nLab particle statistics

For disambiguation see at statistics.

Idea

In quantum physics, what is called particle statistics refers to the symmetry-properties of many-particle wavefunctions under “exchange” of (positions $x_i$ and jointly of all other parameters $p_i$, like spin, of) “indistinguishable” particles.

The term “particle statistics” refers to the fact that the transformation properties of wavefunctions under particle exchange crucially effect the nature of the quantum statistical mechanics of these particles.

Or rather, this is the case for the ordinary Bose/Fermi-statistics satisfied by fundamental particles, namely by bosons and fermions respectively (cf. Bose-Einstein condensates as opposed to the Fermi sea due to the Pauli exclusion principle).

While the richer “para-statistics” and “anyon braid statistics” is often likened to a generalization of Bose/Fermi-statistics, these “exotic statistics” actually arise in practice not as intrinsic symmetries of wavefunctions, but as their “Berry phase” unitary transformations under adiabatic transport along external parameters — which is conceptually quite different.

Boson/Fermion statistics

In the most prominent base case one assumes a single quantum state represented by a wavefunction $\Psi\big((x_1, p_1), \cdots (x_N, p_N)\big)$ and asks for its transformation under a permutation $\sigma \in$ $Sym(N)$ of its arguments. The particles $(x_i, p_i)$ being indistinguishable means (cf. at quantum symmetry) that the corresponding quantum states remain invariant, which means that the wavefunction changes at most by multiplication by a complex number $s(\sigma) \in \mathbb{C}^\times$.

$$\Psi\big((x_{\sigma(1)}, p_{\sigma(2)}), \cdots (x_{\sigma(N)}, p_{p(N)})\big) \;=\; s(\sigma) \cdot \Psi\big((x_1, p_1), \cdots (x_N, p_N)\big) \,.$$

But the symmetric group is generated by pair exchanges $\sigma_{i j} = (i \leftrightarrow j)$ which all square to the neutral element, $\sigma_{i j} \sigma_{i j} = \mathrm{e}$, whence the same must follow for the phase factor with respect to the complex group of units $\mathbb{C}^\times$:

$$s(\sigma_{i j})^2 \,=\, 1 \,\in\, \mathbb{C}^\times \,.$$

The only two solutions to this equation are:

• $s(\sigma_{i j}) = +1$ — in this case the corresponding indistinguishable particles are called bosons,

• $s(\sigma_{i j}) = -1$ — in this case the corresponding indistinguishable particles are called fermions.

In representation theoretic words: The wavefunctions of ordinarily indistinguishable particles must, by definition, span 1-dimensional linear representations of the symmetric group and there are exactly two such, up to isomorphism:

1. the trivial representation (bosons),

2. the alternating representation (fermions).

Para-statistics

More generally, one may ask that the symmetric group of particle exchanges does not necessarily leave the quantum state invariant, but transforms it homomorphically within a linear subspace of the Hilbert space of all states.

This means to ask for any linear representation of the symmetric group (not necessarily 1-dimensional as for bosons and fermions above).

In this case $s(\sigma)$ is not just a complex number but a (unitary) linear operator, and there are many more possibilities for its values.

This generalization of particle statistics is referred to as parastatistics.

Anyon braid statistics

Still more generally, one may consider the case that the behaviour of the wavefunction depends not just on the permutation of the particle labels, but also on the “way” in which the permutation is accomplished.

Concretely, envisioning the $N$ “particles” to be confined to an effectively 2-dimensional surface $\Sigma^2$, and envisioning that the “ways” to permute them are isotopy-classes of motions of $N$ pairwise distinct points in the surface (hence of paths in the configuration space of points), then the group of such exchanges is not just the symmetric group but the braid group $Br_N(\Sigma^2)$ which (surjects onto the symmetric group) or the pure braid group $PBr_N(\Sigma^2)$ (which is the kernel of that surjection):

$$1 \to PBr_N(\Sigma^2) \hookrightarrow Br_N(\Sigma^2) \twoheadrightarrow Sym_N \to 1 \,.$$

In this situation one may hence ask that the quantum states of these $N$ “particles” form a linear representation of the braid group.

In this case one speaks of braid statistics.

Even if the braid representation is 1-dimensional, this is more general than Bose/Fermi-statistics: The (unitary) phase assigned to the motion of a pair of such “particles” past each other may be any complex number of unit absolute value. For this reason the “particles” satisfying 1-dimensional braid statistics have been called any-ons.

More in detail, for such 1-dimensional braid representations one speaks of abelian anyons, while for higher dimensional braid irreps one speaks of non-abelian anyons.

Related entries

• boson, fermion

• spin-statistics theorem

• braid group statistics, anyon

• parastatistics

References

General

Review:

• Allan David Cony Tosta: Review: The theories of non-standard quantum statistics, Chapter 2 in: Quantum information and computation with one-dimensional anyons (2021) [pdf, pdf]

• Wikipedia: Particle statistics

Via AQFT

Discussion via algebraic quantum field theory and in view of causal locality and DHR superselection theory:

• Sergio Doplicher, Rudolf Haag, John E. Roberts: Local observables and particle statistics I, Commun.Math. Phys. 23 (1971) 199–230 [doi:10.1007/BF01877742, euclid:cmp/1103857630]

• Sergio Doplicher, Rudolf Haag, John E. Roberts: Local observables and particle statistics I, Commun.Math. Phys. 35 (1974) 49–85 [doi:10.1007/BF01646454]

Review and further discussion:

• Sergio Doplicher: The statistics of particles in local quantum theories, in: International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics 39 (1975) 264-273 [doi:10.1007/BFb0013339, inSpire:105411, pdf]

• Hans Halvorson: Statistics, permutation symmetry, and identical particles, Section 11.4 in: Algebraic Quantum Field Theory, in Philosophy of Physics, Handbook of the Philosophy of Science (2007) 731-864 [doi:10.1016/B978-044451560-5/50011-7, arXiv:math-ph/0602036]

category: physics