Can Quantum Theory Concepts Shed Light on Biological Evolution Processes?
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theory stems from the exchange of radiation occurring in the form of energy packets,
expressed as a relation between energy and frequency, while relativity theory rests
on the velocity of light remaining invariant in all inertial frames.
At the lower end of the material world, the atomic theory was conjectured as early
as the 6th century BC. It was shown to be consistent with increasing experimental
and theoretical evidence starting from the 17th century AD, and definitely proven
during the 20th century. Today, the ‘standard model’ assumes that all particles are
made from four families of fermions, interacting through four basic forces mediated
by bosons. But here also, the increasing number of unexplained phenomena led to
epicyclic complexities, which makes a new paradigm necessary.
We believe that present-day theories of biological evolution are in a state of elaboration comparable to that of the physical sciences somewhere between Ptolemy
and Newton, and we conjecture that further progress may come from the search of
analogies with the more advanced quantum theories.
3.1 From Lagrange’s Equations to Hamilton’s Principle
‘Nature … always acts through the simplest paths.’
Pierre Louis de Maupertuis (Essai de Cosmologie, 1750)
Newton’s main, second law states that, for a material point of mass m and position r
submitted to a resultant force F, motion is governed by a second-order, differential
equation: F = m(d
2 r/dt
2 ). The first law simply explicits the limiting case F = 0.
From the concepts of force and momentum emerged those of potential and kinetic
energies: V (q) and T (p), q designating the set of 3N Cartesian coordinates (N being
the number of material points in the system) and p that of momentum coordinates.
A century after Newton, Lagrange generalized his formulation by introducing
the function: L = T(v) − V (q, v, t), of n generalized coordinates q i and respective generalized velocities v i and of time t. The Lagrangian coordinates may be
Cartesian, curvilinear, or independent combinations of them if one wished to take
account of c constraints or symmetries between the N particles, i.e.: n = 3N − c.
The potential V may involve conservative (e.g. gravitational) or non-conservative
(e.g. electromagnetic) forces, and may depend explicitly on time t. Newton’s laws
can then be derived from the generalized, n-component, second-order differential
Euler-Lagrange equations:
d[∂ L/∂v]/dt = ∂ L/∂q.
(1)
Still later, Hamilton reformulated the Lagrangian mechanics into a more convenient form by using conjugate momenta: p i = ˆ
oL/∂v i , and introducing a total
energy function: H = T ( p) + V (q, p, t), which is the Legendre transform of the
Lagrangian L. The time evolution of the system is then given by the 2N coupled,
first-order differential Hamilton equations:
443
theory stems from the exchange of radiation occurring in the form of energy packets,
expressed as a relation between energy and frequency, while relativity theory rests
on the velocity of light remaining invariant in all inertial frames.
At the lower end of the material world, the atomic theory was conjectured as early
as the 6th century BC. It was shown to be consistent with increasing experimental
and theoretical evidence starting from the 17th century AD, and definitely proven
during the 20th century. Today, the ‘standard model’ assumes that all particles are
made from four families of fermions, interacting through four basic forces mediated
by bosons. But here also, the increasing number of unexplained phenomena led to
epicyclic complexities, which makes a new paradigm necessary.
We believe that present-day theories of biological evolution are in a state of elaboration comparable to that of the physical sciences somewhere between Ptolemy
and Newton, and we conjecture that further progress may come from the search of
analogies with the more advanced quantum theories.
3.1 From Lagrange’s Equations to Hamilton’s Principle
‘Nature … always acts through the simplest paths.’
Pierre Louis de Maupertuis (Essai de Cosmologie, 1750)
Newton’s main, second law states that, for a material point of mass m and position r
submitted to a resultant force F, motion is governed by a second-order, differential
equation: F = m(d
2 r/dt
2 ). The first law simply explicits the limiting case F = 0.
From the concepts of force and momentum emerged those of potential and kinetic
energies: V (q) and T (p), q designating the set of 3N Cartesian coordinates (N being
the number of material points in the system) and p that of momentum coordinates.
A century after Newton, Lagrange generalized his formulation by introducing
the function: L = T(v) − V (q, v, t), of n generalized coordinates q i and respective generalized velocities v i and of time t. The Lagrangian coordinates may be
Cartesian, curvilinear, or independent combinations of them if one wished to take
account of c constraints or symmetries between the N particles, i.e.: n = 3N − c.
The potential V may involve conservative (e.g. gravitational) or non-conservative
(e.g. electromagnetic) forces, and may depend explicitly on time t. Newton’s laws
can then be derived from the generalized, n-component, second-order differential
Euler-Lagrange equations:
d[∂ L/∂v]/dt = ∂ L/∂q.
(1)
Still later, Hamilton reformulated the Lagrangian mechanics into a more convenient form by using conjugate momenta: p i = ˆ
oL/∂v i , and introducing a total
energy function: H = T ( p) + V (q, p, t), which is the Legendre transform of the
Lagrangian L. The time evolution of the system is then given by the 2N coupled,
first-order differential Hamilton equations: