56
J. Maruani
plus x 0 , and p 4 ≡ mc of the three p i ’s plus p 0 , one may wonder if the Lorentzinvariant x 0 and p 0 do not point to some wrapped dimension related to a different
metric. This is not to be confused with the fifth dimension designed to unify gravitation and magnetism in Kaluza-Klein theories [27], which pointed the way to string
theories.
The Dirac equation was derived in several steps [5]. By analogy, Dirac first wrote:
p i → −i∂/∂x i (i = 1, 2, 3) and p 4 → i∂/∂(ct).
(3.9)
One notices that there is no ‘coordinate’ derivative associated with the fifth, combined, invariant ‘momentum’ p 0 . It has been proposed [26] to formally assign to
this rest-mass momentum a combined operator: id/d(ct) = i[∂/∂(ct) + α.∇].
But this amounts to a tautological reformulation of Eq. (3.11), not to a definition of
a specific fifth dimension.
Substituting Eq. (3.7) into Eq. (3.5) with the generalized momenta p μ replaced
by their respective operators from Eq. (3.9) yields:
p 4 −
p
2
0 + p
2
1 + p
2
2 + p
2
3
1/2
Ψ = 0,
(3.10)
which is linear in p 4 but not in the other p i ’s. Multiplying on the left side by the
conjugate expression yields an equation that is symmetric in all p μ ’s but not linear
in p 4 . Dirac’s feat was to design a relativistic wave equation that was both symmetric
and linear:
p 4 − (α 0 p 0 + α 1 p 1 + α 2 p 2 + α 3 p 3 )
Ψ = 0.
(3.11)
In order to yield the same solutions as Eq. (3.10), the α μ ’s must be 4-D matrices
commuting with the four p μ ’s and satisfying, for μ, ν = 0, 1, 2, 3, the relations:
α
2
μ = 1,
α μ α ν + α ν α μ = 0.
(3.12)
In the original, most common representation of the Dirac 4-D matrices α μ , the Pauli
2-D matrices σ i are used as off-diagonal elements.
A result is that Ψ must be a four-valued wave function or a four-component state
vector. It was already known that, in order to satisfy Pauli’s antisymmetry condition, Ψ had to be endowed with a two-valued internal dynamical variable, which
Dirac interpreted as being the spin angular momentum. But he also discovered that
this number must be doubled because Eq. (3.11) has additional, negative energy
solutions, which he assigned to an ‘antielectron’ with opposite charge [2–5]. The
entanglement of the four components of Ψ when Eq. (3.11) is written in the explicit
form of four coupled equations [6] shows that spin itself is related to the negativeenergy states.
The electron spin first entered quantum mechanics through an intrinsic magnetic
moment interacting with an external field. To have the electron magnetic moment
show up, Dirac made it interact with an external field. And to have its spin momentum appear, he made it combine with an orbital momentum. Equation (3.11) was
thus extended to include interactions with an electromagnetic field, with scalar and
vector potentials A 4 and A:
(p 4 + eA 4 /c) − α 0 p 0 − α.(p + eA)
Ψ = 0.
(3.13)
J. Maruani
plus x 0 , and p 4 ≡ mc of the three p i ’s plus p 0 , one may wonder if the Lorentzinvariant x 0 and p 0 do not point to some wrapped dimension related to a different
metric. This is not to be confused with the fifth dimension designed to unify gravitation and magnetism in Kaluza-Klein theories [27], which pointed the way to string
theories.
The Dirac equation was derived in several steps [5]. By analogy, Dirac first wrote:
p i → −i∂/∂x i (i = 1, 2, 3) and p 4 → i∂/∂(ct).
(3.9)
One notices that there is no ‘coordinate’ derivative associated with the fifth, combined, invariant ‘momentum’ p 0 . It has been proposed [26] to formally assign to
this rest-mass momentum a combined operator: id/d(ct) = i[∂/∂(ct) + α.∇].
But this amounts to a tautological reformulation of Eq. (3.11), not to a definition of
a specific fifth dimension.
Substituting Eq. (3.7) into Eq. (3.5) with the generalized momenta p μ replaced
by their respective operators from Eq. (3.9) yields:
p 4 −
p
2
0 + p
2
1 + p
2
2 + p
2
3
1/2
Ψ = 0,
(3.10)
which is linear in p 4 but not in the other p i ’s. Multiplying on the left side by the
conjugate expression yields an equation that is symmetric in all p μ ’s but not linear
in p 4 . Dirac’s feat was to design a relativistic wave equation that was both symmetric
and linear:
p 4 − (α 0 p 0 + α 1 p 1 + α 2 p 2 + α 3 p 3 )
Ψ = 0.
(3.11)
In order to yield the same solutions as Eq. (3.10), the α μ ’s must be 4-D matrices
commuting with the four p μ ’s and satisfying, for μ, ν = 0, 1, 2, 3, the relations:
α
2
μ = 1,
α μ α ν + α ν α μ = 0.
(3.12)
In the original, most common representation of the Dirac 4-D matrices α μ , the Pauli
2-D matrices σ i are used as off-diagonal elements.
A result is that Ψ must be a four-valued wave function or a four-component state
vector. It was already known that, in order to satisfy Pauli’s antisymmetry condition, Ψ had to be endowed with a two-valued internal dynamical variable, which
Dirac interpreted as being the spin angular momentum. But he also discovered that
this number must be doubled because Eq. (3.11) has additional, negative energy
solutions, which he assigned to an ‘antielectron’ with opposite charge [2–5]. The
entanglement of the four components of Ψ when Eq. (3.11) is written in the explicit
form of four coupled equations [6] shows that spin itself is related to the negativeenergy states.
The electron spin first entered quantum mechanics through an intrinsic magnetic
moment interacting with an external field. To have the electron magnetic moment
show up, Dirac made it interact with an external field. And to have its spin momentum appear, he made it combine with an orbital momentum. Equation (3.11) was
thus extended to include interactions with an electromagnetic field, with scalar and
vector potentials A 4 and A:
(p 4 + eA 4 /c) − α 0 p 0 − α.(p + eA)
Ψ = 0.
(3.13)