Why Current Interpretations of Quantum Mechanics Are Deficient

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Our aim in this note is to provide an ob- where we have assumed that A observed O and we have jective examination of the most prevalent interpretations indicated the final state of A with |Af (i)i. If, on the with the goals of highlighting implicit assumptions, un- other hand, Process 2 is valid for all isolated systems veiling inherent deficiencies, and indicating where future then it must too be valid for the assumed isolated system work may resolve such issues. We mention in passing consisting of A + S, thereby dictating that the final state that this present work is not intended to review argu- is ments which appear elsewhere in the literature, although minor overlaps do occur. We begin in section II A with UAS|Aini ⊗ |Sini = |oj i|aj i (2) the Copenhagen interpretation, which is not only a com- Xj monly held interpretation, it is also the interpretation where U is the A + S system time evolution operator that is taught to each student embarking on the study of AS quantum mechanics. and |aj i corresponds to the apparatus reading eigenvalue “oj ”. The process indicated in equation (2) is referred to as a Von Neumann measurement of the first kind. What is the state of the A + S system after the measurement? II. COPENHAGEN INTERPRETATION/ RELATIONAL QUANTUM MECHANICS/ It is clear that possible final states (1) and (2) are mu- ENSEMBLE INTERPRETATION tually exclusive (if a system is in state (1) then it is not in state (2) and vice versa.) because the final state gen- erated by Process 1 is mixed, while that generated by A. Copenhagen interpretation Process 2 is pure. It is well known that no unitary evo- lution can transform a pure state into a mixed state. We The usefulness of the Copenhagen interpretation is un- demonstrate that a logical inconsistency arises because deniable. It allows one to make probabilistic predictions of incompleteness. Processes 1 and 2 result in physi- for the results of measurements, given the wavefunction cally distinct states. According to Process 2 the state for a system, via the Born rule. However, it is evident of the system of interest remains in superposition. As a that the reference to external observation in the form of consequence, the possibility of interference exists. If Pro- “measurement” raises questions about the universal va- cess 1 occurs then the system of interest, in conjunction lidity of the interpretation. The collection of difficulties with the apparatus, is no longer in superposition and the arising from the inclusion of measurement as fundamen- possibility of interference is eliminated. There is noth- tal is known as the measurement problem. Before we ing inherent to the theory that either rejects the mutual make additional comments, it is helpful to review quan- application of Processes 1 and 2 or selects one over the tum mechanics under the Copenhagen interpretation. other. Since the Copenhagen interpretation fails to make Quantum mechanics under the Copenhagen interpre- a firm prediction about the final state, it is appropriately tation purports the following. Isolated physical systems deemed incomplete. Although performing an experiment are completely described by a Hilbert space element (a that determines the physically realized final state of the ray in a Hilbert space, to be precise) |ψi, referred to A+S system is current an impossibility, eventually it will as the “wavefunction.” The wavefunction evolves in time be possible. We will determine which process, 1 or 2, or according to two distinct processes. We will match the some hitherto unconsidered possibility, is actually real- literature and refer to them as Process 1 and Process ized. It is guaranteed that in this way the Copenhagen 2 [22]. interpretation will be found incomplete in a laboratory setting. Process 1. Discontinuous, indeterministic time evolu- Additional deficiencies arise from attempts at recon- tion which sends |ψi into an eigenstate |o i of ob- i ciling Process 1 and 2. Consider a second observer, ob- servable O with probability |ho |ψi|2 as a result of i server B, who will function as a “Wigner’s friend” for a measurement of O. the A + S system. We assume that observer B is equiva- Process 2. Unitary time evolution — the time depen- lent to observer A in all respects. At time tin observer A dence of |ψ(t)i is governed by the Schr¨odinger equa- interacts with the system of interest S, thereby making a measurement. Assume without loss of generality that tion, i~|ψ(˙t)i = H|ψ(t)i. observer A found definite value oi at time tin. At time Let us now demonstrate that these two processes, as tf (tf >tin) let observer B interact with the A + S sys- stated, are inconsistent. Consider an isolated observer A, tem in an attempt to determine which value observer A determined for S. As a “Wigner’s friend,” assume, with- initially in state |Aini and system of interest S, initially out loss of generality, that observer B discovers that the in state |Sini. Whereby the term “observer” refers to any device capable of inducing Process 1. Since A induces system S is in state |oii. The question arises, what time Process 1 the result of interaction between A and S must should observer B assign to the observation that S is in be a mixed state |oii? If unitary dynamics holds good then, according to B, system S was not in a definite state |oii until tf , in 2 ρ = |hoi|Sini| |Af (i)ihAf (i)| ⊗ |oiihoi| (1) contradiction to the claim of A that it was in |oii at tin. Xi One is either forced to abandon a unitary description of 3 observer A from the standpoint of observer B or one is mechanics under the Copenhagen interpretation must be forced to abandon the equivalence of observers A and B. contextual [30]. That is, the result of a measurement This presents difficulties especially in the context of a rel- must depend on the measuring device used to measure it. ativistic theory. In relativistic theories, the finite speed If the subsystems have properties defined without refer- of light demands that (local) observers make local ob- ence to a measuring device, then one has introduced non- servations, in order to collect information about distant contextuality and it is guaranteed, via Gleason’s theorem, events [21]. We find that consistency with this premise of that the interpretation disagrees with quantum mechan- relativity (that equivalent observers make local observa- ics in an experimentally falsifiable manner. If, in order tions) enforces the abandonment of a unitary description to avoid this aforementioned deficiency, contextuality is of the observer. introduced, then one may charge this interpretation with We refer to the set of inconsistencies that arise from being incomplete. Namely, if contextuality is introduced, the attempted simultaneous application of Processes 1 then systems post measurement are different than the and 2 as the P1 &2 problem. systems immediately before the measurement occurred, but the ensemble approach offers no description of this (collapse-type) process. Requiring a collapse-type pro- B. Relational Quantum Mechanics cess, but without an attempt to reduce it to either purely unitary or possibly fundamental non-unitary processes, The standpoint of relational quantum mechanics is indicates a failure to resolve the measurement problem. that both Process 1 and Process 2 occur and are correct Note that on aesthetic grounds this approach is dissatis- because the state of a system is not an absolute quan- fying because it offers no explanation for or description of tity, but may only be given in reference to a particular the mechanism of statistical correlation among the sub- observer, analogously to velocity [42]. That the state systems. of a system is a “relative” quantity is not in question for our current purposes (although we note in passing that for the relational view to be maintained the quan- D. Quantum Bayesianism tum state equivalent of a Lorentz transformation, which would permit the wavefunction with respect to observer Quantum Bayesianism, or QBism, concerns the A to be transformed into the wavefunction with respect application of Bayesian probability (statistical infer- to observer B, must exist and no such quantity has been ence/updating) to quantum mechanics. In particular, an purported). Instead we concern ourselves with the fact agent (read gambler) acts as a classical observer embed- that relational quantum mechanics demands that 1 and ded in an otherwise quantum world. The state vector rep- 2 are simultaneously correct. As we have indicated, these resents the degree of belief that an agent has for events to two processes yield different experimental results. Thus, occur upon measurement, so that wavefunction collapse to maintain them both simultaneously is a logical fal- is understood as a jump in the knowledge of the observer. lacy (assuming two mutually exclusive propositions to It is an essential point that no nonlocal behavior exists, be true). Since experiment will eventually reveal which including nonlocal correlation [23][24][25]. process trumps the other (or that some other process oc- QBism shares so much in common with the Copen- curs), we see that this view will be falsified. Bluntly, rela- hagen interpretation that it cannot rightfully be called tional quantum mechanics suffers from the P1 &2 prob- a distinct interpretation. In particular, it uses a no- lem (see Section II A). Since it makes no attempt to clar- tion of measurement that corresponds precisely to that ify the measurement process nor provides an alternative, of the Copenhagen interpretation. No refinement in un- it fails to resolve the measurement problem. derstanding of the measurement process is introduced. That is, there is no attempt at describing measurement in terms of more fundamental processes. Just as in the C. Ensemble (Statistical) Interpretation Copenhagen interpretation, the Born rule is assumed without justification. The Copenhagen interpretation is The ensemble or statistical interpretation is often cited notable for a (decidedly implausible) sharp cut between as a minimalist interpretation [6]. It states that the wave- quantum and classical worlds, and such a sharp divide function does not apply to individual systems, but only to indeed remains in QBism. The agents of QBism func- an ensemble E of similarly prepared systems. The hope is tion as a classical observers. As such, the sharp divide that one may resolve the measurement problem, at least between the classical and the quantum occurs “at the with regard to collapse, by imagining that the individual agent.” That is, one cannot apply quantum mechanics to subsystems always have well defined properties, but that describe the function of the agents. This is an odd and by some as-of-yet unknown mechanism they are statis- unappealing conclusion as it is certainly believed that tically correlated in the manner prescribed by the Born physical agents are constructed from protons and elec- rule [6]. An immediate deficiency may be noted. Re- trons interacting in a manner that is not different from call that Gleason’s theorem demonstrates that any inter- other matter and yet QBism forces us to refrain from ap- pretation capable of reproducing the results of quantum plying quantum mechanics to the agents because of their 4 status. It is straightforward to see that the commonali- like a promising solution to the measurement problem be- ties between QBism and the Copenhagen interpretation cause it has introduced, via a unitary interaction alone, imply that they share deficiencies. In particular, QBism perfect correlation between the system and the pointer suffers from the P1 &2 problem. states. However, equation (3), as it stands, suffers from A fundamental premise of QBism is the change in the two difficulties. Firstly, equation (3) is a superposition. definition of probability from the frequentist inference Interaction with the measuring apparatus has failed to (probability is defined as the ratio of the number of ele- select a particular value for the observable, and inter- ments with property X to the total number of elements) ference may still be observed. This may not, at first, to the Bayesian inference (probability is defined as a de- seem so problematic because it is often argued that it gree of belief in a given statement). As a result, the would be too difficult to observe interference among the collapse of the wavefunction is assumed nonphysical. We superposed states because the apparatus is macroscopic. find this view untenable. The wavefunction after col- According to this argument, equation (3) is effectively lapse represents a radically different physical system than 2 before collapse. Consider a gambler betting on a horse ρS = |ki| |oiihoi|, (4) race. Assume she has some (incomplete) data on each Xi horse. Her bets are distributed according to the data. If she is given new information about the horses, her bets which is a mixed state with probabilities dictated by will generically be different. Such is the case with wave- the Born rule. For current technologies it would be function collapse in QBism. However, the gambler’s bets very difficult indeed to observe interference with a (near) have no effect on the outcomes of the races, and as such macroscopic device. Fundamentally speaking, however, the analogy breaks down. Finally, the claim that the equation (3) does permit the observation of interference. collapse is a result of the changing knowledge of the ob- Hence we cannot disregard the superposition in a funda- server (agent) contradicts the well verified dictum that mental description, as future technologies may bring it knowing the wavefunction of a system represents a state within experimental grasp. We refer to this deficiency of complete knowledge of system. In standard quantum as the superposition problem. Secondly, (3) suffers from mechanics and quantum statistical mechanics, states of the so-called preferred basis problem [50][26]. We briefly incomplete knowledge (epistemic information) are repre- state it here. Since nothing at this point has selected sented by density matrices (mixed states). Contrary to a preferred set of basis states of the system S or of the this extremely successful model, QBism claims that pure apparatus A, one might imagine re-expressing (3) using ′ ′ ′ states also represent incomplete knowledge. It is appar- |oii = j fij |oj i and i kifij |aii ≡ kj |aj i (no sum on ent that these definitions are mutually exclusive, and it is j) as P P unclear how the distinction is made manifest in QBism. ′ ′ ′ = ki|oii|aii (5) Xi III. DECOHERENCE ′ th ′ where |oii is the i eigenstate of a new observable O , ′ ′ The decoherence program attempts to bypass the diffi- and |aii is the pointer state corresponding to oi. The pre- culties associated with the Copenhagen interpretation by ferred basis problem is quite serious. It implies that the claiming that while the evolution of a perfectly isolated unitary evolution of quantum mechanics is incapable of system is governed by Process 2 alone, no quantum sys- indicating if a particular observable was “observed.” This tem is truly isolated during the measurement process. It stands in strong contradiction to common experimental is, instead, an open system actively interacting with a procedure. If an experimentalist aims to observe the po- quantum environment. Environmental perturbation de- sition of a particle, for example, then a certain device (a stroys coherence among superposed states by spreading position measuring apparatus) is employed and a posi- the initial coherence throughout the environment, whose tion measurement follows. That is, it seems that during state is inaccessible experimentally. It produces mixed the measurement process a base is indeed selected. The states when the environmental degrees of freedom are preferred basis problem indicates that equation (3) has traced out [59] [60] [61]. More explicitly, consider, as be- not only failed to select a particular eigenvalue, it has even failed to select a particular observable! For these fore, a system S whose initial state is |Sini, and a measur- reasons (3) cannot be referred to as a “measurement” in ing apparatus A with initial state |Aini. The apparatus is assumed to perform the following function any strict sense. The decoherence program proposes the following res-

USA|Sini ⊗ |Aini = ki|oii|aii (3) olution. Let the system and apparatus together interact Xi with an environment E whose initial state is |ǫini. The interaction with the environment is unitary and it is as- Where |oii is an eigenstate of observable O with eigen- sumed that it produces a state value oi, |aii is the pointer state corresponding to eigen- value oi, and we have assumed that |Sini = i ki|oii. USAE|Sini ⊗ |Aini ⊗ |ǫini = ki|oii|aii|ǫii (6) This Von Neumann measurement of the first kindP seems Xi 5

If such an interaction occurs, then there is only one base ing measurements” in the first place. Experimentalists of system and apparatus which maintains perfect corre- must, therefore, “trust” the reading on the pointer of lation even after interacting with the environment. Said their apparatus. Can we know if the observed pointer basis is deemed “robust” against environmental interac- reading accurately represents the “system value” or was tion and is referred to as the environmentally selected it perturbed (perhaps even strongly) by the environment? pointer basis. It is in this fashion that the decoherence Clearly, one cannot know without an appeal to another program claims to solve the preferred basis problem [59] measuring apparatus, for which the same difficulty may [46] [47]. arise. It is universally agreed that only testable state- Does decoherence actually solve the preferred basis ments should be included as part of a theory. Can the problem? We will argue that decoherence does not solve claim of perfect correlation among system, apparatus, preferred basis problem, and more generally, that uni- and environment be tested? Since the interpretation tary dynamics alone cannot do so [26]. First consider a rests on this statement (decoherence fails to be an inter- general argument. Quantum mechanics postulates that pretation if this statement does not hold good), then any each system is fully described by a vector (more prop- tests of this statement, which must proceed by not assum- erly Hilbert space ray) in a Hilbert space, and evolves ing perfect correlation, could not be interpreted. They via unitary action. That “a system is fully described by would be non-interpretable. Consequently, the statement a Hilbert space vector” ensures that all systems that are of perfect correlation must be taken as an untestable fun- described by the same state vector are physically equiv- damental postulate, which, within the context of physical alent. If this were not so, then there must be a physical theories, is highly dissatisfying and strongly suspect. quantity that acts to distinguish between identical state A key point must be highlighted here, we do not, a pri- vectors, which is the definition of a hidden variable. As ori, know the form of the unitary interaction in (6), yet a result, if one is unwilling to accept hidden variables, (6) supposes that the environment “Von Neumann mea- then basis independence must also be assumed. Further- sures” the system and apparatus. We wish to emphasize more, the unitary sector of quantum mechanics (unitary how unlikely it is that this assumption is realized natu- evolution without collapse mechanism) completely lacks rally. The measuring apparatus A is designed via the in- a mechanism to single out a base. That is, there is no teraction exhibited in 3 to produce a perfectly correlated type of interaction that might select a basis from a dy- state. It is obvious to every experimentalist that there is namical origin. One is compelled to concede that despite a great difficulty in building a device which acts on a sys- claims otherwise unitary theories do not have preferred tem to produce perfect (or even near perfect) correlation bases. Let us argue further. One may tie the freedom to between states without introducing uncontrollable phase choose a Hilbert space base to the freedom of coordinate shifts. Yet (6) states that the environment performs this system choice. Thusly, that the choice of coordinate sys- function (idealized Von Neumann Measurement) without tem is nonphysical dictates that basis choice must also trouble. While one might argue that this is possible in be nonphysical. The claim that the inclusion of an en- principle, it is also highly contrived. What prevents, for vironment which interacts with the S + A (system and example, in a number of experiments, the result apparatus) system “selects” a basis is faulty. The full system (S + A + E) remains a unitary quantum system USAE|Sini ⊗ |Aini ⊗ |ǫini = κi|oii|aii|ǫii (7) and without an additional mechanism to endow differ- Xi ent bases with physical significance, all basis choices are (where κi 6= ki, and where we still explicitly assume that valid. |Sini = i ki|oii)? Clearly, states such as (7) do not Given that no mechanism in the unitary sector of quan- follow theP Born rule, and so if they were to occur, deco- tum mechanics selects a base, then in what sense is the herence would immediately be at odds with experiment. base which allows for the decomposition (6) “preferred” The decoherence program has the burden to justify (6), to other bases? The answer put forward in the deco- which is tantamount to deriving the Born rule. What herence program is that the decomposition (6) is pre- level of justification would prove sufficient? To justify ferred because it has a particularly appealing interpreta- the unitary action in (6), adherents to the decoherence tion from the viewpoint of a human observer. Namely, program must model the environment as a system of par- the measuring apparatus has “read” the system and is ticles which interacts with S in known ways (electromag- not further disturbed (altered) by the environment. This netically, or perhaps gravitationally) and from this model is both highly ad hoc and circular logic as the base chosen deduce that the aforementioned unitary action is possi- for decomposition within the interpretation is preferred ble and that it occurs vastly more frequently than any only by the fact that it permits an interpretation. Adher- other unitary transformation (so as to remain in agree- ents to the decoherence program are asking one to aug- ment with the Born rule). Let us highlight just how un- ment unitary quantum mechanic with a non-quantitative natural the assumption of (6) is with an accurate analogy. rule for base selection that is constructed on subjective Let the system of interest and apparatus be considered a grounds. boat immersed in the environment, here conceived of as One does not know, a priori, the state of system. It the ocean. Then (6) indicates that size and shape of the is this lack of knowledge that provokes one into “mak- swells are affected by the presence of the boat, but it ex- 6 plicitly excludes any backaction so that the ocean waves sibility, argue for tracing over the environmental degrees have no effect at all on the boat. This we find to be an of freedom [60]. If the interaction with the environment absurdity! Surely any reasonable model of the environ- is responsible for generating state (6), then what is re- ment must permit a great disturbance to the system so sponsible for transforming (6) into the reduced state, that the initial state is scrambled. T The difficulties faced by (6) do not stop there. It is ρS = T r[USE|Sini ⊗ |ǫinihSin| ⊗ hǫin|USE]? (8) clear that in a normal experimental setup in which ob- environmental servable O is measured, a device of a very particular na- The answer most often cited is that of ignorance ture must be constructed. If one wishes to measure O . Namely, the environment is not measured instead of O′, then two different devices must be con- or the experimenter lacks control over the environment. structed. For example, it is evident from experiment that The mention of “measurement” or “ignorance” in an ar- devices which make position measurements are dramati- gument for the trace procedure is unacceptable. Do- cally different from those devices which perform momen- ing so reintroduces measurement as a fundamental pro- tum measurements. This obvious fact is missing from (6). cess, which is the main difficulty with the Copenhagen Indeed, the correlation introduced by the apparatus alone interpretation that a new interpretation must address. works for all observables, as we have noted, and it is, ac- How can the mere ignorance of the measurer (the ex- cording to (6), the environment which finally selects the perimenter) manifest itself as a physical reduction of the “observed” observable (for lack of a more concise way of system from pure state (6) to mixed state (8)? This stating it). It is suspected that the environment exhibits statement indicates that the mental state of the exper- statistical randomness and furthermore it is assumed that imenter has physical bearing experiments. Surely this the state of environment is outside of the experimenter’s is spookier than the state reduction at the hands of a control. Thusly we should expect, in a best case scenario, measuring device in the Copenhagen interpretation. At that the experimenter does not decide which observable least in the latter case, a physical interaction (between is made manifest in a particular measurement. The no- measuring device and system) may be blamed! tion of “constructing a position measuring apparatus to Tracing is explicitly nonunitary. Recall that, as we measure positions” is well documented but does not have have argued in section II A, nonunitary behavior is (at a place in the decoherence program. Even worse is the least in principle) distinguishable from purely unitary be- possibility that different observables may be selected in havior. Let us extend the argument of section IIA to the different trials of an experiment because of changes in the general case of arbitrary nonunitarity, including decoher- environment. ence from environmental interaction. Consider a total system, which consists of a system of interest S, an ob- In the decoherence program, one tacitly assumes that server A, and the environment E. We will assume that |oii in (6) are the eigenstates of an observable. How- the observer A is able to “measure” system S, but is inca- ever, without an adequate model of the unitary evolu- pable of measuring the environment, as per the argument tion in (6) or (3) one cannot claim to know the states for tracing in the decoherence approach). Equivalently, There is currently no mechanism within the deco- |sii. we may assume that A chooses to ignore the environ- herence program that guarantees that states will be |oii ment (also per the argument for tracing in the decoher- the eigenstates of an observable . Assume at this point ence approach) Regardless of how one wishes to argue that it can be demonstrated that |oii are the eigenstates for properties of A, the final result is that A is assumed of an observable. For something definite and without capable, via interaction, of reducing pure density matri- loss of generality, one might imagine that each |oii corre- ces to mixed density matrices. Since the environment is sponds to a discretized position eigenstate, |xii referring considered part of the system, the total system may be to “box” i. Now, it might seem that the formalism in- regarded as closed (there is nothing more external to the dicates that interaction with the environment has lead total system with which its component systems, S, A, or to the localization of system S to within a particular E interact). If all closed systems evolve unitarily, then box |xii. This conclusion is false because a superposition the state of the total system must be of many eigenstates remains. We conclude that, despite the environmental interaction in (6), the resulting state U |S i ⊗ |A i ⊗ |ǫ i = k |o i|a i|ǫ i (9) still suffers from the superposition problem. Indeed, no SAE in in in i i i i Xi superpositions have been destroyed nor have any states with definite classical properties arisen. Outside of a rel- which is pure. If there is a reduction from pure to mixed ative state or “many-worlds” approach (we will discuss state induced by any of the interactions, either with the this possibility in section IV), the state (6) does not lend observer A or possibly with the environment, then the itself to interpretation. final state will not be pure, but will instead be described As we have previously mentioned, if the environment is by a mixed density matrix ρ. It should be clear that a quantum system (all interactions are unitary), then the the details of how the nonunitarity arises are not impor- final state remains pure, and the interpretation of such a tant for this argument. It will go through as long as some state requires the relative state formalism. Adherents to type of nonunitarity gets introduced in the final state. To the decoherence program, who apparently reject this pos- complete the argument recall from Section I, that pure 7 or mixed final states form mutually exclusive alterna- unnatural and unprecedented. The initial states of clas- tives as they give rise to different experimental results. sical systems are free to explore the whole of phase space. Thus, we find that the assumption of unitary evolution of It is expected that quantum systems should exhibit the closed systems is incompatible with any nonunitary evo- same freedom. The restriction of the initial state to non- lution whatever. One might be tempted to evade this superposed states (i.e. the initial state must be ultra conclusion by arguing that the reduction from (6) to (8) pure)[62] indicates that the initial entanglement entropy is not “real.” That is to say, the system and environ- is low, since there are many more states that correspond ment truly are in state (6), and (8) merely arises as an to entangled states which must be excluded as possible effective description. However, it is evident that (8) can- initial apparatus states. Accordingly, the initial state is a not replicate (6) for all observables, as there is a loss of highly ordered state. To phrase this colloquially, consider phase relationships in the transformation from (6) to (8), a formalism describing the air in the room. While it is and correspondingly a loss in possible interference effects. possible that the initial state of the system corresponds One might hope that for a limited class of observables (8) to a macrostate in which all the air was found in one cor- does replicate (6). Under which circumstances this may ner of the room it is extremely unlikely. If the formalism be the case is certainly of pragmatic interest. It cannot, demanded, on the basis of consistency, that the initial of course, be of fundamental theoretical interest, as the system state be such an unlikely configuration we would fundamental object is the state (6), nor can we be con- hardly give it credence. Without an additional mecha- cerned with a limited class of observables at a fundamen- nism that selects the initial states, this interpretational tal level. Note that without a physical state reduction, requirement enters the theory as an ad hoc, and unjus- the evolution is purely unitary, and thus is an example of tifiable, assumption. We refer to this deficiency as the a relative state/MW theory, the discussion of which we Initial Entropy Problem, and note here that it appears in withhold until the section IV. other formulations of quantum mechanics. A similar question with regard to an initially low en- tropy and the need for fine-tuning arises in cosmological A. Initial Entropy Problem considerations. Namely, the second law of thermodynam- ics suggests that the initial entropy state of the universe is extremely low. If this is so, then fine-tuning is appar- It is evident that a realistic environment has a Hilbert ently needed to account for this low entropy initial state. space with a large number of dimensions; however, in- See [12] for further details. sight can be had by considering much simplified models. In particular, we have introduced an environment whose approximate base may be taken to be Γ = {ǫin,ǫj}. It is evident that Γ functions as an approximate base be- B. Observer Energy Problem cause each state in Γ corresponds to a macroscopically distinguishable situation, |ǫini ⇋ “the environment has An essential point is that the system of interest, the not interacted with S,” |ǫj i ⇋ “the environment recog- measuring apparatus, and the environment must neces- nizes value oj for observable O,” and must therefore sat- sarily form a closed system. Well known is that this de- isfy hǫi|ǫji ∝ δij ; hǫj |ǫini ≈ 0. We now observe that for mands that the evolution of the full system is unitary. Of- the decoherence program to replicate the Born rule, the ten overlooked, however, especially within the context of initial state of the environment must be ǫin, see equa- measurement problem resolutions, is the fact that closed tion (6). Of particular note is that the initial environ- systems must also conserve energy. Does the decoherence ment state cannot involve a superposition of |ǫini and formalism satisfy this well established dictum? Let us |ǫj i, because such an initial superposition immediately demonstrate that it does not. Recall that energy conser- spoils the interpretation of (6), and thereby produces vation in quantum mechanics demands that the Hamilto- a mismatch between the predictions of decoherence and nian is time independent. Notice also that the evolution the Born rule. We conclude that the decoherence in- of quantum systems thus far observed in experiment has terpretation requires a superselection rule on the initial two regimes of evolution. The first regime is the “regular” environment state which filters superpositions in order unitary regime wherein the system of interest S, the ap- to guarantee agreement with experiment. Equivalently, paratus A, and the environment E evolve independently. it may be said that decoherence requires fine-tuning be- In the second regime, herein referred to as the “deco- cause only one state out of many is allowable as an ini- herence regime,” S, A, and E become strongly coupled. tial state. One may na¨ıvely believe that one must model Indeed, the coupling is assumed so strong that the time with realistic environments. However, in any realistic en- evolutions of the individual systems become negligible in vironment, there are a very large number of basis states. comparison. Call time t∗ the decoherence-on time, and As a result, the fine-tuning is exacerbated because one let it be the time that decohering effects become impor- ∗ is selecting ǫin (more precisely the equivalent of ǫin in tant. Then the evolution of the full system before t is a realistic environment model) as the only viable initial governed by HS,HA,HE. The system of interest, the environmental state out of many. apparatus, and the environment evolve independently in The requirement of such a superselection rule is both the regular unitary regime. After t∗ the systems interact 8 via HSAE and the systems enter the decoherence regime. approach, it is assumed that all evolution is unitary. No It is straightforward to see that the full Hamiltonian is collapse, projection, nor ‘trace-type’ behavior ever oc-

∗ ∗ curs. Instead, Von Neumann measurement of the first (HS + HA + HE) ǫ(t − t)+ HSAEǫ(t − t ), (10) kind, which permits the formation of a correlation be- tween a system and an apparatus via unitary evolu- where ǫ(x − y) is the Heaviside step function. It is clear tion alone is taken to be sufficient for describing exper- that equation (10) has explicit time dependence, and iment. Of course, the state of perfect correlation be- therefore does not conserve energy. It is also the Hamil- tween system and apparatus does not select a particu- tonian of a closed system, and thus violates the conserva- lar measured value. The state is one of superposition tion of energy for a closed system. An objection may be among perfectly correlated terms (i.e. it is of the form raised because of the sharp turn on time t∗, but it is easy |ii |ii ). A point of contention arises to see that even if the turn on were smooth, in so much as i system apparatus when one is forced to interpret the standing superpo- two regimes of evolution are present, some explicit time P sition. It seems that the most natural interpretation is dependence is necessary, and thus the argument would go that all allowed correlations persist, thereby generating through. As a matter for future work, it may be possi- a set of “quantum worlds.” While this might be at odds ble to begin with a single time independent Hamiltonian with some innate sense of “simplicity,” the resulting the- whose evolution on the full system exhibits the two ob- ory has fewer assumptions than other approaches, and served regimes. We caution, however, that this approach so is quantitatively simpler. The RS/MW interpreta- for resolution is difficult because one would need to inter- tion seems at first quite promising because of the great pret from the dynamics of the system the decoherence-on elegance in eliminating assumptions about intermittent time t∗, which appears to be a degree of freedom in a nor- non-unitary evolution; however, there are assumptions mal experimental setup (i.e. an experimenter can allow necessary for the interpretation which, upon closer ex- decoherence to occur at any time she wishes). Thus, we amination, are severely ad hoc [51]. Let us briefly review see that this approach has features of superdeterminism. the relative RS/MW interpretation. We once again begin Since the apparatus and environment function effectively by considering a system S in state |S i. The evolution in as an observer and in accord with this, we refer to this in all circumstances is purely unitary and so we need only deficiency as the Observer Energy Problem. focus on measurement for a complete understanding of the interpretation. We introduce an apparatus A, with C. The Tails Problem initial state |Aini. The interaction between the system and the apparatus is as follows The action of tracing out the environmental degrees USA|Sini ⊗ |Aini = kj |oj i ⊗ |aj i, (11) of freedom from the full density matrix produces an ap- Xj proximately off diagonal density matrix. This density matrix, the “system density matrix,” may therefore be where |Sini = j kj |oj i and where |oj i corresponds to interpreted as describing an approximate mixed state. “the system hasP value oj for observable O,” and |aj i cor- However, the off diagonal terms do not vanish completely responds to “the apparatus determined that the system for any finite time. As a consequence, the system density has value oj for observable O.” It is evident how (11) matrix is only mixed to the extent that the off diagonal yields the interpretation that the apparatus has measured terms may be neglected. In a position base (if appli- a particular value for the observable since all the terms cable) the nonzero off diagonal terms extend to spatial demonstrate perfect correlation between the system state infinity and as a result constitute “tails” of the distribu- and the apparatus state. Notice also that the superposi- tion [37]. Although no quantitative difficulty arises from tion survives and so (11) must be interpreted as a set of the presence of tails, they complicate the interpretation quantum worlds [22] [8]. by demanding that one make a judgment of size when It appears, at first glance, that (11) corresponds to a the formalism does not supply a scale. That is, when are measurement of observable O. However, it is evident that the off diagonal terms sufficiently small in comparison to it suffers from the preferred basis problem (see section the diagonal terms so that the density matrix is mixed? III). That is, under a change of base from the eigenstates We mention here in passing that the tails problem is not of observable O to the eigenstates of a different observable restricted to the decoherence formalism, but appears also O′ (11) maintains perfect correlation. Thus, one cannot in objective collapse theories [46] [47]. claim that (11) corresponds to a measurement of any particular observable. This difficulty, of course, is well known and in fact spurred on the decoherence approach, IV. RELATIVE STATE/MANY WORLDS which we have argued in section III fails to adequately INTERPRETATION address these issues. The RS/MW interpretation faces additional deficien- The relative state/many worlds (RS/MW) interpre- cies. In the terminology of the previous section, it is said tation represents a radical departure from either the that the RS/MW suffers from both the initial entropy Copenhagen interpretation or decoherence [22]. In this problem and the observer energy problem. Let us review 9 these problems in the context of the RS/MW interpre- Let us proceed by reviewing the treatment of a single tation. The interpretation requires that a Von Neumann point-like system. The wavefunction satisfies the stan- measurement (of the first kind), such as (11), be realiz- dard Schr¨odinger equation, which we do not state. The able. In turn, such interactions require particular initial guiding equation is states for the measuring apparati. That is, the measur- ing apparatus must be in state |A i. If the initial state dx ∇ψ(x(t),t) in m = ~ Im , (12) of the measuring apparatus is any other state or a linear dt  ψ(x(t),t)  superposition of |Aini and other states, then the interpre- tation fails. Since the initial state must be very precisely where m is the mass of the particle. The guiding equa- fixed, it is in a very low entropy state, yet, no mechanism tion is first order in time. The resulting theory is fully for establishing such low initial entropy is provided (see deterministic. Notably, probability does not appear as a section III). fundamental constituent of the theory. Let us raise our The RS/MW interpretation suffers from the observer first objection. As defined by the Schr¨odinger equation energy problem (see IIIB). The full system, composed of and the guiding equation, (12), this theory is not equiv- apparatus (observer) and system of interest, is a closed alent to standard quantum mechanics. Its predictions system. There is, initially, assumed to be no interaction may differ wildly from those of the Copenhagen interpre- between the system of interest and the apparatus; how- tation, for example, and is as a consequence, strongly at ever, at a finite time t∗ the interaction must be “turned odds with experiment. In order to guarantee agreement on.” This induces a time dependence in the (full) Hamil- with experiment one must impose the quantum equilib- tonian, which implies nonconservation of energy. This rium hypothesis. It states that the probability distribu- 2 is inconsistent with the well verified principle of energy tion for positions must be given by |ψ(x,t0)| . Once this conservation together with the assumption of a closed distribution is fixed at one time t0 it will be preserved system. under time translation [9] [10] [11]. It is uncertain (and Finally, defining probabilities in the RS/MW interpre- not specified by the theory) whether the quantum equi- tation faces challenges. It is, strictly speaking, impossible librium hypothesis is always fulfilled or if nonequilibrium to derive the Born rule and the corresponding probability states exist in nature (if only for short times) [8]. interpretation from the RS/MW formalism because one The guiding equation indicates that for real-valued assumes only unitary evolution on an appropriate Hilbert spatial wavefunctions the velocity of the Bohmian par- space, which contains no inherent notion of sample space, ticle vanishes. This predicts physical phenomena. To frequency, degree of belief, nor any other quantity that demonstrate that this feature makes predictions in con- may be bridged to probability theory without further as- flict with experiment we need only consider the ground sumption. Claims that the Born rule have been “derived” state of the hydrogen atom. The spatial wavefunction from a purely Everettian approach (without additional corresponding to this state is real-valued and the corre- assumptions) are faulty [48] [49]. The possibility of ar- sponding Bohmian particle velocity vanishes. Therefore, guing for the Born rule as a unique norm on quantum the ground state of the hydrogen atom would form a per- worlds remains open [17] [53] [54] [55] [56] [43] [44]. If manent electric dipole. In addition to greatly altering superpositions always remain, then the number of quan- the interactions in (single atom) hydrogen gas, the per- tum worlds forms a continuum. To define probabilities on manent dipoles could be aligned in weak electric fields, the continuum requires an appropriate probability mea- producing photon flux as they de-excite from the higher sure. Whether this can be defined, is unique, and if it energy anti-aligned state. None of these phenomena are follows from the dicta of the approach, are open issues. observed, and thus it must be concluded that Bohmian It is not even clear how the quantum worlds should be mechanics is inconsistent with experiment [40] [36]. interpreted. For example, should the frequentist infer- Like several other interpretations we have considered, ence be used or must some other inference scheme be measurement in Bohmian mechanics relies on the use of used/developed [45]? a Von Neumann measurement of the first kind. That is, it assumes that the system of interest S and the appara- tus A originally have independent wavefunctions Sin(x) V. BOHMIAN MECHANICS (x is the position of a particle) and Ain(y) (y is the po- sition of the pointer), respectively, and that during the measurement process the evolution is as follows Bohmian mechanics attempts to resolve not only tech- nical difficulties concerning measurement, but also the (S) (A) U [S (x)A (y)] = k ψ (x)φ (y) ≡ Ψ(x,y,t) conceptual difficulties facing quantum mechanics by pos- SA in in j j j Xj tulating that particles follow a trajectory (have a posi- (13) tion at all times). The trajectory is dictated by a guid- x (S) x (S) x ing equation, which is written in terms of a wavefunc- where Sin( ) = j kj ψj ( ), and where ψj ( ) is the P (A) tion [9] [10] [11]. The wavefunction satisfies the standard eigenstate of observable O with eigenvalue oj and φj (y) Schr¨odinger equation. This approach is not an interpre- is the wavefunction of the “pointer position” correspond- tation, but instead a competing theory [8]. ing to eigenvalue oj . In the relative state/many worlds 10 interpretation equation (13) would be said to describe a na¨ıve response may be to take the pointer very massive measurement of observable O, where the quantum worlds so that wavefunction spreading is negligible. However, remain in superposition. Within Bohmian mechanics, there is no justification for taking the initial wavefunc- the wavefunction generates the equations of motion for tion to have a single peak, and even with the quantum the particles, hypothesis satisfied many initial pointer positions will not coincide with the peak. Additionally, a massive pointer dx ∇xΨ(x,y,t) mx = ~ Im (14) fails to introduce a distinction (classical versus quantum) dt  Ψ(x,y,t)  between the system and the pointer, because the system dy ∇yΨ(x,y,t) too has nonzero mass. What we mean by this is that the my = ~ Im , (15) dt  Ψ(x,y,t)  mass is continuously variable, and properties that change as functions of the mass are also continuously variable, where mx is the mass of the particle system and my is as a consequence if the pointer is approximately OWD, the effective mass of the pointer. It is then supposed that then the system is also OWD, albeit to a worse approx- (A) φj (y) are sharply peaked and at late times are suffi- imation. We argue more generally as follows. From a ciently widely separated that an observation of y allows fundamental standpoint the system and apparatus are (A) one to deduce which state (packet) φj (y) is occupied, (and indeed must be) treated symmetrically — both are which from equation (13) allows one to learn which sys- fully quantum systems. If the pointer may be observed (S) “by simply looking at it” (is OWD), then by logical con- tem state (packet), ψj (x), is occupied, so that one has effectively “measured” observable O and that the effec- sistency the system must also be OWD. The claim that quantum systems follow trajectories that are OWD is (S) x tive wavefunction for the system is just ψj ( ). Finally, currently outside experimental justification. it is argued that the empty states will not again inter- Bohmian mechanics, as defined by the Schr¨odinger fere because the system and apparatus will have already equation and the guiding equation (12) does not delineate coupled to many other degrees of freedom. See [10] for a a regime for a quantum to classical transition. With- more detailed presentation. out modification (or more likely additional dynamical There are several important objections to this descrip- rules), nothing prevents Bohmian mechanics from pre- tion of quantum measurement. Firstly, equation (13) suf- dicted large scale quantum effects, which are obviously fers from the initial entropy problem — introduced in not observed. The most common remedy is to introduce IIIA. The initial apparatus state cannot be in a super- decoherence into the theory, which allows one to compute position or the interpretation of equation (13) fails as a necessary quantities such as decoherence times. We ar- “measurement.” The initial state is, consequently, a very gue against decoherence as an independent interpretation low entropy state, and without a mechanism to reduce in Section III. A Bohmian-decoherence hybrid theory re- the (entanglement) entropy, such as intermittent nonuni- tains the deficiencies of both, at least to the extent that tary behavior it is exceedingly unnatural that the appa- Von Neumann measurement of the first kind remains an ratus be initially in such a state. Secondly, equation (13) essential constituent. suffers from the observer energy problem — introduced Bohmian mechanics, which incorporates interactions in III B. Namely, it is essential that a Von Neumann mea- among the full configuration space (particles interact di- surement Hamiltonian be induced between the system of rectly even when spatially separated) cannot readily be interest and the apparatus at some finite time t . There- 0 made relativistic. It is unclear how, or importantly “if,” fore, the Hamiltonian is endowed with explicit time de- one can realize a relativistic version of the Bohmian the- pendence and accordingly energy is not conserved. As ory that is not also immediately at odds with experiment. a result, either the system of interest + the apparatus Related to this problem is the question of producing a cannot be regarded as a closed system or we must accept “Bohmian field theory” that accounts for particle cre- a violation of energy conservation. Finally, it is tacitly ation and annihilation. These both constitute matters of assumed that the pointer position y may simply be “ob- future work. served” or, if you will, read from the dial. We say that, in the Bohmian mechanics approach, the pointer position is “observable without disturbance,” or simply, OWD. It is evident that in the treatment there is complete symmetry VI. MODAL INTERPRETATION between x and y. That is, both the system and the appa- ratus are given a full quantum description. If the pointer The modal interpretation, or more accurately, “modal position may be observed without disturbance, then why interpretations” disposes of the standard eigenstate- is the system (particle) not also OWD? It is claimed that eigenvalue link (a system has a definite property oi iff the pointer position is a near classical variable, but it it is eigenstate |oii) and instead propose that a system is not described how this near classical behavior arises always has a set of definite properties, referred to as the nor is it described how the pointer position becomes ob- value state, which is independent of the dynamic state, servable (it is still described by a wavefunction). More which always evolves according to the Schr¨odinger equa- precisely, how does one quantitatively adjust the treat- tion [18] [52] [5]. It is obviously necessary to introduce ment of the subsystem y so that y is near classical? A a mechanism that selects which properties have definite 11 values and at what times. This rule for ascription of def- tradictory to known experimental results. Consider the inite properties is called an actualization rule. Currently double slit experiment. The initial state of each particle this “interpretation” is incomplete because no definitive must be in a (near) momentum eigenstate of the same actualization rule has been agreed upon. Instead, many momentum as an initial state or the interference pattern possible actualization rules exist and many have been will be washed out. If a spontaneous collapse occurred proposed. For example, one may take the possible value with a given frequency, then surely some of the parti- states to be elements in an orthonormal basis of the cles would suffer a collapse during the travel between Hilbert space of the system and a probability density the source and the detection screen. Indeed, some frac- is defined on these values in accordance with the Born tion of them will collapse at or immediately before the rule. How is the orthonormal basis selected? It is not double slits. The interference pattern, after many trials, selected from the unitary formalism. Consequently, we would be disturbed. Notice that the particles need not see that the modal interpretation suffers from a preferred collapse to a position eigenstate, any nonlinear evolution basis problem. One may also permanently select a pre- or stochastic jump, occurring even with low probability, ferred basis, for example, a position base. The result is would manifest itself by deforming the interference pat- a theory resembling Bohmian mechanics; however, the tern. No such deformation has been observed. formalism does not generate an evolution equation for It is possible that spontaneous collapses occur with the position state (the guiding equation, (12), or some an extremely low probability. However, the low proba- equivalent must be supplied). One might take energy bility of their occurrence dictates that they cannot play or momentum as a preferred base, but such possibili- an essential role in resolving the measurement problem. ties have not been explored. This approach is currently Again, consider the double slit experiment. Very few incomplete. Indeed, depending on how one proposes to spontaneous collapses occur en route, yet, an overwhelm- complete it the approach may have an enormous possible ingly large number of particles produced by the source are number of ontologies and subsum many different inter- detected by the detection screen in near position eigen- pretations. It is for this reason that we reluctantly deem states. An additional mechanism must be responsible for the modal approach an “interpretation.” the abrupt collapse that occurs at the detection, which Na¨ıve attempts to complete the modal interpretation is not described at the present time by any objective col- give rise to familiar problems, for example, the preferred lapse theories. basis problem. Similarly, modal interpretations often makes use of the decoherence mechanism, which we argue in Section III gives rise to initial entropy and observer en- VIII. CONSISTENT (DECOHERENT) ergy problems. The resulting modal interpretation would HISTORIES also suffer the aforementioned deficiencies. Finally, elimi- nating the eigenstate-eigenvalue link raises the possibility The consistent (decoherent) histories approach to the of measurements of property O that reveal value oi, but measurement problem proposes that the fundamental on- output states different than |oii. More precisely, without tology of closed systems is a “history.” Specifically, at an eigenstate-eigenvalue link measurements of O that re- each time ti in a series of times ti = {t0,t1, ...tn} one as- veal value o may output states of the form k |o i, (ti) i i i i sociates an exhaustive set of projection operators {P } which indicates that a second measurement may reveal j P (ti) a value for O different than oi. The occurrence of such ( j Pj = I). Observe that for each time ti the set phenomena is easily experimentally distinguished from P(ti) Pj forms a partition of the identity in the index j. quantum mechanics under the Copenhagen interpreta- We highlight this property because it will prove vital for tion, and has hitherto been unobserved. interpretation. One constructs a history, Y , as a time ordered sequence of projection operators

VII. OBJECTIVE COLLAPSE THEORIES (t0) (t1) (t2) (tn) Y = Pj0 Pj1 Pj2 ··· Pjn , (16)

Objective collapse theories purport that collapse is a and ascribes to it the interpretation that the system has physical phenomenon. Often the collapse is assumed property Pj0 at time t0, property Pj1 at time t1,... and spontaneous and may be induced by a nonlinear term in property Pjn at time tn. Since the projection operators the Schr¨odinger equation [27] [28] [29]. These theories do form a partition of the identity, the associated properties not constitute a reformulation or interpretation of quan- are mutually exclusive. It is evident that this interpre- tum mechanics, but instead make dramatically different tation implies that a system has a well defined property experimental predictions [28][29]. It is precisely for this only at the times ti. At intermediary times no informa- reason that these theories are non viable. While more tion is known about the system. One defines a chain advanced searches for specific types of nonlinearities are operator as follows, certainly warranted, it is easily demonstrated that objec- (tn) (t2) (t1) (t0) tive collapse theories with spontaneous collapse, such as, Pjn U(tn,tn−1) ··· Pj2 U(t2,t1)Pj1 U(t1,t0)Pj0 . for example, the Ghirardi-Rimini-Weber theory, are con- (17) 12

So that the system evolves unitarily in between “projec- a mere repackaging of the measurement problem as it tions.” We now wish to assign a probability to a history. arises within the Copenhagen interpretation. The key demand that we require is that the probabilities Call a theory incomplete if there exists a statement should be additive, so that the probability of the system which is undecidable from within the theory (the theory following history Y (≡ P (Y )) or the probability of the makes no prediction), but which is decidable through ex- system following history X (≡ P (X)) is P (Y )+ P (X). periment. Consider a system that has at least two realms It is evident that in general the additivity property fails (we will prove later that all systems have at least two because of the unitary evolution. On what set of histories realms). We will refer to them as realms A and B. Con- will this property hold? It is straightforward to demon- sider the statement “the system of interest follows a his- strate that a sufficient condition for additivity between tory within realm A with probability P(A).” An equiva- two histories Y and X is lent statement holds for realm B. We will refer to this as † statement α. It is clear that this statement is undecid- T r CY ρinCX =0, (18) able through the consistent histories formalism because it h i asks about a relative probability between different realms where CY and CX are the corresponding chain operators and the formalism only assigns probabilities between his- of histories Y and X and ρin is the initial state of the tories in a fixed realm. Statement α is decidable through system. If the condition 18 holds for any two histories in experiment. If experiments are capable of detailing the a set, the set is referred to as a realm. One then defines history that a system follows, then one may compute the probability for a particular history Y to be followed P(A) from the ratio nA/N, where nA is the number of within the realm to be observed histories that lie in realm A and N is the total number of observed histories. Therefore, statement α is † P (Y ) ≡ T r CY ρinCY . (19) decidable through experiment, but undecidable through h i the formalism, and thus the formalism is incomplete. We note in passing, as it will be useful for our purposes, An adherent to consistent histories may object to the that the set of single-time histories (a single projector is aforementioned conclusion on the basis that the realm inserted at time t0) form a realm, as is readily proved is chosen by the experimenter, so that the “probability directly from the consistency condition (18) [33] [31] [32] for a given realm” is meaningless. Let us demonstrate [38] [34] [35]. that the probability for a given realm is necessary for consistency and argue that it is an obvious quantity to be deduced from experiment. Consider as a system, a A. Realm Selection Problem system of interest, an apparatus, and an experimenter. A key point is that this system is closed. There are two A deficiency arises in this formulation because of the notable consequences of closure. Firstly, the consistent dependence on choosing a realm. Consider, as an exam- histories formalism deems that it must follow a partic- ple, a system composed of a single particle. It is straight- ular history (in some realm). Secondly, there is nothing forward to demonstrate that the set of histories such that outside the system to enact realm selection. It is straight- the particle has a well defined position x0 at t0 forms a forward to see that there are only two ways to guarantee realm. In a like manner, one may show that the set of the consistency of these two statements. Since a realm histories such that the particle has a well defined mo- must be selected, either the system states/dynamics must mentum p0 at t0 also forms a realm. These realms are be such that there is a single realm (so that there is no obviously mutually inconsistent because the position and choice of realm) or a relative probability among realms momentum operators fail to commute. This system thus must be included in the theory (so that realms are ran- has (at least two) realms. We now ask a question of the dom but occur with different weights). We will first show formalism “what is the probability that the system has a that there are at least two realms for this system. Con- position (as opposed to a momentum)?” It is clear that sequently, we will be forced to conclude that a relative the formalism alone provides no answer to this question. probability among realms must be included in the the- Immediately we see that the consistent histories formal- ory. As a first step in proving that this system has at ism demands a realm-dependent reality – only after a least two realms notice that if the system of interest is a realm is selected can one compute relative probabilities. quantum system, then it has at least two noncommuting How is a realm selected? It is apparent that realm selec- hermitian (observable) operators, without loss of gener- tion is a physical process (i.e. in practice a realm must ality we will take them to be xˆ and pˆ. Then two realms be selected by a choice of experiment or the design of the may be formed by considering the single time realm with measuring apparatus), yet a description of this process is well defined position and the single time real with well excluded by the the consistent histories formalism. This defined momentum. Let us now consider the full sys- is in very strong analogy with the Copenhagen interpreta- tem with apparatus and experimenter included. Apart tion, which introduces a physical process, measurement, from the placement and choice of projection operators, yet a description of this process is excluded from the the- the evolution type in the chain operators is unitary, and ory. Indeed, the consistent histories formalism contains unitary evolution cannot select a base, as per the pre- 13 ferred basis problem, it is clear that there exists a single IX. TRANSACTIONAL INTERPRETATION time realm where the projector projects onto a definite x0 value of position ( for definiteness) and the states “the The transactional interpretation strongly derives from x0 apparatus has found ,” and “the experimenter has ob- the Wheeler-Feynman absorber theory, which we review x served 0.” Similarly, there exists a second single-time briefly [13]. It is well known that Maxwell’s equations realm where the projector projects onto a definite mo- admit both retarded and advanced solutions, in which ef- p mentum 0. Since the operators fail to commute, these fects are delayed or advanced by the light travel time. are two distinct realms for a generic system, as we wished Classically, the advanced solutions are disregarded as to demonstrate. The claim of incompleteness continues nonphysical. In quantum field theory, one cannot simply to hold. banish the advanced solutions by fiat. Indeed, the correct choice of Green’s function (propagator) is the Feynman propagator, which is fully time symmetric because it is B. Inconsistency with nonexistence of null result composed of an equally weighted sum of retarded and measurements advanced propagators. Naturally one might wonder if a time symmetric choice of solutions is viable even at the classical level? Particularly, if one wishes to gen- There are two features, apparent from experiment, erate a theory in which charged particles interact with that will prove valuable in highlighting a further defi- only the fields created by other charged particles (no self ciency in consistent histories. Firstly, the times at which interaction), then the phenomena of radiation damping measurements occur appear to be determined by the ex- (emission) requires the existence of advanced fields aris- perimenter herself. We do not use the term measurement ing from absorbers. If a source is accelerated, then a here to refer to Copenhagen style measurement, but is in- retarded (and advanced) field is created. The retarded stead just a name for a the physical process that occurs field from the source interacts, at a later time, with the during an experiment. That is to say that there appears absorbers, which causes them to emit both retarded and to be a complete freedom of choice in deciding when the advanced fields. The advanced fields from the absorbers, measurement will occur. It is difficult to proceed if this which are defined on the past light cone, interact with freedom is not exhibited, as the resulting theory is su- the source at the initial moment of emission, and are perdeterministic. In the following discussion we ignore such that if sufficiently many absorbers exist then the the possibility of a superdeterminism. Secondly, a mea- net advanced field at the position of the source exerts a surement of property O always reveals that the system ˆ force on the source particle so as to induce the expected is in an eigenstate of O. We refer to measurements of a amount of energy and momentum loss if the emission property whose result is that the system does not have occurred via the standard picture [57] [58]. the aforementioned property as null result measurements The transactional interpretation concerns the applica- (not to be confused with null measurements). With this tion of “absorber theory” to quantum theory. Just as terminology, one may say that null result measurements the equations of electromagnetism admits both retarded do not exist. We may now state a main result. and advanced solutions, other relativistically invariant equations, in particular those with quantum mechani- The statement “The consistent history fol- cal significance (for example the Klein-Gordon equation), lowed by a system is Y ” is false or null result do so as well. An immediate difficulty presents itself – measurements exist. the Schr¨odinger equation does not have advanced solu- tions. It is proposed that the transactional interpretation Proof : Assume that the consistent history followed by a requires a fully relativistic theory. However, it is well system S has is determined to be Y . An experimenter known that the nonrelativistic limit includes both the makes a measurement at any time tA 6= ti, which is not Schr¨odinger equation and its complex conjugate. The one of the sequence times. If a result is obtained, then the solutions to the former are purely retarded and the so- true consistent history was not Y , since Y indicates that lutions of the later are purely advanced. Making use the system has a well defined property at the sequence of the Schr¨odinger equation and its conjugate equation times, and this system has a well defined property at the gives one access to a set of retarded and advanced solu- additional time tA. If a non result is obtained, then null tions. An emitter is assumed to produce a retarded and result measurements must exist. Being able to determine advanced wave. Consider an emission process located the history of a system is essential for comparing this for- at (x1,t1). The retarded wave from the emitter will be malism to experiment. Indeed, if the history is not known called ψE(x,t>t1) (verbally, the “offer wave”) and the nor is the probability. This result indicates that the con- advanced emitter wave will be called φE (x,t t2) (ver- [39], among others. bally, the “confirmation wave”) and φA(x,tDirac equation? For the Dirac equation has been absorbed. In particular the transactional ap- the appropriate product for constructing a probability proach fails to specify if there is a change in the wavefunc- distribution is tion upon absorption. This has a strong bearing on the “repeatability of measurement” assumption of quantum † ρD(x,t)= ψ (x,t)ψ(x,t) (22) mechanics under the Copenhagen interpretation, which states that a repeated measurement that occurs a short The advanced solutions of the Dirac equation are not, time after a first measurement will produce the same however, “daggered.” Therefore, if 20 did hold for the result. Without an assumption concerning the change Dirac equation then the resulting probability distribution in the wavefunction upon absorption, it seems that the would be wrong. A similar argument holds for the Klein- transactional interpretation is incomplete. However, if Gordon equation because there the appropriate probabil- such an assumption is added to the theory, then it is ap- ity is constructed as parent that observation has a distinguished role, which is the measurement problem within the Copenhagen in- i~ ρ = (ψ∗∂ ψ − ψ∂ ψ∗), (23) terpretation. KG 2m t t It is claimed that a transaction occurs if E = hν which is necessary for local conservation of (signed) prob- and other conservation laws are met. This statement ability. It is unclear how (23) could be arrived at from a is vacuous. The approach has not introduced a quantum transaction. equivalent for conservation laws. Namely, there is no Absorber theory demands the existence of emitters mention of an operator formalism, nor is there a men- (sources) and absorbers. Such objects exist in elec- tion of an eigenstate-eigenvalue link. For example, if tromagnetism because generically the equations of elec- the wavefunction is a (near) plane wave, are we to in- tromagnetism are sourced. Namely, the acceleration terpret that as a system with well defined momentum? of charges sources electromagnetic radiation. However, The statement that a transaction may occur if E = hν equations with quantum mechanical interest are not (and other conservation laws are satisfied) suggests that 15 the term “transaction” refers to transition phenomena of Schr¨odinger’s cat results from attempting to answer between energy eigenstates. However, such transitions when collapse occurs. The main difficulty is that certain do not conserve energy/momentum since the formalism physical processes (measurements) seem to induce col- does not include the electromagnetic field (i.e. the elec- lapse (definite properties) but these processes cannot be tromagnetic field functions as external perturbing agent). described from within the unitary sector of the theory. As a result, it is not clear if one should demand en- The transactional interpretation presents no description ergy/momentum conservation or attempt to account for of the measurement process, and subsequently does not the energy/momentum loss to the electromagnetic field. at all resolve the measurement problem. There are two distinct possibilities and no clear answer In the current state of this formalism, one assumes as to which possibility one should appeal. As an exam- that a single emitter interacts with a single absorber. ple of an objective collapse theory, we see that, as ex- Generically, however, one expects that this condition pected, there is energy/momentum nonconservation at will not be met. Emitted waves should interact with the moment of collapse (read transaction). That con- many future absorbers and each of those absorbers servation laws must be added as additional assumptions should emit an advanced wave that interacts with to this theory makes the transactional approach inele- many emitters “in the past.” That only one emis- gant and implies incompleteness and experimental de- sion/absorption/confirmation process in considered is a ficiency. Finally, we emphasize that without an oper- serious shortcoming of this formalism that needs to be ator formalism/eigenstate-eigenvalue link measurements addressed by either (1) rigorous justification or (2) multi- of common physical quantities (energy, momentum, an- ple emission/absoprtion/confirmation processes must be gular momentum, and others) are formally undefined by considered. the theory [13] [14] [15]. We see that there are difficulties concerning the struc- ture of the transactional interpretation. Most poignantly X. TIME SYMMETRIC QUANTUM MECHANICS for this current work, however, is that it fails to re- solve the measurement problem. Consider without loss of generality the transactional interpretation of the Time symmetric quantum mechanics is introduced in Schr¨odinger’s cat paradox. The claim is made that the part to deal with a paradox concerning state vector col- main difficulty with regard to this paradox is in answering lapse caused by simultaneous, but spatially separated when collapse occurs. Since the transactional interpre- measurements. A prime example of which is an experi- tation is (essentially) atemporal asking when a collapse ment to test the Bell’s inequalities. If the measurements occurs is an ill defined question. Instead, the transac- occur simultaneously in an frame in which both observers tional approach suggests that during the period when (Alice and Bob, presumably) are at rest, then in comov- the cat is sequestered, a transaction may occur or may ing Lorentz frames the order of measurements may be not, with probability provided by the Born rule, regard- seen to be reversed, raising the question as to which ac- less of an act of observation (measurement). However, tion collapsed the wavefunction. The interpretation is there are two dilemmas in this claim. Firstly, the formal- arrived at by first computing the probability for an in- ism does not suggest that a transaction is “atemporal.” termediate measurement result oi at present time t given Indeed, the formalism suggests that transactions occur that the initial state is pre-selected to be |ψini and the within well defined temporal intervals. For example, ex- final state is post-selected to be |ψouti. Namely, plicit in the emission of a (half advanced half retarded) wave from an emitter is the time of emission. The ab- P (oi|ψin,tin; ψfin,tfin)= sorption and emission of a confirmation wave also occur † 2 2 |hψfin|U(t →t)|oii| |hoi|U(tin→t)|ψini| at well defined times so that it seems that the transac- fin (24) † 2 2 |hψfin|U |oj i| |hoj |U |ψini| tion occurs within the time interval, even if the precise j (tfin→t) (tin→t) time is undefined. Secondly, the unitary development of P   quantum mechanics predicts that the until the moment The formalism may be further refined by the use of two of wavefunction collapse the system is in a superposi- states, which are operators of the form tion and interference effects may be generated by mea- suring an appropriate variable. The claim of the trans- ρ = |ψ1(t)ihψ2(t)|, (25) actional approach stands in contradiction to standard quantum mechanics by denying the possibility of inter- or linear combinations thereof [1] [2] [3] [4]. Whether ference. Since superposition is an experimentally well equation (24) is useful in understanding quantum me- verified facet, it is clear that there is a serious experi- chanics is not of concern for this work. We ask, instead, mental shortcoming in the transactional interpretation. whether it may form the basis of an interpretation that In this regard the transactional interpretation is an ex- resolves the measurement problem and other related dif- ample of an objective collapse theory, and subsequently ficulties. It is apparent that (24) makes use of the Born suffers the fwailings of such theories (see section VII). Fi- rule, without an attempt at derivation or explanation. nally, it is incorrect to state that the paradoxical aspect We are only lead to assume that an interpretation based 16 on (24) must also assume the Born rule, and as a re- resolution of the measurement problem. sult, does not provide any insight into measurement as a physical process nor does it obviate the need for collapse- like (more generally nonunitary) behavior. While it may be objected that the possibility of deriving the Born rule XI. CONCLUDING REMARKS from the overlap of retarded and advanced waves, in great similarity to the transactional interpretation, still exists, pursuit of such a “derivation” raises further issues (see We challenge the viability of common interpretations Section IX). Equation (24) still indicates that a nonuni- of quantum mechanics. We have argued, and where pos- tary transformation of the wavefunction occurs within sible, demonstrated, that each interpretation exhibits sig- the interval [ti,tf ]. What spurs such an evolution? The nificant deficiency, in the form of fine-tuning problems/ad formalism does not dictate the mechanism. If, as one hoc assumptions, internal inconsistencies, incomplete- might naturally assume, they occur spontaneously, then ness, disagreement with experiment, among others. We the result is an objective collapse theory, which could eas- raise an obvious question, but do not attempt an answer ily be ruled out experimentally (see section VII). We em- here. Is there an interpretation (more appropriately rein- phasize, in passing, that the measurement problem does terpretation) or minor modification of quantum mechan- not appear to stem from a problem with the treatment ics that will maintain full agreement with experiment of time so that variations of standard quantum mechan- (both current and future) and will resolve the measure- ics with different approaches to time will not lead to a ment problem without introducing new issues?

[1] Y. Aharonov, D. Bohm, Phys. Rev. 122, 1649 (1961), [12] S. M. Carroll, “In What Sense Is the Early Universe Fine- reprinted in Quantum Theory and Measurement, eds. J. Tuned?” arXiv:1406.3057 (2014). A. Wheeler, W. H. Zurek (Princeton University Press), [13] J. G. Cramer, Phys. Rev. D 22, 362 (1980). 1983. [14] J. G. Cramer, Found. Phys. 13, 887 (1983). [2] Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. [15] J. G. Cramer, “The Transactional Interpretation of Rev. 134, B1410 (1964); reprinted in Quantum The- Quantum Mechanics,” Rev. of Mod. Phys. 58, 647–688 ory and Measurement, eds. J. A. Wheeler, W. H. Zurek (1986). (Princeton University Press), 1983, pp. 680-686. [16] G. M. D’Ariano, H. P. Yuen, “Impossibility of Measuring [3] D. Albert, Y. Aharonov, S. DAmato, Phys. Rev. Lett. the Wave Function of a Single Quantum System,” Phys. 54, 5 (1985). Rev. Lett. 76, 2832 (1996). [4] Y. Aharonov, J. Tollaksen, “New Insights on Time- [17] D. Deutsch, “Quantum Theory of Probability and Deci- Symmetry in Quantum Mechanics,” VISIONS OF DIS- sions,” Proceedings of the Royal Society of London A455, COVERY: New Light on Physics, Cosmology, and Con- 31293137 (1999). sciousness, ed. R. Y. Chiao, M. L. Cohen, A. J. Leggett, [18] D. Dieks, “Modal interpretation of quantum mechanics, W. D. Phillips, and C. L. Harper, Jr. Cambridge: Cam- measurements, and macroscopic behavior,” Phys. Rev. A bridge University Press (2007). arXiv:0706.1232. 49, 2290 (1994). [5] G. Bacciagaluppi, M. Hemmo, “Modal Interpretations, [19] F. Dowker, A. Kent, “On the Consistent Histories decoherence, and measurements,” Studies in the History Approach to Quantum Mechanics,” J.Statist.Phys. 82, and Philosophy of Modern Physics 27, 239-277 (1996). 1575-1646 (1996). arXiv:gr-qc/9412067. [6] L. E. Ballentine, “The Statistical Interpretation of Quan- [20] F. Dowker, A. Kent, Phys. Rev. Lett. 75, 3038, (1995). tum Mechanics,” Rev. of Mod. Phys. 42, 4 (1970). [21] A. Einstein “Zur Elektrodynamik bewegter K¨orper,” An- [7] J. A. Barandes, D. Kagan, “A Synopsis of the Mini- nalen der Physik, 17, 891 (1905). mal Modal Interpretation of Quantum Theory,” arXiv: [22] H. Everett III, “Relative State Formulation of Quantum 1405.6754. Mechanics,” Rev. Modern Physics, 29 (1957). [8] J. S. Bell, “The Measurement Theory of Everett and De- [23] C. A. Fuchs, R. Schack, “Quantum-Bayesian coherence,” Broglie’s Pilot Wave” in Quantum Mechanics, Determin- Rev. Mod. Phys. 85, 1693 (2013). ism, Causality, and Particles, M. Flato, et al., (eds.) D. [24] C. A. Fuchs, N. D. Mermin, R. Schack, “An Introduction Reidel, Dordrecht (1976); reprinted in J. S. Bell, Speak- to QBism with an Application to the Locality of Quan- able and Unspeakable in Quantum Mechanics, Cambridge tum Mechanics,” arXiv: 1311.5253 (2013). University Press, Cambridge (1987). [25] C. A. Fuchs, “QBism, the Perimeter of Quantum [9] D. Bohm, “A Suggested Interpretation of the Quantum Bayesianism,” arXiv:1003.5209 (2010). Theory in Terms of “Hidden” Variables. I,” Physical Re- [26] B. Galvan, “On the preferred-basis problem and its pos- view 85, 166–179 (1952). sible solutions,” arXiv:1008.3708 (2010). [10] D. Bohm “A Suggested Interpretation of the Quantum [27] P. Ghose, “Testing Quantum Mechanics on New Theory in Terms of “Hidden” Variables. II,” Physical Re- Ground,” Cambridge University Press (1999). view 85, 180–193 (1952). [28] G. C. Ghirardi, A. Rimini, T. Weber, “Unified dynamics [11] D. Bohm, J. Bub, “A Proposed Solution to the Mea- for microscopic and macroscopic systems,” Phys. Rev. D, surement Problem in Quantum Mechanics by a Hidden 34, 2 (1986). Variable Theory,” Rev. of Mod. Phys. Vol 38, 3 (1966). [29] G. C. Ghirardi, A. Rimini, T. Weber, “A Model for a 17

Unified Quantum Description of Macroscopic and Micro- berg, (2007). scopic Systems”. Quantum Probability and Applications, [48] C. T. Sebens and S. M. Carroll, “Self-Locating Uncer- L. Accardi et al. (eds), Springer, Berlin (1985). tainty and the Origin of Probability in Everettian Quan- [30] A. M. Gleason “Measures on the closed subspaces of a tum Mechanics,” arXiv:1405.7577 (2014). Hilbert space,” Journal of Mathematics and Mechanics [49] C. T. Sebens and S. M. Carroll, “Many Worlds, the Born 6, 885893 (1957). Rule, and Self-Locating Uncertainty,” arXiv:1405.7907 [31] R.B. Griffiths, J. Stat. Phys. 36, 219 (1984). (2014). [32] R.B. Griffiths, Found. Phys. 23, 1601 (1993). [50] H. P. Stapp, “The basis problem in many-worlds [33] R. B. Griffiths, “Consistent Quantum Theory,” Cam- theories,” Can. J. Phys. 80, 10431052 (2002). bridge University Press, (2002). arXiv:quant-ph/0110148v2. [34] M. Gell-Mann and J.B. Hartle, Phys. Rev. D 47, 3345 [51] M. Tegmark, “The Interpretation of Quantum Mechan- (1993). ics: Many Worlds or Many Words?” Fortsch. Phys. 46, [35] M. Gell-Mann, J. Hartle, “Decoherent Histories Quan- 855-862 (1998). arXiv:quant-ph/9709032 tum Mechanics with One “Real” Fine-Grained History,” [52] B. C. vanFraassen, “A Modal Interpretation of Quan- Phys. Rev. A, 85 (2012). arXiv:1106.0767 [quant-ph]. tum Mechanics,” in “Current Issues in Quantum Logic,” [36] K. Jung, “Is the de Broglie-Bohm interpretation of quan- Springer USA, Plenum Press, New York, 229-258 (1981). tum mechanics really plausible?” Journal of Physics: [53] D. Wallace, “Quantum Probability and Decision Theory, Conference Series, 442 (2013). Revisited,” arXiv:quant-ph/0211104 (2002). [37] P. Lewis, “GRW and the tails problem,” Topoi, 14, 1, [54] D. Wallace, “Everettian Rationality: defending Deutschs 23-33 (1995). approach to probability in the Everett interpretation,” [38] R. Omn`es, J. Stat. Phys. 53 (1988) 893; ibid 53 (1988) Stud. Hist. Phil. Mod. Phys. 34, 415438 (2003). 933; ibid 53 (1988) 957; ibid 57 (1989) 357. [55] D. Wallace, “Quantum Probability from Subjective Like- [39] E. Okon, D. Sudarsky, “On the Consistency of the lihood: improving on Deutsch’s proof of the probability Consistent Histories Approach to Quantum Mechanics,” rule,” arXiv:quant-ph/0312157 (2003). Found. of Phys. 44, 19–23 (2014). [56] D. Wallace, “A formal proof of the Born rule from [40] D. Rainer, “Advanced Quantum mechanics: Materials decision-theoretic assumptions,” arXiv:0906.2718 (2009). and Photons,” Springer Science+Business Media, LLC, [57] J. A. Wheeler, R. P. Feynman, “Interaction with the Ab- (2012). sorber as the Mechanism of Radiation”. Rev. of Mod. [41] B. Reznik, Y. Aharonov, “On a Time Symmetric Formu- Phys. 17, 157-161 (1945). lation of Quantum Mechanics,” Phys. Rev. A 52, 2538 [58] J. A. Wheeler, R. P. Feynman, “Classical Electrodynam- (1995). ics in Terms of Direct Interparticle Action”. Rev. of Mod. [42] C. Rovelli, “Relational Quantum Mechanics,” Int. J. of Phys. 21, 425–433 (1949). Theor. Phys. 35, 1637 (1996). [59] W. H. Zurek, “Pointer Basis of Quantum Apparatus: [43] S. Saunders, “Derivation of the Born rule from op- Into What Mixture Does the Wave Packet Collapse?” erational assumptions,” Proc. Roy. Soc. Lond. A460, Phys. Rev. D 24, 1516 (1981). 17711788 (2004). [60] W. H. Zurek, “Environment-Induced Superselection [44] S. Saunders, “What is Probability?” Rules,” Phys. Rev. D 26, 1862 (1982). arXiv:quant-ph/0412194 (2004). [61] W. H. Zurek, “Reduction of the Wave Packet: How Long [45] S. Saunders, J. Barrett, A. Kent, D. Wallace,“Many Does It Take?” Frontiers of Nonequilibrium Statistical Worlds? Everett, Quantum Theory and Reality”, Ox- Physics. Edited by P. Meystre, and M. O. Scully. New ford University Press (2010). York, Plenum (1984). [46] M. A. Schlosshauer, “Decoherence, the measure- [62] States that are described by a Hilbert space element are ment problem, and interpretations of quantum me- referred to as pure. We introduce terminology such that chanics,” Rev. of Mod. Phys.76, 1267-1305 (2004). states that are single basis elements in a preferred basis arXiv:quant-ph/0312059. are ultra pure. [47] M. A. Schlosshauer, “Decoherence: And the Quantum- To-Classical Transition,” Springer-Verlag Berlin Heidel-