Spectral/Spatial Complemetarity

We can also extend the uncertainty principle to information carried in a beam of light after it passes a lens. Depending on the location of a screen behind the lens, we can either have a sharp image of the object in experiment, or we may come up with a fuzzy image or no image at all. On a proper location every portion of image data has a defined spatial location. Whereas in no image locations the information is carried by light rays are at spectral version and they do not exist in spatial location. Interestingly, in spectral form the data are non local. Which means every small portion of space carries all of the information about our object in hand.

Here we have a complimentarity relation between spectral and spatial features. As we come close to focal point we depart from spectral phase and come to locality. With the same token when we leave focal point we enter spectral phase again and the non-locality prevails. Isn't this a good example for understanding the relation between a spatial local world and non-spatial non-local energy-informatics domain? In our model this paradigm is the proposed singularity.

Information/ Interference Compementarity

George Greenstein and Arthur Zajong mention another very interesting complimentarity relation in their book “The Quantum Challenge”. They explained the complimentarity between information and interference pattern in double slit experiment (see the explanation under the same subtitle in this chapter). Referring to moveable slit modification of the experiment, they wrote:

“Wootters and Zorek have returned to Einstein's modification of the classic double-slit interference experiment, and analyzed it from the stand point of partial information…The slits are free to move. After particle has past through, we measure the slit's momentum… if the slits are moving downward, the particle must have past through slit 2…

Wootters and Zorek noted that the above conclusion is not in fact entirely certain. The same motion of the slits would also be observed had the particle passed the wrong slit --if the initial slit momentum had been large and downward…

They evaluated the probability of the initial slit momentum being large enough to yield such an erroneous conclusion, and so obtained an expression of the probability that we had obtained path information. Using the same wave function, they also calculated the resulting pattern of arrival at the final screen. It turned out to be a partially smeared-out interference pattern”¹¹

Reproduced from Ref. #11 page 101

Their experiment showed that if they had certain knowledge about the slit that particle passed through, there where no interference on screen. But as the uncertainty about the slit increased a better interference were developed.

Looking at different complimentary pairs, the elements for each uncertainty relation are different in that, one of them is either spatial or mass type, and the other is either energy type or informational type. The two characteristic that we hypothesized for proposed singularity.  The fuzzy states of fundamental elements in boundaries of space-time were discussed in Boundaries Chapter.

For details check the link: http://www.upscale.utoronto.ca/
GeneralInterest/Harrison/SchrodCat/SchrodCat.html

Objects in classical physics terms are in a well defined state, whereas in quantum mechanics the objects are not in a definite state. For example, in classic physics an object will rotate either clockwise or counterclockwise. It was shown that in microcosm, a particle is spinning simultaneously in both directions. In other words, a particle concurrently exists in any probable state that is possible. Erwin Schrödinger who first explained the super-position of states offered the analogy of a cat being dead and alive at the same time after the probability of being poisoned is famous and is named after him.

In classical physics terms either the poison is released and cat dies or the poison is contained and cat would be alive. But in quantum mechanics arena, the both outcomes exist simultaneously. This is of course before we open the box. As soon as we open the door we just see one outcome (either a dead or a live cat).

The Schrödinger’s cat is simultaneously dead and alive. Such a sentence does not relay any meaningful concept to us. Or does it?

In the observable world a cat is either dead or alive. The series of events lead to just one of the possibilities.

How are we going to explain this quantum mechanical effect.. Multi universe theories claim that there are different universes that accommodate different possible outcomes, of each action. This is hard to accept. There are countless actions in each miniscule of time in the universe and far more possibilities as well. Possibilities are endless. It means we have to have endless numbers of universes and the number is growing every second at a rate that transcends all concepts of infinitudes. It is also grossly against conservation of energy law if we choose to hold it. This concept is not economical either.

On the other hand, Schrodinger's wave equation which represent the super-position of states, also contains the imaginary factor i. Thus quantum state “will always turn out to contain terms that are imaginary…The complex character of the wave function in Schrodinger's wave equation means that what is there in a sense is hidden from us.”⁸ In other words, Somehow in quantum arena we are exposed to out of space-time realm.

Renormalization in Schrödinger’s Equation

The Schrödinger’s probability equation for the position of a particle during its wave function can be written as:

Ψ * Ψ = Probability

If we believe that the particle is somewhere in space but we cannot exactly pin point it, we can normalize the formula and write:

∫ Ψ * Ψ cr = 1

Which means that we have altered the formula in a way that would show the probability of the particle being somewhere in space is %100. We do this because we believe that everything is confined inside the space-time. So, it is natural to assume that the particle is in space somewhere. Therefore, we write a formula to reflect a normal situation by our space-time logic. If we are revisiting the existing concepts then we are allowed to question normalization as well. Normalization is a major parts of quantum mechanical calculations. If we take normalization out of quantum calculations, we tremble the pillars of its existing mathematics.

Questioning the normalization is similar to questioning the attempts to understand the reality on the basis of space-time properties alone. This is a courageous action. But haven't we been very bold so far? As a matter of fact, solving these great mysteries need enormous amount of daring. At the same time revisiting the calculations can confirm or reject the validity of the singularity concept presented above. It seems that if we redefine the particle wave function to include singularity (a mind like informational domain), we will come to a logical explanation for the phenomena that we are facing. I do not see why we shouldn't let our imagination explore that possibility. Aren't we defining the location of each particle with a complex number, which contains imaginary part as inseparable feature?