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Decoherence of Correlation Histories

arXiv:gr-qc/9302013v1 10 Feb 1993Decoherence of Correlation HistoriesEsteban Calzetta ∗B. L. Hu †AbstractWe use a λΦ4 scalar quantum field theory to illustrate a new approach tothe study of quantum to classical transition.

In this approach, the decoherencefunctional is employed to assign probabilities to consistent histories defined interms of correlations among the fields at separate points, rather than the fielditself. We present expressions for the quantum amplitudes associated with suchhistories, as well as for the decoherence functional between two of them.

Thedynamics of an individual consistent history may be described by a Langevin-type equation, which we derive.Dedicated to Professor Brill on the occasion of his sixtieth birthday, August 19931.Introduction1.1.Interpretations of Quantum Mechanics and Paradigmsof Statistical MechanicsThis paper attempts to bring together two basic concepts, one from the foundationsof statistical mechanics and the other from the foundations of quantum mechanics,for the purpose of addressing two basic issues in physics:1) the quantum to classical transition, and2) the quantum origin of stochastic dynamics.Both issues draw in the interlaced effects of dissipation, decoherence, noise, and fluc-tuation. A central concern is the role played by coarse-graining –the naturalness of itschoice, the effectiveness of its implementation and the relevance of its consequences.On the fundations of quantum mechanics, a number of alternative interpretationsexists, e.g., the Copenhagen interpretation, the many-world interpretation [1], theconsistent history interpretations [2], to name just a few (see [3] for a recent review).∗IAFE, cc 167, suc 28, (1428) Buenos Aires, Argentina†Department of Physics, University of Maryland, College Park, MD 20742, USA1

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESThe one which has attracted much recent attention is the decoherent history approachof Gell-Mann and Hartle [4]. In this formalism, the evolution of a physical system isdescribed in terms of ‘histories’: A given history may be either exhaustive (defininga complete set of observables at each instant of time) or coarse-grained.

While inclassical physics each history is assigned a given probability, in quantum physics aconsistent assignment of probabilities is precluded by the overlap between differenthistories. The decoherence functional gives a quantitative measure of this overlap;thus the quantum to classical transition can be studied as a process of “diagonaliza-tion” of the decoherence functional in the space of histories.On the foundational aspects of statistical mechanics, two major paradigms are oftenused to describe non-equilibrium processes (see, e.g., [5, 6, 7, 8]): the Boltzmanntheory of molecular kinetics, and the Langevin (Einstein-Smoluchowski) theory ofBrownian motions.

The difference between the two are of both formal and conceptualcharacter.To begin with, the setup of the problem is different: In kinetic theory one studies theoverall dynamics of a system of gas molecules, treating each molecule in the system onthe same footing, while in Brownian motion one (Brownian) particle which definesthe system is distinct from the rest, which is relegated as the environment.Theterminology of ‘revelant’ versus ‘irrevelant’ variables highlights the discrepancy.The object of interest in kinetic theory is the (one-particle) distribution function (orthe nth-order correlation function), while in Brownian motion it is the reduced densitymatrix. The emphasis in the former is the correlation amongst the particles, while inthe latter is the effect of the environment on the system.The nature of coarse-graining is also very different: in kinetic theory coarse-grainingresides in the adoption of the molecular chaos assumption corresponding formallyto a truncation of the BBGKY hierarchy, while in Brownian motion it is in theintegration over the environmental variables.

The part that is truncated or ‘ignored’is what constitutes the noise, whose effect on the ‘system’ is to introduce dissipationin its dynamics. Thus the fluctuation-dissipation relation and other features.Finally the philosophy behind these two paradigms are quite different: In Brownianmotion problems, the separation of the system from the environment is prescribed:it is usually determined by some clear disparity between the two systems.

Thesemodels represent “autocratic systems”, where some degrees of freedom are morerelevant than others. In the lack of such clear distinctions, making a separation ‘byhand’ may seem rather ad hoc and unsatisfactory.

By contrast, models subscribingto the kinetic theory paradigm represent “democratic systems”: all particles in agas are equally relevant. Coarse-graining in Boltzmann’s kinetic theory appears lesscontrived, because information about higher correlation orders usually reflects thedegree of precision in a measurement, which is objectively definable.2

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESIn the last five years we have explored these two basic paradigms of non-equilibriumstatistical mechanics in the framework of interacting quantum field theory with theaim of treating dissipative processes in the early universe [9, 10] and decoherenceprocesses in the quantum to classical transition issue [11]. Here we have begun toexplore the issues of decoherence with the kinetic model.Because of the difference in approach and emphasis between these two paradigms andin view of their fundamental character, it is of interest to build a bridge between them.We have recently carried out such a study with quantum fields [12].

By delineatingthe conditions under which the Boltzmann theory reduces to the Langevin theory,we sought answers to the following questions:1) What are the factors condusive to the evolution of a ‘democratic system’ to an‘autocratic system’ and vise versa ? A more natural set of criteria for the separationof the system from the environment may arise from the interaction and dynamics ofthe initial closed system [13, 14].2) The construction of collective variables from the basic variables, the descriptionof the dynamics of the collective variables, and the depiction of the behavior of acoarser level of structure emergent from the microstructures.

[13, 15, 16].The paradigm of quantum open systems described by quantum Brownian models hasbeen used to analyze the decoherence and dissipation processes, for addressing basicissues like quantum to classical transitions, fluctuation and noise, particle creationand backreaction, which arise in quantum measurement theory [17, 18, 19], macro-scopic quantum systems [20], quantum cosmology [21] (for earlier work see referencesin [22]), semiclassical gravity [23, 24, 25], and inflationary cosmology [26]. The readeris referred to these references and references therein for a description of this line ofstudy.The aim of this paper is to explore the feasibility for addressing the same set ofbasic issues using the kinetic theory paradigm.

We develop a new approach basedon the application of the decoherence functional [2, 4] formalism to histories definedin terms of correlations between the fundamental field variables. We shall analysethe decoherence between different histories of an interacting quantum field, a λΦ4theory here taken as example, corresponding to different particle spectra and studyissues on the physics of quantum to classical transition, the relation of decoherenceto dissipation, noise and fluctuation, and the quantum origin of classical stochasticdynamics.1.2.Quantum to Classical Transition and Coarse-GrainingOne basic constraint in the building of quantum theory is that it should reproduceclassical mechanics in some limit.

(For a schematic discussion of the different criteriaof classicality and their relations, see [27]). Classical behavior can be characterized3

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESby the existence of strong correlations between position and momentum variablesdescribed by the classical equations of motion [28] and by the absense of interferencephenomena (decoherence).Recent research in quantum gravity and cosmology have focussed on the issue of quan-tum to classical transition. This was highlighted by quantum measurement theory forclosed systems (for a general discussion, see, e.g., [29]), the intrinsic incompatibilityof quantum physics with general relativity [30], and the quantum origin of classicalfluctuations in explaining the large scale structure of the Universe.

Indeed, in theinflationary models of the Universe [31], one hopes to trace all cosmic structures tothe evolution from quantum perturbations in the inflaton field. More dramatically,in quantum cosmology [32] the whole (classical) Universe where we now live in isregarded as the outcome of a quantum to classical transition on a cosmic scale.

Inthese models, one hopes not only to explain the ‘beginning’ of the universe as aquantum phenomenon, but also to account for the classical features of the presentuniverse as a consequence of quantum fluctuations. This requires not only a theoreti-cal understanding of the quantum to classical transition issue in quantum mechanics,but also a theoretical derivation of the laws of classical stochastic mechanics fromquantum mechanics, the determination of the statistical properties of classical noise(e.g., whether it is white or coloured, local or nonlocal) being an essential step in theformulation of a microscopic theory of the structure of the Universe [26].Our understanding of the issue of quantum to classical transition has been greatlyadvanced by the recent development of the decoherent histories approach to quantummechanics [4].

An essential element of the decoherent histories approach is that theoverlap between two exhaustive histories can never vanish. Therefore, the discussionof a quantum to classical transition can only take place in the framework of a coarsegrained description of the system, that is, giving up a complete specification of thestate of the system at any instant of time.As a matter of fact, some form of coarse graining underlies most, if not all, successfulmacroscopic physical theories.

This fact has been clearly recognized and exploited atleast since the work of Nakajima and Zwanzig [33, 6] on the foundations of nonequi-librium statistical mechanics.Like statistical mechanics, the decoherent historiesapproach allows a variety of coarse-graining procedures; not all of these, however, areexpected to be equally successful in leading to interesting theories. Since the pre-scription of the coarse graining procedure is an integral part of the implementation ofthe decoherent histories approach, the development and evaluation of different coarsegraining strategies is fundamental to this research program.When we survey the range of meaningful macroscopic (effective) theories in physicsarising from successfully coarse-graining a microscopic (fundamental) theory, one par-ticular class of examples is outstanding; namely, the derivation of the hydrodynamicaldescription of dilute gases from classical mechanics.

The crucial step in deriving the4

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESNavier-Stokes equation for a dilute gas consists in rewriting the Liouville equationfor the classical distribution function as a BBGKY hierarchy, which is then truncatedby invoking a ‘molecular chaos’ assumption. If the truncation is made at the levelof the two-particle reduced distribution function, the Boltzmann equation results.

Inthe near-equilibrium limit, this equation leads to the familiar Navier-Stokes theory.We must stress that in this general class of theories exemplified by Boltzmann’s work,coarse-graining is introduced through the truncation of the hierarchy of distributionfunctions; i.e., by neglecting correlations of some order and above at some singled-outtime [5, 6]. This type of coarse graining strategy is qualitatively different from thoseused in most of the recent work in quantum measurement theory and cosmology,which invoke a system-bath, space-time, or momentum-space separation.

In most ofthese cases, an intrinsically justifiable division of the system from the environment islacking and one has to rely on case-by-case physical rationales for making such splits. (An example of system-bath split is Zurek’s description of the measurement processin quantum mechanics, where a bath is explicitly included to cause decoherence inthe system-apparatus complex [17].

Space-time coarse graining has been discussedby Hartle [34] and Halliwell et al [35]. An example of coarse-graining in momentumspace is stochastic inflation [36], where inflaton modes with wavelenghts shorter thanthe horizon are treated as an environment for the longer wavelenght modes [37]).1.3.Coarse-Graining in the Hierarchy of CorrelationsIn this paper we shall develop a version of the decoherent histories approach where thecoarse-graining procedure is patterned after the truncation of the BBGKY hierarchyof distribution functions.

For simplicity, we shall refer below to the theory of a singlescalar quantum field, with a λΦ4- type nonlinearity.The simplest quantum field theoretical analog to the hierarchy of distribution func-tions in statistical mechanics is the sequence of Green functions (that is, the expec-tation values of products of n fields) [9]. In this approach, the BBGKY hierarchyof kinetic equations is replaced by the chain of Dyson equations, linking each Greenfunction to other functions of higher order.The analogy between these two hierarchies is rendered most evident if we introduce“distribution functions” in field theory through suitable partial Fourier transforma-tion of the Green functions.

Thus, a “Wigner function” [38] may be introduced asthe Fourier transform of the Hadamard function (the symmetric expectation valueof the product of two fields) with respect to the difference between its arguments. Itobeys both a mass shell constraint and a kinetic equation, and may be regarded asthe physical distribution function for a gas of quasi particles, each built out of a cloudof virtual quanta.

Similar constructs may be used to introduce higher “distributionfunctions” [9, 7].5

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESAs in statistical mechanics, the part of a given Green function which cannot be re-duced to products of lower functions defines the corresponding “correlation function”.Thus the chain of Green functions is also a hierarchy of correlations.To establish contact between the hierarchy of Green functions and the decoherenthistories approach, let us recall the well-known fact that the set of expectation valuesof all field products contains in itself all the information about the statistical stateof the field [9]. For a scalar field theory with no symmetry breaking, we can evennarrow this set to products of even numbers of fields.This result suggests thata history can be described in terms of the values of suitable composite operators,rather than those of the fundamental field.

If products to all orders are specified(binary, quartet, sextet, etc), then the description of the history is exhaustive, anddifferent histories do not decohere. On the other hand, when some products are notspecified, or when the information of higher correlations are missing, which is oftenthe case in realistic measurement settings, the description is coarse-grained, whichcan lead to decoherence.In this work we shall consider coarse-grained histories where the lower field prod-ucts (binary, quartic) are specified, and higher products are not.

Decoherence willmean that the specified composite operators can be assigned definite values with con-sistent probabilities. Higher composite operators retain their quantum nature, andtherefore cannot be assigned definite values.

However, their expectation values canbe expressed as functionals of the specified correlations by solving the correspond-ing Dyson equations with suitable boundary conditions. This situation is exactlyanalogous to that arising from the truncated BBGKY hierarchy, where the molecularchaos assumption allows the expression of higher distribution functions as functionalsof lower ones ( e.g., [5, 6]).For those products of fields which assume definite values with consistent probabili-ties, these values can be introduced as stochastic variables in the dynamical equa-tions for the other quantities of interest (usually of lower correlation order).

Thisapproach would provide a theoretical basis for the derivation of the equations ofclassical stochastic dynamics from quantum fields.It can offer a justification (orrefutation) for a procedure commonly assumed but never proven in some populartheories like stochastic inflation [36, 37]. Moreover, since in general we shall obtainnontrivial ranges of values for the specified products with nonvanishing probabili-ties, it can be said that our procedure captures both the average values of the fieldproducts and the fluctuations around this average.

The statistical nature of thesefluctuations is a subject of great interest in itself [26].There is another conceptual issue that our approach may help to clarify. As we havealready noted, in the system-bath split approach to coarse-graining, as well as in re-lated procedures, it is crucial to introduce a hierarchical order among the degrees offreedom of the system, in such a way that some of them may be considered relevant,6

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESand others irrelevant. While it is often the case that the application itself suggestswhich notions of relevance may lead to an interesting theory, in a quantum cosmo-logical model, which purports to be a “first principles” description of our Universe,all these choices are, in greater or lesser degree, arbitrary.

Since correlation functionsalready have a “natural” built-in hierarchical ordering, in this approach the ‘arbi-trariness’ is reduced to deciding on which level this hierarchy is truncated, and thatin turn is determined by the degree of precision one carries out the measurement. Inmost case one still needs to show the robustness of the macroscopic result against thevariance of the extent of coarse-graining, and exceptional situations do exists (an ex-ample is the long time-tail relaxation behavior in multiple particle scattering of densegas, arising from a failure of the simple molecular chaos assumption).

But in generalterms correlational coarse-graining seems to us a less ad hoc procedure compared tothe commonly used system-bath splitting and coarse-graining.This paper is organized as follows: In Sec. 2 we discuss the implementation of ourprocedure for the simple case of a λΦ4 theory in flat space time.

We then derivethe formulae for the quantum amplitude associated with a set of correlation historiesand the decoherence functional between two such histories. In Sec.

3 we discuss thedecoherence of correlation histories between binary histories and derive the classicalstochastic source describing the effect of higher-order correlations on the lower-orderones, arriving at a Langevin equation for classical stochastic dynamics. In Sec.

4 wesummarize our findings.2.Quantum Amplitudes for Correlation Histories and Effec-tive Action2.1.Quantum Mechanical Amplitudes for Correlation His-toriesIn this section, we shall consider the quantum mechanical amplitudes associated withdifferent histories for a λΦ4 quantum field theory, defined in terms of the values oftime-ordered products of even numbers of fields at various space time points. Let usbegin by motivating our ansatz for the amplitudes of these correlation histories.In the conceptual framework of decoherent histories [4], the “natural” exhaustivespecification of a history would be to define the value of the field Φ(x) at every spacetime point.

These field values are c numbers. The quantum mechanical amplitudefor a given history is Ψ[Φ] ∼eiS[Φ], where S is the classical action.

The decoherencefunctional between two different specifications is given by D[Φ, Φ′] ∼Ψ[Φ]Ψ[Φ′]∗.Since |D[Φ, Φ′]| ≡1, there is never decoherence between these histories.A coarse-grained history would be defined in general through a “filter function” α,which is basically a Dirac δ function concentrated on the set of exhaustive histo-7

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESries matching the specifications of the coarse-grained history. For example, we mayhave a system with two degrees of freedom x and y, and define a coarse-grainedhistory by specifying the values x0(t) of x at all times.

Then the filter function isα[x, y] = Qt∈R δ(x(t) −x0(t)). The quantum mechanical amplitude for the coarse-grained history is defined asΨ[α] =ZDΦ eiSα[Φ](1)where the information on the quantum state of the field is assumed to have beenincluded in the measure and/or the boundary conditions for the functional integral.The decoherence functional for two coarse-grained histories is [4]D[α, α′] =ZDΦDΦ′ei(S(Φ)−S(Φ′))α[Φ]α′[Φ′](2)In this path integral expression, the two histories Φ and Φ′ are not independent; theyassume identical values on a t = T = constant surface in the far future.

Thus, theymay be thought of as a single, continuous history defined on a two-branched “closedtime-path” [39, 40, 41, 42], the first branch going from t = −∞to T, the secondfrom T back to −∞. Alternatively, we can think of Φ = Φ1 and Φ′ = Φ2 as thetwo components of a field doublet defined on ordinary space time [9], whose classicalaction is S[Φa] = S[Φ1] −S[Φ2].

This notation shall be useful later on.Let us try to generalize this formalism to correlation histories. We begin with thesimplest case, where only binary products are specified.In this case a history isdefined by identifying a symmetric kernel G(x, x′), which purports to be the valueof the product Φ(x)Φ(x′) in the given history, both x and x′ defined in Minkowskyspace - time.

By analogy with the formulation above, one would write the quantummechanical amplitude for this correlation history asΨ[G] =ZDΦ eiS Yx≫x′δ(Φ(x)Φ(x′) −G(x, x′))(3)(In this equation, we have introduced a formal ordering of points in Minkowsky space- time, simply to avoid counting the same pair twice. )But this straightforward generalization for the correlation history amplitude is un-satisfactory on at least two counts.

First, it assumes that the given kernel G canactually be decomposed (maybe not uniquely) as a product of c number real fieldsat different locations; however, we wish to define amplitudes for kernels (such as theFeynman propagator) which do not have this property. Second, (which is relatedto the first point,) it is ambiguous, since we do not have a unique way to expresshigher even products of fields in terms of binary products, and thus of applying theδ function constraint.8

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESTo give an example of this, observe that, should we expand the exponential of the ac-tion in powers of the coupling constant λ, the second order termR dx dx′ Φ(x)4Φ(x′)4could become, after integration over the delta function, eitherZdx dx′G(x, x)2G(x′, x′)2,Zdx dx′G(x, x′)4,(4)or any other combination; of course, if G could be decomposed as a product of fields,this would be unimportant.Let us improve on these shortcomings. The general idea is to accept Eq.

(3) as thedefinition of the amplitude in the restricted set of kernels where it can be applied, andto define the amplitude for more general kernels through some process of analyticalcontinuation. To this end, we must rewrite the quantum mechanical amplitude in amore transparent form, which we achieve by using an integral representation of theδ function.

Concretely, we redefineΨ[G] =ZDKZDΦ eiS+ i2Rdxdx′ K(x,x′)(Φ(x)Φ(x′)−G(x,x′))(5)where the filter function in the Gell-Mann Hartle scheme is replaced by an integrationover “all” symmetric non-local sources K. Eq. (5) is not yet a complete definition,since one must still specify both the path and the measure to be used in the Kintegration.

Performing the integration over fields, we obtainΨ[G] =ZDK ei(W [K]−(1/2)KG)(6)where W[K] is the generating functional for connected vacuum graphs with λΦ4interaction, and (∆−1 −K)−1 for propagator (see below). Here ∆−1 = −∇2 + m2 isthe free propagator for our scalar field theory (our sign convention for the flat space- time metric is −+ ++).The path integral over kernels can be computed through functional techniques.

Forexample, for a free field, λ = 0,W[K] = −i ln Det[(∆−1 −K)−1/2] + constant(7)Through the change of variables(∆−1 −K) = κG−1(8)we obtain9

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESΨ[G] = constant [Det G]−1/2 e(−i/2)∆−1G(9)When the self coupling λ is not zero, the evaluation of Ψ[G] is more involved; however,if we are interested in the leading behavior of the amplitude only, we can simplyevaluate the functional integral over K by saddle point methods. The saddle lies atthe solution to∂W[K]∂K= 12G(10)We recognize immediately that the exponent, evaluated at the saddle point, is simplythe 2 Particle Irreducible (2PI) effective action Γ, with G as propagator (see below).Including also the integration on gaussian fluctuations around the saddle, we findΨ[G] ∼[Det{ ∂2Γ∂G2}](1/2)eiΓ[G](11)This is our main result.As a check, it is interesting to compare the saddle method expression with our exactresult for free fields.

For a free field Γ[G] = (−i/2) ln Det(G) −(1/2)∆−1G, andtherefore Γ,G = (−i/2)(G−1 −i∆−1), Γ,G,G = (i/2)G−2, so[Det{ ∂2Γ∂G2}](1/2)eiΓ[G] = [DetG]−1[DetG]1/2e(−i/2)∆−1G(12)which is exactly the earlier result, Eq. (9).2.2.Quantum Amplitudes and Effective ActionsEq.

(11) is the natural generalization to correlation histories of the quantum me-chanical amplitude eiS associated to a field configuration. Let us consider its physicalmeaning.The effective action is usually introduced in Field Theory books [43] as a compactdevice to generate the Feynman graphs of a given theory.

Indeed, all Feynman graphsappear in the expansion of the generating functionalZ[J] =ZDΦei(S+JΦ)(13)in powers of the external source J [here, JΦ =R d4x J(x)Φ(x)]. Z has the physicalmeaning of a vacuum persistance amplitude: it is the amplitude for the in vacuum(that is, the vacuum in the distant past) to evolve into the out vacuum (the vacuum10

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESin the far future) under the effect of the source J. Thus, after proper normalization,|Z| will be unity when the source is unable to create pairs out of the vacuum, andless than unity otherwise.A more compact representation of the Feynman graphs is provided by the functionalW[J] = −i ln Z[J]; the Taylor expansion of W contains only connected Feynmangraphs.

Thus W developing a (positive) imaginary part signals the instability of thevacuum under the external source J.The external source will generally drive the quantum field Φ so that its matrix elementφ(x) = ⟨0out|Φ(x)|0in⟩⟨0out|0in⟩(14)between the in and out vacuum states will not be zero. Indeed, it is easy to see thatφ = ∂W∂J(15)The transformation from J to φ is generally one to one, and thus it is possible toconsider the matrix element, and not the source, as the independent variable.

This isachieved by submitting W to a Legendre transformation, yielding the effective actionΓ[φ] = W[J] −Jφ [J and φ being related through Eq. (15)].

This equation can beinverted to yield the dynamic law for φ∂Γ∂φ = −J(16)Eq. (16) shows that Γ may be thought of as a generalization of the classical action,now including quantum effects.

In the absence of external sources, the in and outvacua agree, so φ becomes a true expectation value; its particular value is found byextremizing the effective action. Indeed, in this case it can be shown that Γ is theenergy of the vacuum.Γ[φ] can be defined independently of the external source through the formula [44]Γ[φ] = S[φ] + (i/2) ln Det (∂2S∂φ2 ) + Γ1[φ](17)where Γ1 represents the sum of all one particle irreducible (1PI) vacuum graphs ofan auxiliary theory whose classical action is obtained from expanding the classicalaction S[φ + ϕ] in powers of ϕ, and deleting the constant and linear terms.

Eq. (17)shows that Γ is related to the vacuum persistance amplitude of quantum fluctuationsaround the matrix element φ.Therefore, an imaginary part in Γ also signals avacuum instability.

This situation closely resembles the usual approach to tunneling11

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESand phase transitions, where an imaginary part in the free energy signals the onsetof instability [45].Observe that each of the transformations from Z to W to Γ entails a drastic sim-plification of the corresponding Feynman graphs expansions, from all graphs in Z toconnected ones in W and to 1PI ones in Γ. Roughly speaking, it is unneccessary toinclude non 1PI graphs in the effective action, because the sum of all one-particleinsertions is already prescribed to add up to φ.

Now the process can be continued:if we could fix in advance the sum of all self energy parts, then we could write downa perturbative expansion where only 2PI Feynman graphs need be considered. Thisis achieved by the 2PI effective action [46].Let us return to Eq.

(13), and add to the external source a space-time dependentmass termZ[J, K] =ZDΦei(S+JΦ+(1/2)ΦKΦ)(18)where ΦKΦ =R dx dx′ Φ(x)K(x, x′)Φ(x′). Also define W[J, K] = −i ln Z[J, K].Then the variation of W with respect to J defines the in-out matrix element of thefield, as before, but now we also have∂W∂K(x, x′) = 12[φ(x)φ(x′) + GF(x, x′)](19)where GF represents the Feynman propagator of the quantum fluctuations ϕ aroundthe matrix element φ.

As before, it is possible to adopt G as the independent variable,instead of K. To do this, we define the 2PI effective action (in schematic notation)Γ[φ, GF] = W[J, K] −Jφ −(1/2)K[φ2 + GF]. Variation of this new Γ yields theequations of motion Γ,φ = −J −Kφ, Γ,GF = (−1/2)K.We can see that the 2PI effective action generates the dynamics of the Feynmanpropagator, and in this sense it plays for it the role that the classical action plays forthe field.

In this sense we can say that Eq. (11) generalizes the usual definition ofquantum mechanical amplitudes.The perturbative expansion of the 2PI effective action reads [46]Γ[φ, GF] = S[φ] + (i/2) ln DetG−1F + (12) Tr(∂2S∂φ2 GF) + Γ2[φ, GF] + constant(20)where Γ2 is the sum of all 2PI vacuum graphs of the auxiliary theory already consid-ered, but with GF as propagator in the internal lines.

As we anticipated, to replaceGF for the perturbative propagator amounts to adding all self energy insertions, andtherefore no 2PI graph needs be explicitly included.12

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESLike its 1PI predecessor, the 2PI effective action has the physical meaning of a vac-uum persistence amplitude for quantum fluctuations ϕ, constrained to have vanishingexpectation value and a given Feynman propagator. Therefore, an imaginary part inthe 2PI effective action also signals vacuum instability.The description of the dynamics of a quantum field through both φ and GF simulta-neously, rather than φ alone, is appealing not only because it allows one to performwith little effort the resummation of an infinite set of Feynman graphs, but also be-cause for certain quantum states, it is possible to convey statistical information aboutthe field through the nonlocal source K. This information is subsequently transferredto the propagator.

For this reason, the 2PI effective action formalism is, in our opin-ion, a most suitable tool to study statistical effects in field theory, particularly forout-of-equilibrium fields [9, 47]. In our earlier studies the object of interest is theon-shell effective action, that is, the effective action for propagators satisfying theequations of motion.

Here, in Eq. (11), we find a relationship between the quantummechanical amplitude for a correlation history and the 2PI effective action whichdoes not assume any restriction on the propagator concerned.2.3.Quantum Amplitudes for More General Correlation His-tories: 2PI CTP Effective ActionOne of the peculiarities of the ansatz Eq.

(3) for the amplitude of a correlation his-tory is that the kernel G must be interpreted as a time - ordered binary product offields. This results from the known feature of the path integral, which automaticallytime orders any monomials occurring within it.

Before we proceed to introduce thedecoherence functional for correlation histories, it is convenient to discuss how thisrestriction could be lifted, as well as the restriction to binary products.The time ordering feature of the path integral is also responsible for the fact thatthe c-number field φ in Sec. 2.2 is a matrix element, rather than a true expectationvalue.

As a matter of fact, the Feynman propagator GF discussed in the previoussection is also a matrix elementGF(x, x′) = ⟨0out|T[ϕ(x)ϕ(x′)]|0in⟩⟨0out|0in⟩(21)Because φ and GF satisfy mixed boundary conditions, the dynamic equations result-ing from the 2PI effective action are generally not causal. This drawback has placedlimitations in their physical applications.Schwinger [39] has introduced an extended effective action, whose arguments aretrue expectation values with respect to some in quantum state.

Because the dynam-ics of these expectation values may be formulated as an initial value problem, the13

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESequations of motion resulting from the Schwinger-Keldysh effective action are causal.Schwinger’s idea is also the key to solving the restrictions in our definition of quantumamplitudes for correlation histories.Schwinger’s insight was to apply the functional formalism we reviewed in Sec. 2.2 tofields defined on a “closed time-path”, composed of a “direct” branch −T ≤t ≤T,and a “return” branch T ≥t ≥−T (with T →∞) [39, 40].

Actually, we havealready encountered this kind of path in the discussion of the decoherence functionalfor coarse - grained histories. Since the path doubles back on itself, the in vacuum isthe physical vacuum at both ends; the formalism may be generalized to include moregeneral initial states, but we shall not discuss this possibility[9].The closed time-path integral time-orders products of fields on the direct branch, anti-time-orders fields on the return branch, and places fields on the return branch alwaysto the left of fields in the direct branch.

To define the closed time-path generatingfunctional, we must introduce two local sources Ja, and four nonlocal ones Kab (asin Sec. 2.1 an index a, b = 1 denotes a point on the first branch, while an index2 denotes a point on the return part of the path).

These sources are conjugatedto c number fields φa and propagators Gab, which stand for ⟨0in|Φa(x)|0in⟩and⟨0in|ϕa(x)ϕb(x′)]|0in⟩. Explicitly, decoding the indices, the propagators are definedas (here and from now on, we assume that the background fields φa vanish):G11(x, x′) = ⟨0in|T[Φ(x)Φ(x′)]|0in⟩(22)G12(x, x′) = ⟨0in|Φ(x′)Φ(x)|0in⟩(23)G21(x, x′) = ⟨0in|Φ(x)Φ(x′)|0in⟩(24)G22(x, x′) = ⟨0in|(T[Φ(x)Φ(x′)])†|0in⟩(25)They are, respectively, the Feynman, negative- and positive- frequency Wightman,and Dyson propagators.

The definition of the closed time-path (CTP) or in-in 2PIeffective action follows the same steps as the ordinary effective action discussed inthe previous section, except that now, besides space-time integrations, one must sumover the discrete indexes a, b.These indexes can be raised and lowered with the“metric” hab = diag(1, −1).Similarly, the “propagator” to be used in Feynmangraph expansions is the full matrix Gab, and the interaction terms should be read outof the CTP classical action S[Φ1] −S[Φ2], discussed in Sec. 2.1.In the case of vacuum initial conditions, these can be included into the path integralby tilting the branches of the CTP in the complex t plane (the direct branch should14

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESacquire an infinitesimal positive slope, and the return branch, a negative one [48]).The CTP boundary condition, that the histories at either branch should fit continu-ously at the surface t = T, may also be explicitly incorporated into the path integralas follows. We first include under the integration sign a termYx∈R3δ(Φ1(x, T) −Φ2(x, T))(26)which enforces this boundary condition; then we rewrite Eq.

(26) asexp{(−1/α2)Zd3x (Φ1(x, T) −Φ2(x, T))2}(27)where α →0. This term has the formexp{iZd4x d4x′Kab(x, x′)Φa(x)Φb(x′)},(28)whereKab(x, x′) = (i/α2)δ(x −x′)δ(t −T)[2δab −1].

(29)In this way, we have traded the boundary condition by an explicit coupling to a nonlocal external source.As before, variation of the CTP 2PI effective action yields the equations of motionfor background fields and propagators. The big difference is that now these equationsare real and causal [42, 9].We can now see how the CTP technique solves the ordering problem in the definitionof quantum amplitudes for correlation histories.

One simply considers the specifiedkernels as products of fields defined on a closed time - path. In this way, we may defineup to four different kernels Gab independently, to be identified with the four differentpossible orderings of the fields (for simplicity, we assume the background fields arekept equal to zero).

If the kernels Gab can actually be decomposed as products ofc-number fields on the CTP, then we associate to them the quantum amplitudeΨ[Gab] =ZDΦa eiSYx≫x′,abδ(Φa(x)Φb(x′) −Gab(x, x′))(30)(where S stands for the CTP classical action) The path integral can be manipulatedas in Sec. 2.1 to yieldΨ[Gab] ∼[Det{∂2Γ∂Gab∂Gcd}](1/2)eiΓ[Gab](31)15

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESwhere Γ stands now for the CTP 2PI effective action.This last expression canbe analytically extended to more general propagator quartets, and, indeed, even tokernels which do not satisfy the relationships G11(x, x′) = G21(x, x′) = G12∗(x, x′) =G22∗(x, x′) for t ≥t′, which follow from their interpretation as field products.Quantum amplitudes for correlation histories including higher order products aredefined following a similar procedure.For example, four particle correlations arespecified by introducing 16 kernels [9]Gabcd ∼ΦaΦbΦcΦd −GabGcd −GacGbd −GadGbc(32)If the new kernels are simply products of the binary ones, then the amplitude is givenbyΨ[Gab, Gabcd]=ZDΦa eiS Yabδ(ΦaΦb −Gab)Yabcdδ(ΦaΦbΦcΦd −GabGcd −GacGbd −GadGbc −Gabcd)(33)(In the last two equations, we have included the space - time index x and the branchindex a into a single multi index). Here, each pair appears only once in the product,as well as each quartet abcd.

Exponentiating the δ functions we obtainΨ[Gab, Gabcd] =ZDKabcdZDKabZDΦ exp{i[S + 12Kab(ΦaΦb −Gab)+ 124Kabcd(ΦaΦbΦcΦd −GabGcd −GacGbd −GadGbc −Gabcd)]}(34)Now the integral over fields yields the CTP generating functional for connectedgraphs, for a theory with a non local interaction term. ThusΨ[Gab, Gabcd]=ZDKabcdZDKab exp{i[W[Kab, Kabcd] −12KabGab−124Kabcd(GabGcd + GacGbd + GadGbc + Gabcd)]}(35)The integral may be evaluated by saddle point methods, the saddle being the solutionto W,Kab = (1/2)Gab, W,Kabcd =124(GabGcd + GacGbd + GadGbc + Gabcd).

To evaluatethe exponential at the saddle is the same as to perform a Legendre transform onW –it yields the higher order CTP effective action Γ[Gab, Gabcd]. Variation of Γ yieldsthe equation of motion for its arguments, which are also the inversion of the saddle16

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESpoint conditionsΓ,Gab =(−1/2)Kab −(1/4)KabcdGcdΓ,Gabcd =(−1/24)Kabcd(36)Thus up to quartic correlations, the quantum mechanical amplitude is given byΨ[Gab, Gabcd] ∼eiΓ[Gab,Gabcd](37)This expression can likewise be extended to more general kernels.As a check on the plausibility of this result, let us note the following point. Sincequantum mechanical amplitudes are additive, it should be possible to recover ourearlier ansatz Eq.

(11) for binary correlation histories from the more general resultEq. (37), by integration over the fourth order kernels.Within the saddle pointapproximation, integration amounts to substituting these kernels by the solutionto the second Eq.

(36) for the given Gab, with Kabcd = 0, and with null initialconditions. (Indeed, since initial conditions can always be included as delta function- like singularities in the external sources, the third condition is already included inthe second.) This procedure effectively reduces the fourth order effective action tothe 2PI CTP one [9], as we expected.A basic point which emerges here relevant to our study of decoherence is that, whilequantum field theory is unitary and thus time reversal invariant, the evolution ofthe propagators derived from the 2PI CTP effective action is manifestly irreversible[9, 49].

The key to this apparent paradox is that, while the evolution equations areindeed time reversal invariant, when higher order kernels are retained as indepen-dent variables, their reduction to those generated by the 2PI effective action involvesthe imposition of trivial boundary conditions in the past. Thus the origin of irre-versibility in the two point functions is the same as in the BBGKY formulation instatistical mechanics [5].

The lesson for us in the present context is that there is anintrinsic connection between dissipation and decoherence [26, 23]. Knowledge thatthe evolution of the propagators generated by the 2PI effective action is generallydissipative leads us to expect that histories defined through binary correlations willusually decohere.

We proceed now to a detailed study of this point.17

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIES3.Decoherence of Correlation Histories3.1.Decoherence Functional for Correlation HistoriesHaving found an acceptable ansatz for the quantum mechanical amplitude associatedwith a correlation history, we are in a position to study the decoherence functionalbetween two such histories. As was discussed in the Introduction, if the decoherencefunctional is diagonal, then correlation histories support a consistent probability as-signment, and may thus be viewed as classical (stochastic) histories.For concreteness, we shall consider the simplest case of decoherence among historiesdefined through (time-ordered) binary products.Let us start by considering twohistories, associated with kernels G(x, x′) and G′(x, x′), which can in turn be writtenas products of fields.

Taking notice of the similarity between the quantum amplitudesEqs. (1) and (3), we can by analogy to Eq.

(2) define the decoherence functional forsecond correlation order asD[G, G′]=ZdΦdΦ′ ei(S[Φ]−S[Φ′])Yx≫x′δ((Φ(x)Φ(x′) −G(x, x′))δ((Φ′(x)Φ′(x′) −G′∗(x, x′))(38)Recalling the expression Eq. (30) for the quantum amplitude associated with themost general binary correlation history, we can rewrite Eq.

(38) asD[G, G′] =ZDG12 DG21 Ψ[G11 = G, G22 = G′∗, G12, G21](39)This expression for the decoherence functional can be extended to arbitrary kernels.In the spirit of our earlier remarks, we use the ansatz Eq. (31) for the CTP quantumamplitude and perform the integration by saddle point methods to obtainD[G, G′] ∼eiΓ[G11=G,G22=G′∗,G120 ,G210 ](40)where the Wightman functions are chosen such that∂Γ∂G120=∂Γ∂G210= 0(41)for the given values of the Feynman and Dyson functions.

These last two equationsare the sought-for expression for the decoherence functional.As an application, let us study the decoherence functional for Gaussian fluctuationsaround the vacuum expectation value (VEV) of the propagators for a λΦ4 theory,18

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIEScarrying the calculations to two-loop accuracy. Gaussian fluctuations means that weonly need the closed time-path 2PI effective action to second order in the fluctuationsδGab = Gab −∆ab0 , where ∆ab0stands for the VEVs.Since the effective action isstationary at the VEV, there is no linear term.

FormallyΓ[δGab] = (1/2){Γ,(aa),(bb)δGaaδGbb + 2Γ,(a̸=b),(cc)δGa̸=bδGcc + Γ,(a̸=b),(c̸=d)δGa̸=bδGc̸=d}(42)so the saddle point equations (41) become{Γ,(a̸=b),(c̸=d)}δGc̸=d0= −Γ,(a̸=b),(ee)δGee(43)The formal Feynman graph expansion of the 2PI effective action is given in Eq. (20).To two-loop accuracy, we find [9]Γ2[Gab]= −λ8habcdZd4x Gab(x, x)Gcd(x, x)+iλ248 habcdhefghZd4x d4x′Gae(x, x′)Gbf(x, x′)Gcg(x, x′)Gdh(x, x′)(44)where hab, habcd = 1 if a = b = c = d = 1, −1 if a = b = c = d = 2, and vanishotherwise.Computing the necessary derivatives, we find∂2Γ∂Gab(x, x′)∂Gcd(x′′, x′′′) =(−12 )[−i(G−1)ac(x, x′′)(G−1)db(x′′′, x′)+(1/2)λhabcdδ(x′ −x)δ(x′′ −x)δ(x′′′ −x)−(i/2)λ2haceghbdfjδ(x′′ −x)δ(x′′′ −x′)Gef(x, x′)Ggj(x, x′)](45)These derivatives are evaluated at Gab = ∆ab0 , where(∆−10 )ab(x, x′)= i[hab(−∇2 + m2 −ihabǫ)δ(x′ −x)+(λ/2)habcdδ(x′ −x)∆cd0 (x, x)−(i/6)λ2haecdhbfgh∆ef0 (x, x′)∆cg0 (x, x′)∆dh0 (x, x′)]+ 12α2δ(x′ −x)δ(t −T)[2δab −1](46)where it is understood that the limits ǫ, α →0, T →∞are taken.

The first in-finitesimal is included to enforce appropiate Feynman/Dyson orderings, the secondto carry the CTP boundary conditions in the far future.19

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESIn computing the Feynman graphs in these expressions, the usual divergences cropup. They may be regularized and renormalized by standard methods, which we willnot discuss here.

The “tadpole” graph ∆cd0 (x, x) can be made to vanish by a suitablechoice of the renormalization point, which we shall assume.Let us narrow our scope to a physically meaningful set of histories, namely, thosedescribing ensembles of real particles distributed with a position-independent spec-trum f(k), k being the four momentum vector. Such ensembles are described bypropagators [9]δG(x, x′) = 2πZ( d4k(2π)4) eik(x−x′)δ(k2 + m2)f(k)(47)The distribution functions f are real, positive, and even in k. We wish to analyzeunder what conditions it is possible to assign consistent probabilities to differentspectra f. To this end we must compute the decoherence functional between thepropagator in Eq.

(47) and another, say, associated with a function f ′.Let us begin by investigating Eqs (43) for the missing propagators G12 and G21.We shall first disregard the boundary condition enforcing terms in these equations,introducing them at a later stage. When this is done, the right hand side of Eqs.

(43) vanishes, since (−∇2 + m2)Gaa(x, x′) ≡0 in the present case.On the other hand, we only need the left hand side to zeroth order in λ, since any otherterm would be of too high an order to contribute to the decoherence functional at thedesired accuracy. With this in mind, Eq.

(43) reduces to the requirement that theunknown propagators should be homogeneous solutions to the Klein-Gordon equationon both of their arguments.To determine the proper boundary conditions for these propagators, we may considerthe boundary terms in Eq. (46), or else appeal to their physical interpretation.

Weshall choose the second approach.To this end, we observe that the physical meaning of the propagators as (non stan-dard) products of fields, Eqs. (22) to (25), entails the identity G12 +G21 = G11 +G22,which is consistent in this case, since both sides solve the Klein - Gordon equation.Actually, this identity is satisfied by the VEV propagators, so it can be imposeddirectly on their variations.Physically, a change in the propagators reflects a corresponding change in the sta-tistical state of the field.

To zeroth order in the coupling constant, however, thecommutator of two fields is a c-number , and does not depend on the state. There-fore, to this accuracy, G12 −G21 should not change; that is, δG12 should be equal toδG21.

We thus conclude that the correct solution to Eq. (43) is20

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESδG12 = δG21 = (12){δG11 + δG22}(48)Consideration of the CTP boundary conditions would have led to the same result.We may now evaluate the second variation of the 2PI CTP effective action, Eq. (42).We should stress that the Klein-Gordon operator annihilates all propaga-tors involved, and that the O(λ) term in ∆−10vanishes because of our choice ofrenormalization point.

Therefore the second (mixed) term in Eq. (42) is of higherthan second order and may be disregarded.

The same holds for terms of the form(∆0)−1ac δGcd(∆0)−1db δGab, disregarding boundary terms.The remaining terms can be read out of Eq. (45), with the input of the “fish” graph[43, 9]Σ(x, x′)= (∆110 )2(x, x′) =iµǫ(4π)2Zd4k(2π)4eik(x−x′)[2ǫ + ln m24πµ2 −ψ(1)−k2Z ∞4m2dσ2σ2(σ2 + k2 + iǫ)s1 −4m2σ2 ](49)where ǫ = d −4 and µ is the renormalization scale.

Clearly, the local terms in Σ canbe absorbed into a coupling- constant renormalization.The important thing for us to realize is that the O(λ) terms in Eq. (45), as well as theimaginary part of Σ, contribute only to the phase of the decoherence functional, andthus are totally unrelated to decoherence.

The only contribution to a decoherenceeffect comes from the real part of Σ. Reading it out of Eq.

(49), we obtain the soughtfor result|D[f, f ′]|∼exp{(−πλ28)Zd4p d4q(2π)8δ(p2 + m2)δ(q2 + m2)(f(p) −f ′(p))(f(q) −f ′(q))θ[−((p + q)2 + 4m2)]vuut1 +4m2(p + q)2}(50)where θ is the usual step function. As expected, we do find decoherence betweendifferent correlation histories.

Moreover, decoherence is related to dissipative pro-cesses, which in this case arise from pair production [49]. Indeed, the real part ofthe kernel Σ is essentially the probability of a real pair being produced out of quantawith momenta p and q, with p + q = k [43].21

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESLet us mention two obvious consequences of our result for the decoherence functional.The first point is that decoherence is associated with instability of the vacuum: thedistribution functions whose overlap is suppressed represent ensembles which areunstable against non trivial scattering of the constituent particles. This scatteringproduces correlations between particles.

Therefore, truncation of the correlation hi-erarchy leads to an explicitly dissipative evolution. This would not be the case ifthere were no scattering.The second point is that |D| remains unity on the diagonal.Thus, at least forGaussian fluctuations, and to two-loop accuracy, all histories are equally likely.

Whatthis means physically is that the two-point functions to be perceived by an observerafter the quantum to classical transition need not be close to their VEV in anystringent sense. Indeed, what is observed will not even be “vacuum fluctuations” inthe proper sense of the word; they are real physical particles whose momenta are onshell, and may propagate to the asymptotic region, if they manage not to collide withother particles.3.2.Beyond Coarse GrainingFor the observer confined to a single consistent history, as is the case for the quan-tum cosmologist, questioning the probability distribution of histories is somewhatacademic.

What would be relevant is one’s ability to predict the future behavior ofone’s particular history. This ability is impaired by the lack of knowledge about thecoarse-grained elements of the theory, which, in our case, are the higher correlationsof the field.As we have already seen, variation of the 2PI CTP effective action, id est, of thephase of the decoherence functional, yields the evolution equations for the VEVs ofthe two-point functions.

These equations should be regarded as the Hartree-Fockapproximation to the actual evolution, since in them the effect of higher correlationsis represented only in the average. Deviations of the actual evolution from this idealaverage may be represented by adding a source term to the Hartree-Fock equation.As the detailed state of the higher correlations is unknown, this right hand side shouldtake the form of a stochastic binary external source.The non-diagonal terms of the decoherence functional represented in Eq.

(50), whilenot contributing to the Hartree-Fock equations, contain the necessary information tobuild a phenomenological model of the back reaction of the higher correlations on therelevant sector. To build this model, we compare the actual form of the decoherencefunctional against that resulting from the coupling of the propagators to an actualgaussian random external source [50].The result of this comparison is that higher correlations react on the propagators as22

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESif these obey a Langevin- type equation∂Γ[δG11 = δG, δG22 = δG′∗, δG120 , δG210 ]∂(δG(x, x′))= −12v F(x −x′)J(x −x′)(51)where, after the variational derivative is taken, we must take the limit δG′ →δG. InEq.

(51) v is (formally) “the space - time volume”, the gaussian stochastic source Jhas autocorrelation ⟨J(u)J(u′)⟩= δ(u −u′), andF 2(u) = λ2Z ∞0ds(4πs)2Z ∞4m2 dσ2 sin(sσ2 −u24s)s1 −4m2σ2(52)Because the limit δG′ →δG is taken, the imaginary terms of the CTP effective actionreproduced in Eq. (50) do not contribute to the left hand side of Eq.

(51); as far asthe “Hartree - Fock” equations are concerned, they could as well be deleted from theeffective action.However, the stochastic source in the right hand side of Eq. (51) modifies the quantumamplitude associated with the correlation history by a factorexp{(i/2)Zd4u F(u)J(u)δG(u)},(53)where G(u) = (1/v)R d4X G(X + (u/2), X −(u/2)).

Correspondingly, the decoher-ence functional gains a factorexp{(i/2)Zd4u F(u)J(u)(δG(u) −δG′(u))}. (54)Upon averaging over all possible external sources, each having a probabilityexp{(−1/2)Zd4u J2(u)},(55)the new factor in the decoherence functional becomesexp{(−1/8)Zd4u F 2(u)(δG(u) −δG′(u))2},(56)which exactly reproduces Eq.

(50). Observe that the assumed form for the righthand side of Eq.

(51), and the requirement of recovering Eq. (50) upon averaging,uniquely determines the function F.In this way, Eq.

(51) yields the correct, if only a phenomenological, description ofthe dynamics of classical fluctuations in the aftermath of the quantum to classical23

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIEStransition. It should be obvious that nonlinearity is essential to the generation ofthese fluctuations.4.DiscussionThis paper presents three main results.

The first is the ansatz Eq. (11) for the quan-tum amplitude associated with a correlation history.

The second is the ansatz Eq. (40) for the decoherence functional between two such histories.

On the basis of thisansatz, we have shown in Eq. (50) that the quantum interference between historiescorresponding to different particle spectra is suppressed whenever these spectra differby particles whose added momenta go above the two particle treshold 4m2, m2 beingthe one-loop radiative-corrected physical mass.

The third result is the phenomeno-logical description in Eq. (51) of the dynamics of an individual consistent correlationhistory.What we have presented in the above, despite its embryonic form, is a framework forbringing together the correlational-hierarchy idea in non-equilibrium statistical me-chanics and the consistent-history interpretation of quantum mechanics.

This frame-work puts decoherence and dissipation due to fluctuations and noise (manifested herethrough particle creation) on the same footing. It suggests a natural (intrinsic) mea-sure of coarse-graining which is commensurate with ordinary accounts of dissipativephenomena, and with it addresses the issue of quantum to classical transition.

It alsoprovides a theoretical basis for the derivation of classical stochastic equations fromquantum fluctuations, and identifies the nature of noise in these equations.It should be noticed that a formal identity exists between the present results and thosepreviously obtained from the influence functional formalism [51, 20, 11]. Indeed, ourdecoherence functional has the same structure as the influence functional, with thenon diagonal terms in Eq.

(50) playing the role of the “noise kernel”. This is morethan an analogy, as it should be clear from the discussions above and elsewhere.While for reasons of clarity and economy of space, we have focused on a simpleapplication from quantum field theory to develop our arguments, the implicationson quantum mechanics and statistical mechanics go beyond what this example canshow.

The theoretical issues raised here in the context of quantum mechanics andstatistical mechanics, as well as the consequences of problems raised in the contextof quantum and semiclassical (especially the inflationary universe) cosmology, whichmotivated us to make these inquiries in the first place, will be explored in greaterdetail elsewhere.This work is part of an on-going program which draws on many year’s worth ofpondering on the role of statistical mechanics ideas in quantum cosmology, usingquantum field theoretical methods while placing the issues in the larger context ofgeneral physics. The project began in 1985, when one of us (EC) was invited by24

CALZETTA AND HU: DECOHERENCE OF CORRELATION HISTORIESDieter Brill to join the General Relativity Group at Maryland. It is therefore anhonour and a pleasure for us to dedicate this paper to him on this happy occasion.AcknowledgmentsE.

C. is partially supported by the Directorate General for Science Research andDevelopment of the Commission of the European Communities under Contract NoC11 - 0540 - M(TT), and by CONICET, UBA and Fundaci´on Antorchas (Argentina).B. L. H’s research is supported in part by the US NSF under grant PHY91-19726 Thiscollaboration is partially supported by NSF and CONICET as part of the Scientificand Technological Exchange Program between Argentina and the USA.References[1] H. Everitt, III, Rev.

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