Can you remember how complex numbers are created?
The numbers we now call “complex” appeared in Renaissance Italy in the first half of the 16th century. Scipione del Ferro, Niccolò Tartaglia and Ludovico Ferrari had obtained general formulas for the solutions of degree 3 and 4 polynomial equations. They were published by Cardan in his Ars Magna of 1545.
For example, for x3=15x+4 with these formulas we get \(\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}\). Cardan knew perfectly well that negative numbers have no roots, called terms like \(\sqrt{-121}\) “challenging” and finally came up with the idea that his results were “as subtle as they were useless”.
An important point is that 4 is an obvious solution to the equation. This presented the problem of understanding the connection between 4 and \(\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}\), that is, the connection between Cardan’s formulas, when they reveal complex numbers, and the real solutions of third- and fourth-degree equations.
However, this also raised another, more thorny question, which is to understand if we can get a Cardan formula to find the actual roots of these equations without using complex numbers. These two questions will be of great interest to Leibniz, for example.
I mention the second one here because it’s actually quite close to quantum mechanics: Quantum theory, in its standard version, uses complex numbers and works very well. But we can ask ourselves the question of a formulation that would structurally only use real numbers.
Why did Descartes speak of “imaginary” numbers?
The case of Descartes is undoubtedly not isolated from the rest of the history of complexes, but it is very interesting. Above all, we must remember that the concept of number, adopted by Greek mathematics and philosophy, was very different from ours at the time. Numbers are still closely linked to counting and measuring; So a priori they are positive integers or positive real numbers, and the status of negative numbers is discussed. For all its modernity, Descartes speaks of “false roots” in his geometry of 1637 with regard to the roots of polynomials that are “less than nothing”, i.e. negative.
In his Geometry published in 1637, René Descartes emphasizes that “both the true and false roots are not always real, but sometimes only imaginary”.
© Gallica
A key idea for him is the factorization of polynomials as products of monomials, formed from the difference between the unknown quantity (which for us would be the variable) and a root, “true or false”. In this context he will speak of “imaginary roots”. It is undoubtedly worth recalling the passage verbatim: “Moreover, both the true and false roots are not always real, but sometimes only imaginary: that is, we can always imagine some of what I have said in each equation , but this.” Sometimes there is no set that matches the ones we imagine. » We find in him this idea of imaginary roots applied to the problems of solving equations of the 3rd and 4th degree, in the manner of Scipione del Ferro and Cardan, and to other geometric problems, such as determining the point of intersection of a parabola and a circle.
Do numbers regularly pose epistemological problems for physicists?
It seems to me that throughout history it was mathematicians and philosophers who understood that numbers can pose a problem, although up until the 19th century it was undoubtedly somewhat artificial to separate mathematicians and physicists. . I talked about negative numbers, but even 0 and 1 caused trouble for a very long time. Their ontology, i.e. their nature and mode of existence, was discussed until the end of the 19th century!
To stick with complex numbers, Leibniz often calls them “fictions,” a word he also uses to describe negative numbers, but also logarithms, the infinitely large and small, and especially the infinitesimals. This word “fictions” is very strong – we also find it used to qualify the mode of existence of mathematical objects in certain contemporary mathematical philosophers, anti-Platonists (i.e. in contrast to the realistic conception of mathematical idealities introduced by Plato and which survives ). , actually the conception that was spontaneously adopted by the majority of mathematicians).
The status of complex numbers emerged in the second half of the 18th century and the beginning of the 19th century. On the one hand there is the fundamental theorem of algebra, or d’Alembert-Gauss theorem, which states that a polynomial of degree n with real or complex coefficients has exactly n complex roots (possibly several). For mathematicians, this theorem corresponds to a mathematical necessity but also, and this is sometimes just as important, to an aesthetic necessity: in particular, it elegantly concludes the uncertainties that have surrounded the problem of solutions to polynomial equations since Descartes and before him, Albert Girard. The theorem contributed significantly to complex numbers being accepted as independent numbers.
The other main reason for this full acceptance of complex numbers was the discovery of the geometric representation with which we are familiar today, which consists of representing \(a+b\sqrt{-1}\) as a point on the coordinate plane (a,b) . This happened at the beginning of the 19th century with the works of Argand, Buée, Carnot, Gauss… The appearance of this geometric representation gave them a concrete existence, perhaps because it made it possible to associate them with an intuition that is not only symbolic and formal.
What is the connection between physics and numbers?
I believe that there are two points of view on this issue.
The first aspect is that of the question of measurements, of what can be observed: what numbers, or more generally, quantities can be obtained as a result of a measurement? This is a less obvious question than it seems, but it plays an important role in the debate about the “real” or “complex” nature of quantum physics, which, for the time being, involves real quantities being taken into account. On the one hand, we can, for example, require that a measurement refers to quantities that can be added (such as energies, velocities, etc.). In other cases, it’s more the idea of order and gradation that matters (it doesn’t necessarily make sense to add temperature observations, for example). If we insist on the idea of order, we must exclude complex numbers from the measurable! The example is undoubtedly a little artificial, but it shows that it is anything but self-evident to give an axiomatic definition of what a measurable physical quantity is – and thus to classify the corresponding ranges of possible numbers. Hermann von Helmholtz sparked some very interesting debates on these questions at the end of the 19th century.
How about the second point of view?
Physics’ other relationship to numbers is of a completely different nature, but just as essential. It adheres to the idea of mathematical models and mathematical tools. We know that modern physics is mathematized, with the Galilean idea that the world is written in mathematical language. We can then ask ourselves whether this language is uniquely defined, whether there is a privileged language, or whether multiple languages can coexist. This is the old conventionalism debate that Poincaré led at the beginning of the 20th century with the idea that the same physical theory can have several mathematically equivalent models between which the physicist is completely free to choose. In the case that interests us, the question arises about the equivalence between real and complex quantum theory and the possibility (or not) of transferring all quantum phenomena from one theory to another.
But the question is more general: there are of course real and complex numbers, but also Hamilton quaternions, matrix algebras, Grassmann algebras, which are very useful in differential calculus… From this point of view, it is not entirely clear where exactly the term stands number should end. However, in almost all cases in which they are used in physics, these ideas of generalized numbers do not seem to capture the core of the theories and phenomena. Rather, it is about the modalities of their mathematical formulation. This is also what makes the problem of the connections between complex numbers and quantum mechanics so interesting.
When and why did physics concern itself with complexes?
We must keep in mind the difference between physical reality and mathematical models.
From a technical point of view, it is often interesting to consider the quantities associated with a wave phenomenon as real parts of complex numbers. Complex numbers are therefore often used to model wave phenomena, for example in optics or fluid dynamics. This use of complex numbers often makes things mathematically easier, but does not change the nature of the theories. This explains why physics quickly adopted complex numbers as basic mathematical tools, but without connecting them to specific epistemological questions.
The special thing about quantum mechanics is that, unlike other physical theories, it is quite artificial to distinguish between the real part and the imaginary part of a complex number.
There was an extraordinary period for the emergence of quantum theory: the years 1925-1926, with the emergence of two competing theories that reveal two a priori very different ways of understanding the quantum world: the matrix approach of Heisenberg and the wave approach of Schrödinger. It is a fundamental textbook case in the history of science with theories of very different formulations, but which are fundamentally equivalent and lead to the same description of phenomena. However, complex numbers play a really essential role in both the matrix approach and the wave approach, as well as in the theoretical unification of the two points of view through von Neumann and the Hilbert complex, which is classically formulated in spaces.
I note in passing that Dirac, another founder of quantum theory, introduced the notion of “q-numbers,” a form of algebraic abstraction of the notions of matrices, operators, and their switching rules, during these debates. Calling these mathematical entities “numbers” reflects the general problem of deciding what a number ultimately is…
If it is confirmed that only a complex number formulation is capable of fully explaining the results of experiments in quantum physics laboratories, what does that tell us about the special role of complex numbers?
This is a difficult question, like all epistemological questions related to quantum physics, which often give rise to endless and fruitless debates. To answer this question, one would have to delve into the details of refuting a true quantum theory and address the question of what exactly such a theory consists of (i.e., the axioms used to justify it).
In any case, it seems clear that these questions are closely linked to the specific wave nature of quantum mechanics. They also raise the question about the relationship between physical reality and mathematical modeling: what exactly would the mathematical necessity of using complexes in theoretical models say about the world? This more philosophical question seems to underlie the fact that physicists have been trying to develop a true quantum theory, while the complex theory works perfectly well, is entirely satisfactory, and is technically easier to use in the first place.
In any case, these ideas will undoubtedly deepen our understanding of the meaning of the basic axioms of quantum theory. What seems interesting to me beyond these specific questions of quantum physics is that they invite us to ask a whole range of questions about the relationships between numbers, physics, mathematics and phenomenal reality.
Google Pixel 8 Pro // Source: OnLeaks x Smartprix
Source: Google
