Add new comment
Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.
What do chocolate and mayonnaise have in common? It's maths! Find out how in this podcast featuring engineer Valerie Pinfield.
Is it possible to write unique music with the limited quantity of notes and chords available? We ask musician Oli Freke!
How can maths help to understand the Southern Ocean, a vital component of the Earth's climate system?
Was the mathematical modelling projecting the course of the pandemic too pessimistic, or were the projections justified? Matt Keeling tells our colleagues from SBIDER about the COVID models that fed into public policy.
PhD student Daniel Kreuter tells us about his work on the BloodCounts! project, which uses maths to make optimal use of the billions of blood tests performed every year around the globe.
Hello. I am an outsider-- an engineer reading to keep brain working. As an outsider I feel compelled to interpret your interpretations. The language in this article about possibility seems to be translated into the language of complex numbers and the language of probability into the language of real numbers. As an outsider here is how that makes sense to me: Say that I want the possibility to win a raffle where one needs to be present to win. The possibility of me winning then involves (at least) two possibilities-- the possibility that I will own a ticket and the possibility that I will be present at the raffle. If either is zero, the possibility of my winning then becomes zero, or impossible. The language here is like the multiplication of two numbers-- if either factor is zero then the multiplicative product is zero. Now say that in the middle of the raffle, the rules change and one need not be present to win. In the language of numbers, this is like dividing the product of the two factors by one of the factors-- the factor translated into natural language that "one needs to be present to win." Factor that away from the multiplicative product, and the one factor that remains is the possibility of owning a ticket.
The natural language translation of words about possibility into the language of numbers therefore means the desired type of number to translate such sentences about possibility must be one of the "division algebras"-- real, complex, quaternion, or octonion. But which of these types?
Point: It doesn't matter which I say first in this "product"-- the possibility of being present or the possibility of owning a ticket. So the translation from natural language into the language of numbers must support multiplicative commutativity.
This rules out mathematical language involving the quaternions and the octonions as the language of choice for translating natural language about "possibility"-- because these types of numbers do not commute, multiplicatively.
So we are left for the candidates of translation of natural language about possibility either mathematical language involving the real numbers or mathematical language involving the complex numbers.
OK here is the next step: Logically, something either is a possibility or it is not a possibility. Whereas something may be more or less probable than something else.
Nothing is more or less possible than something else that is possible. Here are all the cases: (a) Both are possible. (b) One or the other is impossible. (c) Both are impossible. In this way natural language involving "possibility" is logical.
(Note: in natural language there are "connotations." Sentences about possibility "connote" information about probability. In this discussion I ignore these connotations.)
Again without these connotations, something can be more or less probable than something else that is probable. But nothing can be more or less possible than something else that is possible.
In terms of language about number, this translates into the "greater than/ less than" relation. Therefore normalized real numbers are appropriate for translating natural language about probability, since real numbers support the "greater than/less than" relation. And since complex numbers do not necessarily support the "greater than/less than" relation, mathematical language involving complex numbers supports translation into natural language about possibilities.
The Born rule is therefore a translation from language about possibilities (complex numbers, i.e., the wave function) into language about probabilities (the real numbers).
TEST: an interpretation of the Schrodinger equation.
Consider a free particle, where there is no energy potential as a function of configuration. In terms of physical displacement "x," there is in the Schrodinger equation a representation of momentum as the partial derivative of the wave function with respect to "x." And then this partial derivative is multiplied by the complex number "i."
I will attempt a translation of the mathematical language into natural language--
A factor multiplying by "i" -> "a possibility exists that"...
A factor of partial derivative of complex numbered function (the wave function) with respect to "x" -> "given a change in position "x," a change in possibility for existence at each location "x" will occur"
"a possibility exists that given a change in position "x," a change in possibility for existence at each location "x" will occur"
A possibility exists, so it is not necessary. Comparing this to Newton's laws, it is not necessary, but it is a possibility that at some future position "x" the particle in question will hit another particle. In the case of that possibility of hitting another particle, the possibilities for existence for the particle existing at all locations "x" will change.
Or, it is not necessary, but it is a possibility that at some future position "x," the particle in question will enter a field of potential energy depeding on "x." In which case, the possibilities for the particle existing at locations "x" will change.
So far this is just language about momentum. But the Schrodinger equation involves language about energy.
A change in energy, due to an impact with another particle, or due to entering a field of potential energy, means that there is the possibility (depending on changes in "x") of such changes in possibility.
Again, it's not necessary that there be an impact with another particle or that the particle enter a field of potential energy. But according to the Schrodinger equation, these are the possibilities.