---

String Theory Modifies Quantum Mechanics

arXiv:hep-th/9207103v2 29 Jul 1992CERN-TH.6595/92ACT-17/92CTP-TAMU-58/92String Theory Modifies Quantum MechanicsJohn Ellis, N.E. Mavromatos and D.V.

Nanopoulos†Theory Division, CERN, CH-1211, Geneva 23, SwitzerlandAbstractWe argue that the light particles in string theory obey an effective quantum mechanicsmodified by the inclusion of a quantum-gravitational friction term, induced by unavoidablecouplings to unobserved massive string states in the space-time foam. This term is relatedto the W-symmetries that couple light particles to massive solitonic string states in blackhole backgrounds, and has a formal similarity to simple models of environmental quantumfriction.

It increases apparent entropy, and may induce the wave functions of macroscopicsystems to collapse.CERN-TH.6595/92ACT-17/92CTP-TAMU-58/92July 1992† Permanent address : Center for Theoretical Physics, Dept. of Physics, Texas A &M University, College Station, TX 77843-4242, USA, andAstroparticle Physics Group, Houston Advanced Research Center (HARC), TheWoodlands, TX 77381, USA.

1IntroductionThe discoveries of quantum mechanics and general relativity have caused two ofthe greatest revolutions in twentieth-century physics, and their reconciliation re-mains one of its most important pieces of unfinished business. One could expectthat this reconciliation would entail a modification of one or both of these very suc-cessful theories, and hope that it would cast light on the transition between classicaland quantum physics.

Indeed, the only candidate for a consistent quantum theory ofgravity is string theory [1], which is essentially non-local and hence modifies generalrelativity at short distances.It has also been suggested that the usual formula-tion of quantum mechanics and quantum field theory might require modificationsin a consistent quantum theory of gravity. Specifically, studies of field theory intopologically non-trivial space-times such as black hole backgrounds have indicatedthat information loss across an event horizon requires the introduction of mixedquantum-mechanical states, and the possibility, forbidden by conventional quantummechanics or S-matrix theory, that pure states may evolve into mixed states.It has been proposed by Hawking [2] that a full quantum theory of gravity mightbe formulated only in terms of density matrices describing in general mixed states,and that the transitions between initial and final density matrices might not befactorizable as products of S-matrix elements and their hermitian conjugates:ρout = /Sρin:/S ̸= S†S(1)Evidence for this suggestion was inferred from studies of topologically non-trivialsolutions of Einstein’s equations in Euclidean space-times.Two of us (J.E.

andD.V.N.) then suggested together with J.S.

Hagelin and M. Srednicki [3] that theevolution of quantum-mechanical systems over time-scales that are long comparedwith the Planck time should be described by a modified Liouville equation:∂tρ = i[ρ, H] + /δHρ(2)A modification of the Liouville equation of the type (2) is characteristic of openquantum-mechanical systems [4], and represents in our interpretation the intrinsiccoupling of a microscopic system to space-time foam.We derived upper bounds of the order of 1GeV/MP lanck on the magnitude of ma-trix elements of the /δH term in hadrons from the consistency with conventionalquantum mechanics of measurements of the K0 −K0 system [3, 5] and long-baselineneutron interferometry [3]. Subsequently, the same two of us together with S. Mo-hanty [6] demonstrated that the non-quantum-mechanical effects of the /δH would beenhanced in macroscopic systems such as SQUIDs, and could lead to the collapse ofthe wave function of a macroscopic object.

An operationally similar modification ofthe Liouville equation was proposed independently on completely phenomenologicalgrounds by Ghirardi, Rimini and Weber [7], and the required values of their model

parameters were entirely consistent with our upper bounds and order-of-magnitudeestimates.The study of topologically non-trivial space-times, event horizons and singularitiesin string theory was opened up by Witten’s realization that suitable Wick rotationsof a cosmological string theory could be interpreted as black holes in Minkowski orEuclidean space. We then embarked on a series of studies of quantum coherencein such stringy black hole backgrounds, with the explicit motivation of checkingthe conjectured breakdowns (1) and (2) of conventional quantum field theory andquantum mechanics.We found that quantum coherence was maintained in thescattering of light particles on a black hole background by an infinite set of localW1+∞-symmetries [8, 9], which link light asymptotic states to massive string states,and whose associated conserved W-charges could encode all initial-state information.We believe that quantum coherence can be maintained in this situation only in anintrinsically non-local theory such as string with its infinite set of gauge symmetries,and that local field theories are doomed to failure in this respect because they onlyhave a finite set of hair.

We have shown that the infinite set of stringy W-hairis sufficient to label all the black hole states, and thereby quench all the entropyS = M2 of a spherically-symmetric black hole in four dimensions [10]. We havealso argued [11] that black hole decay is a purely quantum-mechanical higher-genuseffect that does not require a thermal description in terms of Hawking radiation,and that all the W-charges are in principle measurable via scattering experiments orAharonov-Bohm phases for massive string states [12].

We have also given a stringydesription of space-time foam in terms of a plasma of defects on the world-sheet,which is intimately related to a Hall conductor [13].The maintenance of quantum coherence in the scattering of light particles offastringy black hole does not, however, mean that conventional quantum mechanicsand quantum field theory are sacrosanct. The essential problem is that althoughall the W-charges are in principle measurable, this would entail observations of themassive string states that are linked to the light states by the W-symmetries, andsuch observations are not in practice possible in realistic laboratory experiments.We argue in this paper that a modification of the Liouville equation of the type (2) isindeed essential for observable systems containing only light particles.

Its origin isexactly the W-linkage between the light and heavy string states, which means thatthe former must be regarded as an open system coupled to the unobserved heavydegrees of freedom. A modification of the form (2) would be forbidden for any ex-actly marginal deformation of the underlying string theory.

However, it is knownfrom explicit examples that in non−trivial space−time backgrounds such exactlymarginal deformations involve massive string states. Therefore, any deformation ofthe effective theory of light particles in which the massive modes are unobservedwill necessarily not be exactly marginal, and will hence be associated with a con-tribution to /δH (2).

The unitarity of the effective field theory of the light degreesof freedom and Zamolodchikov’s c-theorem [14], as proved by one of the authors

(N.E.M.) and Miramontes [15], guarantee that any contribution to /δH can only in-crease the entropy of the light-particle system.

Topologically non-trivial space-timebackgrounds appear intrinsically in the stringy realization of space-time foam, sowe conclude that a /δH term is inevitable in string theory. We argue, moreover, thatthe dominant such modifications are just those associated with the known S-wavedeformations of spherically-symmetric black holes.

A primitive order-of-magnitudeestimate indicates that their contribution to /δH might be suppressed by just one in-verse power of MP lanck, and hence be close to the present experimental upper limits[3, 5], and in the ball-park needed to explain the collapse of the wave function for aclassical system.2Renormalization Group Flow as Quantum Fric-tionLet us consider a general dynamical system whose state is described by a den-sity matrix ρ{qi, pi}, where the qi are generalized coordinates, and the pi are theirassociated conjugate momenta. In conventional quantum mechanics, the qi includeconventional space-time coordinates, which in string theory become parameters ofbackground target spaces corresponding to σ-models on the world-sheet.

Thus theqi can be regarded as couplings in a space of possible σ-models. We consider theevolution of ρ with respect to a renormalization group flow variable t = ln Λ, whereΛ is some covariant cutoff.

We will identify t with a conventional time variable,which may or may not be considered as a string Liouville mode. On dimensionalgrounds, the target time is then measured in units of Planck length.

This in turnimplies that even marginal changes in the renormalisation group scale produce ap-preciable time-variations in target-space. It is clear that renormalisability of thesystem impliesdρdt = 0 = ∂ρ∂qi ˙qi + ∂ρ∂pi˙pi(3)where (.˙..) denotes ∂t(...).It is known [16] that in the Wilson renormalization scheme the renormalizationflow of the coordinates (σ-model couplings) qi is given in the neighbourhood of afixed point S0 by the gradient of Zamolodchikov’s c-function Φ(qi, S0):βi ≡˙qi = Gij(S0)∂Φ(qi, S0)∂qi(4)where Gij is the metric in the space of coupling constants:Gij ≡2|z|2 < Oi(z, z)Oj(0, 0) >(5)with < ... > denoting an average with respect to the deformed σ-model actionI∗+R d2zqiOi, where I∗is the fixed-point conformal field theory action, and the Oiare a complete set of (renormalised) vertex operators.

In view of equation (4), we can regard Φ as a Lagrange function, to which we canassociate a Hamiltonian in the usual way:Φ =Zdt[ ˙qi · pi −H(qi, pi)](6)Again in the usual way, it follows that˙qi = ∂H∂pi(7)and˙pi = −∂H∂qi −Gijβj(8)The second term in equation (8) is a friction term characteristic of open dynamicalsystems [4], which will be at the root of our subsequent modification of quantummechanics 1. Substituting the expressions (7),(8) into the time derivative equation(3), we find the following equation˙ρ = i[ρ, H] + Gijβj ∂ρ∂pi(9)for the classical evolution of the density matrix 2.

When proposing any such modi-fication of the traditional evolution equation for the density matrix, it is importantto check that the total probability P =R dpidqitrρ(pi, qi) is conserved. In our case(9), it is easy to check that∂tP =Zdpldqltr ∂∂pi(Gij(q)βj(q)ρ(p, q))(10)which vanishes if there are no contributions from the boundary of phase space, aswould occur if it has no boundaries, or if ρ vanishes there.The quantum version of this classical equation is obtained by first replacing{, } →1i [, ](11)1In the Wilson renormalisation scheme (4) the friction is linear , which simplifies many of thecomputations.

However in practice, or in certain cases where this scheme is not applicable [16],one can work in schemes where (4) is valid but Gij = Gij(q), in which case one is faced with anon-linear friction problem. For most of our discussion we can stay in the general case.2A similar equation has been considered by Kogan [17], but his formalism is incorrect becausehe considers total time derivatives in places where one should consider partial ones.

In addition,he identifies the Liouville field with the target time coordinate, which clearly does not apply to thetwo-dimensional black hole string theory [18, 19]. We stress again that from our point of view theworld-sheet cutoffis identified with the target time (evolution parameter) independently of any(possible) identification of the Liouville field with time.

The quantum version of the second term in (9) is obtained by recalling Euler’sequation (7), whose quantum version is∂H∂pi= −i[qi, H](12)from which it follows that∂F(H)∂pi= −i[qi, F(H)](13)for any function F of the Hamiltonian, including in particular the density matrixρ. We therefore arrive at the folowing evolution equation for the quantum densitymatrix:˙ρ = i[ρ, H] −iGij[ρ, qi]βj(14)We will not enter here into a discussion of the appropriate quantum ordering of thefactors in the second term in equation (14), which represents the quantum frictioninherent to our open dynamical system.An additional remark we would like to make concerns the connection of the gra-dient flow relation (4) with the problem of a supersymmetric quantum mechanicalparticle moving in a constant magnetic field ( more specifically in a Kahler potentialΦ(q) [20]).

Indeed, it can be shown [20] that whenever relation (4) is valid, the evo-lution of qi in t = lnΛ can be described by a super-quantum mechanics HamiltonianH = 12(Q + Q†), with supercharge Q = ψi(∇i −βi) and its hermitian conjugate Q†,where the ψi are Grassmann variables satisfying canonical commutation relations.The charges are nilpotent : Q2 = (Q†)2 = 0 if and only if the covariant βi arecurl-free, i.e. if relation (4) is satisfied.

In this case the metric in coupling constantspace can be viewed as a Kahler metric, whose potential coincides with the flow-function Φ(q) [20]. This analogy makes clearer the physical interpretation of therenormalisation group flow as a friction problem related to the quantum motion ofa particle in a magnetic field [21].Our previous demonstration that the total probability P is conserved carries overdirectly from the classical case (9) to the quantum case (14).

However, entropy isS = −trρ ln ρ is not necessarily conserved: we see easily that˙S = −trGij[ρ, qi]βjlnρ(15)which does not in general vanish if βi ̸= 0. Our next task is to demonstrate thatindeed βi ≡Gijβj ̸= 0, and in so doing we will be able to argue that the entropy Sincreases monotonically, as one would expect on general physical grounds.

3Light Particle Operators are not Exactly MarginalSince string theory is based on conformal field theory, one might naively expectthat the renormalization coefficients βi introduced above all vanish, and hence thatthe quantum friction term in equation (14) vanishes identically. This is indeed thecase in a flat space-time background, but is not true for operators that create lightparticles in non-trivial backgrounds.

The prototype for this phenomenon is the oper-ator creating a massless particle in a two-dimensional string black hole background,which can also be interpreted as a spherically-symmetric four-dimensional black holebackground. We first present a heuristic argument that βi ̸= 0 in this case, whichis followed by a more complete treatment in section 5.We recall that the black hole solution possesses a W1+∞-symmetry that is globalon the world-sheet, and includes the Virasoro algebra.

It becomes a local symme-try when elevated in target space, providing an infinite-dimensional extension ofthe general coordinate transformations of general relativity. This W-symmetry isresponsible, in our interpretation, for the existence of the S-matrix for scatteringmassless particles, confusingly called “tachyons”, in this black hole background, andhence for the maintenance of quantum coherence in this scattering process.

The fea-ture of these W-symmetries that is crucial for the present discussion is that they linktogether states with different masses. In particular, they relate massless “tachyon”states to massive string levels.

One could therefore suspect that the conformal sub-algebra of the W-symmetry associated with the Virasoro algebra might also combinemassless and massive states in an essential way, and we shall see in a moment thatsuch is indeed the case. Therefore, the β-functions for operators contructed out oflight particle fields alone will not have the full W-symmetry, and hence not be con-formal in general, and hence have some βi ̸= 0.

But physical laboratory experimentsare conducted with light particles, corresponding to states that are massless in theapproximation which we discuss here, and do not measure any properties of the mas-sive string states. Therefore the relevant vertex operators are precisely those thatinvolve only massless states, and hence have non-vanishing β-functions.

Hence theeffective quantum mechanics of observable particles has the extra quantum fric-tion term in (14), even though the density matrix of the full string theory would bedescribed by the usual Liouville equation without such a term. The light observableparticles constitute an open quantum-mechanical system coupled by W-symmetryto unobserved massive states.Examples of the above assertion that in non-trivial backgrounds exactly marginaldeformations generally involve massive string states have been provided in ref.

[22].In two-dimensional flat-space Liouville theory, the following “tachyon” vertex oper-ator is exactly marginal:φc,−c−1/2,0,0 = (g++g−−)−12F(12; 12; 1; g+−g−+g++g−−)(16)

where gab, a, b = +, −represent the components of a generic SL(2, R) element[23]. However, the corresponding exactly marginal operator in a two-dimensionalMinkowski space-time black hole, described by an SL(2,R)/O(1,1) Wess-Zumino(WZ) coset model, isL10L01 = φc,−c−1/2,0,0 + i(ψ++ −ψ−−) + .

. .

(17)whereψ±± ≡: (J±)N(J±)N(g±±)j+m−N :(18)with J± ≡(k−2)(g±∓∂zg±±−g±±∂zg±∓), and J± ≡(k−2)(g∓±∂¯zg±±−g±±∂¯zg∓±),with k the WZ model level parameter [23]. The combination ψ++ −ψ−−generatesa level-one massive string mode, and the dots in equation (17) represent operatorsthat generate higher-level massive string states.An analogous exactly marginalopearator isL20L02 = ψ++ + ψ−−+ ψ−+ + ψ+−+ .

. .

(19)which also involves in an essential way operators for massive string modes. For lateruse we note that the coupling constant, α, corresponding to this deformation of thecoset model is responsible for a global rescaling of the target space-metric [22], andtherefore to a global constant shift by α of the dilaton field.

Thus it produces shiftsin the black hole mass [23].We now remark that in two-dimensional string theory all these massive states arein fact discrete states of the type generated by the W1+∞algebra which maintainsquantum coherence [8, 24]. Each of these discrete solitonic states can be representedas a singular gauge configuration [25], whose conserved W-charges can be measuredin principle by generalized Aharonov-Bohm phase effects [12].

However, laboratorymeasurements of light microscopic objects such as the K0−K0 system or interferingneutron beams do not measure such effects, so do not observe the massive states andare restricted to the light (massless) parts of the exactly marginal operators (17,19).These by themselves are not exactly marginal, hence the corresponding light-field β-functions do not vanish, and therefore make non-zero contributions to the quantumfriction term in equation (14). We shall give order of maginitude estimates of theseeffects in the next section.As we have discussed in ref.

[13] the sum over quantum configurations in the stringpath integral includes a sum over many vortices and spikes on the world-sheet:Z =ZD ˜Xexp(−βSeff( ˜X))(20)where ˜X ≡β12X, β is a ‘temperature ’ variable characterising the topological defectson the world-sheet, andβSeff=Zd2z[2∂˜X∂˜X + 14π[γvǫα2 −2(2q|g(z)|)1−α4 : cos(√2πα[ ˜X(z) + ˜X(¯z)]) :+(γv, α, ˜X(z) + ˜X(¯z)) →(γm, α′, ˜X(z) −˜X(¯z))]](21)

Here γv,m are the fugacities for vortices and spikes respectively, andα ≡2πβq2vα′ ≡e22πβ(22)are related to the conformal dimensions ∆v,m of the vortex and spike creation oper-ators respectively, namelyα = 4∆vα′ = 4∆m∆m = (eqv)216∆v(23)In the low-temperature phase relevant in the present-day Universe, this path-integralis dominated by a plasma of world-sheet spikes corresponding to microscopic Minkowski-space black holes, which constitute a stringy realization of space-time foam [13].Therefore, many configurations with non-zero values of the βi contribute impor-tantly to the string path integral, and one cannot a priori expect the modification(14) of the quantum Liouville equation to be negligible.The reader might worry that although individual configurations contribute to thequantum fiction term in (14), some as yet unknown symmetry principle might causetheir total net contribution to vanish. An immediate intuitive counterargument tothis suggestion is the comment that no-one has ever seen a macroscopic body speedup as a result of friction!

Thus a cancellation would be surprising in the light of ourphysical picture of the renormalisation group flow (of unitary theories) as the motionof a (supersymmetric) particle in coupling constant space, under the influence of anexternal magnetic field [20, 16]. Indeed, as we shall now explain, the unitarity of thetruncated effective light-particle field theory guarantees that all quantum frictionterms tend to increase the entropy, in accord with everyday experience, and hencecannot cancel in the path-integral sum.In our case, it is easy to see that the renormalization group flow is irreversible, ashas also been argued in ref.

[13] on the basis of the isomorphism of the space-timefoam theory of vortices and spikes with a Hall conductor. The rate of change of theLagrange function Φ (6) is given by˙Φ = βi ∂Φ∂qi(24)which can be rewritten using equation (4)˙Φ = −βiGijβj(25)Since the low-energy effective field theory of the light degrees of freedom should beunitary by itself, the metric Gij in the space of coupling constants must be positivedefinite [14].

Hence the Lagrange function decreases monotonically, corresponding

to a monotonic change in the effective central charge. This in turn corresponds toa monotonic increase in the entropy, as can be seen explicitly from the expression(15) for the rate of change of the quantum entropy.

The commutator factor in (14)can be rewritten as−i[ρ(H), qi] = ∂ρ∂pi= ∂ρ∂H ˙qi = ∂ρ∂H βi(26)Substituting this expression into equation (15), we find˙S = TrβiGijβj ∂ρ∂H lnρ(27)In systems that exchange energy with their environment, as is our case, the densitymatrix is actually given byρ = eβ(F −H)(28)where F is the free energy of the system. In the particular case of strings, F isidentified with the effective action Φ [15].

Thus, taking into account the facts thatlnρ < 0, and that for unitary σ-models βiGijβj = 2|z|4 < Θ(z, ¯z), Θ(0) > is positivedefinite [14], one observes that the quantum entropy does indeed increase monoton-ically thanks to the unitarity of the effective low-energy theory. This simple andgeneral argument excludes the possibility of a sneaky cancellation between differentcontributions to the quantum friction.We conclude this section by noting an essential difference between wormhole cal-culus and the type of quantum gravitational physics that we are discussing in thispaper.

Wormholes connect different parts of the target space-time via throats. Clas-sical gauge symmetries that are carried by particles falling into one end of the worm-hole are carried out by particles expelled at the other end, and there is no sign ofinformation loss.

Wormholes can give rise to non-local effects in space-time, whichmay be physically dubious for other reasons, but do not cause obvious problemsfor conventional quantum field theory or quantum mechanics. We do not addresshere the question whether string theory admits such wormholes.

The defects on theworld-sheet that we discuss here correspond to Minkowski space black holes withoutthroats, where, as discussed above, information is transferred monotonically by W-symmetries to unobserved massive string modes, resulting in the above-mentionedmonotonic increase in the apparent entropy of the observable light degrees of free-dom.4Simple Models of Quantum Friction EffectsBefore making an order-of-magnitude estimate of quantum friction effects in thelow-energy effective field theory derived from string, we first remind the reader ofthe standard formalism for microscopic systems interacting with “environmental”oscillator modes. This enables us to make contact with the previous literature on

the transition between the quantum behaviour of microscopic systems and the clas-sical behaviour of macroscopic systems. The standard “environmental” formalismassumes weak perturbations around the equilibrium state of the microscopic sys-tem, which is justified in our case by the suppression of Planck-mass excitations atlow energies.

The effective lagrangian for a single harmonic oscillator coupled to“environmental” oscillators representing the infinite set of massive string modes isL = 12M ˙q2 −V (q) + 12Xj(mj ˙x2j −mjω2jx2j) −XjFj(q)xj −XjFj(q)22mjω2j(29)where the last two terms represent the interactions. The density matrix of the system(29) can be expressed as a path integral in the usual way.

The reduced density matrixdescribing the evolution of the primary microscopic harmonic oscillator from q = qiat time t = 0 to q = qf at time T is then given byK(qi, qf, T) = K0(T)Z q(T)=qfq(0)=qiDq(τ)e−Seff[q(τ)]/¯h(30)where the prefactor K0(T) involves only the environmental oscillator frequencies[21], and can be absorbed in the normalisation of the uncoupled case. The Euclideaneffective action Seff is given bySeff[q(τ)] =Z T0 dτ(12M ˙q2 + V (q)) + 12Z +∞−∞dτ ′Z T0 dτα(τ −τ ′)(q(τ) −q(τ ′))2 (31)whereα(τ −τ ′) ≡XnC2n4mnωne−ωn(τ−τ ′)(32)In deriving these results, we have simplified matters by assuming linearity in theinteractions: Fj(q) = qCj, corresponding to a Wilson renormalisation scheme instring theory [16].

In this case it can be shown that α is given by the asymptoticform of the so-called Drude model [21]α(τ −τ ′) = η1(τ −τ ′)2(33)where η is the “environmental” friction coefficient, given byη ≡XjC2jmjω2jδ(ω −ωj)(34)We see from Eq. (32) that the strengths of the dissipation terms are suppressed forheavy “environmental” oscillators, a feature that we might expect to carry over tomore massive modes in string theory.

The “environmental” dissipation effects (32)are suppressed linearly in the large masses or frequencies. If this result is carriedover to string theory (with the correspondence of the oscillator frequencies ωj in(34) to the string mass levels), the magnitude of the stringy quantum friction effectscould be comparable to the upper bounds established in refs.

[3, 5] for the neutralkaon and neutron systems.

5Order-of-Magnitude Estimate in String TheoryIn previous sections we have seen that in two-dimensional string theory the exactlymarginal operators that turn on backgrounds for the light degrees of freedom in thepresence of a black-hole contain necessarily massive string states which are discretesolitonic states, that are not measured in laboratory experiments.To estimatethe orders of magnitude of the quantum- gravitational friction term in (14) and ofthe rate of change of the entropy (27), we need to discuss the magnitude of therenormalization functions βi. In general, one hasβi =∞XN=1Cii1...iNgi1...giN(35)where the coefficients Cii1...iN are not totally symmetric among covariant and con-travariant indices.

The usual on-shell string N-point amplitudes coincide with theexpansion coefficients γi1...iN = of the covariant βi ≡Gijβj[26, 27]. These coefficients are totally symmetric as follows from the flow equation(4).

Factorization implies that all higher-order coefficients are obtainable from 3-point functions [27]. If one chooses an appropriate Wilson renormalization scheme,all contact terms are eliminated from these 3-point functions if the boundaries ofmoduli space are treated correctly [28], as we assume for the coset black hole model,in which case the metric tensor Gij in coupling constant space is flat with its non-zero entries O(1) 3.For exactly marginal operators with both light- and heavy-state terms: Vi =V (L)i+ V (H)i, such as L10L10 (17), we have0 = βi = Gij(S0)∞XN=1gi1...giN < ΠNn=1(V (L)in+ V (H)in) >(36)Therefore, defining the light-state renormalization coefficientsˆβi =∞XN=1Gij(S0)gi1...giN < ΠNn=1V (L)in>(37)we findˆβi = −∞XN=1NXM=1gi1...giN < ΠMm=1V (M)im ΠN−Mn=1 V (L)in>(38)3 The existence of such a Wilson scheme, and hence the vanishing of such contact terms, is subtlein certain cases such as supersymmetric strings [29], requiring a careful definition of correlationfunctions with colliding punctures [28].

Thus it might not be easy for this scheme to be implementedin practice in the black hole case, whose singularity is associated with a twisted supersymmetricfixed point [30]. If a conventional renormalisation scheme were adopted in this case [31], it wouldgive non-linear friction.

However, we believe that physical observables would not be affected, sincethe corresponding conformal field theory is well-defined even at the singularity [23].

The effective βi appearing in sections 2 and 3 are in fact the ˆβi (37) in the light-field theory, and we now revert to the hatless notation of those earlier sections.As we have seen, these effective βi are non-vanishing to the extent that the mixedlight-heavy N-point functions in (36) are non-zero.To estimate these, we use the Euclidean counterpart of the SL(2, R)/O(1, 1) WZcoset model [23], which can be mapped [32] into a c = 1 matrix model with amodified cosmological constant on the world-sheet 4: in their notation2πµββexp(−2 φα+) ≡µV (−)1,1(39)where β,β are chiral bosons used along with the free field φ and another chiral bosonγ to parametrise SL(2, R) [32]. The scale µ is related to the mass of the black holeby µ = MBH/MP lanck, and the massive string states are represented in general byV (±)r1r2 = [∂r1r2X + ...]ei r1−r2√2±β±r1r2φ(40)The (−) states are often discarded in usual Liouville theory [19], but play a keyrole here, as we shall see.

In particular, following ref. [32], we have the followingrepresentation for the WZ model representing a two-dimensional black hole [23]SW Z = Sc=1,µ=0 + µV (−)1,1(41)The (+) states are massive string modes, so the correlation functions of interest tous can be expressed as+∞Xl=01l!µl < ΠN′i=1V (+)riri+1ΠMj=1V 0j Πlm=1V (−)rm−1rm >1(42)where the superscript (0) denotes a massless “tachyon” state, and the subscript 1 onthe v.e.v.

denotes the fact that it is calculated in a c=1 matrix model with µ = 0.We now seek to understand which correlation functions are in general non-vanishing,and how they scale with µ. The first important observation in this regard is thatcorrelators of V (−) operators alone vanish: only correlators with at least one V (+)operator are non-zero.

This means that the black hole dynamics necessarily turnson physical massive string states, as we had seen previously from the point of view ofthe W-symmetry linking massless and massive modes [8, 12]. The second commentis that the non-zero correlators contain in general logarithmic factors that violatethe expected power-law scaling with µ. Specifically, there are factors of (lnµ)±1 for4Although Euclidean and Minkowski coset models are not physically equivalent in this formalism[11, 33], the Euclidean version will be sufficient for our order-of-magnitude estimates if we assumethat the analytic continuation in target space needed to produce Minkowski black holes does notaffect the scaling arguments we are using below.

insertions of V (±) operators respectively, associated with divergences close to theFermi level and representing physical pole conditions in amplitudes [34, 35]. Thuswe have the following general scaling laws:< Π2N′i=1V (+)ri−1riΠMj=1V 0j Π2Nm=1V (−)rm−1rm >1∝ µs(lnµ)N′−N × F({k}) forN′ −N > 00otherwise(43)where F({k}) is a kinematical factor, and the power s is determined by naive kine-matical scaling considerations.

It is related to the Liouville-energy non-conservationwhich arises from the reality of the coupling of the Liouville field in the expressionfor the vertex operators [36]. For resonant amplitudes, s = 0.

For general s one canonly compute the amplitudes by analytic continuation from the integer s case [37].Such a procedure is assumed in the following.We see in the general scaling law (43) logarithmic factors in the non-zero corre-lators that appear to vanish as lnµ →0, i.e., as µ →1. Therefore we are led toconclude that the contributions to the quantum-gravitational friction are suppressedfor configurations with µ ≃1.

Formally, the expression (43) is ill-defined for µ →0,i.e., lnµ →−∞. This we interpret as a reflection of the well-known ill-definitionof the c = 1 matrix model itself when µ = 0.

Analytic continuation is not safe inthis regime, since if one formally evaluates the amplitudes in the limit µ →0 byanalytically continuing to s = integer > N′−N in (43), then the amplitudes vanish.We interpret these subtleties as follows: as discussed in ref. [13], we believe thatthe appropriate string vacuum is a space-time foam consisting of a plasma of spikeson the world-sheet corresponding to microscopic Minkowski-space black holes.

Theend-state of black hole decay is a microscopic black hole that is indistinguishablefrom this foam, so the latter should be subtracted and serve as a regulator for blackhole physics. This would therefore be dominated in our application by configura-tions with µ ≥1 in the gravitational path integral analogue of the simple model(32).Incorporating the estimate (43) into the expression (38) for the effective light-field renormalization coefficients βi, and then using the path integral (20,21) toperform the sum over microscopic Minkowski black holes in the space-time foam,which is dominated by configurations with µ = MBH/MP lanck ≥1 as discussedabove, we estimate that the coefficients of the quantum-gravitational friction termsin (14,27) are O(1) in Planck units.

Thus their effects are suppressed by dimensionalpowers of MP lanck, and the largest could be O(M−1P lanck) as in the simple model (32).Their detailed evaluation requires more knowledge of the effective low-energy theoryderived from string, and goes beyond the scope of this paper.

6Collapse of the wave-function.It has been argued [6] that the microscopic entropy increase offered by quan-tum gravity effects may cause the collapse of macroscopic wave functions and theirtransition from quantum to classical behaviour. We now discuss whether quantumquantum-gravitational friction (14) and the rate of entropy increase (27) could havethis effect.

We start from the simple model for friction described in section 4. Inthe string case, the renormalized σ-model couplings play the rˆoles of coordinatesq, and τ is the renormalization group scale parameter, identified with target time.Since the q’s are almost but not exactly marginal in the truncated effective light-mode field theory, the dominant contributions to the path-integral in the asymptoticDrude model (33) come from the limit where τ →τ ′.

In this case eq (31) may bewritten asρ(qi, qf, T)/ρS(qi, qf, T) ≃e−η R T0 dτ R τ+ǫτ−ǫ dτ ′ (q(τ)−q(τ′))2(τ−τ′)2≃e−η R T0 dτ Rτ′≃τ dτ ′βiGij(S0)βj ≃e−DT(qi−qf)2+...(44)where D is a small constant, proportional to the sum of the squares of the effectiveanomalous dimensions of the renormalised couplings qi.We reinterpret (31) asrepresenting the overlap, after an elapsed time interval T, between quantum systemslocalized at different values q = qi, qf of the coordinates, and the subscript “S”denotes quantities evaluated in Schroedinger quantum mechanics. Equation (44)exhibits a quadratic dependence on the β-functions in this simple mechanical modelfor friction.

It captures all the essential features of the complicated string case,in which the light-mode effective β-functions are suppressed by inverse powers ofMP lanck, as we discussed in the previous section.The vanishing of off-diagonal terms in the density matrix leads in general tothe collapse of the wave-function. This effect is usually negligible for microscopiclight-mode systems - except possibly for special cases such as the K0 −K0 systemdiscussed elsewhere [3, 5] - but it is enhanced for macroscopic systems.

This was firstderived in ref. [6] in the context of a wormhole model, but the result is more generaland applies here.

We consider a macroscopic body containing many particles withcoordinates qi, i = 1, 2, ...N. Decomposing these into a centre-of-mass coordinateQ and relative coordinates ∆qi, (i = 1, 2, ...N), writing the full density matrix asρq = ρQρ∆q, and assuming friction terms −Dt(qi −q′i)2 (i = 1, 2, ...N), the equationfor the centre-of-mass motion is˙ρQ ≃i[ρQ, H(Q)]ρ∆q + i[ρ∆q, H(∆q)]ρQ −NXi=1D(qi −q′i)2ρ(qi, q′i)(45)Tracing over the relative coordinates ∆qi, we find˙ρQ ≃i[ρQ, H(Q)] −ND(Q −Q′)2ρQ(46)

and henceρ(Q′, Q, t)/ρS(Q′, Q, t) ≃e−NDt(Q′−Q)2(47)where we see an enhanced suppression for bodies containing many particles. Asan illustration, if D ≃m6proton/M3P lanck as in ref.

[6], then the locations of bodieswith N ≥1024 constituent particles (cf Avogadro’s number) are fixed to within aBohr radius within about 10−7 sec. A similar mechanism for wave function collapsewas introduced phenomenologically in ref.

[7] without reference to any microscopicmodel.A final comment we would like to make concerns other possible effects of thespace-time foam, which could also be enhanced macroscopically. As we have arguedin [13], the space-time foam is represented as a plasma of world-sheet spikes, andone should examine the motion of a single light string mode in this plasma whichis not yet feasible.

We notice, however, that taking into account the no-net-forcecondition among the spikes of the plasma [13], we can define a ‘centre-of-mass’coordinate for the black hole moduli and examine the motion of the tachyon in asingle coset model, representing in some sense this ‘collective’ effect. In such a case,following [6], one would have an enhanced macroscopic non-quantum mechanicaleffect if there were microscopic violations of the no-net-force condition, leading toa motion of the centre of mass of the black-hole moduli.

Such effects are left forfuture investigations. However one should keep in mind that they might also leadto “observable” modifications of quantum mechanics.7DiscussionThe following remarks may help the reader to understand intuitively the relationof this work to other studies.

As we have emphasized previously [12], the scatteringof light particles offa black hole resembles the Callan-Rubakov [38] description ofthe scattering of charged particles offa grand unified monopole. There, conservationlaws enforce selection rules on the scattered particles, and here similar selection rulesensure that quantum coherence is conserved by the full light/heavy particle system.The situation we have discussed in this paper is more akin to the ’t Hooft [39] analysisof instanton effects in the Standard Model.

In that case, scattering light particleshave a very small probability of “meeting” an instanton, but then it generates newphysics. In our case, light particles again have a very small probability of “meeting”a microscopic or virtual black hole, but then it also generates new physics.

Sincethe heaavy degrees of freedom are necessarily ignored in a light-particle scatteringexperiment, information and hence coherence are lost in this case.We have laid out in this paper the conceptual basis for the modification of lab-oratory quantum mechanics in string theory. We re-emphasize that the full stringtheory is perfectly quantum-mechanical, but that experiments cannot see this un-less they measure all the massive string states, which are linked to light states by

an infinite set of W-symmetries. Observing these massive modes is an impossibil-ity in realistic laboratory experiments.

Therefore, one must work with an effectivetheory of the light degrees of freedom as an open system, with the correspond-ing modification (14) of the Liouville equation for the density matrix. A simpleextension of this argument will lead to a non-factorizing /S-matrix description ofthe transition between initial and final density matrices in quantum field theory.The quantum-gravitational friction that we have identified in this paper appearsable to explain qualitatively the transition from quantum-mechanical behaviour ofmicroscopic systems to the classical behaviour of macroscopic systems containingAvogadro’s number of elementary particles.AcknowledgementsThe work of D.V.N.

is partially supported by DOE grant DE-FG05-91-ER-40633and by a grant from Conoco Inc.References[1] M.B. Green, J.H.

Schwarz and E. Witten, Superstrings Vol. I & II (CambridgeUniversity Press 1988).

[2] S. Hawking, Comm. Math.

Phys. 43 (1975), 199; ibid 87 (1982), 395.

[3] J. Ellis, J.S. Hagelin, D.V.

Nanopoulos and M. Srednicki, Nucl. Phys.

B241(1984), 381. [4] Y.R.

Shen, Phys. Rev.

155 (1967), 921;A.S. Davydov and A. A. Serikov, Phys.

Stat. Sol.

B51 (1972), 57;B.Ya. Zel’dovich, A.M. Perelomov, and V.S.

Popov, Sov. Phys.

JETP 28 (1969),308;For a recent review see : V. Gorini et al., Rep. Math.

Phys. Vol.

13 (1978), 149. [5] J. Ellis, N.E.

Mavromatos and D.V. Nanopoulos, CERN and Texas A & MUniv.

preprint, CERN-TH.6596/92; CTP-TAMU-47/92; ACT-12/92 (followingpaper). [6] J. Ellis, S. Mohanty and D.V.

Nanopoulos, Phys. Lett.

B221 (1989), 113; ibidB235 (1990), 305. [7] G.C.

Ghirardi, A. Rimini and T. Weber, Phys. Rev.

D34 (1986), 470; G.C.Ghirardi, O. Nicrosini, A. Rimini and T. Weber, Nuov. Cim.

102B (1988), 383. [8] J. Ellis, N.E.

Mavromatos and D.V. Nanopoulos, Phys.

Lett. B267 (1991), 465.

[9] J. Ellis, N.E. Mavromatos and D.V.

Nanopoulos, Phys. Lett.

B272 (1991), 261.

[10] J. Ellis, N.E. Mavromatos and D.V.

Nanopoulos, Phys. Lett.

B278 (1992), 246. [11] J. Ellis, N.E.

Mavromatos and D.V. Nanopoulos, Phys.

Lett. B276 (1992), 56.

[12] J. Ellis, N.E. Mavromatos and D.V.

Nanopoulos, Phys. Lett.

B284 (1992), 27,43. [13] J. Ellis, N.E.

Mavromatos and D.V. Nanopoulos, CERN and Texas A & MUniv.

preprint, CERN-TH.6534/92; CTP-TAMU-47/92; ACT-12/92; CERN-TH.6536/92; CTP-TAMU-48/92; ACT-13/92;[14] A.B. Zamolodchikov, JETP Lett.

43 (1986), 730; Sov. J. Nucl.

Phys. 46 (1987),1090.

[15] N. Mavromatos and J.L. Miramontes, Phys.

Lett. B212 (1988), 33; N. Mavro-matos, Mod.

Phys. Lett.

A3 (1988), 1079; Phys. Rev.

D39 (1989), 1659; A.Tseytlin, Phys. Lett.

B194 (1987), 63; H. Osborn, Phys. Lett.

B214 (1988),555. [16] N.E.

Mavromatos, J.L. Miramontes and J.M.

Sanchez de Santos, Phys. Rev.D40 (1989), 535.

[17] I. Kogan, Univ. of British Columbia preprint UBCTP 91-13 (1991).

[18] I. Antoniadis, C. Bachas, J. Ellis and D.V. Nanopoulos, Phys.

Lett. B211(1988), 393; Nucl.

Phys. B328 (1989), 117; Phys.

Lett. (1991), 278.

[19] J. Polchinski, Nucl. Phys.

B324 (1989), 123. [20] C. Vafa, Phys.

Lett. B212 (1988), 29; S. Das, G. Mandal and S. Wadia, Mod.Phys.

Lett. A4 (1989), 745.

[21] R.P. Feynman and F.L.

Vernon Jr., Ann. Phys.

(NY) 94 (1963), 118;A.O. Caldeira and A.J.

Leggett, Ann. Phys.

149 (1983), 374. [22] S. Chaudhuri and J. Lykken, FERMILAB preprint FERMI-PUB-92/169-T.[23] E. Witten, Phys.

Rev. D44 (1991), 314.

[24] G. Moore and N. Seiberg, Int. J. Mod.

Phys. A7 (1992), 2601.

[25] I. Klebanov and A.M. Polyakov, Mod. Phys.

Lett. A6 (1991), 3273;A.M. Polyakov, Princeton Univ.

preprint PUPT-1289 (1991). [26] N.E.

Mavromatos and J.L. Miramontes, Phys.

Lett. B226 (1989), 291.

[27] R. Brustein, D. Nemeschansky and S. Yankielowicz, Nucl. Phys.

B301 (1988),224.

[28] J. Distler and M. Doyle, Princeton Univ. preprint PUPT-1312 (1992).

[29] M.B. Green and N. Seiberg, Nucl.

Phys. B299 (1988), 559.

[30] T. Eguchi, Mod. Phys.

Lett. A7 (1992), 85.

[31] R. Dijkraaf, H. Verlinde and E. Verlinde, Nucl. Phys.

B371 (1992), 269. [32] M. Bershadsky and D. Kutasov, Phys.

Lett. B266 (1991), 345.

[33] J. Distler and P. Nelson, Nucl. Phys.

B374 (1992), 123. [34] D. Gross, I. Klebanov and M.J. Newman, Nucl.

Phys. B350 (1991), 621.

[35] P. di Francesco and D. Kutasov, Phys. Lett.

B261 (1991), 385; Nucl. Phys.B375 (1992), 119.

[36] A.M. Polyakov, Mod. Phys.

Lett. A6 (1991), 635.

[37] M. Goulian and M. Li, Phys. Rev.

Lett. 66 (1991), 2051.

[38] C. Callan, Phys. Rev.

D25 (1982), 2141;V.A. Rubakov, Nucl.

Phys. B203 (1982), 311;V.A.

Rubakov and M.S. Serebryakov, Nucl.

Phys. B237 (1984),239.

[39] G. ’t Hooft, Phys. Rev.

Lett. 37 (1976), 8; Phys.

Rev. D14 (1976), 3432; ibidD18 (1978), 2199.

---

출처: arXiv:9207.103 • 원문 보기