Mathematical Principles of Theoretical Physics

4.1. PRINCIPLES OF UNIFIED FIELD THEORY 183

based on the Geometric Interaction Mechanism, which focuses directly on the four interaction
forces as in (4.1.1), and does not involve a quantization process.
A radical difference for these two mechanisms is that the Yukawa Mechanism is oriented
toward to computing the transition probability for the particle decays and scatterings, and the
Geometric Interaction Mechanism is oriented toward to fundamental laws, such as interaction
potentials, of the four interactions.

4.1.2 General introduction to unified field theory

Einstein’s unification

The aim of a unified field theory is to establish a set of field equations coupling the four
fundamental interactions. Albert Einstein was the first person who attempted to establish a
unified field theory. The basic philosophy of his unification is that all fundamental forces of
Nature should be dictated by one large symmetry group, whichcan degenerate into a sub-
symmetry for each interaction:

(4.1.9) Unification through an action under a large symmetry

In essence, with the Einstein unification, under the large symmetry, the four fundamental
forces can be regarded as one fundamental force.
Recall that there are four fundamental interactions of Nature: gravitational, electromag-
netic, strong, and weak, whose fields and actions:

1) fieldsgμ ν,Aμ,{Wμa| 1 ≤a≤ 3 }and{Skμ| 1 ≤k≤ 8 }, and

2) their actionsLEH,LEM,LWandLS,

are dictated by the symmetries in (4.1.3), as described in (4.1.5)-(4.1.8).
Basically, the Einstein unification is to search for a large symmetry, which dictates anN
component fieldG:

(4.1.10) G= (G 1 ,···,GN),

and an action

(4.1.11) L=L(G 1 ,···,GN).

The basic requirements for such a unification are as follows:Under certain conditions,

1) the large symmetry degenerates into sub-symmetries: thegeneral invariance, the Lorentz

invariance, and theU( 1 ),SU( 2 ),SU( 3 )invariance;

2) the fieldGof (4.1.10) is then decomposed into the fields of the four fundamental inter-

actions:

(4.1.12) (G 1 ,···,GN)

degenerate

−→ gμ ν,Aμ,Wμa,Skμ,