IICCUB-15-017
Inconstant Planck’s constant
Gianpiero Mangano ∗ , Fedele Lizzi , and Alberto Porzio INFN, Sezione di Napoli, Complesso Univ. Monte S. Angelo,Via Cintia, I-80126 Napoli, Italy Dipartimento di Fisica, Universit`a di Napoli “Federico II”,Complesso Univ. Monte S. Angelo, Via Cintia, I-80126 Napoli,Italy Departament de Estructura i Constituents de la Mat`eria,Institut de Ci´encies del Cosmos, Univ. de Barcelona, Spain CNR-SPIN, Unit`a di Napoli, Complesso Univ. MonteS. Angelo, Via Cintia, I-80126 Napoli, Italy
Abstract
Motivated by the Dirac idea that fundamental constant are dynam-ical variables and by conjectures on quantum structure of spacetime atsmall distances, we consider the possibility that Planck constant (cid:126) isa time depending quantity, undergoing random gaussian fluctuationsaround its measured constant mean value, with variance σ and atypical correlation timescale ∆ t . We consider the case of propagationof a free particle and a one–dimensional harmonic oscillator coher-ent state, and show that the time evolution in both cases is differentfrom the standard behaviour. Finally, we discuss how interferometricexperiments or exploiting coherent electromagnetic fields in a cavitymay put effective bounds on the value of τ = σ ∆ t . [email protected]@[email protected] a r X i v : . [ qu a n t - ph ] N ov Introduction
The idea that fundamental constants in physics may be spacetime varyingparameters was first suggested by Dirac with his “Large Numbers Hypoth-esis” [1]. Since then, many efforts have been devoted to embed this idea incoherent theoretical frameworks. One possibility is that these constants maybe effective quantities. This happens, for example, in higher-dimensional the-ories, after reduction of coupling constants to the four-dimensional subspace.For a critical summary of this subject see [2, 3].The most studied examples of varying fundamental constant are perhapsthe Newton gravitational constant G N and the fine structure coupling α . Inthe latter case there is quite a large number of experimental constraints onits time evolution, spanning several order of magnitude in redshift, or cosmo-logical time, from laboratory experiments, geophysical tests as the Oklo phe-nomenon, the anisotropies of Cosmic Microwave Background, till the earlyepoch of Big Bang Nucleosynthesis, see [3] and references therein. Spatialand time variations of α have been also constrained by observations of ab-sorption spectral lines in intervening clouds along the line of sight of distantQuasars at redshifts z ∼ −
3, [4–8].Evidences for dynamical features of fundamental constants would haveof course, an enormous impact on the general picture of physics laws. Thetypical approach to this theoretical and experimental issue, as in the case of α just mentioned, is to look for signals of variation over long time period,billion years in the case one uses cosmological observables. The underlyingidea is that the dynamical field which describes a particular fundamentalconstant is changing as a classical background field under the action of someeffective potential which drives the field towards its minima. However, if thisputative field Ψ reaches a stable configuration in very short time intervals,so that there is no further evolution in the long time regime, we could onlysee an effect by strongly perturbing this configuration and moving Ψ awayfrom the minimum. This can be achieved by exploiting the Ψ coupling withmatter fields and exciting the field by pumping energy into it, i.e. the typicalaccelerator approach to unveil new degrees of freedom. However, even ifΨ seats at the classical potential minimum, we expect its value to undergosmall random fluctuations. In this case the question is how these fluctuationsmight produce observable effects in experiments.We recall that many theoretical arguments point out that at short timeand distance scales both quantum field theory and gravitational interactions2escribed by general relativity ought to be embedded in some form of quan-tum gravity , a theory not yet known. Whatever is this theory, it will deeplychange the geometry of spacetime and phase space. Examples are theoriesin which spacetime is composed by branes, feature a noncommutative geom-etry or are characterized by the spin foam of loop quantum gravity, see, e.g.[9, 10] and [11] and references therein. The role of a spacetime fuzziness andstochastic Lorentz invariance violation has been recently studied in [12].Since its introduction at the dawn of last century [13], h has been consid-ered one of the fundamental constant ruling physical phenomena. Its valuehas been measured by different methods spanning the spectrum of black bodyradiation, kinetic energy of photoelectrons in photoelectric effect, and morerecently, X-ray density crystal measurement, and watt balance. The lattertwo have recently achieved a relative precision of 1 . [14]. Theactual accepted value is h = 6 . · − J s [15]. The precisemeasurement of h plays a crucial role in establishing both a new SI and arevised fundamental physical constant system.The (reduced) Planck constant (cid:126) enters into the definition of the Heisen-berg uncertainty principle, one of the pillars of the probabilistic nature ofquantum mechanics. In fact, deformations of the quantum phase spacehave been put forward in the form of generalized uncertainty principle, seee.g. [16–19], described by the commutation relation[ x i , p j ] = i (cid:126) (cid:0) δ ij + f ij ( x , p ) (cid:1) , (1)where f ij depend upon some “small parameter”. This is particular case of amost general phase space commutation relation among position and momentawhich also includes [ x i , x j ] = iθ ij [ p i , p j ] = iC ij (2)where f, θ C are such that Jacobi identities are satisfied. In general, theserelations break Lorentz invariance, and a way to restore it to consider f, θ and C to be random variables with microscopic variations, so that for largedistances and time intervals the invariance is restored. The commutationsrelations (1) and (2) mean that the phase space has a non canonical sym-plectic structure, but with a Darboux transformation it is always possible toexpress (locally) the structure as a nontrivial commutator between positionand momenta, all other commuters being zero. Since the transformation3ixes positions with momenta this calls into question the meaning of theposition and momentum. We will consider that at any given instant the ob-servables can be seen as generated by two sets of operators, x i and p j , withcommutation relations as in (1).If one considers that corrections to the commutators are small and dependon the background and fluctuations at some high energy scale, the net resultis that the r.h.s. of commutation relations can be effectively described by arandom function acting as an effective dynamical field. This is tantamountto consider (cid:126) as a fluctuating parameter. In this paper we will consider thatthe space fluctuations are integrated out, and will concentrate on the timevariations. Energy scale dependence of (cid:126) induced by nontrivial commutationrelations has been proposed in [20]. In [21] the possibility that (cid:126) is a fieldwhose expectation value can vanish at high temperatures was considered asa possibility to quantize gravity.A comment about the issue of dimensionality. Planck constant is notadimensional, and claims about its time dependence would be always re-lated to some particular combinations of (cid:126) and other dimensionful quanti-ties, depending on the adopted experimental method. Only for adimensionalcombinations of fundamental constants, such as α , it is meaningful to speakabout spacetime variation unambiguously. The variability of fundamentalconstants is a delicate issue, see e.g. [22]. We will see next that the stochas-tic nature of (cid:126) is encoded in a dimensionful time parameter τ and observableeffects for light propagation or a harmonic oscillator show up through thedimensionless combination ωτ , with ω the light or oscillator frequency.Our aim in this paper is not to develop a specific theoretical frameworkdescribing this scenario. We will only assume that there is no classical back-ground evolution, but the field nature of the Planck constant shows up inrandom fluctuation around the constant measured mean value, which we willcontinue to denote by (cid:126) . Random fluctuations of (cid:126) are introduced via an adimensional gaussian stochas-tic variable ε ( t ), so that the effective Planck constant reads (cid:126) (1 + ε ( t )), with ε ( t ) = 0 , (3) ε ( t ) ε ( t (cid:48) ) = τ δ ( t − t (cid:48) ) . (4)4verline denotes the mean over ε probability distribution. The parametersare as follow: τ = σ ∆ t , σ is the variance and equation (4) states that fluctu-ations are uncorrelated for time differences larger than a typical correlationtime ∆ t , analogous of the strength of fluctuation force in brownian motiondynamics in Stokes–Einstein theory for diffusion coefficient. These relationsare time translation invariants.Planck constant fixes the commutator of canonically conjugated variables,position and momentum for non relativistic particles, and bosonic (fermionic)field and conjugate momentum field commutator (anticommutator). Con-sider a particle in one dimension. If (cid:126) fluctuates , we have[ x , p ] = i (cid:126) (1 + ε ( t )) . (5)The commutator with a Hamiltonian H gives the time evolution of operators A d A dt = 1 i (cid:126) [ A , H ] + ∂ A ∂t . (6)The first term on the r.h.s. now explicitly depends on time via ε ( t ), andthe time evolution of A , the average over the probability distribution of ε ,shows non–standard behaviour, as we will see. Had we defined the Poissonstructure by normalizing the commutator to (cid:126) (1 + ε ( t )), all effects canceland we would be back to standard quantum mechanics. More interesting isthe case of equation (6), i.e. the fundamental Poisson bracket fluctuates. Weinvestigate this possibility.Our approach is different from stochastic quantization [23], where a sys-tem is considered in thermal contact with some external reservoir, and itsquantum properties are obtained in the equilibrium limit. This is achievedby introducing an auxiliary fictious time. Equilibrium is reached in the largefictious time limit. In our case, Planck constant depends upon physical time,leading to novel features.Quantum measurements is itself a probabilistic concept, and it only makessense to compute observable averages over the probability distribution givenby a wavefunction. We denote this averaging by (cid:104) A (cid:105) . The relation betweenstandard quantum mechanical average and average over ε ( t ) distribution isdiscussed later.Commutation relation of equation (5) is represented by defining the action5f position and momentum on wavefunctions ψ ( x ) x ψ ( x ) = A ( t ) x ψ ( x ) = A ( t ) x ψ ( x ) , (7) p ψ ( x ) = − i (cid:126) B ( t ) ddx ψ ( x ) = B ( t ) p ψ ( x ) , (8)with A ( t ) B ( t ) = 1 + ε ( t ), and x and p the canonical pair of standardquantum mechanics. A change of representation A ( t ) → A (cid:48) ( t ) is a timedependent dilation induced by the unitary operator U ( t ) = exp (cid:20) i (cid:126) log A (cid:48) ( t ) A ( t ) ( x p + p x ) (cid:21) . (9)Here we adopt the choice A ( t ) = B ( t ) = (cid:112) ε ( t ) , (10)and assume that in this representation equations of motion are given by (6).The rationale is that effects of any high energy scale theory producing afluctuating (cid:126) are expected to disturb phase-space elementary volumes. Itseems unnatural that coordinates and momenta should be treated differently,and rescaled by different factors.Although in principle, all values of ε ( t ) are possible, we expect ε ( t ) prob-ability distribution to be narrow, so the number of events corresponding tolarge values is exceedingly small. Since ε ( t ) is constant in the time interval∆ t , in a time T its value is randomly extracted T / ∆ t times. For T ∼ . t ≥ . · − s, the Planck time, for agaussian distributed ε , the event number for ε < −
1, i.e. imaginary A ( t ) T ∆ t (cid:90) − −∞ dε √ πσ e − ε σ = T t erfc (cid:18) √ σ (cid:19) , (11)is smaller than unity provided σ ≤ .
06, a value which seems already too largeto be acceptable. It is unlikely that in the whole history of the universe these large fluctuations were ever produced, and we can safely assume A ( t ) ∈ R . If Planck constant is randomly inconstant , the issue is to find experimentalapproaches to constrain the τ , or to find signature of its dynamics. We discusstwo possibilities, interferometric experiments and coherent electromagneticmodes in a cavity. 6 .1 Free particles and long baseline interferometric ex-periments Consider a free particle with mass m . Schr¨odinger equation reads i (cid:126) ∂∂t ψ = 12 m (1 + ε ( t )) p ψ . (12)Since [ p , H ] = 0, fundamental solutions are still plane waves ψ p , eigenfunc-tions of p with eigenvalue p . In the representation L ( R ), with R the x spectrum ψ p ( x, t ) = 1 √ π exp (cid:20) i p x (cid:126) − i p m (cid:126) (cid:18) t + (cid:90) t ε ( t (cid:48) ) dt (cid:48) (cid:19)(cid:21) . (13)Using these solutions we can construct wave packets. Position and momen-tum operators act as follows x ψ ( x, t ) = (cid:112) ε ( t ) x ψ ( x, t ) , (14) p ψ ( x, t ) = − i (cid:126) (cid:112) ε ( t ) ddx ψ ( x, t ) . (15)We want to evaluate position mean value and uncertainty versus time. Asmentioned already, averaging should be performed both in the usual quan-tum sense, and over the ε ( t ) probability distribution. We denote this doubleaverage of some observable A in the state | ψ (cid:105) by (cid:104) A (cid:105) ψ . For ∆ t much smallerthan experimental time resolution, standard quantum mechanics average isalready an average over ε values. Indeed, repeating the measurement on thesame state, which defines a quantum measurement, we also sample ε proba-bility distribution. Actually, in quantum mechanics averaging over repeatedexperiments is the same as an average over simultaneous experiments on alarge number of identical copies of the apparatus. In our case the two aver-ages are different in principle, although they coincide for practical purposes,unless experiments are performed in time intervals shorter than ∆ t . We con-tinue nevertheless, to use the double averaging notation, to underline theconceptual difference.Suppose we prepare the system at initial time t = 0. If (cid:126) fluctuates, po-sition measurements receive additional contributions when averaging over ε distribution. For mean position value this term is √ ε ∼ O ( σ ), whichis time independent, since fluctuation averages are time translation invariant.7he same results holds at some final time t , again by time translation invari-ance. This implies that measurements of the mean distance travelled by aparticle is insensitive to this order σ factor, which cancels in the difference (cid:104) x (cid:105) ψ ( t ) − (cid:104) x (cid:105) ψ (0). The observable effect of averaging over ε is only containedin the ε term in the state time evolution, see equation (13). Same holds forposition uncertainty and momentum measurements.We choose for reference a gaussian profile peaked at p and variance δ ψ ( x, t ) = (cid:90) dp √ π πδ ) / e − ( p − p )22 δ + ip x/ (cid:126) − ip ( t + (cid:82) t ε ( t (cid:48) ) dt (cid:48) ) / (2 m (cid:126) ) . (16)Using equations (3,4), the mean distance travelled by a particle and uncer-tainty read (cid:104) x (cid:105) ψ ( t ) − (cid:104) x (cid:105) ψ (0) = pm t , (17)(∆ x ) ψ ( t ) − (∆ x ) ψ (0) = δ m t + p + δ / m τ t . (18)Squared uncertainty displays a new contribution, linear in time. This is muchalike a Brownian motion with diffusion coefficient D = p + δ / m τ . (19)For δ (cid:28) p , D can be regarded as due to scatterings with mean free path( p/m ) τ .First term in r.h.s. of equation (18) is typically expected to be dominantover the genuine new effect, but considering massless particles, as photons,the linear dispersion relation leads to no O ( t ) wave packet spread. A gaus-sian wave packet reads in this case ψ ( x, t ) = (cid:90) dp √ π πδ ) / e − ( p − p )22 δ + ip x/ (cid:126) − icp (cid:82) t √ ε ( t (cid:48) ) dt (cid:48) / (cid:126) . (20)and to leading (non trivial) order in the expansion of (cid:112) ε ( t ) one finds (cid:104) x (cid:105) ψ ( t ) − (cid:104) x (cid:105) ψ (0) = c t , (21)(∆ x ) ψ ( t ) − (∆ x ) ψ (0) = c τ t . (22)8his random walk behavior leads to interesting effects in interference ex-periments. Consider a light wave packet impinging a plate pierced by twoslits, and then observed on a screen, producing an interference pattern. Sincewaves are detected at some fixed distance L from the plate, the (cid:126) stochas-tic nature can be effectively viewed as a change δt of the travel time, withvariance, from equation (22) δt = τ t τ L c , (23)with t = L/c the time mean value, neglecting the distance between the slitscompared to L . Assuming a plane wave with frequency ω , light intensity I say, at the point on the screen equidistant from the two slits is I ∝ (cid:12)(cid:12) e − iω ( t + δt ) + e − iω ( t + δt ) (cid:12)(cid:12) = 12 (1 + cos [ ω ( δt − δt )]) , (24)where the two contributions come from the two slits and δt , are their (un-correlated) time shift along the path from slits to the screen. In the standardcase the two waves show a constructive interference. Here, averaging over δt , , from equation (23) I ∝ (cid:18) (cid:18) − ω τ L c (cid:19)(cid:19) . (25)The interference term decays exponentially for large L , and asymptoticallyintensity behaves as the two waves were not interfering . This is a genuineeffect of Planck constant fuzziness, which destroys wave coherence on dis-tances ω τ L/ (4 c ) ≥
1. Reasoning in a similar way one finds that the wholeinterference pattern changes.Notice that, using equation (22), the effect can be cast in terms of afluctuating light speed with variance (∆ c ) (∆ c ) c = c τ L , (26)Of course, this holds for relativistic particles only, but not in general.The result of equation (25) could be tested in long–baseline interferomet-ric experiments, like Virgo or Ligo [24, 25], aimed at detecting gravitationalwaves from astrophysical sources. To this end, all noises, mechanical, seismicetc., are kept under an exquisite control, sensitivity being eventually limited9n the relevant frequency range by the irreducible shot noise contribution,due to photon number N (or wave packet phase) Poisson fluctuations inlight bunches, ∆ N = √ N . For a small τ , equation (25) leads to∆ II (cid:39) − ω τ Lc , (27)which assuming no detection of τ related effects, should be smaller than shotnoise (sn) ω τ Lc < (cid:18) ∆ II (cid:19) sn = ∆ NN = 1 √ N = (cid:114) hν ∆ νI , (28)with ν and ∆ ν the light frequency and bunch bandwidth, or τ < · − km L (cid:18) Hz ν (cid:19) / (cid:114) ∆ ν Hz (cid:114)
10 W
I . (29)We normalized to visible light frequency and status of the art values for ∆ ν , L and power I . This bound would translate into a lower limit for an effectiveenergy scale of (cid:126) dynamics Λ = (cid:126) /τ (cid:38) GeV.
For a one–dimensional harmonic oscillator with frequency ω and mass m theHamiltonian reads H = 12 m (1 + ε ) p + mω ε ) x , (30)i.e. a standard harmonic oscillator with time dependent mass and frequency, M = m/ (1 + ε ) and Ω = ω (1 + ε ), [26, 27], with M Ω = mω a constant.The Hamiltonian depends on time via an overall multiplicative factor, andthe commutator [ H ( t ) , H ( t (cid:48) )] vanishes, implying that Dyson series for timeevolution operator can be computed explicitly U ( t ) = exp (cid:18) − i (cid:126) (cid:90) t H ( t (cid:48) ) dt (cid:48) (cid:19) . (31)Defining, with standard normalization, creation operator a = (cid:114) mω (cid:126) x + i √ m (cid:126) ω p , (32)10e have H = (cid:126) ω (1 + ε ) (cid:18) a † a + 12 (cid:19) . (33)Notice that a and a † do not explicitly depend on time and obey (cid:2) a , a † (cid:3) = 1 . (34)The a equation of motion d a ( t ) dt = − iω (1 + (cid:15) ( t )) a ( t ) , (35)has the formal solution a ( t ) = a (0) e − i ωt (cid:88) n ( − iω ) n n ! (cid:90) t dt .... (cid:90) t dt n (cid:15) ( t ) ....(cid:15) ( t n ) . (36)Averaging over (cid:15) ( t ) probability distribution and computing n –point correla-tion functions in terms of two–point correlation `a la Wick a ( t ) = a (0) e − i ωt (cid:88) k ( − ω ) k k ! (2 k − τ t ) k = a (0) e − i ωt (cid:88) k ( − ω τ t ) k k k ! = a (0) e − i ωt e − ω τt/ . (37)Apart from standard oscillatory term, evolution is exponentially damped ontime–scales larger than the characteristic time 2( ω τ ) − .Consider now a coherent state | λ (cid:105) at time t = 0. With no loss of generalitywe take λ real. As discussed already, position/momentum measurements atsome particular time amounts to measure x and p . From equation (36)after averaging over ε distribution (cid:104) x (cid:105) λ ( t ) = (cid:114) (cid:126) mω λ cos( ωt ) e − ω τt/ , (cid:104) p (cid:105) λ ( t ) = −√ (cid:126) mω λ sin( ωt ) e − ω τt/ , (cid:104) x (cid:105) λ ( t ) = (cid:126) mω (cid:20)
12 + λ (cid:16) ωt ) e − ω τt (cid:17)(cid:21) , (cid:104) p (cid:105) λ ( t ) = m (cid:126) ω (cid:20)
12 + λ (cid:16) − cos(2 ωt ) e − ω τt (cid:17)(cid:21) , (38)11o that coherent states do not saturate the lower limit of Heisenberg relationas time flows, unless λ = 0,∆ x λ ∆ p λ = (cid:126) (cid:104) λ (cid:16) − e − ω τt (cid:17)(cid:105) . (39)This effect is in all similar to decoherence processes affecting the oscillatorphase but leaving its energy unperturbed. We show position mean value,squared mean value and uncertainty in Figure 1. The growing behaviour of∆ x λ can be appreciated provided the state remains in a coherent configu-ration for times tω ≥ ( τ ω ) − .These features could be constrained using optical cavities. Excited byan external coherent beam, they can store coherence properties of electro-magnetic field for long times, and consist in spatial confinement between twohighly reflecting surfaces of a well defined propagation mode. Commonlyused in laser/maser physics and in optical experiments where high spectralspatial purities are required (high resolution spectroscopy, interferometry,quantum optics, etc.), their confinement ability can be quantified, as for me-chanical oscillators, in terms of quality factor Q = ωt c , the ratio betweenenergy lost in a cycle to the energy stored in the cavity, with ω the frequencyand t c the cavity decay time. Using supermirrors, at optical frequency, valuesof Q ∼ are accessible [28].A single mode coherent state of an electromagnetic harmonic oscillatoris a minimum uncertainty states for any pair of orthogonal field quadratureoperators [29, 30], the analogue of position and momentum operators. Inphase space this state is represented by a two dimensional Gaussian dis-tribution with equal variances at all direction. To keep a stricter analogywith the harmonic oscillator discussed here, we define field quadratures as X = (cid:112) (cid:126) / (cid:0) a + a † (cid:1) and Y = i (cid:112) (cid:126) / (cid:0) a † − a (cid:1) . Measurements of the uncer-tainty for a given quadrature of an electromagnetic mode can be obtained bya homodyne detector.For a random (cid:126) , coherent configuration of radiation do not saturate thelower bound (cid:126) / X ∆ Y , which monotonically increases and is related to | λ | , the mean photon occupation number, see equation (39). Measurementsin resonant cavities however, are limited by t c . For t > t c the coherentelectromagnetic field escapes from cavity due to unavoidable couplings tothe external thermal bath, and the system evolves towards vacuum state λ = 0. To account for this effect we modify equation (39) by introducing an12
20 40 60 80 100 (cid:45) (cid:45) Ω t Figure 1:
The time evolution of position mean value (cid:104) x (cid:105) λ (solid), (cid:104) x (cid:105) λ (long-dashed) and squared uncertainty ∆ x λ (short-dashed) for a coherentstate with λ = 1 . Values are in units of appropriate powers of the lengthunit (cid:112) (cid:126) / ( mω ) . We have chosen an unrealistic large value ωτ = 0 . toemphasize the non standard time behavior with respect to the case of aconstant (cid:126) . λ ∆ X λ ∆ Y λ = (cid:126) (cid:104) λ e − t/t c (cid:16) − e − ω τt (cid:17)(cid:105) . (40)This approximation is satisfied if t c = Q/ω is much larger than τ , so that λ adiabatically decays on τ time scales. The r.h.s. of equation (40) grows forsmall times, reaches a maximum at t ∗ ∆ X λ ∆ Y λ (cid:12)(cid:12) t ∗ = (cid:126) (cid:34) λ Qωτ (cid:18)
22 +
Qωτ (cid:19) (2+
Qωτ ) / ( Qωτ ) (cid:35) , (41)and eventually decays towards the standard value (cid:126) /
2. For
Qωτ (cid:28) t ∗ (cid:39) t c / − Q τ / (cid:126) /
2, so is a wayto determine Planck constant, with some uncertainty, (cid:126) ± ∆ (cid:126) . If the timebehaviour of equation (40) is undetected, in particular the peak at t ∗ , thismeans that τ should be sufficiently small λ Qωτ (cid:18)
22 +
Qωτ (cid:19) (2+
Qωτ ) / ( Qωτ ) < ∆ (cid:126)(cid:126) , (42)To have an order of magnitude of this bound, we take λ = 1. Choosing Q = 10 , ω ≈ · Hz, [28], and a measurement uncertainty ∆ (cid:126) / (cid:126) ∼ τ < − s , (43)or, in terms of energy scale, Λ = (cid:126) /τ > GeV. For smaller ∆ (cid:126) , the boundson τ scales approximately as (∆ (cid:126) / (cid:126) ) / ( Qω ), see equation (42). We have considered the possibility that Planck constant is a randomly fluc-tuating quantity. These fluctuations are viewed as the manifestation at lowenergies of some fuzzy structure of spacetime and phase space at very highenergy scales
E >
Λ, and are found to change the standard results of quan-tum mechanics, such as the Heisenberg uncertainty relation and interferenceof massive or massless particle beams. The novel effects can be effectively14egarded as due to the coupling of a particular system, such as an harmonicoscillator or a free particle, with some dynamical field. Even if this field hasreached a stable background configuration, we expect its value to undergosmall quantum fluctuations, which introduce a noise in the time evolution ofthe system. In particular, we have studied the case in which this noise showsup in terms of a random Planck constant with a variance times typical cor-relation time parameter τ = σ ∆ t . Interestingly, the model can be stronglyconstrained by future experiments with long baseline interferometers or oncoherent light cavities, and adopting present status of the art parametersfor these kind of experiments, we found that τ could be bound in dedicatedmeasurements to a very tiny value, of order of 10 − − − s. These timevalues translate into energy scales Λ ∼ (cid:126) /τ larger than 10 − GeV, whichare well below the Planck mass scale, 10 GeV, yet above the energy rangewhich can be explored with present accelerators, like LHC at CERN, andpresumably also unaccessible to next generation accelerator programs.Of course, since Planck constant is ubiquitous, there is potentially a verylarge number of different dynamical systems and experimental approacheswhich may be exploited, and which were not covered here. Possibly, some ofthem may be even more powerful than those we discussed in constraining astochastic (cid:126) . To study them all would entail the reconsideration of the basicsof all quantum phenomena, an enterprise which is well beyond the scope ofthis paper.
Acknowledgments
F.L. and G.M. are supported by INFN, I.S.’s GEOSYMQFT and TASP,respectively. This article is based upon work from COST Action MP1405QSPACE, supported by COST (European Cooperation in Science and Tech-nology). F.L. is partially supported by CUR Generalitat de Catalunya un-der projects FPA2013-46570 and 2014 SGR 104, MDM-2014-0369 of ICCUB(Unidad de Excelencia ‘Maria de Maeztu’), and by UniNA and Compagniadi San Paolo under the grant Programma STAR 2013.
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