A Child's Approach to the Quantisation of Space

We will not define space (or the more fundamental concept of position) here, but rather will try to discuss on the issue of whether there are any elements of space that could be eliminated. We know that space is the set of all possible positions of any physical system, and it exists because of the way systems exhibit changes in position as an inherent property or in response to physical forces. Space does exist, because of perceived changes in position. Therefore, we may take a probe-system and see what the possible changes in position are during its motion and thus define space operationally. If, however, we discover some regions of the predicted space that show unobservable changes in position, we may render that as a thing to be eliminated from the theory. Space does not exist, because of unperceived changes in position. Now, the perception of position is made possible by radiation; most commonly, it is the visible range of the electromagnetic spectrum – “light” (a magnificent element of the universe, as fundamental as space and time). However, any radiation such as gravitational waves could be used to measure positions (in common words, just by ‘feeling’ the gravity we could tell that there was something over there). So we see that the existence of space is fundamentally determined by the perception of radiation. Space does exist, because of perceived radiation.

But, according to Plank’s quantum theory, all radiation must be discontinuous consisting of distinguishable individual tiny packets of the radiation, the energy of each packet varying linearly as the frequency of the radiation. This little fact adds a new turn here. The quantization of radiation is perceived, if and only if the time lag between every two consecutive packets of the radiation is perceived to be a non-zero quantity. The time lag being zero implies the radiation being continuous. So the corollary is: this time lag without which radiation is continuous, between two successive quanta of radiation eliminates the space of observation in between. This, it seems, breaks the continuity of space. The space vs. time plot of our probe-system moving across space would exhibit periodic dark splashes in time when the radiations breaks because of quantization and so does our perception of space which is due to radiation. For all practical purposes, this breakdown of the continuum of space would be in so negligibly small scales that we would not notice it in our daily life. It must be noted here that the information of the positions of our probe-system during the said time lags would be completely lost. One of my colleagues argued that radiation is not the only means of conveying the information of positions of objects in motion: we could for instance, measure the entropy change of the universe during these time lags and thus retrieve the information of where the object has been during that time. But the problem here is that even entropy changes are due to energy transactions which are fundamentally quantized. Therefore it is apparent that whatever means is taken to measure changes in position, the dark splashes in time are an intrinsic part of a complete description of motion. Space is kind of quantized in time.

            Here is a rough sketch of what the space vs. time plot of a uniformly moving object would look like at extremely tiny scales.

             Note in the figure that the dark splashes are periodic in nature. And we lose information of the graph in those splashes. It is to be remembered that the dark splashes are the collapse of perception in times when there is no radiation due to quantization.

            Let us now make a quantitative evaluation of some specific case regarding this issue. We have talked on length about time lag between successive quanta of radiation. Let us now determine experimentally a very rough estimate of the time lag between two successive photons of known frequency ‘f’. We take a certain amount of a gas, say ‘N’ molecules of it, which has reasonably ideal behavior and put it in a container that maintains isochoric conditions at all times. We assume that the degrees of freedom of the gas molecules are ‘σ’ and the Newton’s cooling constant associated with it is ‘K’. Now we expose the gas to an environment that has a temperature equal to ‘T_a’, and shine monochromatic light of frequency ‘f’ on it as the only external source of heat transfer to the gas. At some point of time we measure the temperature of the gas to be equal to ‘T₀’, and exactly after a time interval ‘t’ we again measure the temperature of the gas and find it to be equal to ‘T’. These data that we have collected can predict the time lag ‘τ’ between two successive photons according to the following equation.
$$ \tau \approx - \frac{2hf}{\sigma N K k \Delta T} \log \Big( \frac{T - T_a}{T_0 - T_a} \Big) $$

where,              $h$ = Plank’s constant,     $k$ = Boltzmann constant,     $\Delta T = T - T_0$.

If we conduct the experiment in empty space, the equation would reduce to the following.
$$ \tau \approx - \frac{2hf}{\sigma N K k \Delta T} \log \Big( \frac{T }{T_0} \Big)$$

According to Newton’s law of cooling,
$$ T(t) = T_a + (T_0 - T_a) e^{-Kt} \\ i.e. \quad t = -\frac1K \log  \Big( \frac{T - T_a}{T_0 - T_a} \Big) $$

We know from the laws of thermodynamics that during an isochoric change in temperature, the corresponding change in total internal energy of the gas is governed by the following equation.
$$ \Delta E = \frac12 \sigma Nk\Delta T $$

This change in energy is due to the energy delivered by the incoming photons. If ‘n’ photons were required to produce the above energy, then by Plank’s equation we have the following.
$$ \Delta E = nhf $$

Equating the two, we can solve for ‘n’.
$$ \frac12 \sigma Nk\Delta T = nhf \\ i.e. \quad n = \frac12 \frac{\sigma Nk\Delta T}{hf} $$

Now we know how many photons arrived in how much time. Therefore if we distribute the photons evenly over the time interval, we can measure the spacing in time between two successive photons. So we have the following.
$$ \tau = \frac{t}{n} \\ i.e. \quad \tau = \frac{-\frac1K \log  \Big( \frac{T - T_a}{T_0 - T_a} \Big)}{\frac12 \frac{\sigma Nk\Delta T}{hf}} \\ i.e. \quad \tau = - \frac{2hf}{\sigma N K k \Delta T} \log \Big( \frac{T - T_a}{T_0 - T_a} \Big) $$

This is how we arrived at the equation. Please note here that we have derived the above relation based on Newton’s law of cooling which is only an approximate description of the phenomenon and is at most times inaccurate and a merely a very rough estimate. Hence the truest thing to write would be the following. $$ \tau \approx - \frac{2hf}{\sigma N K k \Delta T} \log \Big( \frac{T - T_a}{T_0 - T_a} \Big) $$

Note here that the negative sign has here popped in because we have implemented Newton's law of "cooling" on a "heating" effect. Therefore the negative sign can be ignored and dropped without any logical blunder. That gives the magnitude of the required time. Obviously, negative time has no meaning at all.

I do not have the apparatus or any means to conduct the experiment. Therefore I go no further. In thought experiments, we have seen that it is possible to estimate, although very roughly the time lag that exists between two successive photons and determines the breakdown of the continuity of space.

[This text is an excerpt from a project I completed last year, March 14, 2011]