The magical"Born Rule"&quantum"measurement": Implications for Physics

I. The arena of quantum mechanics and quantum field theory is the abstract, unobserved and unobservable, M-dimensional formal Hilbert space [not equal to] spacetime. II. The arena of observations and, more generally, of all events (i.e. everything) in the real physical world, is the classical 4-dimensional physical spacetime. III. The"Born Rule"is the random process"magically"transforming I. into II. Wavefunctions are superposed and entangled only in the abstract space I., never in spacetime II. Attempted formulations of quantum theory directly in real physical spacetime actually constitute examples of"locally real"theories, as defined by Clauser&Horne, and are therefore already empirically refuted by the numerous tests of Bell's theorem in real, controlled experiments in laboratories here on Earth. Observed quantum entities, i.e. events, are never superposed or entangled as they: 1) Exclusively"live"(manifest) in real physical spacetime, 2) Are not described by entangled wavefunctions after"measurement", effectuated by III. When separated and treated correctly in this way, a number of fundamental problems and"paradoxes"of quantum theory vs. relativity (i.e. spacetime) simply vanish, such as the black hole information paradox, infinite zero-point energy of quantum field theory and quantization of general relativity.

1 Introduction: Quantum to Classical I.Quantum theory lives in abstract Hilbert space: H M , which, more often than not, is infinite-dimensional H ∞ [1].Complex quantum wavefunctions, ψ (= vectors in H M ), for N discrete quantum entities are defined in configuration space of 3N dimensions (if their spins are zero), ψ = ψ(q 1 , ..., q 3N ).The quantum state, i.e. the value of ψ, is determined by simultaneously giving all numerical values of the 3N variables (q 1 , ..., q 3N ).The time-dependence of ψ is only implicit, as t neither is an operator nor a variable in configuration space, but merely a parameter, where ψ t = exp −itH/h ψ 0 , and this "dynamics" occurs in abstract, complex Hilbert space, not in spacetime.There is no spacetime description of ψ, so relativistic causality is not definable.The quantum states are normalized, |ψ(q 1 , ..., q 3N )| 2 dq 1 ...dq 3N = 1 (i.e.ψ lie at a point on the surface of the unit sphere in Hilbert space).
II. Classical physics lives in four-dimensional physical spacetime: the Lorentzian manifold L 4 .III.On "measurement" the "Born Rule" [2] is postulated to irreversibly, instantaneously and randomly map H M into specific points (= events, i.e. particular outcomes) in L 4 , with calculable probabilities.The joint probability density of finding N "particles" in N detectors in real space at time t (i.e. in L 4 ) upon "measurement" is calculated by |ψ(r 1 , ..., rN )| 2 .Observe that there is no real spatial dependence for ψ until this "measurement".The "fundamental" statistical character of "quantum theory" is actually only introduced here in III., due to "Born".Everything we ever measure (e.g. using laboratory "detectors"), perceive or experience occurs in spacetime, but it is only the Eigenvalues that can be observed in L 4 , the quantum Eigenfunctions still reside in H M (they constitute bases there) even after "Born".The "Born Rule" is "magical" in the sense that there is no physical dynamics underlying it and that it transforms unobservable H M into observable events in L 4 .On "measurement", the wavefunction ψ, with complex quantum amplitudes c n = n|ψ in Eigenbase |n , randomly "collapses" ("jumps") to one of the Eigenfunctions |n of the, for observables, allowed Hermitian operators Ôn (assuring real Eigenvalues o n ) Each individual "jump", ψ → |k , resulting in the observed Eigenvalue, o k , is a priori postulated to occur randomly, but with statistical probability (over many identical measurements) given by It is not the operators themselves that are observed, they only operate in Hilbert space not in spacetime.It is the Eigenvalues that are observed, and then only indirectly as a result of "measurement", Eq. ( 1), transforming I. into real events in II.as a result of III.However, we only infer I. and III.indirectly through observations, experiments and experiences in II.-the only world we have direct access to.Observed observables (= events) live in L 4 .Quantum entities solely live in H M and have no classical properties.The "measurement" transforms the infinitely many abstract quantum potentialities in H M into perfectly mundane actual occurrences in L 4 .Consequently, this shows that "decoherence" [3] cannot be a solution to the "measurement problem" [4], [5], [6], [7], [8], in quantum mechanics: i) It does not realize any objective outcome (unlike "Born"), ii) And if it did, it would mean that our classical world would be manifestly non-local (as decoherence is based purely on the wholly deterministic non-local "dynamics" of I.) In pure quantum theory (without "Born") there cannot be any mixed states, as probability presumes prior measurement.
Hence we see that statistically correlated observed "quantum" non-locality in spacetime in practice actually only arises through the magical "Born Rule".However, it still poses serious problems, as the results of measurement (after "Born") are objective classical events in L 4 (e.g.data printouts on paper), but "simultaneous" is not relativistically invariant: In a canonical entangled-pair experiment, with correlated observables A and B at either end, for an observer moving relative to the lab (with any non-zero velocity v, however small, | v |= ǫ > 0) A is prior to B if v > 0 (A is the "cause" of B and the "Born collapse" is not instantaneous), but B is prior to A if v < 0 (B is the "cause" of A and the "Born collapse" is not instantaneous), if A is simultaneous with B in the lab-frame (v = 0) [9].The problem is that the "Born Rule" is formulated in an absolute frame, the one where v = 0.
"Reality" occurs only in the spacetime of events -where the actual events are the fundamental, relativistically invariant and irreducible building blocks of objective reality [10].It is only here, in II., that all experimental results, and everything else we ever perceive, actually occur.That is why Bohr was fond of saying: "There is no quantum world" [11] only an abstract quantum algorithm, I. together with III., allowing us to relate experiences in the real world II.-the only one.There is no "quantum" reality, there are no "quantum" events, only a classical reality and classical events.Every time a "quantum" probability is calculated, it is really a result of "Born" III., not of pure (unmeasured) quantum theory I. Likewise, there are no "quantum" particle reactions in spacetime, only observed consequences in L 4 .So, only II. is really real, I. and III.merely an abstract, and unobservable machinery, very much like a black box we cannot peer into, but with observable inputs and outputs.In a very real sense, the "quantum world" is operationally built up of events in our real world, not the other way around.Reality does not occur in Hilbert space H M .This also means that there are no fundamental quantum entities in spacetime, only in (unobservable) Hilbert space.
The interaction in quantum theory I. is non-local in non-relativistic quantum mechanics (e.g.Schrödinger) and local in relativistic quantum theory (Dirac/quantum field theory), but the wavefunction is non-local in both (but only implicitly, in abstract Hilbert space, not in spacetime).
Wavefunctions are entangled [12] only in Hilbert space, not in real physical spacetime.
The dynamical real spacetime itself (II.) is local.The entanglement superposition (in H M ) is broken by the measurement, I.

III.
−→ II., hence there is never any non-causal entanglement in real spacetime L 4 .Only classical particles and fields are defined directly in L 4 .
III. is non-local [13] in real spacetime.It correlates space-like separated events in our real world.But particles "manifest" as events only as a result of "measurement" (through the "Born rule") -it is therefore fundamentally wrong to assume that they separate and travel, moving apart in real physical space, to the detectors from the source while unobserved.(In Hilbert space they neither separate nor are "far apart" as they are described by the global wavefunction in H M .)The Bohm version [14] of the EPRgedankenexperiment [13] disregards the actual "measurement", as the spatial part (ψ space ) of the total wavefunction is omitted/implicit.As we know today, it cannot be factored, ψ tot = ψ tot (q 1 , q2 , s) = ψ space ψ spin , as ψ tot is global and depends on both quantum entities in an entangled, not factorizable, way, Only quantum entities that do not interact and have never interacted may be factorized.The "measurement" ("Born Rule") collapses both space- [13] and spin-parts [14] of ψ tot (which due to the spin-1/2 degree of freedom in this case lives in H 12 ) at once.
Unobserved quantum entities are always (merely abstract) "waves" in H M , observed quantum entities are always "particles" manifested as events in L 4there is never any "particle-wave-duality" in either space.Specifically, there is never any causal "quantum-wave" propagation in spacetime.This means that classical physics, II., can never be the limit h → 0 of "pure" quantum theory, I.
The quantum description of a system of N entities (for N > 1) cannot be embedded in real spacetime [15] -actually, the very formulation of quantum mechanics precludes its embedding in spacetime for N > 1.For example, the quark-and gluon-fields (M = ∞) interacting "in" a proton never objectively exist as particles in spacetime -only when a "measurement" is made, for example using deep inelastic scattering, the results of an "electron-quark collision", mediated by a photon-or Z 0 -field (not particle), in Hilbert space (H ∞ ) through "Born" becomes translated into some experimental signal in real 4D spacetime (L 4 ).
Abstract configuration space (q 1 , ..., q 3N ) and physical space (x, y, z) can coincide only if there is only one (spinless) quantum entity, actually measured at (x, y, z), in the entire universe (this unfortunately precludes any interactions, experiments and observers), otherwise they are distinct -and actually the origin of most confusion.
"Born", III., is just a random sampling, upon "measurement", of the abstract, globally ever present and completely deterministic Hilbert spacemeaning that the (unobservable) "quantum world" is completely deterministic -determined by all the unobserved variables in configuration space, while the real world is local and uncertain in part due to our ignorance of the global/non-local "hidden variables" of configuration space.
We thus see that even orthodox quantum mechanics already, in a sense, has "hidden variables" in fact always present in configuration space, which globally keeps track both of what has happened, and also of everything that can ever happen -potentially including even the "decisions" of observers.
Even for two free quantum entities, that have ever interacted in the past, measurement on one affects measurement on the other.For example, in energy Eigenbase an energy Eigenvalue measurement on particle one depends on the energy measurement on the other, regardless of their separation in L 4 (this being just a special example of entanglement of two presently non-interacting quantum entities), where |A| 2 + |B| 2 = 1.This entanglement persists indefinitely until "measurement" ⇒ "Born" ⇒ "collapse of the wavefunction" ⇒ probability ensembles in spacetime, L 4 .
In dynamical collapse models, energy is not strictly conserved.There exists no continuous Noether symmetry in time, as the dynamical collapse is irreversible.This is side-stepped in "Born" collapse as it is non-dynamical -"magical".
Even if the actual "measurement" is assumed to take a finite time, we still obtain a causal paradox as the "measurement" at the other end is not connected to the first by a Lorentz transformation if the opposite ends are spacelike-separated in spacetime [9].

Quantum Theory
Quantum theory for N, even non-interacting, spinless quantum entities lives in H 3N , an abstract, complex, linear (vector) space.The evolution in H 3N is continuous, linear [21], reversible, non-local (but merely abstractly/implicitly so) and deterministic (describable by differential equations).The wavefunction is not defined until/unless all points in configuration space (q 1 , ..., q 3N ) are used as input.For N > 1, quantum theory cannot be embedded in real physical spacetime L 4 [15].The spacetime description is only appropriate for our detectors and observations in L 4 -not for the abstract theory supposedly "underlying it all" in H M .No quantum fields ever "permeate" spacetime.

Classical Physics
Events define, and also constitute, dynamical classical spacetime, L 4 .The dynamics is continuous, nonlinear, reversible, local, causal and deterministic (describable by generally relativistic covariant differential equations).This nonlinear dynamics evidently cannot result from "pure" linear quantum theory alone.

The magical "Born Rule"
The "Born Rule" is discontinuous, nonlinear, irreversible (entropy increasing [1]), non-local (explicitly -assumed to be instantaneous in spacetime), noncausal and postulated to be intrinsically/fundamentally random/probabilistic (e.g.giving no possibility of superluminal signalling despite the, now physical, non-locality in spacetime).It is not describable by differential equations, or in any other dynamical way, instead being "magical".Observe that "Born" kills all superpositions (including entanglements) as the end result is a classical probability ∝ |ψ| 2 , not longer any interfering amplitudes/wave functions.This also means that there can be no superpositions in spacetime (or of spacetimes), as probabilities do not interfere, only add, forbidding any "quantum spacetime".It maps H M "Born" − −−− → into specific outcomes (= events) in spacetime, L 4 .Observe that the Eigenvalues are the physical (and random) "observables" in L 4 , never the Eigenfunctions themselves (they perpetually live in abstract, complex Hilbert space).Expectation values, Ô = ψ| Ô|ψ , are statistical averages of many measured Eigenvalues in identically "prepared" systems, |ψ , and are predictable in a statistical sense only.
As Bell showed, all measurement results can ultimately be boiled down to position results [22] which, together with time, are the events in spacetime, L 4 .
A hypothetical free (non-interacting and spinless) single particle can be represented in physical spacetime only when (q 1 , q 2 , q 3 ) = (x, y, z) = the location of the detector in L 4 at real "measurement" of the particle, and then by an infinite wave-train with equal probability (= 0) to be anywhere (at "measurement").If instead regarded as semi-localized wave-packets (infinite superposed sum of different wave-trains) they will: i) disperse, ii) not have a unique energy or propagation speed, meaning that there would be no reason they should arrive at a detector at a calculable time.Hence, even single quantum "particles" cannot travel through spacetime as microscopic "bullets".The momentum "conservation" always assumed, e.g to ensure spatial correlation of entangled pairs, actually occurs in abstract Hilbert space, not physical spacetime.Neither particle in the "pair" exists anywhere in spacetime until/unless "measured".Quantum "particles" have no trajectories in spacetime, and if N > 1 the evolution cannot be embedded in spacetime anyway.
From quantum theory alone, there is thus no reason that detection of both "particles" of an entangled pair should be detected simultaneously, equidistant from the source.This can, at best, hold only in the mean as: i) each individual measurement event is random (postulated so by "Born"), ii) the probability of "Born" are weighted statistical means of very many individual (random) measured events.
The locality assumption only applies to real physical spacetime, not to abstract Hilbert space where obviously everything is non-locally interconnected through the global configurationi.e., "unmeasured" quantum theory does not respect Lorentz invariance -but this is irrelevant as Lorentz invariance is only observed in spacetime and H M itself is unobservable in principle.
4 Quantum Space = Real Spacetime: Some Physical Consequences The non-locality in Hilbert space is an abstract "unphysical", ever-present, global non-locality.But it becomes a non-locality in real physical spacetime through "Born's Rule".The non-locality of measurement is evident already in the 1-particle case, as pointed out already very early on by Einstein [23], but it becomes experimentally testable in N-particle entangled states.Originally, tests had N = 2, [24], [25], and all "locally real" [26] models formulated in real spacetime, L 4 , are soundly falsified by these tests [15], including quantum mechanics and quantum field theory formulated in real spacetime.Hence, a truly relativistically invariant formulation of quantum theory in spacetime, which includes "measurement", could never be compatible with the non-locality of nature already observed in these tests, as correlations in real outputs of real experiments in our real world II.

Consequence 1: No "Quantized" General Relativity
Apart from having completely different mathematical structures, quantum theory and general relativity "live" in completely different spaces, which means that "Quantum General Relativity" and "Quantum Spacetime" are meaningless concepts [27].Quantum theory lives in the abstract mathematical linear vector space H M with perfectly deterministic, and linear evolution.General relativity lives in, and actually constitutes, real physical 4-dimensional spacetime L 4 with non-linear causal evolution of chains of "events" = the actual "happenings" that constitute the fundamental, irreversible, invariant "constituents" of spacetime, which, when warped by classical energy-momentum in spacetime, T µ ν , results in classical gravitation in L 4 through Einstein's equations G µ ν = κT µ ν .

Consequence 2: No "Zero-Point Energy" or Cosmological Constant Problem
Virtual "particles" exclusively live in Hilbert space, not in physical spacetime.They never manifest into L 4 through "Born".That is why they are not real.The same applies for the infinite "zero-point energy" of the quantum vacuum in quantum field theory arising from "virtual particles".Which in turn explains why the cosmological constant, Λ, does not go to infinity, and hence why the physical cosmos (L 4 ) has been able to expand leisurely without ripping itself apart.
As "virtual particles" never physically manifest in spacetime they have no influence at all on the classical energy density T 0 0 , or pressure T i i , in spacetime, so no effect on the expansion of the universe, given by Einstein's equations: In fact, there is no instance where this "vacuum energy" is actually physically needed 1 We have here assumed a Λ that is solely due to the presently very fashionable, albeit completely hypothetical, "Dark Energy", i.e. "quantum vacuum".As Λ in classical general relativity is merely a free parameter we can choose it to have any value whatsoever to comply with cosmological observations (e.g.finite and very small ).Such a Λ classical would give a curvature in spacetime even in the absence of T µ ν (i.e. in the classical vacuum) but Λ classical is not a classical vacuum energy, which by definition is identically zero.It is a geometric curvature of empty spacetime itself.
[29], [30], [31]. 2.3 Consequence 3: Quantum "particle" reactions do not happen in spacetime but in Hilbert space Only the observed ("measured") quantum lives in spacetime, quite contrary to what one might believe when drawing innumerable, linearly superposed Feynman diagrams.A particle, in L 4 , never occupies two (or more) distinct positions at the same time.The quantum superpositions occur in H M , where no classical attributes ever manifest.The same goes for "particle" interactions in particle physics, which by definition have N > 1, the quantum reactions happen in H ∞ , the outcomes happen in L 4 , and then only as a result of "Born".In the canonical two-slit experiment, e.g. using a laser, each "hit" at the detector screen is the result of a quantum (photon-field) interaction in H ∞ manifesting as an event in L 4 where only one discrete small region of the screen randomly lights up, as if by a photon-"particle".It is only after many hits that the superposition (in H ∞ ) becomes manifested in our real world (L 4 ) as real discrete data patterns through the statistical "Born Rule", Eq. (1).

Consequence 4: No Black Hole "information paradox"
As wavefunctions, for N ≥ 2 quantum entities, are objects in Hilbert space with global entanglement through (q 1 , ..., q 3N ), not in L 4 , they are unaffected by causal horizons in spacetime, meaning that quantum entities inside the horizon are always accessible by entangled quantum entities outside the horizon -nullifying, for quantum theory, e.g. the classical one-way membrane of a black hole event horizon -and hence potential information is in principle always accessible across horizons.A causal probability current in spacetime is definable, and conserved, only for a single, non-interacting particle, making it physically irrelevant.For N quantum entities, entangled or not, no conserved probability current is definable in spacetime, and hence can never "flow" causally.(And neither in Hilbert space as "probability" requires that "Born" already has occurred.)The abstract non-locality in Hilbert space binds arbitrarily distant quantum entities into a single global irreducible ψ. "Born" then binds actual events non-locally in real spacetime, regardless of spacetime-interval separation.This resolves the quantum information paradox [32] for black holes, making it a non-question.
5 Some proposed Alternatives to "Orthodox" Quantum Mechanics

Everett/many worlds
Only Hilbert, no collapse [33].i) Linear, cannot give the nonlinear classical world [21].ii) Does not give any probabilities (no "Born"), and never even any outcomes at all, meaning that a classical world is absent in all parallel "universes".

Explicit collapse
i) Dynamical nonlinear collapse, does not give a classical world as explicit non-locality (in principle) persists in real spacetime and energy is not strictly conserved.ii) Collapse time not relativistically invariant, cause-effect for entangled systems ill-defined (depends on frame).

de Broglie-Bohm
No collapse [34], [35], everything is (in principle) completely deterministic.i) Does not give a classical world.ii) Positions for particles always live directly in spacetime, and are guided by an extra equation, simultaneously the guiding "pilot wave", ψ, lives in Hilbert.The de Broglie-Bohm theory has no need for a "Born Rule", as the classical level is objectively real all the time, but the "pilot wave" guiding the (now objectively real) quantum particles is manifestly non-local and lives in unphysical Hilbert space, eternally global in configuration space, but as its predictions are designed to be exactly those of orthodox quantum mechanics it cannot explain the nonlinearity of classical physics.Through the guiding equation (which includes ψ) the positions of particles in spacetime depend on the positions of all other particles (arbitrarily far) making also the dynamics in real spacetime manifestly non-local, i.e. it breaks the relativistic invariance of the real world II.explicitly.In the orthodox theory it is "Born" III., that saves the real world II.from manifestly/deterministically breaking relativistic invariance, as "Born" is only statistically non-local in spacetime.

Summary & Conclusion
"Pure" quantum theory, I., is implicitly non-local, but the non-locality is unphysical (not observable) as it does not "live" in spacetime but in Hilbert space.
The "Born Rule", III., is explicitly non-local for entangled quantum systems -it correlates spacelike separated events in real spacetime, as required by Bell's theorem and its empirically validated requirement of a non-local reality.
"Reality", II., occurs only in the spacetime of events (which are the fundamental "building-blocks" of objective reality) not in quantum Hilbert space.Thus, "quantum information" is a misnomer, information only manifest in spacetime after "measurement" (i.e."Born Rule", III.) has occurred.
The fact that quantum systems, with more than one quantum entity N > 1, cannot be embedded in spacetime has very deep, profound and startling consequences.It means, for instance, that quarks and gluons are not "constituents" of (e.g.) protons in spacetime, only in abstract, infinitedimensional Hilbert space [36] -the proton is not a "bag" (in spacetime) containing quarks and gluons 3 .More generally, fundamental (quantum) "particle" interactions never occur in spacetime: Rather, merely abstract quantum fields in H ∞ result, through the magical "Born Rule", in observed phenomena as objective events in real physical spacetime interpreted as particles.Objects in our real world L 4 thus do not "consist" of fundamental quantum entities.Not even atoms or molecules "consist" of electrons, protons and neutrons in spacetime, rather the entangled electron-proton-neutron wavefunction in Hilbert space can manifest as events (in L 4 ) interpreted as arising from "atoms" and "molecules" upon "measurement", i.e. upon "Born".Even for superfluids and superconductors, macroscopic in size, the quantum properties perpetually live in H M alone.The observations of superfluids/conductors are always perfectly mundane events in our normally perceived world.Also, the 10 57 neutrons in a neutron star live in a configuration space of 3 × 10 57 dimensions in Hilbert (not in spacetime) resulting, again as always through "Born", in the observed physical properties of the neutron star in L 4 .(Quarks in quark(-gluon) stars, if they exist at all, would live in H ∞ .)"Schrödinger's Cat" [12] is dead or alive in our real world II., after "Born" III.("magically") has realized the outcome from its entangled wavefunction in merely abstract Hilbert space I.
The only mystery remaining is why (and how?) the "Born Rule" occurs at all.But then again, maybe nature really is magical.