arXiv:1907.10503v1 [hep-th] 24 Jul 2019 hnuiaiyi auae o rcse ihmomen- with processes for transfer saturated tum is unitarity when rte nafr htdrcl atrsteconnection the captures directly that unitarity, form with a in written rtsprubtv ntrt ttesae1 scale sat- the theory at underlying unitarity the perturbative when urates exactly entropy the on on sdrcl idto mass tied of directly historical is the bound references. for of [3] list see complete to a theory, expected and consistent is development any and gravity in of hold independent is bound This oti nrp.Ti ieo huh a aesm in- possibility some a have [4] can in thought example, For of applied line consequences. is This teresting it entropy. once this induces impli- cor- to bound of the Bekenstein kind color, assigned what the is, to be cations question must respect The it entropy. with Therefore, responding states spin. ap- degenerate and can sym- flavor energy internal many given and in a space-time of pear particle with elementary theories an metries In it satisfy well. to have as states one-particle objects, macroscopic such objects instantons. of and list baryons a solitons, theory. for as explicitly the checked of been renormalizability has the This of unitar- independent and is bound Bekenstein ity the of saturation the tween httemxmlaon fifrainsoe nasys- us a in tells stored energy [2], information of also of tem amount see maximal [1], the bound that entropy Bekenstein The vlae tteeeg cl 1 scale energy the at evaluated where eety[] twspitdotta h Bekenstein the that out pointed was it [4], Recently vntog h on smil ple oconstrain to applied mainly is bound the though Even α .NNGAIAINLBOUND NON-GRAVITATIONAL I. M ersnsadmninesculn constant coupling dimensionless a represents n size and lmnayprilso ag spin large of particles Elementary r ujc oteBknti nrp on.W bev that observe We bound. entropy Bekenstein the to subject are nesoda h lc oeseisbud ..a nue q high induced a an of i.e. emission bound, the species by hole black M violated the is generate as radiation additionally understood gravity hole stringiness le of black of associated inclusion glimpse the a The new as a gravity. regarded of acquires be can particle This the wavelength. unless violated is ∼ M P 1 1 2 / /R nttt o hoeia hsc,UrctUiest,35 University, Utrecht Physics, Theoretical for Institute √ n radius and nttt fPyis nvriyo mtra,19 HAmst XH 1098 Amsterdam, of University Physics, of Institute 4 s orsodnl,tebudcnbe can bound the Correspondingly, . h w onscryqaiaieydffrn information. different qualitatively carry bounds two The . a-lnkIsiu ¨rPyi,Forne ig6 80805 6, F¨ohringer Ring f¨ur Physik, Max-Planck-Institut S R max S auae h eesenbound Bekenstein the saturates max 5 2 = unitarity e okUiest,76Boda,NwYr,N 00,USA 10003, NY York, New Broadway, 726 University, York New etrfrCsooyadPril hsc,Dprmn fPh of Department Physics, Particle and Cosmology for Center = 3 nrp osrit nHg pnParticles Spin High on Constraints Entropy R M . πMR rodSmefl etr Ludwig-Maximilians-Universit Center, Sommerfeld Arnold α 1 sgvnby given is /R , h orlto be- correlation The . hrsesrß 7 03 ¨nhn Germany, M¨unchen, 80333 37, Theresienstraße aey nobject an Namely, . aksDierigl Markus /R s htis, That . tr unu nomto ndgnrt ttsadtherefo and states degenerate in information quantum store Dtd uy2,2019) 25, July (Dated: (1) (2) 1 , 2 n i Dvali Gia and oteU-hoyaoeΛ ic hsi h aefrall for case the is belong this and Since prop- sense Λ. the usual above lose the UV-theory in necessarily the elementary spin to arbitrarily being high an of of That erty have particles bound. cannot Bekenstein the the particle is, violating the without spin case high a such in fteparameters the of Comp- the than larger be to wavelength. has length-scale ton and must This Bekenstein spin spin with length-scale. the grow high must associated of very new violation of a conclude particles a acquire will elementary prevent We the to bound, spin. order high in of that particles elementary to asymptotic, of particles. presence a colored the the as in of bound understood entropy violation Bekenstein be would-be against can large mechanism a confinement prevention with colors, theories QCD-like of for number that out, pointed was na ffcieter elblwti uo.I particu- In stay cutoff. shall assume this must we below we well discussion lar, theory our effective In an in be elementary. cannot as they and treated macroscopic becomes radius tational eodtePac scale way Planck masses the with grav- beyond particles an in elementary have example, of cannot For notion one exist. ity usual not or the does gravity which particle by elementary above induced physics, is other which some Λ, UV-cutoff effective an smttcSmti ttsi omo neeetr par- mass elementary an of of ticle form in states S-matrix asymptotic relevant their with of and connection reasonability application physical in uncertainties. by our control guided energy, be parametric localized will finite exactly bound of no the are and objects there [3] see Since quantum authors, several therein. by references investigation of subject a ewudlk opitotta h rcs definition precise the that out point to like would We reasoning similar a apply shall we note, present the In osdraPicreivratter htalw for allows that Poincar´e-invariant theory a Consider e cl twihtetemlt of thermality the which at scale new a s ghsaedffrn rmisCompton its from different ngth-scale oevr hsbudi independent is bound this Moreover, . 4C teh,TeNetherlands, The Utrecht, CC 84 atmgaiyctffsaegvnby given cutoff-scale gravity uantum pnpril.Ti on a be can bound This particle. spin o ucetylarge sufficiently for 3 , M 4 ¨ nhn emn and M¨unchen, Germany , 5 ra,TeNetherlands, The erdam, n spin and M and M M ≪ ¨ at, R ysics, P s .Orga st hwthat show to is goal Our Λ. neigtebudhsbeen has bound the entering o uhojcstegravi- the objects such For . h hoymyinclude may theory The . s h bound the re 2 the known consistent cases with particles of high spin – Additional corrections due to cutoff-dependent terms are such as, e.g., string theory or QCD – one may argue that more suppressed and hence are negligible. The first or- we are pushing at an open door. Nevertheless, gaining a der correction indicated above can be derived as fol- novel understanding of the subject from the point of view lows. Since we consider an expectation value in a one- of the Bekenstein entropy bound can be of fundamental particle state, only the free part of the Hamiltonian ˆ † importance and is worth to point out. H = k,ǫ ωk,ǫ aˆk,ǫaˆk,ǫ contributes. By Poincar´esym- In order to make our arguments clean we assume that metryP and the fact that we are way below the cutoff Λ, 2 2 the particle in question is stable. However, the argument we have ωk,ǫ √M + k . Taking into account that is equally applicable to unstable particles provided their the momenta effectively≃ contributing to the wave-packet life-time is larger than the relevant time-scales in our satisfy k . 1/R, gives (4). thought experiment. Consequently,| | the Bekenstein bound (1) must be satis- fied with the corresponding time and energy uncertainties Our assumptions imply that a one-particle state 1 k , taken into account. However, the actual entropy of the | i ,ǫ characterized by a definite momentum k and spin- state (3) due to the spin-degeneracy is S = ln(2s + 1). polarization ǫ, represents a well-defined state in the Thus, we arrive at the following bound Hilbert space. This state is obtained by an action of a corresponding creation operator on the S-matrix vac- ln(2s + 1) 6 2πMR . (5) † uum, 1 k =ˆak 0 . If so, we can consistently consider If the particle is a legitimate elementary degree of free- | i ,ǫ ,ǫ | i their superpositions and form a Gaussian wave-packet lo- dom, the relevant quantum length-scale associated with calized within some radius R. it is its Compton wavelength 1/M. In such a case, the Due to the possibly complicated underlying structure right hand side of the above equation is a matter of of the theory, the mass parameter M might of course de- choice. For arbitrary M we can choose the width of the pend on s as well as on an arbitrary number of coupling wave-packet R> 1/M at will. Then, for sufficiently high constants via renormalization. However, we are not mak- s, it is impossible to satisfy the bound. ing any assumptions about this dependence. At the end We can make the fact that the bound is violated exact of the day our ignorance is all summed up in a single by taking the limit effective physical quantity M. Notice, that we are not trying to confine the particles s , 2πMR = ln(2s + 1) . (6) → ∞ p in a cavity and/or to stabilize the state using some ex- In this limit the velocity of the spread of the wave-packet ternal device. Such a situation was analysed in details vanishes and the wave-packet becomes an energy eigen- in e.g. [5]. Of course, the process of localization would state. There is no ambiguity about the definition of en- interfere with the Bekenstein bound since the device nec- ergy and a localization width and the Bekenstein bound essarily contributes to the total energy of the system. can be evaluated exactly. We have, Furthermore, it is impossible to estimate this contribu- S tion without knowing the interaction strength and other = ln(2s + 1) , (7) parameters. However, this does not affect our argument Smax p → ∞ since our approach is different. Our starting point is, and the Bekenstein bound is violated. that the elementary particle of high spin s represents a legitimate asymptotic S-matrix state. This implies that Of course, for finite parameters the values of s that vi- a one-particle momentum state exists and it directly fol- olate the bound are exponentially large and hence are too lows that the Hilbert space must contain arbitrary super- large for being of any relevance for the particles of the positions thereof. We then choose the state vector to rep- Standard Model. However, we arrive at an interesting resent a Gaussian wave-packet of any spin-polarization ǫ conclusion. An object of sufficiently high spin cannot be at some initial time t =0 an elementary particle in the following sense. In order to 2 2 d −R k satisfy the Bekenstein bound on information storage the 1ǫ = C d k e 1 k , (3) | i Z | i ,ǫ object must acquire a notion of a new length-scale which grows with spin. Notice that this is precisely what is with normalization constant C chosen such that happening in known examples with arbitrarily high-spin 1 1 = 1. Being a legitimate state in the Hilbert ǫ ǫ resonances such as QCD and string theory. In these the- space,h | i we require that (3) obeys the Bekenstein entropy ories, the high-spin states represent highly excited string bound (1). vibrations and the role of the new length-scale is played by the string length. To have parametric control, we need to take R some- what larger than the Compton wavelength 1/M since the −1 wave-packet spreads with the velocity v = (MR) . The II. ROLE OF INTERACTIONS energy of the wave-packet is given by the expectation value of the Hamiltonian, In the above reasoning we have at no point used infor- 2 1ǫ Hˆ 1ǫ = M + (1/(MR )) . (4) mation about the strength of the interaction. Indeed, h | | i O 3 the only assumption we have made is that the particle in thermal. One of the signals that the size of a black hole question is a legitimate asymptotic S-matrix state. This reached the scale L∗ is that its Hawking evaporation can- may create possible confusion, since all we have said must not maintain its thermality. The measure of thermality apply to a free theory. It would be an extremely powerful is given by a parameter T/T˙ 2, where T is the Hawking statement if we could rule out a free theory of a high-spin temperature. It is easy to see that in an (approximately) particle, based on violation of the Bekenstein bound (1). thermal regime and for the emission of a single particle Unfortunately, we cannot obtain such a strong statement of spin s 1 this parameter has the following form, for the following reason. As already pointed out in [4], ≫ the information content stored in a quantum state of a T˙ 1 2 (2s + 1) , (9) non-interacting theory is sterile and therefore meaning- T ∼ SBH less. In the present case, in order to read out the informa- where SBH is the black hole entropy [7]. Hence, the tion stored in a state of a high-spin particle, we need a parameter becomes of order one once the black hole tem- probe that is able to distinguish among the different spin- perature reaches a critical value T∗ MP /√s and cor- ∼ states. Thus, the theory must be interacting, since it only respondingly the black hole reaches the size L∗ √sLP . ∼ makes sense to constrain the amount of information that Therefore, one obtains the following bound on the grav- can be, at least in principle, decoded. This simple fact itational cutoff and the spin of a particle gives a hint of why the Bekenstein bound must be tied to unitarity as argued in [4]. Indeed, the unitarity is L∗ & √sLP . (10) directly sensitive to the strength of the interaction and One can straightforwardly generalize this to the presence thus to the speed by which the information stored in the of many massive particle species, for which √s √N = object can be decoded. Therefore, when we apply the → Bekenstein bound to a high-spin particle we must keep j (2sj + 1), where j runs over all massive species of q in mind that some minimal strength of a spin-sensitive spinPsj . The massless fields must also be included with interaction that allows for a read-out of the information the appropriate degeneracies. must be assumed. In a free theory this is not the case. Furthermore, since the particle carries spin the remain- The fact that in a free theory the true information bound ing black hole will in general carry angular momentum can never be violated is clear by the reformulation of the after the Hawking radiation process involving the spin s bound given in (2), which is more convenient in such a particle. This modifies the evaporation. In general, the case. Since for a free theory α = 0 the bound (2) is never maximal angular momentum of a black hole is obtained violated. for a extremal Kerr black hole, at

2 2 Jmax M /MP . (11) III. COMPARING WITH GRAVITY ∼ Therefore, the smallest black hole that can accommodate In the discussion above no assumption was made about an angular momentum of the size of the particle spin s the presence of gravity in the theory. The inclusion of is of the size gravity requires a separate detailed investigation that 2 max √ will not be given here. However, it is useful to compare R M/MP J /MP = sLP , (12) ∼ ∼ p the non-gravitational bound to one immediate constraint which perfectly coincides with the quantum gravity due to gravity, which we shall derive next. For that, it cut-off scale (10) derived above. This means that is enough to remember that different spin-polarizations L does not only indicate a scale at which the ther- from the point of view of black hole physics represent dif- ∗ mal behavior of the black hole evaporation breaks ferent species and are therefore subject to the black hole down, but also the scale beyond which an evaporation species bound [6]. According to this constraint, the num- process involving the spin s particle leads to a naked sin- ber N of elementary particle species of the low energy gularity within the classical gravitational approximation. effective theory and the quantum gravity cutoff length- scale L are related as, ∗ Finally, it is useful to compare the two bounds above. Although the intrinsically gravitational bound (10) L∗ & √NLP , (8) seems much more stringent than the non-gravitational where LP =1/MP is the Planck length. Notice that in N one (5), it carries qualitatively different information. To all the polarization-eigenstates count separately, as they see this, let us discuss some explicit numbers. Using the −1 all contribute equally into the black hole evaporation. current experimental constraint L∗ . TeV , the bound A simple argument for obtaining an analogous bound (10) implies that a single elementary particle of mass 32 on the spin is to consider the thermal evaporation process M 1/L∗ can have a spin as high as s 10 . At in a theory of Einstein gravity with the quantum grav- the≪ same time, such an elementary particle∼ is excluded ity cutoff-scale L∗, see also [6]. If the black hole radius by the bound (5), which demands that an object with is much larger than L∗, the Hawking radiation is nearly such a large spin cannot be an elementary particle in 4 the usual sense and must acquire a notion of a new ACKNOWLEDGEMENTS length-scale. This is necessary in order to make the lo- calization much more energetically expensive than in the We thank Cesar Gomez, Dieter L¨ust, Miguel Mon- case of a Gaussian wave-packet of an elementary particle. tero, Oriol Pujolas, Wilke Van Der Schee, John Stout, Stefan Vandoren, and Nico Wintergerst for discussions. M.D.’s work is part of the D-ITP consortium, a pro- gram of the Netherlands Organisation for Scientific Re- search (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). The work of G.D. was supported in part by the Humboldt Foundation under Humboldt Professorship Award, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) under Germany’s Excellence Strategy - EXC- 2111 - 390814868, and Germany’s Excellence Strategy under Excellence Cluster Origins.

[1] J. D. Bekenstein, Universal upper bound on the entropy-to- G. Dvali, Unitarity Entropy Bound: Solitons and Instan- energy ratio for bounded systems, Phys. Rev. D 23 (1981) tons, arXiv:1907.07332 [hep-th]. no. 2, 287–298. [5] R. Bousso, Bound states and the Bekenstein bound, JHEP [2] H. J. Bremermann, Quantum noise and information, in 0402 (2004) 025, hep-th/0310148. Proceedings of the Fifth Berkeley Symposium on Mathe- [6] G. Dvali, Black Holes and Large N Species Solution to matical Statistics and Probability 4 (1967), 15–20. the Hierarchy Problem, Fortschr. Phys. 58 (2010), 528, [3] D. N. Page, The Bekenstein Bound, arXiv:1804.10623 arXiv:0706.2050 [hep-th]. [hep-th]. G. Dvali and M. Redi, Black Hole Bound on the Number R. M. Wald, Jacob Bekenstein and the Develop- of Species and Quantum Gravity at LHC, Phys. Rev. D77 ment of Black Hole Thermodynamics, arXiv:1805.02302 (2008), 045027, arXiv:0710.4344 [hep-th]. [physics.hist-ph]. G. Dvali and C. Gomez, Quantum Information and Grav- R. Bousso, Black hole entropy and the Bekenstein bound, ity Cutoff in Theories with Species, Phys. Lett. B674 arXiv:1810.01880 [hep-th]. (2009), 303, arXiv:0812.1940 [hep-th]. [4] G. Dvali, Area Law Saturation of Entropy Bound [7] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D from Perturbative Unitarity in Renormalizable Theories, 7 (1973) no. 8, 2333–2346. arXiv:1906.03530 [hep-th].