Mathematical Principles of Theoretical Physics

4.1. PRINCIPLES OF UNIFIED FIELD THEORY 181

We know now that the four interactions are dictated respectively by the following sym-
metry principles:

(4.1.3)

gravity: principle of general relativity,

electromagnetism: U( 1 )gauge invariance,

weak interaction: SU( 2 )gauge invariance,

strong interaction: SU( 3 )gauge invariance.

The last three interactions also obey the Lorentz invariance. As a natural consequence, the
three chargese,gw,gsin (4.1.2) are the coupling constants of theU( 1 ),SU( 2 ),SU( 3 )gauge
fields.
Following the simplicity principle of laws of Nature as stated in Principle2.2, the three ba-
sic symmetries—the Einstein general relativity, the Lorentz invariance and the gauge invariance—
uniquely determine the interaction fields and their Lagrangian actions for the four interac-
tions:

1.Gravity. The gravitational fields are the Riemannian metric defined onthe space-time
manifoldM:

(4.1.4) ds^2 =gμ νdxμdxν,

and then second-order tensor{gμ ν}stands for the gravitational potential. The Lagrangian
action for the metric (4.1.4) is the Einstein-Hilbert functional

(4.1.5) LEH=R+

8 πG

c^4

S,

whereRstands for the scalar curvature of the tangent bundleTMofM.

2.Electromagnetism.The field describing electromagnetic interaction is theU( 1 )gauge
field
Aμ= (A 0 ,A 1 ,A 2 ,A 3 ),

representing the electromagnetic potential, and the Lagrangian action is

(4.1.6) LEM=−

1

4

Aμ νAμ ν,

which stands for the scalar curvature of the vector bundleM⊗pC^4 , with

Aμ ν=∂μAν−∂νAμ.

being the curvature tensor.

3.Weak interaction. The weak fields are theSU( 2 )gauge fields

Wμa= (W 0 a,W 1 a,W 2 a,W 3 a) for 1≤a≤ 3 ,

and their action is

(4.1.7) LW=−

1

4

GabwWμ νaWμ νb,