Physical mathematics vs mathematical physics
TL;DR: We shouldn't make physics be more like math, we should find the math that fits physics
Physics should be more mathematically rigorous. Yet, theoretical physics has been following a mathematization trend that ultimately obscures the physics. These statements may seem contradictory, but they are not. If we take the first to be true, we have two options. The first is to turn physics into math: you take the language and techniques from math and you reformulate physics with them. You get “mathematical physics:” physics that fits the math. The second is to turn math into physics: based on physics requirements, you design tools that model them effectively. You get “physical mathematics:” mathematics that fits the physics. We need more of the second.
An example of something typically lost in mathematical physics are units. It always struck me as odd that in the first years of physics and engineering, a lot of emphasis is put on units and dimensional analysis, only to be completely abandoned when one studies more advanced theories, like quantum mechanics or field theories. The irony is that a lot of mathematical structure is there to capture the physical structure given by the unit system, and this is completely lost. For example, in differential geometry, one can use contravariant components \(v^i\) and covariant components \(v_i\) to specify a vector and a covector. What is the difference? The units. The units of \(v^i\) are proportional to the units of position, the coordinates, while the \(v_i\) are inversely proportional to the units of position. That is why the product of upper and lower indices gives you an invariant: the units are simplified. The units of invariants should be independent of the units chosen for the coordinates. For example, we can define force as infinitesimal work over an infinitesimal distance. The infinitesimal distance \(dx^i\) has an upper index because its units are proportional to the units of position. Conversely, the force \(f_i\) must have a lower index because it is inversely proportional to the distance. We can then write \(dW = f_i dx^i\): infinitesimal work is force times infinitesimal distance. Units are linked to transformation rules. A scalar field, like temperature \(T(x)\), transforms like a scalar (i.e. \(T(x) = T(y)\)) precisely because the units of temperature are independent of the units of distance. A density, like the charge density \(\rho(x)\), does not transform like a scalar, even if it is a number, because its units depend on the inverse of the units of volume, which do depend on the units of distance. This is what physical mathematics should address: math symbols and their properties must all have physical meaning. Nothing is surprising.
Sometimes this just means adding some extra interpretation to the math, but sometimes means looking at the definition and realizing the ones we have are not quite right for physics. For example, the tangent space, the space of vectors, is defined as the space of derivations of scalar functions. In two-dimensional Cartesian coordinates, for example, the basis vectors are \(\frac{\partial}{\partial x}\) and \(\frac{\partial}{\partial y}\). The sum \(\frac{\partial}{\partial x} + \frac{\partial}{\partial y}\) is another vector. Suppose we use polar coordinates. Now \(\frac{\partial}{\partial \rho}\) and \(\frac{\partial}{\partial \theta}\) are the basis vectors. However, \(\frac{\partial}{\partial \rho} + \frac{\partial}{\partial \theta}\) makes no physical sense: inverse distance cannot be summed with inverse angles. One may say, well, just convert everything to distance. This assumes the existence of a global notion of distance in the space, which is not always the case. For example, we use differential geometry also in phase space: how are we supposed to convert units of position into units of momentum? This is not just an academic problem: it really prevents us from having a clear map between physics and math, and therefore really understanding what the math is describing at a physical level. This type of reverse engineering is exactly what ultimately led us to a geometric and physical interpretation of the action principle. It pays off.
It’s not just that the current mathematical structures don’t fit exactly what we need in physics; we are very likely missing the correct structures to precisely formalize and then generalize our physical theories. One of the issues we identified is that Hilbert spaces are unphysical in the sense that they contain objects that cannot represent physical systems. You are forced to include pure states that have infinite or undefined expectations for position, momentum, energy, number of particles, and so on. These are all states that are not realizable. So, what should we use? It is not clear because function spaces, which are infinite-dimensional vector spaces, are tricky. It is tempting from the physics side to simply ignore the problem and let mathematicians handle it, but the issue is ultimately connected to understanding what is a physically well-specified system. This is something that a mathematician cannot solve: it’s part of the physics. So the problem is not solved, and we are still without a solid standard general foundation for field theories.
The point is that, in order to have sound foundations of physics, it is not enough to be mathematically precise. We need to be physically precise as well. In mathematical physics, often the mathematical precision is at the expense of the physical precision, which is the real problem with the mathematization of physics. It shouldn’t be surprising: mathematicians often lack the physical intuition to guide them through these details. It is up to physicists to go through these details and understand what works and what doesn’t.