Quantum Field Theory on the Tangent Bundle

Modern physics rests on two great pillars. The first is our understanding of the very large: gravity, which shapes the motion of stars and the expansion of the universe. The second is our understanding of the very small: the Standard Model, a theory that describes how elementary particles like electrons and quarks interact through forces such as electromagnetism and the strong nuclear force. While both pillars are incredibly successful, they do not fit together easily. The Standard Model explains the behavior of particles with mathematical precision, but it treats the forces that bind them as separate, arbitrary rules rather than natural consequences of geometry. It also leaves unexplained why particles have the specific masses they do, why there are three families of matter, and how gravity fits into the quantum picture. Physicists have long sought a deeper framework where these forces and particles emerge naturally from a single geometric structure, rather than being imposed from the outside.

A new theoretical study by Joachim Herrmann proposes such a framework, building on the idea that the universe might be described not just by the space we live in, but by the "tangent bundle." In simple terms, imagine that at every single point in space, there is not just a location, but also a tiny, internal space attached to it, containing all the possible directions and velocities a particle could have at that spot. This internal space is the tangent fiber. The paper suggests that the complex rules governing particles and forces are actually the result of how fields behave within these tiny internal spaces. By treating the geometry of these internal spaces as the source of physical laws, the author attempts to derive the known forces of nature and the properties of matter from a unified geometric starting point.

The core of this work is a new way of looking at the symmetries that govern the universe. In physics, a symmetry is a transformation that leaves the laws of nature unchanged. The Standard Model relies on specific internal symmetries to explain why particles have electric charge or color charge. Herrmann's approach starts with a geometric symmetry that includes both rotations and translations, similar to the symmetries of space and time, but applies them to these internal tangent fibers. The author separates this geometric structure into two parts. One part, the translational sector, is identified with the geometry of gravity, specifically a version of gravity known as teleparallel gravity, which describes the universe using twisting rather than curvature. The other part, the rotational sector, is used to generate the internal forces that hold atoms together.

From this geometric setup, the paper derives the existence of the weak nuclear force and the strong nuclear force, which are responsible for holding atomic nuclei together and for radioactive decay. The author shows that if you look at the mathematical structure of these internal fibers, a specific type of symmetry naturally emerges that acts like the weak force, which distinguishes between left-handed and right-handed particles. Furthermore, the paper demonstrates that if you assume there are three identical copies of a fundamental particle state within this internal space, a new symmetry arises that behaves exactly like the strong force, known as quantum chromodynamics. This strong force is what binds quarks together to form protons and neutrons. The theory does not just assume these forces exist; it suggests they are inevitable consequences of the geometry of the tangent bundle and the way particles occupy these internal spaces.

One of the most significant findings is how this framework explains the electric charges of particles. In the Standard Model, the electric charges of quarks are fractional, meaning they are one-third or two-thirds of the charge of an electron, a fact that has no obvious explanation in the current theory. Herrmann's model connects these fractional charges to the way particles rotate within the internal tangent space. By carefully matching the rules of rotation in this internal space with the rules of color charge, the theory naturally produces the correct fractional charges for quarks and the integer charges for electrons. This suggests that the strange, fractional nature of quark charge is not an accident, but a geometric necessity arising from the structure of the universe's internal dimensions.

The paper also addresses the stability of the theory and the behavior of the strong force at high energies. It confirms that the strong force derived from this geometric model behaves exactly like the strong force in the Standard Model, becoming weaker at very high energies and stronger at low energies, a property known as asymptotic freedom. This is a crucial test, as it means the new geometric framework does not contradict the well-established experimental results of particle physics. The author also explores the possibility of new, hypothetical particles that could exist within this framework. These include neutral particles that do not interact with light or the strong force but carry a different type of internal charge. Such particles are candidates for dark matter, the invisible substance that makes up most of the mass in the universe. The paper outlines the conditions these particles must meet to be consistent with known physics, offering a potential path to understanding the dark sector.

While the theory provides a unified geometric origin for gravity and the other forces, it also highlights areas that require further investigation. The model suggests that the masses of particles are not fundamental constants but arise from the interaction of particles with a background field in the internal space, similar to how a particle acquires mass by moving through a medium. However, the paper does not calculate the specific masses of the particles, noting that this would require a more detailed study of the dynamics of this internal vacuum. The author also points out that the theory predicts the existence of additional force-carrying particles, which have not yet been observed. These hypothetical particles would mediate interactions between the different internal states of matter and could be detected in future high-energy experiments.

The work stands as a structural proposal rather than a completed theory of everything. It establishes a consistent mathematical framework where the gauge groups of the Standard Model and the geometry of gravity emerge from a single source. It does not claim to have solved every mystery of particle physics, but it offers a new perspective on why the universe is structured the way it is. By placing the forces of nature and the geometry of spacetime on the same geometric footing, the paper provides a fresh starting point for understanding the fundamental building blocks of reality. The next steps involve testing these geometric predictions against experimental data and exploring the dynamical consequences of the internal vacuum structure to see if they can explain the observed masses and families of particles.