Aharonov-Bohm superselection sectors Cortona 2018 AQFT: Where - PowerPoint PPT Presentation

Aharonov-Bohm superselection sectors Cortona 2018 AQFT: Where Operator Algebra meets Microlocal Analysis Ezio Vasselli Roma ezio.vasselli@gmail.com Work in progress with C. Dappiaggi and G.Ruzzi

Contents • Geometry of the Aharonov-Bohm effect • Dirac fields interacting with background AB-potentials • Interacting Dirac fields vs. sectors • Non-abelian phases • Conclusions and outlooks

� � � � Geometry of the Aharonov-Bohm effect γ γ � B • Interference pattern Electron source S γ ′ γ ′ � • B is directed towards you ❀ • No em field outside the shielded region S (S ∼ R , ideally infinite) ❀ the ”spacetime” is M := ( R 3 − S) × R , π 1 ( M ) = Z • the em potential is A ∈ Z 1 • dR ( M ), F = dA = 0 ❀ A | o = dφ o , φ o ∈ C 1 ( o, R ), ∀ o ⊂ M a.s.c. •

AB assumption: φ o = φ o ( t ) for all o ⊂ M , o a.s.c.. ❀ If ψ solves the free Schroedinger eq. with supp ( ψ ) ⊆ o , then ψ o := ψe − iφ o solves the Schroedinger eq. with interaction A . A (homotopic invariant!) phase shift � exp i γ ∗ γ ′ A appears for coherent superpositions of states of the type ψ o , ψ e , with the loop γ ∗ γ ′ ⊂ and homotopic to o ∪ e . The shift disappears whenever the experimenter: switches off � • B (clearly) • or makes S ”finite” (S ⊃ S’ ❀ M ⊂ M ′ , π 1 ( M ′ ) = 0)

How geometers describe the wavefunctions ψ o : sections ς : M → L , where L → M is the flat line bundle with (l.c.) transition maps λ hk := e − i ( φ oh − φ ok ) ∈ U (1) , o h ∩ o k � = ∅ , [BM]. Actually the following objects are equivalent: 1 - L → M • A : π 1 ( M ) → U (1) 2 - e i � ( ← the phase shift) 3 - A ∈ Z 1 dR ( M ) A o ′ o := φ o ′ | o − φ o ∈ R , ∀ o ⊆ o ′ a.s.c. A ∈ Z 1 ( M asc , R ), ˆ 4 - ˆ • 1 ⇔ 2 ⇐ 3 are well-known, [KN] • 2 ⇒ 3 [Freed], folklore • 3 ⇔ 4 [RRV’]. M asc := base (poset) of a.s.c. subsets

The phase shift can be written in terms of ˆ A : • ℓ : [0 , 1] → M loop • poset approximation of ℓ : a finite cover p ℓ = { o k ∈ M asc } ⊃ ℓ , such that there are o k, 0 , o k, 1 ⊂ o k , o k +1 , 1 = o k, 0 , for all k = 1 , . . . , n . ❀ n � � � A o k o k, 0 − ˆ ˆ � ℓ A = A o k o k, 1 . k =1

Dirac fields interacting with background AB-potentials • M glob.hyp. 4d spacetime A ∈ Z 1 • dR ( M ) ( dA = 0) • ∃ a Clifford bundle and a Dirac bundle DM → M • There is a spin connection ∇ Clifford bundle ❀ one can define / ∇ and / • A Task: construct a Dirac field ψ int such that { i / ∇ + / A − m } ψ int = 0 .

Remark: on any o ∈ M asc we have A = dφ o ❀ ψ int ( e iφ o s ) s ∈ S o ( DM ) , , must be a solution of the free Dirac equation ❀ Idea [Vas]: take a free Dirac field ψ : S ( DM ) → B ( H ) ([Dimock]), and for any o ∈ M asc define ψ o ( s ) := ψ ( e − iφ o s ) ψ o : S o ( DM ) → B ( H ) , � One has ψ o (( i / ∇ + / A − m ) s ) = 0 for all s ∈ S o ( DM ) But, ψ o ′ ( s ) = e − i ˆ A o ′ o ψ o ( s ) for s ∈ S o ( DM ) and o ⊆ o ′ •

Let ς ∈ S o ( DM ⊗L ) and π o ′ : L| o ′ → o ′ × C be local • charts for all o ′ ⊇ o . • Set ς o ′ := { id DM ⊗ π o ′ } ς ∈ S o ′ ( DM ). = e i ˆ A o ′ o ⇒ ς o ′ = e i ˆ A o ′ o ς o ❀ π o ′ π − 1 • o • ψ o ′ ( ς o ′ ) = ψ o ( ς o ) ❀ � ψ int : S ( DM ⊗ L ) → B ( H ), ψ int ( ς ) := ψ o ( ς o ) ❀ Theorem. Given A ∈ Z 1 dR ( M ) and a free Dirac field ψ , there exists the interacting field 1 − 1 ↔ { ψ o : ψ o ′ ( s ) = e − i ˆ A o ′ o ψ o ( s ) } . ψ int

Interacting Dirac fields vs. sectors An ”interacting net”: for all o , set F ( o ) := { ψ o ( s ) , s ∈ S o ( DM ) } ′′ = F free ( o ) ⊂ B ( H ) , R ( o ) := F α ( o ) , α : U (1) → Aut F . Inclusion maps: dictated by ψ o ′ = e − i ˆ A o ′ o ψ o , o ⊆ o ′ ❀ • α ( e − i ˆ A o ′ o ) : F ( o ) → F ( o ′ ) ❀ • • ( F , α ( e − i ˆ A )) precosheaf (more general than a net) ❀ • R = R free is a net R is represented as π = ⊕ κ ∈ Z π κ : R → B ( H ) • π 0 fulfils Borchers [dAH] and Haag duality [V] !

Borchers property ❀ π κ ≃ π κ o := ad u κ • o u κ o ∈ U ( F ( o )) a ”phase” of ψ o ( s ) κ • • Charge transport ❀ z κ o ′ o ∈ R ( o ′ ): z κ o ′ o π κ o ( · ) = π κ o ′ ( · ) z κ o ′ o o ′ o phase of ψ o ′ ( s ′ ) κ ψ o ′ ( s ) κ, ∗ = ψ o ′ ( s ′ ) κ e iκ ˆ A o ′ o ψ o ( s ) κ, ∗ • z κ o ′ e iκ ˆ A o ′ o u κ, ∗ z κ o ′ o = u κ • o Theorem. Pairs ( π κ , z κ ) are sectors with s.d.= 1 (=: sect 1 ( R )) in the sense of [BR], with holonomy � ∗ · · · z κ z κ ( p ℓ ) := z κ o 1 o 11 = exp iκ ℓ A . o n o 0 n Better: sect 1 ( R ) ∋ ( π, z ) 1 − 1 • ↔ ( κ ; A, ψ int ) • Sectors as in [GLRV]: A = dϕ ⇒ z ( p ℓ ) ≡ 1

Non-abelian phases ◦ From topology ( π 1 ( M ) non-Ab): ( π ρ , z ρ ) ∈ sect > 1 ( R ) ❀ ρ : π 1 ( M ) → U ( d ) • [Barrett] ❀ A ρ ∈ Ω flat ( M, u ( d )) • o,i · u ρ, ∗ • u ρ o, 1 . . . u ρ o,d Borchers’ isometries, π ρ i u ρ o = � o,i ❀ � � � u ρ o,i u ρ, ∗ z ρ ( p ℓ ) = ℓ A ρ � P exp i o,j ij ij No suitable local primitives φ o of A ρ ❀ ! ! There is no immediate way to construct ψ int M × ρ C d • The test space should be S ( DM ⊗E ρ ), E ρ := ˆ

◦ From gauge symmetry ( G cp Lie non-Ab, G ⊆ U ( n )): ψ G : S ( DM ⊗ C n ) → B ( H ) free field ❀ F G , R G • ( π σ , z σ ) ∈ sect > 1 ( R G ), σ ∈ irr ( G ) • o,i · u σ, ∗ u σ o,i Borchers’ isometries, π σ i u σ • o = � o,i ❀ � � � o,i u σ, ∗ z σ ( p ℓ ) = ℓ A σ u σ � σ P exp i o,j g ij ij ! There is no immediate way to construct ψ G,int

Conclusions and outlooks � Given A ∈ Z 1 dR ( M ), ∃ ψ int s.t. { i / ∇ + / A − m } ψ int = 0 � { κ ; A, ψ int } ↔ sect 1 ( R ) � A ρ ∈ Ω flat ( M, u ( d )) ↔ sect > 1 ( R ) ( ← π 1 ( M ) n.a.) � { σ ; A σ g } ↔ sect ( R G ) Interpretation of A ρ for π 1 ( M ) n.a. • (e.g. two shielded solenoids ⇒ π 1 ( M ) = F 2 ) • A more complete formulation should involve lgt’s • Non-flat background potential A ∈ Ω( M, R ), we should get connections as in [RRV,CRV] • Non-relativistic case, relation with [MS]

References: • [BR] CMP 287 (2009) arxiv 0801.3365 • [RRV] Adv.Math. 220 (2009) arxiv 0707.0240 • [RRV’] IJM 24 (2013) arxiv 0802.1402 • [Vas] CMP 335 (2015) arxiv 1211.1812 • [BM] Baez-Muniain book • [Barrett] : Int.J. Th. Phys. 30 (1991) • [CRV] ATMP 16 (2012) arxiv 1109.4824 • [dAH] CMP 261 (2006) arxiv 0106028 • [Dimock] Trans. Am. Math. Soc. 269 (1982) • [Freed] Adv.Math. 113 (1995) arxiv 9206021 • [GLRV] RMP 13 (2001) arxiv 9906019 • [KN] Kobayashi-Nomizu book • [MS] LMP 82 (2007) arxiv 0707.3357 • [V] RMP 9 (1997) arxiv 9609004