Wednesday, March 5, 2014

Baseball Physics as Explained by Quantum-Geometry Dynamics

Or How Physics at Our Scale Emerges from Subatomic Physics

Note: This article assumes some familiarity with the ideas and concepts of quantum-geometry dynamics. If you are not familiar with QGD, you may want to read the short article titled Quantum-Geometry Dynamics in a Nutshell.

A postulate of quantum-geometry dynamics is that space is fundamentally discrete (quantum-geometrical, to be precise). Of course, proving this using our present technology may appear to be beyond difficult especially if, as QGD suggests, the discreteness of space exists at a scale that is orders of magnitude smaller than the Planck scale. The task of proving that space is made of preon{{s}^{\left( - \right)}}  may even be impossible because, if as discussed in On Measuring the Immeasurable, fundamental reality lies beyond the limit of the observable. That said, in the same article I explain that though preon{{s}^{\left( - \right)}}  , which according to QGD are the discrete and fundamental units of space, and preon{{s}^{\left( + \right)}}  , its predicted fundamental unit of matter, must be unobservable, their existence implies consequences and effects that must be observable at larger scales.

This implies that we already observed consequences of space and matter being quantum-geometrical but only lacked the theory capable of recognize them. It then makes sense to re-examine observations which, when interpreted by QGD, may provide proof of that space is quantum-geometrical. And this is exactly what we will do in the present article. But before doing so, we need to explain how QGD’s explanation of the law of conservation of momentum at the fundamental scale can be used to explain the conservation of momentum at our scale.

According to QGD, the momentum of a particle or structure is given by \left\| {{{\vec{P}}}_{a}} \right\|=\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\| where \left\| {{{\vec{P}}}_{a}} \right\| is the magnitude of the momentum vector of a particle or a structure a, {{\vec{c}}_{i}} the momentum vectors of the component preon{{s}^{\left( + \right)}}  of a and {{m}_{a}} its mass measured in preon{{s}^{\left( + \right)}}  . The speed of particle is defined as {{v}_{a}}=\frac{\left\| {{{\vec{P}}}_{a}} \right\|}{{{m}_{a}}} . We saw that when a structure a absorbs a photon b of mass {{m}_{b}} , then its new momentum \displaystyle \left\| {{{{\vec{P}}'}}_{a}} \right\| is given by \displaystyle \left\| {{{{\vec{P}}'}}_{a}} \right\|=\left\| {{{\vec{P}}}_{a}}+{{{\vec{P}}}_{b}} \right\| . We also saw that when a is subjected to gravitational interaction, \displaystyle \vec{G}\left( a;b \right), the change in momentum \Delta \left\| {{{\vec{P}}}_{a}} \right\| is equal to \displaystyle \left\| \vec{G}\left( a;b \right) \right\| so that \displaystyle \left\| {{{{\vec{P}}'}}_{a}} \right\|=\left\| {{{\vec{P}}}_{a}}+\vec{G}\left( a;b \right) \right\|. This is explained in more details in earlier articles. Now, let us see how QGD’s equations can be applied to explain and predict reality at our scale. To illustrate this, we will apply the QGD’s equations to baseball.

Let a be a baseball and ba baseball bat and let’s look at what happens when the ball, traveling towards the bat at speed {{v}_{a}} is hit by a baseball bat, itself going at speed {{v}_{b}}. Using the definitions above, we know that the momentums of a and b are respectively given by \left\| {{{\vec{P}}}_{a}} \right\|=\left\| \sum\limits_{i=1}^{{{m}_{a}}}{{{{\vec{c}}}_{i}}} \right\|and \left\| {{{\vec{P}}}_{b}} \right\|=\left\| \sum\limits_{i=1}^{{{m}_{b}}}{{{{\vec{c}}}_{i}}} \right\| and their speed by {{v}_{a}}=\frac{\left\| {{{\vec{P}}}_{a}} \right\|}{{{m}_{a}}} and {{v}_{b}}=\frac{\left\| {{{\vec{P}}}_{b}} \right\|}{{{m}_{b}}}. We also know that saw that, if space is quantum-geometrical, any change in momentum of an object must an exact multiple of it mass. That is : \Delta \left\| {{{{P}'}}_{a}} \right\|=x{{m}_{a}}. As a consequence, unless the mass of the bat is an exact multiple of the mass of the ball, it cannot transfer all of its momentum to it. Then x=\left\lfloor \frac{\left\| {{{\vec{P}}}_{b}} \right\|}{{{m}_{a}}} \right\rfloor and \Delta \left\| {{{\vec{P}}}_{a}} \right\|=\left\lfloor \frac{\left\| {{{\vec{P}}}_{b}} \right\|}{{{m}_{a}}} \right\rfloor {{m}_{a}}, where the brackets represent the floor function.

Both bat and ball cannot occupy the same region of quantum-geometrical space, nor can they move through each other; which is prevented by the preonic exclusion (a preo{{n}^{\left( - \right)}}  can be occupied by only one preo{{n}^{\left( + \right)}}  ) and the electromagnetic repulsion between the atomic electrons of the ball and the bat. In short, the ball’s momentum along the axis of impact is not allowed. Similarly, the momentum of the bat along the perpendicular axis that passes through the point of impact is also not allowed. We will show that the momentums of the ball and the bat at impact must momentarily become is zero.

To resolve the impact event and at the same time conserve momentum, the ball and the bat must emit particles that carry with them the forbidden momentums and which bring their momentum along the axis of impact down to zero. If {{a}_{i}} is one of {{n}_{a}} particles emitted by the ball and {{b}_{i}} is one of {{n}_{b}} particles emitted by the bat at impact then \displaystyle \sum\limits_{i=1}^{{{n}_{a}}}{{{{\vec{P}}}_{{{a}_{i}}}}}={{\vec{P}}_{a}}\left( a;b \right)and \displaystyle \sum\limits_{i=1}^{{{n}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}={{\vec{P}}_{b}}\left( a;b \right)where \displaystyle {{\vec{P}}_{a}}\left( a;b \right)and \displaystyle {{\vec{P}}_{b}}\left( a;b \right)are respectively the forbidden momentums of ball and the bat along the axis of impact (in gray in see figure below).

Now we know from observation of such mechanical systems that the ball and the bat will transfer part their forbidden momentums to each other. What happens is that the bat will absorb the particles {{a}_{i}} emitted by the ball and the ball will absorb the particles {{b}_{i}} which have been emitted by the bat at impact. For a perfectly elastic collision, the momentums of the ball and the bat after impact, respectively {{{\vec{P}}'}_{a}}and {{{\vec{P}}'}_{b}} are given by the equations {{{\vec{P}}'}_{a}}={{\vec{P}}_{a}}-{{\vec{P}}_{a}}\left( a;b \right)+{{\vec{P}}_{b}}\left( a;b \right)and {{{\vec{P}}'}_{b}}={{\vec{P}}_{b}}-{{\vec{P}}_{b}}\left( a;b \right)+{{\vec{P}}_{a}}\left( a;b \right). These equations provide a sufficiently precise description of the dynamics of momentum transfer at our scale, but they differs significantly from reality when we examine the impact at the microscopic scale at which, as we have seen in earlier posts, space is not continuous but quantum-geometrical.

If are to remain consistent with the axioms of QGD, then the momentum particles or structures (here the ball and the bat) can only change by discrete values which must be integer multiples of their mass; which QGD defines simply as the number of preon{{s}^{\left( + \right)}}  they contain. For the ball, this means that \displaystyle \Delta {{\vec{P}}_{a}}=x{{m}_{{{a}'}}}, where {{m}_{{{a}'}}}={{m}_{a}}-\sum\limits_{i}^{{{n}_{a}}}{{{m}_{{{a}_{i}}}}}+\sum\limits_{i}^{{{n}_{b}}}{{{m}_{{{b}_{i}}}}}and x=\left\lfloor \frac{\sum\limits_{i=1}^{{{{{n}'}}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}}{{{m}_{{{a}'}}}} \right\rfloor ; the quotient of the Euclidean division of the sum of the momentums of the emitted particles over the mass of the ball after absorption of the particles so that {{{\vec{P}}'}_{a}}={{\vec{P}}_{a}}-{{\vec{P}}_{a}}\left( a;b \right)+\left\lfloor \frac{\sum\limits_{i=1}^{{{{{n}'}}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}}{{{m}_{{{a}'}}}} \right\rfloor {{m}_{{{a}'}}}. This implies that given the remainder of the above Euclidian division must correspond to sum of the momentums of the particles emitted by the bat but which the ball is forbidden to absorb. That is; \displaystyle \sum\limits_{i=1}^{{{{{n}''}}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}=\sum\limits_{i=1}^{{{{{n}'}}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}-\left\lfloor \frac{\sum\limits_{i=1}^{{{{{n}'}}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}}{{{m}_{{{a}'}}}} \right\rfloor {{m}_{{{a}'}}}where i is the unique cardinal number attributed to one of {{{n}'}_{b}} particles that are absorbed or one of the {{{n}''}_{b}}particles which absorption by the ball is forbidden and {{{n}''}_{b}}={{n}_{b}}-{{{n}'}_{b}}.

If the impact preserves the physical integrity of the ball, the momentum that is not transferred to it will be radiated away carried by photons (mostly as infrared). If the impact is such that physical integrity of the ball is not preserved, then the particles could also be electrons, atoms or molecules.

Similarly, the bat will absorb photons from the ball and its momentum after impact will be \displaystyle {{{\vec{P}}'}_{b}}={{\vec{P}}_{b}}-\sum\limits_{i=1}^{{{n}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}-\left\lfloor \frac{\sum\limits_{i=1}^{{{n}_{a}}}{{{{\vec{P}}}_{{{a}_{i}}}}}}{{{m}_{{{b}'}}}} \right\rfloor {{m}_{{{b}'}}} and \displaystyle \Delta {{\vec{P}}_{b}}={{{\vec{P}}'}_{b}}=-\sum\limits_{i=1}^{{{n}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}+\left\lfloor \frac{\sum\limits_{i=1}^{{{n}_{a}}}{{{{\vec{P}}}_{{{a}_{i}}}}}}{{{m}_{{{b}'}}}} \right\rfloor {{m}_{{{b}'}}}.

Using QGD’s definition of speed we find that the speed of the ball after impact is {{v}_{{{a}'}}}=\frac{{{{\vec{P}}}_{{{a}'}}}}{{{m}_{{{a}'}}}} with \Delta {{v}_{a}}=\frac{-{{{\vec{P}}}_{a}}\left( a;b \right)}{{{m}_{{{a}'}}}}+\left\lfloor \frac{\sum\limits_{i=1}^{{{{{n}'}}_{b}}}{{{{\vec{P}}}_{{{b}_{i}}}}}}{{{m}_{{{a}'}}}} \right\rfloor . So if the momentum of the ball along the impact axis is less than that of the bat, then the ball after impact will have greater momentum, hence speed. If the momentum of the ball along the impact axis is greater than that of the bat, then the ball will have less momentum and speed after impact.

The physics of baseball bat hitting a baseball illustrates the fundamental mechanisms responsible for transfer of momentum. It is an example of how the physics at quantum-geometrical scale determines the behaviour at larger scales. For instance, it can be shown that much of the same equations we used to describe the physics of baseball can be used to describe nuclear fission. This is not surprising since, according to QGD, the same forces and laws apply at all scales.

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