The Larson Research Center

Last time I explained a little about the standard model of LST physics and how its underlying mathematical structure, in the form of Lie groups, suggests a relationship to our chart of motions. However, at the time I was thinking that since the groups are synonymous with magnitudes of force, the actual correlation ought to be with the three dimensions of magnitude in the chart, not the three bases of motion. In fact, I even tried modifying the chart in the previous post, but ran into trouble with formatting the concepts in an understandable way and decided to leave it be, due to time constraints.

Now that I have had some time to think about it, It seems I can get the idea across better, by separating the chart of motions into three charts, one for each base of motion. The three charts are:

Mechanical Motion (Base 2)

1. (2/2)⁰ N/A
2. (2/2)¹ U(1)
3. (2/2)² SU(2)
4. (2/2)³ SU(3)

Electrical Motion (Base 3)

1. (3/3)⁰ N/A
2. (3/3)¹ N/A
3. (3/3)² SU(2) + U(1)
4. (3/3)³ SU(3) + SU(2) + U(1)

Scalar Motion (Base 4)

1. (4/4)⁰ N/A
2. (4/4)¹ N/A
3. (4/4)² N/A
4. (4/4)³ SU(3) + SU(2) + U(1)

Figure 1. Lie Groups of the Standard Model and the Chart of Motions

To understand how this works, it’s best to start with the mechanical motion of base 2, the normal vectorial motion of objects. There are two modes of all motion, translation and vibration. In mechanical motion, the vibration mode is harmonic; that is, this motion is the oscillation between two positions, like the motion of pendulum in a gravitational field, or the motion of a crystal in an electrical field. In both of these cases, there is a 90 degree angular relationship between the potential and kinetic energy that corresponds to the sine and cosine of the angle between the horizontal and vertical positions of the pendulum. As the angle of a pendulum between the vertical position of the weight at the end of the arm, and the two possible horizontal positions of the weight, varies with time, so does the potential and kinetic energy of the mechanical motion.

When the 90 degree angle of the rotating arm of the pendulum is maximum, the sine of the angle is maximum, but the cosine is minimum and vice versa, which corresponds to the maximum potential energy at the top of the swing, on either side, and the minimum kinetic energy of the pendulum, since it is motionless at that point. However, when the pendulum is passing through the bottom of the swing, the speed is maximum, the kinetic energy is maximum, and the potential energy is zip. The kinetic energy due to the speed of the mass at the end of the arm also equates to a value of angular momentum that depends upon the weight of the mass, as well as its speed.

Of course, in the case of the harmonic motion of the pendulum, the angular momentum is not constant, because as the speed of the mass goes to zero at the top of the swing, so does the kinetic energy and the angular momentum. This simple physical relationship of potential and kinetic energy in the motion of the pendulum is important because it clearly demonstrates the relation between the principle of symmetry and the law of conservation of energy that is central to the mathematical analysis of physical phenomena. The motion of a pendulum (disregarding the effects of energy loss due to friction), in a gravitational field, is symmetrical; that is, it swings as far to one side of the bottom point, as it does to the other side, and in both cases, energy that is conserved as potential energy is transformed into kinetic energy and vice versa.

This would not be the case, if the symmetry of the pendulum’s swing was disturbed some how. Energy would be lost to whatever intervening force “broke” the symmetry of the swing. For instance, if the swinging motion of the pendulum was exploited to perform work like hammering a nail into a block of wood, the resistance of the wood to the nail’s penetration would constitute an intervening force that breaks the symmetry of the motion.

This same principle can be seen in the translational motion of a rotating pendulum; that is, if a pendulum were to be made to rotate in a plane perpendicular to the gravitational field so that its motion was perpetual in one direction of rotation around one position (again disregarding the effects of energy loss due to friction), rather than oscillating between two positions, the previous transformation of potential energy into kinetic energy, and vice versa, associated with the oscillation, would not occur. Thus, one would think that the kinetic energy and the angular momentum would be constant, and it would be, if it were not for the loss of energy due to the breaking of another symmetry.

The symmetry that is broken in the case of rotation is the symmetry of direction; that is, the angle tangent to the rotational path of the weight represents a continuous change in one direction, a continuous change to the “inside’ direction, if you will, and just as before, the breaking of the symmetry is accompanied by the loss of energy. If the lost energy is not replaced, as it is, for instance, in the case of orbiting planets or moons, by the gravity of the mass being orbited, the rotational motion of the system will eventually stop.

This was, of course, the problem facing the Bohr model of the atom, at the turn of the 19th Century, when quantum mechanics was devised to solve the problem of an electron orbiting the nucleus of an atom. There was obviously no way that the energy associated with the gravitational attraction of the mass of the atomic nucleus can supply the lost energy of the orbiting electron, to keep it orbiting perpetually, in the way stars, planets, and moons maintain their orbits.  At the same time, the energy associated with the Coulomb attraction between the nucleus and the electron acts differently than the gravitational attraction, because, unlike accelerating masses, accelerating charges radiate energy.  Thus, the electron could not orbit the nucleus without continually radiating, or losing, energy due to the changing direction of its path in the circular orbit.

At the same time, however, Bohr’s idea was the only game in town that explained the quantization of energy: that only an integral number of wavelengths fit into a circumference of a given radius around an atomic nucleus. This concept of quantization was needed for explaining the recently discovered fact that the energy of electrons, and the radiation from atoms, are quantized and inter-related. Thus, a new mechanics, or concept of motion, was born. A concept that has been mysterious and enigmatic ever since, because, while its foundations are unknown, it works really well for calculating the frequency of atomic spectra and formulating a theoretical foundation of other atomic phenomena.

The key to the new mechanics was ultimately found to lie in the use of complex numbers, where the ad hoc invention of the imaginary number, the square root of -1, fortuitously came to the rescue. However, when most people think of quantum mechanics they think of Schrodinger’s wave equation, which describes the behavior of a charged particle in a field of force. There is the time-dependant equation, used for describing progressive waves, applicable to the motion of free particles, and then there is the time-independent form, used for describing standing waves. The imaginary number i doesn’t appear directly in the time-independent equation, but solutions to this equation come in two forms: the oscillating and the rotational form of the pendulum that we have been discussing.

In other words, the quantum mechanical wave equation relates a periodic mathematical function (wave function) with the associated physical principle of energy conservation and symmetry, in the form of the potential and kinetic energy transformations of the swinging pendulum, to the rotational form of the same conservation principles. Of course, In the case of the oscillating form, the energy to resupply the lost energy is supplied to the mechanical instance of the pendulum system by a gravitational field, when the plane of oscillation is parallel to the force of the field, while in the case of the rotational form, the energy would hopefully come from the electrical field, associated with the Coulomb charges, in the absence of a suitable gravitational effect, at the center of rotation.  Of course, since the orbiting electron would radiate away this energy in time (10^-9 seconds!), this model is unstable.

Nevertheless, it is still the rotational form of the solution to the time-independent equation that is used in quantum mechanics, but only because the concept of the complex number, in describing the motion of the electron rotation, can be exploited to provide the solution to the energy conservation problem.  Mathematically, the two solutions to the equation have opposite signs, one is positive, the oscillation solution, and one is negative, the rotation solution. Since the electron is negative, and is envisioned as rotating around the atomic nucleus, it is the negative, rotation, solution that is used in QM. Specifically, the Schrodinger equation for ψ(x) can be reduced to the form:

d²ψ(x)/dx² +/- k²ψ(x) = 0 ;

a two-dimensional equation, where k is a real number. The form of the solution is determined by the sign of the k² term. If k² is taken to be positive, then the equation has the same form as the two-dimensional harmonic oscillator:

ψ(x) = A cos(kx) + B sin(kx) ;

but, if the k² term is taken to be negative, then the solution has the two-dimensional general form:

ψ(x) = Ae^kx + Be^-kx

which, if written as a complex number, using the polar form of coordinates, instead of rectangular form, where

z = ρ(cosθ + i sinθ),

it can be rewritten, using Euler’s Formula, where

cosθ + i sinθ = e^iθ,

which is a rotation expressed as a complex number.  In other words, the complex number has the form