Imaginary time

severianx (50)in #science • 6 years ago (edited)

Ink, cream, sugar, and heat

The image shows a drop of ink in water. Imagine how it slowly dilutes, becoming more transparent. This phenomenon is called diffusion.

(CC BY-SA 4.0, by Zvonimir Lončarić)

Diffusion is a physical process by which energy and matter are transmitted from one point in space to another. Not only a drop of ink in the water experiences diffusion

Any substance (matter) that can be dissolved in a solvent is diffused into it. Ink in water, sugar or cream in coffee, and almost any example we can imagine.
Heat (energy) is diffused into bodies. The resistance inside the iron generates heat that diffuses into the metal, and when touching the fabric of the shirt also diffuses into it.

Diffusion works in a very simple way, is a tendency to equate concentrations. It is described with a fairly simple equation, which I will try to write here. Don't panic, it's very easy, and it's nice to understand how equations are used to explain reality.

Suppose we have a region of space with a part that we can call its interior and a surface that acts as a border. It could be a drop inside a glass full of water. Suppose that region is full of something whose diffusion we want to describe. For example, the drop might contain ink.

(PD)

The content leaves the region across its border. The ink comes out of the initially darker area through the surface surrounding that area. We can establish a simple law to describe this phenomenon:

The amount of content that escapes every second from the interior is proportional to the difference between the interior concentration and the exterior concentration, and also to the surface of the border.

Sounds complicated? Read it again, it is very simple: the world tends to homogeneity.

If inside the region the ink is very concentrated, and outside it is very diluted, the amount of ink that crosses the surface per second will be large.
If, on the other hand, the concentration inside is barely greater than the concentration outside, the ink will escape very slowly.
And of course, if the concentration outside is greater than the concentration inside, the ink will enter the region instead of escaping from it.

It's simply the matching tendency that things have. A kind of quest for homogeneity.

Let's write an equation to describe the phenomenon. The concentration of ink is the number of ink molecules divided by the total number of molecules. In other words, it is the probability P that by taking a molecule at random, it is ink and not water.

If N is the number of molecules within the region, and dN its change in a time lapse dt, the equation is

dN = k (Pe - Pi) S dt

where Pe is the outer concentration and Pi the inner concentration, S the surface of the region, and k is a constant that characterizes the process.

This diffusion equation is simply the mathematical expression of what we explained before. If you do not understand the formula, don't worry, it doesn't contain new information beyond what the text says.

This equation controls the physical processes of dissolution of matter or energy. It tells us how ink is diffused in water, sugar or cream in coffee, or heat in the iron.

Electrons, protons and other small objects

The interesting thing is that the diffusion equation resembles very much the Schrœdinger equation, which describes the quantum mechanics of very small objects such as subatomic particles.

(PD)

The Schœdinger equation for a particle can be written as follows

dN = i h (Pe - Pi) S dt

Where N is now a measure of the probability of finding the particle within the region, Pe and Pi of finding it on either side of the surface, i the square root of -1, and h a constant known as Planck's constant.

As we see, the equation is very similar... it almost seems that if we change the name to h calling it k it would exactly be the same... except for that damn i in front... If we wanted to read the equation to understand it in the words that we understood the diffusion equation, what role would that i play?

The caveat here is that in quantum mechanics N and P are not really probabilities but amplitudes, i.e. numbers that in principle can be complex. Then we can identify our diffusion constant with k = i h, and the interpretation in words is the same as before.

The amount of content (amplitude) that leaves the region in each second is proportional to the difference between the interior amplitude and the exterior amplitude, and to the surface of the border.

It doesn't sound bad. The only problem is that, amplitude being a complex number, we lose intuition about things that then to equate on each side of the surface.

However, there is another possible interpretation. Let's change the name k = h and redefine the way we measure time as dT = i dt. Again the equation reads like this:

The amount of content (amplitude) that leaves the region in every imaginary second, is proportional to the difference between the inner amplitude and the outer amplitude, and to the surface of the border.

Now if the amplitude is initially real number, since there is no i in the equation it will remain real over time. In this way, we recover the intuition of things that are homogenized that we had with the diffusion of the ink.

Ok, very nice, but... what the hell is an imaginary second? Here comes the pragmatism of physicists: an imaginary second is a convenient way of transforming the Schrœdinger equation into the diffusion equation, keeping k real.

With this reinterpretation of the Schœdinger equation, we can say that electrons, protons, or as subatomic particles _are diffused into space as if they were ink, only they do so when time runs in the imaginary direction.

-- But wait, time is a real number, my clock does not mark imaginary numbers. An imaginary time means that the universe...
-- Shut up and calculate!

It often happens that very different physical situations become similar if we imagine that time can be an imaginary number. If this is a simple mathematical coincidence or if it has some deeper interpretation, it is something no one knows.

But wow!