The Theory of Causal Fermion Systems

What is Fundamental? And How Can We Understand Things Intuitively?

Prerequisites

What is a causal fermion system?
The causal structure
The causal action principle

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Underlying physical principles
The standard model and general relativity
Why the Dirac sea?

For a physical theory an important question is: which selection of object/fields/particles are fundamental and which objects/fields/particles are emergent, or rather just effective descriptions?

On the fundamental level, the only objects that are relevant in the Theory of Causal Fermion Systems are the Hilbert space $\mathcal{H}$ and the minimizing measure $\rho$ as well as the causal structure defined on $\F$. The physical wave functions are a representation of the Hilbert space in spacetime and can hence be regarded as being fundamental in their nature.

For Minkowski space, the physical wave function representation of the Hilbert space that gives rise to this minimizing spacetime structure is given by the Dirac sea, formed of all the negative-energy solutions to the Dirac equation.

As discussed briefly here and extensively here, the bosonic matter fields describe collective perturbations of these physical wave functions. Fermionic fields and fermionic one-particle states, on the other hand, describe perturbations of a single physical wave function. Hence, the bosonic fields are an effective description of the collective behavior of all physical wave functions. In the same sense, the “vacuum” Minkowski space is an effective description for the unperturbed Dirac sea. Hence our description of the world in terms of a spacetime manifold endowed with a metric and fermionic and bosonic one particle states is an effective description of a weighted causal structure with a physical wave function representation of the underlying Hilbert space.

The physical world and the forces we experience are thus just the perturbations of the Dirac sea. To help intuition, water seems to provide a good analogy (of course, these analogies have to be taken with a grain of salt). When you are in a boat on an ocean, your physical experience does not depend much on the fact that the water beneath you is made up of little atoms in constant motion, all you feel (to a first approximation) is what happens on the surface. You can think of flat space as a sea on a calm day. The water underneath allows you to travel across it, and with a calm surface you feel no force acting (ignoring drag for the moment). Classical bosonic fields correspond to smooth waves. You feel a force but the surface of the ocean can still be clearly identified. Finally, a world full of particles and interactions corresponds to a rough day at sea with spray and foam. The boundary between air and water is not so clearly identifiable any more.

Last but not least, the fact that the surface of the ocean is not a flat plane only matters to you once you travel large distances. This corresponds to our practical experience with gravity that it only matters at macroscopic scales. In the Theory of Causal Fermion Systems, this originates from the fact that curvature only comes in as a next-to-next-to-leading-order effect.

Another analogy that might be helpful to understand the emergent nature of bosonic fields in the setting of causal fermion systems is the following: If you take a walk with somebody on the beach and for some reason (don’t ask me why, it might just be curiosity) you get interested in the shape of their feet, you have two options:
a) you look right at their feet and investigate them or
b) you take a look at the collective shape of the grains of sand where they just stepped before.
In a similar sense, the electromagnetic and other classical fields are just effective descriptions of the collective behavior of the physical wave functions.

Claudio Paganini

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