complete positivity and neutron interferometry

8 April 1999

Ž .Physics Letters B 451 1999 422–429

Complete positivity and neutron interferometry

F. Benatti a,b, R. Floreanini c

a Dipartimento di Fisica Teorica, UniÕersita di Trieste, Strada Costiera 11, 34014 Trieste, Italy`b Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Italy

c Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Dipartimento di Fisica Teorica, UniÕersita di Trieste, Strada Costiera 11,`34014 Trieste, Italy

Received 9 December 1998Editor: R. Gatto

Abstract

We analyze the dynamics of neutron beams in interferometry experiments using quantum dynamical semigroups. Weshow that these experiments could provide stringent limits on the non-standard, dissipative terms appearing in the extendedevolution equations. q 1999 Elsevier Science B.V. All rights reserved.

An open quantum system can be modeled ingeneral as being a small subsystem in interactionwith a suitable large environment. Although theglobal dynamics of the compound system is de-scribed by unitary transformations generated by thetotal hamiltonian, the effective time evolution of thesubsystem usually manifests dissipation and irre-versibility.

This reduced dynamics, obtained by eliminatingŽ .i.e. by tracing over the environment degrees offreedom, turns out to be in general rather compli-cated. However, under some mild assumptions, thatessentially ask for a weak coupling between systemand environment, one obtains a subdynamics freefrom memory effects, that can be realized in terms oflinear maps. Furthermore, this set of transformationspossesses very basic and fundamental physical prop-

Žerties, like forward in time composition semigroup.property , probability conservation, entropy increase

and complete positivity. They form a so-called quan-w xtum dynamical semigroup 1–3 .

This description is rather universal and can beapplied to model a large variety of different physicalsituations, ranging from the study of quantum statis-

w xtical systems 1–3 , to the analysis of dissipativew xeffects in quantum optics 4–6 , to the discussion of

the interaction of a microsystem with a macroscopicw xmeasuring apparatus 7–9 . It has also been proposed

as an effective description for phenomena leading tow xloss of quantum coherence 10–14 , induced by

w xquantum gravity effects at Planck’s scale 15 . Thebasic idea is that space-time should be topologicallynontrivial at this scale, manifesting a complicated‘‘foamy’’ structure; as a consequence, transitions

Žfrom pure to mixed states could be conceivable forw x.more recent elaborations, see 16 .

Analysis based on the study of the dynamics ofstrings also support, from a different point of view,

w xthe idea of loss of quantum coherence 17,18 . In thisrespect, one can show that, in a rather model-inde-pendent way, a non-unitary, dissipative, completelypositive subdynamics is the direct result of the weak

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00177-X

( )F. Benatti, R. FloreaninirPhysics Letters B 451 1999 422–429 423

interaction with an environment constituted by a gasof D0-branes, effectively described by a heat-bath of

w xquanta obeying infinite statistics 18 .These results suggest that it should be possible to

observe dissipative, non-standard effects in manyquantum systems. However, crude dimensional esti-mates show that these effects are in general tiny, andtherefore unobservable in practice.

Nevertheless, a detailed analysis of the dynamicsand decay of the neutral kaon system based onquantum dynamical semigroups shows that the non-standard contributions in the extended time-evolution

w xequations could lead to testable effects 19–23 .These non-standard terms can be parametrized by sixreal phenomenological constants, whose presencemodify the expressions of usual K–K observables,

w xlike decay rates and asymmetries 21,22 . Althoughthe present experimental data are not accurate enoughto detect these modifications, the next generation ofkaon experiments should be able to put stringentlimits on the six non-standard parameters. In thisrespect, particularly promising are the experiments at

w xf-factories 23 , where systems of correlated neutralkaons are copiously produced. One should note thatthe description of the kaon dynamics in terms ofcompletely positive maps is in this case essential inorder to obtain a consistent extension of the standard

Žquantum mechanical time-evolution for a completew x.discussion, see 24 .

Another system in which non-standard quantumevolutions based on dynamical semigroups can be

w xstudied is a neutron interferometer 25–28 . Thecapability of producing very slow neutron beams atreactors, together with the technological ability ofproducing and cutting with high precision macro-scopic silicon crystals have made possible direct,very accurate tests of various physical phenomenaw x25–30 .

In a typical experimental setup, a neutron beam issplit into two components which travel along differ-ent paths and are subsequently brought together tointerfere. The two components pass through a tinyslab of material before interfering; this produce arelative ‘‘phase shift’’ between the two splittedbeams. By varying the relative orientation of the slabacross the two beams, one obtains an interferencefigure. This figure changes under the action of vari-ous external phenomena, produced e.g. by earth

gravity and rotation, or by an external magnetic fieldŽ .earth magnetic field is usually properly screened ;the corresponding theoretically calculated phase shiftsinduced by these phenomena have all been experi-mentally checked with high precision by analyzing

w xthe modified interference patterns 28–30 .In the following, we shall analyze in detail the

dynamics of the neutron beams in such interferomet-ric devices under the hypothesis that the correspond-ing time-evolution be described by a quantum dy-namical semigroup. We shall see that, at least inprinciple, neutron interferometry experiments couldprovide very accurate estimates of the non-standard,dissipative terms appearing in the corresponding evo-lution equations. A preliminary analysis based onrecent published data from one of those experimentswill also be presented.

States of a quantum system evolving in time canbe suitably described by a density matrix r; this is apositive, hermitian operator, i.e. with positive eigen-values, and constant trace. We shall analyze theevolution of neutrons in an abstract interferometer,where the original monoenergetic beam is splitted intwo components that then interfere at the end, givingrise to intensity fringe patterns in two possible exitbeams. We can model this generic physical setup bymeans of a two-dimensional Hilbert space, taking asbasis states those corresponding to the two compo-nents of the split beam inside the interferometer.

With respect to this basis, the density matrix r

describing the state of our physical system can bewritten as:

r r1 3rs , 1Ž .

r rž /4 2

where r 'r ) , and ) signifies complex conjuga-4 3

tion.As explained in the introductory remarks, our

analysis is based on the assumption that the evolu-tion in time of the neutrons inside the interferometeris given by a quantum dynamical semigroup, i.e. by

Ž .a completely positive, one parameter s time familyŽ . Ž .of linear maps: r 0 ¨r t . These maps are gener-

ated by equations of the following form:

Er tŽ .syiHr t q ir t HqL r t . 2Ž . Ž . Ž . Ž .

E t

( )F. Benatti, R. FloreaninirPhysics Letters B 451 1999 422–429424

The first two terms in the r.h.s. of this equation arethe standard quantum mechanical ones. They contain

Ž .the effective time-independent hamiltonian H, thatcan be taken to be hermitian, since the fact that theneutrons are unstable can be neglected in interferom-etry experiments.

w xThe third piece L r is a linear map, whose formis fully determined by the requirement of complete

w xpositivity and trace conservation 1–3 :

1 † † †w xL r sy A A rqr A A q A r A .Ž .Ý Ýj j j j j j2j j

3Ž .

The operators A must be such that Ý A† A is aj j j j

well-defined 2=2 matrix; further, to assure entropyincrease, the A can be taken to be hermitian. Inj

w x Žabsence of L r , pure states i.e. states of the form.:²c c would be transformed into pure states. In-

Ž .stead, the additional piece in 3 produces dissipationand possible transitions from pure to mixed states.

Ž .As already mentioned, equations of the form 2 ,Ž .3 have been used to describe various phenomenaconnected with open quantum systems; in particular,they have been applied to analyze the propagation

w xand decay of the neutral kaon system 21–24 . Al-though the basic general idea behind these treatmentsis that quantum phenomena at Planck’s length pro-duce loss of phase-coherence, it should be stressed

Ž . Ž .that the form 2 , 3 of the evolution equation isindependent from the microscopic mechanism re-

Ž .sponsible for the dissipative effects. Indeed, Eqs. 2Ž .and 3 are the result of very basic physical assump-

tions, like probability conservation, entropy increase,complete positivity, and therefore should be regardedas phenomenological in nature.

Among the just mentioned physical requirements,complete positivity is perhaps the less intuitive. In-deed, it has not been enforced in previous analysis,in favor of the more obvious simple positivityw x10,19,20 . Simple positivity is in fact enough toguarantee that the eigenvalues of the density matrixŽ .r t describing our system remain positive at any

time; this is an unavoidable requirement in view ofthe probabilistic interpretation of r.

Complete positivity is a stronger property, in thesense that it assures the positivity of the density

matrix describing the states of a larger system, ob-tained by coupling in a trivial way the system understudy with another arbitrary finite-dimensional one.

Ž .At first, the requirement of complete positivity of 2seems a mere technical complication. Nevertheless,it turns out to be essential in properly treating corre-lated systems, like the two neutral kaons comingfrom the decay of a f-meson; it assures the absenceof unphysical effects, like the appearance of negativeprobabilities, that could occur for just simply posi-

w xtive dynamics 23,24 . One should also add thatstandard unitary quantum mechanical time evolu-tions satisfies this property in a rather trivial way.For these reasons, in analyzing possible non-stan-dard, dissipative effects even in simpler, non corre-

Ž . Ž .lated systems, the phenomenological Eqs. 2 and 3should be used.

In the particular case of neutron interferometers,as for the K–K system, a more explicit description

Ž . Ž .of 2 and 3 can be given. In the chosen basis, theeffective hamiltonian can be written as:

Eqv 0Hs . 4Ž .ž /0 Eyv

Indeed, the neutron beams inside the interferometercan be assimilated to a two-level system, with E the

Ž .incident kinetic neutron energy. The splitting inenergy 2 v among the two internal beams can beinduced by various physical effects. In the following,we shall consider the case of a thin slab of materialŽ .e.g. alluminium inserted transversally to the twosplit beams. A slight rotation of this slab producesdifferent effective interactions of the neutrons withthe slab material in the two internal paths, yielding anon-vanishing v. Using an eikonal approximation,perfectly suitable for describing slow neutrons, onecan theoretically compute this energy splitting interms of the neutron–nuclear scattering parameters

w xfor the slab material 25–27 . Typically, one findsthat v is of the order of 10y7 eV.

w x Ž .The explicit form of the term L r in 3 can bemost simply given by expanding the 2=2 matrix r

in terms of Pauli matrices s and the identity s :i 0w xrsr s , ms 0, 1, 2, 3. In this way, the map L rm m

can be represented by a symmetric 4=4 matrixL , acting on the column vector with componentsmn


Ž .r ,r ,r ,r . It can be parametrized by the six real0 1 2 3w xconstants a, b, c, a , b , and g 21 :

0 0 0 00 a b c

L sy2 , 5Ž .mn 0 b a b� 00 c b g

with a, a and g non-negative. These parameters arenot all independent; the condition of complete posi-

Ž .tivity of the time-evolution r™r t imposes thefollowing inequalities:

2 R'aqgyaG0 , RSGb2 , 6Ž .2S'aqgyaG0 , RTGc2 ,

2T'aqaygG0 , STGb 2 ,

RSTG2 bcbqRb 2 qSc2 qTb2 .

Ž .As already observed, the dissipative correction 5Ž .to the evolution equation 2 should be regarded as

phenomenological; it is therefore difficult to give anapriori estimate of the magnitude of the non-standard

Ž .parameters in 5 . However, following the idea thatw xthe term L r originates from quantum effects at

Planck’s scale, one expects the values of a, b, c, a ,b and g to be very small, at most of the orderm2rm ,10y19 GeV, where m is the neutron mass,n P n

while m is the Planck scale. The dissipative contri-Pw x Ž .bution L r in 2 is therefore at least three order of

magnitude smaller than the one given by the stan-dard hamiltonian terms: this allows an approximate

Ž .analysis of the evolution equation 2 , in which theŽ .term 5 can be treated as a small perturbation.

For the considerations that follows, it will besufficient to stop at the first order in the perturbativeexpansion. For generic initial conditions, the timedependence of the four components of the corre-

Ž . Ž .sponding solution r t of 2 is explicitly given by:

Cyi v tr t s 1yg t r qg t r y e sin v t rŽ . Ž . Ž .1 1 2 3

v

C )

i v ty e sin v t r , 7aŽ . Ž .4v

Cyi v tr t sg t r q 1yg t r q e sin v t rŽ . Ž . Ž .2 1 2 3

v

C )

i v tq e sin v t r , 7bŽ . Ž .4v

C )

yi v tr t sy e sin v t r yrŽ . Ž . Ž .3 1 2v

By2 i v tq 1yAt e r q sin 2v t r ,Ž . Ž .3 42v

7cŽ .

Ci v tr t sy e sin v t r yrŽ . Ž . Ž .4 1 2

v

B)

2 i v tq sin 2v t r q 1yAt e r ,Ž . Ž .3 42v

7dŽ .

where the following convenient combinations of thenon-standard parameters have been introduced:

Asaqa , Bsayaq2 ib , Cscq ib . 8Ž .Any physical property of the diffracted neutron

beams exiting the interferometer can be extractedŽ . Ž .from the solution 7 for the density matrix r t by

computing its trace with suitable hermitian operators.In particular, the observation of the neutron intensitypattern just outside the interferometer corresponds tothe computation of the mean value of the following

w xprojector operators 10 :

1 eiu1OO s ,q 2 yi už /e 1

Ž .i uqp1 e1OO s , 9Ž .y 2 yi Žuqp .ž /e 1

that refer to the two possible exit beams in which aneutron can be found, having traveled the whole

w xinterferometer 27 ; the parameter u is a phase whichdepends on the specific experimental setup. Then,the intensity I of the interference figure in the two"

exit beams is given by:

² :I t s OO sTr OO r t . 10Ž . Ž . Ž ." " "

In the basis we are using, the initial conditions fora neutron entering the interferometer is given by oneof the following two density matrices:

1 1 1 y11 1Ž1. Ž2.r s , r s . 11Ž .2 2ž / ž /1 1 y1 1

They correspond to the two possible choices oforientation of the incident neutron beam with respectto the interferometer, and give rise to the same final


results; in the following, we shall work with r Ž1..Inserting this initial condition in the time evolution

Ž . Ž .given by 7 , from 10 one obtains the followingtwo interference patterns:

1 yA tI t s 1" e cos uq2v tŽ . Ž ." 2 ½< <B

q sin 2v t cos uyu , 12Ž . Ž . Ž .B 52v

< <where B and u are modulus and phase of B inBŽ .8 ; this formula holds for times such that At<1.Since a neutron having traveled fully inside theinterferometer can only be detected in one of the two

Ž .exit beams, particle conservation requires: I t qqŽ . Ž .I t s1, as it is evident from 12 .y

Ž .The interference figures described by 12 arethose produced by a perfectly monoenergetic neutronincident beam and an ideal interferometer. In prac-tice, the neutron momenta in the incident beam havea finite distribution of magnitude and directions;furthermore, there are always slight imperfections inthe construction of the interferometer, not to mentionresidual strains in the crystal itself. These effects canonly partially be controlled, and produce significantattenuation in the intensity of the interference fig-ures. Detailed calculations based on neutron opticsallow precise estimates of the modifications that are

Ž .needed in the spectra 12 in order to take intow xaccount those effects 25–27 . In keeping with our

phenomenological point of view, we will not usedirectly those estimates, but rather modify the ex-

Ž .pressions 12 by introducing suitable unknown pa-rameters. By denoting with N the actual neutron"

countings at the two exit beams, one generalizes theŽ .spectra in 12 as:

Ž0. yAtN t sN 1"CC e cos uq2v tŽ . Ž ." " "½< <B

q sin 2v t cos uyu . 13Ž . Ž . Ž .B 52v

The parameters CC , the so-called fringe contrast,"

take into account the previously mentioned intensityattenuation, while N Ž0. are just normalization con-"

stants. Clearly, the accuracy of the determination of

the parameters A and B from the measured data willincrease as the fringe contrast gets closer to one. Inactual experiments, one finds that the best values forCC are usually around 0.6. Further, note that now"

particle conservation requires:

N Ž0. CC sN Ž0. CC . 14Ž .q q y y

In order to be able to compare the phenomenolog-Ž .ical predictions 13 with actual experimental data,

further elaborations are required. Although the inten-Ž .sity spectra in 13 are time-dependent, in an inter-

ference experiment, being the paths followed by theneutrons fixed, the evolution time t is also fixed; itcan only be modified by changing the wavelengthŽ .i.e. the energy of the primary neutron beam. Itfollows that the phase w'2 v t that gives the inter-ference figure can be varied only by changing theenergy split v between the two paths inside theinterferometer, i.e. by changing the orientation of thematerial slab with respect to the neutron beams.

Also, since t is fixed, it is not possible a priori toextract the fringe contrast parameters CC from the"

w-dependence of the intensities N . In other words,"

Ž .in comparing the behaviour in 13 with that givenby the experiment, one needs to use the followingform for the two intensity patterns:

N wŽ ."

sinwŽ0.sN 1" P cos uqw qQ ,Ž ." " "½ 5w

15Ž .

where

yA t < <P sCC e , Q sCC B t cos uyu .Ž ." " " " B

16Ž .

A fit with the experimental data will give estimatesfor the parameters N Ž0., P , Q and u ; at least in" " "

principle, this is sufficient to determine the non-standard constants A and B.

We have performed a preliminary x 2 fit of theŽ .formulas in 15 with recent experimental data, pub-

w xlished in Ref. 30 . In that experiment, a so-calledskew-symmetric silicon interferometer and polarizedneutrons were used to study the ‘‘geometrical phase’’of the neutron wavefunction; however, in order to


check the apparatus, also standard interferometricspectra had been taken. In this way, the experimental

w xsetup described in Ref. 30 is an actual realization ofthe abstract interferometer so far discussed in deriv-

Ž .ing 15 . Although the results of other recent inter-w xferometry experiments are also available 28 , in

lacking for an ad-hoc experiment, we find the pre-w xsentation of the data in Ref. 30 the more suitable

for the analysis of the consequences of the evolutionŽ . Ž .equations 2 and 5 . The results of our fit can be

summarized as follows:

N Ž0.s942"6 P s0.17"0.01q q

Q s0.02"0.02 us0.09"0.05 17aŽ .q

N Ž0.s366"4 P s0.46"0.02y y

Q s0.06"0.02 us0.03"0.03 . 17bŽ .y

Extracting estimates for the parameters A and Bfrom these results requires the determination of thefringe contrast CC , characterizing the ‘‘optical’’"

properties of the neutron interferometer. These con-stants are in fact function of the imaginary part ofthe refraction index of the material from which the

w xinterferometer is built 27 . Therefore, the best wayto obtain estimates for CC is to measure the inter-"

ference spectra with two neutron beams of wave-w xlength l and lr2 29 , keeping unchanged the rest

of the experimental setup. By comparing the corre-sponding fit estimates for the coefficients P and"

Q , with the help of neutron optics theory, one is"

then able to extract the values of CC at wavelength"

l.In lacking of a two-wavelength experiment, in the

following we shall estimate CC using directly the"

data. In the standard quantum mechanical case, i.e.for AsBs 0, one can easily obtain the coefficientsCC from the maximum N Žmax . and the minimum"

N Žmin . neutron counts of the experimental interfer-Ž .ence figures. Indeed, from 15 with AsBs 0, one

obtains:

N Žmax .yN Žmin ." "

CC s . 18Ž ." Žmax . Žmin .N qN" "

Although this relation is only approximately validfor nonvanishing A and B, in practice one can still

Ž .use 18 with confidence, since the systematic errorthat one thus makes in the evaluation of the parame-

ters A and B can be estimated at the end to be muchsmaller than the pure experimental uncertainty.

Ž .Using the experimental data and 18 , one ob-tains: CC s0.19"0.02 and CC s0.54"0.03. Asq yan independent test of the correctness of this evalua-tion, one can check that, within the errors, the rela-

Ž .tion 14 is perfectly satisfied. Note that the value ofCC is significally smaller than one, while that ofqCC is close to the best figures that can be attained iny

w xpractice 26–28 . This difference in the fringe con-trast of the two data samples will result in a signifi-cally less accurate determination of the non-standard

Ž .parameters A and B from the the results in 17aŽ .with respects to those in 17b .

The flight time t of the neutrons inside the inter-ferometer can be very accurately determined by us-ing the geometric specifications of the silicon inter-

w xferometer used in Ref. 30 , so that 1rts5.83=

10y21 GeV. Further, on general grounds one expectsŽ . Ž .Re B and IIm B to be of the same order of

Žmagnitude for the case of the neutral kaon system,w x.see 22 ; then, as a working assumption, we shall

Ž .neglect the small phase u with respect to u in 16 ,B

so that only the real part of B can be extracted fromthe estimates of Q . Putting everything together,"

Ž . Ž .one finally obtains from 17a : As 0.71"0.73 =y21 Ž . Ž . y2110 GeV and Re B s 0.76 " 0.49 = 10

Ž . Ž . y21GeV, while from 17b : As 0.84"0.41 =10Ž . Ž . y21GeV and Re B s 0.65 " 0.24 = 10 GeV.

These two estimates are compatible, but, as ex-pected, the second one is much more accurate.

Ž .Alternatively, recalling the definitions 8 , onecan express the previous results as an estimate for

Ž .the parameters a and a of 5 ; using the best valuesŽ . Ž .for A and Re B , one finds: as 0.10"0.24 =

y21 Ž . y2110 GeV, as 0.74"0.24 =10 GeV. Al-though these values should be taken as indicative,they seem to suggest a possible nonvanishing valuefor a , while a is compatible with zero at the presentlevel of accuracy.

If the non-standard parameter a actually vanishes,Ž . w xthe expression 5 of the extra term L r in the

Ž .evolution equation 2 greatly simplifies. Indeed, forŽ .as 0, the inequalities 6 readily imply: gsa ,

bscsbs 0. In this case, the evolution equationŽ .2 gives the most simple extension of ordinaryquantum mechanics, compatible with the conditionof complete positivity.


In this simplified situation, the combinations AŽ .and B in 8 become both equal to a , so that the

Ž .relations 16 are modified as:

P sCC eya t ,CC 1ya t ,Ž ." " "

Q sCC a t cosu . 19Ž ." "

By eliminating a t from these two formulas, one isnow able to determine the fringe contrast factors CC"

from the fitted parameters P and Q without" "

further assumptions: CC s0.20"0.02, CC s0.52"q y0.03. Note that these values are equal within errorsto those determined before. Then, from either of the

Ž .relations in 19 , one obtains two determinations ofthe parameter a , which combined finally give:

as 0.71"0.21 =10y21 GeV . 20Ž . Ž .Although this estimate for a points toward a nonva-nishing value, roughly of the right order of magni-tude for a quantum gravity or ‘‘stringy’’ origin, itshould not be regarded as an evidence for non-stan-dard, dissipative effects in the dynamics describingneutron interferometry. Rather, it should be consid-ered as a rough evaluation of the sensitivity thatpresent neutron interferometry experiments can reachin testing quantum dynamical time evolutions of the

Ž . Ž .form given in 2 and 5 .In closing, we would like to make a few com-

ments on the existing literature on the subject. In ourstudy of the effects of the environment on the propa-gation of the neutrons inside the interferometer, wehave assumed that the refractive phenomena on thevarious silicon blades of the device be described bystandard neutron optics. This theory is the result of aquantum mechanical analysis of the scattering of theneutron beams by the nuclei in the silicon crystal.For slow neutrons, the effects of these scatteringphenomena can be effectively described by a modelthat resemble very closely ‘‘geometric optics’’ of

w xlight propagation theory 25–27 .In principle, non-standard dissipative effects can

also be present in the scattering neutron-nucleusw x31–33 ; these phenomena can be described again by

Ž . Ž .phenomenological equations of the form 2 , 3 , andcould modify the predictions of standard neutronoptics. A precise estimate of these changements re-quires detailed computations that certainly go be-yond the scope of the present investigation. In any

case, it should be stressed that these possible dissipa-tive effects in the interaction neutron-nucleus wouldmostly affect the estimate of the intrinsic interferom-eter parameters, like the fringe contrast or the phaseu , as functions of the wavelength of the incidentneutrons and the nuclear properties of the refractivematerial. In our study, these parameters have beenobtained directly from the experimental data, so thatwe expect little changements in our analysis fromthese extra effects. Nevertheless, the entire topiccertainly deserves further attention and we hope tocome back to these problems in the future.

A study of possible phenomena violating quantummechanics in neutron interferometry has been origi-

w xnally presented in Ref. 10 . There, an equation ofŽ .the form 2 has also been used to describe these

effects, but without imposing the condition of com-plete positivity. Fitting an approximated formula forthe exit beams interference figures with the experi-mental data available at that time, limits on some ofthe parameters violating quantum mechanics weregiven. These limits have been further strengthen by

w xlater analysis 34 , exploiting wavefront splitting in-Žterference experiments the analogs of Young’s two

.slit experiment .These estimates turn out to be rather crude: they

are based on a rough evaluation of the flight-time ofthe neutrons inside the interferometer, rather than adetailed analysis of the interference patterns. Further-more, as already mentioned, lacking of imposing thecondition of complete positivity on the evolution

w xequation could lead to serious inconsistencies 24 .We stress that to avoid these problems, one needs toadopt phenomenological descriptions based on Eqs.Ž . Ž .2 and 3 .

Finally, the neutron interference experiments real-ized so far allows determining at best the values of

Ž .only two of the six non-standard parameters in 5 .The remaining ones could be estimated, at least inprinciple, by studying the behaviour of other observ-

Ž .ables OO, different from those appearing in 9 . ThisŽ .would allow a direct test of the inequalities 6 and

therefore of the hypothesis of complete positivity.In practice, however, the analysis of these new

observables would correspond to the realization ofcompletely different experimental setups. In this re-spect, a detailed study of possible non-standard,dissipative effects in neutron interferometry appears


to be an exiting challenge not only theoretically, butexperimentally as well.

Acknowledgements

R.F. thanks the ‘‘Bruno Rossi’’ INFN-MIT ex-change program for partial support, and the Centerfor Theoretical Physics, MIT, for the kind hospital-ity.

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complete positivity and neutron interferometry

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