3 The Dirac Electron and Basic Physical Concepts
55
some internal motion, involving a ‘momentum’ m 0 c, within a ‘ball’ of width 2r C
determined by the Heisenberg uncertainty relation:
m 0 c · 2r C ∼ → 2r C ∼ /m 0 c = λ C /2π.
(3.4)
This internal motion was identified by Schrödinger [11, 12] as a ‘trembling motion’ (Zitterbewegung), which vanishes when one takes expectation values over
wave packets made up entirely of positive (or negative) energy solutions of the
Dirac equation. This was understood by de Broglie [6] as resulting from a wave
beat between the positive and negative energy states: ±mc 2 , the beat frequency being the difference of the two wave frequencies: ν e = 2mc 2 /h. The amplitude of this
internal motion was shown to be precisely 2r C .
Various authors have seen the electron as a point charge oscillating at velocity
c within a ball of radius r C [13–16, 18–21], and identified the orbital momentum
of this motion to its spin momentum [17–21]. In a previous paper [22], we have
conjectured that the rest mass involved in external motions (inertia) and interactions (gravitation) results from this very internal motion. In the present paper, we
reassess our previous discussion and see how this may lead to some insights into
basic physical concepts.
3.2 The Dirac Equation and the Electron Internal Motion
Today, the Dirac equation can be deduced from more general theoretical frameworks [7, 9, 23–26]. But the inductive derivation originally given by Dirac [2–5]
has shown great heuristic value, and we shall summarize it with only a few notation
changes. He started with the time-dependent wave equation for a free particle in the
Schrödinger representation, the Hamiltonian being given the relativistic expression,
H = mc 2 :
i∂Ψ/∂(ct) = mcΨ,
(3.5)
with the expanded form for mc:
mc =
m
2
0 c
2
+ p
2
1/2 .
(3.6)
Here, p 2 = p 2
1 + p 2
2 + p 2
3 with p i = mv i along x i , and m = m 0 γ . Dimensionwise, one can define an overall ‘momentum’ p 4 ≡ mc corresponding, according
to Eq. (3.5), to the time coordinate x 4 ≡ ct, and an invariant ‘momentum’ p 0 ≡ m 0 c
for a particle at rest. With these notations, Eq. (3.6) can be rewritten:
p
2
4 = p
2
0 + p
2
1 + p
2
2 + p
2
3 .
(3.7)
This expression for the invariant (rest mass) ‘momentum’ p 0 is similar to that for
the invariant (proper interval) ‘coordinate’ x 0 :
x
2
4 ≡ x
2
0 + x
2
1 + x
2
2 + x
2
3 .
(3.8)
The Minkowski 4-D relativistic space-time has a Lorentz (non-Euclidean) hyperbolic metric. However, as x 4 ≡ ct appears as a Pythagorean sum of the three x i ’s
55
some internal motion, involving a ‘momentum’ m 0 c, within a ‘ball’ of width 2r C
determined by the Heisenberg uncertainty relation:
m 0 c · 2r C ∼ → 2r C ∼ /m 0 c = λ C /2π.
(3.4)
This internal motion was identified by Schrödinger [11, 12] as a ‘trembling motion’ (Zitterbewegung), which vanishes when one takes expectation values over
wave packets made up entirely of positive (or negative) energy solutions of the
Dirac equation. This was understood by de Broglie [6] as resulting from a wave
beat between the positive and negative energy states: ±mc 2 , the beat frequency being the difference of the two wave frequencies: ν e = 2mc 2 /h. The amplitude of this
internal motion was shown to be precisely 2r C .
Various authors have seen the electron as a point charge oscillating at velocity
c within a ball of radius r C [13–16, 18–21], and identified the orbital momentum
of this motion to its spin momentum [17–21]. In a previous paper [22], we have
conjectured that the rest mass involved in external motions (inertia) and interactions (gravitation) results from this very internal motion. In the present paper, we
reassess our previous discussion and see how this may lead to some insights into
basic physical concepts.
3.2 The Dirac Equation and the Electron Internal Motion
Today, the Dirac equation can be deduced from more general theoretical frameworks [7, 9, 23–26]. But the inductive derivation originally given by Dirac [2–5]
has shown great heuristic value, and we shall summarize it with only a few notation
changes. He started with the time-dependent wave equation for a free particle in the
Schrödinger representation, the Hamiltonian being given the relativistic expression,
H = mc 2 :
i∂Ψ/∂(ct) = mcΨ,
(3.5)
with the expanded form for mc:
mc =
m
2
0 c
2
+ p
2
1/2 .
(3.6)
Here, p 2 = p 2
1 + p 2
2 + p 2
3 with p i = mv i along x i , and m = m 0 γ . Dimensionwise, one can define an overall ‘momentum’ p 4 ≡ mc corresponding, according
to Eq. (3.5), to the time coordinate x 4 ≡ ct, and an invariant ‘momentum’ p 0 ≡ m 0 c
for a particle at rest. With these notations, Eq. (3.6) can be rewritten:
p
2
4 = p
2
0 + p
2
1 + p
2
2 + p
2
3 .
(3.7)
This expression for the invariant (rest mass) ‘momentum’ p 0 is similar to that for
the invariant (proper interval) ‘coordinate’ x 0 :
x
2
4 ≡ x
2
0 + x
2
1 + x
2
2 + x
2
3 .
(3.8)
The Minkowski 4-D relativistic space-time has a Lorentz (non-Euclidean) hyperbolic metric. However, as x 4 ≡ ct appears as a Pythagorean sum of the three x i ’s