aa r X i v : . [ phy s i c s . g e n - ph ] A ug A Dynamic Histories Interpretation of Quantum Theory
Timothy D. Andersen ∗ Georgia Institute of Technology, Atlanta, Georgia (Dated: September 10, 2020)
Abstract
The problem of how to interpret quantum mechanics has persisted for a century. The disconnectbetween the wavefunction state vector and what is observed in experimental apparati has hadno shortage of explanations. But all explanations so far fall short of a compelling and completeinterpretation. In this letter, I present a novel interpretation called dynamic histories. I show math-ematically how quantum mechanics can be reinterpreted as deterministically evolving dynamicalworld lines in a 5D universe. Quantum probabilities can be then be reinterpreted as stemmingfrom ignorance of the state of our own world line. Meanwhile, the lack of observed superpositionin experimental apparati is explained in that we only live on a single history with a definite set ofproperties. Hence, superposition is not an actual state of a particle but a model of ignorance as inclassical probability theory. This explains nonlocal effects without superluminal communication. Ialso discuss how this relates to 5D Kaluza-Klein theory. NTRODUCTION
Nobel laureate, Steven Weinberg in 2017 published an article [1] called The Troublewith Quantum Mechanics in which he laments the lack of a clear interpretation of quantummechanics. This is, as he says, a debate that has been raging for 100 years with no signs ofstopping.The problem is simply that the current mathematical formalism of quantum mechanicsand quantum field theory includes no model or indication of how measurements arise fromthe predictions of the theory. Rather the theory presents us only with probabilities based ona wavefunction state vector. To quote Erwin Schrdinger [2], describing a radioactive decayexperiment,the emerging particle is described as a spherical wave that impinges con-tinuously on a surrounding luminescent screen over its full expanse. The screenhowever does not show a more or less constant uniform surface glow, but ratherlights up at one instant at one spot .There are a handful of interpretations of this phenomenon. These can be classed intocomplete and incomplete interpretations. Complete interpretations are those that say thatthe quantum wavefunction is a complete description of a quantum system. Incompleteinterpretations are sometimes called hidden variable theories. Albert Einstein favored thesewith Bohmian mechanics being the simplest.The problem of quantum interpretation arises from the incompatibility between the basicassumptions of quantum mechanics and the measurement of quantum phenomena. Quantummechanics makes two basic assumptions:1. The quantum mechanical wavefunction is a complete description of a quantum mi-crosystem such as a particle.2. Interactions between measurement apparati (including lab equipment and people), i.e.macrosystems, and microsystems are governed completely by Schr¨odingers equation.That is, they are linear.Based on these assumptions Von Neumann gave his idealized general description of mea-surement (based on the description found in [3])2onsider a microscopic system S and select one of its observables O. Let o n be the eigen-values of O. Let the spectrum of O be purely discrete and nondegenerate. The correspondingeigenvectors are | o n i . Let M be an apparatus that measures O for the microsystem S. Mhas a ready state | A i where M is ready to measure the property under test and a set ofmutually orthogonal states | A n i that correspond to different macroscopic configurations ofthe instrument. These are often called pointer positions but they could be any recognizablydifferent measurements.Now, invoking our assumptions above, the interaction between M and S is linear. Wealso assume that there is a perfect correlation between the initial state of S and the finalstate of the apparatus, | o n i ⊗ | A i must yield | o n i ⊗ | A n i . Thus, we are sure that if the stateof the apparatus is | A n i , that the state of the particle is | o n i .The measurement problem is then when we have a superposition of states | m + l i =12[ | o m i + | o l i ]. The linearity of Schr¨odingers equation guarantees that the apparatus will beplaced into superposition as well: 12[ | o m i ⊗ | A m i + | o l i ⊗ A l ]. The macroscopic measure-ment apparatus is not in position m or l but a superposition of both. Yet in experiment,macroscopic apparati are always apparently in one or the other state randomly as in a clas-sical mixed state. This leads to the postulate of wavepacket reduction (WPR) that thewavefunction somehow chooses which of its superimposed states to be in.A more realistic description of measurement that does not assume perfect correlation ora lack of interference from outside sources does not resolve the problem [3]. While the in-teraction between the macroscopic measuring device and the quantum particle undoubtedlydisrupts its state in various ways, this disruption is insufficient to explain the paradox.A resolution to the problem can take three approaches: (1) Violate assumption EQUIVALENCE TO STANDARD QUANTUM THEORY
The equivalence between Schr¨odinger’s equation, i ~ d | ψ i /dt = ˆ H | ψ i , and Feynman’spath integral is well known. Suppose ˆ H = ˆ p/ m + V (ˆ q ) is the Hamiltonian operator and ψ the wavefunction in a Hilbert space, ˆ p and ˆ q are the Heisenberg momentum and positionobservables. The classical Lagrangian functional for a closed system is L ( q ( t ) , ˙ q ( t )) = m ˙ q − V ( q ) where q is the classical position. The classical action is S = R T dt L . Then theappropriately normalized probability amplitude for a final free-particle spatial state | F i attime T is h F | ψ ( T ) i = D F (cid:12)(cid:12)(cid:12) exp h − i ~ ˆ HT i(cid:12)(cid:12)(cid:12) E = R Dq ( t ) exp (cid:0) i ~ S (cid:1) where | t i ≡ | ψ ( t ) i .Under a Wick rotation t → iτ , by analytic continuation, the path integral probabil-ity amplitude is equivalent to the partition function for a canonical equilibrium statis-tical ensemble at temperature ~ , R Dq ( t ) exp( iS/ ~ ) → R Dq ( τ ) exp( − E/ ~ ) = Z where E = R dtH ( q ( t ) , ˙ q ( t )) is the energy integrated over time.If the probability amplitude exists, then, because it is a continuum integral over Hilbertspace, it is equivalent to the microcanonical ensemble up to a constant factor, Z = C Ω = C Z Dq ( t ) δ ( A − E ) , for an appropriate choice of A such that the temperature is ~ . This can be shown bya gradient descent method in a thermodynamic limit, e.g., from a discretized q functionto a continuum [17]. Microcanonical quantum field theory derives from this equivalence[18][19][20][21].As in molecular dynamics simulations of quantum lattice gauge theory [22][23], up to aconstant factor we add an additional kinetic term to the Lagrangian in a new dimension y ∈ Y (generally Y = R ) and add a parameter to q ( t ) so that it becomes q ( y, t ). The originalclassical Hamiltonian becomes the potential energy and the new Lagrangian is ¯ L ( q y , ˙ q, q ) = q y / − H ( ˙ q, q ). The new action is ¯ S = R dydt ¯ L . It is easy to show that both the canonicaland microcanonical ensembles in this action, e.g., ¯ Z = R Dq y Dq exp (cid:0) − ¯ S/ ~ (cid:1) , are equivalentup to a constant ¯ Z = C Z . The kinetic term can simply be integrated out of the functionalintegral since it is quadratic. 5ow, we can apply Euler-Lagrange to the new action ¯ S to obtain equations of motion in y , q yy = ∂ ¯ L ∂q − ∂∂t ∂ ¯ L ∂ ˙ q (1)The right hand side are the classical equations. If we solve these for q , we obtain a solution¯ q ( y, t ), given appropriate stochastic initial conditions ¯ q ( y,
0) that satisfy the Born rule andthe given lab preparation. (Thus, while the equations are not stochastic, the initial conditionmakes it impossible to predict. This reflects our ignorance of the exact state of q ( y,
0) atany y .)For example, in the free particle case, the equations of motion (in real time) are theone dimensional wave equation: q yy = q tt /m , the so-called guitar string equation. Thissolution involves two pulses (or moving waves) the right and the left that move forward andbackward in time respectively as they move in the y direction. The velocity in y is inverselyproportional to the square root of the mass of the particle (and the speed of light whichhere is unity), suggesting that macroscopic objects change their position little in the freecase between world lines.By ergodicity, we can obtain the expected value of any classical observable O ( q ( y, t )) byaveraging it over all y , h O ( q ( t )) i = lim Y →∞ Y R Y dyO (¯ q ( y, t )). Thus, for a final configuration F , one need only apply the correct observable for the measurement.Analytic continuation of any predictions back to real time can be done at any point.As I showed in a previous paper, this can be done as well for all of quantum field theory in5D. This has been done since the early 1980s in lattice simulations for QED, electroweak, andQCD using a Hamiltonian-based molecular dynamics approach [24]. (Modern implementa-tions are typically hybrid Monte Carlo which includes an acceptance/rejection step, but theHamiltonian dynamical principle is still used.) My paper simply extends this to perturba-tion theory of continuous fields showing how vacuum loops can arise from the interaction ofvarious particle world lines in the fifth dimension. HOW TO INTERPRET THE EQUATIONS OF MOTION
The key step in interpreting quantum mechanics is understanding the leap from theequations of motion to generating averages of observables or probability amplitudes. Whilethe statistical averaging is equivalent to Schr¨odinger’s equation, the equations of motion6ontain much more information. Likewise, while the wavefunction ψ ( t ) contains sufficientinformation to compute probabilities, the state is determined by the dynamics of q ( y, t ).I.e., it is not a probability amplitude of world lines q ( t ) but an evolving world sheet q ( y, t ).Of course, q need not be position. It could be any variable or field of interest.It is clear that if what is measured in our measurement scheme is not a superpositionbut a single result or set of results, then what we are measuring is not some average over q ( y, t ) but a particular q ( y F , t ). In other words, the equations of motion in 5D give us aparticular world line to measure, but we are not able to determine, because of ignorance ofthe boundary conditions, what the full state is. Instead, we can only access a property of y F which tells us only that y F is from the subset that contains worldlines with that property.With this in mind, let us revisit the Von Neumann measurement scheme. Given thesuperposition of states we know that a subset of M ⊂ Y have property | o m i while anotherequal subset L ⊂ Y have property | o l i . For any given y ∈ Y , a particle on a world line q ( y, t )can have only one property.Upon carrying out an experiment, both particle, experimenter, and apparatus share asingle position in y as they share a position in time t . The fifth dimension is assumed to beorthogonal to time. Thus, the particle world line as well as the apparatus’ worldline are inmotion in y forming world sheets q ( y, t ) and Q ( y, t ). The particle state | m + l i represents allpossible worldlines in Y . Yet, the particle, by equations of motion 1 only takes on a singleworld line with a single property. Thus, the superposition of states represents our ignoranceof which world line is the correct one, not an intrinsic property of the particle.The experimental measurement is made at a particular y F and t F where the particleterminates. (This is an ideal statement since measurement takes some non-zero amount oftime as well to decohere the particle wavefunction.) The apparatus now reflects the property | o m i or | o l i that the wavefunction on y F has. By gaining this knowledge, we now reduceour ignorance of which subset of Y we are on, e.g., given property | o m i we know we are insubset M .Since the measurement occurs at a particular y F ∈ Y it must fall into either subset M or L exclusively. Because we cannot predict from the outset, however, into which it falls, itappears random.Once the measurement is made, as in the consistent histories interpretation [4], it appearsas if the particle had that property all along even though this is not true.7ote that the dynamics in y is for all of history, not just at the present time. The equationsare similar to those of a moving string that propagates through time. While we can onlyperceive history as belonging to the subsets of Y for which we have made measurements,history itself belongs to one and only one y not a combination of different y at different t ,which could lead to historical inconsistencies.This approach resolves quantum paradoxes as arising from ignorance and increasingknowledge about our worldline. For example, in the case of Schr¨odinger’s cat, some worldlines include a dead cat while others include a live cat. Thus, if the experiment is conductedat time t , the cat’s world sheet C ( y, t ) will contain a subset D ⊂ Y and a subset A ⊂ Y for the dead and alive cats. Yet, if our world line is at y F , then it will determine the cat’sultimate fate.The Einstein-Podolsky-Rosen paradox [25], which exposes the nonlocal nature of quantummechanics, can also be dealt with without superluminal communication. In this case, usingDavid Bohm’s version [26][27], an electron positron pair is produced as a spin singlet. Alicemeasures the electron spin while Bob measures the positron spin. Depending on the setup of the measurement apparatus, Bob’s measurement is 100% correlated with Alice’s ornot at all. If Alice measures the property + z she knows that her world line, y A , is in asubset S z ⊂ Y that has that property. When Bob makes his measurement, he can chooseto measure the z-axis as well, applying S z to the wavefunction. What this does is selectsout the subset of worldlines from Y that have the z-property that Bob measures. When hecompares notes with Alice, he and Alice are on the same world line. Therefore, their worldline must have consistent z-spin properties. Hence they know their world line is in a subset S z ⊂ Y from Alice’s measurement alone. Bob contributes no new information.If Bob chooses to measure the x-spin propety, however, Alice is ignorant of this property;thus, Bob has a random chance of measuring + x or − x . When he compares notes withAlice, he will find that between the two of them they have reduced their knowledge of thesubset of Y to be that which has the property Alice measured and the one he measured, S z ∩ S x ⊂ Y . Thus, in this case, their ignorance is further reduced.Alice, however, cannot gain this knowledge by measuring the electron by itself becauseof the non-commutative nature of S z and S x .Non-commutativity is a feature of stochastic processes in general such as Brownian motion[28] as Feynman noted in his development of the path integral [29]. In the dynamical case,8t is guaranteed by Alice and Bob’s uncertainty about their world line state, which causesstate knowledge to diffuse (in imaginary time) stochastically.What this shows is that measurement is a process of applying observable operators toreduce ignorance about what subset of Y we are living on. Randomness is not inherent inthe universe, or, as Einstein would say, God (or Nature) does not play dice. Randomness isguaranteed only by our inability to know the full state of our world line. HOW THE UNIVERSE CAN APPEAR 4D
While the issue of quantum field theory has been largely addressed in my previous pa-per [24], the problem of how this fits in with the 4D general relativity is still an issue.In particular, how does the fifth dimension’s symmetry group appear in quantum theory?Kaluza-Klein theory posits that the universe is 5D and shows how the fifth dimension playsthe role of U(1) symmetry of electromagnetism. I will not go into the mathematical for-malism of non-compactified KK theory; see [30] for a detailed discussion. I will also notaddress how other forces or matter fit into the theory. The problems with going beyond 5Dtheories are substantial [31]. I will only address the direct implication of adding a dimensionto spacetime.Kaluza’s original theory proposed that the universe was constant in the 5th dimension.While this is the classical limit of the quantum theory presented here, in reality there wouldhave to be variations at all scales in the 5th dimension to account for quantum effects, atleast down to the Planck scale. Compatification, however, appears to be unnecessary here.One of the main reasons the fifth dimension is not navigable as spatial dimensions are,even if it is spacelike as in many KK theories, is because of the symmetry breaking betweengravity and electromagnetism in the KK theory. While local Lorentz boosts and rotationsare possible in four of the dimensions, in the fifth dimension de Sitter rotations and boostsbecome phase translations in electromagnetic fields. Thus the local de Sitter symmetry,SO(4,1), has been broken into local Lorentz symmetry, SO(3,1), and U(1).Even if the fifth dimension is timelike, given the statistical results from quantum theory,entropy is constant and maximal in that dimension, i.e., it is in equilibrium. In a constantentropy, equilibrium system there is no way to distinguish which points are past and whichare future [32] and no way to do work because there is no way to flow free energy in a single9irection. There is no arrow.It is impossible even in principle to measure motion through a maximally entropic di-mension without doing work in another dimension that is not in equilibrium such as time.A good example of this is that one can create an equilibrium Brownian clock by measuringthe distance a particle travels to be ∝ √ t , but one cannot record this information in theBrownian system itself. CONCLUSION
I have shown how quantum mechanics is equivalent to averaging over world sheets ina 5th dimension orthogonal to the ordinary four. In doing so, I have demonstrated thatmeasurement is a process of applying observable operators to state vectors to reduce ourignorance of what world line we live on within that dimension. This explains the process ofquantum measurement without wavefunction collapse (only our ignorance collapses). Thetheory is completely deterministic. It also does not invoke many worlds or minds. It allowsfor many different world histories to exist and for the past and future to change in thefifth dimension. Yet, only one history ever needs to exist at a “time” since history itself isdynamical. Unlike Many Worlds, there is no splitting when observations are made. (Whatsplits is our knowledge of what subset we live on). One startling conclusion however is thatin the dynamical history theory history itself is not fixed. Rather history from the Big Bangto the end of time can change all at once as it propagates in the 5th dimension. We wouldhave no knowledge of this change since history always remains consistently on a single worldline. ∗ [email protected][1] S. Weinberg, The trouble with quantum mechanics, The New York Review of Books , 51(2017).[2] E. Schr¨odinger, Die gegenwrtige situation in der quantenmechanik, Die Naturwissenschaften , 807 (1935), english translation by John D. Trimmer, 1980, The Present Situation inQuantum Mechanics: A Translation of Schrdingers Cat Paradox Paper, Proceedings of theAmerican Philosophical Society, 124(5): 323338, reprinted in Wheeler and Zurek 1983: 152167.
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