Can Quantum Theory Concepts Shed Light on Biological Evolution Processes?
445
This integral formulation looks teleological in that sense that the positions and
velocities describing the system can be known at each instant t by minimizing the
action integral S[q] between the initial and final states. Here the future determines
the past since the system will, throughout its motion, ‘choose’ the steps that will
minimize the action integral.
This Hamilton principle is invariant under coordinate transformations and yields
the same results as the Hamilton equations. But it can readily be extended to
deformable systems and to relativistic and quantum theories, where it finds an
interpretation in Einstein’s curved gravitational field and Feynman’s path-integral
formulation.
3.2 From Snell-Descartes’ Laws to Fermat’s Principle
‘The principle … that Nature always acts through the shortest and simplest paths is merely a
moral, not a physical principle, which is not and cannot be the cause of any effect in Nature.’
Claude Clerselier, reply to Fermat’s letter (1662)
The early observations of celestial objects also aroused interest in the propagation of
light. In the 4th century BC, Euclid investigated reflection and, in the 2nd century AD,
Ptolemy studied refraction. In the 11th century, Alhacen gave an early expression of
the ‘principle of least path’ and applied it to both reflection and refraction. Some time
earlier, the law of refraction had been accurately described and used to devise lenses
by Ibn Sahl. In the 17th century, this law was reformulated by Harriot then by Snell.
Descartes gave a different derivation of the law and used it to solve optical problems.
But it was Fermat who made the correct assumptions and used the ‘principle of least
time’ to derive both laws of reflection and refraction.
These laws, which apply to isotropic or specular media, can be written as follows.
If θ 1 is the angle of incidence of a light ray through medium 1 with refractive index n 1
and θ 2 is its angle of refraction (or reflection) through medium 2 (or 1) with refractive
index n 2 (or n 1 ), then the two angles are related by:
sin θ 2 / sin θ 1 = n 1 /n 2 (for refraction);
(6)
θ 2 = θ 1 (for reflection).
(7)
Again these laws are determinist in that sense that the angle of refraction (or
reflection) can be known if the angle of incidence and the properties of the media
are given. Here the internal structure of the system is defined by a set of refractive
indices that, according to Fermat’s principle, determine the velocities of light in the
media: n 1 v 1 = n 2 v 2 = c.
This principle, which was clearly formulated much after the earlier studies on
reflection and refraction, but was used to provide the first exact proof of their laws,
445
This integral formulation looks teleological in that sense that the positions and
velocities describing the system can be known at each instant t by minimizing the
action integral S[q] between the initial and final states. Here the future determines
the past since the system will, throughout its motion, ‘choose’ the steps that will
minimize the action integral.
This Hamilton principle is invariant under coordinate transformations and yields
the same results as the Hamilton equations. But it can readily be extended to
deformable systems and to relativistic and quantum theories, where it finds an
interpretation in Einstein’s curved gravitational field and Feynman’s path-integral
formulation.
3.2 From Snell-Descartes’ Laws to Fermat’s Principle
‘The principle … that Nature always acts through the shortest and simplest paths is merely a
moral, not a physical principle, which is not and cannot be the cause of any effect in Nature.’
Claude Clerselier, reply to Fermat’s letter (1662)
The early observations of celestial objects also aroused interest in the propagation of
light. In the 4th century BC, Euclid investigated reflection and, in the 2nd century AD,
Ptolemy studied refraction. In the 11th century, Alhacen gave an early expression of
the ‘principle of least path’ and applied it to both reflection and refraction. Some time
earlier, the law of refraction had been accurately described and used to devise lenses
by Ibn Sahl. In the 17th century, this law was reformulated by Harriot then by Snell.
Descartes gave a different derivation of the law and used it to solve optical problems.
But it was Fermat who made the correct assumptions and used the ‘principle of least
time’ to derive both laws of reflection and refraction.
These laws, which apply to isotropic or specular media, can be written as follows.
If θ 1 is the angle of incidence of a light ray through medium 1 with refractive index n 1
and θ 2 is its angle of refraction (or reflection) through medium 2 (or 1) with refractive
index n 2 (or n 1 ), then the two angles are related by:
sin θ 2 / sin θ 1 = n 1 /n 2 (for refraction);
(6)
θ 2 = θ 1 (for reflection).
(7)
Again these laws are determinist in that sense that the angle of refraction (or
reflection) can be known if the angle of incidence and the properties of the media
are given. Here the internal structure of the system is defined by a set of refractive
indices that, according to Fermat’s principle, determine the velocities of light in the
media: n 1 v 1 = n 2 v 2 = c.
This principle, which was clearly formulated much after the earlier studies on
reflection and refraction, but was used to provide the first exact proof of their laws,