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The Aharonov-Bohm Effect and the Geometry of Connections
Leo Tzou
Speaker is partially supported by the Academy of Finland
Speaker is partially supported by NSF Grant DMS-3861041
Double Slit Experiment
one shoots electrons through a double slit
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Double Slit Experiment
and obtains a symmetric distribution function
3
The Aharonov-Bohm Experiment
One places a solenoid behind the slit
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The Aharonov-Bohm Experiment
without magnetic potential
5
The Aharonov-Bohm Experiment
and scatter electrons
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The Aharonov-Bohm Experiment
and obtains again a symmetric distribution function
7
The Aharonov-Bohm Experiment
•We turn on the magnetic potential A in such a way that the magnetic
field E = ∇×A is completely contained in the solenoid.
•Observe that electrons only pass through regions where E = 0
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The Aharonov-Bohm Experiment
• Surprisingly, the distribution is no longer symmetrical.
• Remember that electrons only passed through regions of vanishing
magnetic field. So what happened?
• The region where E = ∇×A vanishes is not simply connected.
• The potential A does not vanish up to gauge outside of the solenoid.
• What kind of A gives trivial interference patterns?9
The Magnetic Schrodinger Equation
Quantum mechanical effects involving the magnetism is modeled by
the magnetic Schrodinger equation:
LAu := (d+ iA)∗(d+ iA)︸ ︷︷ ︸=∆ if A=0
u = 0
• A is real valued 1-form represents the magnetic potential.
• The curl dA is the magnetic field.
• How does A effect the boundary behaviour of the solution?
• Classically only dA matters and not A.
• But when there is topology the Aharonov-Bohm experiment suggests
otherwise.
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This question was studied in the setting of Euclidean setting with cavities by Ballestero-Weder. In the
geometric setting the time dependent hyperpolic and boundary spectral data case was done by Kurylev
et al. Furthermore recovering the magnetic field from partial boundary measurement was done recently
by Emanuvilov-Yamamoto-Uhlmann on planar domains.
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Calderon Problem for (d+ iA)∗(d+ iA)
• Let (M, g) be a Riemannian manifold with boundary and f ∈ C∞(∂M)
• Assume well-posedness, there exists a unique uf solving
(d+ iA)∗(d+ iA)uf = 0
uf = f ∂M
• Define the Dirichlet-Neumann map by
ΛA : f 7→ in(d+ iA)uf
where n is the normal vector field along the boundary.
Does ΛA uniquely determine A?
12
Calderon Problem for (d+ iA)∗(d+ iA)
• Let (M, g) be a Riemannian manifold with boundary and f ∈ C∞(∂M)
• Assume well-posedness, there exists a unique uf solving
(d+ iA)∗(d+ iA)uf = 0
uf = f ∂M
• Define the Dirichlet-Neumann map by
ΛA : f 7→ in(d+ iA)uf
where n is the normal vector field along the boundary.
Does ΛA uniquely determine A?
NO
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Gauge Invariance
• Let φ ∈ C∞(M) be a real function with φ |∂M= 0.
• Consider the operator LA+dφ = (d+ iA+ idφ)∗(d+ iA+ idφ)
• If LAu = 0 then LA+dφe−iφu = 0.
• So ΛA = ΛA+dφ.
Natural Conjecture (false in general):
If ΛA1= ΛA2
then A1 −A2 is exact.
This holds only on planar domains
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Simply Connected Domains
Lets suppose M =unit disk.
•Observe that LA = (d+iA)∗(d+iA) =(2nd order elliptic) +idA+︸ ︷︷ ︸magneticfield
|A|2.
• So by analytic techniques we can show that
ΛA1= ΛA2
⇒ dA1 = dA2
• By the fact that M is simply connected,
φ(z) =∫ z
z0
A1 −A2
is path independent and well defined so
A1 = A2 + dφ
Thus ΛA1= ΛA2
⇔ A1 −A2 = dφ
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This corresponds to the double slit experiment with no topology:
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Topological Obstructions
• On a surface M with genus similar analytic techniques will obtain
ΛA1= ΛA2
⇒ d(A1 −A2) = 0
• However, this does not imply A1 −A2 is exact. So,
A1 −A2 is exact ⇒ ΛA1= ΛA2
A1 −A2 is closed ⇐ ΛA1= ΛA2
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Topological Obstructions
• On a surface M with genus similar analytic techniques will obtain
ΛA1= ΛA2
⇒ d(A1 −A2) = 0
• However, this does not imply A1 −A2 is exact. So,
A1 −A2 is exact ⇒ ΛA1= ΛA2
lA1 −A2 is closed ⇐ ΛA1
= ΛA2
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Topological Obstructions
• On a surface M with genus similar analytic techniques will obtain
ΛA1= ΛA2
⇒ d(A1 −A2) = 0
• However, this does not imply A1 −A2 is exact. So,
A1 −A2 is exact ⇒ ΛA1= ΛA2
(cohomology of M)
A1 −A2 is closed ⇐ ΛA1= ΛA2
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Topological Obstructions
• On a surface M with genus similar analytic techniques will obtain
ΛA1= ΛA2
⇒ d(A1 −A2) = 0
• However, this does not imply A1 −A2 is exact. So,
A1 −A2 is exact ⇒ ΛA1= ΛA2
? ⇔ ?
A1 −A2 is closed ⇐ ΛA1= ΛA2
20
A Satisfactory Answer (Guillarmou - LT, GAFA 2011)
ΛA1= ΛA2
⇔ (A1 −A2) ∈ H1(M,∂M ; N)
This means that
• d(A1 −A2) = 0 (ie. (A1 −A2) ∈ H1(M,∂M ; R))
•∫γ(A1 −A2) ∈ 2πN for all closed loops γ.
Corollary
ΛA = Λ0 IFF dA = 0 and∫γ A ∈ 2πN for all loops γ.
What motivated us to this condition?
The answer is in the geometry of connection.
21
Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• Fix v ∈ Ez0
22
Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• Parallel transport v along γ by ∇A
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Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• Parallel transport v along γ by ∇A
24
Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• Parallel transport v along γ by ∇A
25
Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• Parallel transport v along γ by ∇A
26
Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• Parallel transport v along γ by ∇A
27
Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• Parallel transport v along γ by ∇A
28
Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• Parallel transport v along γ by ∇A
29
Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• to obtain v′ ∈ Ez0
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Point of View of Parallel Transport
Let E = C×M be the trivial complex line bundle over M .
• ∇A := d+ iA is a connection acting on this line bundle.
• Let γ be a closed loop and z0 ∈ γ
• to obtain v′ ∈ Ez0
• Solving the ODE for parallel transport yields v′ = (ei∫γ A)v
31
• Therefore that holonomy of d+ iA is equal to that of ∇0 = d iff∫γA ∈ 2πN
for all closed loops γ
• From the geometric point of view, it is not the exactness of A but
rather the isomorphism of the connections ∇A = d+ iA and ∇0 = d that
matters.
32
Proof of Result
ΛA = Λ0 IFF dA = 0 and∫γ A ∈ 2πN for all loops γ.
• Analytic methods show that ΛA = Λ0 ⇒ dA = 0 and A = 0∂M
33
Proof of Result
ΛA = Λ0 IFF dA = 0 and∫γ A ∈ 2πN for all loops γ.
• Analytic methods show that ΛA = Λ0 ⇒ dA = 0 and A = 0∂M
Consider a closed loop γ on M :
34
Proof of Result
ΛA = Λ0 IFF dA = 0 and∫γ A ∈ 2πN for all loops γ.
• Analytic methods show that ΛA = Λ0 ⇒ dA = 0 and A = 0∂M
Consider a closed loop γ on M :
Since dA = 0 we can choose any representative of the homology class.
35
Proof of Result
ΛA = Λ0 IFF dA = 0 and∫γ A ∈ 2πN for all loops γ.
• Analytic methods show that ΛA = Λ0 ⇒ dA = 0 and A = 0∂M
Consider a closed loop γ on M :
so we deform the curve as such so that γ = Γ1 + Γ2 with Γ2 ⊂ ∂M
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Proof of Result
ΛA = Λ0 IFF dA = 0 and∫γ A ∈ 2πN for all loops γ.
• Analytic methods show that ΛA = Λ0 ⇒ dA = 0 and A = 0∂M
Consider a closed loop γ on M :
We need that∫γ A =
∫Γ1A+
∫Γ2
A︸ ︷︷ ︸=0
∈ 2πN
37
Proof of Result
ΛA = Λ0 IFF dA = 0 and∫γ A ∈ 2πN for all loops γ.
• Analytic methods show that ΛA = Λ0 ⇒ dA = 0 and A = 0∂M
Consider z0, z1 ∈ ∂M ∩ Γ1 :
We need that∫γ A =
∫Γ1A+
∫Γ2
A︸ ︷︷ ︸=0
∈ 2πN
38
Proof of Result
ΛA = Λ0 IFF dA = 0 and∫γ A ∈ 2πN for all loops γ.
• Analytic methods show that ΛA = Λ0 ⇒ dA = 0 and A = 0∂M
Consider z0, z1 ∈ ∂M ∩ Γ1Let f ∈ C∞(∂M) such that f(z1) = f(z0) = 1 :
We need that∫γ A =
∫Γ1A+
∫Γ2
A︸ ︷︷ ︸=0
∈ 2πN
39
Proof of Result
• Let L0w = 0 and LAv = 0 such that
v = w = f ∂M
• Since ΛA = Λ0,
∂νw = ∂νv
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Proof of Result
• Let ∆w = 0 and LAv = 0 such that
v = w = f ∂M
• Since ΛA = Λ0,
∂νw = ∂νv
Consider an open neighbourhood U of Γ1
We will show that in U one can transform v to w via parallel transport.41
Proof of Result
• Let ∆w = 0 and LAv = 0 such that
v = w = f ∂M
• Since ΛA = Λ0,
∂νw = ∂νv
Consider an open neighbourhood U of Γ1
We will show that in U one can transform v to w via parallel transport.42
L0w = LAv = 0, w = v = f on ∂M , ∂νw = ∂νv
• For z ∈ Γ1 define
φ(z) =∫ z
z0
A
43
L0w = LAv = 0, w = v = f on ∂M , ∂νw = ∂νv
• Since U is simply connected and dA = 0, we can extend φ to U
with
dφ = A
44
L0w = LAv = 0, w = v = f on ∂M , ∂νw = ∂νv
• Since U is simply connected and dA = 0, we can extend φ to U
with
dφ = A
Therefore, if L0w = 0 then e−iφw solves LA(e−iφw) = 0 in U .
45
L0w = LAv = 0, w = v = f on ∂M , ∂νw = ∂νv
• Since U is simply connected and dA = 0, we can extend φ to U
with
dφ = A
Therefore, if L0w = 0 then e−iφw solves LA(e−iφw) = 0 in U .
46
47
48
49
50
I claim that e−iφw = v in U51
Since A = 0 on ∂M52
Since A = 0 on ∂M53
Since A = 0 on ∂M54
Since A = 0 on ∂M55
Both v and e−iφw solve LA(v) = LA(e−iφw) = 0 in U .56
Unique continuation gives e−iφw = v in U .57
In particular, e−iφ(z1)w(z1) = v(z1).58
In particular, e−iφ(z1)f(z1) = f(z1).59
In particular, e−iφ(z1)1 = 1.60
Therefore, φ(z1) =∫ z1z0A ∈ 2πN
61
Generalization to Higher Rank Bundles
Let E be a complex bundle of rank n over a Riemann surface M with
boundary. Let ∇ be a connection acting on sections of E and V an
endomorphism of E. Define the elliptic operator
L := ∇∗∇+ V
and consider the boundary value problem
Luf = 0 M,u = f ∂M
Let Λ∇,V : f 7→ ∇νuf |∂M be the Dirichlet-Neumann operator.
Thm (Albin-Guillarmou-LT)
If Λ∇1,V1= Λ∇2,V2
then there exists a unitary endomorphism F : E → E
such that
F−1∇1F = ∇2, F−1V1F = V2
62