(μk)

]

φan for 1≤an≤Kn^2 − 1 , 1 ≤n≤N,

iγμ

[

∂μ−

ie

hc ̄

A^0 μ−

ie

hc ̄

Aaμnτan

]

Ψn−

mec

̄h

(6.5.62) Ψn= 0.