The Theory of Causal Fermion Systems

General Introduction

Setting the Scene: What is the Current Situation in Physics?

In order to know where to go, we first need to know where we are. Therefore, let us begin by providing some background information intended to give an orientation to readers who are not so familiar with questions in Fundamental Physics. So, in a nutshell: What is the current situation in Physics?

Our present understanding and description of the Universe rests on two fundamental pillars: Quantum Field Theory and General Relativity. Abstractly speaking, both of them are self-contained webs of principles, mathematical concepts and structures appropriate to describe certain clusters of experimental observations. One such cluster is formed by the elementary particles together with their physical properties, their behavior and mutual interactions. All this can be modelled within the framework of Quantum Field Theory and leads, together with experimental data, to what we refer to as the Standard Model of particle physics. This theory subsumes in mathematical language what we presently know about elementary particles and their interplay via the electromagnetic, weak and strong interactions. The second big cluster does not deal with the relations between particles, but instead asks for the relation between matter as a whole and the surrounding, namely spacetime. The suitable theoretical framework to model this situation is General Relativity. This theory encodes our current knowledge about the shape of the Universe as well as the interplay with its matter and energy content. Briefly summarized, our current understanding of the Universe is a scale-dependent, two-component model: On large scales we neglect the inter-particle relations and merely talk about matter and energy interacting with spacetime, while on small scales we disregard gravitational interactions and only take into account electromagnetic, weak and strong interactions among the particles.

At first glance, this seems a perfectly reasonable approach towards an understanding of our Universe, and indeed: The predictions of the Standard Model of particle physics have been tested to high precision, just as predictions of General Relativity have been experimentally verified again and again. So, why are physicists dissatisfied with this actually very good status quo?

There are several answers to this question. One answer is that there are physical effects which can be explained only by combining the interactions of the Standard Model with General Relativity. Just to give one example from cosmology: From observations we know that the Universe is undergoing an accelerated expansion. Although this fact can be perfectly described within the framework of General Relativity on a theoretical level, we do not understand the underlying physical mechanism causing this expansion. In fact, we do know that and to what extent currently known forms of matter and energy influence the expansion, but in any case they can definitely not account for the measured expansion rate. The quantum-field-theoretic attempt at explanation, namely to introduce additional fields corresponding to yet unknown particle and energy species, seems natural but cannot resolve the problem in a physically fully convincing way.

Maybe the most prominent example of a physical system where all physical interactions come into play are black holes. Near a black hole, the gravitational fields are extremely strong, so strong that they must be described within the framework of General Relativity. But at the same time, the density of matter is very high, so that the particles are very close to each other, which in turns means that the quantum nature of matter must be taken into account. So far, despite many ideas and attempts, there is no physical theory which can explain in a consistent and convincing way what happens near or inside a black hole.

We as mathematically oriented physicists and mathematicians are not so much interested in the phenomenology of the effects just described. What we have in mind points to another answer to the above question why physicists are dissatisfied with the present status: It would be desirable to describe all of physics, from the smallest to the largest length scales, within one consistent mathematical framework. The endeavour is to find a theory which achieves this goal. The hope is that this quest will also give new insight into some of the fundamental questions of physics.

Defining the Objective: What is a Unified Theory?

Having set the scene, it should be clear what we are aiming at: We want to find a theory which allows us to explain phenomena which include both aspects from Quantum Field Theory and concepts from General Relativity. The natural question to ask is: What should such a Unified Theory look like? And even more importantly, how can one find such a Unified Theory? Asking this question to the physics community, one gets a broad variety of strenuously expressed opinions on what the correct approach to find such a theory is and, connected with it, what such a Unified Theory should look like. Being confronted with a plurality of opinions on a seemingly simple question always strongly indicates that the underlying problem has not yet been understood sufficiently well.

To our opinion, it is most important that a Unified Theory provides a unified mathematical framework resulting from a careful analysis of the currently used mathematical objects. This new framework is not simply obtained by generalizing current structures and reformulating them in a general mathematical language until one ends up with a sufficiently abstract and general framework. On the contrary! Our approach is guided by the wish that the framework should be concrete and as simple as possible. It was developed by translating what we consider to be the basic physical principles into mathematical language, and by combining the resulting structures in a minimalistic way to a coherent framework.

What is the Conceptual Core of the Theory of Causal Fermion Systems?

Keeping these considerations in mind, the following basic question will naturally come across one’s mind: Having talked about mathematical structures and principles, which concepts are at the heart of the Theory of Causal Fermion Systems? The short but at the same time probably not too enlightening answer is: A variational principle for a measure on a set of operators on a Hilbert space. Sounds complicated? Honestly speaking: Understanding the Theory of Causal Fermion Systems on a mathematical level needs some time and effort. But one can understand a few general ideas even without entering mathematics, as we will now try to explain.

Working with a variational principle simply means that a certain quantity is minimized. Such variational principles play an important role in physics. We do not really know why Nature behaves this way, but it seems a general principle that in Nature certain quantities are maximized or minimized. Let us illustrate this concept in a few examples. Probably the most intuitive example for such a tendency is the preference of low-energy states over high-energy states. Respecting possibly existing constraints, Nature tries to reach the state of lowest energy in every conceivable setting: Considering a freely falling chain on Earth, it will try to reduce its gravitational potential energy by approaching the source of the gravitational attraction, namely by falling down to Earth. Setting restrictions on this natural behavior by fixing the ends of the chain at a certain height, the chain will form what is referred to as catenary. This is observed in free-hanging bridges, overland cables and bollard chains. In the same way as massive objects try to reduce their gravitational potential energy, also electrically charged objects try to lower their electric potential energy: While oppositely charged objects approach each other, objects with charges of the same sign move away from each other. However, imposing restrictions by confining a bunch of, say, positively charged particles to a bounded region, they will form a configuration which minimizes the electric potential energy of the system as a whole. But Nature not only tends to produce low-energy configurations, at the same time it also tries to realize configurations which maximize a quantity referred to as the entropy: Given a system of particles which, as a whole, contains a predetermined, fixed amount of energy, Nature strives for a configuration in which every particle carries the same fraction of the total energy of the whole system. This phenomenon occurs in everyday life in a broad variety of situations: Pouring hot water into a basin of cold water and measuring the temperature (which is precisely a measure of energy per particle) after some time, it will be constant all over the basin. In simple terms, Nature tries to realize equilibrium configurations, states of maximal uniformity.

Similar to the above examples, at the heart of the Theory of Causal Fermion Systems there is a variational principle, where a certain quantity is minimized. This variational principle also involves constraints.

Which Quantity is Minimized in the Theory of Causal Fermion Systems?

Having motivated the usage of a variational principle as a fundamental ingredient to the Theory of Causal Fermion Systems, several questions arise: How is the physical system described in the Theory of Causal Fermion Systems? Which quantity of the system should be minimized, and what are the configurations of the system which are to be varied?

Answering these questions in a fully convincing way makes it necessary to use the language of mathematics. Without entering math, the best we can do is to make a few general comments. A causal fermion system can be understood as being formed of “quantum matter” described by an ensemble of wave functions living on spacetime points. A-priori, these “spacetime points” are merely points, but there are no relations between them. All the structures which we usually associated to spacetime (like a flow of time, a sequence of causal events, speed of light, etc.) are induced on the spacetime points by the ensemble of wave functions. The causal action is a nonnegative number which tells about “how good” the configuration of wave functions is. By minimizing the causal action, the ensemble of wave functions is brought into an “optimal” configuration. Intuitively speaking, this minimization process means that all the wave functions describing matter “organize themselves.” In the resulting optimal configurations, the ensemble of wave functions encode all the spacetime structures and also describe the distribution of matter in spacetime.

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