Unitarity Principle of Quantum Mechanics


In 1926, Max Born suggested that the probability of finding an elementary particle such as an electron in a given location is |\psi|^2. The  \psi is known as the quantum mechanical wavefunction. You can read more about the quantum mechanical wavefunction here.

Remember, wavefunction is complex-valued. When you square a complex number you can get a negative number. This is why you have to use the “absolute square” to get the probability density. Negative probability is not physically meaningful. Hence the absolute value signs in |\psi|^2.

If you want to find the probability of finding a particle in a specified volume then you sum up (integrate) |\psi|^2 over that volume. If you sum up (integrate) over the entire space you should get Probability=1. This interpretation forms the core of the Copenhagen interpretation of quantum mechanics. Max Born was awarded the Nobel Prize in physics for this interpretation of wavefunction in 1954.

The “unitarity principle”  of Quantum Mechanics refers to the fact that when we sum up the values of the |\psi|^2 over the entire space we must get 1. Unitarity refers to this number 1.

What is the significance of 1? The assumption is that a particle is entirely in this universe. Other possibilities such as the particle being partially in this universe and partially in a different universe is not acceptable according to the Max Born interpretation of the wavefunction.

The wavefunction evolves according to the Schrödinger or Dirac equations but while it is evolving the Born probability interpretation must be obeyed. One year from now or 1 million years from now the probability of finding this particle in this universe must still be 1. This particular particle, if still coasting in the universe, must still be in this universe. But, elementary particles do not live that long. Most of them live for a fraction of a second. It is well understood that the Max Born interpretation is meant to say that the total sum of the values of the |\psi|^2 should be 1 while the particle is in existence.

The significance of the fact the total sum of the values of the |\psi|^2 should be 1 while the particle is in existence is also expressed by saying that no information is lost in a quantum mechanical process.

There is an enhanced (supposedly better) version of Quantum Mechanics known as the Quantum Field Theory (QFT). You can read simplistic descriptions of QFT here. In QFT the “wavefunction” is replaced by the “quantum field.” The “wavefunction” does not create or destroy particles but the “quantum field” does. If that’s the case then what does “unitarity” mean in the context of QFT? That’s a very good question and physicists do not have a good answer for that. Let me know if you find a satisfactory answer.

The “unitarity principle” of Quantum Mechanics played a central role in the so-called “The Black Hole War” which is also the title of a very popular book by Leonard Susskind. In his book Susskind reminds us that the “unitarity principle” also implies the mathematical time-reversibility of Quantum Mechanics.

Going back to my problem as to how QFT explains “unitarity” in the context of “quantum field” which is free to create and destroy particles randomly, Susskind gives an answer as follows: (In my opinion the explanation below is not satisfactory and indicates a weakness in both Quantum Mechanics and Quantum Field Theory)

“Take the photon. When we run the photon in reverse, does it reappear at its original location, or does the randomness of Quantum Mechanics ruin the conservation of information? The answer is weird: it all depends on whether or not we look at the photon when we intervene. By “look at the photon” I mean check where it is located or what direction it is moving. if we do look, the final result (after running backward) will be random, and the conservation of information will fail. But if we ignore the location of the photon – do absolutely nothing to determine its position or direction of motion – and just reverse the law, the photon will magically reappear at the original location after the prescribed period of time.”

Here’s another explanation/clarification by Philippe Jacquet (http://hipercom.inria.fr/~jacquet/retro1/crazy-science-corner/unitarity.html)

“In fact we don’t need to visit a black hole in order to meet non unitary effects. In theory the measurement of a physical parameters, for example the position of a particle is called a wave function collapse and is highly non unitary. In other words basic quantum physics is per essence unitary when we don’t touch it, but it becomes non unitary in its basic connection with reality. But this is far too easy! One way to escape this apparent paradox is the multi-universe (sometimes called multi-verse theory, since the prefix «uni» refers to unicity): when a measurement is done on the spin of a particle the universe forks and one branch contains the universe with the particle spin measured in one direction, and another branch contains the the copy of universe with the particle spin measured on exactly the opposite direction. In other words, if this morning you have selected the color of your socks via a quantum random generator, then there is one branch of the universe where you wear red socks and another branch where you wear blue socks.”


About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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