Fuzzy Topology, Quantization and Long-distance Gauge Fields

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 lp ( 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 Ψis 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 elementsX= { xl } ; xl - points xi xj X

4. pPPppPartial ordered set – P u B Beside , it can be also: xi ~ xj - incomparabilty relation u1 - incomparable point D Dx u1 X xi x ST - total set; ST = X P u P u = { uj } - partial ordered subset X = { x l } - ordered subset; Example: let’s consider interval - i Fu1 ~ xk , u1 ,xk are incomparable (equivalent) ST elements

5. Partial ordering in classical mechanics Beside geometric relations between events there is also causal relations t e2 events - causal set: { e j } Their time ordering corresponds to : t1 ≤ t3 ≤ t2 e3 e1 x

6. Partial ordering in classical mechanics Beside geometric relations between events there is also causal relations t e2 events - causal set: { e j } Their time ordering corresponds to : t1 ≤ t3 ≤ t2 if one event dynamically induces another : e1 → e2 If to describe also causal relations between events: e1 ≤ e2 ; e1 ~ e3 ; e2 ~ e3 So e3is incomparable in causal terms to e1 and e2 e3 e1 x

7. Partial ordering in special relativity The causal relations acquire invariant, principal meaning due to light cone t e2 event set: { e j } Their time ordering is : e1 ≤ e3 ≤ e2 Let’s describe also causal relations between events, then: e1 ≤ e2 ; e1 ~ e3 ; e2 ~ e3 ife3 is outside the light cone, then other causal relations between e3 and e1,2 can’t exist e3 e1 x

8. Fuzzy ordered set (Foset) - F ={ fi } (Zeeman, 1964) Iff j - fuzzy points,then:if Ffi~ fk Ffi , fk , wi,k >0 , F ={f i} - set of fuzzy points; Σ wjk =1 EExample:ST - total set ST = X F; F={ f1 } X – ordered subset w1i f1 . . . . . . _xi X Ffi~ xk , wi,k >0 , Σ wjk =1

9. Continuous fuzzy sets EExample:ST - total set; ST = R1 F; F={ f1 } – discrete subset of fuzzy points X – continuousordered subset; X = { xa } fi , x ; w(x) > 0; w is fuzzy parameter ST is fuzzy space x w(x)dispersioncharacterizesf coordinate uncertainty

10. Classical mechanics: In 1 dimension the particle m is ordered point x(t) its state: | m } ={ x(t), vx(t), … } describesmtrajectory x(t) x(t0) Fuzzy mechanics: w(x,t0 ) w(x,t ) x Massive particle is fuzzy point m(t), its coordinate is fuzzy parameter its state:|g} = {w(x,t), …. ? }

11. Fuzzy evolution in 1-dimension Fuzzy state | g } : | g}= {w(x), q1 , q2 … } w(x,t0 ) X 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 functionsqi(x,t)

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

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

14. v(x1) w(x,t) X x1 in 1-dimensionmflow can be expressed as: where v(x) is w flow velocity, it is supposed to be independent of w(x) mstate g(x) = { w(x), v(x)} – it’s observational g state representation

15. formstateg = { w(x), v(x)},we substitutev(x)by related parameter: ; k is theory constant c is arbitrary real number so m state is:g = { w(x),α(x)}, then g ~ g(x) = g1 (x) + ig2(x) we suppose that g(x) is normalized,so that g(x)g* (x) = w(x), yet g(x) is still arbitrary complex function: 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 Correctg(x) ansatz shouldgive allvalues to be independent of cvalue 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. Evolution Equation for Fuzzy state g(x) We start with linear operator for fuzzy dynamics, later the nonlinear operators will be considered also For free m motion H should be invariant relative to space shifts so: It means that H can be only polinom: jcan be only even numbers, j = 2,4,6,… because noneven j contradict to X -reflection invariance

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

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

20. Nonlinear evolution operator Bargman - Wignertheorem(1964) : g(t0 ) → g’(t) If arbitrary pure state evolves only into the pure state and module of scalar product for all pure states |< ψ|φ >| is conserved, then the evolution operator is linear. Jordan (2006): If for all g(t0 ) : g(t0 ) →g’(t) i.e. ifallpure 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 spaceH Origin of the results : w(x,t) flow continuity is incompatible with the presence of high x - derivatives in g evolution equation

21. 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 II

22. 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: Starting from it and demanding that w(x,t)>0 It results in the linear equation for 4-spinor which is Dirac equation : Its fuzzy space: F ={ } isdouble (entangled) fuzzy structure

23. Fuzzy space: F={ } generatesHilbert spaceH of complex functionsg(x,t) which describe particle’s evolution inF Observables are linear, self-adjoint operators on H particle’s momentum: commutation relations followfrom 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 Example: Fock free quantum field { nk} in 1-dimension - particle numbers can be regarded as fuzzy values each particle possesses fuzzy coordinate - double fuzzy structure e

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

26. Interactions in Fuzzy Space Free Schroedinger equation: is equivalent to system of two equations for real and imaginary parts of I ) - flow continuity equation + Hint 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, becauseis quantum phase

27. Particle’s Interactions in Fuzzy Space - Toy model I ) II ) + Hint Let’s consider interaction of two particles m1 , m2 , their states: g1,2 (x,t) g2 g1 x In geometrical framework it assumed that m1 react just on m2 presence i. e. if , then It means that there is term in Hint = Q1 Q2 F(r 12 )= , where Q1 ,Q2 are particle charges.

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

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: • 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 m1 , m2 repulse each other Deflections of m1 motioninduced by m2 presence at some distance, even if ; yet for and + Hint Hence: this is deformation of m1 fiber bundle H

34. Particle’s Interactions in Fuzzy Space suppose that m1 is described by Klein-Gordon equation with E>0 and m2 is classical particle, i.e. m2 If in initial RF: then: For Lorenz transform RF to RF’ with velocity where : and only such vector field makes transformation consistent So this toy-model is equivalent to Electrodynamics ifAμis massless field

35. Fuzzy state for fermion (iso)doublet Consider fermion (iso)spin doublet of mass m0 Fuzzy parameters: ; particle’s state : Fuzzy representation: g = D z - linear representation - (iso)spin vector from Jordan theorem ψsevolution should be linear

36. Nonabelian case:Yang-Mills Fields Consider interaction of two femions m1 , m2 of doublet and suppose that their interaction is universal, so that at it gives D - isospin vectors To agree with global isotopic symmetry, two isospin vectors in nonrelativistic limit can interact with each other only via this ansatz: Hence: s1 s2 and so : - Yang-Mills For nonrelativistic QCD limit it gives: But it supposes that local gauge invariance follows from global isotopicsymmetry :

37. Yang-Mills Interactions Suppose that m1 is described by Klein-Gordon equation with E>0 and m2 is classical particle, i.e. m2 ; and so m2 generates classical Yang-Mills field in initial RF: For Lorenz transform RF to RF’ with velocity where : and only such vector field makes transformation consistent So we obtain standard Yang-Mills Hamiltonian for particle-field interaction

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 spaceH Analogously to general relativity, in geometry euipped with fuzzy topology the interactions induce the deformation of Hand Hfiber 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 • 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 px 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 ) + Hint Let’s consider interaction of two particles m1 , m2 , their states: g1,2 (x,t) g2 g1 x In geometrical framework it’s natural to assume that Hintis universal, i. e. if , then It means that there is term in Hint ~ Q1 Q2 F(r 12 ), where Q1 ,Q2 are particle’s charges.

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

44. Fuzzy evolution and linearity We don’t assume that w(x,t ) is linear function of w(x’,t0 ) which is standard assumption of classical kinetics w(x,t0 ) Δ1 w(x,t) X Example : Boltzman diffusion

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) ta tb Constraint: each event time t1 in one RF is mapped to the point on time axe in other RF t’1 e1 ~ e3

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

47. mparticle’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) = g1 (x)+ ig2(x) such that g1 (x), g2 (x) = 0, if w(x) = 0 This is symmetric grepresentation example:

48. Generalization of event time mapping one can substitute points by δ-functions ta ~ e3

49. FuzzFuzzy dynamics – space symmetry restoration as the possible fundamental low of dynamics E Global symmetries are important in quantum physics w0 Free particle m: w0(x): space shift symmetry is broken w0 w(t) X Low of fuzzyevolution: space symmetry of m state |g} is restored as fast as possible : So free m evolution U(t) : w0(x) const at very large t Interactions:to restore space symmetry two identical particles e1 , e2 should repulse each other

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