A quantum theory for a total system including a measuring apparatus

Quantum description **for** a **total****system** with respect to an internalreference **system**Wen-ge WangDepartment of Modern Physics, University of Science andTechnology of China, Hefei, China1

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

The type of **theory** we want to find:CopenhageninterpretationMany-worldsinterpretationConsistenthistoriesinterpretationFeatures we want tokeepDefinite properties of**measuring** **apparatus**Universal Schrodingerevolution (and,perhaps, branching)Consistency conditionNew elements we wantto add (features wewant to avoid).Unified description ofmacro and microscopic**system**sCondition of branching(many worlds)Dynamical descriptionof processes.3

Our method:To generalize the standard **for**malism of QM (SFoQM) todescribe such a **total** **system**.Axiom of measurement in SFoQM:Be**for**e 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 **system**Measuring **apparatus**(RS)Measurement results aregiven by recordableproperties of the RS.measureBe able to measure allproperties of the **system**.**system**Be able to experimentally distinguishbetween pure states (vectors) andmixed states (density operators) of the**system**.5

─ reference property of RS,─ measurementrecordable property ofa **measuring** **apparatus**Reference 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 the**total** **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 the**total** **system** arephysically equivalentwith respect to a RS Ʀ,ifthey give the samepredictions **for** results ofmeasurements per**for**medby 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 **total****system** is described by |Φ〉=P μ |Ψ〉,then, | Ψ〉and ρ are physically equivalent descriptions **for** the**total** **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 the**total** **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 per**for**med by ƦThe probability **for** Ʀ to have the value μ of the reference property:|Ψ〉22

A consistency condition (with a **for**m **for**mally 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** **system**25

(2) Irreversible feature of time evolutionVon Neumann entropy of the **total** **system**It 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 **system**The 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 **for**m, |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