Foreword to Conceptual Foundations of Theoretical Physics
First, a foreword to the present FPU faculty in the conceptual foundations of theoretical physics, which is important you understand before considering for application. In fact, theoretical physics has nowadays found so many fields of application and has become so diversified, that there is no unique and coherent definition of what exact curricula it might entail. If you have in mind to become an applied physicist in the domains of material science, nanotechnologies, biophysics, electronics, etc., this course can greatly help too, but will not prepare you for that. What we have in mind here is something that skips the trendy mantra of forced interdisciplinarity at all costs in view of practical applications led by a managerial styled R&D. Here we are not offering you something which is supposed to project you towards a high-impact career which deals with contemporary research lines in fashion supposed to revolutionize the world. It is not about knowledge that focuses on industrial and practical applications preparing you for the competition in a globalized world, as so many universities love to make you believe. What this course prepares to, is therefore not a domain user for applied physics, even though lots of stuff that is studied here is a basic request for that too. If you are looking for that, please understand that most probably you are not in the right place here.
With ‘theoretical physics’ we mean the subjects that prepare you to understand and further research the deeper meaning and fundamental structure of the physical world. It is physics that rises curiosity and the desire to know more about nature, independently from whatever that knowledge might or might not be useful for. It is about something which primarily is supposed to prepare you to tackle first and foremost with the interesting and complex philosophical questions that arise from physics. Moreover, it is intended to prepare the grounds so that you can later eventually research and contribute at a professional level to everything related to what goes beyond the standard model of particle physics (SM) in the view of a future quantum gravity theory. But it does not introduce you (at least not at this stage) to any speculative and yet not experimentally tested theory, like string theory, or canonical quantum gravity, etc., because there are good chances that these theories might not survive the test of history (however, these approaches might eventually become subjects of other courses in the future, once these introductory courses have been activated successfully).
What we are talking about is therefore first of all an understanding of the mathematical necessary basics, then classical mechanics, electromagnetism, quantum mechanics, relativity, quantum field theory (QFT), the SM, especially from the foundational theoretical perspective (but without excluding entirely the experimental aspects neither). With ‘theoretical physics’ we, therefore, do not mean a full-fledged course in solid-state physics and even not astrophysics, cosmology, or other chapters that are often cited as such, even though several aspects will be touched on and might well be included as an appendix or added in one of the given courses (for instance, a brief introduction to cosmology is considered an integral part of the general relativity course).
What instead this course is going to offer you is a preparation from scratch (i.e. beginning all over again from middle school math and physics level) that will make you competent at the highest professional level to understand by your own modern approaches that go beyond the incomplete and now 40 years old SM or tackle with the philosophical issues related to quantum mechanics, relativity and QFT.
Having clarified that, let us now describe the content and structure of the proposed curriculum.
As well known, physics and mathematics have a lot in common. Not only is math absolutely essential to understand physics, but they are both subjects that have a strongly hierarchically build up foundation. You cannot begin to build a house from the roof. You can not write a novel without learning the letters of an alphabet first, the words of the vocabulary, the grammatical rules of the language. Similarly, you will never be able to play a musical instrument without having gone through sufficiently long practice and exercise.
This seems to us quite obvious, and yet there are many who are convinced that this does not hold for physics. They firmly believe that they do not need to learn math and the basics of physics and begin immediately to imagine, speculate, fantasize and build a huge castle of misunderstandings and made of concepts on foundations that do not exist. In some sense, this might indicate a strong desire to discover and inquire about things as soon as possible, but the aspirant physicist of the FPU must learn to balance a good and healthy curiosity with the awareness of one’s own limits. As Newton did when he stated: "If I have seen further it is by standing on the shoulders of Giants", meaning, that one can discover new truths only by building on previous discoveries. In fact, while in a FPU students are even encouraged to express their interests almost from the beginning with the portfolio work which comes at the end of each course, on the other side it is made clear that the foundations of physics are not dispensable.
There is a basis and a conceptual structure that holds in place a sort of conceptual pyramid. For example, you cannot understand how atomic physics works, if you didn’t learn quantum physics. You can’t understand quantum physics if you did not learn classical mechanics first. You can’t understand classical mechanics if you didn’t learn decently calculus. You can’t understand calculus if you have no longer the basic math tools you learned in high school. The very same principle holds also inside the respective subjects. For instance, you can’t understand what a differential equation is and know how to take advantage of it, if you don’t learn what a differential of a function is. You can’t understand what a differential is if you don’t know what a derivative of a function is. You can’t understand what a derivative is if you don’t know what at all a mathematical function is in the first place. And so on...
The result of a careful selection of courses for a curriculum of an FPU faculty in the foundations of physics is illustrated in the pyramid below:
So, you must realize that almost all the concepts in physics and mathematics build upon each other. And it is dangerous to skip any steps in between, otherwise, you might not be allowed to proceed further. Not because there is an authority who dictates what to do, but simply because it is inherent in the nature of things as they are.
A possible source of confusion emerges sometimes when, once you will ask experienced academicians what really is the essential minimum to learn physics, you might get very different answers. Everyone has his/her own point of view and will place an emphasis on their own interests. Some will point out a plethora of things you are supposed to learn but that you might realize (when it is too late) that they were not so relevant to your interests. For instance, nowadays physics is heavily oriented towards experimental and applied science, which means that most physicists work in a sector that, for example, needs deep knowledge of electronics, computational science, or solid-state physics and will try hard to convince you how essential these are even though you might be interested in entirely different aspects of physics, like the physics of black holes or the philosophy of quantum mechanics. While having a good background in these topics might be quite useful too, especially when you will have to interact with experimentalists, during the first run of your apprenticeship in the foundations of theoretical physics, these topics will not turn out to be of such paramount importance, because they are not part of the set of topics contained in a common intersection of the basics you need to acquire in order to be able to proceed later by your own. Whereas, there are other things that are part of this common intersection and will permanently present themselves in the most diverse domains of theoretical physics. If you won’t learn these they will always hunt you and never allow you to fully appreciate the ideas you want to absorb. For instance, Fourier transforms, Euler-Lagrange equations, Maxwell equations or the Schrödinger equation are almost ubiquitous mathematical and physical concepts or tools you will have to learn to be accustomed to because you will find them everywhere in theoretical physics (by the way, in applied physics too). Whereas, for the theoretical physicists who are looking more to the conceptual foundation of theoretical physics, knowing how the semiconductor physics in a transistor or a diode works, is useful but not a priority. General education and an interdisciplinary culture are fine, but since it takes already many years to get acquainted with the kind of stuff we are trying to elaborate here, it is wiser to fix our attention on the necessary basics first and let eventually the student the freedom to look further (for instance, during the portfolio phase).
So, that’s enough for the course's introduction, now let us take a look at each of the courses.