Demystifying Quantum MechanicsThe “ETH Approrach”

“I leave to several futures (not to all) my garden of forking paths”J. L. Borges

Great minds of Quantum Mechanics

A. Einstein W. Heisenberg P.A.M. Dirac R. Haag

PSI, March 21, 20191

1Jurg Frohlich, ETH Zurich

Some quotes – for fun

• “If someone tells you they understand quantum mechanicsthen all you’ve learned is that you’ve met a liar.”

R. P. Feynman

• “Anyone who is not shocked by quantum theory has notunderstood it.”

N. Bohr

•“We have to ask what it means!”K. G. Wilson

In my opinion, only Wilson’s challenge remains significant.

****

In this lecture, I propose to find out which features of quantumtheory might be shocking at first sight. I will then attempt toexplain why they should not shock you, and what it all means.Incidentally, I don’t think I am a liar

Contents and creditsContents:

1. What this lecture will be about

2. Why might we be shocked by Quantum Mechanics?

3. Recap of the “Copenhagen Interpretation” of QM

4. The “ETH Approach” to non-relativistic QM

5. Observations of events in Quantum Mechanics

6. Summary and conclusions

7. [Appendix about relativistic quantum theory]

Credits:

I am indebted to my last PhD student Baptiste Schubnel for anenjoyable collaboration, to M. Ballesteros, Ph. Blanchard andM. Fraas for cooperation, and to these colleagues as well as manyfurther ones, including some of the “Bohmians”, various colleaguesat ETH Zurich, and D. Buchholz for useful discussions on QM.

1. What this lecture will be about

New Foundations of Quantum Mechanics are proposed: the

“ETH - Approach to Quantum Mechanics”

where “E” stands for Events, “T” for Trees, and “H” for Histories.This approach enables us to introduce a precise notion of “events”into Quantum Mechanics ( Haag), explain what it means toobserve an event by recording the value of an appropriate physicalquantity, and to exhibit the stochastic dynamics of states ofisolated open systems featuring events. It adds to the standardformulation of the theory two simple, but fundamental hypothesesconcerning the emergence of “events” and their effect on theevolution of states of such systems.

The “ETH - Approach to QM” results in a “Quantum Theorywithout observers”. It does away with “extensions of QuantumMechanics”, all of which have remained unacceptably vague.

A metaphor of Quantum Mechanics

Some say: “Shut up and calculate!” But let’s abandon this maximand try to open the “quantum black box” !

2. Why might we be shocked by Quantum Mechanics?

Many if not most people appear to be confused about the deepermeaning of Quantum Theory. Asking twenty-five professors of theoreticalphysics to explain their understanding of the Foundations of QM you arelikely to get ten different answers most of which contradict some orseveral of the other ones. – Worse, generations of students areindoctrinated with erroneous claims about QM; e.g., that theSchrodinger-picture dynamics of states and the Heisenberg-picturedynamics of “observables” are equivalent, ...

Soon one hundred years after the discovery of matrix mechanics byHeisenberg, Born, Jordan, and Dirac, this is quite shocking and representsan intellectual scandal, which we had better remove, as soon as possible!

Unfortunately, in 60 minutes I will not reach this goal! (To accomplishsomething valuable and lasting, one would have to organise a one- ortwo-semester course on these matters!)

My discussion will be fairly non-technical, yet somewhat daring. I wouldsay, it represents an exercise in “natural philosophy”.

A lightening review of unusual or strange features ofQuantum Mechanics

I Bell’s Inequalities: Quantum-mechanical correlations between valuesof physical quantities (e.g., polarisations of a “Bell pair” of photons)measured in two independent labs violate Bell’s inequalities, in thesense that the numerical range of the qm correlation matrix exceedsthe one of the corresponding classical correlation matrix; ( Tsirel-son). – Has been checked experimentally!

I Kochen-Specker Theorem: Shows that the following twoassumptions are incompatible with Quantum Mechanics:• All physical quantities have definite values at any given time.• The values of physical quantities are independent of the

measurement context.

Example: Squares of spin-components of a spin-1 particle. –∃ experiments!

Conclusions: 1. 6 ∃ “local” theories of commutative hidden variablesreproducing the predictions of Quantum Mechanics (if dim(H) > 2).2. Bell-type “non-locality” of Quantum Mechanics.

The Schrodinger Equation does not describe the evolutionof states of isolated systems featuring events

I Wigner’s Friend:

Courtesy of Frauchiger & Renner

Agent F measures z-component of spin of a silver atom, pepared instate 1√

2(| ↑〉+ | ↓〉), with outcome s ∈ ↑, ↓; state of spin of silver

atom after measurement given by ψS = |s〉. Instead, W regards thelab containing F as one big quantum system, L, in a state evolvingaccording to a Schrodinger Equation ⇒ Assigns pure state

ψL = 1√2

(|s =↑ ;D+,F+〉L + |s =↓ ;D−,F−〉L)

to L, which could be tested by a suitable measurement applied to L.This leads to a contradiction with the description given by F andshows that W should not use the Schrodinger Eq. to describe theevolution of the state of L! (More sophisticated versions proposedby Lucien Hardy, Popescu et al., Frauchiger-Renner ...)

And Quantum theory cannot be fully predictive, because ...

Setup of a Gedanken-Experiment ( Faupin-F-Schubnel):

↑ ↑

Q = sub-system “confined” to Ω Particle P propagates to the right →

Time evolution of P ess. indep. of Q (cluster props.) → Application:

︸ ︷︷ ︸Q:=spin filter∨ particle P′⊂Ω, shaded area := ess. supp of orbital wave function of P

... Quantum theory is fundamentally probabilistic –in spite of the deterministic nature of the Schrodinger Eq. –Temporary assumptions (leading to a contradiction):

I P and P ′: Spin- 12 particles prepared in a spin-singlet initial state;

spin filter prepared in a poorly known initial state not entangledwith initial state of P ′ and P.

I Dynamics of state of total system fully determined by Schrodingerequation. In particular, initial state of spin filter determines whetherP ′ will pass through it or not, (given that the initial state of P ′ ∨ Pis a spin-singlet state, with P ′ and P moving into opposite cones).

I Correlations between outcomes of spin measurements of P ′ and ofP are as predicted by standard quantum mechanics, (relying on the“Copenhagen interpretation”).

Fact: Heisenberg-picture dynamics of observables (spin, etc.) referring toP is ess. independent of dynamics of Q := P ′ ∨ spin filter. Thisfollows from our choice of initial conditions & cluster properties of timeevolution! Hence spin of P ess. conserved before measurement ⇒

Expectation value of spin of P ≈ 0,∀ times!

But this contradicts the third (last) assumption stated above!

Relativistic theories are not fully predictive, because ...Space-time with an event horizon. (Observer sits at “Present”; is unawareof dangers lurking from outside his past light-cone; he might get killed at†. Events 1 & 2 are space-like separated; event 3 is in the future of 2)

︸ ︷︷ ︸t0: time right after inflation→ event horizon⇒ initial conditions not fully accessible!

Past = History of Events / Future = Ensemble of Potentialities

This fundamental structure must be retained in Quantum Mechanics!

3. Recap of the “Copenhagen Interpretation” of QM

In the Copenhagen Interpretation of QM the state/wave function,Ψ, of a physical system S does not have an “ontological status”.According to Born and Heisenberg : Ψ is merely a mathematicalobject enabling us to predict probabilities of different possiblevalues a physical quantity X can take in case this quantity ismeasured at a certain sharp time t. If the measured value of X attime t is given by ξ then the state of the system immediately aftermeasuring X must be taken to be an eigenstate corresponding tothe eigenvalue ξ of a linear operator X (t) representing X at timet, (“projection postulate”).The probability of measuringξ ∈ spec(X ) ≡ spec(X (t)) is given by Born’s Rule.

In the absence of measurements/observations, the q.m. state Ψof a physical system and its evolution in time mean: Nothing !There is essentially no invariant information encoded in Ψ and itsSchrodinger-pict. time evolution, Ψ(t) = U(t, t ′) Ψ(t ′), t, t ′ ∈ R ,beyond information on energy spectra (of generators of U).

Evolution of states when measurements are made

In the Heisenberg picture, states of a system S do not evolve intime, except at times when a physical quantity, X , is measured,and then they must be changed in accordance with the

“Projection Postulate”:Assume, for simplicity, that X has pure-point spectrum, (possiblyonly finitely many eigenvalues). Suppose that the operator X (t)represents X at time t, and consider its spectral decomposition,

X (t) =∑

ξ ∈ spec(X )

ξ Πξ(t)︸ ︷︷ ︸spect.proj .

Let Ψ be the state of the system S right before X is measured attime t. If ξ ∈ spec(X ) is the measured value of X then the stateto be used to make predictions of measurement outcomes at latertimes (> t) is determined by the following

Critique of the Copenhagen Interpretation

Projection Rule:

Ψ 7→ Πξ(t)Ψ/‖Πξ(t)Ψ‖, (5)

where ‖Πξ(t)Ψ‖2 is the Born probability of measuring the value ξ.∃ natural generalisation of Born’s Rule, due to Luders, Schwingerand Wigner, that promises to determine the probabilities of entirehistories of outcomes in arbitrarily many measurements of differentphysical quantities made at subsequent times, (“LSW rule”) ...

Critique:

I Mysterious/ominous role of observers who performmeasurements. Role of “free will” of observers?

I Act of measuring a physical quantity not described by theory.

I Measurements take place instantaneously → unphysical!

I If measurement-acts of observers are not specified theory istotally un-predictive.

4. The “ETH Approach” to Quantum TheoryNext, we address the question of what is meant by “events” featured byisolated systems, and of how they can be recorded (in direct/projectivemeasurements/observations). I sketch what I call the “ETH Approach”to QM. For simplicity I consider non-relativistic theories:Let S be an isolated physical system. Pure states of S are given by unitrays in a separable Hilbert space HS ; general states by density operators,ω, acting on HS , with ω(A) := Tr(ω · A), for any bd. operator A on HS .

Time is a fundamental quantity in n.r. physics. The time axis is givenby R. Let’s suppose the present time is t0, and let I be an arbitraryinterval of future times, i.e., I ⊂ [t0,∞).

Definition: “Potential future events” in an isolated system S –“potentialities” – are described by certain orthogonal projections actingon HS & associated with future time intervals. The ∗algebra generatedby all “potential future events” associated with an interval, I , of futuretimes is denoted by EI , and we define

E≥t :=∨

I⊂[t,∞)

EI , and E :=∨t∈RE≥t‖·‖, (2)

The “Principle of Diminishing Potentialities”(where the algebras E≥t , t ∈ R, are assumed to be weakly closed!2)

By definition,

EI ⊇ EI ′ if I ⊇ I ′ , E≥t ⊇ E≥t′ if t ′ > t .

The “Principle of Diminishing Potentialities” (PDP) is the statementthat

E≥t ⊃6=E≥t′ , whenever t ′>t ≥ t0 (3)

In Quantum Mechanics, an isolated open system S , including itsHeisenberg time-evolution, can be defined in terms of a “filtration”,E≥tt∈R, of algebras of future potentialities satisfying PDP. –(Examples!)

Given a state, ω, of S , we set

ωt := ω|E≥t, i.e., ωt(A) = ω(A) , ∀A ∈ E≥t . (4)

2Considering weakly closed (von Neumann) algebras is convenient, becausethe spectral projections of any element of the algebra will then also belong tothe algebra!

EventsNote that ω might be a pure state on E . But, since E≥t ⊂

6=E , ∀t <∞,

ωt will generally be a mixed state on E≥t ; (entanglement!). Thisobservation opens our eyes/minds towards a clear notion of what mightbe meant by “events” and to a theory of direct/projective observationsand recordings of “events”.

To render the above definition more precise, we say that a “potentialfuture event” is given by a family, πξ|ξ ∈ X, of disjoint orthogonalprojections contained in an algebra E≥t , for some t ≥ t0, (t0 = time of“present”), with

∑ξ∈X πξ = 1.

In accordance with the “Copenhagen interpretation” of QM, it appearsnatural to say that a potential future event πξ|ξ ∈ X ⊂ E≥t actuallyhappens in the interval [t,∞) of times iff

ωt(A) =∑ξ∈X

ω(πξ Aπξ), ∀A ∈ E≥t , (5)

i.e., no off-diagonal elements appear on the R.S. of (5) → ωt is anincoherent superposition of states in the images of the projections πξ!

Implications of PDP and of entanglementIn RQFT, PDP is a consequence of Huygens’ Principle (Buchholz) andcan be understood from the following drawing (see blackboard):

PDP → Theory of q.m. Measurements:Suppose that a physical quantity X is measured at time t or later. Anoperator X (t) =

∑ξ ξ Πξ(t) ∈ E≥t then represents X at times ≥ t.

Following “Copenhagen”, one would argue that, ∀A ∈ E≥t :

ωt(A) =∑

ξ∈spec(X )

ωt(Πξ(t)AΠξ(t)) (6)

⇒ ωt([Πξ(t),A]) = 0,∀ξ, hence ωt([X (t),A]) = 0 . (7)

The centralizer of a state and its center

Thus, ωt is an incoherent superposition of eigenstates of X (t), and X (t)belongs to what is called the “centralizer” of the state ωt .

Next, we render the meaning of Eq. (5) more precise.

Let M be a von Neumann algebra, and let ω be a state on M. Given anoperator X ∈M, we set

adX (ω)(A) := ω([A,X ]) , ∀A ∈M .

We define the centralizer of a state ω on M by

Cω(M) := X ∈M|adX (ω) = 0

Note that ω is a normalized trace on Cω(M) ... ! The center, Zω(M), ofCω(M) is defined by

Zω(M) := X ∈ Cω(M)| [X ,A] = 0, ∀A ∈ Cω(M) . (8)

We are now prepared to introduce a notion of (actual) “events”.

Events happening around time t

Let S be an isolated open physical system.

Definition: If ωt is the state of S on the algebra E≥t , an “event” ishappening at time t iff Zωt (E≥t) contains at least two non-zero orth.projections, π(1), π(2), which are disjoint, i.e., π(1) · π(2) = 0, and

0 < ωt(π(i)) < 1 , for i = 1, 2 .

For simplicity suppose that Zωt (E≥t) is generated by a family of disjointorthogonal projections πξ|ξ ∈ Xωt, with Xωt = spec[Zωt (E≥t)] acountable set.

“Axiom”: (2nd Law of QMTh) If card(Xωt ) ≥ 2 and ωt(πξ) 6= 0, for atleast two different points ξ ∈ Xωt , then the state ωt must be replaced byone of the states ωt,ξ := ωt(πξ)−1 · ωt(πξ(·)πξ) , for some ξ ∈ Xωt withωt(πξ) 6= 0. The probability, probt(ξ), for the state ωt,ξ to be selected asthe state of S right after time t when the event has happened is given by

probt(ξ) = ωt(πξ) − Born′sRule (9)

Metaphoric picture of time evolution of states in QMIn the “ETH approach”, the time-evolution of states of a phys. system Sis apparently described by a stochastic branching process, with branchingrules as determined by the above “Axiom”, Eq. (9); (6= “decoherencemumbo-jumbo”!) The following figure illustrates this claim:

t: time, ρ: initial state of S

E : “Events”, T : “T rees (of possible states of S), H: “H istories”(of events/states) – probs. of “H istories” deteremined by Born’s Rule

“I leave to several futures my garden of forking paths”Note that, in an autonomous isolated open system S , all the algebras(E≥t)t∈R are isomorphic to one another, with E≥t ' N , where N is a

certain “universal algebra”. We define

XS :=⋃

ω∈ phys states of S

Zω(N ) ,

the “non-commutative spectrum” of the system S .

The above picture “shows” that a continuous-time stochastic branchingprocess, with state space XS and transition probabilities given by Born’sRule, describes the time evolution of states of S . (The Schrodingerequation can only be used to describe the dynamics of states of (veryboring) closed systems not featuring any events !)

A trajectory of states described by such a process is called a “history”.The space of histories of a system S is equipped with probabilitymeasures, µω, where ω is an “initial state” on the algebra E which S hasbeen prepared in; (see blackboard). A measure µω assigns a probabilityto every history of events that may evolve from the initial state ω.(Naive implementation: Lindblad evolution of density matrices)

5. Detection of eventsWe have characterised an isolated open system S in terms of a filtrationof algebras

E≥tt∈R ,

with (see PDP)

E≥t ⊃6=E≥t′ , whenever t ′ > t (10)

The flow of time in S , (i.e., the time evolution of S in the Heisenbergpicture) is encoded in the proper embeddings (10), which, in an auto-nomous system S , are completely determined by its Hamiltonian.

However, the characterisation of S given in (10) is incomplete! Toretrieve physical information from (10) and from our definition of events,we must specify operators representing “phys. quantities” characteristicof S and – when observed/measured – may signal the occurrence ofevents. Let

OS := Xι|ι ∈ IS (11)

be a list/set of abstract linear operators representing physical quantitiescharacteristic of S ; (usually, OS isn’t a linear space, let alone an algebra).

Measurements of physical quantitiesFor any operator Y ∈ OS and any time t, we specify a concrete self-adjoint operator Y (t) ∈ E≥t representing Y at time t; (i.e., ∃ a repr. ofOS by operators on HS , ∀t ∈ R). For an autonomous system S , theoperators Y (t) and Y (t ′) are conjugated to one another by thepropagator of S .

Suppose that, at some time t, an event happens; i.e., ∃ a partition ofunity, πξ|ξ ∈ Xωt ⊆ Zωt ⊂ E≥t , by disjoint (commuting) orthogonalprojections, as above, containing ≥ 2 elements with positive probability ofoccurrence representing possible events (one of which actually happens).

Let Y ∈ OS , and let Y (t) =∑η∈spec(X ) ηΠη(t) (spectral dec. of Y(t))

be the operator epresenting Y at time t. If the “distance”3

dist(Πη(t), 〈πξ|ξ ∈ Xωt 〉

)is “very small” , ∀η ∈ spec(Y ) , (12)

then we say that the physical quantity Y ∈ OS is observed/measured attime t or later, because the state of S just after time t is then an approxi-mate eigenstate of Y (t). The measurement of Y is a signal of an eventhappening at time t. ...

3defined in terms of “conditional expectations”

6. Summary and conclusions

1. So far, the “ETH Approach to QM” is merely a conceptual frame-work. It is not effective (yet?) when one wants to do concrete q.m.computations. However, it clarifies the Foundations of QM and ithopefully helps to dispel those persistent confusions that surroundthem. It leads to plenty of very concrete and interesting-lookingproblems in functional analysis and probability theory (!).

2. I have spent quite a lot of time trying to develop a relativisticversion of the “ETH Approach”.In fact, if we think that, fundamentally, isolated systems should beautonomous we cannot avoid turning towards a relativisticformulation of QM. So far, things look good! But I cannot claim tohave worked out all the details, yet. Tentative conclusions:

I Massless modes (electromagnetic- and gravitational field) playa fundamental role in a quantum theory of “events” and ofdirect (projective) observations/measurements – just like inRelativity Theory !)

I Huygens’ Principle (as formulated by Buchholz), and hence theeven-dimensionality of space-time are crucial ingredients.

Theory of Indirect Measurements

3. The statistics of long sequences of events observed in an isolatedopen system S tends to reveal important information about physicalquantities not accessible to direct observation. This is the basis of a“Theory of Indirect (Weak) Measurements” (pioneered by Kraus,Elizur and Vaidman, Maassen and Kummerer, and others). Thistheory rests on a clever exploitation of ”Entanglement”, combinedwith statistics. It leads to an understanding of phenomena such asthe emission of photons from atoms, radioactive decay of nuclei, orthe emergence of particle tracks in a cloud chamber, ... Muchrecent work!Deeper problems are, however, encountered when one studies what Ihave talked about in this lecture: The Quantum Theory of Eventsand of direct (projective) measurements of physical quantities; andthe embedding of Quantum Theory into Relativity Theory.

I hope I have convinced you that there are reasons to be optimistic aboutbeing able to make progress in this direction.

Thank you for your attention!

7. Appendix about relativistic quantum theoryWhat will a relativistic formulation of quantum theory tell us aboutspace-time?

First I assume that space-time is Minkowski space, M4, and, immodestly,that my own proper time is the time of the Universe.

Worldline of JF ↑

A Theorem of D. Buchholz

TheoremIn an RQFT with massless particles, such as photons, the algebra, E≥Pt ,of all physical quantities (“observables”) potentially measurable in thefuture of the space-time point Pt is of type III1, and E ′≥Pt

∩ E≥Pt0is of

type III1, too, for arbitrary times t0 < t.

This result is a consequence of “Huygens’ Principle” (in the jargon ofBuchholz): Photons from the region O will asymptotically escape alonglightcones in the future, V+

Pt0, of Pt0 but below V+

Pt. We cannot catch up

with them, anymore, if we have missed them just after they have beenemitted. Thus, the “Principle of Shrinking Potentialities” (PSP) holds inthe form proposed in Eq. (3) of the last Section:

E≥Pt0⊃6=E≥Pt , for t > t0 , (8)

and we could now follow the arguments outlined in Sect. 5. However, Idon’t like to be in the center of the Universe; so, let’s take JF out of thepicture! Before knowing better I propose a formulation of relativisticlocal Quantum Theory with roughly the following features:

A tentative formulation of relativistic local quantum theory

Let M be some (Hausdorff) topological space. We consider a fibrebundle, qmF , with base space given by M and fibre above a pointP ∈M given by an ∞-dimensional ∗-algebra E≥P (whose weak closure,also denoted by E≥P , in any representation determined by a “physicalstate” can be expected to be of type III1). All the algebras E≥PP∈Mare assumed to be isomorphic to one another.4

Definition:We say that a point P0 ∈M is in the past of a point P ∈M, written asP0 ≺ P, iff ∃ an injection map ι : E≥P → E≥P0 , (enabling one to identifyE≥P with a subalgebra of E≥P0 ), and(

ι(E≥P))′∩ E≥P0

is inifinite-dimensional. The relation ≺ introduces a partial order on M.If P0 6 ≺P and P 6 ≺P0 then we say that P0 and P are space-likeseparated, written as P0 X P. The relations “≺” and “X” determine a“causal structure” on M.

4This could be generalized by introducing sheaves of algebras

What are “events”?Let ω be a state on the algebra

E≥Σ :=∨

P′∈Σ

E≥P′ ,

where Σ is a space-like hypersurface contained in M containing a pointP ∈M.

Definition:We say that an “event” happens in P iff the center ZωΣ

(E≥P) ≡ ZPωΣ

ofthe centralizer, CωΣ

(E≥P), is non-trivial and contains at least twoprojections, ΠP

1 and ΠP2 with the property that

0 < ωΣ(ΠPi ) < 1, for 1 = 1, 2 .

Let X PωΣ

denote the spectrum of ZPωΣ

.

“Axiom” (compatibility – locality): If two points, P and P ′′, of M arespace-like separated, and “events”, ΠP

ξ and ΠP′′

η , actually happen in Pand P ′′ then

[ΠPξ ,Π

P′′

η ] = 0, ∀ ξ ∈ X PωΣ

and all η ∈ X P′′

ωΣ. (9)

The compatibility axiom

The above axiom is illustrated in the following figure.

However, projections describing events happening in P ′ and P do notcommute in general, since P ′ is in the past of P.Next, we propose to describe histories of events. We choose a space-likesurface Σ in M with the property that some bounded subset of Σ lies inthe past of a point P ∈M, as shown in the following figure.

Histories of events

We suppose that a state ωΣ is prescribed in the past of a space-likesurface Σ (choice of initial conditions). Our task is to find out whetherall events in the past of P but in the future of Σ (so-called “histories”),together with the state ωΣ, uniquely determine a state, ωP , on thealgebra E≥P and, given ωP , to find out whether an event happens at P.

Probabilities of histories of events

For this purpose, we assume inductively that all such events areknown: Let P1,P2, ..., be all points in the past of P but not in thepast of any point on Σ with the property that, given initial condi-tions corresp. to ωΣ, an event has happened at Pi , i = 1, 2, ....With any of these points we can then associate an orthogonalprojection ΠPi

ξi, ξi ∈ XPi

ωPi= spec

(ZPiωPi

). We define “history

operators”

H(P|ωΣ) :=−→∏

i=1,2,...

ΠPiξi, (10)

where Pi is either in the past of Pi+1, or Pi and Pi+1 arespace-like, ∀i = 1, 2, ... Thanks to the compatibility - localityaxiom the operator H(P|ωΣ) is well-defined! We then set

ωP(A) := prob(H(P|ωΣ)

)−1ωΣ(H(P|ωΣ)AH(P|ωΣ)∗) , (11)

∀A ∈ E≥P , where

Events and “geometrical structure”

prob(H(P|ωΣ)

):= ωΣ

(H(P|ωΣ) · H(P|ωΣ)∗

)Generalized Born Rule

We are now able to answer the question whether an event happensin the space-time point P:An event happens in P iff the center ZωP

(E≥p) of the centraliser ofωP is non-trivial and contains ≥ 2 disjoint orthogonal projectionswith strictly positive probabilities in ωP , as predicted by Born’sRule.This completes the induction step.

The “compatibility – locality axiom” can be expected to yieldnon-trivial constraints on the geometry in the vicinity of twospace-like separated points, P and P ′′, if it is known that ∃ eventsin P and P ′′ localised in explicitly known regions in the future of Pand of P ′′, respectively. But this will have to be investigated morethoroughly in the future.

Summary and conclusions

I As in the genesis of Special Relativity, the e.m. field, as wellas Huygens’ Principle play key roles in the genesis of aQuantum Theory solving the “measurements problem” –which may not have been properly appreciated, so far.

I As in the genesis of General Relativity, the causal structure ofspace-time and its curvature play key roles in QuantumTheory:

I The non-commutative nature of Quantum Theory and the“compatibility-locality axiom” determine a “causal structure”(but not a Lorentzian metric) on M.

I Thanks to the “Principle of Shrinking Potentialities” (PSP)and the natural presence of an “arrow of time” in the “ETHapproach” to Quantum Theory, the “Information Paradox”and the “Unitarity Paradox” appear to dissolve.

I thanks you for your attention !