aa r X i v : . [ qu a n t - ph ] M a y Quantum histories without contrary inferences
Marcelo Losada
Instituto de F´ısica Rosario, Rosario, Argentina
Roberto Laura
Instituto de F´ısica Rosario and Facultad de Ciencias Exactas,Ingenier´ıa y Agrimensura, Rosario, Argentina (Dated: January 2014)
Abstract
In the consistent histories formulation of quantum theory it was shown that it is possible toretrodict contrary properties. We show that this problem do not appear in our formalism ofgeneralized contexts for quantum histories. . INTRODUCTION. In the consistent histories formulation of quantum theory [2] [3] [4], the probabilisticpredictions and retrodictions depend on the choice of a consistent set. It was shown thatthis freedom allows the formalism to retrodict two contrary properties [1]. This is not aproblem for the defenders of the theory, because each retrodiction is obtained in a differentconsistent sets of histories, i.e. in different descriptions of the physical system not to beconsidered simultaneously [5] [13]. However, this fact is considered by some authors as aserious failure of the theory of consistent histories [1] [6] [14].We are going to analyze this problem with our formalism of generalized contexts [7] [8],developed to deal with expressions involving properties at different times. The formalismis an alternative to the theory of consistent histories, which has proved to be useful for thetime dependent description of the logic of quantum measurements [9], the decay processes[10] and the double slit experiment with and without measurement instruments [8]. Morerecently [11] we have discussed the relation of our formalism with the theory of consistenthistories.In section II we show that there is no possibility for contrary inferences in ordinaryquantum mechanics. In section III we discuss the retrodiction of contrary properties inthe theory of consistent histories. In section IV we show that there are no retrodiction ofcontrary properties in our formalism of generalized contexts. The main conclusions are givenin section V.
II. CONTRARY PROPERTIES IN AN ORDINARY QUANTUM CONTEXT.
In quantum mechanics, a property p is represented by a projector Π p in the Hilbert space H , or alternatively by the corresponding Hilbert subspace V p = Π p H . By definition [1], twoquantum properties p and q are said to be contrary if they satisfy the order relation p ≤ q ,which can also be expressed in terms of the inclusion of the corresponding Hilbert subspacesin the form Π p H ⊆ ( I − Π q ) H . (1)The inclusion of subspaces is equivalent to the following relation between the correspond-2ng projectors (see [12], section 1.3)Π p ( I − Π q ) = ( I − Π q )Π p = Π p , from which we easily deduce that Π p Π q = Π q Π p = 0, that means the projectors Π p and Π q are orthogonal.As they also commute, p and q are compatible properties. The projectors Π p , Π q andΠ p ∨ q = I − Π p − Π q form a projective decomposition of the Hilbert space, i.e. they areorthogonal and their sum is the identity operator. Therefore, the properties p , q and p ∨ q can be considered the atomic properties generating a context of quantum properties withwell defined probabilities [8].For any state of the system represented by a state operator ρ , the probability of anyproperty p ′ in the context is obtained with the Born rule, i.e. Pr ρ ( p ′ ) =Tr( ρ Π p ′ ). For theatomic properties p , q and p ∨ q we obtainPr ρ ( p ) + Pr ρ ( q ) + Pr ρ ( p ∨ q ) = 1 (2)From this equation we easily deduce that if Pr ρ ( p ) = 1 then Pr ρ ( q ) = 0 and if Pr ρ ( q ) = 1,Pr ρ ( p ) = 0.We conclude that in ordinary quantum mechanics it is impossible for any state ρ that twocontrary properties p and q have probability equal to one. These results are the stochasticversion of contrary proposition in ordinary logic. They can be interpreted by saying thatwhenever the property p ( q ) is true, the property q ( p ) is false. By the way, this result alsojustify to have given the name contrary to quantum properties p and q satisfying equation(1).More generally, it is easy to see that if p and q are contrary properties, it is not possibleto have a state ρ and another property r for whichPr ρ ( p | r ) = 1 , Pr ρ ( q | r ) = 1 . (3)Taking into account that p , q and r should be represented by commuting projectors, sothat the conditional probabilities be well defined, we would havePr ρ ( p | r ) = Tr( ρ Π p Π r )Tr( ρ Π r ) = Tr( ρ ∗ Π p ) = Pr ρ ∗ ( p ) , Pr ρ ( q | r ) = Tr( ρ ∗ Π q ) = Pr ρ ∗ ( q ) , where ρ ∗ ≡ Π r ρ Π r Tr(Π r ρ Π r ) . Taking into account equation (2) with ρ = ρ ∗ we conclude that thereare no state ρ and property r for which equations (3) can be both valid.3 II. CONTRARY PROPERTIES IN THE THEORY OF CONSISTENT HISTO-RIES.
In the theory of consistent histories n different contexts of properties at each time t j ( j = 1 , ..., n ), satisfying a state dependent consistency condition, can be used to define a family of consistent histories , i.e. a set of n times sequences of properties with well definedprobabilities [2] [3] [4]. According to the theory, each possible family of consistent historiesis an equally valid description of the quantum system. In general it is not possible to includetwo different families in a single larger one. Different families of this kind are complementarydescriptions of the system, which the theory excludes to be considered simultaneously.A discussion on the logical aspects of the theory was opened by Adrian Kent [1], who firstpointed out that it is possible the retrodiction of contrary properties in different families ofconsistent histories, i.e. Pr ρ t ( p, t | r, t ) = 1 , Pr ρ t ( q, t | r, t ) = 1 , (4)where ρ t is the state of the system at time t , p and q are contrary properties at time t > t and r is a property at time t > t (see references [1] and [5] for explicit expressions of p , q , r and ρ t ).The first equation above is valid for the consistent family that includes p and p at time t together with r and r at time t . It gives the retrodiction of property p at time t conditionalto property r at time t . The second equation is valid for the consistent family including q and q at time t together with r and r at time t , and it gives the retrodiction of property q at time t conditional to property r at time t .From the point of view of the theory of consistent histories equations (4) cannot beinterpreted as the retrodiction of two contrary properties, because they are valid in twodifferent and complementary descriptions, which cannot be included in a single consistentfamily [13] [5]. However, some authors have considered the results given in equations (4) asa serious objection for the internal consistency of the theory of consistent histories [1] [14][6]. 4 V. CONTRARY PROPERTIES IN THE FORMALISM OF GENERALIZEDCONTEXTS.
In this section contrary quantum properties will be considered from the point of view ofour formalism of generalized contexts. We start with a brief description of the formalism,which was presented in full details in our previous papers [7] [8].Quantum mechanics do not give a meaning to the joint probability distribution of observ-ables whose operators do not commute. It can only deal with a set of properties belongingto a context.A context of properties C i at time t i is obtained starting from a set of atomic properties p k i i ( k i ∈ σ i ) represented by projectors Π k i i corresponding to a projective decomposition ofthe Hilbert space H , i.e. verifying P k i ∈ σ i Π k i i = I, Π k i i Π k ′ i i = δ k i k ′ i Π k i i . (5)Any property p of the context C i is represented by a sum of the projectors of the projectivedecomposition, Π p = P k i ∈ σ p Π k i i , σ p ⊂ σ i . (6)The context C i is an orthocomplemented distributive lattice, with the complement p of aproperty p defined by Π p ≡ I − Π p and the order relation p ≤ p ′ defined by Π p H ⊆ Π p ′ H .A well defined probability (i.e. additive, non negative and normalized) is defined by theBorn rule Pr t i ( p ) ≡ Tr( ρ t i Π p ) on the context C i . In Heisenberg representation, the probabilityof a property p at time t i can be written in terms of the state at a reference time t , i.e.Pr t i ( p ) = T r ( ρ t Π p, ) , Π p, ≡ U ( t , t i )Π p U ( t i , t ) , U ( t i , t ) = e − i ℏ H ( t i − t ) . (7)Taking into account equations (6) and (7), the Heisenberg representation of the property p of the context C i at time t i is given byΠ p, = P k i ∈ σ p Π k i i, , (8)where the projectors Π k i i, = U ( t , t i )Π k i i U ( t i , t ) represent the time translation of the atomicproperties p k i i from time t i to the time t . The projectors Π k i i, also satisfy equations (5).The Heisenberg representation of the context C i at time t i suggest a generalization ofquantum mechanics for including the joint probability of properties belonging to differentcontexts C ,..., C i ,..., C n corresponding to n different times t < ... < t i < ... < t n .5y extending what is a common assumption in ordinary quantum mechanics, we proposedto give a meaning to the joint probability of properties at different times if they correspondto commuting projectors in Heisenberg representation. This will be the case if the atomicproperties generating each of the n contexts are represented by projectors satisfying[Π k i i, , Π k j j, ] = 0 , i, j = 1 , ..., n, k i ∈ σ i , k j ∈ σ j . If these projectors commute the projectors Π k ≡ Π k , ... Π k i i, ... Π k n n, , with k = ( k , ..., k n )and k i ∈ σ i , form a projective decomposition of the Hilbert space H , as they satisfy P k Π k = I, Π k Π k ′ = δ kk ′ Π k , k , k ′ ∈ σ × ... × σ n In our formalism we postulate that an expression of the form “ property p k at time t and ... and p k n n at time t n ” is an atomic generalized property p k with the Heisenbergrepresentation given by the projector Π k . A generalized context is defined by all the gen-eralized properties p having a Heisenberg representation given by an arbitrary sum of theprojectors Π k , i.e. Π p = P k ∈ σ p Π k , where σ p is a subset of σ × ... × σ n . The generalized context is an orthocomplemented dis-tributive lattice, with the complement p of p defined by Π p = I − Π p , and the order relation p ≤ p ′ defined by the inclusion of the corresponding Hilbert subspaces (Π p H ⊆ Π p ′ H ).An extension of the Born rule provides a definition of an additive, non negative andnormalized probability on the generalized context, given byPr( p ) ≡ Tr( ρ t Π p ) . (9)We are now going to analyze the retrodiction of contrary properties in the formalism ofgeneralized contexts. We consider a state ρ t at time t , two contrary properties p and q attime t > t and another property r at time t > t , and we search for the possibility to obtainfor both conditional probabilities the results Pr ρ t ( p, t | r, t ) = 1 and Pr ρ t ( q, t | r, t ) = 1.The projectors Π p and Π p, = U ( t , t )Π p U ( t , t ) are respectively Schr¨odinger andHeisenberg representations of the property p at time t . Analogously, Π q and Π q, = U ( t , t )Π q U ( t , t ) are representations of the property q at time t . Moreover, Π r andΠ r, = U ( t , t )Π r U ( t , t ) are representations of the property r at time t .6he conditional probabilities are meaningful in our formalism if the following compati-bility conditions are satisfied[Π p, , Π r, ] = 0 , [Π q, , Π r, ] = 0 , (10)while the contrary properties p and q are represented by orthogonal projectors, and therefore[Π p, , Π q, ] = 0 . (11)The commutation relations given in equations (10) and (11) are the compatibility con-ditions required to consider a two times generalized context including the contrary proper-ties p and q at time t and property r at time t , in which both conditional probabilitiesPr ρ t ( p, t | r, t ) and Pr ρ t ( q, t | r, t ) are meaningful.In our formalism, the required retrodictions would have the explicit formsPr ρ t ( p, t | r, t ) = Tr( ρ t Π p, Π r, )Tr( ρ t Π r, ) = 1 , Pr ρ t ( q, t | r, t ) = Tr( ρ t Π q, Π r, )Tr( ρ t Π r, ) = 1 . Taking into account the commutation relations given in equations (10), the previousequations are equivalent toTr( ρ ∗ t Π p, ) = 1 , Tr( ρ ∗ t Π q, ) = 1 , ρ ∗ t ≡ Π r, ρ t Π r, Tr(Π r, ρ t Π r, ) . (12)As Π p, and Π q, represent contrary properties at the same time t , we can follow thearguments given at the end of section II to show that there is no ρ ∗ t for which both equationsgiven in equations (12) can be valid. Therefore we conclude that the problem of retrodictionof contrary properties do not arise in our formalism of generalized contexts for quantumhistories V. CONCLUSIONS.
In ordinary quantum mechanics, contrary properties are represented by orthogonal sub-spaces of the Hilbert space associated with the physical system. In section II, we provedthat given two contrary properties p and q , there is no state ρ and property r for whichthe probability of p conditional to r and the probability of q conditional to r can be bothequal to one. Therefore, there is no possibility of contrary inferences in ordinary quantummechanics. This result corresponds to a state and properties considered at a single time.7s we discussed in section III, this is not the case for the theory of consistent histories,where a state at time t , two contrary properties p and q at time t > t and another property r at time t > t can be found in such a way that the probability of p conditional to r andthe probability of q conditional to r are both equal to one. Although these conditionalprobabilities are defined in different sets of consistent histories [5] [13], some authors haveconsidered this fact as a serious problem for the logical consistency of the theory [1] [6] [14].The main purpose of this paper was to analyze the problem of contrary inferences in theframework of our formalism of generalized contexts. In this formalism, as it was explainedin section IV, ordinary contexts of properties at different times can be used to obtain avalid set of quantum histories if they satisfy a compatibility condition. This condition isgiven by the commutation of the projectors corresponding to the time translation of theproperties to a single common time. These compatibility conditions are state independent,an important difference with respect to the state dependent consistency conditions of thetheory of consistent histories. Each quantum history has a Heisenberg representation givenby a projection operator, and each valid set of quantum histories is generated by a projectivedecomposition of the Hilbert space. As a consequence, a generalized context of quantumhistories has the logical structure of a distributive orthocomplemented lattice of subspacesof the Hilbert space, i.e. the same logical structure of the quantum properties of an ordinarycontext. It is because of this logical structure that in our formalism there is no place for theretrodiction of contrary properties.Recently we have annalyzed the relations of our formalism with the theory of consistenthistories [11]. Our formalism was also successful in describing the time dependent logic ofquantum measurements [9], the quantum decay process [10] and the double slit experimentwith and without measurement instruments [8]. The results of this paper encourages us tocontinue our future research considering more applications of the formalism of generalizedcontexts. [1] A. Kent, Phys. Rev. Lett. , 2874-2877 (1997)[2] R. Griffiths, J. stat. Phys. , 219 (1984)[3] R. Omn`es, J. Stat. Phys. , 893 (1988)
4] M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of Information, W. Zurek(ed.). Addison-Wesley, Reading (1990)[5] J. B. Hartle, J. Phys. A , 3101-3121 (2007)[6] E. Okon, D. Sudarsky, Found. Phys. , 19-33 (2014)[7] R. Laura, L. Vanni, Found. Phys. , 160-173 (2009)[8] M. Losada, L. Vanni, R. Laura, Phys. Rev. A , 052128 (2013)[9] L. Vanni, R. Laura, Int. J. Theor. Phys. , 2386-2394 (2013)[10] M. Losada, R. Laura, Int. J. Theor. Phys. , 1289-1299 (2013)[11] M. Losada, R. Laura, Annals of Physics (in press)[12] P. Mittelstaedt, Quantum Logic , D. Reidel Publishing Company, Dordrecht, Holland (1978)[13] R. Griffiths, J. B. Hartle, Phys. Rev. Lett. , 1981 (1998)[14] A. Kent, Phys. Rev. Lett. , 1982 (1998), 1982 (1998)