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Fermi-Walker transport, second-order vector bundles and EPR correlation experiments

Final Presentation Joel Ong, A0098750U

National University of Singapore

April 14, 2016

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 1 / 30

Introduction

Physical motivations — quantum physics in curved spacetimes

Curved Spacetimes Thomas precession: spin-orbit interaction correction term owing to special relativistic efgects, per Newburgh (1972). Emerges from noncommutative nature of Lorentz group. Generalisation to curved spacetimes is Fermi-Walker transport Quantum Physics Prototypical quantum-mechanical observable: spin Physical (active) symmetry transformations represented as operators

• n Hilbert space under the usual (nonrelativistic) phenomenology.

Same underlying structure group SO(3)! Possible correspondence?

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 2 / 30

Introduction

Phenomenology: Thomas precession

Figure 1: Transport of a spin vector in a circle under (a) Galilean and (b) special

• relativity. From Alamo and Criado (2009)

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 3 / 30

Introduction

Outline of project

Formulation of Fermi-Walker transport as a connection

requires higher-order bundle structures for gauge covariance

Evaluation of holonomy along closed path in T M: vertical closure

Active vs. Passive

Interpretation of SO(3)-valued geometric phase as spin precession Comparison against literature predictions

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 4 / 30

Introduction

Mathematical preliminaries

Fibre bundles Generalise the notion of a direct set product to topological spaces with nontrivial global structure (just as manifolds generalise Rn)

notationally: projection π : E → X, with fjbre F and structure group G if ϕα : X × F → Uα ⊂ E, then ϕ−1

β

• ϕα acts on F with group action in

G.

Special cases:

Principal bundles: fjbre is the structure group G itself Vector bundles: real vector spaces for fjbres, with the structure group being GL(n).

For each fjbre bundle, there is an associated principal bundle.

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 5 / 30

Introduction

Fibre bundle: Example

The Möbius strip as line bundle over S1: Local trivialisations as (θ, z), and transition functions as group action in Z2 (z → −z).

Figure 2: Two difgerent line bundles over S1, with difgerent structure groups. From Penrose (2007).

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 6 / 30

Introduction

Parallel Transport and Gauge Transformations

(Ehresmann) connection on principal bundle P: separation of TpP into horizontal and vertical subspaces: TpP = HpP ⊕ VpP

Connection form: Lie-algebra-valued one-form on T P such that ker ω = HpP. Parallel transport along γ : [0, 1] → M via horizontal lift ˜ γ : [0, 1] → P satisfying ω(˙ ˜ γ) = 0

Gauge transformations: right-equivariant vertical automorphisms (i.e. group actions on P). F(p) = pτ(p) for p ∈ P, τ ∈ G Example (Tangent bundle over Riemannian manifold) Group action in GL(4) locally preserving the metric gp: ι ˙

˜ γ

(

τ −1dτ + ω

)

= 0 ⇐ ⇒ ˙ xν (∂νuµ + Γµ

σνuσ) ∂µ = 0

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 7 / 30

Second-order structures

Motivation and Construction strategy

Gauge transformations on T M break Fermi-Walker transport condition because of manifest dependence on fjber values. Ansatz: Tangent bundle as base manifold of second-order bundle?

Fundamental Theorem of Riemann Geometry: L-C connection uniquely defjned

Construction strategy: induce metric on T M, and construct induced connection once submanifold identifjed.

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 8 / 30

Mathematical Results

Given a choice of connection ∇, a metric g on a manifold M uniquely induces a metric ˜ g on T M — “Sasaki Metric” Sasaki metric induces a Levi-Civita connection ˜ ∇ on T M Restricting image of ˜ ∇ to X yields restricted connection ˆ ∇. T M X ⊂ T (T M) T M M π π∗ id πT π

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 9 / 30

Mathematical Results

Fermi-Walker transport condition on the tangent bundle: hypersurface defjned by f(v) = g(v, v) = c, a constant — “mass-shell submanifold” Connection D induced by ˆ ∇ via Gauss-Codazzi equation, projecting

• nto tangent spaces of this hypersurface

In coordinates, we recover classical Fermi-Walker transport equation from the vertical parts of the parallel transport equation!

This is nontrivial

Yields constraint on possible gauge transformations:

must leave tangent vector invariant

Yields new condition for gauge invariance

In particular: closure in T M

For massive particles, holonomy group is SO(3)!

From Lorentz isotropy subgroup considerations upon bundle reduction Relevant to further discussion

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 10 / 30

Vertical Closure

γ1 γ2

(a) Closed in M but not T M

γ1 γ2

(b) Closed in T M

If closed, holonomy evaluated as Tγ = T−1

2

• T1.

For FW transport, we evaluate vertical parallel transport maps.

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 11 / 30

Vertical Closure

Closure enforced by instantaneous accelerations.

Uniqueness?

Second-order geodesic equation: ∇ ˙

γ∇ ˙ γ ˙

γ + g(∇ ˙

γ ˙

γ, ∇ ˙

γ ˙

γ) g(˙ γ, ˙ γ) ˙ γ = 0 Example (Flat spacetime) Constant-acceleration curves satisfy the 2nd-order geodesic equation; the vertical parallel transport maps along them are Lorentz boosts. They are vertical curves in the limit of infjnite acceleration. Hypothesis: evaluate vertical parallel transport maps V(0)

21 along

vertical geodesics γ(0)

21 : ˙

γ2(0) → ˙ γ1(0) Evaluate holonomy as Tγ = T−1

γ2 V(1) 12 Tγ1V(0) 21 .

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 12 / 30

EPR Correlations

For massive particles, Bell’s inequality maximally violated for particular choices of measurement axes Holonomy as SO(3)-valued geometric phase

Appears to reduce extent of Bell’s inequality violation Restored by rotating one set of measurement axes relative to the other.

Exactly which element of SO(3)?

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 13 / 30

EPR Correlations

Flat spacetime

Terashima and Ueda (2003), Alsing, Stephenson Jr, and Kilian (2009): Rotation predicted if entangled-pair centre-of-mass and measurement apparatus are moving relative to each other. Element of SO(3) computed from composing Lorentz boosts Electron 1’s rest frame: k Centre-of-mass frame: p+ Measurement frame: Λp+ +δ L(p+) Λ L(Λp+)−1

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 14 / 30

EPR Correlations

Flat spacetime

Active vs. Passive:

In fmat spacetime, Lorentz boosts are parallel transport maps along vertical geodesics

• vs. Lorentz boosts as change of basis — same matrix representation,

difgerent physical processes

Vertical closure:

Evaluate holonomy along paths taken by both electrons Measurements at difgerent locations!

OK because global parallelism in fmat spacetime. Still require closure on fjbers! No horizontal closure required in fmat spacetime.

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 15 / 30

EPR Correlations

Flat spacetime: maps on tangent bundle

Centre-of-mass: k1 Electron 1: p+ Electron 2: p− Measurement frame: k2 +2δ V+ T1 T−1

2

V−−1

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 16 / 30

EPR Correlations

Curved spacetime

We use the Schwarzschild geometry for illustrative purposes. We choose circular paths (not necessarily geodesics) for convenience: p = γ1(0) = γ2(0) q = γ1(1) = γ1(1) γ1 γ2

Figure 4: Two circular trajectories from p to q forming a closed curve on M, but again not T M

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 17 / 30

EPR Correlations

Curved spacetime

Proposed generalisation per Terashima and Ueda (2004), Bakke, Furtado, and Carvalho (2015): Electron 1’s rest frame: k Tetrad frame: p+ Measurement frame: Λp+ 1 + r δτ L(p(τ)) Λ = 1 + ω δτ L(Λp(τ))−1 r ∈ so(3); R1 = P exp [

∫ r dτ].

Holonomy as R = R−1

2

• R1

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 18 / 30

EPR Correlations

Curved spacetime: double bundle

Again, we go from passive → active:

Emission centre-of-mass momentum k1 Electron 1: p+(0) Electron 2: p−(0) Electron 1: p+(1) Electron 2: p−(1) Measurement apparatus momentum k2 Tγ V(0)

+

T1 = Tγ1 V(1)

+

V(1)

− −1

T−1

2

= Tγ2

−1

V(0)

− −1

V(0)

21

V(1)

12

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 19 / 30

EPR Correlations

Curved spacetime: double bundle

Replace boosts with vertical geodesic parallel transport maps

Constant-acceleration curves in curved spacetimes are not simple boosts!

explicit dependence on r and M in this case

Ambiguity as to which vertical curve to follow!

e.g. could take the dashed path at endpoints of trajectories recourse to experiment?

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 20 / 30

Conclusion

FW transport as vertical parallel transport on double tangent bundle Vertical parallel transport maps along second-order geodesics generalise boosts in curved spacetimes A priori: non-unique SO(3)-valued holonomy

Difgerent ways of restoring vertical closure: proposed experimental discrimination

Further work:

Better trajectories Second-order geodesics in detail Other geometries Photons Torsion Modifjed gravity?

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 21 / 30

The End

Questions?

Slides available at http://hyad.es/fyp

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 22 / 30

Projection Operators

Massive Particles P ˙

γv = v − g(v, ˙

γ) g(˙ γ, ˙ γ) ˙ γ (1) Photons Pl,nv = v − g(v, n) g(l, n) l − g(l, v) g(l, n)n (2) Constrained Transport P

[ D

dt (Pv)

]

= 0 (3)

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 23 / 30

Fermi-Walker Transport Conditions

Massive Particles 0 = ∇uv − g(v, u) g(u, u)∇uu + g(v, ∇uu) g(u, u) u ? = Duv (4) Photons 0 = ∇ ˙

γv − g(v, ˙

γ) g(l, ˙ γ) ∇ ˙

γl + g(v, ∇ ˙ γl)

g(l, ˙ γ) ˙ γ − g(l, v) g(l, ˙ γ)∇ ˙

γ ˙

γ + g(∇ ˙

γ ˙

γ, v) g(l, ˙ γ) l. (5)

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 24 / 30

Double Tangent bundle

Defjnition (Natural Projection) T M X ⊂ T (T M) T M M π π∗ πT π For π : T M → M as (x, u) → (x), we have π∗ : T T M → T M as (x, u, h, v) → (x, h). Second-order structure This is distinct from the secondary projection πT : T T M → T M as (x, u, h, v) → (x, u).

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 25 / 30

Double Tangent bundle

Defjnition (Connection Map) T (T M) T M M T M u K(v∗u) = ∇uv v∗ K The connection map K : T (T M) → T M associated with the connection ∇ satisfjes K(v∗(u)) = ∇uv.

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 26 / 30

Double Tangent Bundle

Defjnition (Vertical and Horizontal Lifts) The horizontal (resp. vertical) lift uh (resp. uv) of u ∈ X(M) is the unique element of X(T M) such that π∗uh = u, and K(uh) = 0 (resp. K(uv) = u, π∗uv = 0). This sets up a separation of the double tangent bundle into vertical and horizontal bundles as T (T M) = H(T M) ⊕ V(T M).

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 27 / 30

Double Tangent Bundle

Double lifts

I = [0, 1] T I M T M

d dτ

γ γ∗

˙ γ = γ∗ d

dτ

π = ⇒ I T I T M X

d dτ

˙ γ ˙ γ∗

¨ γ = ˙ γ∗ d

dτ

π∗ If γ : [0, 1] → M is an integral curve of a vector fjeld u ∈ X(M), then ˙ γ : [0, 1] → T M = u ◦ γ = γ∗ d

dt.

¨ γ satisfjes π∗¨ γ = πT¨ γ; i.e. ¨ γ : [0, 1] → X ⊂ T T M.

T M X ⊂ T (T M) T M M π π∗ id πT π

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 28 / 30

Double Tangent Bundle

From Kappos (2001): Defjnition (Sasaki Metric) With K as the connection map for a connection ∇ on a Riemannian manifold (M, g), then for U, V ∈ X(T M) the Sasaki metric ˜ g on T M (itself a Riemannian manifold) is given by ˜ g(U, V)

def

= g(π∗U, π∗V) + g(KU, KV). From Bowman (1972): Defjnition (Second-order Connection) The Levi-Civita connection ˜ ∇ on T (T M) induced by the Sasaki metric in turn induces a metric-compatible, torsion-free connection ˆ ∇ : X(M) × ˜ X(M) → ˜ X(M)

• n the vector subbundle X ⊂ T (T M), X ≃ T M ⊕ T M, uniquely determined by

ˆ ∇ ˙

γU = (∇ ˙ γπ∗U)h + (∇ ˙ γKU)v

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 29 / 30

Works Cited

Alamo, Nieves, and Carlos Criado. 2009. “Parallel Transport and Thomas-Wigner Rotation.” American Mathematical Monthly 116 (5). Mathematical Association of America: 439–46. Alsing, P. M., G. J. Stephenson Jr, and Patrick Kilian. 2009. “Spin-induced non-geodesic motion, gyroscopic precession, Wigner rotation and EPR correlations of massive spin 1/2 particles in a gravitational fjeld.” ArXiv Preprint ArXiv:0902.1396. Bakke, K., C. Furtado, and A. M. de M. Carvalho. 2015. “Wigner rotation via Fermi–Walker transport and relativistic EPR correlations in the Schwarzschild spacetime.” International Journal

• f Quantum Information 13 (02). World Scientifjc: 1550020.

Bowman, Robert H. 1972. “Second Order Connections.” J. Difgerential Geom. 7 (3-4). Lehigh University: 549–61. http://projecteuclid.org/euclid.jdg/1214431172. Kappos, Elias. 2001. “Natural Metrics on Tangent Bundles.” Master’s thesis, Lund University. Newburgh, R. G. 1972. “Thomas Precession as Evidence for the Non-Euclidean Geometry of Atomic Orbits.” Lettere Al Nuovo Cimento (1971–1985) 3 (5). Springer: 173–74. Penrose, R. 2007. The Road to Reality: A Complete Guide to the Laws of the Universe. Vintage

• Series. Vintage Books. https://books.google.com.sg/books?id=coahAAAACAAJ.

Terashima, Hiroaki, and Masahito Ueda. 2003. “Relativistic Einstein-Podolsky-Rosen correlation and Bell’s inequality.” International Journal of Quantum Information 1 (01). World Scientifjc: 93–114. ———. 2004. “Einstein-Podolsky-Rosen correlation in a gravitational fjeld.” Physical Review A 69 (3). APS: 032113.

Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 30 / 30