Mathematical Principles of Theoretical Physics

290 CHAPTER 5. ELEMENTARY PARTICLES

Usually, we regardmas a static mass which is fixed, and energy is a function of velocityv.
Now, taking an opposite viewpoint, we regard energyEas fixed, massmas a function of
velocityv, and the relation (5.3.8) is rewritten as

(5.3.9) m=

√

1 −

v^2

c^2

E

c^2

.

Thus, (5.3.9) means that a particle with an intrinsic energyEhas zero massm=0 if it moves
at the speed of lightv=c, and will possess nonzero mass if it moves with a velocityv<c.
All particles including photons can only travel at the speedsufficiently close to the speed
of light. Based on this viewpoint, we can think that if a particle moving at the speed of light
(approximately) is decelerated by an interaction force~F, obeying

d~P

dt

=

√

1 −

v^2

c^2

~F,

then this massless particle will generate mass at the instant. In particular, by this mass gener-
ation mechanism, several massless particles can yield a massive particle if they are bound in
a small ball, and rotate at velocities less than the speed of light.

Interaction charges

In the unified field model introduced in the last chapter, we derived that both weak and
strong interactions possess charges, as for gravity and electromagnetism; see Section4.3.4:

(5.3.10)

gravitation: mass chargem,

electromagnetism: electric chargee,

weak interaction: weak chargegw,

strong interaction: strong chargegs.

IfΦis a charge potential corresponding to an interaction, thenthe interacting force generated
by its chargegis given by
F=−g∇Φ,

where∇is the spatial gradient operator.
It is very crucial to introduce both weak and strong interaction charges for us to develop
the weakton model. The charges in (5.3.10) possess the following physical properties:

1) Electric chargeQe, weak chargeQw, strong chargeQsare conservative. The energy

is a conserved quantity, but the massMis not a conserved quantity due to the mass

generation mechanism as mentioned in (5.3.9).

2) There is no interacting force between two particles without common charges. For

example, if a particleApossesses no strong charge, then there is no strong interacting

force betweenAand any other particles.

3) Only the electric chargeecan take both positive and negative values, and other charges

take only nonnegative values.