Anyons in Algebraic Quantum Field Theory (AQFT) Jakob Yngvason University of Vienna NORDITA, Stockholm, March 12, 2019 Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 1 / 23 Topics The general framework of AQFT Superselection sectors Fields and statistics There are no “free” anyon fields Scattering theory Conclusions Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 2 / 23 The general framework of AQFT AQFT, also called Local Quantum Physics (monograph of R. Haag, 1992), incorporates basic principles of quantum physics and the (special) theory of relativity in an elegant and mathematically rigorous framework. (R. Haag, D. Kastler, H. Araki, S. Doplicher, J. Roberts; H. Borchers; D. Buchholz; K. Fredenhagen,...) Important aspect: No position operators for “particles”. Localization by means of local observables. Particles emerge asymptotically via scattering theory. In the 80’s and 90’s the framework was adapted to include anyonic statistics in D=2+1 and 1+1 space-time dimensions. (K. Fredenhagen, H. Rehren, B. Schroer, R. Schrader, J. Mund,...; J. Frohlich,¨ P. Marchetti,...) Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 3 / 23 Assumptions Basic object: Net O 7! A(O) d of von Neumann algebras, O ⊂ R (“algebras of local observables”). Subalgebras of a global C∗-algebra A (“quasi-local algebra”). Assumptions: Isotony: O1 ⊂ O2 )A(O1) ⊂ A(O2) 0 Locality (Micro-causality): O1 ⊂ O2 ) [A(O1); A(O2)] = f0g " Covariance: Automorphisms αg of A for g 2 P+ with αgA(O) = A(gO) Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 4 / 23 Assumptions (cont.) Vacuum representation: Irreducible representation π0 of A on a Hilbert space H0, and a unitary representation P µ d U0(x) = exp(i µ P xµ) of the translation group R implementing αx such that µ (i) spec fP g ⊂ V+ (ii) There exists a translationally invariant Ω0 2 H0 (vacuum). 0 0 Haag duality: π0(A(O)) = π0(A(O )) for a suitable class of d O ⊂ R . Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 5 / 23 Superselection sectors; spectrum condition Besides the vacuum representation there are in general other unitarily inequivalent representations π (superselection sectors) of A of interest. DHR (Doplicher-Haag-Roberts) criterion: There is a bounded O such that π is unitarily equivalent to the vacuum representation π0 when restricted to A(O0). B (Borchers) criterion: Space-time translations αx are unitarily µ implemented on Hπ and spec fP g ⊂ V+. (Spectrum condition.) It can be shown that DHR implies B, but the converse may not hold in general. Important: The spectrum condition implies im particular that x 7! Uπ(x) , 2 Hπ, can be extended to a holomorphic function on d R + iV+. Crucial for many results! Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 6 / 23 Localization in spacelike cones Special class of representations fulfilling B: Massive particle representations: Translation group is unitarily implemented and the lower boundary of of spec fP µg is an isolated mass shell. By a theorem of Buchholz and Fredenhagen (1982) it is always possible to localize massive particle representations in spacelike cones. These are regions of the form [ C = a + λCH λ>0 d H with a 2 R , C a double cone in the spacelike hyperboloid x · x = −1. Localization means here, by definition, that π is unitarily equivalent to 0 π0 when restricted to A(C ). Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 7 / 23 ....,· eo H Charges; statistics Parameters labelling (equivalence classes of) of superselection sectors are often called “charges”. In the DHR situation (localization in bounded regions) one speaks of simple charges. If cone-localization is the tightest possible one speaks of gauge- or topological charges. On the set ∆ of (equivalence classes of) superselection sectors there is an intrinsic definition (based on locality, Haag duality, and covariance) of composition of charges: (π; σ) ! π × σ. This leads to the definition of statistic operators, which are unitary operators "πσ : Hπ×σ !Hσ×π The family of statistic operators satisfies the relations of the braid group. In D ≥ 3 + 1 this leads to Bose- or Fermi representations of the permutation group as the only possibilities. Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 8 / 23 2. Let π be localized in some C. There there is a unitary U such that π(A) = UAU −1 for A 2 A(C0). 3. Define a localized morphism on A by ρ(A) := UAU −1 for all A. 0 0 −1 4. If C1 ⊃ C and A 2 A(C1), B 2 A(C1) ⊂ C , then B = UBU and −1 0 0 ! hence [ρ(A);B] = U[A; B]U = 0. Thus ρ(A) 2 A(C1) = A(C1) by Haag duality. Hence ρ is an endomorphism of A. 5. If π1 corresponds to ρ1 and π2 to ρ2 then π1 × π2 is, by definition, given by ρ1ρ2. 6. If the ρi are localized in spacelike separated regions (achieved by means of “charge transporters” Ui 2 A) the statistics operator ∗ "12 = (U2ρ2(U1)) (U1ρ1(U2)) intertwines between ρ1ρ2 and ρ2ρ1. Statistics (cont.) Sketch of the composition of charges 1. Represent the algebra on the vacuum Hilbert space, i.e., identify A and π0(A). Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 9 / 23 3. Define a localized morphism on A by ρ(A) := UAU −1 for all A. 0 0 −1 4. If C1 ⊃ C and A 2 A(C1), B 2 A(C1) ⊂ C , then B = UBU and −1 0 0 ! hence [ρ(A);B] = U[A; B]U = 0. Thus ρ(A) 2 A(C1) = A(C1) by Haag duality. Hence ρ is an endomorphism of A. 5. If π1 corresponds to ρ1 and π2 to ρ2 then π1 × π2 is, by definition, given by ρ1ρ2. 6. If the ρi are localized in spacelike separated regions (achieved by means of “charge transporters” Ui 2 A) the statistics operator ∗ "12 = (U2ρ2(U1)) (U1ρ1(U2)) intertwines between ρ1ρ2 and ρ2ρ1. Statistics (cont.) Sketch of the composition of charges 1. Represent the algebra on the vacuum Hilbert space, i.e., identify A and π0(A). 2. Let π be localized in some C. There there is a unitary U such that π(A) = UAU −1 for A 2 A(C0). Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 9 / 23 0 0 −1 4. If C1 ⊃ C and A 2 A(C1), B 2 A(C1) ⊂ C , then B = UBU and −1 0 0 ! hence [ρ(A);B] = U[A; B]U = 0. Thus ρ(A) 2 A(C1) = A(C1) by Haag duality. Hence ρ is an endomorphism of A. 5. If π1 corresponds to ρ1 and π2 to ρ2 then π1 × π2 is, by definition, given by ρ1ρ2. 6. If the ρi are localized in spacelike separated regions (achieved by means of “charge transporters” Ui 2 A) the statistics operator ∗ "12 = (U2ρ2(U1)) (U1ρ1(U2)) intertwines between ρ1ρ2 and ρ2ρ1. Statistics (cont.) Sketch of the composition of charges 1. Represent the algebra on the vacuum Hilbert space, i.e., identify A and π0(A). 2. Let π be localized in some C. There there is a unitary U such that π(A) = UAU −1 for A 2 A(C0). 3. Define a localized morphism on A by ρ(A) := UAU −1 for all A. Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 9 / 23 5. If π1 corresponds to ρ1 and π2 to ρ2 then π1 × π2 is, by definition, given by ρ1ρ2. 6. If the ρi are localized in spacelike separated regions (achieved by means of “charge transporters” Ui 2 A) the statistics operator ∗ "12 = (U2ρ2(U1)) (U1ρ1(U2)) intertwines between ρ1ρ2 and ρ2ρ1. Statistics (cont.) Sketch of the composition of charges 1. Represent the algebra on the vacuum Hilbert space, i.e., identify A and π0(A). 2. Let π be localized in some C. There there is a unitary U such that π(A) = UAU −1 for A 2 A(C0). 3. Define a localized morphism on A by ρ(A) := UAU −1 for all A. 0 0 −1 4. If C1 ⊃ C and A 2 A(C1), B 2 A(C1) ⊂ C , then B = UBU and −1 0 0 ! hence [ρ(A);B] = U[A; B]U = 0. Thus ρ(A) 2 A(C1) = A(C1) by Haag duality. Hence ρ is an endomorphism of A. Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 9 / 23 6. If the ρi are localized in spacelike separated regions (achieved by means of “charge transporters” Ui 2 A) the statistics operator ∗ "12 = (U2ρ2(U1)) (U1ρ1(U2)) intertwines between ρ1ρ2 and ρ2ρ1. Statistics (cont.) Sketch of the composition of charges 1. Represent the algebra on the vacuum Hilbert space, i.e., identify A and π0(A). 2. Let π be localized in some C. There there is a unitary U such that π(A) = UAU −1 for A 2 A(C0). 3. Define a localized morphism on A by ρ(A) := UAU −1 for all A. 0 0 −1 4. If C1 ⊃ C and A 2 A(C1), B 2 A(C1) ⊂ C , then B = UBU and −1 0 0 ! hence [ρ(A);B] = U[A; B]U = 0. Thus ρ(A) 2 A(C1) = A(C1) by Haag duality. Hence ρ is an endomorphism of A. 5. If π1 corresponds to ρ1 and π2 to ρ2 then π1 × π2 is, by definition, given by ρ1ρ2. Jakob Yngvason (Uni Vienna) Anyons in Algebraic QFT 9 / 23 Statistics (cont.) Sketch of the composition of charges 1. Represent the algebra on the vacuum Hilbert space, i.e., identify A and π0(A). 2. Let π be localized in some C.