# Quantum and Fractional Statistics

Quantum statistical mechanics is the study of probability potential of subatomic particles and bulk properties. Subatomic particles don’t obey the classical counting rules and anyons take the bizarre world of quantum theory a step further–disobeying the “usual” rules for subatomic particles including fractional charges.

Quantum statistics governs the way things work in the three-dimensional microscopic world from subatomic particles to the behavior of matter at room temperature. Special rules have been created for the behavior of these particles, which do not obey the classical counting rules. Bosons and fermions exhibit different behaviors: bosons like to stay in the same ground state while fermions stay in different states. A third particle has been hypothesized: Anyons are fractional statistics of quasiparticles in two dimensions (Wilczek, 1982) that have characteristics not seen in bosons or fermions. They are called Anyons because particles may obey any statistics, adopting any quantum phase when exchanging position [1].

Anyons are found in two dimensions and not in three-dimension or higher spaces (Khare, 2005). We live in a three-dimensional world, so in the “real” world, anyons don’t exist: all particles are either fermions or bosons. However, some specialized condensed-matter systems (e.g. thin layers at different semiconductor interfaces) can be treated as though they were two-dimensional [1]—as long as the two-dimensional system has quasiparticles which act like charges and fluxes.

## Measurable Fractional Statistics

Fractional statistics is a field under debate; there isn’t a consensus about their status (Hansson, 2013) and they may or may not exist. Statistics that can, theoretically at least, be measured:

• Fabry-Pérot interferometer
• Mach-Zehnder interferometer

Although mostly theoretical, fractional statistics shows promise for the development of an anyonic quantum computer.

## References

[1] (1992) Introduction to Fractional Statistics in Two Dimensions. In: Anyons. Lecture Notes in Physics Monographs, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-47466-1_2
Glen, S. (2020). Abelian Functions .
Hansson, T. (2013). Fractional quantum statistics. Retrieved January 20, 2021 from: http://users.physik.fu-berlin.de/~pelster/Anyon1/hansson.pdf
Horowitz, J. (2009). From Path Integrals to Fractional Quantum Statistics. Retrieved January 20, 2021 from: http://joshuahhh.com/projects/fractional.pdf
Khare, A. (2005). Fractional Statistics and Quantum Theory. World Scientific.
Wiczek, F. (1982). “Quantum Mechanics of Fractional-Spin Particles” (PDF). Physical Review Letters. 49 (14): 957–959.