aa r X i v : . [ qu a n t - ph ] D ec Time translation of quantum properties
Roberto Laura
Facultad de Ciencias Exactas, Ingenier´ıa y Agrimensura(UNR) and Instituto de F´ısica Rosario (CONICET - UNR),Av. Pellegrini 250, 2000 Rosario, Argentina ∗ Leonardo Vanni
Instituto de Astronom´ıa y F´ısica del Espacio (UBA - CONICET). Casilla de Correos 67,Sucursal 28, 1428 Buenos Aires, Argentina. † (Dated: July 2008) Abstract
Based on the notion of time translation, we develop a formalism to deal with the logic of quantumproperties at different times. In our formalism it is possible to enlarge the usual notion of contextto include composed properties involving properties at different times. We compare our resultswith the theory of consistent histories. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION. The absence of determinism is one of the main differences of the quantum theory withthe classical one. Nevertheless, it is necessary in quantum theory to deal in a consistent waywith expressions involving different properties of the system at different times. For example,it is necessary to relate a property of a microscopic system at a given time, previous to ameasurement, with a property of an instrument when the measurement is finished. Moreover,in the famous double slit experiment it is argued about the impossibility to say which slit aparticle passed before producing a spot on a photographic plate [1].In a series of papers starting in 1984, an approach to quantum interpretation knownas consistent histories, or decoherent histories, has been introduced by R. Griffiths [2], R.Omn`es [3], M. Gell-Mann and J. Hartle [4]. In their approach, the notion of history is definedas a sequence of properties at different times. The probability for a history is also definedthrough a formula motivated by the path integral formalism, but with no direct relationwith the Born rule. A consistency condition should be satisfied by the possible differenthistories which can be included in a legitimate description of the system, therefore the nameof consistent histories. For a given physical system the possible sets of consistent historiesdepend on the state. This is not entirely satisfactory, because in axiomatic theories ofquantum mechanics the state is usually considered as a functional on the space of observables,and it appears after these observables in a somehow subordinate position. The importanceof the notion of state functionals acting on a previously defined space of observables wasstressed by one of us in references [5] and [6].In this paper we explore a different approach to define the probability of a conjunc-tion of properties at different times, and to discriminate which are the properties that canbe simultaneously considered in a description of the system. Essentially, we consider thetime translation of a property, already at our disposal in the dynamic generated by theSchr¨odinger equation. When properties corresponding to different times are translated to acommon single time, we can apply to them the usual rules of compatibility between differentobservables, and compute the probabilities using the Born rule.In section II we present a brief summary of the theory of consistent histories, and wediscuss its application to a spin system. In section III we give a short review of the notionsof quantum logic, contexts and probabilities. The notion of time translation of quantum2roperties is introduced in section IV, where we also obtain the non distributive lattice oftime dependent properties. This definitions are used in section V to implement expressionsinvolving the conjunction of properties at different times, and to define the compatibility ofthese type of expressions. Distributive lattices of time dependent properties, called general-ized contexts, are also obtained in this section. The generalized contexts are compared withthe theory of consistent histories in section VI. The conclusions are given in section VII.
II. THE THEORY OF CONSISTENT HISTORIES.
In what follows we present the main features of the theory of consistent histories, followingessentially the approach given by R. Omn`es [7], [8], [9], and we also discuss the applicationof this theory to the case of a spin system.Let us consider a state of a system at time t , represented by the state operator b ρ t . Anobservable represented by an operator b A j with spectrum σ j is considered for each time t j ( j = 1 , ..., n ) of a sequence verifying t < t < ..... < t n . Each spectrum σ j is partitioned bya family { ∆ ( µ ) j } of mutually exclusive sets ( ∪ µ ∆ ( µ ) j = σ j , ∆ ( µ ) j ∩ ∆ ( µ ′ ) j = ∅ ( µ = µ ′ )).The operator b E ( µ ) j is the projector onto the subspace of the Hilbert space corresponding tothe subset ∆ ( µ ) j of the spectrum σ j , and it represents the property ”the value of the observable A j is in the set ∆ ( µ ) j at time t j ”. These projectors satisfy the equations b E ( µ ) j b E ( µ ′ ) j = δ µµ ′ b E ( µ ′ ) j and P µ b E ( µ ) j = b I .A history a is defined by the property ”(the value of A is in ∆ ( k )1 at t ) and (the valueof A is in ∆ ( k )2 at t ) and ..... and (the value of A n is in ∆ ( k n ) n at t n )”. It is represented bythe history operator b C a = b E ( k n ) n ( t n ) ..... b E ( k )2 ( t ) b E ( k )1 ( t ) , b E ( k j ) j ( t j ) = e i b H ( t j − t ) / ~ b E ( k j ) j e − i b H ( t j − t ) / ~ , (1)where b H is the Hamiltonian operator of the system.The probability of the history a is defined by the expressionPr( a ) = T r ( b C a b ρ t b C † a ) . (2)As the probability should verify positivity, normalization and additivity, the possiblehistories to be included in a valid description of the system should verify some consistencyconditions . Sufficient conditions, given by Gell-Mann and Hartle, are T r ( b C a b ρ t b C † b ) = 0 , a = b. (3)3he theory of consistent histories is a framework suitable to include properties of a systemat different times in the language of quantum theory. Moreover, these properties at differenttimes are given a well defined probability by Eq. (2), provided we consider properties within aconsistent family of histories. Each family of consistent histories generate a possible universeof discourse about a quantum system. In general, it is not possible to include two familiesin a single larger one. Different sets of consistent histories are considered complementarydescriptions of the system.For simplicity we are going to consider the case n = 2, involving histories with only twodifferent times t and t . For the time t the spectrum of the observable b A is partitionedby two complementary sets ∆ and ∆ with projectors b E and b E , while for the time t the spectrum of the corresponding observable b A is partitioned by the sets ∆ and ∆ withprojectors b E and b E .For this special case the necessary and sufficient consistency condition to obtain welldefined probabilities is Re T r ( b E ( t ) b ρ t b E ( t ) b E ( t )) = 0 , (4)which is called the Griffiths condition.In this case, the sufficient Gell-Mann and Hartle condition is T r ( b E ( t ) b ρ t b E ( t ) b E ( t )) = 0 . (5)Logical operations and relations are well defined on a family of consistent histories. IfΣ = ∆ ∪ ∆ is the spectrum of b A and Σ = ∆ ∪ ∆ is the spectrum of b A , the elementaryhistories are represented in Σ × Σ by the sets ∆ × ∆ , ∆ × ∆ , ∆ × ∆ and ∆ × ∆ . Allthe properties of the family are represented by the unions of these four sets. In this way, thenotions of conjunction , disjunction and negation are obtained. According to the approachof R. Omn`es [7], [8], a property a is said to imply another property b of the same family ifPr( b | a ) = Pr( b and a ) / Pr( a ) = 1. The conventional axioms of formal logic are satisfied withthese definitions [8]. We notice that in this theory the implication relation is defined trougha previous definition of probability. This makes a big difference with the usual approaches toquantum mechanics, where the logical relations and operators on the properties have theirown definition independent of the probability, which is later on given by the Born rule. Ourown construction of the logical operators and relations, to be developed in sections IV and4, will be established in a way which do not depend on the definition of probability or onthe state of the system.R. Omn`es [8] applied the theory of consistent histories to the case of a spin system,prepared at the time t in a pure state with S x = + ~ , represented by the vector | x + i ( b ρ t = | x + ih x + | ). The description of the system include the two possible values of the spin alongthe z axis direction for the time t , which may be obtained by an Stern-Gerlach experiment.With the simplifying assumption of the vanishing of the Hamiltonian, he searched for thepossibility to include in the description the two possible values of the spin along a direction n ( | n | = 1) for a time t after the preparation of the spin along the positive directionof the x axis, and before the time t corresponding to the measurement along the z axis( t < t < t ). The state vectors | n + i and | n −i correspond to the values + ~ and − ~ along the direction n . If the sufficient Gell-Mann and Hartle condition (5) is applied with b E ( t ) = | n + ih n + | , b E ( t ) = | n −ih n − | , b ρ = | x + ih x + | and b E ( t ) = | z + ih z + | , twopossibilities are obtained:(i) A set of histories including the two possible spin values along the x axis at time t ,represented by the projectors b E = | x + ih x + | and b E = | x −ih x − | , together with the twopossible spin values along the z axis at time t , represented by the projectors b E = | z + ih z + | and b E = | z −ih z − | . Therefore in this case n = (1 , , z axis at time t ,represented by the projectors b E = | z + ih z + | and b E = | z −ih z − | , together with the twopossible spin values along the z axis at time t , represented by the projectors b E = | z + ih z + | and b E = | z −ih z − | . In this case n = (0 , , n is the preparation spin direction at the time t , and if n isthe spin direction at the time t , the equation ( n × n ) · ( n × n ) = 0 is obtained for thepossible n spin directions at the time t (see reference [9], page 161). For n = (1 , ,
0) and n = (0 , , x and the z directions, the direction n could be any vectorin the planes xy or yz .Up to now, we have considered well known facts of the theory of consistent histories.We now analyze in more detail the consistent family obtained in the case (i). Well definedprobabilities can be obtained for all the members of the family applying Eq. (2). Let usconsider the probability of the spin to be + ~ along the x axis at time t and to be + ~ z axis at time t ,Pr(( x + , t ); ( z + , t )) = T r ( b E b E b ρ t b E b E ) = |h x + | z + i| = 12 . Provided that t < t < t , any value of t gives a consistent family of histories, andtherefore a valid description of the spin system. As the time t can be chosen very close tothe time t we have lim t −→ t Pr(( x + , t ); ( z + , t )) = 12 . (6)The property giving simultaneously well defined values to different components of thespin is forbidden in the universe of discourse of ordinary quantum mechanics, due to theuncertainty principle (operators representing two different components of the spin do notcommute). Therefore the limit of Eq. (6) cannot be interpreted as the probability for theconjunction of the spin values x + and z + at the single time t . The theory of consistenthistories is discontinuous in its property ascriptions for different times.In the following sections we present our own approach to the description of time dependentproperties of a quantum system. We are going to prove that in our formalism only the case(ii) survives as a valid description for the two times properties of the spin system. Only forthis case, the limit t −→ t of Eq. (6) can be interpreted as a single time probability. III. QUANTUM LOGIC, CONTEXTS AND PROBABILITIES.
We shall give in this section a brief summary of the logical structure of quantum mechanicsaccording to G. Birkhoff and J. von Neumann [10], which is one of the standard approachesto quantum logic [11]. This summary is a preparation for our own approach to the problemof time dependent properties, to be developed in section IV.A Hilbert space H is associated with each isolated physical system. Every quantumproperty p is represented by a subspace H p of the Hilbert space H . For each subspace H p there exists a projection operator b Π p such that H p = b Π p H , and therefore a property p canalso be represented by the projector b Π p .The implication relation between two properties is defined by the inclusion of the cor-responding subspaces ( p ⇒ p iff H p ⊆ H p ). The conjunction of two properties p and p ′ is represented by the greatest lower bound of the two corresponding subspaces H p and H p ′ ( H p ∧ p ′ = Inf ( H p , H p ′ ) = H p ∩ H p ′ ), while the disjunction is the least upper bound6 H p ∨ p ′ = Sup ( H p , H p ′ )). Moreover, the negation of a property p is represented by theorthogonal complement of the corresponding subspace ( H − p = H ⊥ p ).Endowed with these implication, conjunction, disjunction and negation, the set of allproperties of a physical system is an orthocomplemented nondistributive lattice. Accordingto quantum theory, not all the properties can simultaneously be considered in a descriptionof a physical system. Different descriptions involve different sets of properties.Each possible description is called a context , and it is defined through a set of atomicor elementary properties p j (where j belongs to a set σ of indexes). The properties p j ,represented by projectors b Π j , are mutually exclusive and complete, i.e. b Π i b Π j = δ ij b Π i , X i ∈ σ b Π i = b I. By disjunction of these elementary properties it is possible to generate all the propertiesof the context, obtaining a distributive lattice . Each property p of the context can berepresented by a projector which is a sum of some of the projectors b Π i , b Π p = X i ∈ σ p b Π i , σ p ⊂ σ. A state of the system is represented by the statistical operator b ρ , which is self adjoint,positive and with trace equal to one. In the state represented by b ρ , the probability for eachproperty p of a context is given by the Born rulePr( p ) = T r ( b ρ b Π p ) . This rule gives a well defined probability (i.e. it is positive, normalized, and satisfy theadditivity property) when it is applied to the properties of a given context.Moreover, the probability of a property b conditional to a property a , defined byPr( b | a ) = Pr( b ∧ a )Pr( a ) , (7)is also a well defined probability within a context. The conditional probability can be usedto give a statistical meaning to the implication relation. It is not difficult to prove that aproperty a implies a property b ( a ⇒ b ) if and only if Pr( b | a ) = 1 for all the states of thesystem.We emphasize that the conjunction and the disjunction of the logical structure of thequantum properties are obtained from the previous notion of implication, defined by the7nclusion of subspaces. On this lattice of properties, the Born rule is applied to obtainthe probabilities, which are only meaningful for subsets of properties within a context.Properties belonging to different contexts do not have simultaneous physical meaning.In the next section we are going to present an extended notion of context to deal withdescriptions involving time dependent properties. IV. TIME TRANSLATIONS AND THE LATTICE OF TIME DEPENDENTPROPERTIES.
The time enters quantum theory through the Schr¨odinger equation generating the evo-lution of the state of an isolated system. The evolution of a vector of the Hilbert space H ,representing a pure state, is given by | ϕ t ′ i = b U ( t ′ , t ) | ϕ t i , b U ( t ′ , t ) = e − i b H ( t ′ − t ) / ~ , (8)where b H is the Hamiltonian operator of the system.Let us consider the physical system at the time t , in a pure state represented by the vector | ϕ t i . Moreover, assume that | ϕ t i is an eigenvector with eigenvalue one of the projector b Π p corresponding to the property p b Π p | ϕ t i = | ϕ t i . (9)We can say that the physical system has the property p at the time t , because it hasprobability equal to one, according to the Born rule:Pr( p ) = h ϕ t | b Π p | ϕ t i = h ϕ t | ϕ t i = 1 . At a later time t ′ , the time evolved state is represented by the vector | ϕ t ′ i given byequation (8). Using equations (8) and (9) it is easy to prove that b Π p ′ | ϕ t ′ i = | ϕ t ′ i , b Π p ′ = b U ( t ′ , t ) b Π p b U − ( t ′ , t ) . Therefore, if the system has the property represented by the projector b Π p at time t , italso has the property represented by the projector b Π p ′ at time t ′ . We have obtained in thisway a procedure for the time translation of properties .It is easily proved that the obtained relation between p at time t and p ′ at time t ′ istransitive, reflexive and symmetric. Therefore, it is an equivalence relation, that we shallindicate by the expression ( b Π p , t ) ∽ ( b Π p ′ , t ′ ).8he expression ( b Π p , t ) is a symbol indicating the property p at time t . We shall alsoindicate by [ b Π p , t ] the class of properties equivalent to the property p at time t .If a property p at time t is equivalent to p ′ at time t ′ , the Born rule gives for them thesame probability,Pr( b Π p ′ , t ′ ) = T r ( b ρ t ′ b Π p ′ ) = T r ( b U ( t ′ , t ) b ρ t b U − ( t ′ , t ) b Π p ′ ) = T r ( b ρ t b Π p ) = Pr( b Π p , t ) , (10)and therefore a single probability is obtained for the properties of the same class of equiva-lence. In physical terms, all the properties of a given class are essentially the same property,as they can be obtained one from the other trough time evolution.The just considered time translation, and the implication of ordinary quantum mechanicspresented in the previous section, suggest that we define that the equivalence class [ b Π , t ] implies the equivalence class [ b Π , t ] if the representative elements at a common time t verify the implication of the usual formalism of quantum mechanics, i.e. b Π , H ⊂ b Π , H , b Π , ≡ b U ( t , t ) b Π b U − ( t , t ) , b Π , ≡ b U ( t , t ) b Π b U − ( t , t ) H . It is not difficult to prove that if two projectors b Π and b Π verify this condition for agiven time t , they verify the condition for all possible values of t . The implication relationis transitive, reflexive and antisymmetric, and therefore it is a well defined order relation onthe equivalence classes.Having defined an order relation on the equivalence classes, the conjunction ( disjunction )of two classes [ b Π , t ] and [ b Π ′ , t ′ ] can be obtained as the greatest lower (least upper) bound,i.e. [ b Π , t ] ∧ [ b Π ′ , t ′ ] = Inf { [ b Π , t ]; [ b Π ′ , t ′ ] } = [ lim n →∞ ( b Π b Π ′ ) n , t ] , (11)[ b Π , t ] ∨ [ b Π ′ , t ′ ] = Sup { [ b Π , t ]; [ b Π ′ , t ′ ] } = [( b I − lim n →∞ { ( b I − b Π )( b I − b Π ′ ) } n ) , t ] , (12)where b Π = b U ( t , t ) b Π b U − ( t , t ) and b Π ′ = b U ( t , t ′ ) b Π ′ b U − ( t , t ′ ) are the translations of theproperties b Π and b Π ′ to the time t . The projectors lim n →∞ ( b Π b Π ′ ) n and ( b I − lim n →∞ { ( b I − b Π )( b I − b Π ′ ) } n ) generate the greatest lower and the least upper bounds of the subspacesgenerated by b Π and b Π ′ [12].The negation of an equivalence class [ b Π , t ] is defined by[ b Π , t ] = [ b Π , t ] = [( b I − b Π) , t ] . With the implication, disjunction, conjunction and negation previously obtained, the setof equivalent classes has the structure of an orthocomplemented nondistributive lattice.9 . THE GENERALIZED CONTEXTS.
The usual concept of context was briefly reviewed in section III as a subset of all possiblesimultaneous properties which can be organized as a meaningful description of a quantumsystem at a given time, and endowed with a boolean logic with well defined probabilities.The definitions and notations given in the previous section will be useful to our purposeof representing valid descriptions involving properties at different times , which we are goingto call generalized contexts . In what follows, we shall only consider descriptions involvingproperties at two times t and t , but our formalism has an immediate extension to casesinvolving more than two times.Let us consider a context of properties at time t , generated by atomic properties p (1) j represented by projectors b Π (1) j verifying b Π (1) i b Π (1) j = δ ij b Π (1) i , X j ∈ σ (1) b Π (1) j = b I, i, j ∈ σ (1) . (13)Let us also consider a context of properties at time t , generated by atomic properties p (2) µ represented by projectors b Π (2) µ verifying b Π (2) µ b Π (2) ν = δ µν b Π (2) µ , X µ ∈ σ (2) b Π (2) µ = b I, µ, ν ∈ σ (2) . (14)We wish to represent with our formalism a universe of discourse capable to incorporateexpressions like ”the property p (1) j at time t and the property p (2) µ at time t ”. With thispurpose, we note that the properties associated to different times t and t can be translatedto a common time t , by using the equivalence relations previously defined( b Π (1) i , t ) ∽ ( b Π (1 , i , t ) , b Π (1 , i ≡ b U ( t , t ) b Π (1) i b U − ( t , t ) , ( b Π (2) µ , t ) ∽ ( b Π (2 , µ , t ) , b Π (2 , µ ≡ b U ( t , t ) b Π (2) µ b U − ( t , t ) . (15)The conjunction of the equivalence classes [ b Π (1) i , t ] and [ b Π (2) µ , t ] can be obtained applyingEq. (11) [ b Π (1) i , t ] ∧ [ b Π (2) µ , t ] = [ b Π (1 , i , t ] ∧ [ b Π (2 , µ , t ] = [ lim n →∞ ( b Π (1 , i b Π (2 , µ ) n , t ] . The conjunction of the classes with representative elements b Π (1) i at t and b Π (2) µ at t ,is also the conjunction of the classes with representative elements b Π (1 , i and b Π (2 , µ at the10ommon time t . Moreover, the conjunction is a class with the representative elementlim n →∞ ( b Π (1 , i b Π (2 , µ ) n at the time t . The conjunction of properties at the same time is al-ready defined in quantum mechanics, for the particular case in which they are representedby commuting projectors . The usual quantum theory do not give any meaning to the con-junction of simultaneous properties represented by non commuting operators.To make contact with the usual formalism of quantum theory, it seems natural to considerquantum descriptions of a system, involving the properties generated by the projectors b Π (1) i at the time t and b Π (2) µ at the time t , only for the cases in which the projectors b Π (1) i and b Π (2) µ commute when translated to a common time t , i.e. b Π (1 , i b Π (2 , µ − b Π (2 , µ b Π (1 , i = 0 (16)If this is the case, we have lim n →∞ ( b Π (1 , i b Π (2 , µ ) n = b Π (1 , i b Π (2 , µ , and for the equivalenceclass of composed properties h iµ , representing ”the property p (1) j at time t and the property p (2) µ at time t ” we obtain h iµ = [ b Π (1) i , t ] ∧ [ b Π (2) µ , t ] = [ b Π (1 , i b Π (2 , µ , t ] = [ b Π (0) iµ , t ] , b Π (0) iµ ≡ b Π (1 , i b Π (2 , µ . As we can see, the conjunction of properties at different times t and t is equivalent toa single property represented by the projector b Π (0) iµ at the single time t .If the different contexts at times t and t produce commuting projectors b Π (1 , i and b Π (2 , µ at the common time t , it is easy to prove that b Π (0) iµ b Π (0) jν = δ ij δ µν b Π (0) iµ , X iµ b Π (0) iµ = b I. (17)Therefore, we realize that the composed properties h iµ , represented at the time t by thecomplete and exclusive set of projectors b Π (0) iµ , can be interpreted as the atomic propertiesgenerating a usual context in the sense already described in the previous section. Moregeneral properties are obtained from the atomic ones using the disjunction operation definedin Eq. (12). For example, taking into account the commutation relation (16), we obtain h iµ ∨ h jν = [ b Π (0) iµ + b Π (0) jν , t ] . More generally, we can represent the property p (1) j at time t and the property p (2) µ attime t , with j and µ having any value in the subsets ∆ (1) ⊂ σ (1) and ∆ (2) ⊂ σ (2) , in the11orm h ∆ (1) , ∆ (2) = [ X i ∈ ∆ (1) X µ ∈ ∆ (2) b Π (0) iµ , t ] (18)As a consequence of Eqs. (17), the set of properties obtained in this way is an orthocom-plemented and distributive lattice.As we proved in Eq. (10), the Born rule defines a single probability to all elements of anequivalence class. If the state of the system at time t is represented by b ρ t , the probabilityof the class of properties h ∆ (1) , ∆ (2) has the following expressionPr( h ∆ (1) , ∆ (2) ) = X i ∈ ∆ (1) X µ ∈ ∆ (2) T r ( b ρ t b Π (0) iµ ) . (19)As we already mentioned in the previous section, a description of a physical system shouldnot involve properties belonging to different contexts. As a natural extension of the notionof context, we postulate that a description of a physical system involving properties at twodifferent times t and t is valid if these properties are represented by commuting projectorswhen they are translated to a single time t . We will call generalized context to each of thesevalid descriptions. On each generalized context, the probabilities given by the Born rule arewell defined (i.e. they are positive, normalized and additive), and therefore they may havea meaning in terms of frequencies.In summary, our formalism is based on the notion of time translation, allowing to trans-form the properties at a sequence of different times into properties at a single common time.A usual context of properties is first considered for each time of the sequence. If the projec-tors representing the atomic properties of each context commute when they are translatedto a common time, the contexts at different times can be organized forming a generalizedcontext of properties. A generalized context of properties is a distributive and orthocom-plemented lattice, a boolean logic with well defined implication, negation, conjunction anddisjunction. This logic can be used for speaking and reasoning about the selected propertiesof the system at different times. Well defined probabilities on the elements of the lattice ofproperties are obtained using the well known Born rule.12 I. COMPARISON OF THE GENERALIZED CONTEXTS WITH THE SETS OFCONSISTENT HISTORIES.
In our opinion, the generalized contexts seem to be a natural generalization of the usualcontexts of quantum mechanics. They are suitable to deal with the logic of properties atdifferent times. But to be a ”natural generalization” may have no any scientific value, andperhaps may only reflects our confidence in the usual form of quantum theory. Therefore, itis necessary to compare our new approach with the theory of consistent histories, designedto deal with the same kind of problems, and also to apply the new formalism to physicallyrelevant situations. The main relation between both theories is given by the followingTheorem:
A generalized context obtained with our formalism, is also a consistent set of histories,with the same probabilities .We give the proof for a generalized context with two times. The probability for theproperty p (1) j at time t and the property p (2) µ at time t , is given by Eq. (19)Pr(( p (1) j , t ) ∧ ( p (2) µ , t )) = T r ( b ρ t b Π (0) jµ ) = T r ( b ρ t b Π (1 , j b Π (2 , µ )= T r ( b Π (2 , µ b Π (1 , j b ρ t b Π (1 , j b Π (2 , µ ) , where the last equality is a consequence of the commutation relation (16) and the cyclicpermutation of the operators in the trace. Taking into account the definitions (15), weobtain for the probability the same expression which is obtained with Eq. (2) for consistenthistories. Moreover, the consistency conditions T r ( b C a b ρ t b C † b ) = 0, for a = b , are satisfieddue to the commutation relations (16). A simple generalization of this proof is obtained fora generalized context with n times. In simple words, the meaning of this theorem is thatour formalism put more restrictions than the theory of consistent histories on the numberof valid descriptions of a physical system.A search of the sets of consistent histories which are forbidden by our formalism, andtheir physical relevance, is unavoidable.We can analyze with our formalism the spin system, already described using the theoryof consistent histories at the end of section II. Once again we consider a description includingthe two possible values of the spin along the z axis for the time t , and we ask which propertiescan also be considered at the time t ( t < t < t ), in such a way that they are compatible13ith the properties chosen at the time t .The atomic properties for the time t are represented by the projectors b E z + = | z + ih z + | and b E z − = | z −ih z −| , while the atomic properties at t are represented by b E n + = | n + ih n + | and b E n − = | n −ih n − | , for an unknown direction n of the spin. These projectors areinvariant under time translations, due to the vanishing of the Hamiltonian. Therefore, theyare invariant when translated to any common time. We may choose this common time as t ( t < t < t ), where the initial state b ρ t is given. If the commutation conditions (16) aresatisfied, we should have b E n ± b E z ± − b E z ± b E n ± = b E n ∓ b E z ± − b E z ± b E n ∓ = 0, which gives the z direction ( n = (0 , , z components of the spin at time t isthe only choice compatible with the z components at the time t , and it corresponds to thecase (ii) obtained with consistent histories in section II. Moreover, this is the only possiblechoice for any initial state b ρ t .The case (i) for the Gelmann and Hartle condition, and all the possibilities for the Griffithscondition are ruled out by our formalism of generalized contexts. Only the case (ii), whichis time continuous with respect to property ascriptions, remains.It is also necessary to verify if the postulated compatibility condition for time translatedproperties is successful to give a good description of well established physical processes.We only mention here what are the results for the well known double slit experiment.R. Omn`es [13] proved that with no measurement instruments there is no place for a setof consistent histories including in its universe of discourse through which slit passed theparticle before reaching a zone in front of the slits. Therefore, as a consequence of thetheorem given at the beginning of this section, there is also no room for such a descriptionwith our generalized contexts. For the case of the double slit with a measurement instrumentsrecording through which slit passed the particle, and another instrument recording theparticle in different zones of a plane in front of the double slit, we found with our approachthe existence of a generalized context suitable for the description of the registration ofthe instruments (but not of the particle positions). As a consequence, the theorem of thebeginning of this section can be used to deduce that such a description has also a placein a set of consistent histories. These are preliminary results which will be included in aforthcoming paper.The version of the theory of consistent histories given by R. Omn`es [7], [8], [9] emphasizesits role as a logical construction, i.e. as a tool for obtaining valid descriptions and reasonings14bout properties of the system. As this is also the case in our formalism, it is interesting tocompare both logical structures.As we briefly summarized in section II, in the theory of consistent histories there areordinary contexts on each time of the sequence. The conjunction, disjunction and negationof properties at different times are defined through the union, intersection and complementof the corresponding spectrums, as shown for the two times case in the paragraph below Eq.(5). In this theory, a history a implies a history b when Pr( b | a ) ≡ Pr( b ∧ a ) / Pr( a ) = 1. Asthe probabilities depend on the state, the implication of the theory is also state dependent.If the set of histories verify the state dependent consistency conditions given by Eqs. (3), (4)or (5), it is named a set of consistent histories, and within this set the conventional axiomsof formal logic are satisfied. Therefore, the possible universes of discourse provided by thistheory have a very special entanglement with the state of the system.This situation is not entirely satisfactory, because in the usual axiomatic theories ofquantum mechanics the state is considered as a functional on the space of observables, andit appears after these observables in a somehow subordinate position. The importance of thenotion of state functionals acting on a previously defined space of observables was stressedby one of us in references [5] and [6]. In our approach of sections IV and V, the logicalstructure of the properties is an orthocomplemented lattice defined independently of thestate of the system and of the probability definition. Moreover, the conditions to have ageneralized context are commutation relations, also state independent (see the conditiongiven by Eq. (16) for the two times case). Probability is later on introduced on the alreadyconstructed logical structure, trough the usual Born rule. VII. CONCLUSIONS.
We have introduced in this paper a formalism suitable to deal with descriptions andreasonings about physical systems involving quantum properties at different times. Thedynamic generated by the Schr¨odinger equation provides a natural definition for the timetranslation of quantum properties. Time translations generate a partition in equivalenceclasses of the set of properties and times. From a physical point of view, properties atdifferent times which are connected by a time translation are essentially the same property,on which the Born rule gives the same probability value.15ime translation also provide the possibility to define an implication between classes. Weused this implication to obtain through infimum and supremum the definitions of conjunctionand disjunctions of classes. The orthogonal complement of Hilbert spaces is immediatelygeneralized to obtain the negation of a class. In this way we construct a non distributiveorthocomplemented lattice of classes of properties and times, and we obtain what in ouropinion is a natural extension of the logical structure of quantum mechanics given by Birkhoffand J. von Neumann [10], one of the standard approaches to quantum logic.As the lattice is non distributive, the Born rule do not provide a well defined probabilityon the whole of it. Therefore, we extended the usual notion of context to the notion ofgeneralized context, which is a subset of the whole set of classes, organized in a distributiveand orthocomplemented lattice. On each generalized context, the Born rule provides a welldefined probability. A generalized context is a boolean logic which can be used for speakingand reasoning about properties of the system at different times. It is interesting to notethat our formalism allows to define the logic of quantum properties without referring to anystate of the system under consideration.Our approach impose more restrictions than the theory of consistent histories on thepossible valid descriptions of a physical system. For a spin system, we proved that ourmore restrictive conditions eliminate the sets of consistent histories which do not satisfytime continuity for the property ascriptions. This continuity is in our opinion a desirableproperty, and it is a direct consequence of the fact that our formalism only allows quantumproperties represented by commuting projectors when translated to a common time.We also obtained good preliminary results with our approach describing the double slitexperiment with and without measurement instruments detecting the particle passing troughthe slits. This open the possibility to apply our formalism to the description of the mea-surement process and to the classical limit, and moreover to explore in this framework therole of the environment induced decoherence. The work in this direction is in progress. [1] R. Feynmann, R. Leighton, M. Sands.
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