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TWO QUANTUM THEORIES THAT BELL LIKED Nino Zanghì Dipartimento di Fisica dell'Università di Genova, INFN sezione di Genova, Italy [email protected] Summer School on the Foundations of Quantum Mechanics dedicated to John Bell, Sesto, July 2014 John Bell, 1989 i; : I s =47 gF i 3E iE i i i: E i3.E a! z'*-g 1e,F =63-=:Sl=Ii = a':d =6?''Z==1i F+g 7 f-+toF- 7 , q# i + d ! 3 rei ,i733aEi;9i gz. i3? iifsE9Tg E:gE r3 ia [E;F?rq 1 eS l4.=.'e r! r o 6 ? i. i;^=r-=5a'^ I - = i I rE 1=i-1==V;i= ie= : i 3 = i;' z;'fr7e ? + + i** ; Es g * Fi + izi,V: f ! a =i ;-.-2-.? I I 17#=fi-= e-?;,E I 476 93+: i'3 E d+zH E I F I c q :'= -:I =. = " E i^ T-.<.2 6 a'11 ='T = = va'7P{==*-r=i116- i ; P i ;cl Lrl =3q. =. = = = = = =. = E i R = =n V -: ^ 6'5a-:l= 6x;)?.^o!!o-o 5 :- * - a'; ts i. + H g -. E=' =' + T t iE g I + aE sa Towards An Exact Quantum Mechanics What is the problem of quantum mechanics? the measurement problem? Erwin Schrödinger (1887 – 1961) the deeper problem (of which the measurement problem is just a manifestation)... 1 ':ili = Ei*;u l'*E =s illtt1l* t, ::i;= ; John Stewart Bell (1928 – 1990) a This is the problem :JJ ,l::tJ *-J* All the variants of Quantum Field Theory (Cut-offs, Algebraic, etc.) \:t/ Non-Relativistic Quantum Mechanics Are there solutions of the the problem? Yes! There are many, indeed! o t(t= o ! 9 V 6>; -l ?6 (= -# !-3- I -el9 /o-' e8; c-a- 3I= *lii* 2*. ; s +':' -p_1, =alggla'F 3r-t- leF ,"n rut.'=A;;" * 9 3 Beables “The terminology, be-able as against observ-able, is not designed to frighten with metaphysic those dedicated to realphysic. It is chosen rather to help in making explicit some notions already implicit in, and basic to, ordinary quantum theory. For, in the words of Bohr, 'it is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be e x p re s s e d i n c l a s s i c a l terms'.” A algebra of operators on H operators as observables A∈A Z macroscopic variables classical variables !ψ, Aψ" < Z >ψ = !ψ, ψ" “The concept of 'observable' lends itself to very precise mathematics when identified with 'selfadjoint operator'. But physically, it is a rather wooly concept. It is not easy to identify precisely which physical processes are to be given status of 'observations' and which are to be relegated to the limbo between one observation and another. So it could be hoped that some increase in precision might be possible by concentration on the beables, which can be described in 'classical terms', because they are there. The beables must include the settings of switches and knobs on experimental equipment, the currents in coils , and the readings of instruments. 'Observables' must be made, somehow, out of beables. The theory of local beables should contain, and give precise physical meaning to, the algebra of local observables.” (ψt , Z) (ψt , X) LOCAL BEABLES EXACT (on all scales) When von Bortkewitch collected statistics on the kicking of soldiers to death by horses, in the Prussian army, in different years, he found a Poisson distribution. Now, you don’t go out into the world looking for the Poisson distribution, you go out looking for soldiers and horses and kicks. BM universal wave function Ψ up to the universal scale ↑ (ψ, Z) down to the microscopic scale ↓ Q universal wave function Ψ up to the universal scale ↑ (ψ, Z) down to the microscopic scale ↓ Q Q variables: what the theory is fundamentally about for example, particles, fields, strings, et. expressing the laws which govern the behavior of the Q variables in a simple and natural way. (Q, Ψ) universal wave function Ψ up to the universal scale expressing the laws which govern the behavior of the Q variables in a simple and natural way. ↑ (Q, Ψ) (ψ, Z) down to the microscopic scale ↓ Q Q variables: what the theory is fundamentally about for example, particles, fields, strings, et. Q = (X, Y ) X system Y environment ψ(x) = Ψ(x, Y ) Z = F (Q) universal wave function Ψ up to the universal scale expressing the laws which govern the behavior of the Q variables in a simple and natural way. ↑ (Q, Ψ) (ψ, Z) down to the microscopic scale ↓ Q Q variables: what the theory is fundamentally about for example, particles, fields, strings, et. Q = (X, Y ) X system Y environment ψ(x) = Ψ(x, Y ) Z = F (Q) !ψ, Aψ" Z −→ A < Z >ψ = !ψ, ψ" universal wave function Ψ up to the universal scale expressing the laws which govern the behavior of the Q variables in a simple and natural way. ↑ (Q, Ψ) (ψ, Z) down to the microscopic scale ↓ Q = (X, Y ) Q X system Q variables: what the theory is fundamentally about for example, particles, fields, strings, et. Y environment |Ψ| 2 ψ(x) = Ψ(x, Y ) Z = F (Q) !ψ, Aψ" Z −→ A < Z >ψ = !ψ, ψ" plug the actual configuration Y of the environment into the second slot of Ψ (x, y) to obtain a function of x, ψ(x) = Ψ (x, Y ) (Almost) all implications of BM follow from this formula 2 PΨ (X ∈ dx | Y ) = |ψ(x)| dx GRW ψ !→ ψ = set {1, . . . , N}. with uniform distribution. (6) is chosen at random in the Th Mechanics 1/2 ψ # T12 Bohmian T1 + #Λ (X ) 1/2 I 1 T1 succinctly su experiments that may distinguish this theory 1We 3shall #ΛI1 (Xin ψT1 # space. ation, a stochastic jump process this process as follows. 1 ) Hilbert 1/2 3 chosen randomly with probability distribution Λ (X ) ψ I 1 T the collapse X is chosen randomly with probability distribution 1 1 [6]. According to the way in which this theory 1 ψ → ! ψ = . uniform distribution ( Consider a quantum system described (in the standard language) by an N-“p process as follows. T1 in the T1 + set {1, . . . , N} I is chosen at random with 1/2 1 3 Ghirardi, Rimini, and Weber sen at random in theinstead set {1, of . . .Schrödinger’s , N } with uniform#Λdistribution. ψT1 # The center I1 (X1 ) wave function follows, 3 3 3 wave function ψ = ψ(q , ..., q ), q ∈ R , i = 1, . . . , N; for any point x in R (the 3 1/2 2 of the collapse X is chosen randomly with probability distribution 1 N i Consider a quantum system described (in the standard language) by an N-“ 1/2 2. . . 1 probability ollapse X is chosen randomly with distribution 3.1 GRWm . . . . . . . . . . . . . . . . . . . . . 1 |ψ , I = i ) %ψ |Λ (x )ψ & dx #Λ (x ) ψ # dx . (7) P(X ∈ dx |ψ , I = i ) %ψ |Λ (x )ψ & dx #Λ (x ) ψ # dxcen Hilbert space. We shall succinctly summarize T1 1 I11 is1 chosen i11 1 in T1 set 1i1. . . ,1iN} 1 1 T1at T1random 1 the T1 {1, 1 T1i1distribution. 1 1 T1 The 1. 1 T 1 with uniform 3 3 sy the collapse that,will be defined next), define on the Hilbert space of the e of function ψ = ψ(q ..., q ), q ∈ R , i = 1, . . . , N; for any point x in R (the 3.2 GRWf . . . . . . . . . . . . . . . . . . . . . . #. 2 3 1 N i 1/2 1/2 2 of the collapse X is chosen randomly with probability distribution P(X ∈ dx |ψ , I = i ) %ψ |Λ (x )ψ & dx #Λ (x ) ψ 1 1 i (xT11) i1ψT 1# dx T1 1 . 1 i1 1 T1 P(X dx #Λ (7) 1 ∈ dxoperator 1 |ψT1 , I1 = i1 ) %ψT11|Λi1 (x11 )ψTT11 & 1 collapse 1 1 1 1 he collapse that will be next), define on the Hilbert space of the sy (in the standard language) by defined anEmpirical N-“particle” 3.3 Equivalence Between GRWm and GRWf 2 " −x) ( Q 1 i en the algorithm is iterated: ψ evolves unitarily until a random tim 1/2 2 T T + − 1 is iterated: ψ evolves unitarily until a random time = 3 2 −7 P(X dxR1 |ψ(the , Ii“center” = i= dx1 #Λ, i1 (x1σ ) ∼ψ10 . ( T T1Λ 1(x) 1 ) %ψT1 |Λi1 (x1 )ψ T1 & 2σ T1 # dx 12 1 +1x∈in e = 1, . .operator . , N; for3.any point m apse 2until 3/2 Then is theaψTalgorithm is iterated: ψ evolves unitarily until a random T + he algorithm is iterated: evolves unitarily a random time T = 1 (2πσ ) + 2 +∆T , where ∆T random time (independent of ∆T ) distributed ac 1 2 2 2 1 " ( Q −x) define on the Hilbert space of the system the 4 Primitive Ontology 1 i a random time (independent of ∆T ) distributed according − 2 , is 1 T +∆T , where ∆T is a random time (independent of ∆T1 ) distribut 2 1 algorithm 2 timeΛ 2= where ∆T is a random (independent of ∆T ) distributed according 2σ 2the 2 1 (x) e , 3. Then the is iterated: ψ evolves unitarily until a random time T2 i with T1 + "i is the position exponential distribution rate Nλ, and so on. 2 3/2 where Q operator of “particle” i. Here σ is a new constant of 4.1 Primitive Ontology and Physical Equivalence . . . (2πσ ) 2 to the exponential distribution with rate Nλ, and so on. " distribution with rate Nλ, and so on. (Q −x) xponential distribution with rate λ, andtime so on. T2 1−7 +∆T ∆T is the aNrandom (independent of ∆T1 ) distributed accordi 1 2 , where 2is − ithe evolution of ψ Schrödinger evolution 2σ order of 10 , m. e (4) 4.2 Symmetry Properties and Relativity . . . . . . . 2 3/2 to the exponential distribution with rate Nλ, and so on. σre) Q "words, her the evolution of the waveoffunction is the evolu interrupted by collapses is the position operator of “particle” Here σ is Schrödinger a isnew ofe In other words, the evolution thei.wave function the constant Schrödinger • • • words,iLet the ψevolution the wave function is i.e., the Schrödinger evolution in-. . . . at initial wave function, the normalized wave function so t0 be the of 4.3 Without Primitive Ontology . . . . . . . . volution of the wave function is the Schrödinger evolution in−7 d by collapses. When the wave function is ψ a collapse with center x a ticle” i. Here σ is a new constant of nature of terrupted by collapses. When the wave function isthe ψxSchrödinger afollowing collapse with center In other words, the evolution of the wave function is evolution er 10 m. collapses. When the wave function is ψ a collapse with center and label • tof arbitrarily chosen as initial time. Then ψ evolves in the way: 0 by collapses. When the wave function is ψand a collapse with center xlabel and lab When the function is ψ a collapse with center x and iwave occurs at rate te at rate Let ψtterrupted be the initial wave function, i.e., the normalized wave function at s 5 Differences between BM GRW 0 e., the1.inormalized wave function at some time occurs at rate It evolves unitarily, according to Schrödinger’s equation, until a rand r(x, i|ψ) = λ %ψ | Λ (x)ψ& r(x, i|ψ) = λ %ψ | Λ (x)ψ& (8) i i −15 −1 5.1 r(x, Primitive and Quadratic Functionals . . rbitrarily chosen as initial time. in the following way: i|ψ) Then = Ontology λ %ψψ |evolves Λi (x)ψ& λ ∼ 10 s ψ evolves the following way: r(x, i|ψ) = λ %ψ | Λi (x)ψ& ( T1 in = t + ∆T , so that 1/2 1/2 0 1 1/2 and 1/2 to Λ (x) . ψ/#Λ and when this happens, the wave function changes 5.2 Primitive Ontology Equivariance . .1/2.i(x) .(8) . .ψ s happens, the wave function changes to Λ (x) ψ/#Λ (x) ψ#. i i i 1/2 r(x, i|ψ) = λ %ψ | Λ (x)ψ& i 1/2 ψ/#Λ (x) 1/2 n this happens, the wave function changes to Λ (x) ψ#. ψ = U ψ , and when this happens, the wave function changes to Λ (x) ψ/#Λ (x) ψ#. i i • i i T ∆T t . It evolves unitarily, according to Schrödinger’s equation, until a rand 1 1 0 Schrödinger’s until random time any time if between time and > t0 , n collapses have occurred etween time tequation, and any time at > t0 , nt0collapses have toccurred at the times 0 Thus, Thus,to if emphasize between time t0that and any time t01/2 > t0 , collapses n collapses have occurred at “partic the tim 1 1/2 ,he if between time t and any time t > t , n have occurred at 6 A Plethora of Theories 0 We wish here there are no particles in this theory: the word T = t + ∆T , so that < t,T11with < T2centers < . . .ψ < t, inwith andwave labels I1 , . . .th , < .wave X1T , nto . .< . ,Λ X and centers labels IX .. .. ,. I, nX, nthe 1. . < 0Ttnfunction 0< changes (x) ψ/#Λ ψ#. → 1 ,1i.,(x) tfor <convenience T . .order . < Tto t,able withtocenters X Iterminology. 0 ψ 1 < T2 <in n < 1 , . . . , Xnotation n and labels 1 , . . . , In , the wa only be use the standard and = U , (5) t.0function 6.1 Particles, Fields, and Flashes . . . . I.1 ,. . .. .., .In., .t < Tt 2∆T < . . < Tnat<time t, with centers X .∆T . ,1X and labels t will be 1 1 be me will 1 , .U n ψ = ψ , T t 1 0 F t0 and any attime > be t0L,Ft,tnnψcollapses have occurred at the times n function time ttwill L ψ t F t,t0 0 and Many-Worlds . . n Functions oat particles intthis theory: the6.2 word “particle” 0 t0 is usedWave Schrödinger L ψ time will be t ψt,tt 0= 0 in ψt = thatFnthere are (9)word “parti( We wish to emphasize here no particles this theory: the F n F ψ = n Flabels he< standard notation and terminology. t and t, with centers X#L . .0 ψ. t,0 X I , . . . , I , the wave #L ψ # # nL n 1 , t,t 1 n t ψ t,t 0 0 t00 ψt0 # notation and terminology. t,t#L 0 standard t,t for convenience in order to be able to use the ψ = t time location of that collapse. Accordingly, equation (11) gives the joint distribution of GRWf the first n flashes, after some initial time t0 . The flashes form the set F = {(X1 , T1 ), . . . , (Xk , Tk ), . . .} (with T1 < T2 < . . .). In Bell’s words: [...] the GRW jumps (which are part of the wave function, not something else) are well localized in ordinary space. Indeed each is centered on a particular spacetime point (x, t). So we can propose these events as the basis of the ‘local beables’ of the theory. These are the mathematical counterparts in the theory to real events at definite places and times in the real world (as distinct from the many purely mathematical constructions that occur in the 13 working out of physical theories, as distinct from things which may be real but not localized, and distinct from the ‘observables’ of other formulations of quantum mechanics, for which we have no use here). A piece of matter then is a galaxy of such events. (Bell, 1987a) That is, Bell’s idea is that GRW can account for objective reality in three-dimensional GRWm e m function is given by the following equation: m(x, t) = N � i=1 mi � � � 2 3 3 3 � � d x1 · · · d xN δ (x − xi ) ψt (x1 , . . . , xN ) . ψt is a wave function as in quantum mechanics, a function on R3 ψt is a GRW process ng to the usual Schrödinger equation N 2 � ∂ψ � 2 i� =− ∇i ψ + V ψ , ∂t 2m i i=1 denotes the mass of particle i, i = 1, . . . , N . m function (1) is basically the natural density function in 3-spac Lorenz invariance Relativistic (Lorentz) Invariance Those paradoxes are simply disposed of by the 1952 theory of Bohm, leaving as the question, the question of Lorentz invariance. So one of my missions in life is to get people to see that if they want to talk about the problems of quantum mechanics – the real problems of quantum mechanics – they must be talking about Lorentz invariance. Bell (1990) The big question, in my opinion, is which, if either, of these two precise pictures [GRW and Bohm] can be redeveloped in a Lorentz invariant way. Bell (1990) [N]either of these theories [GRW and Bohm] is Lorentz invariant, and ... it is pretty clear that no theory in ei- associated with the GRW evolution and the fact that GRWf, as discussed in Section 5.2, is microscopically equivariant relative to that evolution. In fact, it follows from the GRW warming that there is, for GRWf, no equivariant association ψ !→ Pψ with ψ a Schrödinger-evolving wave function.17 What is a precise quantum theory? 8 What is a Quantum Theory without Observers? To conclude, we delineate the common structure of GRWm, GRWf, and BM: (i) There is a clear primitive ontology, and it describes matter in space and time. (ii) There is a state vector ψ in Hilbert space that evolves either unitarily or, at least, for microscopic systems very probably for a long time approximately unitarily. (iii) The state vector ψ governs the behavior of the PO by means of (possibly stochastic) laws. 17 Since the GRWf flash process is non–Markovian, the formulation of the notion of equivariant as(iv) Thegiven theory provides a not notion of a typical history Pofψ should the POnow (ofbethe universe), fora sociation in Section 5.2 is appropriate here; instead, understood to be example by aonprobability distribution on the space probability measure the space Ω of possible histories of the PO of for all all possible times, buthistories; one whose from conditional this probabilities the future ofthe anyprobabilistic time given its past are as prescribed, notion for of typicality predictions emerge. here by the formula (33). " ψ The association is equivariant if T−t P = Pψt , with Tτ now the time translation mapping on Ω. (v) The predicted probability distribution of the macroscopic configuration at time t determined by the PO (usually) agrees (at least approximately) with that of the 31 quantum formalism. The features (i)–(v) are common to these three theories, but they are also desiderata, presumably even necessary conditions, for any satisfactory quantum theory without quantum probability P ( = α|B̂ = β) = | < α|β > | < ψ|χ > | < ψ|χ > | probability amplitude 2 probability non- Kolmogorovian ? 2 momentum probability to find the value p of P̂ if the system is (initially) in the the state ψ � | < ψ|p > | = ψ(p) 2 Fourier transform time of flight measurement of momentum (Heisenberg, Bohm52, Feynman & Hibbs) ψ wf at time 0 free evolution measure X̂ at large time T 1 X̂(T ) = P̂ T + X̂ m → mX̂(T ) X mX̂(T ) P̂ = + ≈ T T T Bohm The particle has a well defined position X whose evolution is guided by ψ � ψt∗ ∇ψt (X(t)) Ẋ(t) = Im ∗ m ψt ψt P (X(t) = x|ψ, t = 0) = |ψt (x)|2 P (X̂(t) = x|ψ, t = 0) = |ψt (x)|2 mX(T ) T Thus the asymptotic momentum P = (T large) is a RANDOM VARIABLE on the space on initial conditions with probability distribution 2 � P (P = p|ψ, t = 0) = |ψ(p)| spin σx = � | ↑>= 0 1 1 0 � � 1 0 | < ψ| ↑> |2 | < ψ| ↓> | 2 � , σy = and � | ↓>= 0 i −i 0 � � 0 1 � , σz = probability spin up along z probability spin down along z Classically Non-Describable Two-Valuedness (Pauli) � 1 0 0 −1 � ascio di atomi produrrebbe sullo schermo una singola ra). Quello cheStern invece Gerlach si osserva measurement sono due macchioline of spin ra a destra) chermo Schermo Magneti � ≈ (b + az)σz HI = µ�σ · B nella direzione z del campo magnetico, a seinitial Ψ = ψ ⊗ Φ(z) entazione NORD-SUD dell’atomo. o vengono preparati con un forno e poi colli- P̂z a 2 Ẑ(T ) = Ẑ + T+ σz T m 2m in the space of initial positions with prob. distribution |Φ(z)|2 Thus the RANDOM VARIABLE FT (Z(T )) , 2mz FT (z) = aT 2 in the limit of T large has values +1 with probability < ψ| ↑> |2 −1 with probability < ψ| ↓> |2 FT (z) is the calibration function of the experiment assignment of num. values to the outcome of the exp. Morals • Association between random variable Z (numerical result of the experiment) and operators • Operators compactly express the statistics of the experiment Ā =< ψ|Aψ > mean value of Z < ψ|(A − Ā)2 ψ > variance of Z < ψ|An ψ > higher moments of Z • One can completely understand what's going on in the experiments measuring momentum or spin. • No need of invoking any putative property of the electron such as its actual z-component of spin that is supposed to be revealed in the experiment. • There is nothing the least bit remarkable about the nonexistence of this property. • Measurements are “active.”