Imaginary numbers could be needed to describe reality, new studies find

Imaginary numbers could be needed to describe reality, new studies find

Ben Turner

In fact, even the founders of quantum mechanics themselves thought that the implications of having complex numbers in their equations was disquieting. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrödinger — the first person to introduce complex numbers into quantum theory, with his quantum wave function (ψ) — wrote, "What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Ψ is surely fundamentally a real function."

"The early founders of quantum mechanics could not find any way to interpret the complex numbers appearing in the theory," lead author Marc-Olivier Renou, a theoretical physicist at the Institute of Photonic Sciences in Spain, told Live Science in an email. "Having them [complex numbers] worked very well, but there is no clear way to identify the complex numbers with an element of reality."

To test whether complex numbers were truly vital, the authors of the first study devised a twist on a classic quantum experiment known as the Bell test. The test was first proposed by physicist John Bell in 1964 as a way to prove that quantum entanglement — the weird connection between two far-apart particles that Albert Einstein objected to as "spooky action at a distance" — was required by quantum theory...

Whenever a test of QM is performed, it always comes out in favour of QM - that gets rather boring in the end

I remember a discussion had with EthanT, over at Skeptiko. I said it was strange that imaginary numbers come into QM and Special Relativity.

He replied that there was no role for imaginary numbers in SR. We discussed this (with several twos and fros) until he realised that the old way of formulating SR used a time coordinate of I c t - i.e. the speed of light times time times sqrt(-1).

This makes Pythagorus' equation look the same in 4 dimensions - distance between 2 4-dimensional points =  Δx^2 +  Δy^2 + Δz^2 +  Δw^2 where w=i c t.

It turns out that the modern rendering of SR uses something called a 'metric tensor', which obscures the fact that imaginary numbers enter SR too!

I think the metric tensor came in with General Relativity, and it is something of an overkill to use tensor algebra in SR.

It always seemed to me extremely strange that sqrt(-1) has this role in both these theories. Normally sqrt(-1) enters mathematics just because it simplifies a lot of algebra. I think fundamentals like this should be given a lot more thought.

A couple of extra thoughts. The idea that we live in a four-dimentional universe, is a stranger concept than it at first appears. For example the concept of a 'block universe' isn't what it seems, because if you conceive of the thing conventionally (i.e. without a metric tensor, as I suspect most people do) then the its time dimension is imaginary! Conversely, if you think of the thing in terms of the metric tensor, then the space they occupy is not analogous to ordinary space at all.

One example of this, is that if you go out and look at the stars, you may be looking at a star as it was 1000 years ago - because it is 1000 light years away. However if you look at the expression for the 4-d distance between you and the star - you get the answer zero!.

This leaves me with the feeling that physics ideas of higher dimensional spaces may be quite misleading.

Incidentally, it is possible to conceive of a 'conventional' 4-d cube - known as a tesseract - and its properties have been studied. It just isn't the same as a cubical chunk of a block universe.

I was kind of hoping to get a bit of discussion going! I mean the concept of the block universe features quite a lot as in connection with the concept of Akashic records.