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Noname manuscript No. (will be inserted by the editor) Donatello Dolce Quantum Mechanics from Periodic Dynamics: the bosonic case the date of receipt and acceptance should be inserted later Abstract Enforcing the periodicity hypothesis of the semiclassical formulation of Quantum Mechanics we show the possibility for a new scenario where Special Relativity and Quantum Mechanics are unified in a deterministic field theory[1]. From this follows a novel interpretation of AdS/CFT.1 1 Periodic Dynamics In the old, semiclassical approach to QM, free bosonic waves are supposed to have angular frequencies ω̄; the energies of the related quanta Ē = ~ω̄ = h/Tt depend on the inverse of the time periods Tt = 2πRt . To solve a relativistic differential system (ds2 = c2 dτ 2 = c2 dt2 − dx2 ) we must impose Boundary Conditions such to minimize the action on the boundaries. For this reason, in ordinary QFT one assumes fixed values of the fields on the time boundaries, let’s say at t and t + Tt . For free bosonic fields, we note that it is possible to satisfy the variational principle, and thus all the relativistic symmetries of the action, by imposing Periodic BCs (PBCs) Φ(x, t) ≡ Φ(x, t + 2πRt ) [3]. Since in this case the whole information is contained in a single period, we say that time is compactified on a circle t ∈ S1Rt with radius Rt . Thus, imposing PBCs as a constraint, the resulting Φ(x, t) is decomposed as a tower of normal modes with quantized frequencies ωn = nω̄ = n/Rt . In analogy with the Matsubara and KK theory, it is naturally interpreted as a tower of energy eigenmodes with energies proportional to the frequencies En = n~ω̄ = n~/Rt . Since we assume on-shell fields, in the massless case (ds2 = 0) there is an induced periodicity λx = 2πcRt on the modulo of the spatial dimensions (c2 dt2 = dx2 ) and thus a quantization of the modulo of the Institut für Physik, Johannes-Gutenberg Universität, D-55099 Mainz, Germany E-mail: [email protected] MZ-TH/09-23 1 This letter (readable with [2]) summarizes results of [1], where details of the demonstrations, further relevant issues and physical interpretations are reported. 2 momentum as well |pn | = ~|kn | = n|p̄| = n~/Rt c. For dimensional reasons the proportionality constant is the reduced Planck constant ~ and thus we naturally get the de Broglie relation Rt (p̄) ≡ ~/Ē(p̄) = 1/ω̄(p̄). Since the period is related to the inverse of the fundamental energy Ē, and not to the inverse of an invariant mass as in the KK theory, the compactification must be regarded as dynamical. The periodicities are described by the four-vector Rµ = (Rt c, Rx ) and the fundamental mode (n = 1) has four-momentum p̄µ = (Ē/c, p̄) ≡ ~/Rµ . For this massless field ω̄(p̄) = |p̄|c/~. Thus every mode of the energy tower has a massless dispersion relation. At small fundamental momentum |p̄| → 0 the compactification radius tends to infinity Rt → ∞, i.e. the PBCs can be neglected and the continuos energy spectrum Ē → 0 is restored. Indeed this is the usual relativistic field which can also be obtained by putting ~ → 0. On the other hand, in analogy with the relativistic Doppler effect, increasing the momentum |p̄| → ∞ through interaction or changing the reference system, one obtains a Lorentz deformation of all the spacetime intervals and thus of the compactification radiuses as well. Since now Rt → 0, the PBCs are important and we get a well quantized energy spectrum Ē → ∞ at high frequencies. Actually, this avoids the UV catastrophe in the blackbody radiation. This limit can also be obtained by putting ~ → ∞ and therefore can be addressed as the quantum limit of the massless case. Concerning massive fields (ds2 > 0), it is possible to choose a reference frame where the proper time is equal to the real time: dτ 2 = dt2 . This means that also the proper time has induced periodicity Tτ = Tt (0) = h/M̄ , where we define M̄ to be the mass. By treating the proper time τ = s/c as a virtual extradimension (XD) for a 5D field with zero 5D mass (dS 2 = c2 dt2 − dx2 − ds2 ≡ 0), we obtain again a regular energy tower En (p̄) = n~ω̄(p̄) but now every quantized level has the dispersion relation of a massive relativistic particle ω̄(p̄) = (p̄2 c2 + M̄ 2 c4 )1/2 /~, just as in the normal ordered second quantization [4,5] and in perfect agreement with Lorentz transformations. We define covariant notations: p̄µ ≡ ~/Rµ and M̄ 2 c4 = ~2 /Rτ2 = p̄µ ~/Rµ = (~/Rµ )(~/Rµ ). In this way we see that the time periodicity Tt (p̄) has an upper limit Tτ in the rest frame proportional to the Compton wavelength λs ≡ Tτ c = h/M̄ c, which indeed fixes the mass of the field. Even for a light boson with the rest mass of an electron, this periodicity is already extremely fast, 10−20 s, if compared for instance with the characteristic period of the Cs133 atom which by definition is of the order of 10−10 s. Remarkably, evidences of this intrinsic periodicity of massive particles conjectured by de Broglie [6] have been indirectly observed in a recent experiment for electrons [7].2 We point out that this theory respects relativistic causality. In fact, the compactification radius Rt (p̄) is not static but local and dynamical as the energy: Rt (p̄) ≡ ~/Ē(p̄). Since the given theory is build upon relativistic waves, we can solve the Green function as usual. Therefore, by turning on a source in a point, a retarded variation of the energy in a different point is induced and, just by energy conservation, the field in that interaction point passes from a periodic regime to another periodic regime. In this way it is possible to give a chronological order to events and the resulting notion of time formally fulfils the requirements of SR. 2 Here fermions can be thought of as fields with antiperiodicities Tt (p̄) = h/Ē(p̄). 3 2 Quantum Mechanics We now study the mechanics of these periodic fields, for simplicity we assume only one spatial dimension. The first thing we note is that these fields are a sum of on-shell standing waves with Fourier coefficients An X X X Φ(x, t) = An φn (x, t) = An φn (x)un (t) = An e−i(ωn t−kn x) . (1) n n n Actually, we have the typical case where a Hilbert space can be defined. In fact, the energy eigenmodes φn (x) form a complete set with respect to Rλ the inner product hφ|χiH ≡ 0 x dxφ∗ (x)χ(x)/λx . Furthermore, the bulk EoMs (∂t2 + ωn2 )un (t) = 0, that describe the evolutions along the compact time, can be reduced to first order differential equations which are nothing else than the Schrödinger equation i~∂t φn (x,√t) = En φn (x, t). Formally, from the Hilbert eigenstates hx|φn i ≡ φn (x)/ λx we can build the Hamiltonian operator Ĥ |φn i ≡ ~ωn |φn i and thus a time evolution operator Q −1 ′ U (t′ ; t) = exp[−iĤ(t − t′ )/~] which is Markovian U (t′′ ; t′ ) = N m=0 U (t + ′ ′′ ′ tm+1 ; t + tm − ǫ), where ǫ is s.t. N ǫ = t − t [8]. Finally, without any further assumption than periodicity, we plug the completeness relation in between the elementary time evolutions of the Markovian operator obtaining a path integral (PI) which is formally the Feynman PI for a time independent Hamiltonian ! N −1 Z λx NY −1 Y i ′′ ′′ ′ ′ U (x , t ; x , t ) = lim dxm hφ| e− ~ (Ĥ∆ǫm −p̂∆xm ) |φi . (2) N →∞ 0 m=1 m=0 Here ∆xm = xm+1 − xm and ∆ǫm = tm+1 − tm . In fact, also in the usual Feynman PI the elementary evolutions are supposed to be on-shell [9] i U (xm+1 , tm+1 ; xm , tm ) = λx hφ| e− ~ (Ĥ∆ǫm −p̂∆xm ) |φi = X i X e− ~ (Enm ∆ǫm −pnm ∆xm ) = 2π = δ ω̄(p̄)∆ǫm − k̄∆xm + 2πn′ . (3) nm n′ P To P write the last expression we have used the Poisson summation n e−inα = 2π n′ δ(α + 2πn′ ), so that the PI eq.(2) - or similarly the periodic field itself eq.(1) - can be explicitly written as a sum over periodicon-shell paths P U (x′′ , t′′ ; x′ , t′ ) = 2π n′ δ ω̄(p̄)(t′′ − t′ ) − k̄(x′′ − x′ ) + 2πn′ . Because the initial and final points are now defined modulo space-time periods, they are paths with different winding numbers. These on-shell Marcovian and periodic paths can be ideally cut, translated by periods and combined in such a way to form paths with the same initial and final points, obtaining the analogous of the variations around the classical path of the usual Feynman formulation. However, in the original Feynman PI there is a single classical path linking the initial and final point, so that the variational principle must be relaxed to have interference. In our case the field can self-interfere because the PBCs and the variational principle is preserved since the sum over variations can be reduced to a sum over periodic on-shell paths. Nevertheless, as in the usual 4 QM, we obtain that the square modulo of the periodic field |Φ(x, t)|2 has a maximum along the relativistic free particle path where the periodic on-shell paths have constructive interference. Taking the non relativistic limit |p̄| ≪ M̄ c, the field can be thought of as at low intensity so that only the first level is largely populated. Thus (nm ≡ 1; ∀m = 0, . . . , N − 1) we find that the sum over periodic paths eq.(2) reduces to the usual non-relativistic free particle FPI (plus the de Broglie internal periodicity). This is a Dirac delta distribution along the particle path. Hence we have obtained a consistent interpretation of the wave/particle duality and of the double slit experiment. The uncertainty relation can be easily obtained (we have standing waves) by absorbing the periodic invariance of the phase factor Ēt/~ + 2π either as a time or as an energy variation: ∆E × ∆t = (2π~)2 /Ēt ≥ h, where t ≤ Tt . This is a direct consequence of the periodic conditions En Rt = n~ which can be written in a Bohr-Sommerfeld form: the allowed solutions are those with an integer number of cycles. Following this approach it is straightforward to obtain the usual solution3 of the quantum harmonic oscillator (QHO) as well as many others non-relativistic Schrödinger problems. Moreover, generalising the symmetry breaking mechanism by BCs [3] to a periodic EM field at low temperature [10], we have an effective quantization of the magnetic flux [11] which reproduces collective quantum phenomena such as superconductivity. A further analogy with canonical QM emerges by noting that, from the given definition of Hilbert space, the expectation value of O(x) is Z λx dx X ∗ χ (tf , xf )e−ikχm x O(x)eikφn x φn (ti , xi ) . (4) hχf |O(x)|φi i ≡ λx n,m m 0 Now we suppose that the observable is O(x) ≡ ∂x F (x). Integrating by parts taking into account the PBCs, and then imposing F (x) ≡ x, we get hχ(xf , tf )|1|φ(xi , ti )i = ~i hχ(xf , tf )|p̂x − xp̂|φ(xi , ti )i, where the operator p̂ is defined analogously to the Hamiltonian operator Ĥ. For arbitrary initial and final periodic states φ(xi , ti ) and χ(xf , tf ), we finally get nothing else than the usual commutation relation [x, p̂] = i~. Indeed we have a theory of relativistic waves where QM emerges as a consequence of information lost due to fast periodic dynamics. When the periodicity is too fast, the system can only be described statistically since at every observation it turns out to be in a random phase of its apparently aleatoric evolution. This is just like observing a timekeeper under a stroboscopic light [12] or a dice spinning too fast to predict the result. In fact, we know from the ’t Hooft determinism [4] that there is a deep connection between a particle moving with high frequency on a circle and the QHO. It is important to note that, in our case the are no local-hidden-variables being the time a physical variable that can not be integrate out, and being the PBCs an element of non locality. Therefore we conclude that the present theory is not constrained by the Bell’s or similar theorems, and we can talk about determinism. 3 We are not considering the unimportant phase factors in front of the wave functions [4]. They shift the energy spectrum giving rise to the vacuum energy [5]. 5 3 AdS/CFT interpretation To extend our theory to interacting fields we could, for instance, develop a perturbation theory from the PI eq.(2). However, in simple cases, an approximative description of interactions can be given in an easier way. The trivial case is Compton scattering e′ + γ ′ → e′′ + γ ′′ where we just rewrite the conservation of the four-momentum among the quanta p̄µγ′ + p̄µe′ = p̄µγ′′ + p̄µe′′ as the conservation of the reciprocal of the four-radiuses 1/Rγµ′ + 1/Reµ′ = 1/Rγµ′′ + 1/Reµ′′ . We see from this example that during interaction the compactification radiuses are subject to deformations. Therefore we face the interaction problem using classical field theory in curved space-time. For instance, we know from the Bjorken model that the Quark-Gluon Plasma (QGP) can be approximated as a volume of massless fields at high energy, and that the energy decays exponentially radiating hadronically and electromagnetically [13,14]. Using our terminology we get that the periodicities of the fields have an exponential dilatation described by dt2 → dt̃2 ≃ e2ks̃ dt2 and dx2 → dx̃2 ≃ e2ks̃ dx2 , so the QGP freeze-out is encoded in a AdS metric dS 2 ≃ e−2ks̃ (dt̃2 − dx̃2 ) − ds̃2 . Assuming dS 2 ≡ 0 the proper time acts as a virtual XD. The propagation of 5D YM fields with a 5D coupling g5 in such a warped background gives a classical correlator Π(q 2 ) which, in first approximation4 and assuming 1/kg52 = Nc /12π 2 [15,16], can be matched to the −q2 2 2 quantum vector-vector two point function of QCD: Π(q 2 ) ≃ 2kg 2 log q /Λ . 5 Hence, we obtain classically the quantum runnings of the strong coupling in agreement with the interpretation [17] of the AdS/CFT conjecture [18]. 4 Conclusions In conclusion, by imposing to a bosonic field the local and dynamical spacetime periodicities 2πRµ ≡ h/p̄µ conjectured by de Broglie, we find that the theory is in agreement with SR. Remarkably, from the consequent property of recursivity which is induced to the relativistic time -whose flow as we know depends on the energy-, the usual QM emerges under many of its different formulations and for several nontrivial phenomena [1]. This could open a new scenario where SR and QM can be unified in a deterministic wave theory. After all, the notion of time is strictly related with the assumption of periodicity: time is defined by counting the number of cycles of phenomena supposed to be periodic. Therefore, “we must assume, by the principle of sufficient reason”5 , periodicity to define a relativistic clock. In conclusion, since the whole information is contained in a single period, every single elementary field can be regarded as in intrinsically compactified space-time dimensions. 4 To evaluate the low energies effective propagator we use the holographic method with Neumann BCs at the UV scales Λ and boundary field Aµ (q) at the IR scale µ, in the hypothesis of Euclidean momentum Λ ≫ |q| ≫ µ. 5 “By a clock we understand anything characterized by a phenomenon passing periodically through identical phases so that we must assume, by the principle of sufficient reason, that all that happens in a given period is identical with all that happens in an arbitrary period.” A. Einstein [19]. 6 References 1. D. Dolce, “Compact Time and Determinism for Bosons,” 0903.3680. 2. D. Dolce, “Quantum Mechanics from Periodic Dynamics: Bosonic case,” http://wwwthep.physik.uni-mainz.de/˜dolce/talk.pdf (2009). 3. C. Csaki, C. Grojean, H. Murayama, L. Pilo, and J. Terning, “Gauge theories on an interval: Unitarity without a higgs,” Phys. Rev. D69 (2004) 055006, hep-ph/0305237. 4. G. ’t Hooft, “Determinism in free bosons,” Int. J. Theor. Phys. 42 (2003) 355–361, hep-th/0104080. 5. R. L. Jaffe, “The Casimir effect and the quantum vacuum,” Phys. Rev. D72 (2005) 021301, hep-th/0503158. 6. L. de Broglie, “Recherches sur la thèorie des quanta,” Ann. Phys 3 (1925) 22. 7. P. Catillon, N. Cue, M. J. Gaillard, R. Genre, M. Gouanère, R. G. Kirsch, J.-C. Poizat, J. Remillieux, L. Roussel, and M. Spighel, “A Search for the de Broglie Particle Internal Clock by Means of Electron Channeling,” Found. Phys. 38 (July, 2008) 659–664. 8. H. B. Nielsen and M. Ninomiya, “Intrinsic periodicity of time and non-maximal entropy of universe,” Int. J. Mod. Phys. A21 (2006) 5151–5162, hep-th/0601021. 9. R. P. Feynman, “Space-time approach to nonrelativistic quantum mechanics,” Rev. Mod. Phys. 20 (1948) 367–387. 10. J. Zinn-Justin, “Quantum field theory and critical phenomena,” Int. Ser. Monogr. Phys. 113 (2002) 1–1054. 11. S. Weinberg, “The quantum theory of fields. vol. 2: Modern applications,”. Cambridge, UK: Univ. Pr. (1996) 489 p. 12. H.-T. Elze and O. Schipper, “Time without time: A stochastic clock model,” Phys. Rev. D66 (2002) 044020, gr-qc/0205071. 13. V. K. Magas, A. Anderlik, C. Anderlik, and L. P. Csernai, “Non-equilibrated post freeze out distributions,” Eur. Phys. J. C30 (2003) 255–261, nucl-th/0307017. 14. H. Satz, “The Thermodynamics of Quarks and Gluons,” 0803.1611. 15. A. Pomarol, “Grand unified theories without the desert,” Phys. Rev. Lett. 85 (2000) 4004–4007, hep-ph/0005293. 16. N. Arkani-Hamed, M. Porrati, and L. Randall, “Holography and phenomenology,” JHEP 08 (2001) 017, hep-th/0012148. 17. E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,” Adv. Theor. Math. Phys. 2 (1998) 505–532, hep-th/9803131. 18. J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, hep-th/9711200. 19. A. Einstein, “Principe de relativité,” Arch. Sci. Phys. Natur. 29 (1910).