The Principle of Least Action Revisited

Deep within the theory of Special Relativity lies a minor
theorem concerning the relation between four-vectors and the Lorentz
Transformation: given any two four-vectors pertaining to a specific phenomenon
in a specific inertial frame of reference, multiplying those two four-vectors
together as an inner product (analogous to the dot product of three-vector
algebra) yields a number, called a Lorentz scalar, that remains invariant under
a Lorentz Transformation between the assumed inertial frame and any other. If we
measure the accidental properties of a particle of mass m and organize those
measurements into a minuscule change in the distance-duration four-vector({dx}=(dx,dy,dz,icdt),
denumerating the change in the particle’s location in space and time) and the
particle’s momentum-energy four-vector ({p}=(mv_{x},mv_{y},mv_{z},imc)),
then we can calculate the inner product of those four-vectors,

(Eq’n 1)

and get what we call the action that the particle enacts in going between the
endevents (analogous to the endpoints) of the distance-duration four-vector. In
this situation we have dx=v_{x}dt, dy=v_{y}dt, and dz=v_{z}dt,
so we can rewrite that equation as

(Eq’n 2)

in which T represents the particle’s kinetic energy and E=T+U represents the particle’s total energy, the sum of its kinetic energy and its potential energy U. We thus have

(Eq’n 3)

in which L represents the Lagrangian function. If we represent the particle’s
rest mass as m_{0} and substitute the algebraic description of the
relativistic mass into Equation 2, we get

(Eq’n 4)

which gives us, upon comparison with Equation 3, the relativistic Lagrangian.

If the particle, on its trajectory, touches two events, then there necessarily exists an inertial frame (let’s call it the preferred frame) in which those events occur at the same point in space (which we will call the boundary point). In that frame, if a particle follows a straight line between the two events in all other frames, then the particle enacts

(Eq’n 5)

in which t_{1} and t_{2} represent the times at which the two
events occur in the preferred frame. According to the theorem stated above, the
minuscule elements of action from Equation 4 (with v=0) remain invariant under
the Lorentz Transformation, so their sum (the integral) must also have the same
value in all inertial frames.

If a force acts on the particle, then in the preferred
frame the particle passes through the boundary point at the instant t_{1},
then traces out through space a trajectory that loops around and passes through
the boundary point at t_{2}. In order to make the particle move on the
loop the force must point to the interior of the loop, a fact that we can
exploit in the next step. With an algebraic description of that trajectory as a
function of elapsed time we can derive a description of the particle’s velocity
and other dynamic properties and with those we can calculate the amount of
action that the particle enacts between t_{1} and t_{2}. We
associate that action with the trajectory, which we call the particle’s true
path.

Imagine that we can make the particle follow an alternate
path that differs from the true path by a minuscule vector distance
∂**x**,
with ∂**x**=0
at the boundary point. To make the particle follow the alternate path we must
add an increment ∂**F**
to the force that drives the particle along the true path. If the alternate path
follows the inside of the loop of the true path, then
∂**F**
must add to the applied force and thus points inward, as does
∂**x**;
if the alternate path follows the outside of the loop of the true path, then
∂**F**
must subtract from the applied force and thus points outward, as does
∂**x**:
in both cases the dot product ∂**x•**∂**F**
yields a positive number. We know that

(Eq’n 6)

so we can calculate the variation in the action that the alternate path adds to the action of the true path,

(Eq’n 7)

Note that the product of the incremental distance and the incremental force yields an energy that adds to the Lagrangian of the proper action integral.

Because ∂S adds only positive numbers to the action of the true path, the true path thus represents the path of minimum or least action relative to adjacent alternative paths. That fact necessitates that on the true path we must have

(Eq’n 8)

which gives us the mathematical expression of Hamilton’s principle. By applying the standard method to that equation we can derive the Euler-Lagrange equations, which yield up the equations of motion governing the particle.

As a Lorentz scalar, the action remains invariant under a Lorentz Transformation, so the mathematical form of the laws of motion must also remain invariant. But we can only observe the motions of bodies and particles (either directly or indirectly), so we can only encode the fundamental laws of physics in equations of motion. Thus we come back, as we must, to Einstein’s postulate that the laws of physics all have the same mathematical form for all observers, regardless of how those observers move relative to each other.

Beyond providing equations of motion describing the evolution of simple mechanical systems, both classical and relativistic, the principle of least action gives us even more information about the operations of Reality.

The Law of Entropy

Consider a system made up of a vast number of particles. We then have Equation 8 as

(Eq’n 9)

in which the Lagrangian consists of the sum of all of the particles’ potential energies subtracted from the sum of all of the particles’ kinetic energies (measured relative to the system’s stationary center of mass). If we differentiate that equation with respect to the volume occupied by the system, we get the density of the relevant dynamic quantities,

(Eq’n 10)

In an ideal gas the particles interact with each other only through instantaneous, perfectly elastic collisions, so in that case we can set U=0. Thus the state that achieves the least action density corresponds to the state of least kinetic energy density.

Assume that a rigid container holds a certain amount of an
ideal gas that contains a certain amount of randomized kinetic energy (which we
call heat). Assume also that initially the kinetic energy density varies from
place to place within the container. The average kinetic energy density in the
container doesn’t change, regardless of how the energy gets redistributed within
the gas, so the proposition emanating from Equation 10 doesn’t lead to any new
knowledge. But if we look at the gas in more detail, we __do__ find something
new.

Imagine two small neighboring volumes within the gas and assume that one contains kinetic energy at a higher density than the other does. Between any two given instants the gas in the first volume enacts a greater action density than does the gas in its neighboring volume. Relative to the gas in that neighboring volume, then, the gas in the first volume does not enact the minimum possible action density. In the absence of constraints, though, energy can migrate from the gas in the first volume into the gas in the second volume, thereby bringing the gas in the first volume into conformity with the principle of least action density. But that migration makes the kinetic energy density of the gas in the second volume increase, in apparent violation of the principle of least action density.

We can resolve that dilemma by noticing that energy will migrate out of any parcel of gas and will do so at a rate proportional to the energy density of the gas. Thus, in our present example, heat will migrate at a faster rate from the gas in the first volume than from the gas in the second volume and that fact will remain true until the gas in both volumes has the same energy density.

Look at that analysis in another way. Within any minuscule volume of the gas we can, in concept, determine the average kinetic energy carried by a particle of the gas. That number gives us a measure of the tendency of the energy in the gas to migrate. Unable to measure that number directly, we use thermodynamic tricks to measure another number proportional to it, a number that we call temperature. If we conceive heat as a kind of fluid, then we can say that net heat flows from a higher temperature gas to a lower temperature gas. We can also extend the above analysis and the concepts of heat and temperature to liquids and solids and say that in general heat will not, of itself, go from a colder body to a hotter body. That statement gives us Rudolf Clausius’ form of the second law of thermodynamics, so now we know that the principle of least action density necessitates the law of entropy.

The Quantum Theory

The statement that δS=0 tells us that in no way can anything, particle or field, enact an infinite quantity of action. But that statement obliges us to ask a simple question: an infinite quantity of action relative to what fundamental unit? We know that the fundamental unit of action cannot be an infinitesimal: it takes an infinite set of infinitesimals to add up to a finite value, which value comes out of the summation as indefinite and thereby conflicts with the above statement of the principle of least action. Thus we must infer the necessary existence of a finite, albeit minuscule, natural unit of action. That fact necessitates that phase space divide naturally into an array of those fundamental action units. But rather than fixed units, such as those that we inscribe onto a ruler that we use to subdivide space into an array of subvolumes, the fundamental action unit gives an indeterminate subdivision of phase space in accordance with, for example, the statement that

(Eq’n 11)

in which aitch represents the fundamental action unit, which we call Planck’s constant.

That equation expresses the content of Werner Heisenberg’s Indeterminacy Principle. It tells us that no particle can have a definite existence at a point in phase space. But a particle is a definite entity: it has a definite mass, a definite spin, a definite electric charge, and so on. Therefore, around any given point in phase space, the particle must have an indeterminate existence within the phase area defined by Equation 11, an area that itself has an indeterminate shape.

Indeterminacy necessitates that we base any mathematical description of the particle’s accidental properties (such as location, momentum, etc.) upon the concepts and mathematical expositions of probability. But we must note that probability has two aspects, only one of which we can use in this case. If I hold a blue die and a red die in my hand, those dice exist in an indeterminate state relative to the thirty-six possible determinate states of the dice lying motionless on a flat surface. At that instant each of those determinate states has a certain probability of coming manifest after I toss the dice onto a flat surface: thus we have the probability of indeterminacy. If I then toss the dice onto a flat surface and they roll under an opaque cover before they come to rest, they will manifest one of the definite states, but we can’t see which one it is. We can only state the probability of what we will see when we remove the cover: thus we have the probability of uncertainty.

To express the indeterminacy associated with a particle we assign to the particle a probability density at each point in phase space. Multiplying that probability density by a minuscule phase volume around the point gives us the probability of the particle existing in that miniature volume. Of course, if we integrate those minuscule elements of probability over all of phase space, we must obtain

(Eq’n 12)

in which rho represents the probability phase density. Because multiplying the rho-function by an element of phase-space volume (which always takes a positive value) yields a minuscule element of probability, that function must take only positive values between zero and one. That output, the domain of the rho-function, comes from an input that ranges over the entire set of numbers that we use to describe phase space.

With a range that extends over the entire set of positive
and negative real numbers and a domain that extends over the set of positive
real numbers that lie between zero and one, the rho-function has several
properties that we can discern immediately. First, the fact that rho maps both
positive and negative numbers onto a set of positive-only numbers necessitates
the existence of a function ψ(**p**,**x**)
such that ρ=ψ^{2}.
And second, the fact that rho maps numbers of any possible magnitude onto a set
of numbers whose magnitudes come equal to or less than one necessitates that
ψ
have the mathematical nature of a sine, a cosine, or, more generally, an
imaginary exponential of a third function f(**p**,**x**). Thus, we can, in
concept, calculate the probability of finding the particle in any region
Ω
of phase space through the use of two equations,

(Eq’n 13)

and

(Eq’n 14)

That latter equation expresses the content of Born’s theorem, one of the fundamental theorems of the quantum theory.

We usually convert the double integral in Equation 14 into a single integral by implicitly integrating over all of momentum space, thereby creating a location-space representation of the probability. On rare occasions physicists implicitly integrate over all of location space to create a momentum-space representation of the probability.

What can we say, at last, about the function f(**p**,**x**)?
To answer that question we note that we normally don’t calculate probabilities
but, rather, we use the probability calculation as a means of obtaining an
expectation value, a kind of average value for one of the variables that we can
measure of the particle. If we have a variable g that represents an accidental
property of the particle (as distinct from inherent properties, such as mass or
spin), then it must participate in the state function (ψ=ψ(g));
that fact necessitates the existence of an operator G that will extract the
variable from the state function (Gψ=gψ).
Because applying G to the state function must leave the state function
unaltered, G can only take one of two forms; direct multiplication of the state
function by the variable g or differentiation of the state function’s argument
with respect to a variable conjugate in phase space to the variable g. With
those facts in mind, we can rewrite Equation 14 in the location-space
representation as

(Eq’n 15)

Thus we deduce the mathematical foundation of the quantum theory.

As for the form of the function f(**p**,**x**), we
can say immediately that in order to serve as the argument of an exponential it
must give us a pure number, not a number attached to some system of units of
measurement. Because we have a natural unit of action we can divide that unit
into the action associated with the particle and get a pure number as required:
we have then

(Eq’n 16)

In that equation the factor of 2π ensures that each increment of action brings the state function back to its original value.

If we use Equation 15 to calculate the expectation value of a particle’s location as a function of time, we get an equation of motion. But the right side of that equation describes the evolution of a field through the propagation of waves. Thus in Equation 15 we see a basic mathematical expression of the famous wave-particle duality of the quantum theory.

General Relativity, Et Cetera

In 1915 and 1916 David Hilbert and Albert Einstein used the principle of least action in their development of General Relativity. By setting the action integral invariant with respect to variations in the metric tensor they deduced the relationship between the curvature of space and time and the presence of mass-energy in the mathematical expression that we call Einstein’s Equation.

One interpretation of Einstein’s Equation called into question the validity of the law of conservation of energy. Amalie Emmy Nöther accepted the challenge of that question and in 1918 published the result of her study. Using the principle of least action, she deduced a pair of theorems, which she used to answer the question. In addition, Nöther’s theorems express a relationship between symmetries in the coordinates and conservation laws governing the dynamic variables associated with those coordinates by way of the action function.

In addition we can create a Lagrangian density based on the electromagnetic field and the associated electric current density. Putting that Lagrangian density into the action integral and then varying that integral with respect to the electromagnetic potentials gives us Maxwell’s Equations, the fundamental equations of electricity and magnetism. In this case instead of equations of motion of a particle we get the equations of evolution of a field.

The fact that so many fundamental laws of physics emerge
from the principle of least action strongly implies that the principle of least
action stands as the most fundamental of the laws of physics. As more
connections emerge from our analyses we may be able to deduce Hamilton’s
principle as __the__ fundamental law of physics.

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