$N$-Body Noncollision Singularities For Particle Physics

Originally written: Oct 12, 2024 by Noah Everett as part of the Ideas Blog

Introduction To Noncollision Singularities

The $N$-Body problem in mathematics addresses the challenge of predicting the motions of several particles under mutual gravitational interaction. In its classical form, it involves point masses influenced only by Newtonian gravity, where the force between two bodies follows an inverse square law, $F_g \propto \frac{1}{r^2}$. This leads to collision singularities, where bodies can theoretically reach a point where their distance becomes zero, resulting in an infinite force.

A more fascinating aspect of the $N$-Body problem arises in systems where $N \geq 4$. Poincaré suggested the possibility of noncollision singularities—situations where, although the bodies do not physically collide, one or more of them can exhibit extreme behavior, such as moving off to infinity in finite time. This was partially confirmed by Jeff Xia in 1988, who demonstrated a scenario where five bodies exhibit such behavior [Xia, 1992]. Subsequent work by Xue extended this finding to a four-body case [Xue, 2014].

Implications For Particle Physics

The potential crossover of these noncollision singularities into particle physics presents exciting theoretical possibilities. Particle accelerators, such as the Large Hadron Collider (LHC), currently rely on accelerating particles to high velocities/energy and causing them to collide, facilitating the discovery of fundamental particles. If noncollision singularities could be leveraged, they might lead to the development of accelerators that can surpass existing technologies. The key idea would be the ability to manipulate particle trajectories to extreme energies without physical collisions, perhaps uncovering novel phenomena.

Additionally, these singularities might serve as tools to probe small-scale forces. In particle physics, forces operate at vastly different ranges and strengths—gravity, electromagnetism, the strong nuclear force, and the weak nuclear force all play roles. At extremely small distances, quantum mechanical effects become significant, and classical descriptions like the gravitational $n$-Body problem are no longer valid. However, noncollision singularities might provide a way to indirectly study interactions at small scales, offering insights into force behaviors that might not be easily accessible otherwise.

Challenges And Limitations

1. Singularities and Physical Reality

First, singularities in the mathematical sense are not physically possible. Particles cannot actually reach zero distance or experience infinite forces, as that would violate the laws of physics, particularly the conservation of energy and momentum. If we wanted to use these singularities for particle acceleration, energy would still need to be supplied. The process wouldn’t be “free”; particles cannot be accelerated to infinite velocities without an external energy input. This basic conservation principle presents a fundamental obstacle to realizing noncollision singularities as a practical mechanism for acceleration.

2. Quantum Uncertainty and Mechanics

At very small distances, quantum mechanics introduces further challenges. The Heisenberg Uncertainty Principle prevents precise knowledge of both the position and momentum of particles at the same time. As particles approach very small separations, reducing their positional uncertainty increases their momentum uncertainty, introducing unpredictability into their paths. The deterministic nature of noncollision singularities in classical mechanics does not carry over to the quantum regime, making it impossible to precisely control particle behavior at these scales.

3. Multiple Interacting Forces

In particle physics, particles interact via more than just gravity. The electromagnetic, weak nuclear, and strong nuclear forces all play roles, and they behave very differently depending on the distance between particles. For example, while the electromagnetic force also scales as $1/r^2$, the strong force behaves very differently, becoming stronger as quarks are pulled apart. These multiple interactions make the $n$-body problem in particle physics much more complex than the classical version, which considers only gravitational forces between point masses.

4. Breakdown of Classical Mechanics at Small Scales

Classical mechanics breaks down at very small distances. At quantum scales, particles behave not as point masses but as quantum objects governed by quantum field theory (QFT). In QFT, particles are excitations in underlying fields, and their behavior is probabilistic rather than deterministic. Additionally, at very small distances—approaching the Planck scale—gravity is expected to unify with quantum mechanics, but we lack a complete theory of quantum gravity. This makes it impossible to apply the classical $n$-body problem accurately in this regime.

5. Practical Feasibility

Finally, even if these theoretical issues were resolved, using noncollision singularities in particle accelerators would still be practically difficult. Current particle accelerators, like the LHC, rely on precise control of particle trajectories, but noncollision singularities are inherently chaotic. Small changes in initial conditions can lead to vastly different outcomes, making control difficult. The complexities of managing such a system in practice, along with the unpredictability from quantum effects, present major engineering challenges that would require entirely new technologies.

Conclusion

In conclusion, while noncollision singularities offer fascinating theoretical ideas, they face significant challenges when it comes to practical application, especially in particle physics, due to the constraints of quantum mechanics, force interactions, and conservation laws.

Image credits: DALL-E by OpenAI