First Week BeyondResearch Insights, Year 1

Hi! My name is Berenika and I am a first-year BeyondResearch scholar. Here I would like to share some insights from last week’s session on differential geometry.

In the beginning, as is often the case, we went through our problem attempts and received valuable feedback from Dr. Filip Bar. One such problem led us to a fascinating discussion on continuity!

The original question was on the law of excluded middle (LEM) and why it is incompatible with infinitesimals and the continuum. You might have heard that in logic a proposition must either be true or false, nothing in between. This is the idea behind LEM. We can also translate this to set theory, where it would mean that an element either does or does not belong to a set, therefore the union of a set with its complement gives us the universal set.

All of this seems obvious, but we can arrive at deep insights when considering it carefully. Let’s take the case of the continuum. It can be modelled by the real numbers, which makes intuitive sense. After all, if you imagine the number line, you will find that it is continuous since you can draw it without lifting your pen from the page! If we apply LEM, the set of real numbers should be equivalent to the union of the set containing the number 0 with the set containing all real numbers except for 0. But hold on, this would no longer be continuous! It is like cutting a string (here the number line), laying the two halves next to each other, and claiming that nothing changed. But, of course, the string is no longer in one piece. The law of excluded middle introduced a sense of discontinuity to the continuum!

Then Dr. Bar posed the question that intrigued me the most, as an aspiring theoretical physicist: Is nature continuous?

What do you think? I shall share more about this tomorrow!

Welcome to my second part of this week’s BeyondResearchInsights! Let’s continue where we have left off: Is nature continuous?

At first glance, the answer might seem obvious. If you look around, you can make out discrete objects that have a beginning and an end. But if you change your perspective, you’ll find that all of those objects are also made up of particles of the Standard model. Electrons are the closest to a point particle and thus it is safe to say that they are discrete. The other particles inside the atom, protons and neutrons, are each made up of three elementary particles called quarks. I fell into the trap of assuming that the case with quarks is just the same as with electrons. However, the fact is that we can never isolate a single quark. They always exist in pairs (in mesons) or triples (in baryons) and if we try to pull them apart, new pairs of quarks and anti-quarks appear in between thanks to the strong force. So keeping in mind that they can never exist as a single particle, are quarks truly discrete? Well, I wouldn’t say so. Would you?

Now you see that nature actually has both discrete and continuous aspects. This seems a little paradoxical and it also poses some problems in our theories of physics. Think about a point charge creating an electric field around it. The field gets stronger and stronger as we near the charge, which ultimately leads to a singularity. This means the point charge would have to store an infinite amount of energy. Singularities are bad news, as they tell us that our theory breaks down!

An alternative angle from which we could look at the question is considering Quantum field theory (QFT). As the name suggests, QFT is concerned with fields, the excitations of which are particles, so it is inevitably continuous. However, on top of the mentioned particle vs. field problem, QFT has other issues of its own; first and foremost that, mathematically, it doesn’t exist. But let’s leave that for some other time!

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