The Larson Research Center

The basics of quantum mechanics, the main feature that distinguishes it from classical mechanics, is Heisenberg’s uncertainty principle; that is, in quantum mechanics, unlike in classical mechanics, the position and the momentum of a moving object cannot both be determined precisely, at any given moment in time. This is understood in terms of wave (uncertain position) and particle (certain position) duality, and quantum superposition: Superimposing the quantum states of many waves realizes the particle.

For example, in the case of classical mechanics, the x,y coordinates of a point moving uniformly around the circumference of a circle can be determined precisely, at any given moment in time.  Moreover, if the x,y momentum of an orbiting classical object is plotted against its x,y position on the orbit, at a given time, the plot obtained (phase-space) is also precise (see illustration here).  However, in quantum mechanics, this is not possible, because the more precisely the momentum is known, the less precisely the position is known and vice-versa. So, the idea of probability amplitude was born.  The probability amplitude is associated with the wave function in the phase-space of quantum phenomena, the famous quantum state of the microcosmic world.

In this case, the probability amplitude of the wave function differs from the combined probability distribution of a classical phase-space, where a series of measurements on identical oscillators defines the probability that the definite point will be found at a given location, with a given momentum.  In contrast, in the case of the probability amplitude of the wave function, a given location, for a given momentum, does not even exist. It is not defined, not just unobservable.

At one time, this idea was easily dismissed by skeptics, but not today.  The reason for this is that, today, quantum optics permits these things to be visualized experimentally (see explanation here). The visualizations rely on a few mathematical principles like Wigner functions, Fock states, etc.  Wigner functions (WFs) are the analogs of classical phase-space probability distributions.  In other words, they are like the probability distributions of classic phase-space in some ways, but they are different in other, important, ways. 

The most important distinction is that while WFs are similar, they are not phase-space probability distributions.  This is because they are measurements of only one, or the other, of the two, reciprocal, aspects of the phase-space. They are a measurement of either the momentum or the position aspect of the quantum state, never both.  The “probability” distribution of only one of the reciprocal pair of quantum parameters of phase-space is called a phase-space quasiprobability distribution, or density.  It is the WF of a given parameter in a quantum state.

Another, important, distinction between classical probability distributions and WFs is that, since the WF is not a probability density per se, its magnitude doesn’t have to be a positive definite magnitude; that is, it can be a negative quasiprobability!  This leads to the concept where, no matter what the energy of the Fock state is, the phase-space has regions where the WF, or quasiprobability, of the quantum state is negative. 

Using these two properties of WFs, the states of quantum systems can be studied in real experiments with lasers and optics.  A most interesting experimental result is reported by Alex Lvovsky et al in a paper entitled “Quantum State Reconstruction of the Single-Photon Fock State” (see here).  In the introduction to this paper, they write:

States of quantum systems can be completely described by their Wigner functions (WF), the analogs of the classical phase-space probability distributions. Generation of various quantum states and measurements of their WFs is a central goal of many experiments in quantum optics [1–3]. Of particular interest are quantum states whose Wigner function takes on negative values in parts of the phase space. This classically impossible phenomenon is a signature of highly nonclassical character of a quantum state.

That is to say, in contrast to the classical state, such as the coherent state of laser emissions, the quantum state can be both positive and negative. But how can a probability be both positive and negative?  What exactly is a negative probability?

Rebecca Slayton, writing in the Physical Review Focus, called Lvovsky’s experiment a “strange result” that goes way beyond the strangeness of Schrödinger’s cat.  In her article, entitled “Golfing with a Single Photon”, she explains

…in the quantum world, where a photon’s position and momentum cannot be determined simultaneously, this “elevation” [the WF] can only be understood as an approximation to probability [i.e. quasiprobability]. The new experiments by Alex Lvovsky and his colleagues at the University of Konstanz in Germany show that the photon’s phase space contains a circular ridge, where the photon is likely to be found, and a deep crater in the center, where your chances of finding the photon seem to be negative.

A graphic, depicting the result, is included in the article, as shown below:

As can be seen by the illustration above, the probability density of a single photon goes way below zero, like a volcanic pipe, extending below ground level. 

Slayton explains how Lvovsky’s team exploited the WF of the single photon, by using pairs of photons generated by down-conversion.  The down-converted photons share the same quantum state, so the wave property (the WF of non-local position) of one set of photons can be measured, while the particle property (WF of local momentum) of their corresponding twins can be measured separately, and then the WFs are combined together mathematically, like the slices of a CAT scan, to give the resulting phase-space probability distribution, depicted in the graphic above.

This has to be startling to LST physicists, whose only available interpretation of this is simply as a visualization of the uncertainty principle.  In reality, however, it’s much more than this; It’s tantamount to a CAT scan of a single photon, where we can visualize the photon as a combination of positive and negative oscillations, plotted in the reciprocal relation of phase-space, as probabilities. Clearly, this is a perfect picture of an S|T unit composed of positive and negative 3D oscillations.

But this is not the first time we’ve seen this.  Recall the discussion of Harry Swinney’s “oscillons” in his experiments at the University of Texas, at Austin.  Oscillons are oscillating groups of small brass beads, formed in a bed of such beads, vibrated at a given frequency. The peaks and valleys of oscillons are called crater (or dish) states and peak states.  Two crater states, or two peak states, repel each other, but unlike states attract.  Moreover, when a oscillon in a crater state collides with an oscillon in a peak state, they don’t cancel each other, but rather form a bound sytem of combined crater and peak states, which is stable over time.

I don’t know if anyone in the LST community has connected these two experimental results together yet, as I am doing here, but I think it would be unlikely without the understanding of the RST-based concepts of SUDRs, TUDRs, and the combination of these two theoretical entities into the theoretical SUDR|TUDR (S|T) units, which then are grouped as preon elements of the standard model entities. We’ve discussed, in several of the previous posts below, how combining S|T units into triplets yields the bosonic and fermionic configurations of standard model entities.

Nevertheless, we have yet to discuss the uncertainty principle in connection with these S|T units, and, as we are seeing from the work of Lvovsky and others, the uncertainty principle plays a crucial role in observing and calculating the values of quantum properties.  Clearly, this seems to present a problem, at first blush, because the LST uncertainty principle is expressed in terms of momentum and position, the phase-space of quantum mechanics, and the associated wave function of quantum states, which are quantities within the purview of M₂, or vector, motion, but not in that of M₄, or scalar, motion.  How, then, can we understand these seemingly weird ideas in RST-based theories?

The answer is surprising, because not only does it tell us how to work with these things, in the new system, but it also explains why they exist, which is, of course, something the LST community can’t offer, much to its chagrin. To understand the uncertainty principle, we first need to understand how it arises fundamentally from the particle/wave duality of the discrete world, and also that this duality is, in reality, a principle of reciprocity.  In the LST community, the reciprocal relation of two quantities is a central pillar of the science of physics.  Called conjugate quantities, the relationship between them in quantum mechanics is described in the Wikipedia article on the “Uncertainty principle.”  In part, it reads:

In quantum mechanics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity…. It therefore mathematically limits the accuracy with which it is possible to measure (actually even define) such pairs.

In classical mechanics, the most familiar conjugates (literally joining two entities together) are the concepts of potential and kinetic energy, regarded as two, reciprocal, aspects of the total energy of a dynamic system, such as the swinging pendulum, or bouncing spring.  These two, reciprocal, aspects of the pendulum’s, or spring’s, energy, express the law of energy conservation, as they transform into one another, because no matter what the quantity of either is, at a given point in time, they always add up to the total energy, in a lossless system (e.g. see here).

The same principle is expressed by the sine and cosine numbers of the changing angle that the arm of the pendulum makes with the vertical norm at the bottom of the swing: Their sum is always equal to 1, at any given angle of rotation. The identical relation can also be seen in the two, reciprocal, aspects of radiation, frequency and wavelength, where the total energy can be expressed in terms of either. Consequently, to increase, or decrease, the radiative energy of a system, the frequency must increase, or decrease, but the wavelength must decrease, or increase, and vice versa, because frequency and wavelength are reciprocals in the radiation energy equation.

However, in de Broglie’s hypothesis, this familiar relation of reciprocal quantities starts to seem very strange, because, as he reasoned, if the wave energy of light is discrete, then the discrete entities of matter must be waves.  Of course, this is exactly how it turns out to be, but the big question immediately becomes, how can matter (or anything) be both discrete and continuous at the same time, knowing that waves are continuous, and matter is discrete? 

Modern physics’ initial answer to this perplexing question was found in the Born statistical interpretation of the wave function, where the squared amplitude of the wave at a certain time yields the probability of finding an electron at a certain location in the vicinity of the nucleus of an atom. However, this initial interpretation shortly evolved into the form of Schrodinger’s wave equation, based on a so-called dispersion relation, where the concepts of energy, momentum and velocity are manipulated to yield a reciprocal relation between momentum and position; that is, the greater the momentum of a particle (locality), the more its wave is spread out (non-locality).  Thus, de Broglie’s relation of wavelength, Planck’s constant and momentum (velocity of mass), as discrete units of energy per unit of momentum, leads to a reciprocal relationship between momentum and position that doesn’t exist in classical mechanics (see online article here).  Fundamentally, however, it is clearly a reciprocal relation of the value of “locality” versus the value of “non-locality” that we are dealing with here.

Calling this unclassical relationship (all classical objects are always local objects) the “uncertainty principle” is therefore misleading, in a way, because it’s really only a reciprocal relation between locality and non-locality.  And, as we know, these two properties just happen to be the two, reciprocal, aspects of volume: As we’ve seen time and time again in these posts, the point and the sphere are reciprocals, both numerically and geometrically, where the numerical magnitude, corresponding to the three-dimensional, geometric, unit point is 1³ = 1, and where the numerical magnitude, corresponding to the three-dimensional, geometric, unit sphere is 2³ = 8. 

These are the scalar and pseudoscalar of the octonions, respectively, that exhibit the well known reciprocal symmetry, 1331, in the number of its linear spaces, as generated in the binomial expansion. In the reciprocal system of mathematics (RSM), the magnitude of the M₁ sphere is the (1|1)³ point, and the magnitude of the M₂ sphere is the (2|2)³ cube, and, in the theoretical development of the LRC, these numerical magnitudes correspond to the physical magnitudes of the SUDR and TUDR, based on the reciprocal system of physical theory (RST).

More relevant, however, when one visualizes the point expanding to a sphere, it’s easy to visualize the local vs. non-local nature of these two, reciprocal, aspects: The outward expansion of the point turns its locality (1³) into non-locality (2³), as the expansion is measured along one of the eight diagonals of the (2|2)³ = 8 discrete cubes (or continuous sphere). To measure the expansion of the point into the sphere requires a selection of a definite (local) point on the surface of the sphere, or a specific corner of the cube (a wave function collapse), which can be expressed in terms of these eight diagonals, or octutures.

Of course, motion in the opposite “direction,” a contraction of the sphere to the point, is a transition from non-locality to locality.  In the case of the S|T unit, two of these volume oscillations are bound together, one positive and one negative.  If we plot these changing values of volume, as changing degrees of locality, constituting a change of quantum state, then the result implies the scalar motion of a SUDR|TUDR combo, just as surely as plotting the changing values of kinetic and potential energy, constituting a change of mechanical state, implies the vectorial motion of a pendulum, or bouncing spring.

Obviously, we can observe the motion of a pendulum, or spring, that constitutes its changing state of kinetic|potential energy, but we can also describe it mathematically, in terms of the sine and cosine of the changing angle of the arm, or changing length of the spring.  Unfortunately, we can’t observe the motion of the S|T unit that constitutes its changing state of locality|non-locality, but we can describe it mathematically, in terms of the space and time magnitudes of the changing volume of the SUDR and TUDR.

However, the first order of business is to describe what is being conserved in the oscillations themselves. In the case of the moving pendulum, or the bouncing spring, the motion, or momentum, of the weight at the end of the arm, or spring, changes, but the total energy is conserved;  that is, at the top of the swing, or bounce, the motion, and thus the kinetic energy, is zero, but the total energy of the system is conserved, in the potential energy of the weight’s position, which is relative to the pivot point and the length of the arm, in the case of the pendulum, and relative to the tension and length of the spring, in the case of the bouncing spring.  On the other hand, the motion, or momentum, at the bottom of the swing, or the middle of the bounce, represents the total energy of the system, at that point, and the potential energy of the weight’s relative position is zero, at this point. In between these two points, a portion of the total energy is potential energy, and a portion is kinetic energy, which, when added together, invariably equals the total energy of the system.

In the case of the S|T unit, an analogous situation exists, only there is no mass, and, thus, there is no momentum, and there is no position involved, in the dynamics of the system. The conserved quantity of this system, therefore, is not a given quantity of mechanical energy, expressed in terms of the changing energy of momentum and a changing reciprocal of this, the energy of position, but the conserved quantity is a given value of scalar quantity, expressed in terms of a changing volume of space and a changing reciprocal of this, the “volume” of time.

In this way, the oscillating, or pulsating, volume is analogous to the swinging pendulum, or the bouncing spring, but the two, reciprocal, aspects of the change of the volume are the space|time point and the space|time sphere; that is, the change is from purely local space|time (point) to purely non-local space|time (sphere).  Because space is the reciprocal of time, in the RST, more space is equivalent to less time, and vice-versa. Hence, as the unit space point is transformed into the unit space sphere, its reciprocal aspect, the unit time sphere, is transformed into a unit time point, corresponding perfectly to the reciprocal relation of conjugate quantities.

The constant, unchanging, quantity that is being conserved in the transformation of these two, reciprocal, aspects of volume, is what we can call the volume’s “locality density;” that is, as one space|time aspect of the unit point increases to a unit sphere, its “locality density” decreases, but at the same time, the reciprocal aspect’s “locality density” increases, as it transforms from a unit sphere to a unit point. Hence, as the point’s locality value (probability density) becomes less “dense” as a sphere, the reciprocal sphere’s locality value (probability density) becomes more dense as a point, and the sum of the two “locality densities” is always constant, throughout the cycle.

Consequently then, the analog of the SUDR is the swinging pendulum, or bouncing spring, while the analog of the TUDR is a negative swinging pendulum, or negative bouncing spring, which, though conceptually straightforward, in an abstract sense, is impossible to actually implement with moving masses. 

If we plot the oscillation of the SUDR, and the oscillation of the TUDR, in terms of transitions from point to sphere (local to non-local), and from sphere to point (non-local to local), we get the expected result of two, reciprocal, aspects of the total motion, constantly changing, but, unlike in the case of the pendulum, a coordinate position has no role to play at all in the system, even though the property of non-locality|locality could be expressed in terms of an increasingly/decreasingly “uncertain” position, if that’s what we wanted to do.

Of course, we have no need to do this in our scalar system.  What we want to do instead is to express the local|non-local transitions of the SUDR|TUDR combo in terms of the radius of its spheres, the pulsating spatial and temporal spheres, which constitute the unit space and unit time expansions/contractions of the SUDR and TUDR. These are the two parameters of the scalar quantum system, or scalar phase-space, of the S|T unit. Unfortunately, like the momentum|position phase-space of the LST’s quantum mechanics, these two parameters of the RST-based scalar phase-space are mutually exclusive, as far as measurement is concerned, because, even though, now, non-locality will not be expressed in terms of the probability of defining a location, it will still be expressed in terms of two reciprocal quantities; that is, the magnitude of a radius of a space volume, is the inverse of the radius of a time volume, and as one grows, the other shrinks, and it’s impossible to measure something that is both growing and shrinking at the same time.  The more accurately you measure the change in the space volume, the less accurately you can know the change in the time volume, and vice versa.

Fortunately, however, each of the inverse quantities can be measured separately, as Lvovsky et al and others in the laser physics field have demonstrated. What we end up with, in the final analysis, is simple harmonic motion; that is, the S|T unit is a unit of propagating, simple, harmonic, motion.