Let us play detective and do some physics Standard Model (SM) forensics. The main research question is the mystery of the paucity of basic progress in physics for some six decades. This is our crime scene, so to speak, and we seek some clues concerning how about two generations of physicists could be fooled into an almost religious belief in continuous space-time? Our journey has three parts: first, reminder on the definition of a geometric point; second, the delta function of the great physicist Paul Dirac; and third, usage of this nonsensical math in the current physics SM.
The scientific merit of the present research question is enormous, since it hopes to find the cause of a huge waste of intellectual resources in the physics community over some six decades, not to mention billions in misdirected research funding from indebted governments.
We may have found the answer and will use several Wikipedia mathematics articles to present the "crime scene" evidence. Note that critical points from Wikipedia sources need to be verified from references or other reliable sources; however, for present purposes these articles will do just fine.
First, let's check what we learned in elementary school geometry. And wonder of wonders, we find that what we learned way back in childhood is still true today, namely that as an axiom, a point is defined as an object that does "not have any length, area, volume, or any other dimensional attribute." And "In all of the common definitions, a point is 0-dimensional."
This article mentions several other math adventures beyond the simplest Euclidean geometry, and the point is a zero volume, zero-dimension object in all cases.
In brief, the physical interpretation of a point is nothing, other than a position in a space.
Second, some two generations ago, Paul Dirac showed how nothing can be portrayed as something. "In mathematics, the Dirac delta function, or δ function, is a generalized function, or distribution, on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line." The references are from 1950, 1958 and 1968, some 50-60 years ago.
This is exactly where magic and miracles entered main stream physics. In the real number scale, a point is infinitely small and merely a location, not something physical, unless one wants to add entertainment (magic) and religion (miracles) to science.
Space, time and energy are quantized in binary mechanics (BM), which was derived in part from a pair of relativistic Dirac spinor equations of opposite handedness. So we have nothing but praise for physicist Paul Dirac. Indeed, BM gives him more credit than seen generally in physics literature, because his spinors actually referred to events in quark structures in BM which proved to enable his equations to make impressive predictions regarding electron behavior.
With that caveat, though, we find that the Dirac delta function may be the guilty suspect we are looking for, because it provided a sort of quasi-mathematical justification for saying that nothing is really something. Books like the Bible describe miracles where something is made from nothing. Strangely, we have the same thinking in modern physics.
Third, the standard model strives to define fields for all points in space-time similar to the pre-QM illustrations you may have seen in science books of electric or magnetic fields. With an infinite number of points in any volume, we are asked to believe that some physical mechanism to "encode" the field values exists at each point and that this physical field encoding device must be infinitely small.
If that were not enough nonsense, in addition to multiple fields, the SM lists a number of "operators" which relate to how physical events actually take place. And unbelievably, these operators are represented in SM equations as "acting" at all points. Thus, we have the totally dumb and dumber conclusion that the physical mechanisms that produce events in this world are infinite in number in any volume and infinitely small. Of course, everybody but physicists know that a thing which is infinitely tiny is not a thing at all; it is nothing.
In brief, the physics community may need some tutorials on the difference between mathematics and physics as human intellectual endeavors. The criteria that mathematicians use for a credible concept or proof are different than criteria required in physics. Incidentally, by many usual criteria, the Dirac delta function is not a real or bona fide mathematical function anyway, but just credible enough to seduce physicists into swallowing pure nonsense, to continue support for some two generations of the discredited assumption of continuous space-time and to present this (the SM) as something to be taken seriously.
Perhaps another way to express the relative lack of physical models is that investigators may prefer to refrain from saying, "There is in fact no physical mechanism showing how operator X (pick one) works. It is infinitely small, so it is really nothing. Science tells us that nothing is something. In short, the SM asserts that miracles do exist."
In summary, the currently accepted textbook theory in physics called the SM is based on mathematical absurdities and its mathematical foundations are not self-consistent either logically or physically. Luckily, this problem has been fixed by quite simply defining fields and operations at some selected points, and not at all points. As a result, there are no infinities (also known as singularities) and no need to fix errant calculations by renormalization. The foregoing is equivalent to saying that space, time, and energy need to be quantized for physics theory to mature to adulthood, so to speak. Research along these lines is presented in this Journal of Binary Mechanics.
© 2015 James J Keene