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PHYSICAL REVIEW D, VOLUME 63, 024024
How large are dissipative effects in noncritical Liouville string theory?
John EllisTheoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland
N. E. MavromatosTheoretical Physics Group, Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom
D. V. NanopoulosDepartment of Physics, Texas A & M University, College Station, Texas 77843,
Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, Texas 77381,and Chair of Theoretical Physics, Academy of Athens, Division of Natural Sciences, 28 Panepistimiou Avenue, Athens 10679,
~Received 1 August 2000; published 29 December 2000!
In the context of non-critical Liouville strings, we clarify why we expect non-quantum-mechanical dissipa-tive effects to beO(E2/M P), whereE is a typical energy scale of the probe, andM P is the Planck scale. InLiouville strings, energy is conservedat bestonly as a statistical average, as distinct from Lindblad systems,where it isstrictly conserved at an operator level, and the magnitude of dissipative effects could only be muchsmaller. We also emphasize the importance of nonlinear terms in the evolution equation for the density matrix,which are important for any analysis of complete positivity.
DOI: 10.1103/PhysRevD.63.024024 PACS number~s!: 04.50.1h, 04.62.1v
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I. INTRODUCTION
Motivated by speculations concerning quantum gravsome years ago a modification of conventional quantumchanics was proposed@1#, which admits dissipative phenomena such as transitions from pure to mixed states. Thisproach postulates the appearance of a non-Hamiltonianin the quantum Liouville equation describing the evolutiof the density matrix:
r~ t !5 i @r,H#1d”Hr. ~1!
A derivation of this equation based on a string-inspired trement of quantum fluctuations in space-time has also bpresented@2#. This formalism was applied initially to simpletwo-state systems such as the neutral kaon system@1,3,4#.Arguments have been presented that the magnitude ofdissipative effects could be suppressedminimallyby a singlepower of the gravitational scaleM P;1019 GeV, i.e.,
d”H5OS E2
M PD ~2!
whereE is a typical energy scale of the probe system whexperiences quantum-gravitational induced decoherenceing its propagation through the quantum-gravity ‘‘medium@5#. In particular, the estimate~2! was argued to apply withina non-critical Liouville string model for this foamy spactime medium. If the estimate~2! is indeed correct, suchmodifications of quantum mechanics might be accessiblexperiment in the foreseeable future, for example in the ntral kaon system, which offers one of the most sensitivecroscopic tests of quantum mechanics@1,3,4#.
A new arena for testing quantum mechanics has now bopened up by neutrino oscillations, and atmospheric-neutdata have recently been used@6# to constrain dissipative ef
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fects within a Lindblad formalism@7#. However, it has beenpointed out@8# that the order-of-magnitude estimate~2! isnot applicable to modifications of quantum mechanicsscribed by the Lindblad formalism@7#, because the energscaleE appearing in Eq.~2! is no longer the absolute energof the probe, but ratherthe energy varianceDE[uE12E2uof the two-state system:
d”H5OS DE2
M PD . ~3!
It is therefore crucial@6,8# to know whether the Lindbladformalism @7# applies to the type of of open quantummechanical system provided by a probe propagating throspace-time foam.
There are three key requirements for deriving the Linblad formalism@7#, namely energy conservation at the opetor level, unitarity and entropy production. However, whave pointed out previously@2# within our stringy approachto quantum gravity that, although unitarity and entropycrease follow from generic properties of the world-sherenormalization group for the Liouville string, energy coservation should be interpretedat bestas astatistical prop-erty of expectation values, and even this may be violated icases with potential physical interest, such as a D-brmodel for space-time foam. Thus we do not believe thatLindblad formalism@7# is directly applicable to propagatiothrough space-time foam, since energy conservation isimposed at the operator level. It is this difference that pmits the magnitude of the dissipative effects to be of orgiven in Eq.~2!, larger than that, Eq.~3!, suggested in@8#.We also argue that Eq.~1! contains important nonlinear effects, which are potentially significant@9# for the analysis@10# of complete positivity.
©2000 The American Physical Society24-1
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JOHN ELLIS, N. E. MAVROMATOS, AND D. V. NANOPOULOS PHYSICAL REVIEW D63 024024
II. DISSIPATIVE DYNAMICS IN THE LINDBLADFORMALISM
We first give various definitions of the concepts usedthe following. We consider a quantum-mechanical systwith a density matrixr, in dissipative interaction with anenvironment. In the case of a pure state described by a wvector uC&, the von Neumann density matrix operatorgiven by
r5uC&&Cu, ~4!
and in a representation$a% the matrix elements ofr are
r~a,a8![^a8uC&^Cua&. ~5!
In general, for open systems one cannot always define avector, in which case the density matrix is defined overensemble of theoriesM:
r[TrMuC&^Cu. ~6!
In the coordinate representation,$a%5$xW%, the diagonal ele-ment r(x,x;t) is the probability densityP(x,t), which isgiven in the pure-state case by the wave function at the tt:
r~x,x;t ![P~x,t !~5uC~x,t !u2!>0. ~7!
For well-defined representations$a% one must have the postivity property
r~a,a![ra.0. ~8!
Therefore, to define an appropriate density matrix, the optor ra must have positive eigenvalues in the state space$a%at any timet. Even if the matrixr(t50) has positive eigenvalues, it is not guaranteed that the time-evolved mar(t)5v(t)r(0) necessarily also has positive eigenvaluThis requirement ofsimple positivity~SP! has to be imposedand restricts the general form of the environmental entanment. Further restrictions apply when one considers$a% ascoordinates of a subsystem within a larger system interacwith the environment, namely the requirements ofcompletepositivity ~CP! @10#.
In the particular case of a linear dissipative system wenergy conservation, the Lindblad formalism@7# applies; i.e.,the quantum evolution is governed by a Markov-type procdescribed by an equation of the form
r~ t !5 i @r,H#1d”H@r#,
d”H[2(i
D i(d)bir~ t !bi
†
2(i
D i(d)~bi
†bir~ t !1rbi†bi !,
D i(d)>0, ~9!
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which maintains SP. HereH denotes the appropriate Hamitonian operator of the subsystem, the positive coefficieD (d) are associated to diffusion, and it can be shown thatextra ~non-Hamiltonian! terms in the evolution equation~9!induce complete decoherence, as recalled by the index (d).1
Now suppose one demands that energy be conserveatthe Hamiltonian operator level, so that one not only requirethe statistical average Tr(rH) to be independent of time, bualso the absence of explicit time dependence of the operH:
]H
]t505
d
dtTr~rH !. ~10!
This requirement, together with the monotonic increasethe von Neumann entropyS52Tr r lnr, implies, for theenvironmental operatorsbi ,bi
† ,
@bi ,H#50, bi5bi† . ~11!
Then, the environmental partd”H@r# of the evolution ~9!assumes a double-commutator form@13,8#
d”H5(i
D i(d)@bi ,@bi ,r##. ~12!
Clearly, for a two-state system, with energy levelsEn ,n51,2, the only non-trivial Lindblad operators satisfying E~11! are of the formbi}H.
This simplified case with only one operator suffices four purposes@8#, and we restrict our discussion to this casOne may then estimate the magnitude of the dissipativefects by considering a statistical average ofd”H with respectto a complete basis of states, which can be taken asenergy eigenstates$um&% of the Hamiltonian H, Hum&5Emum&, m51,2. In such a case one has@8#
^^d”H&&52 (m,n51
2
^murun&^nurum&En~En2Em!
52^1uru2&^2uru1&~E22E1!2 ~13!
and thus one gets the order of magnitude~3! for the possibledissipative effects@8#. A similar conclusion is reached if onuses position eigenstatesuq& as a basis, which is the casespontaneous localization models@12,13,8#, instead of energyeigenstates. In this case, the dissipative effects are protional to the position varianceDq, i.e., the separationuq12q2u between the centers of the wave packets.2 These
1It should be noticed that the density-matrix evolution equat~9! can be recast, using Eq.~6!, as a state-vector evolution equatioof stochastic Ito type@12#, if one wishes. We prefer the more general density-matrix formalism, because the concept of a state veis not always well defined, particularly in our quantum-gravity foacontext@1,2#.
2For instance, of the corresponding neutrino probes in theample discussed in@6#.
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HOW LARGE ARE DISSIPATIVE EFFECTS IN . . . PHYSICAL REVIEW D 63 024024
estimates are considerably smaller than our estimate~2!, so itis important to address carefully the question how the Liville formalism differs from the generic Lindblad formalism
III. DISSIPATIVE DYNAMICS IN NON-CRITICALSTRINGS
For the benefit of the non-expert reader, we now revihow a dissipative evolution equation of the form~1! arises inthe context of non-critical string, and how energy isnot con-served atan operator level, but at bestas astatistical aver-age. We also discuss the special circumstances under wEq. ~9! may be obtained, in the hope of clarifying the essetial differences between the two approaches.
A. Non-critical strings in flat world sheets
Critical string theory is described by a conformal fietheory S* on the two-dimensional world sheetS. We de-scribe non-critical string in terms of a generic non-conformfield theory (s model! on S. This is perturbed away fromconformal symmetry~criticality! by deformations that areslightly relevant ~in a world-sheet renormalization-grousense!, with vertex operators$Vi% and couplings$gi%, whichparametrize the space of possible theories:
S5S* 1ESd2sgiVi . ~14!
The renormalization-groupb function for a couplinggi on aflat world sheet reads
b i52«gi1b i , b i5Cjki gjgk1••• ~15!
where «→01 is a regularizing parameter; e.g., in dimesional regularization, the dimensionality of the world sheeassumed to bed522«. As is clear from Eqs.~15!, « playsthe role of a small anomalous dimension that makes theeratorVi slightly relevant. According to standard renormaization theory, counterterms can be expanded in poles in«,and we make the standard dimensional-regularizationsumption that only single poles matter, while higher-ordpoles cancel among themselves.
The fact that the world sheetS is in general curved im-plies that one has to choose a regularization scalem which islocal on the world sheet@2#, as is standard in stringys mod-els @14#. The crucial next step in our approach@2# is to pro-mote the scalem(s) to a dynamical world-sheet fieldfwhich should appear in a world-sheet path integral, takthe form of Liouville string theory@15#. Because of itstarget-space signature, which isnegativefor the supercriticalstrings@16# we consider, the world-sheet zero mode of thfield is identified with target time.
The conformal invariance of the stringys model may berestored by Liouville dressing@15#, leading to the followingequation for the dressed couplingsl i(f,gi):
l i1Q l i52b i ~16!
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whereQ2 is the central-charge deficit of the non-conformtheory@15,16#.3 When one identifies the Liouville field equation ~16! with a curved-world-sheet renormalization-grouflow @2#, the Liouville-dressed couplingsl i may be identi-fied with appropriately renormalizeds-model couplingsgion curved world sheets.
As was shown in@2#, the summation over possible topologies of the world sheet induces canonical quantization oftheory-space coordinates$gi%, much as the couplings in locafield theories are quantized in the presence of wormhole fltuations in space-time@17#. This canonical quantization follows from a certain set of Helmholtz conditions in theospace@2#, which are obeyed provided theb i functions obeythe gradient-flow property
b i5Gi j]C@g#
]gj~17!
where C@g,t# is the ZamolodchikovC function @18#, arenormalization-group invariant combination of averag@with respect to Eq.~14!# of components of the world-sheestress tensor of the deformeds model,Gi j is the inverse ofthe Zamolodchikov metric in theory space,
Gi j 5^ViVj&, ~18!
and the angular brackets denote any expectation valuethe partition function of the deformeds model~14!, summedover world-sheet genera. The ZamolodchikovC functionplays the role of the off-shell effective target-space actionstring theory, which allows the identification
C@g#5E dt~pigi2E! ~19!
whereE is the Hamiltonian operator of the string matter.critical string theories, the couplingsgi are exactly marginal,b i50, and this formalism has trivial content, but this is nlonger the case@2# when one goes beyond critical strings.
Renormalizability of the world-sheets model implies thescale independence of physical quantities, such as the ption function or the density matrixr(gi ,pj ,t) of a stringmoving in the background parametrized by the$gi%, viewedas generalized coordinates in string theory space, with thpjthe canonically conjugate momenta in this space, whichassociated with the vertex operatorsVj in a subtle sense@2#.Let t5 lnm0 be the ~zero mode of! the world-sheetrenormalization-group scale. Since the elements of the dsity matrix are physical quantities, and hence independenthe world-sheet scale, they must obey@2# the followingrenormalization-group equation:
d
dtr~gi ,pj ,t !5
]
]tr1gi
]
]gir1 pi
]
]pir50 ~20!
3We used this approach in@2# to derive a stochastic FokkerPlanck equation with diffusion for the corresponding probabildistribution in the theory space$l i%.
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JOHN ELLIS, N. E. MAVROMATOS, AND D. V. NANOPOULOS PHYSICAL REVIEW D63 024024
where an overdot denotes a partial derivative with respect. The totalt derivative above incorporatest dependence bothexplicitly and implicitly, through running couplings and generalized momenta.4
As an example how non-critical string dynamics enteone may consider thereduced density matrixrs(gs
i ,ps, j ,t)of an effective string theorydescribing lowest-level stringmodes characterized by asubset$gs
i %P$gi% of couplings tooperators$Vs
i % that are not exactly marginal. These deformtions are remnants of mixtures of these lowest-level mowith higher ~Planckian! modes of the string, which are initially exactly marginal. However, this property is lost whethe higher-level modes are integrated over, so that the rnant $Vs
i % provide a non-trivial background ‘‘environmentfor the observable lowest-level modes to propagate throuThis ‘‘gravitational environment’’ of higher-level statesquantized when one sums over world-sheet genera. Wemade case studies of such systems, based on stringy bholes in two space-time dimensions@19#,5 and higher-dimensional analogues provided by the recoil@11# ofD-branes@20# when struck by a light propagating strinstate. At present, we only discuss some generic propertiethis Liouville approach to space-time foam@2#.
The non-critical subsystem$gs% acquires non-trivial dy-namics through Eq.~20!, since theb i functions are non-trivial. Summing over genera, taking into account thenonical quantization of the theory space mentioned aboand identifying the Liouville renormalization-group scale athe target time variable@2#, one arrives at the following evolution equation for the reduced density matrixoperatorrs ofthe observable, propagating, localized, lowest-level modethe effective string theory:
] trs~gs ,ps ,t !5 i @rs ,H#1 i bsi Gi j @gs
j ,rs# ~21!
whereH is a Hamiltonian operator for the subsystem cosisting of propagating string modes, and the second termthe right-hand side of Eq.~21! is an explicit string representation for d”H in Eq. ~1!. It is clear that our fundamentaequation~21! stems from the requirement of renormalizabity ~20!.
This description has several important properties, whwe now describe. For notational convenience in what flows, we omit the sub-indexs in the couplings and conjugatmomenta, i.e.,gs→gi , etc., but we always imply quantitiein the subsystems, unless otherwise stated. We remind t
4Here we study the general consequences of Eq.~20!, although itis possible that only the diagonal elements of the density matrithe coordinate representation, which may be interpreted as probity densities, are in fact measurable physical quantities, in whcase the requirement~20! should be applied only to the world-shepartition function of the stringys model.
5In this case the environment is provided by discrete delocalisolitonic states, which mix explicitly with lowest-level propagatinmatter states of the two-dimensional string in marginal deformtions of the two-dimensional black hole, as a reflection of infinidimensionalW` gauge symmetries@2#.
02402
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reader that in the non-critical string theory model discushere the full system is conformal, and as such the ti~renormalization-group! evolution ~20! acquires trivial con-tent.
Probability conservation. The total probability P5*dpldgl Tr@r(gi ,pj )# is conserved, because
P5E dpldgl TrF ]
]pi~Gi j b
jr!G ~22!
can receive contributions only from the boundary of phaspace, which must vanish for an isolated system.
Entropy growth. The entropyS52Tr(r lnr) is not con-served, to the extent that relevant couplings withb iÞ0 arepresent,
S5~b iGi j bj !S, ~23!
implying a monotonic increase for unitary world-sheet theries for whichGi j is positive definite. We see from Eq.~23!that any running ofany coupling will lead to an increase inentropy, and we have interpreted@2# this behavior in terms ofquantum models of friction. The increase~23! in the entropycorresponds to a loss of quantum coherence, which isknown in models.
Statistical conservation of energy. The most importantproperty for our purposes here is that energy isat bestcon-served statisticallyon the average@2#. In other words, onehas explicit time dependence~dissipation! in the Hamiltonianoperator of the subsystem, thereby allowing flow of eneto the environment at an operator level. Then, it isat bestonly the right-hand equality in Eq.~10! that is valid, depend-ing on the specific model.6
To see this, we recall, as discussed above Eq.~20!, thatthe renormalizability of thes model impliesd/dt Trr50.Since we identify the target time with the renormalizatiogroup scalet on the world sheet@2#, ] t Trrs may be ex-pressed in terms of the renormalized couplingsgi by meansof the evolution equation~21!. We now compute
]
]t^^E&&5
]
]tTr~Er!5^^] t~E2b iGi j b
j !&& ~24!
where E is the Hamiltonian operator, and ^ . . . &&[Tr@r(•••)#. In deriving this result, we took into accounthe evolution equation~21! and the quantization rules intheory space@2#,
@gi ,gj #50, @gi ,pj #52 id i j , ~25!
as well as the fact that in strings models the quantum operatorsb iGi j are functionals only of the coordinatesgi , andnot of the generalized momentapi . We also note that, in ouapproach, the total time derivative of an operatorQ is givenas usual by
inil-h
d
-- 6See the next section for a question mark that hangs over evenstatistical equality.
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HOW LARGE ARE DISSIPATIVE EFFECTS IN . . . PHYSICAL REVIEW D 63 024024
d
dtQ52 i @Q,E#1
]Q
]t. ~26!
We recall that total time derivatives incorporate both expland implicit renormalization-scale dependence~via runningcouplings!, while partial time derivatives incorporate onthe explicit dependence.
Using theC-theorem results~17!,~19! @18,2# and the for-malism developed in@2#, it is straightforward to arrive at
]
]t^^E&&5
]
]t~pib
i !. ~27!
In conventional stringys models, as a result of world-sherenormalizability with respect to a ‘‘flat’’ world-sheet cutofany dependence on the renormalization group scale in thb i
functions is implicit through the renormalized couplings, ahence] tb
i50. Moreover, the quantitypi appearing in Eq.~27! may be written in the form
pi5Gi j bj;(
nCii 1••• i n
gi 1•••gi n ~28!
where theCi j ••• are the~totally symmetric! correlators ofvertex operatorsViVj . . . &. In the usual case, as a resultthe renormalizability of thes-model theory, there is no explicit dependence on the world-sheet scalet in such correla-tors or on b i , and hence the right-hand side of Eq.~27!vanishes, implyingenergy conservation on the average.
In this derivation, renormalizability replaces thetime-translation invariance of conventional target-space fieltheory.
B. Curved world-sheet renormalizationand generic Liouville correlators
An additional feature appears in certain non-critical strtheories that involve solitonic structures in their bacgrounds, such asD particles@21,11,22#. There are deformations in the set of$Vi% that obey a logarithmic conformaalgebra@23#, rather than an ordinary conformal algebra.discussed in detail in@24#, the field correlatorsCi 1••• i m
insuch logarithmic conformal field theories exhibitexplicit de-pendences on the world-sheet renormalization~time! scalet.This, in fact, is essential in guaranteeing the gradient-flproperty~17! of the correspondingb functions@24#, which iscrucial for canonical quantization in theory space@2#, as wediscussed previously. As a result, our previous argumentenergy conservation on the average breaks down in theowith logarithmic deformations, because the right-hand sof Eq. ~27! is no longer non-zero. Physically, this is eplained by the flow of energy from the propagating susystem to the recoilingD-particle background@22#.
In standard~critical! string theory, the correlatorsCi 1••• i mare associated with elements in theSmatrix for particle scat-tering, and as such should not exhibit any explicit depdence on the time coordinate. In view of their explicit depedence on the world-sheet scale in logarithmic conformal fitheories, which in our approach@2# is identified with the
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target time, such correlators cannot be interpreted as contional scattering amplitudes in target space, but rathernon-factorizableS” matrix elements. The time dependenalso means that, while the initial formulations of quantugravitational dissipation@1,3,4# assumed energy conservtion, this can no longer be guaranteed in string soliton mels of space-time foam@11#.
There is one more formal reason for relaxing strict eneconservation in Liouville strings, which we review belo@25#. The discussion of the previous subsection pertainedflat world sheets, but the situation is different when one csiders generic correlators in Liouville strings, becauseworld sheet has curvature expressed essentially by thenamical Liouville mode. A more correct approach is to cosider renormalization in curved space@14#, which leads tonew types of counterterms. It is just this feature that mlead to violations of the energy conservation, as happexplicitly in the D-brane case@22,24#, which is a particularcase of Liouville strings. Here we briefly review the sitution, concentrating on those aspects of the formalismevant to energy conservation, referring the reader interein more details to the literature@25,2#.
We consider theN-point correlation function of vertexoperators in a generic Liouville theory, viewing the Liouvilfield as a local renormalization-group scale on the wosheet@2#. Standard computations@26# yield the followingform for anN-point correlation function of vertex operatorintegrated over the world sheet:Vi[*d2zVi(z,z):
AN[^Vi 1•••Vi N
&m
5G~2s!msK S E d2zAgeaf D s
Vi 1•••Vi NL
m50~29!
where the tilde denotes removal of the zero mode ofLiouville field f. The world-sheet scalem is associated withcosmological-constant terms on the world sheet, whichcharacteristic of the Liouville theory, and the quantitys is thesum of the Liouville anomalous dimensions of the operatVi :
s52(i 51
Na i
a2
Q
a, a52
Q
21
1
2AQ218. ~30!
The G function appearing in Eq.~29! can be regularized@27,2# for negative-integer values of its argument by analycontinuation to the complex plane using the the Saaschcontour of Fig. 1.
To see technically why the above formalism leads tobreakdown in the interpretation of the correlatorAN as astring amplitude orS-matrix element, which in turn leads tthe interpretation of the world-sheet partition function asprobability density rather than a wave function in targspace, one first expands the Liouville field in~normalized!eigenfunctions$fn% of the LaplacianD on the world sheet
f~z,z!5(n
cnfn5c0f01 (nÞ0
cnfn , f0}A21/2,
~31!
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JOHN ELLIS, N. E. MAVROMATOS, AND D. V. NANOPOULOS PHYSICAL REVIEW D63 024024
whereA is the world-sheet area, and
Dfn52enfn , n50,1,2,. . . ,
e050, ~fn ,fm!5dnm . ~32!
The correlation functions~without the Liouville zero mode!appearing on the right-hand side of Eq.~29! take the form@25#
AN}E )nÞ0
dcnexpS 21
8p (nÞ0
encn22
Q
8p (nÞ0
Rncn
1 (nÞ0
a ifn~zi !cnD F E d2jAg expS a (nÞ0
fncnD G s
~33!
whereRn5*d2jR(2)(j)fn . We can compute Eq.~33! if weanalytically continue@26# s to a positive integers→nPZ1. Denoting
f ~x,y
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HOW LARGE ARE DISSIPATIVE EFFECTS IN . . . PHYSICAL REVIEW D 63 024024
We now obtain a non-zero contribution to Eq.~27! froman apparent non-trivial explicitt dependence in]gib i throughthe coefficientsCi 1••• i m
i of the b i functions, associated with
the breakdown of their identification as well-defineS-matrix elements, in contrast to the conventional strcase. As discussed in@2#, one can explicitly verify this pic-ture in the case ofD-particle recoil. The scattering of a lowenergy~string! matter particle off aD particle conserves energy only in the complete system, when the recoil of theDparticle is taken properly into account@22#. The recoil de-grees of freedom are entangled with the subsystem of lenergy string matter, and their neglect leads to an expviolation of the energy conservation condition at an operalevel. Energy conservation canat bestbe imposed in asta-tistical average sense.
C. Can one restore the Lindblad formalismin non-critical string?
We have seen that the non-critical-string time evolut~21! is nothing other than a world-sheet renormalizatiogroup evolution equation in coupling constant space$gi% forthe s model and, in view of the failure of energy conservtion, discussed above, cannot in general be written inLindblad form~9!. There are, however, some special casewhich the equation can indeed be put in a Lindblad form,even in such cases, as we shall explain below, the ordemagnitude of the dissipative effects is still given by Eq.~2!andnot Eq. ~3!.
We first recall from our above discussion that the nocritical string evolution equation~21! represents diffusiveevolution in theory space of the non-critical string$gi%.Hence, as explained in detail in@2#, there is spontaneoulocalization in such a space@12,13#. Thus, even if the situation resembled that of Lindblad, it would not have beensociated with energy-driven diffusion, as was the casecussed in@6#, but with spontaneous localization models@8#.This can be seen straightforwardly from the form~21! for thetime evolution of the density matrix in non-critical strintheory. In this case, there is no ‘‘environment operator’’ thcommutes with the Hamiltonian, for the simple reason tthe role ofbi is played in Liouville strings by various partitions ~within a generalized definition of the quantumordering prescription for the operatorsgi) of Gi j b
j
5Cii 1••• i mgi 1
•••gi m. The operatorsgi play the role of posi-tion operators in a generalized coupling-constant space,as such, thegi do not commute withH in general, whichdepends on the generalized momenta in theory spacepi .Hence our non-critical string model for decoherence doesrespect the Lindblad criterion~11!.
It can easily be seen that a double-commutator struccould only arise in a situation whereonly the linearanomalous-dimension terms are kept inb i5yigi1•••, withno sum overi. In that case, making the antisymmetric ordeing prescription denoted by :•••:, the diffusive term in theLiouville string evolution equation~21! does acquire adouble-commutator structure
:d”H: ; yi@gi ,@gi ,r##. ~40!
02402
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In the two-state case studied in@6,8#, this leads to an estimatof the dissipative effects as being of orderO(yi(Dgi)2),whereDgi is the variance in theory space, which shoulddistinguishedfrom real position variances. Indeed, in respace,Dgi may be considered of order 1, and this leavesanomalous dimension factoryi to determine the order of theeffect. This is due to the fact that, as discussed in@2#, theLiouville string case is characterized by the appearancepointer states@13# in theory space, namely wave packewith Dgi;hs , wherehs is the ‘‘Planck constant’’ in theoryspace, which is found@22# to be proportional to the stringcoupling gs . This latter property can easily be understofrom the fact that quantization in theory space is inducedour approach by higher-genus topologies on the world-shand, hence, string-loop interactions@2#. In conventionalstring models,gs is of order 1, namelygs
2/4p51/20. How-ever, one may consider more general cases in whichgs is aphenomenological parameter, to be constrained by data,those on neutrinos@6#.
In generic non-critical string models, the operators corsponding to thegi are (1,1) on the world sheet, so thatyi
50, but not exactly marginal, which implies that only thethree- and higher-point-function contributions are non-zin b i5Cjk
i gjgk1•••. The latter terms clearly do not lead ta double commutator structure~40,12!. The order of the cor-responding effects can, however, be estimated by thethat the correlatorsCi 1••• i m
i are viewed as~non-factorizable!
S” -matrix elements in target space, and as such can bepanded in a power series ina8k2 in the interesting closedstring case, wherek is a typical target-space four-momentuscale,a851/Ms
2 the Regge slope, andMs the string scale.This yields once more the estimate~2! for string-induceddissipative effects.
This estimate is supported by many specific exampFor instance, in the context of the two-dimensional blachole model of@19#, the analysis of@2# showed that theex-actly marginal world-sheet correlators involvenecessarilythe coupling of low-energy propgating modes with Plancksolitonic modes. The world-sheet correlators involving tlatter are suppressed to leading order by a single powethe Planck massM P .7 This coupling is necessitated bstringy gauge symmetries, specificallyW` symmetries,which were argued in@2# to be responsible for maintaininquantum coherence at the microscopic string level, but nothe level of the low-energy effective theory relevant to oservation. This in turn implies that the splitting between loenergy propagating modes and quantum-gravitational mois suppressed by a single inverse power ofMs or M P , lead-ing again to the above estimate for the magnitude ofCi 1••• i n
i ,
where thegi n refer to low-energy propagating matter modeIn a similar spirit, the recoil approach to D-brane and striscattering@21,11#, which is another example of a gravita
7Strictly speaking, the string scaleMs , but we do not draw thedistinction here.
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JOHN ELLIS, N. E. MAVROMATOS, AND D. V. NANOPOULOS PHYSICAL REVIEW D63 024024
tional medium, also suggests that dissipative effects arepressed by a single inverse power of the gravitational mscale, as in Eq.~2!.
It goes without saying, however, that there is alwayspossibility of a cancellation in some specific case, so thatsuppression is by some higher power of the gravitatioscale as in Eq.~3!, but we see no reason why this shouldgeneric.
IV. NONLINEARITIES AND COMPLETE POSITIVITY
We conclude with some remarks on nonlinearitiesLiouville string dynamics. It has been pointed out@10# thatthe naive extension of a single-particle approach to twparticle systems may not respect CP, and constraints onparametrization of dissipative effects in single-particle stems have been proposed. However, as we argue belomay be an oversimplification to ignore the likelhiood of nolinearities in the quantum-gravitational framework, whiwould require the issue of complete positivity@10# to bere-evaluated@9#.
An important indication of the possible importancenonlinear environmental effects comes from the form ofevolution equation~21! for the reduced density matrix in thcontext of Liouville strings. The appearance of theGi j;^ViVj& term given by Eq.~18! in the dissipative part of Eq~21! is a signature ofnonlinearHartree-Fock evolution, sincethe expectation values & are to be evaluated with respeto a world-sheet partition functionC@gi ,t# of the string thatis resummed over genera. According to the detailed discsion in @29#, such a resummed world-sheet partition functimay be identified as the trace of the density matrixr, with aprobability distribution in theory space. As such, the dissitive aspects of the evolution exhibit a nonlinear integdifferential form
] trs{~TrgirsViVj !b
i@gsj ,rs# ~41!
where Trgidenotes a partial trace over quantum fluctuatio
about the string background$gi% @2,22#,
Trgi~••• 
rietson
s-e
.
HOW LARGE ARE DISSIPATIVE EFFECTS IN . . . PHYSICAL REVIEW D 63 024024
If only the refractive index effect~43! is present and thequantum-gravitational mass parameterM is flavor indepen-dent, there would be no practical consequences for neutoscillation physics. However, consequences would ensuM is flavor dependent or if there are also diffusive effecWe plan to return to these issues in a future publicati
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ACKNOWLEDGMENTS
We thank Eligio Lisi for arousing our interest in this quetion. The work of N.E.M. is partially supported by thP.P.A.R.C.~U.K.!. The work of D.V.N. is partially supportedby DOE grant DE-F-G03-95-ER-40917. N.E.M. and D.V.Nalso thank H. Hofer for his interest and support.
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