On Non-linear Quantum Mechanics and the Measurement Problem I. Blocking Cats
aa r X i v : . [ qu a n t - ph ] O c t On Non-Linear Quantum Mechanicsand the Measurement ProblemI. Blocking Cats
W. David Wick ∗ October 11, 2017 ∗ email: [email protected] LQM and the MP I Abstract
Working entirely within the Schr¨odinger paradigm, meaning wavefunctiononly, I present a modification of his theory that prevents formation of stateswith macroscopic dispersion (MD; “cats”). The proposal is to modify theHamiltonian based on a method introduced by Steven Weinberg in 1989, aspart of a program to test quantum mechanics at the atomic or nuclear level.By contrast, the intent here is to eliminate MD without affecting the predic-tions of quantum mechanics at the microscopic scale. This restores classicalphysics at the macro level. Possible experimental tests are indicated andthe differences from previous theories discussed. In a second paper, I willaddress the other difficulty of wavefunction physics without the statistical(Copenhagen) interpretation: how to explain random outcomes in experi-ments such as Stern-Gerlach, and whether a Schr¨odingerist theory with arandom component can violate Bell’s inequality.
LQM and the MP I In the 30-year-long debate over the interpretation of quantum mechanics(QM) by its founders and others, Schr¨odinger maintained that his 1926 for-mulation had been correct and moreover the wavefunction must contain anelement of reality. He based that view on de Broglie’s insight that particlesappeared to have wave properties; that his equation correctly predicted thespectrum of the hydrogen atom; and other observations [1], [2] such as “G. B.Thomson’s beautiful experiments on the interference of de Broglie waves (ofelectrons), diffracted by crystals.” He rejected the statistical interpretation ofthe wavefunction supported by Max Born, Niels Bohr, John von Neumann,Werner Heisenberg and others, as well as the individual identity of “parti-cles” which he likely believed are epiphenomena (appearing in consequence ofthe presence of a macroscopic measuring apparatus), as explained by NevillMott in 1929, [3].However, in 1935 [4] Schr¨odinger described a difficulty with his theory whenapplied to measurement situations: it gives rise to macroscopic dispersion(MD) which he illustrated in the famous metaphor of a cat in a box containinga “diabolical device”: a bit of uranium, a Geiger counter, a hammer and aflask of cyanide, rendering the cat both alive and dead. To avoid life-and-death issues here, I revise the metaphor by replacing the hammer and poisonby a buzzer; if it goes off, the cat is startled and leaps, say, one meter to theleft. Thus the cat becomes smeared out between leaping and resting. I willcall any similar situation appearing in quantum theory “a cat”.Mathematically, we can describe MD as follows. Define the (spatial) dis-persion of a macroscopic wavefunction to be: D N ( ψ ∗ , ψ ) = < ψ | ( (cid:18) N (cid:19) N X j =1 x j ) | ψ > − ( < ψ | (cid:18) N (cid:19) N X j =1 x j | ψ > ) . (1)Here N is the number of “particles” or degrees of freedom, and I have pre-tended that space has one dimension, purely for ease of writing equations; as LQM and the MP I D N is larger than its spatial extent (squared). Suchcat states arise in wavefunction theory in measurement situations.Consider the simplest such situation, that of measuring the “spin of a singlemagnetic particle,” which I assume takes only two values, ±
1. I write thecorresponding “spin states” as | + > and |− > . The measurement proceedsby sending the “magnetic particle” through an inhomogeneous (external)magnetic field, which deflects “the particle” up or down by a certain amountin a specified time. Suppose the deflection is in the positive y -direction (whilethe “particle” moves also in the x -direction, which isn’t important for thediscussion). Thus an initial state of form ψ ( y ) | + > (2)where ψ ( y ) is a wave packet of small spread centered at the origin, evolvesinto another state: ψ + ( y ) | + > (3)with ψ + ( y ) displaced some distance in the + y -direction. But this is not yeta measurement, because we have not yet included an apparatus capable ofdetecting “the particle” arriving up the y -axis. I will denote the apparatuslocation (say, its center-of-mass, COM) by X ; think perhaps of a needle on ascale that can move in response to the “particle’s intrinsic magnetism.” So,initially, we should have written the combined state of everything as ψ ( y ) θ ( X ) | + > (4)where θ ( X ) is a wavepacket for the apparatus coordinates centered aroundthe initial needle position, and the state after the measurement as ψ + ( y ) θ + ( X ) | + > (5)where θ + ( X ) is a wavepacket of the needle displaced by an amount we inter-pret as “particle detected.” (Better: ψ + ( y, x , ...x N ), whose reduced densityfor X is concentrated at the displaced position.) Similar results hold with +spin replaced by − , with the needle moving down rather than up. LQM and the MP I / √ | + > + (1 / √ e i γ |− > . (6)(Here γ is a real phase.) Then, by the unrestricted linearity of QM, thewhole “particle”-plus-measuring-apparatus state can only evolve into anothersuperposition:(1 / √ ψ + ( y ) θ + ( X ) | + > + (1 / √ e i γ ψ − ( y ) θ − ( X ) |− > . (7)These developments of course represent simplified theory of the celebratedStern-Gerlach experiment, performed in 1922,[5], in which the superpositionwas generated from an initial atomic state of rotational symmetry. (Exactsolutions of Schr¨odinger’s equation for S-G can be found in [6], [7].) It ispresented in textbooks as the most distinctively “quantum” outcome andwhich refuted the classical picture of the atom as a little randomly-orientedmagnet. (I note here that the claim that Stern and Gerlach observed two dotsis a myth; it was actually a lip-print. What this may imply for measurementtheory will be discussed in paper II of this series.)In my opinion, this scenario represents the fundamental conundrum arisingin the QM theory from the 20th Century, often called the MeasurementProblem. In the first place, no definite result has appeared, assuming we areunwilling to believe that “the particle” divided and went both ways. Second,as the needle is a macroscopic object, this version of measurement producesa Schr¨odinger’s (bidirectional) leaping cat. Finally, nothing has appeared to“collapse the wavefunction” after the detection of the “particle” up or downthe y-axis. Related to the Measurement Problem is the issue of the microscopic-to-macroscopic, quantum-to-classical, transition. On the macroscopic side ofthe boundary, Newton’s equations hold (at velocities small with respect tothe speed of light); on the microscopic side, it is Schr¨odinger’s equation.What defines the boundary? The conventional answer is: “¯ h → LQM and the MP I h → G →
0. Presumably whatis meant is that ¯ h is small in some sense. But in what sense, and how wouldthat fix the classical/quantum boundary? The units of ¯ h are [energy]x[time]or [momentum]x[distance]; thus, the statement “¯ h is small” does not in itselfimplicate a distance scale, a time scale, an energy or momentum scale, or anycombined scale that obviously distinguishes macroscopic from microscopic.Second, and somewhat more questionable, one can object based on the ex-istence of the phenomenon called “chaos.” Non-linear Newtonian equations incertain situations yield so-called “chaos,” and as one consequence very com-plex orbital configurations in phase space may appear. By contrast, linearQM produces a quasi-periodic flow incapable of generating complexity of thissort [15]. (Some physicists reject this argument by invoking “Heisenberg’suncertainty principle,” which in pure Schr¨odingerism we interpret simply asa theorem about forming wave packets. In the usual probability interpre-tation of QM, this principle can be interpreted to mean that phase space iscoarse-grained; i.e., observing a phase-space trajectory with a precision belowthat permitted is impossible or meaningless. Therefore the chaos objectionis invalid.)Third: in my opinion, the most convincing reason to reject the ¯ h → X = < ψ | (cid:18) N (cid:19) N X j =1 x j | ψ > (8)and take two time-derivatives. There results: m ¨ X = − (cid:18) N (cid:19) N X j =1 < ψ | V ′ ( x j ) | ψ > . (9)where V denotes an external (scalar) potential; a dot denotes time-derivative;and a superscript ′ denotes space-derivative. LQM and the MP I h dropped out. Two:suppose that in the right-hand side the average over j and the integrals overspace can be pushed inside the function V ′ , up to some negligible error. Theresult is Newton’s equation for the COM: M ¨ X = − N V ′ ( X ) ≡ − V ′ E ( X ) . (10)( M ≡ N m and the factor of N on the right is absorbed into a “macroscopic”external potential energy, rendering it extensive.) In particular, for harmonicpotentials, where V ′ is linear, Newton’s law holds without qualifications, asmany authors have noted previously. I conclude from this calculation that weshould search for the classical limit where the dispersion of the COM of theapparatus pointer is very small relative to the scale on which the higher-orderterms in the external potential vary spatially. One way to describe the MP is to note that QM produces an ‘and’—thewavepackets of S-G propagate up and down—when what we observe is an ‘or’:a detection up or down. Rather esoteric proposals have been made to resolvethis paradox: the mind of the observer collapsing the wavefunction (vonNeumann, 1932, [16]); the universe splitting into branches in the “multiverse”(Everett 1957, [17]), and many others. However, if macroscopic dispersioncan be eliminated at the level of apparatus, the ‘and’ must necessarily betransformed into the ‘or’. Extra-physical postulates will no longer be requiredto explain measurements. That is the motivation for the theory proposed inthe subsequent sections. I will adopt Weinberg’s approach, [8], to defining a class of nonlinear, deter-ministic evolution equations for a complex wavefunction ψ , which is simpleand elegant. (Weinberg’s results lead to experimental tests of whether QMhas non-linear contributions at the nuclear level, with negative results; see[9].) The starting point is a minor reformulation of Schr¨odinger’s theory, es-sentially a change of language. Traditionally the Hamiltonian, or energy ob- LQM and the MP I H , and Schr¨odinger’sequation takes the form: i ¯ h ∂ ψ / ∂ t = H ψ. (11)Instead, from now on I will use “Hamiltonian” to mean the functional H QM = < ψ | H | ψ > . (Which in the statistical interpretation of QM is called the“expected energy.” But Schr¨odingerists have abandoned that probabilitisticconception.) We rewrite Schr¨odinger’s equation as: i ¯ h ∂ ψ / ∂ t = ∂ /∂ψ ∗ H QM . (12)Note that here we regard the Hamiltonian to be a functional of ψ and ψ ∗ rather than the real and imaginary parts of ψ , but the two views areequivalent. (There is no problem about varying ψ and ψ ∗ independently, asin the usual definition of partial derivative; I define it as if by the chain-rule,see Mathematical Appendix.) Now Weinberg’s idea is simply that, givenany other real-valued function H ( ψ ∗ , ψ ), which need not be quadratic in the ψ -variables, we can postulate a new equation: i ¯ h ∂ ψ / ∂ t = ∂ /∂ψ ∗ H . (13)For its application to the MP I will assume an additive form: H = H QM + H NL . (14)where H NL will be small in a sense to be discussed later.It is easy to check (see Mathematical Appendix) by separating real andimaginary parts that (13) is simply a way of packaging Hamiltonian me-chanics, albeit for a pair of “position” and “momentum” variables which arenot the physical quantities. All of the Hamiltonian machinery goes through;for instance, the Poisson bracket takes the form, for any pair of functionals F ( ψ ∗ , ψ ) and G ( ψ ∗ , ψ ): { F , G } = < ∂ F ∂ ψ ∂ G ∂ ψ ∗ − ∂ F ∂ ψ ∗ ∂ G ∂ ψ > . (15)which reduces, in the case that both functionals are quadratic, defined bytwo self-adjoint operators F, G , to the usual “commutator” form: { F , G } = < ψ | [ F, G ] | ψ ∗ > . (16) LQM and the MP I
F, G ] =
F G − G F . In terms of this bracket, the time-derivativeunder the evolution defined by equation (13) is given by: d / dt F = ( i ¯ h ) − { F , H } . (17)In particular, energy is conserved: d / dt H = ( i ¯ h ) − { H , H } = 0 . (18)What about conservation of norm, interpreted in the statistical paradigmas “conservation of probability?” (A Schr¨odingerist might not care; on theother hand, we could worry about conservation of matter or charge.) Innonlinear theory we have: ∂∂ t < ψ | ψ > = − i ¯ h (cid:26) < ψ | ∂ H ∂ ψ ∗ > − < ∂ H ∂ ψ ∗ | ψ > (cid:27) , (19)which is not obviously zero. It is in special cases, in which ∂ H ∂ ψ ∗ = O ( x , ... ; ψ ) ψ, (20)and O ( x , ... ; ψ ) happens to be a self-adjoint operator possibly depending on ψ in some fashion. (Weinberg apparently rejected this approach. Hence,in order to remain consistent to the statistical interpretation, he had to di-vide by the normalization when discussing “expected values” in his nonlinearmodels.) That will be the case here; see next section and the MathematicalAppendix. To motivate the introduction of nonlinear terms into Schr¨odinger’s equation,consider first two wavefunctions that can represent states of a macroscopicsystem. Let φ r ( x ) be a wavefunction mostly confined to an interval [ − r, r ]with r fairly small, say a few centimeters. Let R denote a larger distance,say a few meters. Compare: N Y j =1 n (cid:16) / √ (cid:17) φ r ( x j + R ) + (cid:16) / √ (cid:17) φ r ( x j − R ) o . (21) LQM and the MP I (cid:16) / √ (cid:17) N Y j =1 φ r ( x j + R ) + (cid:16) / √ (cid:17) N Y j =1 φ r ( x j − R ) . (22)Wavefunction (21) can represent a rather peculiar extended object with“particles appearing at two locations” (simultaneously), a situation that canarise with so-called macroscopic quantum phenomena, see section 5. (If Ihad left out the superpositions and simply replaced φ r by a φ R with width2 R , we probably would not worry about this issue.) But it is not a cat.Wavefunction (22) is a (bidirectional) leaping cat. How can we distinguishthe two, mathematically?Here we rely on the dispersion D N introduced earlier. It is easy to seethat, for the non-cat extended state of (21), D N = O( R /N ), while for theleaping cat state in (22), D N = O( R ), differing by a factor of N (perhaps10 ). (If we believed the statistical interpretation of the wavefunction, thisis the statement of the Central Limit Theorem. For Schr¨odingerists, it is afact about a mass distribution with independent contributions.)I therefore propose to exploit conservation of energy in a nonlinear Schr¨odinger’sequation to prevent the formation of the cat. Let H NL ( ψ ∗ , ψ ) = w < ψ | N X j =1 x j ! | ψ > − < ψ | N X j =1 x j | ψ > ! = w N D N ( ψ ∗ , ψ ) . (23)Here w is a (very) small coupling constant with units [energy]/[distance] .(Alternatively, one can construct a momentum version of H NL for which w has different units; see section 4.)Adding H NL to the usual “N-particle” quantum Hamiltonian H QM gener-alized from equation (14), the latter perhaps including internal energy termsholding the apparatus together, external fields, and connection to a microsys-tem, yields my proposed NLQM. (Another difference with Weinberg appearshere: Weinberg assumed that all observables must be homogeneous of degreeone in ψ and ψ ∗ , because, if ψ is a solution of a nonlinear Schr¨odinger’sequation, he required that Z ψ be also, for any complex number Z . I requireonly that this property hold for complex numbers with | Z | = 1, i.e., whichmaintain wavefunction normalization.) LQM and the MP I H NL does not scale “exten-sively,” proportionally to N , as is assumed with other kinds of energy. If wesuppose that, in a measurement situation where the apparatus is coupled toa microsystem, the initial state is of form: N Y j =1 φ r ( x j ) θ ( y ) (24)(where y stands for some microsystem coordinates), then the initial energyconsists of apparatus internal and external energies (scaling like N ), a mi-croscopic energy from coupling to the microsystem, plus a contribution fromthe NL term of order w r N . However the total system evolves, it cannotbecome a leaping cat, because there is not enough energy available to reachO( w R N ) (provided w is not too small).For illustrative purposes, I invent a value for w and see how the numberscome out. Assuming that all apparatus energies are positive (or at leastbounded below by something times N ), the final nonlinear energy term afterthe measurement cannot exceed the initial energy. But if the measurementproduces a leaping cat, the nonlinear term will be of order H NL [final] ∼ w [1 m ] . (25)If we postulate w ∼ − Joule /m , (26)Then H NL [final] ∼ Joule . (27)However, if the apparatus consists of a pointer of size 1mm (plus externalfields) initially H NL [initial] ∼ − − Joule ∼ − Joule . (28)Clearly, the energy of the leaping cat in (27) cannot be supplied by any-thing (apparatus internal energy, external fields, the microsystem) in thissituation. For perspective: if the cat weighed 1kg and the potential en-ergy of (27) were converted into kinetic energy (ignoring relativistic mass LQM and the MP I meters per second, i.e.,near the speed of light.By contrast, the nonlinear energy of a hydrogen atom would be around10 − (cid:2) × − (cid:3) ∼ − Joules , (29)which is about 10 − times the ground state energy (1 . × − Joules), alevel which could never be observed.
One of the requirements for claiming to have a solution of the MP is to restoreNewton’s Laws at the macroscopic side of the classical/quantum boundary.How would this work in the NLQM proposed in the previous section?First, let us futher examine how an energy bound on the nonlinear termscontrols the mass distribution of the apparatus. For concreteness, visualizea pointer suspended initially at an unstable extrema of an applied, external,potential; the pointer will move upon absorbing energy from the microsystemobserved. I will need the reduced density functions of the pointer. The k –threduced density function is defined as: ρ k ( x , ..., x k ) = Z ... Z dx k +1 dx k +2 ...dx N | ψ | ( x , ..., x N ) . (30)(I assume symmetry under interchange of variables.) In terms of reducedfunctions the macroscopic dispersion can be written: D N = (cid:20) N ( N − N (cid:21) ( Z Z ρ ( x , x ) x x dx dx − (cid:18) Z ρ ( x ) x dx (cid:19) ) + (cid:20) N (cid:21) ( Z ρ ( x ) x dx − (cid:18) Z ρ ( x ) x dx (cid:19) ) = (cid:20) − N (cid:21) C + (cid:20) N (cid:21) L . (31)where the third line defines the “correlation function” C and the squaredsize of the system, L . The nonlinear energy becomes, omitting a small term: LQM and the MP I H NL = w N D N ∼ w (cid:8) N C + N L (cid:9) . (32)Thus an energy bound on H NL implies a strong bound on C but a weakerbound on L .However, to transition to Newton, a bound on L is crucial. Consider acubic external potential: V E ( x ) = a x + b x + c x . (33)By an easy computation using (9), M ¨ X = − V E ′ ( X ) − c L . (34)where X is the pointer COM and I have omitted the interaction term withthe microsystem, which can be thought of here as simply giving the pointeran initial (small) kick. Thus Newton for the pointer is restored as long as thesecond term on the right of (34) is small, which will be true if the size of thepointer remains smaller than the variation scale of the cubic term. I note thatno additional terms coming from H NL appear in this computation, becauseadding H NL does not affect the derivation of (9), the macroscopic evolutionequation. (See Mathematical Appendix. Of course, a cubic is not an idealexample, especially for the energy bound, as it is not bounded below.)Now let us assume that the observation does not produce a phase transi-tion (i.e., the pointer doesn’t melt or vaporize) or an explosion. Moreover,the pointer is held together by internal forces, say corresponding to pairpotentials with energy: − N X j = k < ψ | u ( x j − x k ) | ψ >, (35)with u ( x ) becoming small as soon as | x | is larger than some microscopiclength, say a few lattice spacings. Then the internal (negative) binding en-ergy will be extensive (proportional to N ) and much larger than energy sup-plied from the observed microsystem. Now L is the squared average distancebetween atoms and the centroid of the pointer; if it increased, atoms wouldbe farther apart, and the internal energy would increase proportionally.I postulate, as part of measurement theory within Schr¨odingerism, that X , D N and L are observables, and “restoring Newton” for the apparatus pointermeans that X evolves by his equation while D N remains very small and L LQM and the MP I H NL ensures that D N remains small,while internal energies ensure that L doesn’t increase. Then, at least for thecubic case, (34) yields Newton.What about higher-order terms in the external potential? (A quartic wouldbe more realistic.) I do not have a theorem here, but I postulate that thetail thickness of realistic examples of the reduced density function ρ arecontrolled by the second moment, as for Gaussians, implying that higher-order terms in (34) are also small. I remarked that energy is conserved in the nonlinear extension of QM; whatabout momentum? Define the total momentum of a many-body system tobe: P = < ψ | N X j =1 i ¯ h ∂∂ x j | ψ >, (36)Then (easy exercise): { P , H NL } = 0 . (37)It follows that if H QM contains only internal forces (no external fields)plus the usual momentum terms, { P , H } = 0 , (38)so total momentum is also conserved. (Apparently Weinberg adopted a stricthomogeneity condition in the nonlinear Hamiltonians he considered becauseit was necessary to ensure Galilean invariance. As H NL is obviously invariantunder overall rotations, space translations, and boosts, I see no difficulty here.Lorenz invariance is another matter.)How does the NL energy behave for a composite system? If the two subsys-tems are independent, in the sense that they have never interacted, or neverinteracted with the same other system, and the wavefunction factorizes, thenthis energy is additive: if the subsystems are labeled A and B, H NL = H NL ; A + H NL ; B . (39)In particular, if LQM and the MP I ψ = N Y j =1 θ r ( x j ; A + R ) N Y k =1 θ r ( x k ; B − R ) , (40)where θ r ( · ) is a wavepacket of width r centered at the origin, then H NL is proportional to r , not R . So this peculiar form of energy cannot bethought of as a “confining potential” (as for quarks I suppose) preventingthe separation of the two subsystems. For independent systems combinedmentally into one larger system, H NL scales extensively, as usual.If macroscopic subsystems A and B would become correlated by interactingwith a third (microscopic) system, the NL energy kicks in and prevents cat-formation as usual, except for one purely-theoretical instance: EPRB withperfect detectors, in the perfectly-anticorrelated case. (EPRB is discussedin the second paper of this series.) If the detectors are needles that movein opposite directions, the dispersion of the COM of the combined systemcould remain zero, abrogating the cat-blocking mechanism proposed here.However, this case requires exact alignment ( a = b ), identical detectors,and identical initial conditions. With any discrepancy the nonlinear termswould scale again as N R and energy conservation would prevent cats.Additionally: perfect anticorrelations are a consequence of assuming a perfectvon Neumann measurement (in fact, Stern and Gerlach did not observe twodots; see paper II) and may be unrealistic.Thus far, I have only considered two macrosystem wavefunctions: a prod-uct form (“independent particles”) and the cat. For the only “normal” state(the former), the quantity C is exactly zero. But for most states of matter,it will not vanish. This raises the issue of “correlations” in Schr¨odingerism.First, C should not be called a “correlation function,” as that language isonly suitable for the statistical interpretation of the wavefunction (as repre-senting, in some complicated way, an ensemble of particles). C can only beregarded as an aspect or component of the dispersion of the system. Second,to construct a Schr¨odingerist version of “quantum statistical mechanics,” or“quantum thermodynamics,” the only consistent approach is to introduce anensemble of wavefunctions, perhaps describing the system interacting with aheat bath at temperature T . But there is little agreement in the literatureon how this is done. Some authors adopt a priori an ensemble weightedby Gibbs factors: exp( − E/kT ), where E is what I have written H and k isBoltzmann’s constant. Justifying such an assumption, whether the dynamicsis classical or quantum, is famously hard. Then there is von Neumann’s con- LQM and the MP I ρ (not a reduced density function as I used earlier), which subsumes both “purestates” (single wavefunctions) and “mixtures” (as in probability theory). Ittakes the form: ρ = exp( − H/kT ) / tr exp( − H/kT ) . (41)(“tr” stands for matrix trace.) Then one defines the entropy as: − k tr ρ log( ρ ) . (42)Unfortunately, under linear (“unitary”) matrix evolution using Heisen-berg’s rule: ddt ρ = − i ¯ h [ ρ, H ] , (43)such an entropy is constant! This violates the Second Law of Thermodynam-ics, at least as formulated: “outside of equilibrium, entropy always increases”.Many authors have addressed this problem; some introduce nonlinear, dissi-pative evolution to restore the Second Law. It has also been proposed thatstatistical mechanics and thermodynamical behavior can be derived fromlinear QM either because the system has been subjected to a random Hamil-tonian [10] (I make a similar suggestion in paper II to explain the randomoutcomes of certain measurements), or because the classical analog of thesystem is chaotic, [11]; see [12] for a recent review of this program. ButI find no consensus about the proper formulation of “quantum statisticalmechanics.”Third, thermodynamic equilibrium reflects a balance of energy and en-tropy. Thus introduction of a new form of energy that involves correlations(in some sense), which are usually thought of as an aspect of entropy, mayalter the nature of phase transitions or the location of critical points. (Beforethis topic can be addressed, it will be necessary for experiments to fix thefree parameter called “ w .”) However, as phase transitions do not implicatecats, this is unlikely. Finally, even having adopted a thermodynamic theory,we receive no guidance as to the form of C for an individual wavefunction.I make a few conjectures about the form of C for more general cases.Writing: ρ ( x , x ) ρ ( x ) = ρ ( x ) [ 1 + f ( x − x ) ] , (44) LQM and the MP I f ( x − x ) as representing the fractional increase or decreasein the one-dimensional (“single particle”) density at x , conditional on “aparticle at x .” We can imagine that f ( x ) falls falls off rapidly with | x | , sayover a microscopic distance. Then we have: C = Z Z dx dx ρ ( x ) ρ ( x ) f ( x − x ) x x , (45)(where again I have written this equation as though space has one dimension).I worked out a simple example (see the Mathematical Appendix) and found. C ≈ L ǫ max( f ) . (46)where L denotes the size of the system and ǫ is a microscopic length (the“correlation length”). Now the issue becomes: how large can max( f ) be?It represents the maximal increase (or decrease) in density at some point,due to knowledge that there is “a particle nearby.” If we assume that thisis small, of order N − , then we have the usual (“Central Limit”) behavior;this would result in a contribution of the nonlinear energy for a “normal”object that is very small. On the other hand, if max( f ) ∼
1, then with myillustrative value for w , the 1mm needle, and nanoscale correlations, we geta contribution of order: w N L ǫ ≈ − − − ≈ Joules . This is roughly, for example, the total thermal energy of a 0.1g mass withthe heat capacity of water, at room temperature. That is large; but I suspectthat, in realistic models of pointers, what I have written here as max( f ) issmall enough that the contribution to the energy of the object is very small. Can a theory of the type considered here be compatible with the RelativityPrinciple? Although modern Schr¨odingerists need not follow the historicalpath, it is useful to recall the order of 20th Century developments in physics.After Heisenberg and Schr¨odinger published in 1925-6, it must have seemedunlikely that the new quantum mechanics could ever be made compatible
LQM and the MP I H NL by replacing x k by p k = i ¯ h ∂/∂x k , the momentum operator, in equation(23): H NL ( ψ ∗ , ψ ) = w < ψ | N X k =1 p k ! | ψ > − < ψ | N X k =1 p k | ψ > ! . (47)Here the parameter w has units of reciprocal mass; alternatively, if I incorpo-rate such a factor explicitly, w is unit free. (Neither choice seems as felicitiousfor locating the classical/quantum boundary as the spatial formulation. Buteither choice does the job: blocking cats.)The relativity problem with the Schr¨odinger’s equation was that it includesthe energy operator ( i ¯ h ∂/∂ t ) on the left side but the square of the momen-tum operator on the right; relativity requires that energy and momentumbe treated equivalently, as components of the “energy-momentum 4-vector.”In 1926 Oskar Klein and Walter Gordon (and many others) proposed a newequation similar to a classical wave equation:1 c ∂ ψ∂ t = △ ψ − m c ¯ h ψ. (48)The obvious generalization of the Klein-Gordon equation to N “particles”and including the nonlinear energy is to write:¯ h m c ∂ ψ∂ t = ∂∂ ψ ∗ { H QM + H NL } − N m c ψ. (49) LQM and the MP I H QM = 12 m N X k =1 < ψ | p k | ψ > (50)is the usual total kinetic energy and ψ remains as a (scalar) complex functionof all the “particle” coordinates and time.Although K and G proposed their equation as representing a relativisticelectron, Dirac rejected it, for two reasons. (More recently, K-G seems tohave been resurrected as the correct equation to describe the pion and theHiggs boson.) First, it did not incorporate the intrinsic “spin” of an elec-tron. And, as a second-order equation, two initial values could be arbitrarilyprescribed: for ψ ( x , ...,
0) and ∂ /∂ t ψ ( x , ..., ψ ( x ; t ) ψ ( x ; t ),even if each factor satisfies a one-particle KG equation, because of the twotime derivatives on the left side, the product will not satisfy a two-particleKG equation. Assuming the product form at time zero, it will not propagate.Thus, even lacking external fields or any other interactions, particles wouldbecome entangled.Dirac’s clever solution to these dilemmas was to increase the number ofcomponents of ψ to four, and then to factorize K-G by matrix tricks. Hisequation, using the celebrated gamma-matrices, reads (for this section I re-store space to its rightful number of dimensions, namely three): γ µ n p µ + ec Φ µ o ψ − m c ψ = 0; or: γ µ n i ¯ h ∂ µ + ec Φ µ o ψ − m c ψ = 0 . (51)Here µ = 0 , , , γ µ are 4x4 matricies satisfyingthe anti-commutation relations: γ µ γ ν + γ ν γ µ = 2 g µ,ν I ; LQM and the MP I g µ,ν denotes the space-time metric, diagonal(1 , , , − ∂ µ = ( c ∂/∂ t, ∂/∂ x, ∂/∂ y, ∂/∂ z );Φ µ is the 4-potential of the external electromagnetic field; m is the electronmass; c is the speed of light; and ψ = ( ψ , ψ , ψ , ψ ).An obvious approach is to treat the momentum terms in (49) as Dirac did,granted a sufficient supply of anti-commuting γ -matrices to factorize them.For Lorenz covariance of either a nonlinear K-G or Dirac equation we wouldalso need to know that: < ψ ′ | P ′ µ | ψ ′ > = Λ νµ < ψ | P ν | ψ >, (52)where primes denote a second reference frame; P and P ′ stand for momentumoperators; Λ νµ is the matrix of a Lorenz transformation, denoted Λ, connectingthe frames; and, in the Dirac case, in terms of an invertible 4x4 matrix S depending on Λ: ψ ′ ( x ′ , ... ) = S (Λ) ψ (Λ( x , ... )) . (53)In words: energy-momentum transforms as a covariant 4-vector. (In theDirac case it is necessary to define in equation 52: < ψ | · | ψ > = Z d x ψ †∗ β [ · ] ψ (54)where β = i γ , because the matrix S (Λ) is not unitary but “pseudouni-tary”: β S (Λ) †∗ β = S (Λ) − , see [13], p. 218. The probability interpretationfor the electron’s position is then lost, because β has eigenvalues ±
1; butSchr¨odingerists do not require that interpretation to hold.)So let us assume given a set of matrices acting on some vector space inwhich ψ takes values and satisfying: γ µj γ νk + γ νk γ µj = 2 δ j,k g µ,ν I, (clearly such a set of generalized gamma-matrices requires a high-dimensionalvector space on which the matrices act, so we might as well have assumedan infinite-dimensional ψ at the outset), for j, k = 1 , . . . N , and define γ µ = N X j =1 γ µj . (55)We can then postulate (omitting the external potentials for simplicity): LQM and the MP I X k,µ γ µk i ¯ h ∂ µ,k ψ + w X µ γ µ X k [ i ¯ h ∂ µ,k − < ψ | P µ,k | ψ > ] ! ψ = √ N m c ψ. (56)Next, “squaring” the equation (applying the operator on the left a secondtime), using the anticommutation relations, and ignoring terms of order w yields: − X k,µ g µ,µ ¯ h ∂ µ,k ψ + w X µ g µ,µ ( X j i ¯ h ∂ µ, j ) ψ − w X µ,j,k g µ,µ i ¯ h ∂ µ,j < ψ | P µ,k | ψ > ψ = N m c ψ. (57)Here the first term represents the usual total KE and the second and thirdyield a nonlinear KG equation similar to (49) but with additional centeringterm for the energy (from terms with µ = 0). The mutual repulsions ofelectrons could be incorporated through external potentials as in (51), usingthe Dirac current as source terms for Maxwell’s equations.It is well known that Dirac’s theory suffered from anomalies. The ex-istence of states with negative kinetic energies was the primary difficulty;Dirac proposed re-interpreting these states as representing positrons (nega-tive electrons). But then external potentials might induce transitions fromelectron to positron states or vice versa , violating conservation of charge; soDirac assumed the negative-energy states were all filled. There are difficul-ties with observables. The usual position observable mixes the two kind ofstates, and the usual velocity operator produces a peculiar motion (the Zitter-bewegung). Restricting to positive states or introducing some novel positionoperator leads to a violation of Einstein causality (wave packets spread fasterthan the speed of light).[18] R. Jost concluded: “The unquantized Dirac fieldhas therefore no useful physical interpretation.”[19]These conundrums suggest that it may be a mistake to start the search fora relativistic, nonlinear Schr¨odingerist theory with Dirac. Should we followthe historical sequence and progress to Quantum Field Theory, even if it isconsidered the quintessential theory of particles? QFT employs “creation and LQM and the MP I N another argument). Thus QFT, if reinterpreted,may be compatible with Schr¨odingerism. On the other hand, A. O. Barutand colleagues showed in the 1980s (see [20] and references therein) thatmany successful calculations in QFT—including of the Lamb shift, sponta-neous emmission, vacuum polarization, and the anomalous magnetic momentof the electron—can also be carried out in a purely Schr¨odingerist setting. Anonlinear Dirac-type equation, derived from integrating out the Maxwellianpotentials, appeared in their program (but I am not aware that they relatedit to the Measurement Problem). What do we need to test the theory presented here? Before describing sev-eral candidate systems, let’s ask: have observations already falsified the the-ory? Certain systems studied by quantum physicists are said to exhibit“macroscopic quantum phenomena” (MQP). Do these phenomena includeMD (cats)? Back in 1980,[21], Anthony Leggett, after discussing cats etc.,had this to say:To sum up the point crudely and schematically, “macroscopicquantum phenomena” require a many-particle wave function ofthe form, ψ = ( a φ + b φ ) N , (58)while the states of importance in the quantum theory of measure-ment are of form ψ = a φ N + b φ N . (59)As we have seen, the second equation represents a cat, but not the first. ThusMQP is not (necessarily) relevant to the MP.In addition, systems exhibiting MQP are very complex, require coolingto extremely low temperatures, and theory describing them often relies onheuristic or uncontrolled assumptions. Leggett has stated that the exemplar LQM and the MP I –10 ; I understand that astelling us that SQUIDS lie on the quantum side of the classical/quantumboundary.For a test, we need a scalable quantum/classical system—one that exhibitsquantum phenomena at the microscale but transitions to classical behavior atlarger size, keeping all else fixed. Given such an experimental set-up, testingwould proceed in two stages. First, the elimination of macroscopic center-of-mass (COM) dispersion (“cats”) must be used to fix the free parameterI have denoted by “ w ”. Then the now-rigid theory should predict, say, theshape of the transition curve, or the disappearance of some other quantumphenomena (such as interference).Unfortunately, the SQUID is not suitable as it does not produce a separa-tion of the COM of the electrons in the circuit, [22]. A more promising choicemay be the “micromechanical oscillator”: a nearly-atomic-sized “beam” orbridge fixed at both ends and observed at low temperatures, for which quan-tum phenomena such as discrete energy states have been observed, [24]. Asuperposition of macroscopic motional states may be possible soon in such asystem, [25], although achieving large displacements (greater than the size ofthe object) is limited at present to the nanoscale, due to quantum and ther-mal decoherence effects. Another candidate might be a suspended mirror,[26].Conceivably, the toy set-up (a pointer in an external potential) I used inprevious sections for illustrative purposes might be relevant. Perhaps it isworth spelling out how testing would go if it could actually be realized. Soconsider a two-well (quartic) external potential of macroscopic width 2 R ,and imagine a cloud of “particles” initially in a very small band centeredat the origin, which is also the local maxima (unstable critical point) ofthe potential. Let’s regard the cloud as the measuring device and assume acoupling with some microscopic quantum system initially in a superpositionthat implies forces on the cloud “particles” that can send them on a journeyto right and/or left. In the quantum regime for the cloud-plus-particle, a LQM and the MP I − R ) or position (+ R ) where the potential has itsminima. Let the drop in potential energy between the origin peak and theminima be ∆ V . Assuming the energy contributed by the microsystem issmall and other internal energies (e.g., attractions between the “particles”if they are not independent) are also small or do not change, then, in thenearly-macroscopic case, for a cat to form the nonlinear energy has to besupplied by the external potential. Let N c be the point at which the catstate has just broken up. Then we can write, approximately, N c ∆ V ≈ w N c R , (60)which we can take as estimating the free parameter ‘ w ’ by: w ≈ ∆ VN c R . (61)Of course, if no such transition was observed for the range of N we can pro-duce experimentally, then instead of an estimate of w we could only deducean upper bound. The theory proposed here differs from previous wavefunction theories thateither incorporate non-linear terms or otherwise aim to eliminate MD. Forexample, in 1966, Bohm and Bub, [27], proposed adding non-linear termsto QM that generate “basins of attraction” into which the microsystem fallsafter a measurement. This can explain discrete outcomes and wavefunc-tion collapse. However, dissipative dynamics is not Hamiltonian and doesnot conserve energy. The theory presented here is Hamiltonian (and does).Moreover, any theory of the Bohm-Bub type may be ruled out by experimentsof the type motivated by Weinberg’s 1989 papers, if it implies non-linear dy-namics of small systems. (Differences from Weinberg’s formulation of NLQM,particularly concerning the wavefunction normalization, are mentioned in therelevant sections.)In 1986, Ghirardi, Rimini, and Weber proposed a stochastic “spontaneouscollapse” theory that rapidly eliminates MD after it appears, [28]. By con-trast, the theory of this paper is purely deterministic: nonlinear terms pre-
LQM and the MP I N , meaning that itcould not cancel a positive energy proportional to N . Of course, it mustbe checked that incorporating the nonlinear energy does not abrogate thesetheorems.At present, I cannot describe the final state of microsystem-plus-apparatusin detail. Solving high-dimensional, non-linear wave equations analyticallyappears infeasible; nor, because of the exponential growth of dimensionalityof a quantum system with degrees-of-freedom (‘ Q qubits’ requires a Hilbertspace of dimension 2 Q ), is it possible to simulate the theory on a digitalcomputer. Thus I have relied for the claims made about measurement ontwo facts: (a) energy is conserved, making cat-formation impossible; and(b) the non-linear terms do not alter the equation of motion of the COM ofthe apparatus pointer. Conceivably, this theory is “chaotic”; i.e., possessesunstable dependence on initial conditions or measurement parameters, as iscommon for high-dimensional, nonlinear systems.Thus, the question of what determines the final state, as well as the ulti-mate cause of random outcomes, remains open. These topics are addressedin the second paper in this series. Here is the derivation of equation (9). We begin with an apparatus N-degree-of-freedom (“N particle”) plus microscopic-degree-of-freedom Schr¨odinger’sequation, pretending as usual that space has one dimension to avoid cluttered
LQM and the MP I i ¯ h ∂ ψ / ∂ t = − (cid:20) ¯ h m (cid:21) N X j =1 △ j ψ − (cid:20) ¯ h m (cid:21) △ y ψ + V a ( x , x , ... ) ψ + V int.( x , ..., x N ; y ) ψ. (62)where △ j = ∂ /∂x j ; △ y = ∂ /∂y ; x j stands for an apparatus coordinate; y stands for some microscopic system coordinate; V a stands for apparatuspotential energies, possibly including an external field and internal forces;and V int. is the interaction potential of macroscopic and microscopic systems.I’ll discuss forms for the potentials later.The problem: study¨ X = d /dt < ψ | (cid:20) N (cid:21) N X k =1 x k | ψ > . (63)We have d/dt < ψ | x k | ψ > = (cid:20) ¯ h m (cid:21) { < i N X j =1 △ j ψ + i △ y ψ | x k | ψ > + < ψ | x k | i N X j =1 △ j ψ + i △ y ψ > } + (cid:20) h (cid:21) { < − i (cid:0) V a + V int. (cid:1) ψ | x k | ψ > + < ψ | x k | − i (cid:0) V a + V int. (cid:1) ψ > } (64)In (64), the terms involving the V ’s cancel, while the terms containing △ j or △ y and x k with k = j can be integrated by parts and disappear, leaving: d/dt < ψ | x k | ψ > = (cid:20) ¯ h m (cid:21) {− i < △ k ψ | x k | ψ > + i < ψ | x k |△ k ψ > } . (65) LQM and the MP I i ¯ h ∂ /∂ x , this equation just says that the velocity is equal tothe momentum divided by the mass): d/dt < ψ | x k | ψ > = i (cid:20) ¯ hm (cid:21) < ∂∂x k ψ | ψ > . (66)Next: d /dt < ψ | x k | ψ > = (cid:20) im (cid:21) { < ∂∂x k i (cid:20) ¯ h m (cid:21) " N X j =1 △ j ψ + △ y ψ − i (cid:2) V a + V int. (cid:3) ψ (cid:1) | ψ > + < ∂ψ∂x k | i (cid:20) ¯ h m (cid:21) " N X j =1 △ j ψ + △ y ψ − i (cid:2) V a + V int. (cid:3) ψ > } (67)IBPs twice on the terms containing △ j and △ y , they vanish, as do theterms containing derivatives of ψ times V ’s; that leaves only the terms with V ′ ’s. Averaging over k we attain, finally: m ¨ X = − (cid:20) N (cid:21) N X k =1 < ψ | (cid:20) ∂V a ∂x k + ∂V int. ∂x k (cid:21) | ψ > . (68)Just to see what this looks like in a special case, we might take: V a = N X j =1 V ( x j ) + N X j = k u ( x j − x k ); V int. = α ( N X j =1 x j ) y, (69)where u ( · ) stands for internal energy holding the apparatus needle togetherand α is an interaction constant. It is easily seen that the term involving u disappears from (68) and we get: LQM and the MP I m ¨ X = − (cid:20) N (cid:21) N X k =1 < ψ | V ′ ( x k ) | ψ > − α < ψ | y | ψ > . What about the nonlinear term? It adds a term to the right side ofSchr¨odinger’s equation of form: ∂ H NL ∂ψ ∗ = w N X j =1 x j ! − w < ψ | N X j =1 x j | ψ > N X j =1 x j ! ψ ;= V NL( x , ... ; ψ ) ψ. (70)Moreover, V NL can be treated (despite containing a dependence on ψ )exactly as the other potentials in the derivation of (68), and therefore con-tributes a term:2 w < ψ | ( N X j =1 x j − < ψ | N X j =1 x j | ψ > ) | ψ >, (71)which vanishes.Concerning the NLQM Schr¨odinger’s equation, (13), I define the right sideas if the chain rule holds; i.e., if ψ = Q + i P , so that Q = (1 /
2) ( ψ + ψ ∗ ) and P = ( − i/
2) ( ψ − ψ ∗ ), then: ∂ H∂ ψ ∗ = ∂ H∂ Q ∂ Q∂ ψ ∗ + ∂ H∂ P ∂ P∂ ψ ∗ = 12 ∂ H∂ Q + i ∂ H∂ P , (72)from which (13) can be written as the pair: ∂ Q∂t = 12 ¯ h ∂ H∂ P ; (73) ∂ P∂t = −
12 ¯ h ∂ H∂ Q ; (74)
LQM and the MP I / h , has the form of Hamilton’s equations.(Of course, Q and P are fake “generalized coordinates” without units, andNOT the physical position and momentum.) In particular, Schr¨odinger’soriginal equation can be described mathematically as being the special case ofan infinite-dimensional, linear Hamiltonian system that commutes (as flows)with the simplest case with Hamiltonian P [ Q j + P j ], which generates theoverall phase-rotation leaving the quantum state invariant. Whether youwant to insist on that last part in a nonlinear extension depends on yourgoals and philosophy.For the one-dimensional example of “more-realistic correlations” cited insection 3.2, I assumed a two-level function: f ( x ) = ( f , for − ǫ ≤ x ≤ + ǫ ; − f , for ǫ ≤ x ≤ η or − η ≤ x ≤ − ǫ ; (75)where f , f , η, ǫ are positive constants with f < η > ǫ . The reasonthat there are both “attractive” and “repulsive” regions (meaning where theconditional density increases or decreases) is because necessarily Z Z dx dx ρ ( x ) ρ ( x ) f ( x − x ) = 0 . (76)Using flat densities, this necessitated by explicit calculation: f f = η − ǫǫ . (77)Another calculation yields, plugging in the above: Z Z dx dx ρ ( x ) ρ ( x ) f ( x − x ) x x = (cid:20) L f ǫ (cid:21) (cid:26) − f f + 2 ( η − ǫ ) ǫ (cid:27) = L f ǫ. (78)which was cited in the text. References [1] Schr¨odinger, E. “What is an elementary particle?” Annual Reports of theBoard of Regents of the Smithsonian Institute, pp. 183-196. (1950).
LQM and the MP I α -Ray Tracks,” Proceedings of theRoyal Society of London , A126, 79 (1929), reprinted in J.A. Wheelerand W. H. Zurek, Eds.,
Quantum Theory and Measurement , PrincetonUniversity Press, Princeton, NJ, 1983.[4] Schr¨odinger, E. “Die geganw¨artige Situation in der Quantenmechanik”.Die Naturwissenschaften 23, pp. 807-12, 823-828, 844-849. (1935), trans-lated into English and reprinted in
Quantum Theory and Measurement , op cit. [5] Gerlach, W. and Stern, O. “Der experimentelle Nachweis der Rich-tungsquantelung,” Zeitschrift fur Physik 9:349-352 (1922).[6] Scully, M.O., Lamb, W.E., and Barut A. O.“On the Theory of the Stern-Gerlach Apparatus,” Foundations of Physics 17, 575 (1987).[7] Gondran, M. and Gondran, A. “A complete analysis of the Stern-Gerlachexperiment using Pauli spinors”. Preprint on arxiv, 2005.[8] Weinberg, S. “Testing Quantum Mechanics,” Annals of Physics vol. 194,336-86 (1989); a shorter report appeared in Phys. Rev. Letters, “PrecisionTest of Quantum Mechanics,” 62, 485 (1989).[9] Bollingeri, J.J. et. al , “Testing the linearity of quantum mechanics by rfspectroscopy of the Be + ground state”. Phys. Rev. Letters vol. 63, 1031(1989); see also the “News and Views” article by Richard Thompson inNature, vol 344, 571 (19 October 1989).[10] Deutsch, J. M. “Quantum statistical mechanics in a closed system,”Phys. Rev. A, 43: 2046 (1991).[11] Srednicki, M. “Chaos and quantum thermalization,” Phys. Rev. E.,50:888. (1994).[12] Polkovnikov, A. and Sels, D. “Thermalization in small quantum sys-tems,” Science, 353: 752. (2016). LQM and the MP I
The Quantum Theory of Fields , Volume I. Cambridge Uni-versity Press, Cambridge, 1995, 2005.[14] Brauer, R. and Weyl, Hermann. “Spinors in n dimensions”. Am. J. Math57: 425-449. (1935). See also the Wikipedia page (as of 7/3/2017) “Higherdimensional gamma matrices”.[15] Ford, J. and Ilg, M.“Eigenfunctions, eigenvalues, and time evolution offinite, bounded, undriven quantum systems are not chaotic,” Phys. Rev.A, vol. 45, 6165 (1992).[16] Von Neumann, J.
Mathematical Foundations of Quantum Mechanics ,English translation of the original German text of 1932, Princeton Uni-versity Press, Princeton, NJ. (1955).[17] Everett III, Hugh. “Relative state formulation of quantum mechanics”.Rev. Mod. Physics 29, 454-462. (1957), reprinted in Wheeler and Zurek.[18] Thaller, B.
The Dirac Equation , Springer-Verlag, Berlin, 1992.[19] Jost, R.
The General Theory of Quantized Fields , Am. Math. Soc. Prov-idence, 1965, quoted by Thaller, p. v.[20] Barut, A. O. “Combining relativity and quantum mechanics:Schr¨odinger’s interpretation of ψ .” Foundations of Physics 18(1): 95-105.1988.[21] Leggett, A. J. “Macroscopic quantum systems and the quantum theoryof measurement”. Supplement to the Progress of Theoretical Physics,69,(1980); see p. 90.[22] Leggett, A. J. “Testing the limits of quantum mechanics: motivation,state of play, prospects”. Phys. Conden. Matter 14 R415. (2002); see p.442.[23] Leggett, A. J. in Physics Today, August 2000.[24] Teufel, J. D. et. al , “Sideband cooling micromechanical motions to theground state”, preprint on arxiv, (2011).[25] Arndt, M., Aspelmeyer, M., and Zeilinger, A. “How to extend quantummeasurements”. Forscht. Phys. 1-10 (2009). LQM and the MP I