The quantum mechanics of experiments

The Quan tum Mec hanics of Exp erimen ts Jürg F röhlic h 1 and Alessandro Pizzo 2 Marc h 2026 De dic ate d to Isr ael Michael Sigal and Barry Martin Simon on the o c c asion of their eightieth birthdays Abstract This note starts with a recapitulation of what p eople call the “Measurement Problem” of Quan tum Mec hanics (QM). The dissip ative natur e of the quantum-mec hanical time-ev olution of a verages of states o v er large ensembles of iden tical isolated systems consisting of matter in teract- ing with the radiation field is discussed and sho wn to corresp ond to a sto chastic time-ev olution of states of individual systems. The imp ortance of dissipation for the successful completion of measuremen ts is highligh ted. T o conclude, a solution of the “Measuremen t Problem” is sk etched in an idealized mo del of a double-slit exp erimen t. 1 What is the “Measuremen t Problem” of QM? In this note we summarize some w ork done in discussions and collab oration with Carlo Alb ert; a more detailed accoun t of our findings will app ear in a pap er in preparation. Among n umerous problems w e ha v e debated is the so-called “Measurement Problem” of QM, in particular the ques- tion of how to come up with a logically coherent description of quan tum-mechanical measurement pro cesses. W e ha ve b een studying such problems within a certain completion of QM dubb ed E T H - Approac h to (or “ E T H -Completion” of ) QM [ 1 , 2 , 3 , 4 ]. T o give aw a y the punc hline of our efforts, w e are convinced that, in a non-relativistic regime of QM (with the velocity of light tak en to 8 ), the “Measurement Problem” of QM can be solv ed in an entirely satisfactory w ay . T o b egin with, w e indicate what, in our view, the “Measuremen t Problem” of QM is, and wh y people ha ve great difficulties in solving it. 1 W e will, how ev er, not engage in an y discussion of so-called “interpr etations” of QM; our attitude tow ards them is summarized in the following v erdict due to P .A.M. Dirac: “The interpr etation of quantum me chanics has b e en de alt with by many authors, and I do not want to discuss it her e. I want to de al with mor e fundamental things. ” As an example of our general approac h tow ards dev eloping a precise quantum theory of ex- p erimen ts and measuremen ts an idealized model of a double-slit exp erimen t will b e discussed in Sect. 4. 1 F or some time-honored ideas ab out measurements in quantum mechanics, see also [ 5 , 6 , 7 ] and references given there. 1 1.1 Some basic notions and premises In this section we summarize v arious well kno wn notions, facts and premises, the purp ose b eing to in tro duce conv enien t language and to clarify the starting p oin t of our analysis. An isolate d system, S, is a ph ysical system that has only negligibly tiny in teractions with its complemen t (the rest of the univ erse). Only for an isolated system, quan tum-mechanical time ev o- lution of op erators representing ph ysical quan tities c haracteristic of the system can be form ulated in a universal and precise w ay as Heisenb er g evolution of op erators acting on the Hilb ert space of the system. In this note we will only consider isolated systems. An isolated system is op en if – to b e concrete – it can release massless mo des (photons and gra vitons) escaping to the even t horizon of the universe (or of black holes). Suc h mo des can carry a wa y energy and information that, for fundamental r e asons, cannot b e recov ered in the future by any material devices b elonging to the system. (A more general, abstract notion of op en systems will b e describ ed later and is based on what has b een called the “Principle of Diminishing Potentialities,” see [ 3 ].) In this note, we fo cus our attention on non-r elativistic QM. W e consider a system, S , of matter p ossibly in teracting with the quantized radiation field with the property that all characteristic v elo cities of matter mo des are tin y as compared to the velocity of light, c . In the description of suc h a system one ma y consider the limiting regime where c Ñ 8 , and this is what we will be concerned with in the following. Let H denote the Hilb ert space of pure state v ectors of S . General states of S are given by densit y matrices acting on H , and physical quan tities, p X , p Y , . . . , of S are represented b y selfadjoint b ounded op erators, X , Y , . . . , on H . Let H S denote the Hamiltonian generating the time evolution of op erators acting on H . The Heisenb er g e quation of motion [ 8 , 9 ] for the time-dep endence of a selfadjoin t, bounded operator, X p t q , t P R , represen ting a ph ysical quantit y , p X , of S is given b y 9 X p t q “ i ℏ “ H S , X p t q ‰ , (1) where the dot indicates differen tiation with resp ect to time t , and ℏ is Planc k’s constan t. This is a linear deterministic equation for X p t q , t P R . It is a basic ingredient of the following analysis. Remark: In equation (1), it is assumed that the Hamiltonian H S is a densely defined self-adjoin t op erator on H . Many imp ortan t results concerning (Schrödinger) Hamiltonians of non-relativistic QM, e.g., pro ofs of self-adjoin tness, etc. hav e been contributed b y , among n umerous authors, Sigal and Simon; see, e.g., [ 10 , 11 ]. In Classical Mechanics , the dynamics of an isolated system is go v erned by deterministic (most often non-linear) equations of motion for the state , ξ p t q , of the system as a function of time t . The state, ξ , is a point in the state space, X , of the system; ( X is a top ological space, most often a manifold; in Hamiltonian mechanics, X is a symplectic manifold, the phase sp ac e of the system). Under suitable hypotheses, the trajectory ␣ ξ p t q P X ˇ ˇ t P R ( of the system as a function of time t is completely determined by its initial condition ξ 0 . A strict L aw of Causality holds. Physical quan tities (“observ ables”) are describ ed by real-v alued bounded con tinuous functions on X ; they ha ve a precise v alue in ev ery pure state ξ P X . Equiv alently , the state space X of the system can b e thought of as the sp ectrum of the ab elian algebra generated by all its “observ ables,” and time ev olution can be viewed as giv en b y a one-parameter family of ˚ -automorphisms of the algebra of “observ ables,” which (under natural h yp otheses) turns out to b e generated by a v ector field on X ; in Hamiltonian mechanics, this vector field is a Hamiltonian v ector field, hence generated by a real-v alued (Hamilton) function on X . In Quantum Mechanics , the situation is remarkably differen t. Physical quan tities (“observ- ables”) of a system S generate a non-ab elian C ˚ -algebra. In general, they do not hav e precise 2 v alues in pure states: 2 If ω denotes a state of S , and X “ X ˚ is an op erator on H represen ting a ph ysical quantit y p X at an arbitrary , but fixed time, we define X ω : “ ω p X q to b e the exp ectation v alue of p X in the state ω (i.e., the av erage of v alues of p X measured in a long sequence of identical exp erimen ts, with S prepared in the state ω in every exp erimen t), and we take ` ∆ ω X ˘ 2 : “ ` X ´ X ω ˘ 2 ω (2) to b e the square of the unc ertainty of the v alue of p X in the state ω . The celebrated K o chen-Sp e cker the or em [ 12 ] tells us that, under some natural algebraic h yp otheses, the exp ectation v alues and uncertain ties of the op erators represen ting physical quan tities of a system S in QM cannot b e repro duced b y classical random v ariables. 3 Con ven tionally , it is claimed that, in the Heisenb er g pictur e , the time-dep endence of an op erator, X p t q , representing a physical quan titiy , p X , of S at time t is describ ed b y the Heisenberg equations ( 1 ), while states, ω , of S are given b y time-indep endent densit y matrices, Ω , acting on the Hilb ert space H of S . One then goes on to claim that the Heisen b erg picture is e quivalent to the Schr ö dinger pictur e : The exp ectation of a ph ysical quantit y p X of S at time t in the state ω is giv en by ω ` X p t q ˘ “ T r “ Ω ¨ X p t q ‰ “ T r “ Ω p t q ¨ X ‰ , X “ X p 0 q . (3) where 9 Ω p t q “ ´ i ℏ “ H S , Ω p t q ‰ , Ω p 0 q “ Ω . (4) This is the Schr ö dinger-von Neumann e quation describing the time-dep endence of the density ma- trix Ω p t q representing the state of S at time t in the Schrödinger picture. W e note that, under natural hypotheses (see, e.g, [ 11 ]), equation ( 3 ) is a linear, deterministic equation for the time- dep endence of the density matrix Ω p t q , for all times t P R . Ho wev er, ever since Einstein’s analysis (1917) of sp on taneous and induced emission and absorp- tion of photons b y excited atoms [ 14 ] and Born’s analysis (1926) of collision processes [ 15 ], it is claimed that Quan tum Mechanics is fundamen tally probabilistic . How can this b e reconciled with the deterministic nature of the Heisenberg- and the Schrödinger-von Neumann equations? This is the k ey question to b e answered in this note. 1.2 The confusion introduced b y the Sc hrö dinger-v on Neumann equation and the “measuremen t p ostulates” of von Neumann and Lüders In text-b o oks, the tension b et w een the deterministic nature of the Schrödinger-von Neumann equa- tion for the quantum-mec hanical time ev olution of states of isolated systems and the fundamentally probabilistic nature of non-relativistic Quantum Mechanics usually culminates in the following tw o (rather confusing and misleading) P ostulates form ulated by v on Neumann and Lüders [ 16 , 17 ]. I. The time-ev olution in the Schrödinger picture of states of an isolated system, S , is describ ed b y the time-dep enden t (linear, deterministic) Sc hrö dinger-von Neumann equation – except when some physical quantitiy , p X , characteristic of S is measured, in whic h case the state, ω i , of S immediately before the measurement begins makes a jump to a state, ω f , o ccupied b y the system immediately after the conclusion of the measurement of p X , with the following prop erties: 2 Pur e states are states that c annot b e represented as conv ex combinations of several distinct states, while mixed states are conv ex combinations of several distinct pure states. 3 The K o c hen-Sp ec ker theorem can b e deriv ed from Gleason’s theorem [ 13 ] and, just like the latter, requires assuming that the dimension of H is ě 3 . 3 (a) the measured v alue, ξ , of p X (with p X assumed to hav e discrete p oint spectrum) coincides with the exp ectation v alue, X ω f , of p X in the state ω f ; ξ b elongs to the sp ectrum of p X (= sp ectrum of the self-adjoin t op erator X representing p X at the time of measurement); (b) the uncertaint y , ∆ ω f X , of the v alue, ξ , of p X in the state ω f of S vanishes . I I. If the uncertain t y ∆ ω i X of the v alue of p X in the state ω i of S immediately b efore the measuremen t of p X begins is strictly p ositive then the measured v alue ξ of p X cannot b e predicted; QM only predicts the probabilit y (or frequency) of measuring the v alue ξ in the state ω i if the same measurement of p X , with S prepared in the same state ω i , is rep eated man y times. This probability is giv en b y the well-kno wn Born R ule . Since it is claimed that the uncertaint y ∆ ω f X of the v alue of p X in the state ω f of S immediately after the conclusion of the measuremen t of p X vanishes , one concludes that the tw o states ω i and ω f m ust b e distinct . The stochastic transition from state ω i to state ω f happ ening in the course of a measuremen t of p X is usually called “c ol lapse of the wave function. ” These tw o p ostulates lead to the infamous “Measurement Problem” of QM, which is con- sidered to be fundamen tal and op en by many physicists. It can be summarized in the follwoing questions and remarks. (i) What exactly characterizes a “Measurement” ? Why is the time-evolution of the state of a system S during a “measuremen t” not deterministic , more precisely not describ ed b y a time-dep enden t Schrödinger-von Neumann equation; wh y is it describ ed b y a sto chastic jump pro cess, i.e., a “wa v e function collapse,” satisfying Born’s R ule? (ii) Ho w is it possible that the v alue of a ph ysical quantit y of S observed in a “measurement” is sharp (namely given by an eigenv alue of the op erator representing the measured quantit y) if the uncertain ty of the quantit y in the state of S immediately b efore the measuremen t takes place is strictly p ositiv e? (iii) What causes wa ve functions to collapse; namely what causes the state of S immediately b efore a measurement sets in to jump to a state of S righ t after the conclusion of the measurement in which the uncertain ty of the measured quantit y v anishes? (iv) What determines the time when a measuremen t b egins? Ho w long do es a measurement last? As far as w e can tell, there aren’t any go o d answers to these questions within something lik e the Cop enhagen Interpr etation of QM! One gets the impression that Postulates I and I I are, at b est, reasonable heuristic guidelines for the extraction of exp erimen tally v erifiable consequences of a quan tum-mec hanical description of ph ysical systems. This do es, ho w ever, not eliminate their rather confusing character! Here is what we might or should be confused ab out: 1. “Measuremen ts” are physical pro cesses, too, inv olving in teractions betw een a subsystem of in terest and some (often macroscopically large) measuring devices. If the latter are taken to b elong to the system considered in our description then, to a v ery goo d approximation, this system is isolate d. Hence its time-evolution ought to b e gov erned by a deterministic Sc hrö dinger-v on Neumann equation. – Y et, apparently it isn’t! 2. In the Cop enhagen in terpretation, the physic al quantity to b e measured and the time of its measuremen t are usually pre-determined by an “observ er” (whatev er this notion actually means), who is not part of the system. This aspect depriv es the theory of muc h of its predictiv e pow er, as the theory do es not predict actions of “observers. ” 4 3. If there were only one measuremen t made at “the end of time,” the Cop enhagen “conv en- tional wisdom” would b e quite satisfactory . Ho wev er, we would lik e to describe time-or der e d se quenc es of man y measurements made, one after the other, on one and the same system. This leads to the w ell known problems caused b y interference effects. Cures ha ve b een prop osed for them; e.g., a formulation of testable predictions of QM in terms of “consistent histories” [ 18 , 19 ]. But we doubt that these ideas represen t a satisfactory solution of the “measurement problem. ” 4. In most con ven tional form ulations (or “interpretations”) of QM, measuremen ts can b e carried out in an arbitrarily short in terv al of time. By time-energy uncertain ty relations, this suggests that measurements should b e accompanied by arbitrarily large energy-fluctuations, whic h, in practice, they usually aren’t. – Etc. If QM, and in particular quantum-mec hanical pro cesses aimed at measuring physical quantities, cannot be formulated more clearly and coheren tly than sk etched here then one cannot help agreeing with P .A.M. Dirac, who said: It se ems cle ar that the pr esent quantum me chanics is not in its final form. In the remainder of this note w e outline an approac h to non-relativistic QM (dubb ed E T H - Approac h [ 2 ]) that should b e considered to b e an attempt to cast QM in a final form and, in particular, to solve the “Measurement Problem. ” W e apply our general ideas and results to a sp ecific example of a measurement. W e exp ect that most of the concepts developed b elow can b e extended to relativistic quantum theory [ 20 ]; but man y details remain to b e w orked out more precisely . 2 The “Principle of Diminishing P oten tialities” and the Dissipa- tiv e Nature of the Time-Ev olution of States in Quantum Me- c hanics As announced, the following discussion is limited to non-relativistic QM. W e consider a regime where the velocity of light c is taken to 8 ; but a caricature of the quan tized electromagnetic field is included implicitly in our description. F or isolated systems, w e regard the Heisen b erg equations ( 1 ) of motion of op erators as fundamental, but the initial form ulation of Matrix Mechanics [ 8 , 9 ] (text-b ook QM) must b e extended to a more complete framework, in order to enable one to describ e the sto c hastic time-evolution of states of individual systems, including measuremen t pro cesses. In the following, it is adv antageous to emphasize the concept of “ev ents” , rather than talk ab out physical quantities and “observ ables” (whic h, in the E T H -Approach to QM, hav e the status of derive d quantities ). In non-relativistic QM, a potential ev ent , 4 e , is giv en b y a partition of unit y , e : “ ␣ p Π k ˇ ˇ k “ 1 , 2 , . . . ( , consisting of abstract, mutually disjoint, orthogonal pro jections, p Π k , with p Π j ¨ p Π k “ δ j k p Π k , @ j, k, ÿ k p Π k “ 1 . (5) Giv en an isolated system S , let E S denote the family of all p oten tial even ts. A t ev ery time t , there is a representation on the Hilbert space H of S , p Π ÞÑ Π p t q , (6) 4 more precisely , a family of “mutually exclusiv e p oten tial even ts” 5 of all pro jections p Π b elonging to a p oten tial even t e by orthogonal pro jections Π p t q acting on H , for all p otential even ts e Ă E S . The argumen t, t , of Π p t q , with p Π P e , is the time at whic h the p oten tial ev ent e might set in. W e assume that the time-dep endence of the pro jection op erators Π p t q on H is determined by the Heisenb er g e quations 9 Π p t q “ i ℏ “ H S , Π p t q ‰ , for t P R , (1’) see Eq. (1). The family of mutually disjoint orthogonal pro jection op erators ␣ Π p t q ˇ ˇ p Π P e ( on H is in terpreted as the p otential event (corresp onding to e Ă E S ) p ossibly setting in at time t P R . W e denote b y E ě t the (weakly closed ˚ -) algebra generated b y all the op erators ␣ Π p t 1 q ˇ ˇ p Π P e Ă E S , t 1 ě t ( , (7) where t P R is an arbitrary time. The Heisen b erg equation ( 1’ ) implies that all these algebras are isomorphic to one another; E ě t 1 “ e i p t 1 ´ t q H S { ℏ E ě t e i p t ´ t 1 q H S { ℏ , for arbitrary t, t 1 in R . (8) F rom the definition of these algebras it follows immediately that E ě t 1 Ň E ě t , whenev er t 1 ą t . It turns out that, in an isolated op en system, a stronger prop ert y holds: E ě t 1 Ř E ě t , whenev er t 1 ą t . (9) This prop ert y , called “Principle of Diminishing/Declining P otentialities” (PDP), w as first in tro duced (under the name of “loss of (access to) information”) in [ 1 ], with the aim of solving the “Measuremen t Problem. ” It can b e view ed as a general c haracterization of isolated op en systems, more precisely of dissipation in isolated op en systems. It turns out that in relativistic Quan tum Electro dynamics on ev en-dimensional Minko wski space-time, with t the time of some “observer,” and in limiting theories on Newtonian space-time, E 3 ˆ R , obtained b y letting the v elo cit y of light, c , tend to 8 , PDP is a theorem . It follows from results of D. Buchholz [ 21 ], who established a fundamen tal algebraic prop ert y under the name of “Huygens Principle” that implies PDP , and it con tinues to hold in the limit c Ñ 8 , as verified, e.g., in [ 4 ]. It can also be argued to hold on more general space-times in the presence of black holes. But, for systems of massive matter not coupled to any quan tum field with massless mo des PDP fails , the algebra E ě t is indep enden t of time t , and the acquisition of information ab out suc h systems in “measurements” turns out to b e imp ossible. 2.1 What are states? Next, w e in tro duce a natural notion of states . A state of a system S at time t is supp osed to b e a mathematical ob ject enabling one to predict the likelihoo d or frequency of an arbitrary p oten tial ev ent e Ă E S to set in at some time t 1 ě t . This suggests to define a state of S at time t to b e a normalized p ositiv e linear functional on the algebra E ě t (or, equiv alently , a quan tum probabilit y measure, in the sense of Gleason [ 13 ], on the family of all p oten tial ev ents of S p ossibly setting in at some time t 1 ě t ). It turns out to b e necessary to also introduce a notion of “ensem ble 6 states. ” Let E S denote an ensem ble of very many systems all of which are isomorphic to a system S and prepared in the same initial state at some initial time t 0 “ 0 . An ensemble state of S at a time t ą 0 is defined as the av erage o ver the ensemble E S of states at time t of systems in E S that ha ve evolv ed from the same initial state prepared at time t “ 0 . Supp ose the systems in E S are all prepared at time t “ 0 in a state ω 0 on the algebra E 0 : “ E ě 0 . In the Heisenberg picture, the ensemble state, ω t , at some time t ą 0 , giv en the initial state ω 0 , is defined to b e the r estriction of the state ω 0 to the subalgebra E ě t of E 0 ; i.e., ω t : “ ω 0 ˇ ˇ E ě t , for an arbitrary t ą 0 . (10) In non-relativistic QM (in the limit where c Ñ 8 ), the algebra E ě t is isomorphic to the algebra of all b ounded op erators on a Hilb ert subspace, H t , of H , E ě t » B p H t q , (11) for an arbitrary t , and H t 1 Ď H t , for t 1 ą t . One has that H t 1 is the image of H t under the propagator e ´ i p t 1 ´ t q H S { ℏ . The isomorphism in Eq. ( 11 ) is called Property P (see [ 4 ]). 5 Supp ose no w that ω 0 is a pure state on E 0 : “ E ě 0 . Assuming that PDP holds, we conclude that, in general, the ensemble state ω t , for t ą 0 , defined in ( 10 ) is a mixed state on the algebra E ě t Ř E 0 , a manifestation of en tanglement . This is a crucial observ ation. It means that the Heisenberg-picture time evolution of ensemble states introduced in ( 10 ) is dissipative , conv erting pur e states in to mixtur es accompanied by en- trop y pro duction. The underlying ph ysical reason for dissipation is that an op en system satisfying PDP can release massless modes to the outside w orld at some time t that, for fundamental r e a- sons, escap e from the system for go o d and b ecome unobservable at any later time t 1 ě t , causing de c oher enc e. In the non-relativistic limit, with c Ñ 8 , massless mo des emitted by a system escap e to spatial infinity infinitely rapidly . It app ears to b e a gener al fact 6 – not only in QM, but also in classical physics – that the dissipativ e nature of the time-evolution of ensemble states implies that the states of individual systems belonging to the ensemble exhibit a sto c hastic time-ev olution, obtained by “unrav eling” the dissipative (but deterministic) evolution of ensem ble states. It ma y be appropriate to men tion an example of this fact in classical physics. Imagine that w e consider a large ensemble – a “gas” – of p oin t-like particles susp ended in a thermal liquid and exhibiting Brownian motion (caused b y collisions of the particles with lumps in the liquid), but not in teracting with one another. An ensem ble state of this system at time t is given by the density , ρ t , of particles in physical space E 3 . The time ev olution of ρ t is given b y a diffusion e quation, whic h is a line ar, deterministic equation known to b e dissip ative and to pro duce entrop y . The ev olution of the state of a single particle in the ensemble (“gas”), namely its p osition ξ p t q P E 3 , as a function of time t is, ho wev er, sto chastic; it is giv en by a Wiener pr o c ess, which can b e constructed b y “unrav eling” the diffusion equation; (see, e.g., [ 22 ]). More details ab out this example and the analogy with QM can b e found in [ 3 , 4 ]. 2.2 Time-ev olution of ensemble states as Lindblad evolution [ 24 ] Next, we should ask whether the time ev olution of ensemble states in tro duced in ( 10 ) can b e c haracterized explicitly . F or this purp ose, it is con v enient to switc h from the Heisen b erg picture 5 Eq. ( 11 ) is not true in relativistic theories with c ă 8 , in which case the algebras E ě t are typically von Neumann algebras of type I II 1 . 6 related to the dissipation-fluctuation theorem 7 to the Schr ö dinger pictur e. Let S b e an isolated system of the kind considered ab ov e, and let Ω 0 b e the density matrix on the Hilb ert subspace H 0 Ă H representing the initial state ω 0 of S . By Eq. ( 10 ), the density matrix represen ting the ensem ble state ω t on the Hilb ert subspace H t of H 0 is then given b y P t Ω 0 P t , where P t is the orthogonal pro jection onto the subspace H t , for an arbitrary t ą 0 . Ev ery op erator Y in the algebra E ě t Ř E 0 can b e written as Y “ e itH S { ℏ X e ´ itH S { ℏ , for some op erator X P E 0 . W e may then write ω t p Y q p 10 q “ ω 0 p Y q “ T r “ P t Ω 0 P t ¨ Y ‰ “ T r “ P t Ω 0 P t ¨ e itH S { ℏ X e ´ itH S { ℏ ‰ “ : T r “ Ω t ¨ X ‰ , (12) for all Y P E ě t , hence for all X p“ e ´ itH S { ℏ Y e itH S { ℏ q in E 0 , where Ω t : “ e ´ itH S { ℏ P t Ω 0 P t e itH S { ℏ “ P 0 e ´ itH S { ℏ Ω 0 e itH S { ℏ P 0 , (13) is again a density matrix on H 0 . The map Ω 0 ÞÑ Ω t defined in Eq. ( 13 ) is line ar, tr ac e-pr eserving and p ositivity-pr eserving . It is actually c ompletely p ositive . The arguments sketc hed here can b e rep eated to show that the maps Γ t 1 ,t : Ω t ÞÑ Ω t 1 , t 1 ą t, (14) on the space of densit y matrices on H 0 are line ar, tr ac e-pr eserving, p ositivity-pr eserving and c om- pletely p ositive . One easily v erifies that Γ t 1 ,t 2 ˝ Γ t 2 ,t “ Γ t 1 ,t , Γ t,t “ id . The adv antage of the Schrödinger picture is that time-evolution can b e viewed as an evolution of densit y matrices on the fixed Hilb ert space H 0 under linear, trace-preserving, p ositivit y preserving, completely p ositiv e maps. Such maps hav e been c haracterized by Kraus [ 23 ]. F or simplicity , w e henceforth consider a system S of matter interacting with the quantized radiation field in the limiting regime where c “ 8 and with the radiation field prepared in its vacuum state, whic h is time-translation inv ariant. Since, for such a system, “photons” created at some time t escap e to an even t horizon infinitely rapidly , the state of the radiation field at any time t 1 ą t is again the v acuum state. It then turns out that the maps Γ t 1 ,t only dep end on the difference v ariable t 1 ´ t , i.e., Γ t 1 ,t “ Γ p t 1 ´ t q , and that the radiation field can be eliminated from our description; see [ 4 ]. Assuming that Γ p t q conv erges strongly to the identit y map, as t Œ 0 , one concludes that ␣ Γ p t q ˇ ˇ t ě 0 ( is generated by a Lindblad op er ator, L , i.e., Γ p t q “ e t L , for t ě 0 . Th us, the time-ev olution of the densit y matrices Ω t , t ě 0 , is go verned by a Lin blad equation (see [ 24 ]) 9 Ω t “ L “ Ω t ‰ . (15) The general form of a Lindbladian, L , is given b y L “ Ω ‰ “ ´ i ℏ “ H 0 S , Ω ‰ ` α ÿ σ “ 1 , 2 ,... ” T σ Ω t T ˚ σ ´ 1 2 ␣ Ω , T ˚ σ T σ ( ı . (16) where H 0 S is a self-adjoin t “Hamiltonian” on H 0 , α ě 0 is a coupling constant, ␣ T σ ˇ ˇ σ “ 1 , 2 , . . . ( is a family of arbitrary (b ounded) op erators on H 0 , and ␣ ¨ , ¨ ( denotes an anti-comm utator. If S is a system of matter interacting with the radiation field, the op erator H 0 S is the Hamiltonian of S when it is decoupled from the radiation field; α is prop ortional to the square of the feinstructure constan t, and the op ertors ␣ T σ ˇ ˇ σ “ 1 , 2 , . . . ( are determined by r adiative tr ansition amplitudes of matter; see, e.g., [ 4 ]. 8 Remark: If the velocity of light is kept finite the notion of (ensemble) states must be refined; (ensem ble states then turn out to b e asso ciated with p oints in space-time, rather than just dep end on time). While, as a consequence of “Huygens’ Principle” [ 21 ], the evolution of ensembles states is sho wn to b e dissipative, Prop ert y P (see ( 11 )) fails, and one will encoun ter memory effects. Some general ideas ab out the E T H -Approac h to relativistic quan tum theory are describ ed in [ 20 , 3 ]; but plen ty of details remain to b e understo od more precisely . 3 The sto chastic time-ev olution of states of invidual systems In this section w e substan tiate the claim that whenev er the time-ev olution of ensem ble states is dissipativ e , with pure states evolving into mixtures, then the evolution of states of individual systems is sto c hastic . T o b egin with, we describ e some further ingredients of the E T H -Approac h to QM that will turn out to b e essen tial. On tology: In the Schrödinger picture of non-relativistic QM, as describ ed in Sect. 2 , states o ccupied by an individual isolated system S at an arbitrary time t are assumed to b e given by densit y matrices prop ortional to finite-dimensional orthogonal projections, Π t , in the algebra E 0 , i.e., ω t p X q “ 1 T r r Π t s T r H t “ Π t ¨ X ‰ , @ X P E 0 . (17) (In the Heisen b erg picture, the state corresp onding to Π t w ould b e a pro jection in the algebra E ě t .) Generically , the pro jection Π t has rank 1, i.e., it defines a pur e state. The assumption in Eq. ( 17 ) is analogous to the one made in the theory of diffusion of a gas of non-interacting particles alluded to at the end of Subsect. 2.1: The state at an arbitrary time t of an individual particle in the gas is giv en b y its p osition ξ p t q P E 3 , corresp onding to a densit y ρ t p x q “ δ ξ p t q p x q , x P E 3 , which is a pur e state. Let Ω b e a density matrix on H 0 represen ting a state ω of S at some fixed p ositiv e time. The sp ectral theorem says that Ω “ ÿ δ “ 0 , 1 , 2 ,... p δ Π δ , p 0 ą p 1 ą ¨ ¨ ¨ ě 0 , ÿ δ “ 0 , 1 , 2 ,... Π δ “ 1 , (18) with T r r Ω s “ ÿ δ “ 0 , 1 , 2 ,... p δ dim p Π δ q “ 1 , dim p Π δ q : “ T r r Π δ s , where the operators ␣ Π δ ˇ ˇ δ “ 0 , 1 , 2 , . . . ( are the spectral pro jections of Ω . They generate a partition of unit y by mutually disjoin t orthogonal pro jections and can therefore be interpreted as represen ting a p otential event. W e note that ω p X q “ ÿ δ “ 0 , 1 , 2 ,... ω p Π δ X Π δ q , with ω p Π δ X Π δ q “ T r “ Ω Π δ X ‰ “ p δ T r “ Π δ X ‰ , @ X P B p H 0 q . (19) If, at some time t , the state av eraged ov er a large ensemble E S of systems, all iden tical to a system S , is giv en by ω then, according to ( 17 ), the state of an individual system in E S is assumed to b e giv en by the densit y matrix r dim p Π δ qs ´ 1 Π δ , with a frequency giv en by p δ dim p Π δ q , for some δ “ 0 , 1 , 2 , . . . , corresp onding to Born ’s Rule. In the E T H -Approac h to QM, this assumption is called “State-Selection P ostulate” [ 2 , 3 ]. Besides PDP , this p ostulate represents a natural 9 requiremen t for isolate d op en systems. One then says that the p oten tial even t ␣ Π δ ˇ ˇ δ “ 0 , 1 , 2 , . . . ( actualizes at time t , and one calls the pro jection Π δ corresp ondig to the state dim p Π δ q ´ 1 Π δ o ccupied b y some individual systems in E S an “actual event” or “actuality;” see [ 3 , 4 ]. 3.1 Deriv ation of the sto chastic time-evolution W e no w apply the State-Sele ction Postulate to deduce the stochastic time-evolution of an indi- vidual system b elonging to the ensem ble E S from the time-evolution of ensem ble states giv en in Eq. ( 15 ); (w e follow the presentation in [ 4 ] and use notations similar to those in that pap er). Sup- p ose that, at some time t , all systems in E S o ccup y a state Π t (for simplicity assumed to b e a pure state, i.e., given by a rank-1 orthogonal pro jection). Equation ( 15 ) then tells us that the ensem ble state at a later time t ` dt (where dt is a small time-increment) is giv en by a density matrix Ω t ` dt “ Π t ` L “ Π t ‰ dt ` O p dt 2 q , (15’) where L is the Lindbladian of Eq. ( 15 ). The sp ectral decomp osition of Ω t ` dt tak es the form Ω t ` dt “ p nj r t, t ` dt s Π 0 t ` dt ` ÿ δ “ 1 , 2 ,... p δ r t, t ` dt s Π δ t ` dt , (20) where Π 0 t ` dt , Π 1 t ` dt , Π 2 t ` dt , . . . , are mutually disjoin t orthogonal pro jections (the sp ectral pro jections of Ω t ` dt ), with ř δ “ 0 , 1 , 2 ,... Π δ t ` dt “ 1 ˇ ˇ H 0 , and, assuming that dt is “small,” Π 0 t ` dt « Π t , furthermore, p nj r t, t ` dt s ” p 0 r t, t ` dt s “ 1 ´ O p dt q ą 0 , p nj ą p 1 ą p 2 ą ¨ ¨ ¨ ě 0 , with p δ “ O p dt q , @ δ ě 1 , p nj r t, t ` dt s ` ÿ δ “ 1 , 2 ,... p δ r t, t ` dt s dim p Π δ t ` dt q “ 1 , (21) where “ nj ” stands for “no jump” . According to the state-selection p ostulate of the E T H -Approac h, one of the density matrices ␣ r dim p Π δ t ` dt qs ´ 1 Π δ t ` dt ˇ ˇ δ “ 0 , 1 , 2 , . . . ( , r andomly chosen with a fre- quency giv en b y p δ r t, t ` dt s dim p Π δ t ` dt q , is the state of an individual system in E S at time t ` dt . The transition from the state Π t to one of the states ␣ dim p Π δ t ` dt q ´ 1 Π δ t ` dt ˇ ˇ δ “ 0 , 1 , 2 , . . . ( is the ev ent actualizing at time t in an individual system. If Π t ev olves to Π δ t ` dt in the time in terv al r t, t ` dt q , for some δ ě 1 , w e sp eak of a “quantum jump” . 3.2 Explicit form ulae for the sto chastic time-ev olution In [ 4 ], explicit formulae for the pro jections Π δ t ` dt and the coefficients p δ r t, t ` dt s , δ “ 0 , 1 , 2 , . . . , ha ve b een derived from Eq. ( 15’ ), using a form of analytic p erturbation theory d ubbed infinitesimal p erturb ation the ory ; see App endix A of [ 4 ]. Here w e only summarize the results of the analysis in that pap er. (i) Supp ose that, during an interv al of times r t 1 , t 2 q there are no quantum jumps, in the sense that, for all t P r t 1 , t 2 q , the pro jection Π 0 t “ : Π t describ es the state of some individual system 10 in E S , prepared at time t “ t 1 in a state Π t 1 . The probability for this tra jectory of states to b e observ ed is giv en by p nj r t 1 , t 2 s “ exp ! ż t 2 t 1 T r “ Π t L r Π t s ‰ dt ) ď 1 , (22) where it is used that T r ` Π t L r Π t s ˘ “ T r ` L r Π t s ˘ ´ T r ` Π K t L α r Π t s ˘ “ ´ α ÿ σ T r “ Π K t T σ Π t T ˚ σ Π K t ‰ ď 0 , with Π K t : “ 1 ´ Π t ; the first term on the right side v anishes, while the second term is ď 0 , b y Eq. ( 16 ). (ii) In these formulae, the pro jection Π t ” Π 0 t is a solution of the cubic ordinary differential equation d Π t dt “ Π K t L r Π t s Π t ` Π t L r Π t s Π K t , t 1 ď t ă t 2 . (23) (iii) The probabilities p δ r t 2 , t 2 ` dt s , δ “ 1 , 2 , . . . , for observing a “quantum jump” during the time in terv al r t 2 , t 2 ` dt q are giv en b y the eigen v alues of the non-ne gative matrix Π K t 2 L r Π t 2 s Π K t 2 ¨ dt ( 16 ) “ α ÿ σ Π K t 2 T σ Π t 2 T ˚ σ Π K t 2 dt , i.e., ␣ p δ r t 2 , t 2 ` dt s ˇ ˇ δ “ 1 , 2 , . . . ( “ sp ec ` Π K t 2 L r Π t 2 s Π K t 2 dt ˘ , (24) up to corrections of order O p dt 2 q , with p δ r t 2 , t 2 ` dt s ě 0 , @ δ “ 1 , 2 , . . . . Remark: Consider form ulae ( 22 ) and ( 23 ) in the sp ecial case where matter is decoupled from the radiation field, i.e., for α “ 0 . Then T r “ Π t L r Π t s ‰ “ 0 , hence, by ( 22 ), p nj r t 1 , t 2 s ” 1 , and p δ r t 2 , t 2 ` dt s “ 0 , @ δ “ 1 , 2 , . . . , for an arbitrary time interv al r t 1 , t 2 q ; there are no quantum jumps! F urthemore, d Π t dt “ ´ i ␣ Π K t r H 0 S , Π t s Π t ` Π t r H 0 S , Π t s Π K t ( “ ´ i r H 0 S , Π t s , i.e., for α “ 0 , the time-evolution of the state of the system is describ ed by the Schrödinger-von Neumann equation with Hamiltonian H 0 S , as exp ected. The form ulae in Eqs. ( 22 ) through ( 24 ) uniquely determine a probability measure on the space, P , of state tra jectories (as functions of time) of individual systems analogous to the Wiener me asur e on the space of Brownian paths. Explicit expressions can b e found in [ 4 ]. The theory dev elop ed in this section has b een applied to the phenomenon of fluorescence in [ 4 ]. It can also b e used to describ e “measurements” in quan tum mec hanics, in particular the r andom time when a measuremen t sets in and the pro cess of the system’s state to jump into a state corresp onding to a precise v alue of the physical quan tity that is b eing measured. F or mo dels with a discrete time, results on a description of measurements within the E T H -Approach ha ve b een rep orted in [ 25 ]. 11 4 A simple mo del of a double-slit exp erimen t W e consider a cavit y o ccupying a cubical domain Λ Ă E 3 that is comp osed of tw o adjacen t v acuum c hambers separated b y a thin w all, Ξ , pierced b y t wo parallel slits and parallel to t w o opp osite faces of Λ . One face of Λ , denoted b y Σ , parallel to Ξ corresp onds to a light-emitting scr e en (a scin tillation screen), the opp osite face, denoted b y Γ , of the cavit y parallel to Ξ con tains the nozzle of a gun emitting quantum particles (electrons or any other kind of quantum particles, e.g., Buc kminsterfullerenes, in teracting with the radiation field), henceforth called “electrons. ” When an electron hits the light-emitting screen Σ a flash of light emanates from a sensitiv e sp ot (“pixel”) on Σ , the location where the electron has hit the screen. It is assumed that electrons are emitted by the gun on Γ sufficiently rarely that, with very high probability , there is either no “electron” in the ca vity Λ or only one electron moving through Λ at all times; hence electron-electron interactions (man y-b o dy effects) can and will b e neglected. The motion of a single electron de c ouple d fr om the quantize d ele ctr omagnetic field and not in teracting with the pixels on the screen Σ is describ ed b y a Sc hrö dinger equation for the time- dep endence of its state (w av e function), with a Hamiltonian giv en by ´ ℏ 2 2 M ∆ Λ , where M is the mass of the particle, and ∆ Λ is the Laplacian with suitable b oundary conditions imp osed: On all faces of Λ , except Σ , and on the wall Ξ with the double slit, 0-Dirichlet b oundary conditions are imp osed on ∆ Λ . This determines a densely defined, self-adjoin t operator on the Hilb ert space L 2 p R 3 , d 3 x q . Neglecting spin, the total electron Hilb ert space, H , is given by H “ L 2 p R 3 , d 3 x q ‘ h N , h N » C N , (25) where N is the n umber of states that can b e o ccupied b y an electron b ound to the light-emitting screen Σ ; these states corresp ond to wa v e functions lo calized in the vicinit y of the different pixels of the screen Σ . The idea is that an electron that comes close to one of the pixels of Σ is attracted to it and tends to jump into a bound-state in h N lo calized near that pixel by emitting a photon that may subsequently b e recorded. By P w e denote the orthogonal pro jection onto the subspace L 2 p R 3 , d 3 x q of H , and P K : “ 1 ´ P is the pro jection onto the subspace h N . The Hamiltonian of an electron decoupled from the quan tized electromagnetic field and the attractiv e p otentials of the pixels on Σ is defined b y H el : “ ´ ℏ 2 2 M P ∆ Λ P , (26) In teractions of an electron with the ligh t-emitting screen Σ are tak en into accoun t by the follo wing somewhat idealized interaction Hamiltonian H I : “ g ÿ σ P S ´ T σ b A ˚ σ ` T ˚ σ b A σ ¯ , (27) acting on the Hilb ert space H b F , where F is the F o c k space of the quan tized electromagnetic field, g is a coupling constan t (prop ortional to the elemen tary electric charge), S is a family of N symbols lab eling pixels in a rectangular array contained in Σ , where photons are emitted from when they are hit b y an electron, and, for every σ P S , the op erator T σ is a radiativ e transition matrix from the subspace P H “ L 2 p R 3 , d 3 x q of the electron Hilb ert space to a state ˇ ˇ σ  P h N in the range of the pro jection P K , with ˇ ˇ σ  the state of an electron b ound to the pixel lab elled by σ P S . Let B σ,r b e the ball of radius r in R 3 cen tered at the lo cation of the pixel lab elled b y σ , and let P σ,r b e the pro jection onto the subspace of P H of electron wa v efunctions with supp ort in 12 the complemen t of the ball B σ,r . W e assume that, for all σ P S , T σ is a b ounded op erator from P H to P K H with the prop ert y that } T σ P σ,r } ď K e ´ r { R , @ r ą 0 , (28) where K and R are some p ositiv e constants indep enden t of σ . The states ˇ ˇ σ  , σ P S , are long-lived meta-stable states; for simplicity , we henceforth assume that they can b e treated as stationary states. Let Q σ b e the orthogonal projection on to ˇ ˇ σ  ; without loss of generality we assume that the projections ␣ Q σ ˇ ˇ σ P S ( are mutual ly disjoint, with ř σ P S Q σ “ P K . F urthermore, A ˚ σ p t q is an op erator acting on the F o c k space F that creates a “photon” at time t ; ( A ˚ σ : “ A ˚ σ p 0 q ). W e assume that, b efore an electron is released b y the electron gun into an initial state, Ψ 0 , con tained in the range of the pro jection P and supp orted in the region Λ , the electromagnetic field is prepared in its v acuum state , denoted by ˇ ˇ H  . W e contin ue to describ e the electromagnetic field in a limiting (non-relativistic) regime where the velocity of ligh t, c , approac hes 8 . In this regime, one has that  H ˇ ˇ A σ p t q ¨ A ˚ σ 1 p t 1 q ˇ ˇ H  F “ δ σ σ 1 δ p t ´ t 1 q , with A σ p t q ˇ ˇ H  “ 0 , (29) for all σ, σ 1 in S . When a photon is emitted by an electron hitting a pixel σ at some time t (corresp onding to replacing the v acuum state ˇ ˇ H  of the electromagnetic field by the state A ˚ σ p t q ˇ ˇ H  ) it is recorded b y the pixel σ ; subsequently the state of the electromagnetic field immediately relaxes bac k to the v acuum state . 7 Under these assumptions, the basic p ostulates of the ETH-A ppr o ach to QM enable one to completely eliminate the electromagnetic field from the description of the experiment (see [ 4 ]); and the effective dynamics of electron ensemble states is given b y a quantum Mark ov semi-group describ ed, more explicitly , b y the Lindblad equation, 9 Ω “ ´ i “ H el , Ω ‰ ` α ÿ σ P S ´ T σ Ω T ˚ σ ´ 1 2 ␣ Ω , T ˚ σ T σ ( ¯ , (30) where Ω is a density matrix on H describing an electronic ensem ble state, and α “ g 2 ą 0 ; see App endix A in [ 4 ]. (In this pap er one also finds a detailed discussion of how the “Principle of Diminishing/Declining Poten tialities” (PDP) follows from “Huygens’ Principle” [ 21 ], and what this principle says in the limit where c Ñ 8 .) 4.1 The fate of individual electrons Equation ( 30 ) is linear, deterministic and disspative (for α ą 0 ), due to the coupling of electrons to the electromagnetic field, which en tails the v alidity of PDP . Quite generally , the dissipative nature of the time evolution of ensemble states of a system is crucial for the emergence of actual ev ents, and, in particular, for the success of measuremen ts in the system. W e recall that electron states in the subspace h N ” Ran P K » C N are assumed not to ev olve under the electronic Hamiltonian H el ; moreov er, they do not evolv e under the Lindblad evolution of Eq. ( 30 ), either (i.e., they are constant in time), as is easy to verify using that P ¨ P K “ 0 and T σ ˇ ˇ Ran P K “ 0 , for all σ P S . In particular, a pure state giv en by a pro jection Q σ “ ˇ ˇ σ  σ ˇ ˇ , σ P S , do es not evolv e; Eq. ( 30 ) implies that 9 Q σ “ 0 , for all σ P S . 7 In ( 29 ), one ma y b e tempted to replace δ σσ 1 b y matrix elements, g σσ 1 , of a more general p ositiv e-definite matrix g ; but this generalization is without interest. 13 A ccording to the ETH-A ppr o ach to QM, the evolution of an individual electron is sto c hastic , for α ą 0 , and the law of this evolution is obtained by “unr aveling” the Lindblad evolution of ensem ble states describ ed in ( 30 ) in accordance with the State-Selection Postulate of the E T H - Approac h; (see Subsect. 3.2). T o determine the unrav eling concretely , w e use that the ranges of the op erators T σ , σ P S , are mutually orthogonal to one another and orthogonal to the range of the pro jection P . The State-Selection P ostulate says that, at al l times , an individual electron is in a state prop ortional to a finite-rank pro jection, in the case at hand in a pure state, and that, after each time step dt , the pure state is c hosen according to a generalized Born Rule : The probabilit y that, in a time in terv al r t, t ` dt q , an electron jumps from a state, Π t , in the range of the pro jection P to the state P σ “ ˇ ˇ σ  σ ˇ ˇ (with a photon emitted by the electron from the sp ot corresp onding to the pixel lab elled b y σ and subsequently recorded on the screen Σ ) is giv en by α T r ` T σ Π t T ˚ σ ˘ ¨ dt, (31) while the probability for the state of the electron to remain in the subspace P H is given b y Prob ␣ Ran p Π t ` dt q Ă Ran p P q ( “ 1 ´ α ÿ σ P S T r ` T σ Π t T ˚ σ ˘ ¨ dt , (32) for dt small. This result agrees with the prejudices of the Copenhagen in terpretation of QM. As long as the electron do es not jump into a state ˇ ˇ σ  , for some σ P S , the dep endence on time t of the electron state Π t “ : Π p nj q t can b e found as follo ws. Consider op erators (proportional to rank-1 pro jections), denoted by π t , that solve the equation 9 π t “ ´ i ℏ “ H el , π t ‰ ´ 1 2 α ÿ σ P S ␣ π t , T ˚ σ ¨ T σ ( , with π t “ 0 “ Π 0 “ ˇ ˇ Ψ 0  Ψ 0 ˇ ˇ , (33) where Ψ 0 is the initial w av e function of an electron when it is released by the electron gun. Then Π p nj q t : “ π t } π t } , t ą 0 . The total probability , p esc , of an electron to esc ap e from the cavit y without hitting a pixel, as time t tends to 8 , can b e calculated by integrating the rate of decay of the probability , p p t q , for the state of the electron to remain in the range of the pro jection P , (thus p erp endicular to the range of the pro jection, P K , onto stationary states | σ ą , σ P S ), whic h is determined b y 9 p p t q p p t q “ ´ α ÿ σ P S T r ` T σ Π p nj q t T ˚ σ ˘ , t ě 0 , with p p t “ 0 q “ 1 . (34) One then finds that p esc “ lim t Ñ8 p p t q . (35) Electrons escaping from the cavit y will not emit photons and hence will not be detected on the ligh t-emitting screen Σ . But if α ą 0 then, with a p ositive probability 1 ´ p esc ą 0 , an electron jumps in to one of the states ˇ ˇ σ  , σ P S , emitting a photon that produces a flash of light on the screen Σ near the pixel marked with σ . After man y electrons hav e b een released by the electron gun to propagate through the ca vity Λ , the light-emitting screen Σ shows the in terference pattern exp ected in a double-slit exp erimen t; indeed, our results agree with what one finds in actual exp erimen ts. 14 F urthermore, the time when an electron arrives at a pixel σ P S to jump into a state ˇ ˇ σ  , emitting a photon, is a random v ariable whose la w is determined by ( 31 ) - ( 34 ). Equations ( 30 ) through ( 35 ) determine a pr ob ability me asur e on the space, P , of trajectories of states of electrons emitted b y the electron gun (somewhat analogous to the Wiener measure in the theory of Brownian motion). The treatment of the double-slit experiment presen ted here is an example of a quantum Poisson pr o c ess, as studied in more generalit y in [ 4 ]. One can go on to refine the exp eriment by , for example, calculating the effect of shining laser ligh t into the cavit y resulting in the emergence of ele ctr on tr acks (see [ 26 ]) and in the gradual disapp e ar anc e of interfer enc e p atterns on the light emitting screen. It deserv es to b e emphasized that the suc c ess and actual outc ome of this exp eriment is a c on- se quenc e of the dissip ative evolution of ensemble states. Without dissipation, i.e., for α “ 0 , electrons would not emit any photons, and the screen Σ w ould remain dark. A general treatment of measuremen ts in the non-relativisitic regime, within the E T H -Approac h to QM, will b e presen ted in a forthcoming pap er; (see also [ 25 ]). R ichar d F eynman famously claimed that nob ody understands Quantum Mec hanics and that the double-slit exp erimen t represents the “only mystery” at the heart of this w onderful theory . W e hop e that this note is a mo dest con tribution to a b etter understanding of Quan tum Mechanics and to unrav eling this “mystery . ” A c ko wledgements : W e are indebted to Carlo Alb ert and Henri Simon Zivi for plent y of useful discussions related to things discussed in this note and to Jakob Y ngv ason for asking many go od questions ab out our treatemen t of measuremen ts and challenging us to produce a sketc h of our ideas and results. JF gratefully and joyfully remembers his inn umerable encoun ters, interactions and joint efforts with his mentors and friends Israel Mic hael and Barry . — References [1] J. F röhlich and B. Sch ubnel, Quantum pr ob ability the ory and the foundations of Quan- tum Me chanics, arXiv:1310.1484, in: “The Message of Quantum Science,” Ph. Blan- c hard and J. F röhlich (eds.), Springer-V erlag, Berlin, Heidelb erg, New Y ork, 2015 [2] Ph. Blanc hard, J. F röhlic h and B. 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