Conjugates, Statistics Operators, and Statistical Dimension

Conjugates express charge–anticharge annihilation, statistics operators express exchange, and statistical dimension measures the intrinsic multiplicity carried by a sector. For a finite-statistics DHR endomorphism $\rho$, these notions fit together: a conjugate $\bar\rho$ solves explicit evaluation–coevaluation equations, the self-exchange operator $\varepsilon(\rho,\rho)$ represents particle permutations, and a canonical left inverse assigns a statistics parameter whose magnitude is $1/d(\rho)$. The conclusion applies to the selected finite-statistics category; it does not turn every braided or infrared sector into an ordinary Bose/Fermi charge.

Required background. Superselection Sectors and DHR Reconstruction supplies the DHR category; Endomorphisms, Intertwiners, and Tensor Products supplies object and arrow products.

Helpful background. Monoidal, Rigid, and Braided Language supplies duality and braiding notation.

Conjugate equations and dimension

Let $\rho$ be an object in a rigid C*-tensor category with simple unit $\iota$. A conjugate is an object $\bar\rho$ together with arrows

$$R\in(\iota,\bar\rho\rho),\qquad \bar R\in(\iota,\rho\bar\rho)$$

such that

$$\bar R^*\rho(R)=1_\rho, \qquad R^*\bar\rho(\bar R)=1_{\bar\rho}.$$

Diagrammatically, a charge line bent into a charge–anticharge pair can be straightened without residue. A standard solution minimizes $\|R\|\,\|\bar R\|$; the minimum is the statistical dimension $d(\rho)$. It obeys

$$d(\rho\sigma)=d(\rho)d(\sigma),\qquad d(\rho\oplus\sigma)=d(\rho)+d(\sigma),\qquad d(\bar\rho)=d(\rho).$$

For irreducible finite-index sectors, $d(\rho)^2$ equals the Jones index of the corresponding local subfactor. The index–statistics relation and the construction of conjugate endomorphisms are established in Longo 1989, §§2–5, pp. 221–240.

The conjugate is not merely an inverse: usually $\bar\rho\rho$ contains $\iota$ as one summand along with neutral excitations. Only an invertible sector has $d(\rho)=1$ and $\bar\rho\rho\simeq\iota$.

Exchange and the statistics parameter

Transport two copies of $\rho$ to spacelike-separated double cones. Comparing the two orders gives a unitary $\varepsilon(\rho,\rho)\in(\rho^2,\rho^2)$. Repeating adjacent exchanges yields a representation of the permutation group in at least two spatial dimensions; naturality and locality imply the braid relations, and symmetry adds $\varepsilon^2=1$ for the appropriate exchanged pair. A standard solution of the conjugate equations defines a normalized left inverse

$$\Phi_\rho(A)=d(\rho)^{-1}R^*\bar\rho(A)R, \qquad \Phi_\rho\!\circ\rho=\operatorname{id}.$$

For irreducible $\rho$, applying $\Phi_\rho$ to the self-exchange produces a scalar statistics parameter

$$\lambda_\rho=\Phi_\rho\bigl(\varepsilon(\rho,\rho)\bigr) =\frac{\omega_\rho}{d(\rho)}.$$

In the symmetric DHR setting $\omega_\rho=+1$ or $-1$, distinguishing para-Bose and para-Fermi type, while the integer $d(\rho)$ is the order of parastatistics. This theorem is stated already in the first DHR analysis Doplicher, Haag, and Roberts 1971, §§4–5, pp. 217–228. In braided categories, $\omega_\rho$ can be a nontrivial phase and the full braid representation contains information not recoverable from $d(\rho)$ alone.

A finite-group fixed-point calculation

Let a complete field net carry a faithful action of $S_3$, and take the observable fixed-point net. Its irreducible finite-statistics DHR sectors correspond to the irreducible representations $\mathbf1$, $\mathrm{sgn}$, and the two-dimensional standard representation $V$. All are self-conjugate, and

$$d(\mathbf1)=d(\mathrm{sgn})=1,\quad d(V)=2,$$ $$V\otimes V\cong\mathbf1\oplus\mathrm{sgn}\oplus V.$$

The dimension equation independently checks the decomposition: $d(V)^2=4=1+1+2$. The evaluation arrow selects the invariant line in $V\otimes V$; the other neutral channels show why a conjugate is not a strict group inverse. This concrete calculation is the sector counterpart of the generalized fusion structure described in Non-Invertible Topological Defects and Fusion.

Licensed result and nonconverse

For an irreducible transportable DHR sector with finite statistics, the existence of a conjugate licenses finite statistical dimension, a canonical left inverse, and permutation statistics in sufficiently high spacetime dimension. With direct sums, these quantities extend additively. Without finite statistics, conjugates need not exist and $d$ may be infinite.

The converse fails in several ways. A positive number satisfying fusion-dimension equations does not construct a conjugate solution. The value $d=1$ does not by itself specify the exchange phase. Fusion rules and dimensions do not determine associators or braiding. Finally, a cone-localized anyon can have $d=1$ yet possess a nontrivial braid phase, so “dimension one” does not mean boson or fermion.

Adversarial failure: forcing permutation statistics on anyons

In $2+1$ dimensions, clockwise and counterclockwise exchanges of two cone-localized charges lie in different homotopy classes. If an Abelian anyon has braiding phase $e^{i\theta}$, the opposite exchange has $e^{-i\theta}$ and a full winding has $e^{2i\theta}$. Replacing both exchanges by a sign erases winding data unless $e^{2i\theta}=1$. Thus an argument that imports the DHR permutation relation into a cone-localized theory has silently changed the topology of the localization problem.

Independent checks

Solve both conjugate equations, not just the inclusion of $\iota$ in $\bar\rho\rho$. Check additivity and multiplicativity of $d$ against every proposed fusion decomposition. Verify that $\lambda_\rho d(\rho)$ has the phase allowed by the actual exchange topology. When a subfactor realization exists, compare $d(\rho)^2$ with the independently computed minimal index.

Exercises

1. Dimension test. Use the $S_3$ fusion rule above to verify multiplicativity of statistical dimension.

Solution

The left side is $d(V\otimes V)=d(V)^2=4$. Additivity on the right gives $d(\mathbf1)+d(\mathrm{sgn})+d(V)=1+1+2=4$.

2. Invertible sectors. Show that if $\rho\bar\rho\simeq\iota$, then $d(\rho)=1$.

Solution

Multiplicativity and conjugation invariance give $1=d(\iota)=d(\rho)d(\bar\rho)=d(\rho)^2$. Statistical dimension is positive, so $d(\rho)=1$.

3. Braid warning. For a phase $q=e^{i\pi/3}$, compute a double exchange and explain why no permutation sign reproduces it.

Solution

The double exchange is $q^2=e^{2\pi i/3}$, whereas squaring either permutation sign gives $1$. Hence the braid does not factor through the permutation group.

References

• Doplicher, Sergio, Rudolf Haag, and John E. Roberts. “Local Observables and Particle Statistics I.” Communications in Mathematical Physics 23 (1971): 199–230. DOI.
• Longo, Roberto. “Index of Subfactors and Statistics of Quantum Fields. I.” Communications in Mathematical Physics 126 (1989): 217–247. DOI.