36 7 The Invariant Form of the Dirac Equation and Dirac Theory
2. The vector part D
II
, four equations implying eight real scalars R,β,ρ,implies
in addition the density ρ which has a probabilistic (or, following the authors,
statistical) meaning.
So one can deduce a particularity of the Dirac theory which may be extended to
the other ones: these theories may be expressed by means of equations implying the
observable physical entities, but these equations contain too many real parameters
with respect to the numbers of equations, to be solved and other equations implying
probabilistic (or statistical) parameters appear as a necessity.
About the link between equations D
I
and D
II
we have established in [2]the
following theorem:
D
II
is implied by D
I
and the three conservation relations
∂
μ
(ρv
μ
) = 0,∂
μ
(T
μν
) = ρ f
ν
,∂
μ
(S
μνξ
) = (T
ξν
− T
νξ
)
where T is the Tetrode tensor, f ∈ M the Lorentz force and S = ρv ∧σ.
In particular cases of the choice of the potential A, particular solutions of the
equation D
I
, may lead to the expression of phenomena directly observable (see
Sects. 7.4, 7.5).
7.2 The Passage from the Equation of the Electron
to the One of the Positron
The conditions of the invariance of the Dirac equation when one considers the
positron associated with an electron whose orientation of the spin is given (here
in Eq. 7.1 the one of the bivector n
2
n
1
= n
2
∧ n
1
) are well known with the stan-
dard operations on the usual presentation of the Dirac equation. These conditions are
obtained by the CPT transforms.
It is easy to obtain the CPT invariance by using Eq. 7.2.
(a) C (Charge) changes q in −q = e > 0.
(b) P (Parity) changes (e
2
, e
1
) into (e
1
, e
2
) and so n
2
n
1
into n
1
n
2
.
(c) T (Time reversion) changes e
0
into −e
0
and so v in −v.
The l eft hand part of Eq. 7.2 is unchanged by (a), (b) and (c):
– As a consequence of (b), σ
0
is changed into −σ
0
, so −q(−σ
0
) = qσ
0
and the
charge term in Eq. 7.2 is unchanged.
– As a consequence of (c), −v(−σ
0
) = vσ
0
, and the mass term in Eq. 7.2 is
unchanged.
So the right hand part of Eq. 7.2 is unchanged. The left hand part is unchanged
by any of t hese transforms.
But the T transformation seems imply that the positrons come from the future.
In order to explaining this particularity of the T transformation, where the
positrons could be considered as coming from the future, Stückelberg (1941), then