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The Riddle of Infinities

In the Introduction section, I questioned if re-normalizing within the mathematics of theoretical physics is actually blocking path to exploring reality. Steven Weinberg explains the origin of re-normalization:

“This term goes back to the 1940’s when physicists where learning how to use the first quantum field theories to calculate small shifts of atomic energy levels. They discovered that calculations using quantum field theory kept producing infinite quantities.[4]

The infinities that emerged in these calculations were considered flaws. Given that the finite universe cannot include infinities, their appearance seemed to indicate that the process had been pushed beyond the limits of its validity. in calculus we insert limits to avoid zeros and infinities.

Physicists consider infinities, which occur at the scale of the ultra-short, a major problem. Infinity in this model is defined. It refers to energy and information in the proposed singularity. As mentioned before, re-normalization is the act of ignoring and bypassing zero and infinity in calculations. Theoretical physicists use the Bekenstein bound (mentioned above) to get rid of the infinities in the calculations.

Dirac's Delta Function

The delta function is a generalized function, on the real number line where it is zero everywhere else except at the zero point. At the origin it can spike to infinity. One can consider delta function as a model for the singularity containing infinite energy or information. 

In Algebra we write,

X/0 = infinity

In mathematics, the above equation is called undefined. In this model infinity is defined, therefore the equation is valid.

If we confine ourselves to space-time physics, we have to frequently re-normalize our abnormal(!) findings. More and more evidence compels us to leave our ken and believe that actual physics extends beyond the familiar space-time universe. Once we can accept the possibility, maybe there will be no further need to re-normalize.

In conclusion, zero and infinity cannot be ignored and avoided. They should also be considered a part of reality. Zero and infinity have profound effects on our world. We just need to speculate on an identity for them and include them in our theories, not dismiss them as meaningless entities. Fifteenth-century mathematicians had to expand the mathematical arena to include negative numbers. It took us until the nineteenth century to describe a physical meaning to them, for example the negative charges in electromagnetism. We should keep in mind that there are also imaginary numbers for which we have not yet found an exact physical meaning. The actual physics extends to unkown territories, which are vast, active and effective.
The presence of constants in our calculations points to unknown factors that affect our universe. It is not humanlike to accept that the meanings of these factors are beyond our reach. We are hunters in the dark, and so far we have been finders.
3.) Craig Hogan, “Is Space Digital,” Scientific American (February 2012): 32–37

4.) Steven Weinberg, “A Unified Physics by 2050” Scientific American 13, no. 1.

Holographic Universe

As mentioned before, holographic principle asserts that the information of any region exists at the boundaries of that region. We don't need to look at the far away boundaries. According to the holographic universe theory,the region can be as small as we choose, even as small as a Planck cube (about 10-99 cm3). This is the only way that boundaries of any region can contain the information of that region.Therefore the entity at the boundaries of Planck cubes are supposed to accommodate the data. Supposedly, within the Planck cubes space is not defined and we are facing with absolute void. This void accommodates the information. In this view this entity is the space-free singularity.

Fabric of the Universe

Many physical theories consider space as a continuum. But others, like loop quantum gravity, picture time and space as discrete entities. Mathematical discontinuity represents holes in the fields. Dividing a real number by zero creates discontinuity in the math domain. This same problem arises in the general theory of relativity. Because of this, many theoretical physicists believe there should be holes in space-time.

The Planck discovery mentioned above led to the discovery of other fundamental constants. The Planck constant is the origin of a system of natural units known as Planck units. The Planck length is the “atom of length,” or the smallest length possible. It equals roughly 1.6 x 10-35 m, or about 10-20 times the size of a proton. This is the scale at which classical ideas about gravity and space-time cease to be valid. Inside the Planck length, the notion of space is no longer valid. 


2.) Gordon G. Globus, Brain and Being (City: John Benjamins Publishing Company, 2004).

Planck time is the time it would take a photon travelling at the speed of light to cross a distance equal to the Planck length. This is the “atom of time,” the smallest unit of time, equal to 10 -43 seconds. No smaller division of time has any meaning. We can therefore think of the Planck time as the building block of time. Outside these boundaries, the meaning of space and time break down.Can we then suppose that at the boundaries of each Planck distance, our matter-space-time universe ends?

If a distance less than Planck length and a time less than Planck time have no meaning, can we assume that whatever lies inside Planck unit limits is out of our space-time universe, because it does not contain meaningful space or time? 

Steven Weinberg, “A Unified Physics by 2050?” Scientific American 13, no. 1. 
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