Physics Letters A 303 (2002) 135–139

www.elsevier.com/locate/pla

Lax–Phillips evolution as an evolutionof Gell-Mann–Hartle–Griffiths histories and emergence

of the Schrödinger equation for a stable history

D. Bara,∗, L.P. Horwitza,b

a Department of Physics, Bar Ilan University, Ramat Gan, Israelb Raymond and Beverly Sackler Faculty of Exact Science, School of Physics, Tel Aviv University, Ramat Aviv, Israel

Received 25 March 2002; received in revised form 26 August 2002; accepted 26 August 2002

Communicated by P.R. Holland

Abstract

Using the Gell-Mann–Hartle–Griffiths formalism in the framework of the Flesia–Piron form of the Lax–Phillips theory weshow that the Schrödinger equation may be derived as a condition of stability of histories. This mechanism is realized in amathematical structure closely related to the Zeno effect. 2002 Elsevier Science B.V. All rights reserved.

PACS:03.65.-w; 03.65.Xp; 03.65.Db; 03.65.Bz

Keywords:Schrödinger equation; Zeno effect; Functional analysis

The possibility of obtaining physical effects due toa large number of repetitions of the same measurementand interaction has been discussed both at the theo-retical [1–3] and experimental [4] levels. Moreover, ithas been argued [5] that these repetitions not only pro-duce quantum effects, which is the reason for callingthem quantum Zeno effect [1,2], but they may also bea source of macroscopic and classical effects [6].

We show here, using the functional Lax–Phillips(LP) [7] generalization of quantum mechanics [8,9]and the histories formalism of Gell-Mann, Hartle [10,

* Corresponding author.E-mail address:bardan@mail.biu.ac.il (D. Bar).

11] and Griffiths [12] (GMHG) that for a specialchoice of evolution of histories one obtains stabilityof the GMHG state by a mechanism that appears to bethe limit of a large number of repetitive measurementsin a finite total time. The stability of the GMHGstate is associated with the Schrödinger relation on thestructure of the histories, and hence characterizes theconsistent subset.

The theory of Lax–Phillips [7] which was orig-inally formulated to describe resonances and semi-group evolutions (i.e., irreversible processes) for theclassical scattering of electromagnetic waves on a fi-nite target has been generalized to the quantum levelby Flesia and Piron [8], and Horwitz and Piron [8] (seealso [9]). The appropriate space is the “direct integral

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)01231-8

136 D. Bar, L.P. Horwitz / Physics Letters A 303 (2002) 135–139

Hilbert space” [8,14]

(1)H =∫t

⊕Ht dµ(t),

where we take the measuredµ(t) to be of Stieltjes–Lebesgue type, andt corresponds to a foliation para-meter playing the role of the time observable. An ele-ment ofH is the sequence [9,14] (which we representhere as countable)

(2)φ = (ht1, ht2, ht3, . . .),

so thatφti = hti andhti ⊂ Hi where the sequenceHi

corresponds to a set of isomorphicauxiliary Hilbertspaces [9,14] at the timesti . Thus, we see thatφcorresponds to a virtual history [9]. The evolutionoperatorU(τ) on H is defined (for convenience here,on the continuum)

(3)(U(τ)φ

)t+τ

= φτt+τ = Vt(τ )φt ,

whereVt(τ ) is an operator which is unitary onHt .The superscriptτ on φ is the laboratory time whichis a parameter, whilet is a dynamical variable. Thesubscript(t + τ ) signifies that the original elementfrom Eq. (2) has been translated along thet axis byτ .That is, the action of the operatorU(τ) has produceda (auxiliary) Hilbert space unitary evolution combinedwith translation along thet axis by the amountτ . Notethat the operatorU(τ) forms a one-parameter group[8], that is,(U(τ2)U(τ1)φ

)t+τ1+τ2

= (U(τ1 + τ2)φ

)t+τ1+τ2

.

According to Eq. (3)

U(τ)φ = (Vt1(τ )ht1,Vt2(τ )ht2, . . .)

(4)= (h′t1+τ , h

′t2+τ , . . .

),

where the primes indicate that the evolution consti-tutes a translation along thet axis as well as a uni-tary evolution on the auxiliary Hilbert spaces. We maythink that the elements on the right hand side of thepreceding equation have undergone a specific dynam-ical evolution which is only one of a great number ofpossible alternatives which may be distinguished byappropriate projections. In that case we can representeach such element by projection operators in the aux-iliary space at specific times. Let us assume that the

initial stateφα corresponds to a projected chain (de-noted asCαφ), i.e.,

(5)φα = (Pα1(t1)ht1,Pα2(t2)ht2, . . .

).

Then

U(τ)φα = (Vt1(τ )Pα1(t1)ht1,Vt2(τ )Pα2(t2)ht2, . . .

)(6)

= (Pα1(t1 + τ )ht1+τ ,Pα2(t2 + τ )ht2+τ , . . .

),

where

(7)Pαi (ti + τ )= Vti (τ )Pαi (ti )V−1ti

(τ ).

The unitary evolution of the projected history remainsa projected history, i.e.,

(8)U(τ)Cαφ = Cα(τ)U(τ)φ,

where one may write

(9)Cα(τ)=U(τ)CαU−1(τ ).

Thus, the Flesia–Piron generalized state of Eq. (2)may be considered as a GMHG history developingdynamically under aunitary evolution of histories.

In the GMHG histories formalism [10–12] onedeals with a set of alternative histories [10], whichare defined in the most simple example by givingsequences of projections at definite moments of timet1, t2, t3, . . . , tn. The sequences are denoted [10] byP 1α1(t1),P

2α2(t2),P

3α3(t3), . . . ,P

nαn(tn). The projections

may be different at different times, for example, inthe two slit experiment [13]P 2

α2(t2) could distinguish

whether the electron went through the upper slit or thelower one at timet2, while P 3

α3(t3) might distinguish

various places of arrival at the final screen at timet3.In general,αi corresponds to eigenstates of a set ofobservables atti . Each set ofP ’s satisfies [10]∑αK

PKαK(tK)= I,

(10)PKαK(tK)P

Kα′K(tK)= δαKα′

KPKαK(tK),

indicating that theα’s represent an exhaustive setof exclusive alternatives. An individual history corre-sponds to a particular sequence:α = (α1, . . . , αn) andfor each history there is a corresponding chain of timeordered projection operators [10]Cα = Pn

αn(tn) · · ·

P 1α1(t1). Such histories are termed coarse grained [10]

when theP ’s are not projections onto a basis (a com-plete set of states), and when there is not a set ofP ’s at

D. Bar, L.P. Horwitz / Physics Letters A 303 (2002) 135–139 137

each and every time, otherwise, they are fine grained.When the initial state is pure one can resolve it, by us-ing the previous equations, into branches correspond-ing to the individual members of any set of alternativehistories. That is, denoting in the Heisenberg picturethe initial state by|Ψ 〉 one obtains [10]

(11)

|Ψ 〉 =∑α

Cα|Ψ 〉 =∑

α1,...,αn

P nαn(tn) · · ·P 1

α1(t1)|Ψ 〉.

The vectorCα|Ψ 〉 is the branch of|Ψ 〉 that corre-sponds to the individual historyα.

If φ, ξ correspond to the GMHG historiesφk , ξkrespectively, the scalar product between them is givenby [10,11] (we shall use this definition below)

(12)(φ, ξ)=∏k

(φk, ξk)Htk.

If the setα, α differ by one projectionPαk in the se-quence for whichPk

αk(tk)P

kαk(tk)= 0, then〈Ψα|Ψα〉 =

0 where|Ψα〉 is the branch of|Ψ 〉 that correspondsto the historyα (see Eq. (11)). Following the defini-tion given by Isham [11], embedding the space of his-tory filters in the orthocomplemented lattice of historypropositions, we may define a density matrixρ in termof a priori probabilities over the arbitrary histories thatform, in this space, a complete set. The density opera-tor associated with such a state is

(13)ρ =∑α

pα

k=n∏k=1

Pkαk(tk),

where the sum is taken over all histories andpα isthe probability for the occurrence of the historyα.Such histories do not necessarily satisfy the GMHGconsistency requirement. Thus, defining the trace asthe diagonal sum of expectations over all histories(which are conventionally denoted inside the tensorproduct asφα), in the sense of Isham’s completeness,

tr(ρ)=∑α

(φα, ρφα

).

If we take a subset of historiesα to correspond to theα’s that occur inρ, and the remainder orthogonal tothese, we see that

(14)tr(ρ)=∑α

pα∏i

∣∣(φαi , φαi )∣∣2 =∑α

pα = 1.

Such a density operator is consistent with the notionof the Lax–Phillips state, since in this theory eachpure state corresponds to a quantum mechanical his-tory. A decoherence functional for such state definedbyd(α, α′)= tr(CαρC

†α) satisfies the consistency con-

ditions for any pair of histories satisfying the condition(10), i.e.,

tr(CαρC

†α

)(15)=

∑β

∑α′′

pα′′∏i

(φβi ,PαiPα′′

iPαi φ

βi).

Choosing a subset of historiesβ to coincide withα,and the remainder orthogonal to these, we see that

tr(CαρC

†α

) =∑α′′

pα′′∏i

δαiα′′δα′′αi = pαδαα,

i.e., the analog of the GMHG condition for consistenthistories is formally satisfied with these definitions.Furthermore, for an observableA defined on the spaceof histories (as for operators on the Lax–PhillipsspaceH )

(16)tr(Aρ)=∑α

∑α

pα

∏i

(φαi ,APαi φ

αi),

and again taking a subset of historiesα to coincidewith the set occurring inρ, and the remainder orthog-onal to these, one finds

(17)tr(Aρ)=∑α

pα∏i

(φαi ,Aφαi

).

We remark that the Lax–Phillips Hilbert space con-tains elements that represent resonances, and whichevolve according to exact semigroup laws (and hencecorrespond to irreversible processes). This propertycannot be achieved in the framework of the usualquantum theory using the Wigner–Weisskopf formu-lation [15] of the description of an unstable system.

The projection operatorsCα are proper projectionoperators inH . With suitable conditions, identifyingincoming and outgoing subspaces inH which are sta-ble under the action ofU(τ) for τ positive and neg-ative, respectively, the quantum Lax–Phillips theory[7] provides a rigorous framework for describing irre-versible processes (semigroup evolution of a subspaceof H ) here seen as an evolution of the GMHG histo-ries.

138 D. Bar, L.P. Horwitz / Physics Letters A 303 (2002) 135–139

The action ofU(τ) is generated by a self adjointgenerator

(18)K = s- limτ→0

1

τ

(U(τ)− I

).

For example, the Flesia–Piron model [8] is

(19)K =H − i∂t ,

whereH is a (possiblyt-dependent) Hamiltonian op-erator defined on the auxiliary spaces. This general-ized generatorK(q,p, t,E) (which may depend onthe variablet but not on the laboratory parameter timeτ ; E is represented byi∂t ) satisfies the following gen-eralized equation [8]

i∂

∂τ

(U(τ)φ

)t= (

U(τ)Kφ)t= Vt−τ (τ )(Kφ)t−τ

(20)= Vt−τ (τ )(Ht−τφt−τ − i∂tφt−τ ).

Using Eqs. (3)–(6) we write the left hand side ofEq. (20) as

i∂

∂τ

(U(τ)Cαφ

)

(21)

= i∂

∂τ

(Vt1(τ )Pα1(t1)ht1,Vt2(τ )Pα2(t2)ht2, . . .

).

In the case that the stateCαφ is stationary under theevolutionU(τ), so that

(22)Vti (τ )Pαi (ti )hti = Pαi (ti )hti ,

one obtains, in the special case of Eq. (19), theSchrödinger equation

(23)Htiφti = i∂tiφti

at everyti . We now show that for a special choice ofevolution relating successive sequences

(24)htk+δτ = Vtk(δτ )htk = htk (δτ ),

with δτ = Tn

, T is the total spant1, t2, . . . , tn, weobtain the stability of the correspondingCαφ by amechanism analogous to the dynamical Zeno effect[2,3]. SinceVtk(δτ ) is generated on the auxiliary spaceHtk by a self adjoint operatorHtk , for n large Eq. (24)may be written to second order inδτ as

(25)

(1− iδτHtk − δτ2

2H 2tk

)htk = htk+δτ ,

so that forδτ → 0 the Schrödinger equation (23) issatisfied (here we takeδτ = δt). Applying definition

(12) of the tensor product we see that, identifyinghtk+δτ ≡ htk (δτ ) as associated with the componenttk ,⟨U(δτ)φ,φ

⟩(26)=

∏k

⟨(1− iδτHtk − δτ2

2H 2tk

)htk , htk

⟩.

It then follows that

(27)∣∣⟨U(δτ)φ,φ

⟩∣∣2 =∏k

‖htk‖4(1− δτ2!H 2tk

),

where,

(28)!H 2tk

= 〈htk ,H 2tkhtk 〉Htk

‖htk‖2−

〈htk ,Htkhtk 〉2Htk

‖htk‖4.

The product to second order is∣∣⟨U(δτ)φ,φ⟩∣∣2

(29)=(∏

k

‖htk‖4)(

1+ T 2

n2

∑k

!H 2tk

).

The second term vanishes in the limit ofn → ∞ ifthe dispersions!Htk are finite at eachk. The GMHGscalar product for this evolution constructs the se-quence corresponding to the well-known Zeno phe-nomenon in the stabilization of states by successivemeasurement. Moreover, this result demonstrates thatthe set of consistent histories constructed by succes-sive measurement is associated with the Schrödingerrelation between successive states in the Lax–Phillipsfoliation.

In this calculation we have concluded that theGMHG evolution is stationary if the sequencehtk isdetermined by Schrödinger evolution in the Flesia–Piron model [8]. However, if we choose anarbitrarysequencehtk andU(δτ) induces Schrödinger evolu-tion at each stepδτ , the relation (26) remains valid,independently of whether the sequencehtk itself cor-responds to a Schrödinger sequence. The conclusion(29) remains, and we see the resulting GMHG stabil-ity corresponds to the dynamic Aharonov–Vardi [2]Zeno effect. The static Zeno effect follows if the se-quencehtk is chosen to be a set of identical states, i.e.,htk = ht0.

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