Instructions for use

quantum theory and mathematical language

In the immortal words of Galileo Galilei, the greatest Italian writer and intellectual of all time, “the book of nature is written in mathematical language”, anticipated by Leonardo da Vinci, “O scholars, study mathematics, and do not build without foundations”. One cannot help but admire “the unreasonable effectiveness of mathematics in the natural sciences” that Nobel laureate Eugene Wigner always marveled at. And to quote Feynman again (The Physical Law, ch.2): “To those who do not know mathematics, it is difficult to get an idea of the beauty, the deeper beauty, of nature. … If you want to know nature, to appreciate nature, you must understand the language in which it speaks.”

Just as an illiterate person cannot read a book, and you cannot understand a book written in a language you do not know, you cannot understand the world, without understanding its language: numbers, data, graphs algorithms, algebra, topology, derivatives and integrals. This applies to both the “hard” sciences (mathematics, physics, chemistry, computer science,…) and the natural and life sciences, the social sciences, economics, history and politics. Theoretical physics, queen of the sciences (“what is not physics is stamp collecting” Ernest Rutherford used to exaggerate quite a bit), with the study of quantum mechanics, statistical mechanics, and aggregate states of matter, of chaotic dynamical systems and complex systems in general provides the basis for the interdisciplinary study of all other branches of knowledge.

So for the concepts of a physical theory like quantum mechanics, words are never enough; they cannot be fully understood without very advanced and complicated math:

mathematical analysis of functions of one or more real variables, and of complex variables, series, transforms, derivatives, integrals, ordinary and partial differential equations, calculus of variations
linear algebra, vector spaces, Hilbert and Banach spaces of infinite dimensions, functional analysis, spectral theory of operators on Hilbert spaces
Representations of Lie groups and their algebras, differential geometry, fibers, topology
…

All this mathematics, which requires many years of study, is indispensable for reading the original works, university textbooks on the subject, and in the daily work of specialists, of physicists dealing with quantum theories, in the precise calculations from the theory from which experimentally verifiable predictions are derived, and in technological applications.

But it is a big mistake to expect to understand everything right away, and to study from the very beginning in a rigorous and overly thorough way, is equivalent to plummeting down an endless rabbit hole, with no certainty of hitting bottom, as Alice succeeds, but only in Wonderland. Much better to proceed in steps, and those who do not aspire to become subject matter specialists may even stop earlier.

first look: compulsory school math

At first it is useful to try to give an idea of the topic, separating physical meaning from mathematical formalisms, both to the curious, the man in the street who does an entirely different job (layman for English speakers), than to future students of physics, mathematics, engineering or other scientific subjects. The first step is a conceptual exposition, with images, analogies, and without requiring any math prerequisites beyond compulsory schooling:

the four arithmetic operations, decimal numbers, percentages
powers of 10, power elevation
dimensions and units of measurement,
diagrams, graphs, and little else, smoothing out all complications and avoiding calculations.

In a first look we will try to introduce quantum theory by chasing the following metaphors:

landscape: review a theory with a bird’s eye view of the vast landscape of mountains to climb and well-hidden valleys
gemma: comparing theory to a rough gemstone that gradually reveals in patient processing brilliance and colors of a thousand facets
trailer: telling a theory as in a movie trailer, which through a series of images entices you to see the whole movie. A complicated goal because things need to be explained as simply as possible, but one must be careful not to be too simple, and maintain some difficulty. otherwise it only creates the illusion of understanding and a lot of misunderstanding.

Avoid popularized introductions of fairy-tale physics and curious math tricks, full of topics bordering on science fiction on the big unanswerable questions, peppered with historical gossip, the deplorable “fairy-tale physics” (“fairy-tale physics, cited Jim Baggott), an example of bad science and bad disclosure. Many popularizers, even eminent scholars, often use language that is too simple, completely devoid of formulas and algorithms, trivializing the most important concepts too much. Resulting in selling many more copies and entering the non-fiction bestsellers, but creating in lay readers much confusion, misunderstandings, mental pippe and the false illusion that they have finally understood something when they have understood nothing. Worse, there are so many wafflers (journalists, philosophers, literati, …) with no serious scientific training, who publish meaningless filth about things they know nothing about and will never understand, stuffed with words that seem fashionable to him such as quantum, indeterminacy and relativity.

He wrote, again Feynman, in the preface to his little popularized masterpiece “QED. The Strange Theory of Light and Matter”: “Many scientific “vulgarizations” achieve apparent simplicity only at the cost of describing something other than what they claim to describe, and indeed significantly distorted. Respect for the subject matter did not allow us to do the same. Through many hours of discussion, we strived to achieve maximum simplicity and transparency, while renouncing any compromise that would lead to a distortion of the truth.”

second step: high school math

The second step toward a basic understanding is to review all elementary introductions doing some calculations using high school math, but nothing more: elementary algebra, analytic geometry, trigonometric functions, linear and exponential trends, estimates and approximations, elementary definitions of derivative, integral, probability … Expositions using only ordinary language, without formulas, graphs, tables, only lead to endless misunderstandings, to the misconception that one has understood something when one has understood nothing, and to a lot of mental pizazz about concepts that cannot be put into words.

third step: early college math

Finally, in a third stage, and last for those who do not aspire to become professional scientists, with a little more effort and different skills, the ordinary curious man can begin to understand the most important ideas and basics of physics. But for that you have to have a higher mathematical education than what is taught in high schools, never be frightened by formulas, know at least the basic concepts of topics such as:

complex numbers, vectors, vector spaces and linear algebra
probability and statistics
differential and integral calculus

In addition, he should have heard, at least on an elementary and intuitive level, of:

abstract algebra and symmetry groups
topology and non-Euclidean geometries Of course, not at the level of a math graduate, but for example at that of an interdisciplinary bachelor’s degree in the humanities or social sciences in a U.S. interdisciplinary liberal arts college (Liberal Arts College).

These are notions that should be part of everyone’s cultural background, of every good citizen of the world, for their endless applications in all branches of knowledge: mathematics, physics, chemistry, computer science, engineering, biology, ecology, natural science, economics, history, social sciences and humanities,…. But unfortunately, this is often not the case, due to a major limitation of today’s training systems. I discuss this at greater length here:

Math for All

annotated bibliography

For support to those who really want to understand quantum physics, paths of recommended readings have been prepared in a special section of the site: