29 February
1996
PHYSICS
ELSEVIER
LElTERS
B
Physics Letters B 369 (1996) 295300
Entropy, quantum decoherence and pointer states scalar “parton” fields * HansThomas Elze CERNTheory,CH1211 Geneva 23, Swirzerland1 Received 3 May 1995 Editor: R. Gatto
Abstract Entropy arises in strong interactions by a dynamical separation of “partons” from longwavelength “environment” modes due to confinement. We evaluate the time dependent VORNeumann entropy for a general model of two interacting scalar fields representing, for example, partons and their environment, respectively. The relevant density functional is diagonalized in timedependent HartreeFock approximation based on the functional Schriidinger picture. This yields “field pointer states” and their probabilities in terms of Wightman functions. Our approach can be applied to a variety of related nonequilibrium problems. The results also indicate how to calculate a finite geometric enfropy proportional to a surface area.
1. Introduction The longstanding “entropy puzzle” of highmultiplicity events in strong interactions at high energy has been analysed from a new point of view [ 11. The problem dates back to Fermi and Landau and is related to understanding the rapid thermalization of high energy density (>> 1 GeV/fm3) matter [ 21. Why do thermal models work so well? Why do they work at all? Or, why does highenergy scattering of pure initial states lend itself to a statistical description characterized by a large apparent entropy? Effectively, unitary time evolution of the observable part of the system breaks down in the transition from a quantum me* Work supported by the Heisenberg Forschungsgemeinschaftft). ’ Email:
[email protected] Present address: Department of Physics, Bldg.81, Tucson, AZ 85721, USA. 03702693/96/$12.00
Programme
(Deutsche
University
of Arizona,
@ 1996 Elsevier Science B.V. All rights reserved
.SSDf03702693(95)015388
chanically pure initial state to a highly impure (more or less thermal) highmultiplicity final state. Based on analogies with studies of the quantum measurement process (“collapse of the wave function”) [ 33 and motivated by related problems in quantum cosmology and by nonunitary nonequilibrium evolution resulting in string theory [4], one may argue that
environmentinduced quantum decoherence solves the enfropy puzzle of strong interactions. In particular, it is well known that a complex purestate quantum system may show quasiclassical behaviour, i.e. an impure density (sub)matrix together with decoherence of the associated socalled pointer states of the observable subsystem [3]. Furthermore, the induced decoherence, i.e. loss of quantum phase information, seems essential for entropy production and selfthermalization in systems which show deterministic chaos in the classical limit [ 51, such as YangMills fields. In line with those investigations, first steps were undertaken in Ref. [l] to analyze the parton model
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together with its strongcoupling environment of almost constant QCD field configurations. Induced quantum decoherence and entropy production were studied in a nonrelativistic singleparticle model resembling an electron coupled to the quantized electromagnetic field, however, with an enhanced oscillator spectral density in the infrared. The FeynmanVernon influence functional technique for quantum Brownian motion [6] provided the remarkable result that in the shorttime strongcoupfing limit the model parton behaves like a classical particle [ 1 ] : a Gaussian singleparton wave packet experiences ftiction and localization, i.e. no quantum mechanical spreading, and coherent superpositions of these states decohere. If confirmed in QCD, the existence of such partonic pointer states, which in general we anticipate not to be simple singleparticle states, will have important consequences for partonmodel applications to complex hadronic or nuclear reactions [ 71. There, the phase space picture involving structure functions and multiple parton scattering in a strongly coupled environment of infrared QCD fields clearly needs more study. Presently, our purpose is to derive general results for the real time dependent nonequilibrium entropy in a generic model of two interacting scalar jields. Thus, we study an observable field (open subsystem) interacting with a dynamically hidden one (environment), which may be called quantum field Brownian motion. Our approach and results should be applicable to other interesting situations, such as i) nonequibbrium phase transitions or ii) systems which are geometrically subdivided. We indicate the application of our results to the geometric entropy problem in the following. In any case, our present study is rather independent of our motivation from the parton model and meant to provide new general results on entropy and decoherence effects in interacting scalar quantum fields. However, our present model may also represent schematically, of course, the open subsystem of shortwavelength partons interacting with longwavelength environment modes, which are described by one and the other scalar field, respectively. In QCD both, partons and environment, consist of quark and gluon fields which are, however, separated by characteristic momentum scales as discussed in Ref. [ 11. We keep the designations “partons” and “environment” in the fol
lowing for convenience. 2. Nonequilibrium fields
entropy in scalar quantum
Formally, we work in the functional Schriidinger picture employing Dirac’s timedependent variational principle. Using Gaussian trial wave functionals, this amounts to the timedependent HartreeFock approximation (TDHF) in quantum field theory [ 81. In general, these are not single particle states of the fields. The most general normalized Gaussian wavefunctional for two interacting real scalar fields can be written as
 41 (t)l
x exp{~[ch
x 142  42Wl)
IG12(t)
 &2(t)
1
9
(1)
with (j = 1,2)
X [icy’(t)
i8j(t)]
[4j
CJj(t)]
+i*j(t)[4j4j(t)l}.
(2)
We do not write explicitly all spatial integrations involved in expressions like the exponents in ( 1) or (2). The normalization factors are = det{2Gj(t)}“4,
Nj(t)
Nlz(r) = det{l  GI(t)G,2(t)G2(t)G12(t)}“4, (3) discarding an irrelevant constant factor in Nj. Thus, TDHF is embodied here in the real timedependent variational parameter onepoint functions t) , (mean fields) and symmetric twopoint functionsGj(x,y,t),~j(x,Y,t),G12(X,Y,t),Z12(x,y,t) (related to Wightman functions). Their meaning was previously discussed in [ 1,8]. All physical quantities of the complex system can be calculated with *12, expressing inner products by functional integrals. The functional density submatrix lip for the observable parton subsystem (4, ) is obtained by tracing over the environment degrees of freedom (4~)~ f$j
ffj
(x,
t)
(x,
H.T. Elze/Physics FP(~)
=
Tr2 Pdt))(*dt)l
Letters B 369 (1996) D300
(4)
9
where we omit the label P henceforth. Thus, we study in the following particularly the decoherence effects induced by quantum field 42 on quantum field 41. The explicit calculation is straightforward due to the Gaussian nature of the functional integral. The matrix elements of p contain all the information about the subsystem. Our aim is to obtain a general result for its von Neumann entropy, S E Trt b
[email protected] Before, we calculated only the simpler linear entropy directly, which provides a lower bound for the von Neumann entropy
[ 1I,
“”
iTrln
1  Gl(~)G12(~)G2(f)G12(f)
’
(
1 + Gl(t)Z12(t)G2(t)212(t)
> ’
a = $G,‘A
 i[ZI  i(G,2G&,2
r FL41 ev{#4
x
+ LW) = N2 exp{&4
Dc#’ F[#]
exp{+‘[a*
+ W)
+ aI&
.I + Ib* +p+&*l&},
+X,2G2G,2)]
= .I , c 3 ;G,‘B
 $[G,zG&
with the combinations
B = iG,G,?GzG,z
 XnG2G121 = ct.
of twopoint
A = 1  ;GtG,zG2G,2
functions
+ $G,&G&,
+ ~G,C,2G&.
Choosing p = b in (6)) completing the square, shifting I#‘, and requiring resulting Gaussians in 4 to cancel yields the eigenvalue problem: r F[4]
= N*
(5)
tracing over coordinates. Eq. (5) is also valid for nontranslation invariant systems, which is relevant for calculating the geometric entropy related to spatial boundaries separating observable and unobservable subsystems. Geometric entropy is intimately connected to blackhole entropy [ 91. Here, one identifies 41 as the part of a scalar field I# with support outside a given spatial region and q!~ 3 Cp  41, which has its support inside the complement. Our results (5)) ( 17), and (20) below indicate that the geometric entropy comes out finite, once a renormalization of the equations for the twopoint functions, G’s and Z’s in (5), is performed or a UV regularization introduced to provide sufficient integrability constraints. We proceed by diagonalizing jj. Determining its eigenstates and eigenvalues is equivalent to constructingfleld pointer stuates [ 1,3] within TDHF and their probabilities. The eigenvalue problem to be solved, p/r) = rlr), is of the form
291
J Z+b’
F[&’ + Y$l exp{#X4’}, (7)
with X = a* + (Y = X’, Y = $X‘c, and where a = (Y’, by (6), is determined to solve the equation a LY= $c* [a* + (u] ‘c. Note the similarity to the finitedimensional oscillator problem of Srednicki [ 91. Equivalently, replacing F [ 4’1 + F [ S/6( &* ) + 6/S( cq5) ] and +*4’ + {[+c*+’ + 4’41 in (6), we obtain by integration r F[4]
= N2 det{X}‘/2exp{
[email protected]*X‘c$}
s
6 1 + S(4)
x F[s(&*)
7
exp{+i+c*X‘c+)
(8) which is more convenient than (7). Looking for polynomial functional solutions of (8), we find first of all a constant, Fo[~]
= 1
=+ r0 = N2 det{X}“2.
(9)
Secondly, instead of a general linear functional, Fourier transform is sufficient,
S[dl
=
.I
ddx eikx&x)
= & ,
the
(10)
since the problem is linear in F. Then, from (8)(
(6) rt+k = irc[X‘c+
usingtheansatz ((d+&)Ir) E F[4] exp{
[email protected]$+ /?+} with unknown one and twopoint functions p and cr and a nonexponential functional F. From the explicit calculation of 3, we obtain N  Nt Nt2, b E iii] (& does not appear in (6) ) , and
+ @*X‘]k
lo),
= roYkk’@__k’, (11)
summing over indices occurring twice. For a translationinvariant system, ( 11) could immediately be solved. Generally, however, denoting eigenvalues
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and eigenvectors of (Yk.V ) by & and &, one obtains a set of linear eigenvalues rk =
[email protected]$k. Due to the Gaussian structure in (8), the higherorder eigenfunctivnals can be built up as linear combinations of products of &‘s and lowerorder ones. For example, Fkk’[ $t’] = C$k&’ + Ckk’, which yields a Set Of quadratic eigenvalues r&f = ra[&k~ @( k’  k) . Note the constraint k’ >_ k; interchange of k and k’ does not lead to a new eigenfunctional due to the scalar (bosonic) character of the fields. The constant Gkk’ follows with the help of the matrix diagonalizing (&‘). We do not construct explicitly the higherorder eigenfunctionals. However, the nrh order set of eigenvalues, n rkI . ..k., = r06kl
II i=2
(k,@(ki

ki1)
9
=
eigenvalues = ro + 2
IWO
c
rkl...k,,
(15) with zi z ( c/2)‘j2a( (15) into (14), S’“(t) = 1 det
{
c/2) ‘j2. Finally, inserting
A(t) B(t) A(t) + B(t)
(13)
which resembles the evaluation of a bosonic partition function. In the last step we used r0 = det{X‘Re[2a  c]}‘/~ = det{[l  ixtc*][l $X’ c] }lj2, which follows from the equation determining LYor X. Similarly, we obtain the linear entropy, Siin z Trr {/ilj2}=
1 Tri .
1’2 >
(16)
,
which confirms our earlier result, employed in (5). Finally, we calculate the vvn Neumann entropy using the “replica trick”:
= $det{
(; I ;;“} ln(lY)+&lnY}.
= $Tri
b” In=i
InA (17)
kt...k,
k
t0det{ 1  Y}’ = 1 ,
= 1 det{&$}
[n+ (2i2 1)‘/2]t (c/2)1/2,
S(r) = Trt P(t) In/i(t)
n=l
k
Y = (c/2)‘/2
(12)
is easily found, similarly to rkk’ above. To check the result ( 12), we calculate Tr c(t) = c
real parts, a = aG;‘A and c = iG;‘B, simplifying the equation for a, X, or Y, a  a = &z[a I a]‘~. The solution (for integrable eigenfunctionals) is
p2 ( 14)
In order to express Y in terms of A and B, we observe that in a direct calculation [ 1] of Trt 3’ (and in the nfold functional integral for Tri p) imaginaryparts of a and c cancel. Therefore, we replace a and c by their
Together with ( 15), Q. ( 17) presents our main result. It generalizes Eq. (6) of Srednicki [ 91. Basically, the TDHF approximation for interacting quantum fields preserves a Gaussian structure of the wave functionals, see ( 1)  (3), which is exact in the noninteracting case and can be reduced to a coupled harmonic oscillator problem. To evaluate the entropy ( 17) is still a formidable task for any realistic situation. Before trying, it seems worth while to draw some general conclusions: I. Neither mean fields $1~~ nor their conjugate momenta 5ri,2, nor imaginary parts C 1.2of the parton and environment twopoint functions contribute to S. II. Vanishing correlations between partvns and environment, i.e. Gi2 = 212 = 0 (independent subsystems),implyA=1,B=O,i.e.Y=0,andS=0. III. Vanishing widths of patton or environment wave functionals, i.e. Gi ,2 + 0 (one or the other subsystem classical/reversible [ 1 ] ) , imply Y = 0 and S = 0. This presumably holds for any field theory of “partons” coupled to an “environment” in TDHF approximation. The detailed timeevolution of the entropy in particular cases is obtained by solving the equations of motion for the one and twopoint functions, which
H.T. Elze/Physics
Letters B 369 (1996) 295300
follow from the relevant effective action derived previously [ I], see also Ref. [ 81 for a simple example. We stress that the real and imaginary parts, Gt2 and 212, of the correlation function between subsystem and environment, cf. ( 1) and (5), for example, are essential. The main results, ( 15) and ( 17)) are general enough to cover nonequilibrium evolutions with strictly increasing as well as temporarily decreasing entropy. At present, no general criteria are known which distinguish one from the other on the level of a specific dynamical model with given initial conditions [ 3,561. However, our considerations confirm for interacting quantum fields the idea that quantum decoherence and entropy production in a subsystem is induced by an active environment. This includes naturally the possibility of quantum revivals (recoherence) in the observed subsystem, i.e. a temporary recovery of offdiagonal interference terms in its density functional together with a temporary entropy decrease. Equivalently, a backflow of quantum phase information through correlations from the environment may happen and has actually been observed in cavities in quantum optics
[IO]. We do not expect recoherence to play a role in highmultiplicity scattering events in QCD, since the information contained in higherorder multiparticle correlations cannot flow back into typically measured singleparticle observables. Once the correlations are established and the system decouples due to the shortrange interaction, they do not evolve dynamically any more. The above diagonalization of the parton density functional yields timedependent field pointer states as a byproduct, the simplest one of largest probability ro being
+i~~(~)[44~(t)l)~ cl‘. (6)(9),
(18)
with (Y = [(c/2)(5*  1)(~/2)]‘/~. Higherorder eigenfunctionals are less probable, since their eigenvalues, i.e. probabilities, decrease with increasing order, see ( 12) (all & < 1). They also have higher kinetic energy, since their wave functionals have additional nodes, e.g. ( lo), analogous to excited oscillator states.
299
3. Conclusions As a first application of ( 17) we consider the largeentropy limit, i.e. Y x 1 or A x B. Then, using (14) and ( 16), we find: ZTrln(lY)/(l+Y)
S(t)
= i
Tr ln(A  B)/(A
= i cz,
i Tr
+ B)
{[GG2G2G21n
 [GZnG~d”)~
(19)
i.e. (5). If we assume a spatial surface of area A dividing the system into two, which is flat on the scale of the assumed shortranged correlations in (19), then Tr [. . .I” can be interpreted as a sum of closed loops of strings of G’s or Z’s intersecting the surface 2n times: once for each factor Gt2 or Ct2 correlating inand outside fields as discussed after (5) above. The dominant contribution to the trace comes from small loops (let 2 ,?* < G,?*) .Regularizing their contribution by a shortdistance cutoff &, it arises effectively from tracing over small cubes of size 0( Ls), which cover the spatial surface. Parallel to the surface the system is locally translationinvariant. Therefore, the geometric entropy is approximately s(t)
m$TrL
ln(
1  GtGt2G~Gt2
1 + GII:I~G~ZIZ 13
(20)
where Trc is evaluated locally on the scale of C. A dimensional analysis led Srednicki to propose S 0: A before [9]. Eq. (20) can be applied to the moving mirror model; following Kabat et al. [ 91, this approximates the thermal entropy outside a black hole of radius much larger than C. Secondly, coming back to partons interacting with their infrared environment, the rate of entropy production, which follows from ( 17)) is most interesting. We define a dynamical decoherence time r, r‘(t)
=
f
In S(t)
= jd”kji,lllYk s dk Yk Ill Yk ’
X
TrYlnY TrYlnY
(21)
with dk E dd k/( 27r)d. For simplicity we assumed small Y or S and a translation invariant system; the Fouriertransformis~=B~[A~+(A~B~)’~2]1,
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H.T. Eke /Physics
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since A, B are convolutions of twopoint functions now. Generally, two limits are particularly important: T( t + 0) gives the time scale for the decay of a Gaussian partonic field state, cf. ( I)( 3)) into an incoherent superposition of pointer states, e.g. ( 18)) with impure density matrix and nonzero entropy; T( t >>0) reflects the approach to a stationary state (thermdization), if it exists. Using the equations of motion from [ I], the decoherence time can be calculated for phenomenologically interesting situations. In conclusion, we derived here for the first time the von Neumann entropy of a scalar quantum field interacting in a general nonequilibrium situation with an environment consisting of a second scalar quantum field. Our formal results obtained in the timedependent HartreeFock approximation, which is based on Gaussian wave functionals in the functional Schrodinger picture, are in accordance with recent ideas about environmentinduced quantum decoherence. * They present a further step towards a theory of quantum field Brownian motion, which is particularly motivated by the problems of entropy production and thermalization in strong interactions. Naturally, it can be expected to find many other interesting applications, similarly as the well known nonrelativistic theory of singleparticle quantum Brownian motion. I thank P. Carruthers, L. Pesce, and J. Rafelski for stimulating discussions.
’ Decoherence effects have recently also been studied in scalar electrodynamics, however, in the semiclassical WKB approximation [II].
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