Why "Reality Conditions"?

As a first try at explaining a mathematical or physical concept, I provide here a clarification on the meaning of the name of my blog. Feedback on the intelligibility of the explanation will be welcome.

In physics, all quantities that have a definite physical meaning must be real, i.e. take values on real numbers. This is simply because the result of a measurement is always a real number. However, it often happens in physics that we need to use in our calculations complex numbers, which have a real and an imaginary part (the latter proportional to i, the square root of -1). How do we relate then the result of the calculation to a measurable, physical quantity?

With the just one exception to which I shall return later, the general fact is that complex numbers are not essential to the calculation but just a device to make it simpler. At the beginning there is a set of real quantites we want to calculate, but the equations for them are complicated and difficult to solve. Often the equations become simpler, or more useful mathematical techniques for solving them become available, if we define complex variables instead of the real ones. After solving the problem, we need to remember to go back from the complex variables to the real ones. The "reality conditions" are a set of equations that tell us how to do so. They specify which of the many quantities we may be working with are real, and how they are related to the complex ones.

As a trivial example, suppose we are trying to solve equations for two physical quantities x and y. We may notice that the equations become simpler if we use instead of them the complex variable z = x + i y. After solving them and finding z, we use the following two equations to "go back":

x = (z + z*) /2

y = (z - z*) /2 i

Here z* means the complex conjugate of z, which amounts (at this level) to replacing i by - i. These equations are the reality conditions.

In a more complicated case, with many variables of different kinds and satisfying all kinds of complicate equations, the reality conditions may be highly nontrivial. They play, for example, an important role in the approach to quantum gravity known as "loop quantum gravity" (LQG). (Googoling the phrase "reality conditions" gives you for the moment mostly papers on LQG or references to Alex Kasman's book linked to in the sidebar... but we'll see in a few months!)

So why did I choose this as the blog title? Well, on one hand it is a notion in mathematical physics, which will be one of the topics of this blog; but it also can be given a philosophical meaning (conditions set by reality on a theory or idea, or the conditions something must satisfy to be real) and philosophy will be another topic much discussed here, and finally one could read it in a concrete sense as referring to the real conditions of my present life. So I thought it would be an ideal title. Don't you agree?

So, what was the one exception to which I would return later, you must be wondering now? (Or had you forgotten? Shame on you!) Well, complex numbers play an essential and not merely an auxiliary role in that fundamental, strange and wonderful branch of physics called quantum mechanics. Quantum mechanics (QM) is all about calculating some complex quantities called "transition amplitudes", which are defined between any two states of a (quantum) physical system. To calculate the probability that the system will "jump" from state A to state B, the recipe is: use QM to calculate the transition amplitude between A and B, then multiply the complex number you have obtained by its complex conjugate. The result of this is a real number, which is the probability of the jump! I will write more on quantum mechanics another day.