A quantum theory for a total system including a measuring apparatus

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A quantum theory for a total system including a measuring apparatus

Quantum description for a totalsystem with respect to an internalreference systemWen-ge WangDepartment of Modern Physics, University of Science andTechnology of China, Hefei, China1


I. IntroductionThe systems to be studied.The total systemreferencesystem(RS) ƦenvironmentƐ of the RSA RS: a system with respect towhich the state of other systemscan be described.The necessity of addressing RS:there exists no absolute state.Assumption: the totalsystem can be describedwith respect to an internalRS.We’ll consider a type ofRS, which may play theessential role of ameasuring apparatus.2


The type of theory we want to find:CopenhageninterpretationMany-worldsinterpretationConsistenthistoriesinterpretationFeatures we want tokeepDefinite properties ofmeasuring apparatusUniversal Schrodingerevolution (and,perhaps, branching)Consistency conditionNew elements we wantto add (features wewant to avoid).Unified description ofmacro and microscopicsystemsCondition of branching(many worlds)Dynamical descriptionof processes.3


Our method:To generalize the standard formalism of QM (SFoQM) todescribe such a total system.Axiom of measurement in SFoQM:Before measurement: |ψ> =Σ a C a |a>.After measurement: →ρ= Σ a |C a | 2 |a>= Σ μ | Ψ μ >?→ρ= Σ μ |Ψ μ >


An analysis : Measuring apparatus as an external systemMeasuring apparatus(RS)Measurement results aregiven by recordableproperties of the RS.measureBe able to measure allproperties of the system.systemBe able to experimentally distinguishbetween pure states (vectors) andmixed states (density operators) of thesystem.5


─ reference property of RS,─ measurementrecordable property ofa measuring apparatusReference propertyof a RS Ʀmain featureA property of Ʀ that hasa definite value when Ʀis employed as the RS.measurementA process of interaction between a RSand a measured system, in whichsome reference property of the RSmay change.A RS can play the essentialrole of a measuring apparatus(without considering amplification process)We considerthis type ofRS here.9


(III) Basic assumptionsPostulate I:The state of the total system Ʀ+Ɛ can be associated with avector or a density operator in the total Hilbert space H.10


Postulate II:When the state of the total system is described by a vector |Ψ(t)〉in the total Hilbert space, its time evolution obeys SchrődingerequationH is the Hamiltonian of the total system,11


Mathematical expression of propertyA property of a RS, which is indicated by a quantity with discretevalues, μ=μ 1 ,μ 2 , …, is related to an observable A {μ} ,P μ are projection operators in the Hilbert space of the RS Ʀ,is used to indicate the subspace related to P μ .We also use P μ to indicate corresponding projectionoperators in the total Hilbert space.12


Condition for a property to be a reference propertyConsider a state of thetotal system at a time t withrespect to a RS ƦDescribed by |Φ〉afternormalizationThe RS Ʀ has a reference property with a definite value μ,which is related to an observable A {μ} .Finite τ s : stableness of the reference property.13


Stability conditionFor the simplicity in discussion, we assume that a referenceproperty is absolutely stable when the interaction HamiltonianH I can be neglected,orIn this case, H Ʀμ are energy eigen-subspaces of H Ʀ ,the Hamiltonian of the RS.14


Physically equivalent descriptionsTwo descriptions of thetotal system arephysically equivalentwith respect to a RS Ʀ,ifthey give the samepredictions for results ofmeasurements performedby Ʀ.The two descriptionsare experimentallyindistinguishable15


Assumption of RS (the third assumption):For a system Ʀ which is employed as the RS,if an observable A {μ} of Ʀ is a reference property when the totalsystem is described by |Φ〉=P μ |Ψ〉,then, | Ψ〉and ρ are physically equivalent descriptions for thetotal system, where16


RemarksIt is important to note that the assumption of RS addressesdescriptions given with respect to the RS Ʀ, in particular,A {μ} is an observable of Ʀ.|Ψ〉and ρare usually not physically equivalent, when thetotal system is described with respect to another RS.17


(IV) Physically allowed H-regionBranching picture of time evolutioninitial state|Ψ(t 0 )〉P μ(0) |Ψ(t 0 )〉has areference property A {μ(0)} .|Ψ(t 0 )〉18


At t 1having a referenceproperty with avalue μ (1) at time t 1 .19


A picture of evolution similar to that in many-worlds interpretationof quantum mechanics.tree Υ20


Paths and tree expansion of a vectorPath of splitting|Ψ(t)〉21


Predictions for results of measurements performed by ƦThe probability for Ʀ to have the value μ of the reference property:|Ψ〉22


A consistency condition (with a form formally similar to theconsistency condition in consistent-histories interpretation of QM)consistency condition:Explicit expression of thephysical equivalence of|Ψ〉and ρ.23


Restriction in the Hilbert space: allowed H-regionThe allowedH-regionthe set of vectors in H, for which theconsistency condition is satisfied for allpossible trees.The allowed H-region of is the state space when Ʀ isemployed as the RS, which is usually smaller than thanthe total Hilbert space.vectors not in theallowed H-regioncan not be associatedwith any physical state24


(V) Some applications(1)collapse ofstate vectora manifestation ofphysical equivalence ofvector and densityoperator descriptions ofthe total system25


(2) Irreversible feature of time evolutionVon Neumann entropy of the total systemIt keeps constant when no splitting happens, as a result of unitaryevolution, but increases at each splitting time t i along paths.S(Υ,t) may increase but never decrease with time.Time reversalsymmetry is brokenThe allowed H-region of W d issmaller than the total Hilbert spaceMacroscopic irreversibility ?26


(3) A model: RS as a two-level systemThe observable A {k} is the only non-trivial observable thatmay represent a reference property of the RS.When it indeed represents a reference property,Assume: A {k} is a reference property in time intervals [t i ,t ie ]and is not a reference property within (t ie ,t i+1 ).t 0 yes t 0eno t 1 yes t 1e27


ResultsFor initial vectors of product form, |Ψ〉=|Ʀ〉|Ɛ〉, theconsistency condition is satisfied under the followingconditions:(i) The environment is sufficiently large and irregular.(ii) There is enough difference between(iii) The time intervals, within which the observable A {k} is areference property, are much longer than a decoherencetime τ d ,(τ s >>τ d )28


Some problems for the future(1)Further understanding for decoherence due todifference in paths.(2) Size of allowed H-region of a given W d .(3)Relation between descriptions with respect todifferent RS.29


Thank you!30

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