Maximal boost and energy of elementary particles as a manifestation of the limit of localizability of elementary quantum systems
aa r X i v : . [ h e p - ph ] M a r Maximal attainable boost and energy of elementary particles as a manifestation of thelimit of localizability of elementary quantum systems
George Japaridze
Department of Physics, Clark Atlanta University, Atlanta, Georgia, 30314, USA (Dated: April 15, 2019)I discuss an upper bound on the boost and the energy of elementary particles. The limit is derivedutilizing the core principle of relativistic quantum mechanics stating that there is a lower bound forlocalization of an elementary quantum system and the assumption that when the localization scalereaches the Planck length, elementary particles are removed from the S -matrix observables. Thelimits for the boost and energy, M Planck /m and M Planck c ≈ . · eV, are defined in terms offundamental constants and the mass of elementary particle and does not involve any dynamic scale.These bounds imply that the cosmic ray flux of any flavor may stretch up to energies of order 10 GeV and will cut off around this value.
I. INTRODUCTION
This letter presents the scenario establishing the ultimate upper bound on the Lorentz boost and energy of ele-mentary particles with a non-zero mass. The cosmic ray detectors measure higher and higher energies; recently theIceCube collaboration released data containing extra terrestrial neutrino events with E ν ∼ PeV (10 eV) [1]. Theregion of energies probed is steadily increasing: cosmic ray experiments as HiRes, Telescope Area, Auger report eventswith unprecedented energies of up to 10 eV [2].It is natural to ask whether there is no upper bound on Lorentz boosts/energies as it follows from the classicalrelativity. Accounting for the quantum feature of elementary particles and gravity may result in halting the increaseof boost and energy of elementary particles. Though no deviation from Lorentz invariance is observed, which impliesthat space-time symmetry is described by a non compact group and that boosts and energies can acquire arbitrarilylarge values, there have been numerous attempts to modify the untamed growth of the boost and energy, see e.g. [3],[4]. Present note addresses this question.The suggested mechanism for the maximal attainable boost and the energy of elementary particles is based on thefundamental concept of the limit of localizability of elementary quantum system in relativistic quantum mechanics[5]. This limit is combined with the conjecture that when the localization scale of the elementary particle reachesthe critical value, defined by the maximum between the Planck length and Schwarzschild radius, it can not beobserved as a quantum of an asymptotic in, out filed and thus elementary particle is not S -matrix observable anymore. For brevity, let us call this assumption the quantum hoop conjecture, since it can be viewed as a quantumcounterpart to the well-known hoop conjecture, which suggests that when the system is localized inside the volumeof size of classical gravitational length, Schwarzschild radius, system undergoes gravitational collapse [6]. Combiningthe quantum hoop conjecture with the limit of localizability of the elementary particle leads to the upper bound onLorentz boost and energy of elementary particles. Assuming that the mass of the elementary quantum system m isless than Planck mass M P , this model predicts value of maximal attainable boost Γ max = M P /m and of maximalattainable energy E max = M P c . When the boost reaches Γ max , the contracted localization scale reaches the criticalvalue and elementary particle is not the S -matrix observable any more. The limits on boost and energy depend onlyon the fundamental constants and the mass and therefore can be considered as the ultimate bounds for the boost andenergy of an elementary particle. II. MAXIMAL ATTAINABLE BOOST AND MAXIMAL ATTAINABLE ENERGY
To begin with, let us recall fundamental units. In this letter Planck mass M P is defined as the value of a mass pa-rameter at which the Compton wave length of a system with the mass m , λ q ( m ) = ~ /mc , is equal to the Schwarzschildradius, λ gr ( m ) = 2 Gm/c (throughout a four dimensional space-time, no extra dimensions, is considered) λ q ( M P ) ≡ ~ M P c = λ gr ( M P ) ≡ GM P c , (1)where ~ is the Planck’s constant, G is the gravitational constant and c is the speed of light. This results in M P = r ~ c G ≃ . · − kg ≃ . · eV /c . (2)Planck length λ P is defined as a Compton wave length of system with the Planck mass, evidently coinciding with theSchwarzschild radius of a system with the Planck mass. From Eq. (1) it follows λ P ≡ ~ M P c = 2 GM P c = r G ~ c ≃ . · − m . (3)These values differ by a factor of √ λ q ( m ) and λ gr ( m ): λ q ( m ) λ gr ( m ) = λ . (4)We say that system is localized inside volume with a linear size L when the probability of finding the system insidethe ball of linear size L is 1. Observation is described by exchanging quanta of in, out fields between the localizedsystem and the outside observer, separated from the system by distance much greater than L .In classical physics the feature of rearranging the physical degrees of freedom when the localization scale reachesits critical value is formulated by the hoop conjecture, which states that a black hole forms whenever the amountof energy mc is compacted inside a region that in no direction extends outside a circle of circumference (roughly)equal to 2 πλ gr ( m ) [6]. In other words, no signal from the ball radius a can reach external observer when a < λ gr ( m ).Originally, hoop conjecture was put forward for astrophysical bodies, macroscopic objects which can be reasonablydescribed by classical theory of gravity [6].To establish the maximal attainable boost we introduce and utilize a quantum counterpart to the hoop conjecture,which we call throughout the quantum hoop conjecture. Quantum hoop conjecture states that whenever the localiza-tion scale of elementary quantum system approaches the Planck length λ P , the system is not observed as elementaryparticle, as the quantum of asymptotic in, out fields. The idea that below the Planck length the notion of space-timeand length ceases to exists is not new, see [7]-[10]; we just express it in terms of elementary particles, quanta of in,out fields.In the solution of the Heisenberg equations of motion of quantum field theory [11] h α | ˆΨ | β i = h α |√ Z ˆΨ in, out | β i + h α | ˆ R | β i , (5)the term h α |√ Z ˆΨ in, out | β i describes an incoming/outgoing free particle, the quantum of asymptotic field and ˆ R standsfor the rest of the solution, which in case of a classical source j ( x ) is ˆ R ( x ) = R d y j ( y ) △ ret, adv ( x − y ). In terms ofquantum field theory, the quantum hoop conjecture is translated into the statement that when the localization scalereaches the Planck length, the matrix element h α |√ Z ˆΨ in, out | β i vanishes,lim Γ → Γ max h α (Γ) |√ Z ˆΨ in, out | β (Γ) i = 0 . (6)Note that the flux of the incoming/outgoing particles and consequently, the S -matrix elements are defined by ˆΨ in, out ;if matrix element of ˆΨ in, out vanishes, corresponding S -matrix element vanishes as well [11]. According to quantumhoop conjecture the S -matrix element, corresponding to scattering of particle with the boost exceeding the maximalattainable boost, Γ( m ) > Γ max ( m ), should vanish. The simplest way to realize this is to modify the expression forthe operator of in field:ˆΨ in = X k a in ( k ) e − ikx + a † in ( k ) e ikx k → X k e a in ( k ) e − ikx + e a † in ( k ) e ikx k , (7)where k = √ k + m , e a in ( k ) ≡ a in ( k ) Θ( E max − k ), a in ( k ) , a † in ( p ) are the annihilation and creation operatorsof elementary particle (quantum of in field) satisfying the usual commutator relation [ a in ( k ) , a † in ( p )] = δ ( k − p ),Θ( E max − k ) is the Heaviside step function and E max is the energy of particle mass m boosted with Γ max ( m ).From (7) it follows that the matrix element of asymptotic field ˆΨ in ( x ) between the vacuum and the one particle state | k i = a † in ( k ) | i is h | ˆΨ in ( x ) | k i ∼ √ Z e ikx Θ( E max − k ) , (8)which vanishes when k ≥ E max . As far as k ≤ E max , i.e. when Γ ≤ Γ max ( m ), the vacuum-one particle matrixelement of an in, out field exists, i.e. a particle can be observed as a quantum of ˆΨ in, out and the flux and the S -matrixexist. Phenomenological condition (7) has to be derived from future theory of quantum gravity at a regime whenan elementary particle having energy of the order of maximal attainable energy E max in some reference frame wouldpresumably interact with the degrees of freedom of the unknown underlying theory of quantum gravity.Eqs. (7) and (8) are written in a preferred reference frame. As any other scenario speaking of the maximum attainableboost and energy and thus postulating the existence of preferred reference frame, the present model also requires theexistence of the reference frame to which the maximum boost is compared. In this work, motivated by the cosmologicalconsiderations, for such a reference frame the cosmic rest frame is chosen. This is the reference frame where the cosmicmicrowave background (CMB) is at rest, and its temperature is homogeneous and is 2.73 K . The Earth rest referenceframe moves relative to CMB with peculiar velocity ∼
370 km/s ≈ . c , as is follows from CMB dipole anisotropymeasurements [12]. Because of the low value of Γ Earth − CMB ∼
1, with a good approximation the Earth rest referenceframe can be identified with the preferred reference frame, the one where as Γ → Γ max , no elementary particle canbe observed.Maximal attainable boost for the elementary particle with mass m , Γ m ax ( m ), is derived from the requirement thatthe Lorentz-contracted localization scale is still larger than either Schwarzschild radius or Planck length and therefore,elementary particle is still observable as a quantum of in, out fields. From the classical special relativity it followsthat when boosted with Γ, the localization scale is spatially contracted and becomes L = L / Γ. This relation holds ina relativistic quantum mechanics as well - position operator in relativistic quantum mechanics acquires overall factor1 / Γ when the reference frame is boosted with Γ [13]. From the requirement that the Lorentz-contracted size of thesystem is still larger than the threshold value L thr , given by either gravitational radius or Planck scale L = L Γ ≥ L critical ≡ max( λ P , λ gr ( m )) , (9)it follows that Γ ≤ Γ max ( m ) = L L critical = L λ q ( m ) λ q ( m ) L critical . (10)We utilize the well-known fact that in the framework of relativistic quantum mechanics, a particle at rest can notbe localized with accuracy better than its Compton wave length λ q ( m ), i.e. min( L ) = λ q ( m ) [5], [11]. Then theinequality (10) turns into Γ ≤ Γ max ( m ) = λ q ( m ) L thr . (11)We assume that the mass of the elementary quantum system is bounded from above, m ≤ M P . This constraint isrealized as an inequality ordering spatial scales as follows λ gr ( m ) ≤ λ P ≤ λ q ( m ) , (12)and consequently, L critical = max( λ P , λ gr ( m )) = λ P . Therefore, the maximal attainable boost for a particle with mass m ≤ M P is Γ max ( m ) = λ q ( m ) λ P = λ P λ gr ( m ) = M P m . (13)The hoop conjecture which may serve as a physical insight for the rearrangement of the space of physical degreesof freedom, appeals to the Schwarzschild radius λ gr , as the minimal localization length. Connection between Plancklength and gravity effects was established long ago, and it states that in presence of gravity it is impossible to measurethe position of a particle with error less than λ P [7], [8], [9]. We have assumed that similar to the hoop conjecture,suggesting that object collapses into black hole when localized in area with size less than classical gravitational radius,2 Gm/c , elementary particle is removed from S -matrix observables when localized in volume with size less than λ P ,the latter determined by both G and ~ . When the mass of elementary particle approaches M P , as seen from Eq. (4),the Planck length and the Schwarzschild radius coincide λ P = λ gr ( M P ) (14)i.e. the quantum and the classical hoop conjectures merge into the same supposition.According to Eq. (13), the value of maximal attainable boost varies from particle to particle, e.g. Γ max (proton) = M P /m proton ≈ . · , Γ max (electron) ≈ . · . Not so for the maximal attainable energy: E max is the same forall particles and is given by the Planck energy E P ≡ M P c : E max = Γ max ( m ) mc = M P c ≈ . · eV (15)It is important to note that the above results are obtained and are valid for the systems with m ≤ M P which weconsider as elementary particles, i.e. quanta of in, out fields.The disappearance of in, out fields from the solution of Heisenberg equations of motion, in other words, removingelementary particles from the S -matrix observables when Γ > Γ max , seems to violate the S -matrix unitarity. Indeed,asymptotic completeness, according to which the in and out states span the same Hilbert space, which is also assumedto agree with the Hilbert space of interacting theory [11] H in = H out = H interacting (16)is not satisfied. Condition of asymptotic completeness is not trivial already in the framework of standard quantumfield theory: if particles can form bound states, the structure of space of states is modified, and the S -matrix unitarityis restored only after bound states are accounted for in the unitarity condition [11]. Accounting for gravity bringsin another reasoning for the violation of the S -matrix unitarity. It has previously been observed that in and out states, which are related by unitary transformation, can not be be defined in the presence of an arbitrary metric [14].Applying the quantum hoop conjecture to elementary particles drives this observation to the extreme, stating that assoon as the localization region becomes smaller than λ P , the space of physical degrees of freedom is rearranged andelementary particles, quanta of in, out fields are not observables any more. In this case, the unitarity condition hasto be formulated not in terms of elementary particles, but in terms of new physical degrees of freedom of quantumgravity, task which is beyond the scope of this letter.As for the predictions of a suggested scenario, the only clear one is the cut-off of beam of particles/flux of cosmicrays at limiting value E P ∼ GeV, energy, which is not accessible by modern accelerators and the cosmic raysobservatories. Up to E P , i.e. up to Γ max when the localization region is still larger than the Planck length, physicaldegrees of freedom, observables, are elementary particles. When E ≥ E P , elementary particles are removed fromobservables (in full analogy of collapsing system into a black hole according to a hoop conjecture); thus cosmic rayflux should vanish when E → GeV. As mentioned above, maximum attainable boost varies from particle toparticle and is M P /m , but maximum attainable energy for any type of elementary particle is the same E P . Thesebounds are “kinematical” in a sense that no dynamic scale related with any particular interaction is involved inestablishing the bound. Of course, some concrete conditions may alter the maximal observed energy. e.g the well-known GZK limit on the energy of cosmic rays from distant sources, caused by the existence of the omnipresenttarget - cosmic microwave background [15]. However the presented bounds are ultimate, derived from the quantumhoop conjecture and the basic principles of relativistic theory of quantum systems; in other words statement is thatindependently of dynamics the kinematic parameters describing elementary particles can not exceed these bounds.This is in contrast with the scenario for the boost and energy cut-off which was recently put forward [16]. In [16]it is suggested that the maximum attainable boost and maximum attainable energy for the neutrino areΓ max ν = M P M weak ; E max ν = m ν M P M weak (17)where M P is a Planck scale and M weak is a scale of weak interactions ( ∼
100 GeV). The main point of work [16]is that the upper bound on energy is defined by weak scale; it follows from (17) that neutrino spectrum cuts off atenergies ∼ few PeV. Our prediction is Γ max ν = M P /m ν , i.e. much higher than one from Eq. (17). Regard neutrinoenergies, as an example, we quote the estimate from analysis of energetics of gamma-ray bursts - maximum neutrinoenergies may reach 10 − · eV [17]. This value does not contradict Eq. (15) - E max ν = 8 . · eV, and exceedsthe upper bound of a few PeV on neutrino energies suggested in [16]. III. DISCUSSION
We have combined: the lower limit of localizability of an elementary quantum system, Lorentz contraction, and thequantum hoop conjecture to derive the upper bound on Lorentz boost and energy for massive particles. When theupper bound is reached, elementary particles - the quanta of asymptotic in, out fields disappear from a spectrum of S -matrix observables, presumably replaced by the local physical degrees of freedom of quantum gravity. In derivationof this upper bound, we used the property of Lorentz contraction, i.e. validity of the theory of relativity applied toelementary particles up to Γ max is assumed. Though the limiting values of boost and energy are Lorentz invariant,assigning the physical meaning to Γ max and E max implies the existence of a preferred reference frame. In this workit is postulated that the preferred reference frame is the CMB rest reference frame. Since the Earth rest referenceframe almost coincides with the CMB rest frame, we predict that in the Earth rest reference frame the cosmic rayspectrum continues all the way till the Planck energy ∼ GeV, where the cut-off of the flux occurs.Lastly, let us note that as it is assumed that the boost is bounded from above, i.e. for a massive particle the(classical) limit v = c can not be reached, the classical hoop conjecture remains intact. This is because the metricremains of that of boosted Schwarzschild space-time and can not be approximated by a plane impulsive gravitationalwave as it happens for states moving with c [18].I thank V.A. Petrov for fruitful discussions and pointing out to me works regarding boosted gravitational fields ofmassless particles [18]. [1] M. G. Aartsen et al. [IceCube Collaboration], Phys. Rev. Lett. , , 101101, (2014); arXiv:1405.5303.[2] P. Blasi, Plenary talk at at the 33rd International Cosmic Ray Conference, 2013, Rio de Janeiro, Brazil; arXiv:1312.1590[astro-ph.HE].[3] D. Mattingly, Living Rev. Rel. , , 5, (2005); arXiv:gr-qc/0502097; S. Liberati, Class. Quantum Grav. , , 133001, (2013).[4] A. Kostelecky and N. Russell, Rev. Mod. Phys. , , 2011.[5] T.D. Newton and E.P. Wigner, Rev. Mod. Phys. , , 400, (1949).[6] K.S. Thorne, in J.R. Klauder, Magic Without Magic , Freeman, S. Francisco, 231, (1972).[7] C. A. Mead,
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