1. Fuzzy Topology, Quantization and Long-distance Gauge Fields SSss S.Mayburov Hep-th 1205.3019 Quantum space-time and novel geometries: Discrete (lattice) space-time – Fundamental Plank length l p ( Snyder, 1947; Markov,1948) Noncommutative geometry (Connes, 1991) Supergeometry (Rothstein,1986) Hypothesis: Some geometries can induce also novel formalism of quantization (S.M.,2008)

2. SSss of List of Quantization Formalisms: i) Schroedinger quantization: S state Ψ is vector (ray) in Hilbert space H ii) Operator algebra (C* algebra) : S observables constitute nonassociative operator algebra U ( Segal, 1948) iii) Symbols of operators (Berezin, 1970) … etc., …) Geometric Formalism - why not ? Possible applications : Quantum Gravity

3. S Sets, Topology and Geometry Example: Fundamental set of 1-dimensional Euclidian geometry is (continious) ordered set of elements X = { x l } ; x l - points X x i x j

4. S T - total set; S T = X P u P u = { u j } - partial ordered subset X x x i u1u1 D Dx Fu 1 ~ x k,, it can be also: x i ~ x j - incomparabilty relation pPPppPartial ordered set – P u B Beside u 1,x k are incomparable (equivalent) S T elements X = { x l } - ordered subset; i Example: let’s consider interval - u 1 - incomparable point

5. Partial ordering in classical mechanics Beside geometric relations between events there is also causal relations t x events - causal set : { e j } Their time ordering corresponds to : t 1 ≤ t 3 ≤ t 2 e3e3 e1e1 e2e2

6. Partial ordering in classical mechanics Beside geometric relations between events there is also causal relations t x events - causal set: { e j } Their time ordering corresponds to : t 1 ≤ t 3 ≤ t 2 if one event dynamically induces another : e 1 → e 2 If to describe also causal relations between events: e 1 ≤ e 2 ; e 1 ~ e 3 ; e 2 ~ e 3 So e 3 is incomparable in causal terms to e 1 and e 2 e3e3 e1e1 e2e2

7. Partial ordering in special relativity The causal relations acquire invariant, principal meaning due to light cone t x event set: { e j } Their time ordering is : e 1 ≤ e 3 ≤ e 2 Let’s describe also causal relations between events, then: e 1 ≤ e 2 ; e 1 ~ e 3 ; e 2 ~ e 3 if e 3 is outside the light cone, then other causal relations between e 3 and e 1,2 can’t exist e3e3 e1e1 e2e2

8. If f j - fuzzy points, then: if Ff i ~ f k Ff i, f k, w i,k >0, F ={f i } - set of fuzzy points; Σ w jk =1 Fuzzy ordered set (Foset) - F ={ f i } (Zeeman, 1964)...... _ x i f1f1 X EExample: S T - total set S T = X F; F={ f 1 } X – ordered subset w 1i Ff i ~ x k, w i,k >0, Σ w jk =1

9. Continuous fuzzy sets EExample: S T - total set ; S T = R 1 F; F={ f 1 } – discrete subset of fuzzy points X – continuous ordered subset; X = { x a } w f i, x ; w(x) > 0; x is fuzzy parameter S T is fuzzy space w(x) dispersion characterizes f coordinate uncertainty

10. x Classical mechanics: In 1 dimension the particle m is ordered point x(t) its state: | m } = { x(t), v x (t), … } describes m trajectory Massive particle is fuzzy point m(t), its coordinate is fuzzy parameter its state: |g} = {w(x,t), …. ? } x(t 0 ) x(t) Fuzzy mechanics: w(x,t 0 )w(x,t )

11. X Fuzzy evolution in 1-dimension Fuzzy state | g } : | g}= {w(x), q 1, q 2 … } Beside w(x,t), another |g} parameter is - average m velocity we shall look for | g} free parameters assuming that they can be represented as real functions q i (x,t) w(x,t 0 )

12. hhh Fuzzy Evolution Equations Let’s consider free w(x,t) evolution : Suppose that w(x,t) evolution is local and so can be described as: where f - arbitrary function then from w(x) norm conservation:

13. hhh Fuzzy Evolution Equations Let’s consider free w(x,t) evolution : where f - arbitrary function then from w(x,t) norm conservation: if to substitute : then : If this condition is fulfilled, then : - 1-dimensional flow continuity equation Suppose that w(x,t) evolution is local and so can be described as:

14. x 1 X w (x,t) v(x 1 ) in 1-dimension m flow can be expressed as: where v(x) is w flow velocity, it is supposed to be independent of w(x) m state g(x) = { w(x), v(x)} – it’s observational g state representation

15. ; k is theory constant c is arbitrary real number so m state is: g = { w(x), α(x)}, then g ~ g(x) = g 1 (x) + ig 2 (x) we suppose that g(x) is normalized, so that g(x)g * (x) = w(x), yet g(x) is still arbitrary complex function: for m state g = { w(x), v(x)}, we substitute v(x) by related parameter: where is arbitrary real function for any the set of g(x) constitutes complex Hilbert space H

16. We suppose that observables are linear, self-adjoint operators on H Correct g(x) ansatz should give all values to be independent of c value To derive it it ‘s enough to consider flow (current) operator Then : It follows that the only g(x) ansatz which give value to be independent of c value is written as : This is standard quantum-mechanical ansatz

17. We start with linear operator for fuzzy dynamics, later the nonlinear operators will be considered also For free m motion H should be invariant It means that H can be only polinom : j can be only even numbers, j = 2,4,6,… because noneven j contradict to X -reflection invariance Evolution Equation for Fuzzy state g(x) relative to space shifts so :

18. Equation for operator ffr From flow equation: it follows: hence: and

19. ffr From flow equation : it follows: hence: and and: From comparison of highest α derivatives it follows n=2; fuzzy state g(x,t) obeys to free Schroedinger equation with mass m = k ! Equation for operator

20. Nonlinear evolution operator Bargman - Wigner theorem (1964) : g(t 0 ) → g’(t) If arbitrary pure state evolves only into the pure state and module of scalar product for all pure states | | is conserved, then the evolution operator is linear. Jordan (2006): If for all g(t 0 ) : g(t 0 ) → g’(t) i.e. if all pure states evolve only to pure ones, and is defined unambiguously, then is linear operator Yet fuzzy pure states g(x,t) evolve only to pure ones, so their evolution should be linear g(x,t) are normalized complex functions, we obtained free particle’s quantum evolution on Hilbert space H Origin of the results : w(x,t) flow continuity is incompatible with the presence of high x - derivatives in g evolution equation

21. II We didn’t suppose that our theory possesses Galilean invariance, in fact, Galilean transformations can be derived from fuzzy dynamics, if to suppose that any RF corresponds to physical object with

22. Dirac equation : Its fuzzy space: F = { } is double (entangled) fuzzy structure Starting from it and demanding that w(x,t)>0 It results in the linear equation for 4-spinor which is Assuming the formalism extension to relativistic invarince, Let’s assume that the formalism can be extended to relativistic invariance In this case the flow continuity equation is fulfilled:

23. Fuzzy space: F = { } generates Hilbert space H of complex functions g(x,t) which describe particle’s evolution in F Observables are linear, self-adjoint operators on H particle’s momentum: commutation relations follow from geometrical and topological properties, so in this approach fuzzy topology induces such operator algebra What about and ? Modified fuzzy space F supposedly can be underlying structure for them

24. Fuzzy States and Secondary Quantization e Example: Fock free quantum field { n k } in 1-dimension - particle numbers can be regarded as fuzzy values each particle possesses fuzzy coordinate - double fuzzy structure

25. Free Schroedinger equation: is equivalent to system of two equations for real and imaginary parts of Interactions in Fuzzy Space - flow continuity equation I ) II ) here I) is kinematical equation, it describes w(x,t ) balance only

26. Interactions in Fuzzy Space I ) II ) here I) is kinematical equation, it describes w(x,t ) balance only Hence particle’s interactions can be included only in II) But it supposes their gauge invariance, because is quantum phase + H int Free Schroedinger equation: is equivalent to system of two equations for real and imaginary parts of - flow continuity equation

27. Particle’s Interactions in Fuzzy Space - Toy model I ) II ) Let’s consider interaction of two particles m 1, m 2, their states: g 1,2 (x,t) + H int g 1 g 2 In geometrical framework it assumed that m 1 react just on m 2 presence i. e. if, then It means that there is term in H int = Q 1 Q 2 F(r 12 )=, where Q 1,Q 2 are particle charges. x

28. ttt t H (t) H (t+Δt ) H (t+2Δt ) α Hence this is shift of α (x,t) subspace of E t = H - fiber bundle So that each H (t i ) is flat but their connection C(t i, t k ) depends on H int Hence:

29. (Non)locality of interactions on fuzzy space We supposed that in our model the interactions are local, yet such theory is generically nonlocal, two particles with fuzzy coordinates can interact nonlocally; Fermions with fuzzy coordinates can nonlocally interact with bosonic field.

30. Conclusions 1.1. Fuzzy topology is simple and natural formalism for introduction of quantization into any physical theory 2. Shroedinger equation is obtained from simple assumptions, Galilean invariance follows from it. 3. Local gauge invariance of fields corresponds to dynamics on fuzzy manifold

31. Fuzzy Topology, Quantization and Gauge Fields S. Mayburov hep-th 1205.3019 To explore quantum space-time it can be important to describe Quantum physics and space-time within the same mathematical formalism Proposed approaches: i)To extend quantum operator algebras on space-time description: Noncommutative Geometry (Connes, 1988), etc.; Applications to Quantum Gravity (Madore, 2005) ii) To extend some novel geometry on quantum physics: Possible approach: Geometry equipped with fuzzy topology (S.M., 2007)

32. Fuzzy Correlations We exploit fuzzy state : Yet more correct expression is : here: Correlation of g(x) state between its components x, x’ appears due to particle fuzziness on X axe, it defines g(x) free evolution: Such ansatz corresponds to quantum density matrix More complicated correlations can appear in fuzzy mechanics.

33. Particle’s Interactions on fuzzy manifold let’s suppose that two identical particles m 1, m 2 repulse each other Deflections of m 1 motion induced by m 2 presence at some distance, even if ; yet for and this is deformation of m 1 fiber bundle H + H int Hence:

34. Particle’s Interactions in Fuzzy Space So this toy-model is equivalent to Electrodynamics if A μ is massless field suppose that m 1 is described by Klein-Gordon equation with E>0 and m 2 is classical particle, i.e. m 2 If in initial RF: then: For Lorenz transform RF to RF’ with velocity only such vector field makes transformation consistent where : and

35. D Fuzzy state for fermion (iso)doublet Consider fermion (iso)spin doublet of mass m 0 Fuzzy parameters: ; particle’s state : Fuzzy representation: g = - (iso)spin vector - linear representation z from Jordan theorem ψ s evolution should be linear

36. D Nonabelian case: Yang-Mills Fields Consider interaction of two femions m 1, m 2 of doublet and suppose that their interaction is universal, so that at it gives - isospin vectors To agree with global isotopic symmetry, two isospin vectors in nonrelativistic limit can interact with each other only via this ansatz: Hence: - Yang-Mills s1 s1 s2 s2 For nonrelativistic QCD limit it gives: But it supposes that local gauge invariance follows from global isotopic symmetry : and so :

37. Yang-Mills Interactions So we obtain standard Yang-Mills Hamiltonian for particle-field interaction Suppose that m 1 is described by Klein-Gordon equation with E>0 and m 2 is classical particle, i.e. m 2 ; and so m 2 generates classical Yang-Mills field in initial RF: For Lorenz transform RF to RF’ with velocity only such vector field makes transformation consistent where : and

38. We derived free particle’s dynamics from some geometrical and set-theoretical axioms. Is it possible to describe particle’s interactions in the same framework ? We exploited m states: g = { w(x), α(x)} defined in flat Hilbert space H Analogously to general relativity, in geometry euipped with fuzzy topology the interactions induce the deformation of H and H fiber bundle

39. Fuzzy Mechanics and Gauge Invariance In QFT local gauge invariance is postulated. We shall argue that in Fuzzy mechanics it can be derived from global isotopic invariance

40. Such formalism need only three axioms, whereas standard QM is based on seven axioms i)Particles are points of fuzzy manifold, their states are characterized by two parameters |g } = { w(x,t), v(x,t) } ii) Observables Q are supposed to be linear, self-adjoint operators on H iii) Reduction (Projection) postulate for result of Q measurement In this model Plank constant is just coefficient, which connects x and p x scales, so Relativistic unit system gives its natural description

41. Content 1. Geometric properties induced by structure of fundamental set 2. Partial ordering of points, fuzzy ordering, fuzzy sets 3. Fuzzy manifolds and physical states 4. Derivation of Shroedinger dynamics for fuzzy states 5. Interactions on fuzzy manifold and gauge fields For simplicity only 1+1 geometry will be considered

42. Particle’s Interactions in Fuzzy Space - Toy model I ) II ) Let’s consider interaction of two particles m 1, m 2, their states: g 1,2 (x,t) + H int g 1 g 2 In geometrical framework it’s natural to assume that H int is universal, i. e. if, then It means that there is term in H int ~ Q 1 Q 2 F(r 12 ), where Q 1,Q 2 are particle’s charges. x

43. Relativistic Evolution Equation we suppose that fuzzy state: and: || w || =1 If we look for 1 st order evolution equation: where: If twIII then for the unique solution is: n=2, i.e. Schroedinger evolution

44. Δ 1 X Fuzzy evolution and linearity We don’t assume that w(x,t ) is linear function of w(x’,t 0 ) which is standard assumption of classical kinetics Example : Boltzman diffusion w(x,t 0 ) w(x,t )

45. Clocks, proper times and partial ordering In special relativity each dynamical system has it is its own proper time, there is no universal time, so proper time connected with clock dynamics of given reference frame (RF) t a t b Constraint: each event time t 1 in one RF is mapped to the point on time axe in other RF t’ 1 e 1 ~ e 3

46. It can be transformed to symmetric complex g representation in H It can be proved that only such g(x) representation corresponds to consistent dynamics g(x) → { w(x), v(x)} is unambiguous map α(x) is analogue of quantum phase

47. m particle’s state |g} = { w(x), α (x) } The problem : to find dynamical |g} representation g(x,t) : is dynamical operator (Hamiltonian) We have two degrees of freedom: g = { w(x), α(x) }, it can be transferred to: g(x) = g 1 (x)+ ig 2 (x) such that g 1 (x), g 2 (x) = 0, if w(x) = 0 This is symmetric g representation example:

48. Generalization of event time mapping one can substitute points by δ-functions t a ~ e 3

49. FuzzFuzzy dynamics – space symmetry restoration as the possible fundamental low of dynamics X w0 w0 w(t) Low of fuzzy evolution : space symmetry of m state |g} is restored as fast as possible : So free m evolution U(t) : w 0 (x) const at very large t Interactions: to restore space symmetry two identical particles e 1, e 2 should repulse each other E Global symmetries are important in quantum physics Free particle m: w 0 (x): space shift symmetry is broken w0 w0

50. m motion can be described as the space symmetry violation for |g} w(x) α (x) X m velocity: = X d α(x) is analog of quantum phase fffffff

51. Δ 1 Δ 2 X v(x 1 ) v(x 2 ) henc hence w flow equation can be written as:

53. hello ggggggkggghggghfh FuzzFuzzy mechanics – space symmetry restoration as the general low of free motion