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 on 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 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 Evaluation of holonomy along closed path in T M : vertical closure
Introduction Special cases: April 14, 2016 Fermi-Walker transport, second-order vector bundles and EPR correlation experiments Joel Ong, A0098750U (NUS) For each fjbre bundle, there is an associated principal bundle. Vector bundles: real vector spaces for fjbres, with the structure group Principal bundles: fjbre is the structure group G itself G . Mathematical preliminaries Generalise the notion of a direct set product to topological spaces Fibre bundles 5 / 30 with nontrivial global structure (just as manifolds generalise R n ) notationally: projection π : E → X , with fjbre F and structure group G if ϕ α : X × F → U α ⊂ E , then ϕ − 1 ◦ ϕ α acts on F with group action in β being GL ( n ) .
Introduction Fibre bundle: Example Figure 2: Two difgerent line bundles over S 1 , 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 The Möbius strip as line bundle over S 1 : Local trivialisations as ( θ, z ) , and transition functions as group action in Z 2 ( z �→ − z ).
Introduction Gauge transformations: right-equivariant vertical automorphisms April 14, 2016 Fermi-Walker transport, second-order vector bundles and EPR correlation experiments Joel Ong, A0098750U (NUS) Parallel Transport and Gauge Transformations Example (Tangent bundle over Riemannian manifold) 7 / 30 (Ehresmann) connection on principal bundle P : separation of T p P into horizontal and vertical subspaces: T p P = H p P ⊕ V p P Connection form: Lie-algebra-valued one-form on T P such that ker ω = H p P . Parallel transport along γ : [ 0 , 1 ] → M via horizontal lift ˜ γ : [ 0 , 1 ] → P satisfying ω (˙ γ ) = 0 ˜ (i.e. group actions on P ). F ( p ) = p τ ( p ) for p ∈ P , τ ∈ G Group action in GL ( 4 ) locally preserving the metric g p : x ν ( ∂ ν u µ + Γ µ ( ) τ − 1 d τ + ω ι ˙ = 0 ⇐ ⇒ ˙ σν u σ ) ∂ µ = 0 ˜ γ
Second-order structures Motivation and Construction strategy 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 connection once submanifold identifjed. Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 8 / 30 Gauge transformations on T M break Fermi-Walker transport Construction strategy: induce metric on T M , and construct induced
Mathematical Results id April 14, 2016 Fermi-Walker transport, second-order vector bundles and EPR correlation experiments Joel Ong, A0098750U (NUS) 9 / 30 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 ∇ . π ∗ X ⊂ T ( T M ) T M π T π π T M M
Mathematical Results Fermi-Walker transport condition on the tangent bundle: hypersurface April 14, 2016 Fermi-Walker transport, second-order vector bundles and EPR correlation experiments Joel Ong, A0098750U (NUS) Relevant to further discussion From Lorentz isotropy subgroup considerations upon bundle reduction For massive particles, holonomy group is SO(3)! Yields new condition for gauge invariance must leave tangent vector invariant Yields constraint on possible gauge transformations : This is nontrivial equation from the vertical parts of the parallel transport equation! In coordinates, we recover classical Fermi-Walker transport onto tangent spaces of this hypersurface 10 / 30 defjned by f ( v ) = g ( v , v ) = c , a constant — “mass-shell submanifold” Connection D induced by ˆ ∇ via Gauss-Codazzi equation, projecting In particular: closure in T M
2 Vertical Closure 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 γ 1 γ 1 γ 2 γ 2 (a) Closed in M but not T M (b) Closed in T M If closed, holonomy evaluated as T γ = T − 1 ◦ T 1 .
Vertical Closure Closure enforced by instantaneous accelerations. April 14, 2016 Fermi-Walker transport, second-order vector bundles and EPR correlation experiments Joel Ong, A0098750U (NUS) vertical curves in the limit of infjnite acceleration. vertical parallel transport maps along them are Lorentz boosts. They are Constant-acceleration curves satisfy the 2 nd -order geodesic equation; the Example (Flat spacetime) 12 / 30 Uniqueness? Second-order geodesic equation: γ + g ( ∇ ˙ γ ˙ γ, ∇ ˙ γ ˙ γ ) γ ˙ γ = 0 ˙ ∇ ˙ γ ∇ ˙ g (˙ γ, ˙ γ ) Hypothesis: evaluate vertical parallel transport maps V ( 0 ) 21 along vertical geodesics γ ( 0 ) 21 : ˙ γ 2 ( 0 ) �→ ˙ γ 1 ( 0 ) γ 2 V ( 1 ) 12 T γ 1 V ( 0 ) Evaluate holonomy as T γ = T − 1 21 .
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 April 14, 2016 Fermi-Walker transport, second-order vector bundles and EPR correlation experiments Joel Ong, A0098750U (NUS) 14 / 30 Electron 1’s rest frame: k Element of SO(3) computed from composing Lorentz boosts apparatus are moving relative to each other. Rotation predicted if entangled-pair centre-of-mass and measurement Terashima and Ueda (2003), Alsing, Stephenson Jr, and Kilian (2009): L ( p + ) Centre-of-mass frame: p + + δ Λ L (Λ p + ) − 1 Measurement frame: Λ p +
EPR Correlations Measurements at difgerent locations! April 14, 2016 Fermi-Walker transport, second-order vector bundles and EPR correlation experiments Joel Ong, A0098750U (NUS) No horizontal closure required in fmat spacetime. Still require closure on fjbers! OK because global parallelism in fmat spacetime. Evaluate holonomy along paths taken by both electrons Flat spacetime Vertical closure: difgerent physical processes vs. Lorentz boosts as change of basis — same matrix representation, vertical geodesics In fmat spacetime, Lorentz boosts are parallel transport maps along Active vs. Passive: 15 / 30
EPR Correlations Flat spacetime: maps on tangent bundle Centre-of-mass: k 1 Measurement frame: k 2 T 1 2 Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 16 / 30 V −− 1 V + Electron 1: p + Electron 2: p − + 2 δ T − 1
EPR Correlations Curved spacetime We use the Schwarzschild geometry for illustrative purposes. We choose circular paths (not necessarily geodesics) for convenience: Joel Ong, A0098750U (NUS) Fermi-Walker transport, second-order vector bundles and EPR correlation experiments April 14, 2016 17 / 30 γ 1 p = γ 1 ( 0 ) = γ 2 ( 0 ) q = γ 1 ( 1 ) = γ 1 ( 1 ) γ 2 Figure 4: Two circular trajectories from p to q forming a closed curve on M , but again not T M
Recommend
More recommend
