The role of mathematics in physics

Recently, the number of preprints that contain theorems with proofs in the arXiv under quantum physics has increased drastically. I’ve also noticed that some journals in this field tend to publish more such papers, even though they are not ostensibly mathematical physics journals. It seems to suggest that theoretical physics needs to look like mathematics in order to be taken seriously.

Theorems with proofs are not science. Physics, which is a science, is about getting agreement between predictions and experimental observations. So, what is the role of mathematics in physics?

Mathematics

For the physicist, mathematics is a tool, often an indispensable tool, but still, just a tool. When Feynman invented his version of quantum field theory in terms of the path integral, he provided a means to compute predictions for the scattering amplitudes in particle physics that can be compared with the results from high energy particle physics experiments. That was the whole point of this formulation. From a mathematical perspective, the path integral formulation was a bit crude to say the least. It presented a significant challenge to come up with a rigorous formulation of the measure theory that would be suitable for the notion of a path integral.

These days, there seems to be much criticism against quantum field theory. The Haag theorem indicates some inconsistencies in the interaction picture. I also saw that Ed Witten is taking issue with the process of quantization that is used in quantum field theory because of some inconsistencies and he tries to solve these problems with some concepts taken from string theory.

I think these criticisms are missing the point. The one thing that you can take from quantum field theory is this: it works! There is a very good agreement between the predictions of the standard model and the results from high energy physics experiments. So, if anybody thinks that quantum field theory needs to be reformulated or replaced by a better formulations then they are missing the point. The physics is only concerned with having some mathematical procedure to compute predictions, regardless of whether that procedure is a bit crude or not. It is just a tool. Mathematicians may then ask themselves: why does it work?

Mathematics is extremely flexible. There are usually more than one way to represent physical reality in terms of mathematical models. Often these different formulations are completely equivalent as far as experimental predictions are concerned. For this reason, one should realize that physical reality is not intrinsically mathematical. Or stated differently, the math is not real (as Hossenfelder would like us to believe). Mathematical models exist in our minds. It is merely the way we represent the physical world so that we can do calculations. If we come up with a crude model that serves the purpose to perform successful calculations, then there are probably several other less crude ways to do the same calculations. However, it is the amusement of the mathematician to ponder such alternatives. As far as the physicist is concerned, such alternatives are of less importance.

Having said that, there is one possible justification for a physicist to be concerned about the more rigorous formulation of mathematical models. That has to do with progress beyond the current understanding. It may be possible that a more rigorous formulation of our current models may point the way forward. However, here the flexibility of mathematics produces such a diverse array of possibilities that this line of argument is probably not going to be of much use.

Consider another example from the history of physics. Newtonian mechanics was developed into a very rigorous format with the aid of Hamiltonian mechanics. And yet, none of that gave any indication of the direction that special and general relativity took us in. The mathematics turned out to be completely different.

So, I don’t think that we should rely on more rigor in our mathematical models to point the way forward in physics. For progress in physics, we need to focus on physics. As always, mathematics will merely be the tool to do it. For that reason, I tend to ignore all these preprints with their theorems and proofs.

Discreteness

Demystifying quantum mechanics V

Perhaps one of the most iconic “mysteries” of quantum mechanics is the particle-wave duality. Basically, it comes down to the fact that the interference effects one can observe implies that quantum entities behave like waves, but at the same time, these entities are observed as discrete lumps, which are interpreted as particles. Previously, I explained that one can relax the idea of localized lumps a bit to allow only the interactions, which are required for observations, to be localized. So instead of particles, we can think of these entities as partites that share all the properties of particles, accept that they are not localized lumps. So, they can behave like waves and thus give rise to all the wave phenomena that are observed. In this way, the mystery of the particle-wave duality is removed.

Now, it is important to understand that, just like particles, partites are discrete entities. The discreteness of these entities is an important aspect that plays a significant role in the phenomena that we observe in quantum physics. Richard Feynman even considered the idea that “all things are made of atoms” to be the single most important bit of scientific knowledge that we have.

Model of the atom

How then does it happen that some physicist would claim that quantum mechanics is not about discreteness? In her blog post, Hossenfelder goes on to make a number of statements that contradict much of our understanding of fundamental physics. For instance, she would claim that “quantizing a theory does not mean you make it discrete.”

Let’s just clarify. What does it mean to quantize a theory? It depends, whether we are talking about quantum mechanics or quantum field theory. In quantum mechanics, the processing of quantizing a theory implies that we replace observable quantities with operators for these quantities. These operators don’t always commute with each other, which then leads to the Heisenberg uncertainty relation. So the discreteness is not immediately apparent. On the other hand, in quantum field theory, the quantization process implies that fields are replaced by field operators. These field operators are expressed in terms of so-called ladder operators: creation and annihilation operators. What a ladder operator does is to change the excitation of a field in discrete lumps. Therefore, discreteness is clearly apparent in quantum field theory.

What Hossenfelder says, is that the Heisenberg uncertainty relationships is the key foundation for quantum mechanics. In one of her comments, she states: “The uncertainty principle is a quantum phenomenon. It is not a property of classical waves. If there’s no hbar in it, it’s not the uncertainty principle. People get confused by the fact that waves obey a property that looks similar to the uncertainty principle, but in this case it’s for the position and wave-number, not momentum. That’s not a quantum phenomenon. That’s just a mathematical identity.”

It seems that she forgot about Louise de Broglie’s equation, which relates the wave-number to the momentum. In a previous post, I have explained that the Heisenberg uncertain relationship is an inevitable consequence of the Planck and de Broglie equations, which relate the conjugate variables of the phase space with Fourier variables. It has nothing to do with classical physics. It is founded in the underlying mathematics associated with Fourier analysis. Let’s not allow us to be mislead by people that are more interested in sensationalism than knowledge and understanding.

The discreteness of partites allows the creation of superpositions of arbitrary combinations of such partites. The consequences for such scenarios include quantum interference that is observed in for instance the Hong-Ou-Mandel effect. It can also lead to quantum entanglement, which is an important property used in quantum information systems. The discreteness in quantum physics therefore allows it to go beyond what one can find in classical physics.

This image has an empty alt attribute; its file name is 1C7DB1746CFC72286DF097344AF23BD2.png