An Analysis on Randomness and Probability Theory in Quantum Physics

In 1654, a dispute between gamblers led to the invention of Probability theory[1] by Blaise Pascal and Pierre De Fermat. Since then probability is being used extensively in many aspects of life. The chances of rainfall or the odds of survival from a disease, the probability of a team winning a cricket match or the possibility of you winning in Roulette - Probability theory can be made to fit almost everywhere when one isn't certain about the outcome of an event. The merging of Physics with probability theory was complimentary in the development of Statistical Mechanics and Thermodynamics. In the 20th century, the discoveries of physicists like Max Planck, Albert Einstein, Werner Heisenberg, and Erwin Schrodinger gave us what we call today - "Quantum Mechanics". The Quantum theory is said to be inherently indeterminate and random. Along with quantization of physical properties like charge, spin, and energy, the results of any quantum experiments are totally random. In 1926 - A German physicist by the name of Max Born introduced the "Statistical Interpretation" [2] of Quantum Physics. The fusion of Probability theory with the mathematical formalism of Quantum Physics was quite revolutionary. However, the result was like a rose with thorns. Physicists started doubting whether the randomness in Quantum Physics is inherent or is there a set of hidden observables which when unlocked can rectify this quite irritable uncertainty. Eventually, this created a divide amongst them and produced a plethora of different interpretations each favoring their own views. The differences still pertain, but overall the interpretations can be fit into two characteristics : First category states that the quantum realm is truly random and probabilistic and that there is nothing that we can do to get rid of it; Second category holds the belief that the randomness is because of our own ignorance and that it is possible to get rid of it by finding the "hidden variables". There is a third category that includes all the "miscellaneous" but equally important interpretations.[3]

The real question remains - "Which side is correct?". Many like to shake it off by stating that the interpretation we choose has no effect on reality and that the consequences of Quantum Mechanics still hold. Even though this statement is indubitably true, a correct interpretation is significant in making further discoveries. For example, if the Hidden Variable Theory is true (which it isn't) then the indeterminacy totally vanishes and Quantum Mechanics just becomes another extension of Classical Physics. In order to ponder upon the validity of statistical interpretations, it is important first to revisit and analyze some of our preconceived notions of the Probability Theory.

The Classical Probability Theory[4] consists of an event and a set of all the possible outcomes from that event. The probability of occurrence of each outcome is then the ratio of the number of that outcome to the number of total outcomes from the set. For example, a simple event of tossing a coin comprises of two total outcomes: Heads and Tails. Therefore, the set of total outcomes is {Heads, Tails}. The probability of the coin landing on Heads or Tails each is 1/2 or 50%. The set of total outcomes is commonly labeled as "Sample Space" of that event. In the rolling of a dice, the probability of getting each number is 1/6 or in a deck of cards the probability of drawing any card is 1/52. However, by introducing a constraint on the event one can get more specific about the probabilities. For example, the probability of drawing a queen from the deck of cards then becomes 4/52 or the probability of getting an even number on the dice becomes 3/6 etc. There is an increase in the probability if we introduce certain constraints or conditions. Whereas, vaguely stating the probability of occurrence of any outcome in a large enough sample space is miniscule. Let us try to crank up the constraints and become more specific. What is the probability of getting 5 if I throw the dice such that its top face flips through a complete 180 degrees? - Inspite of the subtle detail, the probability still remains 1/6, because we have incomplete knowledge about the number at the top face when the dice is being flipped. If the number at the top is six, which say is exactly on the opposite side of the face having number 5, then we are certain to get the number 5 after a 180 degree flip. However, we don’t have complete knowledge about the number on top face. What if I know that the number at top is 6 and then flip the dice to make a 180 degree turn? - The probability of getting 5 then jumps to 1 or 100%. Here the "Hidden Variables" are - the manner of flipping of dice and the number on the top face. Such conditional probabilities are not new and already exist in Quantum Mechanics as well. This reduction of probability to 100% after introducing proper constraints corresponds to the collapse of wavefunction[5] after a measurement is performed. The original problem still persists which is, whether this probability can be reduced by introducing some constraints other than an obvious measurement or that the randomness is actually inevitable.

We can resort again to the mechanics of macroscopic scales. Revisiting the dice example - When I said that flipping the dice by 180 with 6 on the top face would certainly give us the number 5, it came to us almost naturally. The certainty is never doubted upon - "What if the dice decides to speed up on its own?" or "What if it decides to stop suddenly?". Of course this shouldn't be possible. For a layman, the reasoning behind this might be just intuition. But, we really have Newton's Laws of Motion at play here. The First Law of Motion which states that a body at rest stays at rest and a body in motion stays in motion, unless acted upon by an external force. Given that, there are no external forces of wind or anything else, the dice will remain in motion and will only stop after it lands on the ground. In general, if I am provided with all the possible details like - the rotational torque applied on the dice, the mass of dice, air resistance, force of gravity, initial alignment of the dice faces, etc. I could use Newton’s Laws of Motion to determine which face I will get at with a hundred percent certainty. Thus, we need not one but two general constraints to eliminate uncertainty. Firstly, all the possible details about the system in use and secondly, a set of rules obeyed by these details that enables us to know with certainty how these details will govern the outcome. Here, a surprisingly odd truth surfaces up – Randomness is nothing but an illusion. An illusion that arises due to uncertainty and lack of complete knowledge. Any event outcome which is thought to be random is actually governed by the two generalized constraints mentioned earlier. Now, if I provide two examples in front of you – The rolling of a dice and generating a random number from a computer. The latter event uses an algorithm to give out any number. In both cases, having the complete knowledge of how the dice is thrown and the exact algorithm used by the computer, it is evident that none of the events are “truly” random. But, suppose I am allotted the task of convincing my customers who are willing to buy the computer that – “The numbers generated by this computer are truly random”. I can choose a completely bizarre algorithm, which is nothing but a mathematical formula that takes one number at a time say from 1 to 1000 and feeds it into the formula, giving out another number. For example, take any number, multiply it by 2, add 19239 to it, and then multiply it by another number and so on. This algorithm fairly serves the purpose of generating a random number. I can “randomize” it even more by introducing another complicated arithmetic operation and then choosing the first algorithm one time and the second algorithm other time. Maybe, add another complicated formula and cycle between those three formulas, etc. This becomes similar to tying knots on a rope until it becomes so tangled and complicated that it is almost impossible to untangle it back. The more complications you add, the more random a thing “appears” to be and the harder it becomes to figure out its constraints. Note that in the example of the computer, we are messing around with the second general constraint (algorithm) – which is the set of rules obeyed by the input numbers.

Going quantum again, where does the randomness or uncertainty arises from? The uncertainty arises from the non-commutativity of any two operators[6]. The most famous example is the position and momentum uncertainty – which became known as the Heisenberg Uncertainty principle. Other examples include – the spin operators for any two axes, or the Energy and time uncertainty etc. The fact that you can’t know the position and momentum of any system simultaneously renders it impossible to predict the future state of that system using Newtonian laws. Another form of randomness in Quantum Physics is observed prior to the measurement or observation of any physical property. A particle can exist in multiple positions until it’s observed. This observation collapses the wave function of that particle and it is observed at a certain position. But the position that it takes is completely random. If you pick an ensemble of particles all in the same state i.e. having the same wavefunction then it’s not necessary for all of them to be observed at the same position after measurement. For example, if one of the particles is observed at position A, then the other might be observed at some other position B even though both of them have completely identical probability distributions and wavefunctions. What is the origin of this randomness? – If randomness is nothing but an illusion then it maybe possible to eradicate it from Quantum Physics. One of the solutions given by a “Hidden Variable Theory” was disproved and became incompatible with observations by John Bell in 1964. His work is known as the – “Bell’s Theorem”[7]. With this, the dream of certainty in Quantum Physics went down the drain. However, the Hidden Variables may be only one of the many possible attempts to make quantum theory non-random. The disproving of this theory meant that we should work with what’s available – either the position or the momentum. In other words, the first general constraint is only restricted to position or momentum. Therefore, the use of a different second constraint may be helpful. To see this clearly let us go back to the very first example of flipping the dice by 180 degrees. It was seen that if we know the number on top face is 6, then we are certain to get the number 5 after a complete 180 degree flip. The two general constraints here were the knowledge of number 6 being on top face and Newton’s law of motion assuring nothing fishy will happen after you throw the dice to make a 180 flip. But suppose now, inspite of knowing all the possible details, we still don’t get the number 5 on top face. Let us say we get the number 1. Flip the dice again and we get 3, flip it again and we get 1, then 2 and so on. This soon becomes a frustrating problem because now the numbers we are getting appear to be random. Every time you flip the dice, it is in the same state (same number on top and same 180 degree flip). Just like the quantum particle is in the same state every time before measurement. Our first school of thought obviously goes to the “hidden details” solution. There is an additional detail about the dice which we don’t know and is hidden from us. This hidden detail is responsible for the randomness in the number that we get. Unfortunately, the John Bell of ‘dice throws’ comes along and very elegantly disproves this just like in Quantum Physics and says – “There are no hidden details!”. Since, the first constraint cannot be changed, we can move on the second general constraint, which are the rules obeyed by the details – Newton’s Law of Motion. The laws of motion should be altered such that they are able to predict the outcome of dice with the available details.

The above scenario was of course imaginary. Newton’s Laws of Motions are correct and so repeating the experiment again and again with the same conditions should give us predictable outcomes. Nevertheless, we dont observe the same situation in quantum physics. In order to achieve certainty with the available details, a change in the second constraint should be made. The entire framework and postulates of Quantum theory should be altered to predict the outcomes with available details. This article is a counter-argument to the belief that randomness in quantum physics is inherent and inevitable but gives no elaboration on how the randomness in quantum physics is different from ordinary randomness which can be removed by knowing the constraints. The bottom line is that quantum world may or may not be truly random, in either case – a redefinition of some of our most fundamental notions on randomness and uncertainty is necessary.

- Thank you.

References

[1]https://en.wikipedia.org/wiki/Probability_theory

[2]https://people.physics.anu.edu.au/~cms130/phys2013/fundamentals/Stat.htm

[3]https://www.youtube.com/watch?v=mqofuYCz9gs

[4]https://www.statisticshowto.com/classical-probability-definition/

[5]http://www.quantumphysicslady.org/glossary/collapse-of-the-wave-function/

[6]https://quantummechanics.ucsd.edu/ph130a/130_notes/node188.html

[7]https://plato.stanford.edu/entries/bell-theorem/