aa r X i v : . [ qu a n t - ph ] M a r Quantum Mechanics and imprecise probability
Bruno Galvan ∗ Loc. Melta 40, 38100 Trento, Italy.November 2007
Abstract
An extension of the Born rule, the quantum typicality rule , has recently been proposed[B. Galvan: Found. Phys. 37, 1540-1562 (2007)]. Roughly speaking, this rule states that ifthe wave function of a particle is split into non-overlapping wave packets, the particle staysapproximately inside the support of one of the wave packets, without jumping to the others.In this paper a formal definition of this rule is given in terms of imprecise probability . Animprecise probability space is a measurable space (Ω , A ) endowed with a set of probabilitymeasures P . The quantum formalism and the quantum typicality rule allow us to definea set of probabilities P Ψ on ( X T , F ), where X is the configuration space of a quantumsystem, T is a time interval and F is the σ -algebra generated by the cylinder sets. Thus, itis proposed that a quantum system can be represented as the imprecise stochastic process ( X T , F , P Ψ ), which is a canonical stochastic process in which the single probability measureis replaced by a set of measures. It is argued that this mathematical model, when used torepresent macroscopic systems, has sufficient predictive power to explain both the results ofthe statistical experiments and the quasi-classical structure of the macroscopic evolution. Stochastic processes are the standard tools provided by probability theory to represent systemssubjected to random evolution. In spite of the fact that quantum mechanics is a probabilis-tic theory, the presence of quantum interference prevents the standard formalism of quantummechanics to represent a quantum system as a stochastic process.In order to understand this, let us consider the most general version of a stochastic process,namely the canonical stochastic process. Let X be the configuration space of a system ofparticles (for example R N ) and B its Borel σ -algebra. Moreover, let T be a suitable timeinterval including the origin, and let X T denote the set of all the trajectories λ : T → X . Given∆ ∈ B and t ∈ T , let ( t, ∆) denote the set { λ ∈ X T : λ ( t ) ∈ ∆ } . The sets of this kind will bereferred to as s-sets (an abbreviated name for single-time cylinder-sets); let S denote the class ∗ Electronic address: [email protected]
1f the s-sets. A canonical stochastic process is the triple ( X T , F , P ), where F is the σ -algebragenerated by the s-sets (or equivalently, by the cylinder sets) and P is a probability measureon F . According to the Kolmogorov reconstruction theorem, the probability P is univocallydetermined by its finite dimensional distributions , i.e. by its values at the finite intersections ofs-sets: P [( t , ∆ ) ∩ ... ∩ ( t n , ∆ n )] . (1)Since P is a probability measure, this expression is additive, i.e., if ∆ i ∩ ∆ ′ i = ∅ , we have P [ . . . ∩ ( t i , ∆ i ∪ ∆ ′ i ) ∩ . . . ] = P [ . . . ∩ ( t i , ∆ i ) ∩ . . . ] + P [ . . . ∩ ( t i , ∆ ′ i ) ∩ . . . ] . The physical interpretation of a canonical stochastic process is simple: the evolution of thesystem of particles during the time interval T is represented by a trajectory chosen at randomfrom X T .Let us now attempt to represent a system of quantum particles as a canonical stochasticprocess. In the quantum case as well, we can take X as the configuration space and, for thetime being, let us assume that also in the quantum case the particles follow a trajectory of X T .The problem is then to define the probability P , i.e. to find a quantum expression for the finitedimensional distributions (1).We know that a normalized wave function Ψ( t ) = U ( t )Ψ is associated with the quantumsystem, where U ( t ) is the unitary time evolution operator and Ψ is the wave function of thesystem at the time t = 0. According to the Born rule, the probability of finding the particles inthe region ∆ ∈ B at the time t is || E (∆)Ψ( t ) || , where E ( · ) is the projection valued measure on B . In other words, we can say that || E (∆)Ψ( t ) || is the probability that a trajectory chosen atrandom from X T belong to the s-set ( t, ∆). Thus for n = 1 we have a valid quantum expressionfor (1), namely P [( t, ∆)] = || E (∆)Ψ( t ) || . (2)The problems arise when n >
1. A tentative quantum expression for (1) could be || E (∆ n ) U ( t n − t n − ) E (∆ n − ) . . . U ( t − t ) E (∆ )Ψ( t ) || , (3)where the assumption is made that t ≤ . . . ≤ t n . According to the Born rule and to thereduction postulate, (3) is the probability of finding the particles in the regions ∆ i at the times t i , for i = 1 , . . . , n . However (3) is not an admissible expression for the finite dimensionaldistributions, because it is not additive, i.e. if ∆ i ∩ ∆ ′ i = ∅ , in general we have || . . . E (∆ i ∪ ∆ ′ i ) . . . Ψ( t ) || = || . . . E (∆ i ) . . . Ψ( t ) || + || . . . E (∆ ′ i ) . . . Ψ( t ) || . The non additivity of (3) corresponds to the well known interference phenomena of quantummechanics. Another possible expression, namely: Re h Ψ( t n ) | E (∆ n ) U ( t n − t n − ) E (∆ n − ) . . . E (∆ ) | Ψ( t ) i ; (4)2s also not admissible, because, although it is additive, it is non-positive definite. We musttherefore conclude that a valid expression for the finite dimensional distributions, and thereforea probability measure for X T , cannot be extracted from the standard quantum formalism.Two possible solutions to this problem can be proposed, corresponding to different formu-lations/interpretations of quantum mechanics. The first is to simply remove the set X T , andto let the wave function be the only mathematical entity representing a quantum system. Thisis, for example, the position of the Copenhagen and of the Many Worlds interpretations [5].For example, in [8] Heisenberg explicitly connects the quantum interference phenomena and thenecessity to renounce a description of the motion of the particles in terms of trajectories. Thesecond solution is to add a new element to the standard quantum formalism which allows usto define the required probability measure. This is the case, for example, of Nelson’s stochasticmechanics [11], which introduces a stochastic differential equation, and of Bohmian mechanics[1], which introduces the guidance equation . For various reasons that we do not discuss here,for many physicists none of these formulations is satisfying.In this paper a third solution is proposed, which maintains the set X T and does not addnew elements to the quantum formalism, but rather admits the possibility that the quantumformalism defines a set of probability measures on X T instead of a single probability. For example,if M is the set of all the probability measures on ( X T , F ), the Born rule defines the set ofprobabilities P Ψ B := { P ∈ M : P [( t, ∆)] = || E (∆)Ψ( t ) || for all ( t, ∆) ∈ S} . (5)This paper therefore proposes that, instead of a “precise” stochastic process ( X T , F , P ), the cor-rect mathematical model of a quantum system is an “imprecise” stochastic process ( X T , F , P Ψ ),where P Ψ is a suitable set of probability measures. The word “imprecise” has been used inten-tionally, because imprecise probability is a generic term which also includes the theory of sets ofprobabilities [14]. The meaning of this representation is that a possible evolution of the system isrepresented by a trajectory chosen at random from X T according to any one of the probabilitiesof P Ψ . In general, in an imprecise stochastic process there are events without a well definedprobability, and therefore the predictive power of the process is limited to those events A forwhich P ( A ) has, at least approximately, the same value for all the probabilities P ∈ P Ψ .A fundamental element of the proposed solution is the use of the quantum typicality ruleinstead of the Born rule to define the set P Ψ . Let us explain this. The set P Ψ B defined by theBorn rule arguably explains the results of the statistical experiments, because it attributes awell defined probability to any set of configurations at a given time. However, the Born ruledoes not establish any correlation between the positions of the particles at different times, andtherefore it cannot define a dynamical structure for the trajectories. Therefore, if the imprecise In [12] it is shown that the guidance equation is the limiting case of a general class of stochastic differentialequations which also include Nelson’s theory. X T , F , P B Ψ ) is used to represent a macroscopic system, for example the universe, itcannot explain the quasi-classical structure of the macroscopic evolution. This is also a wellknown problem in connection with the Copenhagen and the Many Words interpretations.In a recent paper we proposed a new quantum rule, the quantum typicality rule , accordingto which, roughly speaking, the particles follow the branches of the wave function [6]. In thepresent paper a set P Ψ of probabilities corresponding to the quantum typicality rule is defined,and some of its properties are studied. This set is contained in P Ψ B , i.e. the quantum typicalityrule implies the Born rule. Moreover, since the quantum typicality rule establishes a correlationbetween the positions of the particles at two different times, the set P Ψ arguably explains themacroscopic quasi-classical structure of the trajectories.The paper is structured as follows: in section 2 the quantum typicality rule is reviewed. Insection 3 a short review of the theory of imprecise probability is given. In section 4 a formaldefinition of the Born rule in terms of imprecise probability is given. In section 5 a formaldefinition of the quantum typicality rule in terms of imprecise probability is given, and the quantum process , i.e. the proposed mathematical model of a quantum system, is defined. Insection 6 two properties of quantum processes are studied. Section 7 presents a concludingdiscussion about the formulation of quantum mechanics based on the quantum typicality rule. Let us first introduce the notion of typicality in a probability space. Let (Ω , A , P ) be a probabilityspace, and let A and B be two measurable subsets of Ω, with P ( B ) = 0. The set A is said tobe typical relative to B if P ( A ∩ B ) P ( B ) ≈ , (6)where ≈ ≥ − ǫ , with ǫ ≪
1. If A is typical relative to B , then theoverwhelming majority of the elements of B also belongs to A . From the empirical point ofview, the consequence of the typicality of A is that a single element chosen at random from B will also belong to A . Two sets A and B are said to be mutually typical if A is typical relativeto B and vice-versa. Mutual typicality can be expressed by the condition P ( A ∩ B )max { P ( A ) , P ( B ) } ≈ . (7)The notion of typicality is used in Bohmian mechanics in order to prove the quantum equilib-rium hypothesis [3] and it is also (implicitly) the basis for Boltzmann’s derivation of the secondlaw of thermodynamics [7].Let us now introduce the quantum typicality rule. In its simplest and most intuitive form,the quantum typicality rule states: suppose that the wave function of a particle is the sum4f two non-overlapping wave packets. Then, during the time over which the wave packets arenon-overlapping, the particle stays inside the support of one of the two wave packets, withoutjumping to the other one.For example, let us consider the following simple experiment: S D R S R D T S T Fig.1The source S emits photons towards a beam splitter, the reflected (transmitted) photonsare detected by the detector D T ( D R ), and S R and S T are two slits. The quantum typicalityrule states that the photons detected, for example, by the detector D R , cross the slit S R . Evenif this assumption is very reasonable, it cannot be deduced from standard quantum mechanics,which does not predict the trajectory of a quantum system between the preparation and themeasurement times. This feature of quantum mechanics is essentially the origin of its difficultyin explaining the emergence of a quasi-classical world.We can easily express the rule in a mathematical form. Let Ψ( t ) = U ( t )Ψ be the wavefunction of a particle (or of a system of particles). Let us suppose that at a time t the wavefunction can be expressed as the sum of two non-overlapping wave packets φ and φ ⊥ = Ψ( t ) − φ ,and that at a time t > t the two wave packets are still non-overlapping, i.e. U ( t − t ) φ and U ( t − t ) φ ⊥ are non-overlapping. This implies that two subsets ∆ and ∆ of the configurationspace of the particle exist, such that φ ≈ E (∆ )Ψ( t ) and U ( t − t ) φ ≈ E (∆ )Ψ( t ) . (8)Due to the unavoidable spreading of the wave function, the wave packets can be only approx-imately non-overlapping. This is the reason for using the approximate equality symbol in (8).The sets ∆ and ∆ can be considered as the supports of φ and U ( t − t ) φ , respectively. Theconditions (8) can be combined to give the condition U ( t − t ) E (∆ )Ψ( t ) ≈ E (∆ )Ψ( t ) . (9)This reasoning can also be reversed: given two subsets ∆ and ∆ satisfying condition (9), thewave packet φ := E (∆ )Ψ( t ) satisfies the conditions of (8).The quantum typicality rule states that, if the particle is in ∆ at the time t and condition(9) holds, then the particle was in ∆ at the time t . Since the two times are symmetric, it isnatural to assume also the reverse conclusion: if the particle is in ∆ at the time t then it willbe in ∆ at the time t . 5et us take a further step and assume, as proposed in the introduction, that a quantumparticle follows a trajectory belonging to X T . Note that this assumption is implicitly containedin the intuitive formulation of the quantum typicality rule, because such a rule assumes thatthe particle has a position at a suitable time even if no measurement is performed at that time.Then the quantum typicality rule can be expressed by stating that condition (9) implies thatthe two s-sets ( t , ∆ ) and ( t , ∆ ) are mutually typical.A more compact notation can also be introduced. If S denotes the s-set ( t, ∆), let Ψ( S )denote the state U † ( t ) E (∆) U ( t )Ψ . With this notation, condition (9) assumes the form || Ψ( S ) − Ψ( S ) || ≈ , (10)where S = ( t , ∆ ) and S = ( t , ∆ ). The norm has been squared for reasons that will becomeclear in section 5. Due to the presence of the approximate equality, condition (10) must benormalized. In order to simplify the normalization, we impose the further natural conditionthat || Ψ( S ) || = || Ψ( S ) || . In conclusion, the quantum typicality can be expressed as follows: Quantum Typicality Rule: if S and S are two s-sets such that || Ψ( S ) || = || Ψ( S ) || and || Ψ( S ) − Ψ( S ) || || Ψ( S ) || ≪ , (11)than S and S are mutually typical.The constraint || Ψ( S ) || = || Ψ( S ) || , which was not present in the first formulation of the rule[6], will be discussed in section 5.The problem is now to correlate the definition of mutual typicality given by the probabilisticexpression (7) with the one given by the quantum expression (11). This correlation is concep-tually similar to the correlation P ( S ) = || E (∆)Ψ( t ) || given by the Born rule, and it will berealized by means of the theory of imprecise probability. A very short review is given here of the theory of sets of probabilities, which is a part of thetheory of imprecise probability [9].Let (Ω , A ) be a measurable space and M be the set of all the probability measures on (Ω , A ).Let P be an arbitrary non-empty subset of M . The upper and the lower probability induced by P are the two set functions P ∗ , P ∗ : A → R + defined by: P ∗ ( A ) = inf P ∈P P ( A ); P ∗ ( A ) = sup P ∈P P ( A ) . (12)6e can easily see that P ∗ and P ∗ satisfy the following properties:0 ≤ P ∗ ( A ) ≤ P ∗ ( A ) ≤ P ∗ ( ∅ ) = P ∗ ( ∅ ) = 0; P ∗ (Ω) = P ∗ (Ω) = 1; (14) P ∗ ( A ) + P ∗ ( A c ) = 1; (15) P ∗ ( A ∪ B ) ≥ P ∗ ( A ) + P ∗ ( B ) for A ∩ B = ∅ ; (16) P ∗ ( A ∪ B ) ≤ P ∗ ( A ) + P ∗ ( B ) for A ∩ B = ∅ ; (17) P ∗ ( A ) ≤ P ∗ ( B ) and P ∗ ( A ) ≤ P ∗ ( B ) for A ⊆ B. (18)Equation (15) states that P ∗ and P ∗ are conjugate; equations (16) and (17) state that P ∗ issuperadditive and P ∗ is subadditive; equation (18) states that P ∗ and P ∗ are monotone.The triple (Ω , A , P ) will be referred to as an imprecise probability space . In the case in whichΩ = X T and A = F , the more specific term imprecise stochastic process will be used. Thepredictive power of an imprecise probability space is limited to those events A for which P ( A )assumes approximately the same value for all P ∈ P . Such a condition is satisfied for exampleif a positive number a exists such that | P ( A ) − a | a ≪ P ∈ P . (19)Let us now study how sets of probability measures can be defined. Let D be an arbitrarysubset of F , and f ∗ : D → R + a non negative set function. Let us define a set of probabilitymeasures P as P := { P ∈ M : P ( A ) ≥ f ∗ ( A ) for all A ∈ D} . (20)Alternatively, the set P can be defined as P := { P ∈ M : P ( B ) ≤ f ∗ ( B ) for all B ∈ D c } , (21)where D c := { B : B c ∈ D} , and f ∗ ( B ) := 1 − f ∗ ( B c ).We now have the following lemma [9]: the set P defined by (20) is not empty iff, for any pairof finite sequences { a , . . . , a n } and { A , . . . , A n } of non negative numbers and of sets of D , thecondition n X i =1 a i A i ( ω ) ≤ ∀ ω ∈ Ω (22)implies the condition X a i f ∗ ( A i ) ≤ , (23)where A i is the characteristic function of the set A i (see note ). Actually, the proof in the given reference applies only to the case of a finite set Ω. The Born process
Let us first apply the theory of imprecise probability to the Born rule. As mentioned in theintroduction, the Born rule defines the set of probabilities on ( X T , F ) satisfying the condition P ( S ) = || Ψ( S ) || for any s-set S .By referring to the notation used in the previous section, we have the condition that (Ω , A ) =( X T , F ), D = S and f ∗ ( S ) = || Ψ( S ) || . Thus the set P Ψ B defined by the Born rule is P Ψ B := { P ∈ M : P ( S ) ≥ || Ψ( S ) || for all S ∈ S} . (24)Note that S c = S and f ∗ = f ∗ . We therefore have P ∗ ( S ) = P ∗ ( S ) for all S ∈ S , which impliesthat P ( S ) = || Ψ( S ) || for all S ∈ S and for all P ∈ P Ψ B , as required. The class P Ψ B is not emptybecause it contains at least the probability P defined by the finite dimensional distributions P ( S ∩ . . . ∩ S n ) := || Ψ( S ) || . . . || Ψ( S n ) || , where the assumption is made that t i = t j for i = j .The imprecise process ( X T , F , P Ψ B ) will be referred to as the Born process . Let us now attempt to define a set of probabilities P Ψ corresponding to the quantum typicalityrule. The most natural definition appears to be the following: the set D is D := { S ∩ S : S , S ∈ S , S ∩ S
6∈ S , || Ψ( S ) || = || Ψ( S ) ||} , (25)and the set function f ∗ is f ∗ ( S ∩ S ) = || Ψ( S ) || − || Ψ( S ) − Ψ( S ) || , (26)The s-sets have been excluded from D because f ∗ is not a well defined set function for S ∩ S ∈ S .The set P Ψ is then defined as P Ψ := { P ∈ M : P ( S ∩ S ) ≥ || Ψ( S ) || − || Ψ( S ) − Ψ( S ) || for all S ∩ S ∈ D} . (27)Let us introduce the conjugate elements f ∗ and D c : D c := { S ∪ S : S , S ∈ S , S ∪ S
6∈ S , || Ψ( S ) || = || Ψ( S ) ||} ; (28) f ∗ ( S ∪ S ) = || Ψ( S ) || + || Ψ( S ) − Ψ( S ) || . (29)We can easily see that sup { S : S ∩ S ∈D} f ∗ ( S ∩ S ) = || Ψ( S ) || , (30)inf { S : S ∪ S ∈D c } f ∗ ( S ∪ S ) = || Ψ( S ) || . (31)8ince f ∗ ( S ∩ S ) ≤ P ( S ∩ S ) ≤ P ( S ) ≤ P ( S ∪ S ) ≤ f ∗ ( S ∪ S ) , (32)we obtain the condition P ( S ) = || Ψ( S ) || for all S ∈ S and for all P ∈ P Ψ . Thus P Ψ ⊂ P Ψ B ,i.e. the formal quantum typicality rule implies the Born rule. For this reason the condition P ( S ∩ S ) ≥ || Ψ( S ) || − || Ψ( S ) − Ψ( S ) || for || Ψ( S ) || = || Ψ( S ) || (33)is also satisfied for S ∩ S ∈ S . Indeed in this case we have P ( S ∩ S ) = || Ψ( S ∩ S ) || ≥ || Ψ( S ∩ S ) || − || Ψ( S ) || = || Ψ( S ) || − || Ψ( S ) − Ψ( S ) || . The imprecise process ( X T , F , P Ψ ) will be referred to as a quantum process , and the definingcondition (33) will be referred to as the formal quantum typicality rule. The adjective “formal”has been adjoined in order to distinguish condition (33) from the quantum typicality rule asexpressed in section 2, which will be referred to as the physical quantum typicality rule. Theseadjectives will, however, be omitted when not required for reasons of clarity.The physical quantum typicality rule can be derived trivially from the formal rule. Thus, if S and S be two s-sets such that || Ψ( S ) || = || Ψ( S ) || and || Ψ( S ) − Ψ( S ) || || Ψ( S ) || ≤ ǫ ≪ , than P ( S ∩ S )max { P ( S ) , P ( S ) } = P ( S ∩ S ) || Ψ( S ) || ≥ || Ψ( S ) || − || Ψ( S ) − Ψ( S ) || || Ψ( S ) || ≥ − ǫ for all P ∈ P Ψ .The definition of the formal quantum typicality rule given by (33) appears to be the simplestand most natural one, corresponding to the physical quantum typicality rule. However, somevariants of this definition are possible, which we will now examine.Let us first discuss the constraint || Ψ( S ) || = || Ψ( S ) || . (34)Such a constraint can probably be removed from both the formal and the physical formulationsof the rule. In this case condition (33) must be replaced by the condition P ( S ∩ S ) ≥ min {|| Ψ( S ) || , || Ψ( S ) || } − || Ψ( S ) − Ψ( S ) || . (35)If P ′ Ψ is the corresponding set of probabilities, we have P ′ Ψ ⊆ P Ψ . Thus, the constraint (34)actually gives rise to a more general set of probabilities. This fact, together with the fact thatthis simplifies both the formal and the physical formulations of the quantum typicality rule,suggests that the constraint (34) is appropriate.9ere it should be noted that the formal and the physical formulations are not totally equiv-alent, because the latter acts only in the typicality regime, i.e. when || Ψ( S ) − Ψ( S ) || || Ψ( S ) || ≪ . (36)On the contrary, condition (33) also imposes a constraint on the probability when (36) doesnot hold true. In order to eliminate such a difference, condition (33) can be replaced by P ( S ∩ S ) ≥ || Ψ( S ) || − || Ψ( S ) − Ψ( S ) || for || Ψ( S ) || = || Ψ( S ) || and (37) || Ψ( S ) − Ψ( S ) || ≤ ǫ || Ψ( S ) || , where ǫ is a suitable “small” positive number. If P ′′ Ψ is the set of probabilities defined by thiscondition, then P Ψ ⊆ P ′′ Ψ . The problem with this definition is its vagueness, because a precisevalue for ǫ cannot be provided. Note also that also a condition of the type P ( S ∩ S ) ≥ || Ψ( S ) || − α || Ψ( S ) − Ψ( S ) || for || Ψ( S ) || = || Ψ( S ) || , (38)where α is positive number not “too small” and not “too big”, could be consistent with thephysical quantum typicality rule. Also this definition is vague.There are conceptual reasons which suggest that the definition of the set P Ψ is necessarilyvague, in the sense that slightly different definitions of P Ψ are empirically indistinguishable.These reasons are connected with the fact that we have access to the past structure of thetrajectories only through the memories of the past which are encoded in the present configurationof our recording devices. See [6] for a discussion of this point. This subject will be furtherdeveloped in a future paper.Unfortunately we cannot make any statement about the consistency of the quantum typi-cality rule, i.e. we cannot prove that the set P Ψ is not empty. The problem of the consistencyof the quantum typicality rule was also discussed in [6], where some inequalities making sucha consistency plausible were proposed. Here the problem has not yet been solved in a rigorousway, but at least it has been formulated in a precise way. Let us study two properties of quantum processes.In general, the probability of the intersection of two non equal time s-sets S and S is notwell defined by P Ψ , i.e. P ( S ∩ S ) may have different values for different P ∈ P Ψ . For example,let the wave function Ψ( t ) be a single wave packet, with ∆ and ∆ such that || E (∆ )Ψ( t ) || = || E (∆ )Ψ( t ) || = 1 /
2. If | t − t | is large enough, we have in any case (that is also in the casein which ∆ ≈ ∆ ) that || E (∆ )Ψ( t ) − U ( t − t ) E (∆ )Ψ( t ) ||
0, and therefore there is noconstraint preventing P [( t , ∆ ) ∩ ( t , ∆ )] from assuming a wide range of values.10here is however a typical situation in which P ( S ∩ S ) has (approximately) the same valuefor all P ∈ P Ψ . Let φ ( t ) := U ( t − t ) φ be a wave packet which does not overlap φ ⊥ ( t ) := Ψ( t ) − φ ( t ) at the times t and t and let ∆ and ∆ be the supports of φ ( t ) and φ ( t ), respectively,with || E (∆ )Ψ( t ) || = || E (∆ )Ψ( t ) || . According to the quantum typicality rule, S = ( t , ∆ )and S = ( t , ∆ ) are mutually typical. Thus, if S ′ is another s-set such that S ′ ∩ S ∈ S and || Ψ( S ∩ S ′ ) || is not “too small” relative to || Ψ( S ) || , we expect that P ( S ∩ S ′ ) ≈ P ( S ∩ S ′ ) = || Ψ( S ∩ S ′ ) || .This result can be proven rigorously. In fact, for all P ∈ P Ψ , we have the inequality: || Ψ( S ∩ S ′ ) || − || Ψ( S ) − Ψ( S ) || ≤ P ( S ∩ S ′ ) ≤ || Ψ( S ∩ S ′ ) || + || Ψ( S ) − Ψ( S ) || , (39)for || Ψ( S ) || = || Ψ( S ) || and S ∩ S ′ ∈ S . Thus, if S , S and S ′ are defined as above, we have || Ψ( S ) − Ψ( S ) || || Ψ( S ∩ S ′ ) || ≪ , (40)and therefore (cid:12)(cid:12) P ( S ∩ S ′ ) − || Ψ( S ∩ S ′ ) || (cid:12)(cid:12) || Ψ( S ∩ S ′ ) || ≪ . (41)Let us prove inequality (39). We have P ( S ∩ S ′ ) ≥ P ( S ∩ S ∩ S ′ ) = P ( S ∩ S ) − P ( S ∩ S ∩ S ′ c ) ≥|| Ψ( S ) || − || Ψ( S ) − Ψ( S ) || − P ( S ∩ S ′ c ) = || Ψ( S ∩ S ′ ) || − || Ψ( S ) − Ψ( S ) || . Moreover P ( S ∩ S ′ ) = P ( S ∩ S ′ ∩ S ) + P ( S ∩ S ′ ∩ S c ) ≤ P ( S ′ ∩ S ) + P ( S ∩ S c ) = || Ψ( S ∩ S ′ ) || + P ( S ) − P ( S ∩ S ) ≤ || Ψ( S ∩ S ′ ) || + || Ψ( S ) − Ψ( S ) || . Let us now refer to another property. In [6] and in section 1 we mentioned that the trajectoriesfollow approximately the branches of the wave function. We can now give a precise mathematicalformulation of this assertion.Let us consider again the wave packets φ ( t ) and φ ⊥ ( t ) defined above, and let us supposethat they do not overlap during the entire time interval [ t , t ]. The wave packet φ ( t ) in thetime interval [ t , t ] is what we view as a branch of the wave function. For t ∈ [ t , t ] let ∆ t be the support of φ ( t ), with || E (∆ t )Ψ( t ) || = || E (∆ t )Ψ( t ) || for t ∈ [ t , t ], and let S t and S denote the s-sets ( t, ∆ t ) and ( t , ∆ t ) respectively. According to the reasoning of section 2, wethen have || Ψ( S ) − Ψ( S t ) || || Ψ( S ) || ≤ ǫ ≪ t ∈ [ t , t ] , (42)and therefore, according to the quantum typicality rule: P ( S ∩ S t ) P ( S ) ≥ − ǫ for all t ∈ [ t , t ] and P ∈ P Ψ . (43)11et { s , . . . , s n } be any sequence of times in the time interval [ t , t ]. Moreover, for any P ∈ P Ψ , let ( S , F ∩ S , P ( ·| S )) be the probability space obtained from ( X T , F , P ) by restricting X T to S . On this space let us introduce the random variable Y : S → [0 ,
1] defined by: Y ( λ ) := 1 n n X i =1 ∆ si [ λ ( s i )] . (44)One can show that E P ( Y ) ≥ − ǫ and P ( Y ≤ − δ ) ≤ ǫδ for all P ∈ P Ψ , (45)where E P ( Y ) is the expectation value of Y (the dependence on the probability measure hasbeen explicitly shown) and δ is a suitable “small” positive number. Indeed we have E P ( Y ) = 1 P ( S ) Z S Y dP = P i P ( S ∩ S s i ) n P ( S ) ≥ − ǫ. As to the second inequality (45), let a be a given point of the interval [0 ,
1] and 0 ≤ P a ≤ { Y : P ( Y ≤ a )= P a } { E P ( Y ) } = aP a + (1 − P a ) = 1 − P a (1 − a ) . Indeed the supremum of the expectation value is obtained when the probability density ρ ( y ) of Y is shifted as much as possible on the right of the interval [0 ,