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An Intuitive Overview of the Theory of Quantum Knots Samuel Lomonaco

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An Intuitive Overview of the Theory of Quantum Knots Samuel Lomonaco Quantum Knots ??? An Intuitive Overview of the Theory of Quantum Knots Samuel Lomonaco University of Maryland Baltimore County (UMBC) Email: [email protected] WebPageWebPage:: www.csee.umbc.edu/~lomonaco LL--OO--OO--PP • Lecture I: A Rosetta Stone for Quantum Computing This is work in collaboration with Lecture II: Quantum Knots and Mosaics, Louis Kauffman • • Lecture III: Quantum Knots & Lattices, • Lecture IV: Intuitive Overview of the Theory of Quantum Knots This talk is based on the paper: Lomonaco and Kauffman, Quantum Knots and PowerPoint Lectures and Exercises can be Lattices, to appear soon on quant-quant-phph found at: This talk was motivated by: www.csee.umbc.edu/~lomonaco Lomonaco and Kauffman, Quantum Knots and Mosaics, Journal of Quantum Information Processing, vol. 7, Nos. 22--3,3, (2008), 8585-- 115. An earlier version can be found at: http://arxiv.org/abs/0805.0339 1 This talk was also motivated by: Kauffman and Lomonaco, Quantum Knots, SPIE Proc. on Quantum Information & Computation II (ed. by DonkorDonkor,, Pirich,Pirich, & Brandt), (2004), 5436-5436-30,30, 268268-- 284. http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0403228ph/0403228 Throughout this talk: Lomonaco, Samuel J., Jr., The modern legacies of Thomson's atomic vortex theory in classical electrodynamics, AMS PSAPM/51, Providence, RI “Knot” means either a knot or a link (1996), 145 - 166. KitaevKitaev,, Alexei Yu, FaultFault--toleranttolerant quantum computation by anyonsanyons,, http://arxiv.org/abs/quanthttp://arxiv.org/abs/quant--ph/9707021ph/9707021 Rasetti, Mario, and Tullio Regge, Vortices in He II, current algebras and quantum knots, Physica 80 A, NorthNorth--Holland,Holland, (1975), 217217--2333.2333. Thinking Outside the Box Quantum Mechanics Preamble is a tool for exploring Knot Theory Objectives Rules of the Game • We seek to create a quantum system that simulates a closed knotted physical Find a mathematical definition of a quantum piece of rope. knot that is • We seek to define a quantum knot in such • Physically meaningful, i.e., physically a way as to represent the state of the implementable, and knotted rope, i.e., the particular spatial configuration of the knot tied in the rope. • Simple enough to be workable and • We also seek to model the ways of useable. moving the rope around (without cutting the rope, and without letting it pass through itself.) 2 Aspirations Themes Formal We would hope that this definition will be Knot Theory = Rewriting useful in modeling and predicting the System behavior of knotted vortices that actually occur in quantum physics such as Formal • In supercooled helium II Group Rewriting = Representation System • In the BoseBose--EinsteinEinstein Condensate • In the Electron fluid found within the fractional quantum Hall effect Group Quantum Representation Mechanics = Theory Overview • Preamble • Mosaic Knots • Quantum Mechanics: Whirlwind Tour • Quantum Knots & Quantum Knot Systems via Mosaics Mosaic Knots • Preamble to Lattice Knots • Lattice Knots Transforming Knot Theory into • Q. Knots & Q. Knot Systems via Lattices a formal Rewriting System • Future Directions & Open Questions Mosaic Tiles ()u Let T denote the following set of 11 symbols, called mosaic (unoriented) tiles: Lomonaco and Kauffman, Quantum Knots and Mosaics, Journal of Quantum Information Processing, vol. 7, Nos. 22--3,3, (2008), 8585-- 115. An earlier version can be found at: http://arxiv.org/abs/0805.0339 Please note that, up to rotation, there are exactly 5 tiles 3 Mosaic Knots Figure Eight Knot 55--MosaicMosaic A 4-mosaic trefoil Hopf Link 44--MosaicMosaic Let K(n) = the set of n-mosaic knots A Cut & Paste Move ←P1→ SubSub--MosaicMosaic Moves 4 A Cut & Paste Move A Cut & Paste Move ←P1→ ←P1→ Planar IsotopyMoves We will now re-express the as standard moves on knot diagrams as sub-mosaic SubSub--MosaicMosaic Moves moves. 11 Planar Isotopy (PI) Moves on Mosaics Planar Isotopy (PI) Moves on Mosaics ←P1→ It is understood that each of the above moves depicts all moves obtained by rotating the 22 × P subsub--mosaicsmosaics by 0, 90, 180, or 270 degrees. ←2 → ←P3 → P ←P4→ ←P5 → For example, ←1→ P6 represents each of the following 4 moves: ←→ ←P7 → ←P1→ ←P1→ ←P8 → ←P9 → ←→P10 ←→P11 ←P1→ ←P1→ 5 Planar Isotopy (PI) Moves on Mosaics Each of the PI 2--submosaicsubmosaic moves represents any one of the (n(n--2+1)2+1)2 possible moves on an n--mosaicmosaic Reidemeister Moves as SubSub--MosaicMosaic Moves Reidemeister (R) Moves on Mosaics Reidemeister (R) Moves on Mosaics R ' ←3 → ←R3 → R1 ' ←→ ←R1→ R'' ''' ←→3 ←→R3 R ' ←2 → ←R2 → '' ''' R2 R2 ←→ ←→ ()iv R()v ←R3 → ←3 → PI & R Moves on Mosaics The Ambient Group An() Each PI move and each R move is a permutation of the set of all (n) knot n--mosaicsmosaics K We define the ambient group An()as the subgroup of the group of all permutations of the set K(n) generated by the all PI moves and all Reidemeister moves. In fact, each PI and R move, as a permutation, is a product of disjoint transpositions. 6 The Mosaic Injection ι : MM()nn→ (+ 1) ()nn (+ 1) We define the mosaic injection ι : MMK(n) → K(n+1) K ()n if 0≤<i , j n (1)n+ = ij, Knot Type K ij, otherwise ι→ Mosaic Knot Type Def. Two n--mosaicmosaic knots K and K ' are of the same knot n--typetype, written n KK∼ ' provided there exists an element of the ambient group An () that transforms K into K ' . Two n--mosaicsmosaics K and K ' are of the same knot type if there exists a nonnon--negativenegative integer k such that nk+ ι kkKiK∼ ' 1 2 3 7 4 5 6 7 8 9 8 10 11 12 13 14 15 9 n KK' ∼ 16 17 n KK' ∼ Conjecture: The Mosaic Knots formal rewriting system fully captures tame knot theory. Recently, Takahito Kuriya has proven this conjecture, T. Kuriya,Kuriya, On a LomonacoLomonaco--KauffmanKauffman Conjecture, arXiv:0811.0710. State of a Quantum System The state of a Quantum System is a vector ψ ψ (pronounced ket ) in a Hilbert space H . Quantum Def. A Hilbert Space is a vector space H over together with an inner Mechanics product −− ,: HH × → such that += + += + 1) uuvuvuv12,,, 1 2 & vu ,,, 12 u vu 1 vu 2 2) uv,,λλ= uv Whirlwind Tour 3) uv,,= vu ∈ 4) ∀ Cauchy seq uu ,, … in , lim un H 12 H n→∞ 10 Dynamic Behavior of Q. Sys. The dynamic behavior of a quantum system is An observable Ω is a Hermitian operator determined by Schroedinger’s equation. on the state space H , i.e., a linear Dynamic Initial State transformation such that Schroedinger’s State Equation T ψ OUT † 0 ∂ ψ Ω=Ω=Ω IN = ψ ψ = ψ iH Ut() 0 H ∂t Hamiltonian Observable Observable where t is time, and where Ut () is a curve in the group UH () of unitary transformations on the state space H . Quantum Measurement Eigenvalue Observable MacroWorld λ Ω j Physical =ψψ Out Prob Pj In Reality Quantum Knots Philosopher P ψ & Turf ψ = j ψ j ψψP BlackBox j Quantum Knot Systems Q. Sys. Quantum Q. Sys. State World State Ω= λ P where ∑ j jj Spectral Decomposition The Hilbert SpaceK ()n of Quantum Knots An Example of a Quantum Knot Recall that K (n) is the set of all n-mosaic knots. For Q.M. systems, we need an underlying Hilbert space. So we define: + The Hilbert space K()n of quantum knots is the Hilbert space with the set K (n) of K = n--mosaicmosaic knots as its orthonormal basis, 2 i.e., with orthonormal basis { KKK: ∈ K()(n)n } 11 The Ambient Group An () as a Unitary Group An Example of the An () Group Action We identify each element g ∈ An () with the linear transformation defined by ()nn () + KK→ Kg K K = 2 Since each element g ∈ An () is a permutation, it is a linear transformation that simply R2 permutes basis elements. + Hence, under this identification, the ambient = group An()becomes a discrete group of R2 K unitary transfs on the Hilbert space K () n . 2 The Quantum Knot System ()K()n ,()An The Quantum Knot System ()K()n ,()An ()n ι ιι ι ()K ,()An + Def. A quantum knot system is a ()()()KKK(1),(1)AAnAn→→ (nn ) ,() → ( 1) ,( + 1) → quantum system having K ()n as its state space, and having the Ambient group An () as its set Physically Physically Physically of accessible unitary transformations. Implementable Implementable Implementable ()n The states of quantum system () K ,() An are Choosing an integer n is analogous to quantum knots. The elements of the ambient choosing a length of rope. The longer the group An () are quantum moves. rope, the more knots that can be tied. ι ιι ι (1) (nn ) (+ 1) ()()()KKK,(1)AAnAn→→ ,() → ,( + 1) → The parameters of the ambient group An () are the “knobs” one turns to spacially manipulate Physically Physically Physically the quantum knot. Implementable Implementable Implementable Quantum Knot Type Two Quantum Knots of the Same Knot Type Def. Two quantum knots K 1 and K 2 are of the same knot n--typetype, written + KK12∼ n , provided there is an element g ∈ An () s.t. K = = 2 g KK12 R2 They are of the same knot type, written KK∼ , 12 + R K = provided there is an integer m ≥ 0 such that 2 ι mmι K12∼ nm+ K 2 12 Two Quantum Knots NOT of the Same Knot Type Hamiltonians = K1 of the Generators of the Ambient Group = + K 2 2 Hamiltonians for An () Hamiltonians for An () 10 Each generator g ∈ An () is the product of σ = Also, let 0 , and note that disjoint transpositions, i.e., 01 π ()σσσ=+−i ()() ∈ gKKKK= ()()()αβ,, αβ KK αβ , ln101 2ss 1 , 11 22 2 = Choose a permutation η so that For simplicity, we choose the branch s 0 .
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