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

• Electron source
• γ
• γ′
• B

S

Interference pattern

γ′

• B is directed towards you ❀
• No em field outside the shielded region S

(S ∼ R, ideally infinite) ❀

• the ”spacetime” is M := (R3 − S) × R, π1(M) = Z
• the em potential is A ∈ Z1

dR(M), F = dA = 0 ❀

• A|o = dφo, φo ∈ C1(o, R), ∀o ⊂ M a.s.c.

AB assumption: φo = φo(t) for all o ⊂ M, o a.s.c.. ❀ If ψ solves the free Schroedinger eq. with supp (ψ) ⊆ o, then ψo := ψe−iφo 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 ψo, ψe, with the loop γ ∗ γ′ ⊂ and homotopic to

• ∪ e.

The shift disappears whenever the experimenter:

• switches off

B (clearly)

• or makes S ”finite” (S ⊃ S’ ❀ M ⊂ M′, π1(M′) = 0)

How geometers describe the wavefunctions ψo: sections ς : M → L, where L → M is the flat line bundle with (l.c.) transition maps λhk := e−i(φoh−φok) ∈ U(1) ,

• h ∩ ok = ∅ ,

[BM]. Actually the following objects are equivalent: 1 - L → M 2 - ei

• A : π1(M) → U(1)

(← the phase shift) 3 - A ∈ Z1

dR(M)

4 - ˆ A ∈ Z1(Masc, R), ˆ Ao′o := φo′|o − φo ∈ R, ∀o ⊆ o′ a.s.c.

• 1 ⇔ 2 ⇐ 3 are well-known, [KN]
• 2 ⇒ 3 [Freed], folklore
• 3 ⇔ 4 [RRV’]. Masc := 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

pℓ = {ok ∈ Masc} ⊃ ℓ , such that there are ok,0, ok,1 ⊂ ok, ok+1,1 = ok,0, for all k = 1, . . . , n. ❀

• ℓ A =

n

• k=1
• ˆ

Aokok,0 − ˆ Aokok,1

• .

Dirac fields interacting with background AB-potentials

• M glob.hyp. 4d spacetime
• A ∈ Z1

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 ψint such that {i / ∇ + / A − m}ψint = 0 .

Remark: on any o ∈ Masc we have A = dφo ❀ ψint(eiφos) , s ∈ So(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 ∈ Masc define ψo : So(DM) → B(H) , ψo(s) := ψ(e−iφos) One has ψo((i / ∇ + / A − m)s) = 0 for all s ∈ So(DM)

• But, ψo′(s) = e−i ˆ

Ao′oψo(s) for s ∈ So(DM) and o ⊆ o′

• Let ς ∈ So(DM⊗L) and πo′ : L|o′ → o′ × C be local

charts for all o′ ⊇ o.

• Set ςo′ := {idDM ⊗ πo′}ς ∈ So′(DM).
• πo′π−1
• = ei ˆ

Ao′o ⇒ ςo′ = ei ˆ Ao′oςo ❀

• ψo′(ςo′) = ψo(ςo) ❀

ψint : S(DM ⊗ L) → B(H), ψint(ς) := ψo(ςo) ❀

• Theorem. Given A ∈ Z1

dR(M) and a free Dirac field ψ,

there exists the interacting field ψint

1−1

↔ {ψo : ψo′(s) = e−i ˆ

Ao′oψo(s)} .

Interacting Dirac fields vs. sectors An ”interacting net”: for all o, set F(o) := {ψo(s), s ∈ So(DM)}′′ = Ffree(o) ⊂ B(H) , R(o) := Fα(o) , α : U(1) → AutF . Inclusion maps:

• dictated by ψo′ = e−i ˆ

Ao′oψo, o ⊆ o′ ❀

• α(e−i ˆ

Ao′o) : F(o) → F(o′) ❀

• (F, α(e−i ˆ

A)) precosheaf (more general than a net) ❀

• R = Rfree is a net
• R is represented as π = ⊕κ∈Zπκ : R → B(H)

! π0 fulfils Borchers [dAH] and Haag duality [V]

• Borchers property ❀ πκ ≃ πκ
• := aduκ
• uκ
• ∈ U(F(o)) a ”phase” of ψo(s)κ
• Charge transport ❀ zκ
• ′o ∈ R(o′): zκ
• ′oπκ
• (·) = πκ
• ′(·)zκ
• ′o
• zκ
• ′o phase of ψo′(s′)κψo′(s)κ,∗ = ψo′(s′)κ eiκ ˆ

Ao′o ψo(s)κ,∗

• zκ
• ′o = uκ
• ′ eiκ ˆ

Ao′o uκ,∗

• Theorem. Pairs (πκ, zκ) are sectors with s.d.= 1

(=: sect1(R)) in the sense of [BR], with holonomy zκ(pℓ) := zκ

• no0n

∗ · · · zκ

• 1o11 = exp iκ
• ℓ A .
• Better: sect1(R) ∋ (π, z) 1−1

↔ (κ; A, ψint)

• Sectors as in [GLRV]: A = dϕ ⇒ z(pℓ) ≡ 1

Non-abelian phases

• From topology (π1(M) non-Ab):
• (πρ, zρ) ∈ sect>1(R) ❀ ρ : π1(M) → U(d)
• [Barrett] ❀ Aρ ∈ Ωflat(M, u(d))
• uρ
• ,1 . . . uρ
• ,d Borchers’ isometries, πρ
• =

i uρ

• ,i · uρ,∗
• ,i ❀

zρ(pℓ) =

• ij
• P exp i
• ℓ Aρ
• ij

uρ

• ,iuρ,∗
• ,j

! No suitable local primitives φo of Aρ ❀ ! There is no immediate way to construct ψint

• The test space should be S(DM ⊗Eρ), Eρ := ˆ

M ×ρCd

• From gauge symmetry (G cp Lie non-Ab, G ⊆ U(n)):
• ψG : S(DM ⊗ Cn) → B(H) free field ❀ FG , RG
• (πσ, zσ) ∈ sect>1(RG), σ ∈ irr(G)
• uσ
• ,i Borchers’ isometries, πσ
• =

i uσ

• ,i · uσ,∗
• ,i ❀

zσ(pℓ) =

• ij

σ

• P exp i
• ℓ Aσ

g

• ij

uσ

• ,iuσ,∗
• ,j

! There is no immediate way to construct ψG,int

Conclusions and outlooks Given A ∈ Z1

dR(M), ∃ ψint s.t. {i /

∇ + / A − m}ψint = 0 {κ; A, ψint} ↔ sect1(R) Aρ ∈ Ωflat(M, u(d)) ↔ sect>1(R) (← π1(M) n.a.) {σ; Aσ

g} ↔ sect(RG)

• Interpretation of Aρ for π1(M) n.a.

(e.g. two shielded solenoids ⇒ π1(M) = F2)

• 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