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Donatello Dolce E− Γ the University of Melbourne, Australia Elementary Cycles of Time Donatello Dolce e− γ The University of Melbourne, Australia µ− µ− µ− 18/10/2012 γ Turin “We proceed with the assumption of existence of a Periodic phenomenon: certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated energy parcel [elementary particle]”. The interpretation of physics in terms of elementary cycles provides unified description of Relativistic and Quantum Mechanics Gravitational and gauge interaction String theory and Quantum Field Theory (AdS/CFT) Time Flow and Time Symmetry ... Based on: Classical to quantum correspondence in a VXD D.D.; Ann.Phys.,327, (2012); arXiv:1110.0316 Gauge interaction as periodicity modulation D.D.; Ann.Phys., 327, (2012); arXiv:1110.0315 Compact Time and determinism: foundations D.D.; Found.Phys., 41, (2011); arXiv:0903.3680 Towards a geometric description of Gauge Symm. Break. : Superconduct. and CM. D.D., Submitted to Ann.Phys. Elementary Time Cycles D.D., FQXi essay 2012. On the intrinsically cyclic nature of space-time in elementary particles D.D.,JPCS. 343 (2012); arXiv:1206.1140 Clockwork Quantum Universe D.D., IV Prize FQXi essay contest 2011; arXiv:1205.1788 De Broglie deterministic dice and emergent relativistic quantum mechanics D.D.,JPCS. 306 (2011); arXiv:1111.3319 Deterministic Quantization by Dynamical Boundary Conditions D.D., AIP Conf.Proc. 1246 (2010); arXiv:1006.5648 Quantum Mechanics for Periodic Dynamics: the bosonic case D.D., AIP Conf.Proc. 1232 (2010); arXiv:1001.2718 Periodic phenomenon: Compton periodicity: “We proceed with the assumption of existence of a certain periodic phenomenon of a yet to be determined character, which h T = is to be attributed to each and every isolated energy parcel τ ¯ 2 [elementary particle]”. deBroglie:1924 Mc topology S1 4-momentum p¯ = E/c,¯ ¯p e− µ { − } µ 4-periodicity T = T , λ /c { t x } Lorentz transformation rest frame cT γcT γβ λ generic frame p¯ τ → t − · x ¯ 2 µ e− Tτ Mc = h cp¯µT = h γ µ− HYPOTHESIS: Every elementary particle has γµ− intrinsic space-time periodicity µ Local− and retardedγ variations of 4-momenta in a generic relativistic interaction are equivalently described by local and retarded modulations of 4-periodicities. Relativistic clocks Every elementary particle has γ T ,hµν + intrinsic space-time periodicity ∞ Cs 133 10 T − 10− s Observed µ e− 20 HYPOTHESIS: cp¯µT = h Experimentally T 10− s Gouanere:2008 Z0 25 T 10− s Every elementary particle can be TeV 27 T 10− s imagined as a reference clock. e− ... t R Planck 44 ∈ T 10− s ∼ Einstein relativistic clock “by a clock we understand anything characterized by a phenomenon passing periodically through identical phases, [...] e− all that happens in a given period is γ identical with all that happens in an arbitrary period. Einstein:1910 t0 µ− Time unit [SI] µ− A “second” is the duration of 9,192,631,770 periods of the γ radiation corresponding to [...] µ− Cs 133 γ the Cs 133 atom. Time & Periodicity Intrinsic periodicity T γ ,h + µν ∞ Cs 133 10 T − 10− s Observed e− 20 Experimentally T 10− s Every particle is a reference clock, Gouanere:2008 Z0 25 T 10− s the local nature of relativistic time is enforced T TeV 10 27s − Relational and effective description of ... t R the arrow of time in terms of the “ticks” (ritarded and local modulations of periodicity)Planck 44 e− T 10− s ∈ of the de Broglie internal clocks Interaction ∼ Relativistic causality and simultaneity, time ordering e− The problem of time symmetry. γ t1 The inversion of helicity in a single clock does not mean to invert all the other clocks. This corresponds to describe an anti-particle µ− µ− Time unit [SI] γ A second is the duration of µ− 9,192,631,770 periods of γ Cs 133 the radiation corresponding to [...] the Cs 133 atom. µ Vibrating cyclic-string p¯ = E/c,¯ ¯p T = Tt, λx/c µ { − } { } intrinsic periodicity µ topology S1 cp¯µT = h Φ(x,t)= A a (¯p)φ (x)u (t) n n n n Energy quantization Tt(¯p ) n 7 6 ¯ En(¯p )=nω¯(¯p ) 5 = nE¯(¯p ) t 4 h 3 = n T (¯p ) 2 t de Broglie internal clock 1 t T b Tt(¯p )=h/E¯(¯p ) Einstein:1910 Physical information contained in a single period T µ 4 Compact space-time dimension ¯ = d x ¯(∂ Φ¯, Φ¯) Sλs L µ with Periodic Boundary Conditions 0 String vibrating with four periodicity T µ Φ(xµ)=Φ(xµ + T µ) we have a semi-classical quantization condition, relativistic generalization of a “particle in a box” µ µ Covariance cp¯ T = h cp¯ T = h µ → µ Lorentz transformation of reference frame a transformation of 4-momentum p¯ p¯ = Λ p¯ µ → µ µ a transformation of (tangent) 4-periodicity µ T T µ ¯ 4 ¯ ¯ ¯ 4 ¯ ¯ ¯ transformation of boundaryλs = d x (∂µΦ, Φ) d x (∂µΦ, Φ) S L L 0 0 µ 1 1 1 T is contravariant, ¯ 2 2 µ M c =¯pµp¯ 2 = µ tangent four-vector. Tτ Tµ T relativistic modulation of periodicity spectrum dispersion relation R p E p Tt(¯p ) E(¯p ) nh ! " λs h ! " En(¯p )= Tτ = Tt(0) = = T (¯p ) c Mc¯ 2 ¯ . t . Mn = nM ¯ 2 h M = h/Tτ c E¯(¯p )= . Tt(¯p ) . p p ¯p ¯p Same energy spectrum 2 4 2 2 as in second quantization: (after normal ordering) En(¯p )=nEn(¯p )=n M¯ c + ¯p c Interactions as Boundary Geometrodynamics Generic interaction free case interaction a xµ variation of 4-momentum p¯ p¯ (x)=e (x)¯p µ → µ µ a µ µ a µ modulation of 4-periodicity T T (x) T ea (x) → ∼ µ a b T (X) η g (x)=e (x)e (x)η x = X deformation of the metric µν → µν µ ν ab µ µ a µ transformation of ref. frame x x (x)=x Λa (x) µ a →µ T T ea (X) 4 d4x√ g (e µ∂ Φ , Φ ) . d x (∂µΦ, Φ) λs a µ λ Tt L S ∼ 0 − L x 0 Holography: kinematics encoded in the boundary. x i persistent x modulated p x x i dx p(x) Φ(x)=e− · Φ(x) e− · ∼ R Weak newtonian inter. VN (r)= GM /r 1 − ⊙ GM GM T1 Redshift E E(1 ⊙ ); p p(1 + ⊙ ) L1 ∼ − r ∼ r GM GM Time dilatation T T (1 + ⊙ ); λx λx(1 ⊙ ) ∼ r ∼ − r 2 2GM 2 2GM 2 λx Tt ds 1 ⊙ dt 1+ ⊙ dx Schwarzschild ∼ − r − r T L “What is fixed at the boundaries of GR?” York,84 Newton Canonical QM, Φ(x,t)= Anan(¯p)φn(x)un(t) n in every interaction point (locally) 7 6 Vx dx3 λx Vx = Nλx Inner Product: φ χ = φ(x)χ(x) → | V N Z. 5 0 x ∈ 4 Hilbert Eigenstates:φn s.t.: x φn = φn(x)/ Vx | | 3 En Hamiltonian Operator: (x) φ = E (x) φ Nielsen:ʼ06 H | n n | n H2 pn Momentum Operator: (x) φn = p (x) φn 1 P | n | P T ψ φn ψ = c φ b | | Generic Hilbert State: | n | n n 2 2 Bulk EoMs: (∂ + ω )φn(x,t)=0 i∂tφn(x,t)=Enφn(x,t) t n → Carena:ʼ03 Schrödinger equation: i∂t ψ(x) = (x) ψ(x) | H | Commutation relation Feynamn:ʼ42 Expectation Value: ∂x (x) χf ∂x (x) φi = i χf (x) (x) φi F | F | |PF −F P| Commutation relations: (x) x [x, ]=i [ (x), ]=i∂x (x); F ≡ Implicit! F P F P Without any further assumption than intrinsic periodicity we have all the ingredints to build a FPI: Complete set of eigenfunctions in Hilbert space, with Marcovian time evolution: Complete set of eigenfunctions Feynman Path Integral in Hilbert space, with unitary time evolution: V N 1 N 1 V − dx − i i m ( ∆m ∆xm) [tf ,ti] =lim φ e− H −P φ = xe S Z N V (x ) | | →∞ 0 m=1 m m=0 Z D 0 tb (t ,t ) dt (x, t) (xm,tm) x˙ m (xm,tm) S b a ≡ L L ≡ P − H ta Set of classical paths with different winding number: . topology S1 2.5 . 5 2.5 2.0 4 2.0. p1 p1 1.5 1.5 . 3 1.0 1.0. 2 0.5 0.5 . 1 0 1 2 3 4 5 .0 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Feynamn:ʼ42 Composition of periodic paths variation around a give path Path Integral as interference of classical paths! --> Gauge interaction as periodicity modulation D.D.; Ann.Phys., 327, (2012); arXiv:1110.0315 --> Compact Time and determinism: foundations D.D.; Found.Phys., 41, (2011); arXiv:0903.3680 d Waves and Particles Particle limit arises at high frequencies 2 ˆ 0 Multiparticles or wave U(∆xTot, ∆tTot)=Φ(∆xTot, ∆tTot) ∼ 2 δ[periodic path] E ∼ 0.3 0.2 0.1 0.0 2 λ0 s 2 Single Particle p¯ Mc¯ !3 !2 2 2 Mc¯ M¯ (x x) U (∆x , ∆t ) exp i (t t)+i − !1 pt Tot Tot ∼ − − 2(t t ) 0 − 3 2 λ ¯ h s M = 1 δ(classical path) T c2 τ 0 ∼ 2 0 !2 Heisenberg Uncertain Princ. p ¯ ¯ i( Et +π) i E(t+∆t) i (E+∆E)t If the phase can not e− = e− = e− : be observed π π ∆t = , ∆E = ∆E¯ t ∆t E¯ t (π)2 (π)2 h w ∆E ∆t = = π = × Et¯ ≥ ET¯ 2 Non-relativistic Schrödinger equations Bohr-Sommerfeld: p (x)dx =2π (n + v) pn = 2m(En V (x)) n − 2 Schrödinger problems: 2 ∂xφn(x)+V (x)φn(x)=Enφn(x) Atomic orbitals −2m Quantum Numbers: {n} and {l,m} Hydrogen Atom: topology S1 S2 ⊗ Quantum Numbers {n,l,m}: ⊗ QM is the full relativistic generalization of the Theory of Sound Elementary particles are vibrations of the space-time dimensions [Rayleigh:1900] QM is the full relativistic generalization of the Theory of Sound Elementary particles are vibrations of the space-time dimensions [Rayleigh:1900] Indirectly observed ‘t Hooft determinism or Stroboscopic quantization Experimentally ‘t Hooft:’01 Elze:’02 e− 20 electron: Tt 10− s ZHz Fast Periodic Dynamics ∼ Gouanere:2008→ Cs133 10 Cs 133: T 10− s “Zitterbewegung [“trembling motion”, Schrödinger 1930] t ∼ models explain the electron's spin, magnetic moment, grassmann variables as generated by de Broglie die the de Broglie internal clock of the electron.” In dice rolling too fast the outcomes can only be predicted statistically QM.
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