Non-Associative Quantum Mechanics from Non-Geometric Strings. Peter Schupp

Transcription

1 Non-Associative Quantum Mechanics from Non-Geometric Strings Peter Schupp Jacobs University Bremen Workshop on Non-Associativity in Physics and Related Mathematical Structures, IGS, PennState May 1-3, 2014 with Dyonysis Mylonas and Richard Szabo inspired by Blumenhagen, Deser, Lüst, Plauschinn,...

2 Motivation Planck scale quantum geometry Heuristic argument: quantum + gravity The gravitational field generated by the concentration of energy required to localize an event in spacetime should not be so strong as to hide the event itself to a distant observer. fundamental length scale, spacetime coarse graining x G c cm need to generalize usual notion of geometry microscopic non-commutative/non-associative spacetime structures

3 Motivation Massless bosonic modes open strings: A µ, φ i gauge and scalar fields closed strings: g µν, B µν, Φ background geometry, gravity Closed string effective action Weyl invariance (at 1 loop) requires vanishing beta functions: d D x g 1 2 β µν (g) = β µν (B) = β(φ) = 0 equations of motion for g µν, B µν, Φ closed string effective action ( R 1 12 e Φ/3 H µνλ H µνλ 1 ) 6 µφ µ Φ +... equivalent noncommutative/nonassociative version of this?

4 Non-geometric Backgrounds Non-geometric flux backgrounds T-dualizing a 3-torus with 3-form H-flux gives rise to geometric and T non-geometric fluxes H a abc f a bc Q ab T c c R abc Hull (2004), Shelton, Taylor, Wecht (2005) T b Q-flux: T-duality transitions between local trivializations T-folds R-flux: metric and B-field not even locally defined; non-geometric strings non-commutative non-associative structures Lüst (2010), Blumenhagen, Plauschinn (2010) Blumenhagen, Deser, Lüst, Plauschinn, Rennecke (2011) Mylonas, PS, Szabo (2012)

5 Non-geometric Backgrounds Geometrized non-geometry I: membrane sigma model Courant sigma model: C = TM T M with natural frame (ϱ i, χ i ), metric ϱ i, χ j = δ i j TFT with 3-dimensional membrane world volume Σ 3 S (2) AKSZ = Σ 3 ( φ i dx i h IJ α I dα J P I i (X ) φ i α I T IJK (X ) α I α J α K ) with embeddings X : Σ 3 M, 1-form α, aux. 2-form φ, fibre metric h, anchor matrix P, 3-form T (e.g. H-flux, f -flux, Q-flux, R-flux). AKSZ construction: action functionals in BV formalism of sigma model QFT s for symplectic Lie n-algebroids E Alexandrov, Kontsevich, Schwarz, Zaboronsky (1995/97)

6 Non-geometric Backgrounds R-space sigma-model action S (2) R Σ = ξ i dx i Σ 3 6 Rijk (X ) ξ i ξ j ξ k + Σ 2 Σ g ij (X ) ξ i ξ j. for constant backgrounds, using Stokes leads to boundary action: S (2) R = 1 2 Θ 1 IJ (X ) dx I dx J 1 + Σ 2 2 g IJ dx I dx J, with Θ 1 = ( ( ) Θ 1 ) 0 j δi IJ = δ i j R ijk p k, ( ) ( ) 0 0 gij = 0 g ij and X = (X I ) = (X 1,..., X 2d ) := (X 1,..., X d, P 1,..., P d ). effective target space = phase space

7 Non-geometric Backgrounds Linearized action Generalized Poisson sigma-model = S (2) R Σ 2 (η I dx I ΘIJ (X ) η I η J ) + Σ G IJ η I η J, with auxiliary fields η I, Θ = ( Θ IJ) ( ) R = ijk p k δ i j j δ i 0, ( G IJ ) ( ) g ij 0 = 0 0 and X = (X I ) = (X 1,..., X 2d ) := (X 1,..., X d, P 1,..., P d ).

8 Non-geometric Backgrounds Geometrized non-geometry II: Non-associative phase space Θ is an H-twisted Poisson bi-vector: [Θ, Θ] S = 3 Θ (H), where H = 1 6 Rijk dp i dp j dp k = db, and B = 1 6 Rijk p k dp i dp j. Twisted Poisson brackets {x i, x j } Θ = R ijk p k, {x i, p j } Θ = δ i j and {p i, p j } Θ = 0. Corresponding Jacobiator: {x i, x j, x k } Θ = R ijk, after quantization: nonassociative star product

9 Deformation quantization Kontsevich formality and star product U n maps n k i -multivector fields to a (2 2n + k i )-differential operator U n (X 1,..., X n ) = Γ G n w Γ D Γ (X 1,..., X n ), where the sum is over all possible diagrams with weight 1 w Γ = (2π) k i n H n i=1 ( dφ h e 1 i dφ h e k i i The star product for a given bivector θ is: ). f g = n=0 ( i ) n U n (Θ,..., Θ)(f, g) n!

10 Deformation quantization Deformation quantization For our Example Θ: constant θ: The graphs Theand graphs hence and hence the integrals the integrals factorize. The Thegraph basic graph θ 1 ψ 1 p 1 yields theyields weight the weight w Γ1 = 1 2π ψ1 (2π) 2 dψ 1 dφ 1 = 1 [ 1 2π 0 0 (2π) 2 2 (ψ 1) 2 w Γ1 = 1 2π ψ1 0 2 (2π) 2 dψ 1 dφ 1 = 1 [ ] 2π (2π) 2 2 (ψ 1) 2 = and the star product turns out to be the Moyal-Weyl one: and the star product f g = turns out (i ) n ( ) to n be: 1 θ µ1ν1... θ µnνn ( µ1 n! 2... f )(... g) µn ν1 νn f g = (i ) n ( ) n 1 Θ µ1ν1... Θ µnνn ( µ1 n! 2... f )(... g) µn ν1 νn

11 Deformation quantization Formality condition The U n define a quasi-isomorphisms of L -DGL algebras and satisfy d U n (X 1,..., X n ) = i<j I J =(1,...,n) I,J = ε X (I, J ) [ U I (X I ), U J (X J ) ] G ( 1) α ij U n 1 ( [Xi, X j ] S, X 1,..., X i,..., X j,..., X n ), relating Schouten brackets to Gerstenhaber brackets. Kontsevich (1997) This implies in particular d Φ(Θ) = i Φ(d Θ Θ), i.e. (non-)associative θ (non-)poisson

12 Nambu-Poisson 3-brackets Remarks on Nambu-Poisson structures The trivector Π = 1 6 Rijk i j k is an example of a Nambu-Poisson tensor. Nambu mechanics: multi-hamiltonian dynamics with generalized Poisson brackets; e.g. Euler s equations for the spinning top : d dt L i = {L i, L 2 2, T } with {f, g, h} ɛijk i f j g k h more generally: {f, h 1,..., h p } = Π i j1...jp (x) i f j1 h 1 jp h p {{f 0,, f p }, h 1,, h p } = {{f 0, h 1,, h p }, f 1,, f p } {f 0,..., f p 1, {f p, h 1,, h p }} Our construction may be useful to quantize these objects.

13 Nambu-Poisson 3-brackets The task of formulating a nonassociative version of quantum mechanics is closely related to the quest of quantizing Nambu-Poisson brackets. Nambu-Heisenberg (NH) bracket introduced by Nambu as half of a Jacobiator [A, B, C] NH = ABC + CAB + BCA BAC ACB CBA in the nonassociative case we choose the convention [A, B, C] NH = [A, B]C + [C, A]B + [B, C]A for a triple of coordinate functions [x i, x j, x k ] -NH = i l ( R ijl p l x k + R jkl p l x i + R kil p l x j)

14 Nambu-Poisson 3-brackets Nambu also suggested to consider nonassociative algebras 3-bracket for nonassocitative algebra Jacobiator [[A, B, C]] [[A, B]C] + [[C, A]B] + [[B, C]A] for a triple of coordinate functions we find [[x i, x j, x k ]] = i l ( R ijl [p l, x k ] + R jkl [p l, x i ] + R kil [p l, x j ] ) = 3 2 R ijk a candidate for a quantized Nambu-Poisson bracket

15 Magnetic sources and 3-cocycles Non-associativity in electrodynamics with magnetic sources A charged particle (charge e, mass m) experiences a magnetic field B (with sources) only via the Lorentz force p = e m p B. The Hamiltonian H = 1 2m p2 is purely kinetic, but p = i[h, p] must imply the Lorentz force momenta cannot commute [r i, r j ] = 0, [r i, p j ] = iδ i j [p i, p j ] = ieɛ ijk B k translations are generated by U( a) = exp(i a p) U( a 1 )U( a 2 ) = e ieφ12 U( a 1 + a 2 ), Φ 12 = flux through triangle ( a 1, a 2 ) (non)associativity: [U( a 1 )U( a 2 )]U( a 3 ) = e ieφ123 U( a 1 )[U( a 2 )U( a 3 )], where Φ 123 is the flux through the tetrahedron ( a 1, a 2, a 3 ).

16 Magnetic sources and 3-cocycles infinitesimally: [p 1, [p 2, p 3 ]] + [p 2, [p 3, p 1 ]] + [p 3, [p 1, p 2 ]] = e B p can be realized as a linear operator i e A only in the associative regime, i.e.: for B = 0: no sources, no flux, associativity; for B 0: non-associativity, unless Φ 123 /(2π) is an integer. 1 fluxes are quantized point sources (magnetic monopoles) so that the fluxes do not change continuously as the a vary; Dirac quantization Jackiw (1985) homogeneous magnetic charge density, nonassociative (Malcev) algebra Günaydin, Zumino (1985) Günaydin, Minic (2013) appropriate nonassociative version of QM needed 1 Taking the into account that the electron is spin 1/2 fermion with double-valued wave function, this becomes Φ 123 /π Z.

17 Nonassociative quantum mechanics Phase space formulation of quantum mechanics Hip Groenewold, Joe Moyal, Hermann Weyl, Eugene Wigner,... treats positions and momenta on equal footing Wigner (real phase space quasi-probability distribution) function f W (x, p) = 1 dy x + y/2 ψ ψ x y/2 e iyp 2π Â = dxdp f W (x, p)a(x, p) observables: real functions on phase space operator product star product enticing opportunity to study nonassociative QM in situ Problems: positivity, lots of parentheses,...

18 Nonassociative quantum mechanics f g = f g + total derivative 2-cyclicity d 2d x f g = d 2d x g f = d 2d x f g f (g h) = (f g) h + total derivative 3-cyclicity d 2d x f (g h) = d 2d x (f g) h inequivalent quartic expressions f 1 ( f 2 (f 3 f 4 ) ) ((f1 ) = (f 1 f 2 ) (f 3 f 4 ) = f 2 ) f 3 f4 f 1 ( ) (f1 (f 2 f 3 ) f 4 = (f 2 f 3 ) ) f 4

19 Nonassociative quantum mechanics Notation and convention compositions (A B) C := A (B C), C (A B) := (C A) B (A 1 A 2... A n ) C = A 1 (A 2... (A n C)...)) A 1 = A = 1 A A B is typically not a function; some notable exceptions: x i x i = x i x i = (x i ) 2 p i p i = p i p i = (p i ) 2 is evaluated before (A B) (C D) := [(A B) C] D = [A (B C)] D

20 Nonassociative quantum mechanics A state ρ is an expression of the form n ρ = λ α ψ α ψα with α=1 ψ α 2 = 1 λ α are probabilities and ψ α are phase space wave functions: Expectation value of an operator : n A = A ρ = λ α A (ψ α ψα) = λ α (A ψ α ) ψα α=1 α = λ α ψα (A ψ α ) = λ α ψα (A ψ α ). α α Normalization 1 = α λ α ψ α 2 = 1,

21 Nonassociative quantum mechanics Expectation values of observables (= real functions) are real A = λ α (A ψ α ) ψ α = λ α ψα (A ψ α ) = A α α Expectation value of compositions A B... C = (A B... C) ( λ α ψ α ψα) α = λ α [A (B... (C ψ α )] ψα α

22 Nonassociative quantum mechanics Positivity A A = α λ α ψ α [A (A ψ α )] = α λ α (ψ α A ) (A ψ α ) = α λ α (A ψ α ) (A ψ α ) = α λ α A ψ α 2 0 semi-definite, sesquilinear form (A, B) := A B = α λ α (A ψ α ) (B ψ α ) Cauchy-Schwarz inequality (A, B) 2 (A, A)(B, B).

23 Nonassociative quantum mechanics State function The expectation value A = λ α (A ψ α ) ψα = α α λ α A (ψ α ψα) = A S ρ can be expressed in terms of a real-valued normalized state function S ρ = λ α ψ α ψα, S ρ = λ α ψ α 2 = 1. α α

24 Nonassociative quantum mechanics Eigenfunctions and eigenstates star-genvalue equation A f = λf with λ C complex conjugation implies f A = λ f real functions have real eigenvalues f (A f ) (f A) f = (λ λ )(f f ) (λ λ ) f f = (λ λ ) f 2 = 0. eigenfunctions with different eigenvalues are orthogonal eigen-state functions eigen-wave functions

25 Nonassociative quantum mechanics Noncommutative nonassociative phase space coordinates with commutator and associator X I X J X J X I = i Θ IJ (p) (X I X J ) X K X I (X J X K ) = 2 2 RIJK where X I {x 1,..., x d, p 1,..., p d } and R IJK K Θ IJ is constant Commutator and common eigen state functions if X I S = λ I S and X J S = λ J S then [X I, X J ] = [X I (X J S) X J (X I S) ] = [λ I, λ J ] = 0 contradiction if Θ IJ (p) 0

26 Nonassociative quantum mechanics Associator and common eigen state functions if X I S = λ I S and X J S = λ J S and X K S = λ K S then [(X I X J ) X K ] S = (X I X J ) (X K S) = λ K (X I X J ) S = λ K X I (X J S) = λ K λ J λ I likewise [X I (X J X K )] S = (X J X K ) (S X I ) = λ I λ K λ J. taking the difference implies 2 2 RIJK = λ K λ J λ I λ I λ K λ J = 0 Nonassociating observables do not have common eigen states spacetime coarse graining

27 Nonassociative quantum mechanics Uncertainty relations uncertainty in terms of shifted coordinates X I = X I X I ( X I ) 2 = (X I ) 2 X I 2 = X I X I = X I X I = ( X I, X I ) we can proof ( X I ) 2 ( X J ) 2 ( X I, X J ) 2 = 1 4 [X I, X J ] { X I, X J } 2 ignoring the last term yields a Born-Jordan-Heisenberg-type uncertainty relation X I X J 1 [X I, X J ] 2

28 Nonassociative quantum mechanics Position-momentum uncertainty [p i, p j ] = [p i, p j ] = 0 and [p i, x j ] = [p i, x j ] = i δ j i and therefore p i p j 0 and x i p j 2 δi j Position-position uncertainty [x i, x j ] ψ x i (x j ψ) x j (x i ψ) = [x i, x j ] ψ 2 R ijk k ψ and therefore = i R ijk( p k ψ i 2 kψ + i k ψ ) = i R ijk ψ p k x i x j R ijk p k, 2 featuring the opposite (!) state ρ = n α=1 λ α ψ α ψ α

29 Nonassociative quantum mechanics Area and volume operators A IJ = Im ([ X I, X ) ( J ] = i X I X J X J X ) I V IJK = Re ([ X I, X J, X ) K ] -NH = 1 2 [[ X I, X J, X K ]]. expectation values of these (oriented) area and volume operators: A IJ = Θ IJ ( p ) and V IJK = R IJK with three interesting special cases A (x i,p j ) = δ i j, A ij = R ijk p k, V ijk = R ijk quantized spacetime with cells of minimal volume R ijk

30 Jordan Algebra Noncommutative Jordan Algebras (1) x(yx) = (xy)x flexible (2) x 2 (yx) = (x 2 y)x properties (1) and (2) imply (3) x m (yx n ) = (x m y)x n power associative and are necessary and sufficient conditions for x y := 1 (xy + yx) 2 to be (commutative) Jordan. P. Jordan (1933), A.A. Albert (1946), R.D. Schafer (1955)

31 Jordan Algebra Noncommutative Jordan Algebras (1) x(yx) = (xy)x flexible (2) x 2 (yx) = (x 2 y)x Question: Are we dealing with a Jordan algebra? x I (x K x I ) = (x I x K ) x I but: (x I ) 2 (x K x I ) = ((x I ) 2 x K ) x I x 2 ( x 2 x 2 ) ( x 2 x 2 ) x 2 = 2iR 2 p x 0 for R ijk Rɛ ijk. Answer: no Alexander Held, PS

32 Günaydin-Zumino Model Exchange x and p, replace R ijk by H ijk... [x i, p j ] = iδ i j [x i, x j ] = 0 [p i, p j ] = ih ijk x k algebra of coordinates and physical (gauge invariant) momenta in a constant homogeneous magnetic charge density background coarse graining in momentum space three copies of p 2 do not associate: p 2 ( p 2 p 2 ) ( p 2 p 2 ) p 2 = 2ieρ 2 magnetic x p 0 cannot diagonalize? no free stationary states?? eigenfunctions: just need to make sure that x p = 0, in fact: p 2 i ψ = λ i ψ ψ(x, p) exp(2ix i (p i λ i )), λ i R

33 Magnetic monopoles in the lab spin ice pyrochlore and Dirac monopoles Castelnovo, Moessner, Sondhi (2008) Fennell; Morris; Hall,... (2009) frustrated spin system huge degeneracy of classical ground state frustration is lifted but pyrochlore spin ice property survives quantization Lieb, PS (1999)

34 Conclusion non-geometric flux quantization leads to nonassociative structures nonassociative quantum mechanics can be formulated in a modified phase-space approach nonassociative versions of uncertainty relations spacetime coarse graining D. Mylonas, PS, RJ. Szabo, Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds, JHEP 1209 (2012) 012 D. Mylonas, PS, RJ. Szabo, Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics, arxiv: D. Mylonas, PS, RJ. Szabo, Nonassociative geometry and twist deformations in non-geometric string theory, arxiv: