54
J. Maruani
However, further developments in quantum mechanics (both in the Schrödinger
and Heisenberg representations) used classical kinetic and potential energy expressions (with spin added as an ansatz), and were not Lorentz invariant. Contrary to
the equations for the electric and magnetic components of the light waves, the
Schrödinger equation for matter waves contains a first-order time derivative with
second-order space derivatives. A relativistic energy was used in the Klein-Gordon
equation, bringing time and space on the same footing but in a quadratic form which,
while being compatible with the quantum superposition principle, led to a nondefinite positive probability density and proved unsuitable for the description of
particles endowed with rest mass and spin momentum.
Dirac made a breakthrough [2–5] by designing a relativistic equation which was
linearized by introducing anticommutative 4-D matrices, which he expressed in
terms of 2-D Pauli matrices. This implied 4-component state vectors or 4-valued
wave functions with double-valued spin and mass as extra coordinates. It was noticed by de Broglie [6] that the process leading from the Klein-Gordon equation
to the Dirac equation is similar to that leading from the second-order equations for
the electric and magnetic fields E and B to the four coupled, first-order, Lorentzinvariant Maxwell equations for the scalar and vector potentials A 4 and A.
Further consistency was reached by quantizing the electromagnetic field, which
led to quantum electrodynamics [7] for the electron and other leptons. This served
as a model to quantum chromodynamics [8] for particles involving quarks, such as
nucleons and other baryons. Modern quantum field theory [9] encompasses the various cases. In these theories, the entanglement of matter and antimatter is expressed
by the necessity to include particles and antiparticles on the same footing to cope
with infinities.
De Broglie’s wavelength λ B was an outcome of a (theoretical) encounter of light
and matter. Another outcome of a (physical) encounter of light and matter was
Compton’s wavelength λ C [10]. When X-rays hit electrons (relatively) at rest, the
wavelength λ 2 of a scattered photon differs from that λ 1 of the incident photon by a
value: λ 2 − λ 1 = λ C (1 − cos θ), θ being the angle between the two X-rays and λ C
being given by:
λ C = h/m 0 c,
(3.2)
where m 0 is the rest mass of the electron and c, the velocity of light.
From the definitions of λ B and λ C it appears that the former depends on the
particle velocity v (and may thus vary from ∞ to 0) while the second depends solely
on its rest mass (and universal constants). The relation between the two involves the
Lorentz ‘boost’ transformation factor:
λ C /λ B = βγ,
(3.3)
where β ≡ v/c and γ ≡ (1 − β 2 ) −1/2 are coefficients of the Lorentz proper transformations.
De Broglie’s wavelength is linked to the external motion, with momentum mv,
of the particle wave packet, whereas Compton’s wavelength seems to be linked to
J. Maruani
However, further developments in quantum mechanics (both in the Schrödinger
and Heisenberg representations) used classical kinetic and potential energy expressions (with spin added as an ansatz), and were not Lorentz invariant. Contrary to
the equations for the electric and magnetic components of the light waves, the
Schrödinger equation for matter waves contains a first-order time derivative with
second-order space derivatives. A relativistic energy was used in the Klein-Gordon
equation, bringing time and space on the same footing but in a quadratic form which,
while being compatible with the quantum superposition principle, led to a nondefinite positive probability density and proved unsuitable for the description of
particles endowed with rest mass and spin momentum.
Dirac made a breakthrough [2–5] by designing a relativistic equation which was
linearized by introducing anticommutative 4-D matrices, which he expressed in
terms of 2-D Pauli matrices. This implied 4-component state vectors or 4-valued
wave functions with double-valued spin and mass as extra coordinates. It was noticed by de Broglie [6] that the process leading from the Klein-Gordon equation
to the Dirac equation is similar to that leading from the second-order equations for
the electric and magnetic fields E and B to the four coupled, first-order, Lorentzinvariant Maxwell equations for the scalar and vector potentials A 4 and A.
Further consistency was reached by quantizing the electromagnetic field, which
led to quantum electrodynamics [7] for the electron and other leptons. This served
as a model to quantum chromodynamics [8] for particles involving quarks, such as
nucleons and other baryons. Modern quantum field theory [9] encompasses the various cases. In these theories, the entanglement of matter and antimatter is expressed
by the necessity to include particles and antiparticles on the same footing to cope
with infinities.
De Broglie’s wavelength λ B was an outcome of a (theoretical) encounter of light
and matter. Another outcome of a (physical) encounter of light and matter was
Compton’s wavelength λ C [10]. When X-rays hit electrons (relatively) at rest, the
wavelength λ 2 of a scattered photon differs from that λ 1 of the incident photon by a
value: λ 2 − λ 1 = λ C (1 − cos θ), θ being the angle between the two X-rays and λ C
being given by:
λ C = h/m 0 c,
(3.2)
where m 0 is the rest mass of the electron and c, the velocity of light.
From the definitions of λ B and λ C it appears that the former depends on the
particle velocity v (and may thus vary from ∞ to 0) while the second depends solely
on its rest mass (and universal constants). The relation between the two involves the
Lorentz ‘boost’ transformation factor:
λ C /λ B = βγ,
(3.3)
where β ≡ v/c and γ ≡ (1 − β 2 ) −1/2 are coefficients of the Lorentz proper transformations.
De Broglie’s wavelength is linked to the external motion, with momentum mv,
of the particle wave packet, whereas Compton’s wavelength seems to be linked to