Multiple Worlds, One Universal Wave Function

Evan Hubinger Harvey Mudd College (2017 Harvey Mudd College Rojansky Writing Award winner for most outstanding paper in quantum mechanics.) (Dated: April 30, 2017)

We seek to present and defend the view that the interpretation of quantum mechanics is no more complicated than the interpretation of plate tectonics: that which is being studied is real, and that which the theory predicts is true. The view which holds that the mathematical formalism of quantum mechanics—without any additional postulates—is a complete description of reality is known as the Everett interpretation. We seek to defend the Everett interpretation of quantum mechanics as the most probable interpretation available. To accomplish this task, we analyze the history of the Everett interpretation, provide mathematical backing for its as- sertions, respond to criticisms that have been leveled against it, and compare it to its modern alternatives.

I. INTRODUCTION sical mechanics. Whenever quantum systems encounter the cutoff point, the theory One of the most puzzling aspects of quan- stated, they collapse down into a single state tum mechanics is the fact that, when one with probabilities following the squared am- measures a system in a superposition of mul- plitude, or Born, rule. Thus, the theory pre- tiple states, it is only ever found in one of dicted that physics just behaved differently them. This puzzle was dubbed the “mea- at different length scales. This traditional in- surement problem,” and the first attempt at terpretation of quantum mechanics is usually a solution was by Werner Heisenberg, who referred to as the Copenhagen interpretation. in 1927 proposed his theory of “wave func- From the very beginning, the Copenhagen tion collapse.”[1] Heisenberg proposed that interpretation was seriously suspect. Al- there was a cutoff length, below which sys- bert Einstein was famously displeased with tems were governed by quantum mechanics, its lack of determinism, saying “God does and above which they were governed by clas- not play dice,” to which Niels Bohr quipped 2 in response, “Einstein, stop telling God II. THE EVERETT what to do.”[2] As clever as Bohr’s an- INTERPRETATION OF QUANTUM swer is, Einstein—with his famous physi- MECHANICS cal intuition—was right to be concerned. Though Einstein favored a hidden variable Enter the Everett Interpretation. In interpretation[3], which was later ruled out 1956, Hugh Everett III, then a doctoral by Bell’s theorem[4], the Copenhagen in- candidate at Princeton, had an idea: if terpretation nevertheless leaves open many you could find a way to explain the phe- questions. If physics behaves differently at nomenon of measurement from within wave different length scales, what is the cutoff mechanics, you could do away with the ex- point? What qualifies as a wave-function- tra postulate of wave function collapse, and collapsing measurement? How can physics thus many of the problems of the Copen- behave differently at different length scales, hagen interpretation. Everett worked on this when macroscopic objects are made up of idea under his thesis advisor, Einstein-prize- microscopic objects? Why is the observer winning theoretical physicist John Wheeler, not governed by the same laws of physics as who would later publish a paper in support the system being observed? Where do the of Everett’s theory.[6] In 1957, Everett fin- squared amplitude Born probabilities come ished his thesis “The Theory of the Uni- from? If the physical world is fundamen- versal Wave Function,”[7] published as the tally random, how is the world we see selected “‘Relative State’ Formulation of Quantum from all the possibilities? How could one ex- Mechanics.”[8] In his thesis, Everett suc- plain the applicability of quantum mechan- ceeded in deriving every one of the strange ics to macroscopic systems, such as Chan- quirks of the Copenhagen interpretation— drasekhar’s insight in 1930 that modeling wave function collapse, the apparent ran- neutron stars required the entire star to be domness of measurement, and even the Born treated as a quantum system?[5] rule—from purely wave mechanical grounds, as we will do in section III. Everett’s derivation relied on what was at the time a controversial application of quantum mechanics: the existence of wave functions containing observers themselves. Ev- erett believed that there was no reason to re- 3 strict the domain of quantum mechanics to the postulates of the Copenhagen interpre- only small, unobserved systems. Instead, Ev- tation and still end up with a theory that erett proposed that any system, even the sys- works. tem of the entire universe, could be encom- passed in a single, albeit often intractable, A. DeWitt’s Multiple Worlds “universal wave function.” Modern formulations of the Everett inter- Removing the Copenhagen postulates had pretation reduce his reasoning down to two some implications that did not mesh well fundamental ideas:[9][10][11][12][13] with many physicists’ existing physical in- tuitions. If one accepted Everett’s univer- 1. the wave function obeys the standard, lin- sal wave function, one was forced to ac- ear, deterministic Schrodinger wave equa- cept the idea that macroscopic objects— tion at all times (the relativistic variant, cats, people, planets, stars, galaxies, even to be precise), and the entire universe—could be in a superpo-

2. the wave function is physically real. sition of many states, just as microscopic objects could. In other words, multiple Specifically, the first statement precludes different versions of the universe—multiple wave function collapse and demands that we worlds, so to speak—could exist simultane- continue to use the same wave mechanics for ously. It was for this reason that Einstein- all systems, even those with observers, and prize-winning physicist Bryce DeWitt, a sup- the second statement demands that we ac- porter of the Everett interpretation, dubbed cept the physical implications of doing so. Everett’s theory of the universal wave func- The Everett interpretation is precisely that tion the “multiworld” (or now more com- which is implied by these two statements. monly “multiple worlds”) interpretation of Importantly, neither of these two princi- quantum mechanics.[9] ples are additional assumptions on top of tra- While the idea of multiple worlds may ditional quantum theory—instead, they are at first seem strange, to Everett, it was simplifications of existing quantum theory, simply an extension of the normal laws of since they act only to remove the prior ad- quantum mechanics. Simultaneous superpo- hoc postulates of wave function collapse and sition of states is something physicists al- the non-universal applicability of the wave ready accept for microscopic systems when- equation.[11][14] The beauty of the Everett ever they do quantum mechanics—by virtue interpretation is the fact that we can remove 4 of the overwhelming empirical evidence in whelming variety of problems, but rather to favor of it. Not only that, but evidence supply a new, more general and complete keeps coming out demonstrating superpo- formulation, from which the conventional in- sitions at larger and larger length scales. terpretation can be deduced.”[8] Thus, it is In 1999 it was demonstrated, for example, not surprising that Stephen Hawking and No- that Carbon-60 molecules can be put into a bel laureate Murray Gell-Mann, supporters superposition.[15]. While it is unlikely that of the Everett interpretation, have expressed a superposition of such a macroscopic object reservations with the name “multiple worlds as Schrodinger’s cat will ever be conclusively interpretation,” and therefore we will con- demonstrated, due to the difficulty in isolat- tinue to refer to the theory simply as the Ev- ing such a system from the outside world, it is erett interpretation instead.[16] likely that the trend of demonstrating superposition at larger and larger length scales will B. The Nature of Observation continue. It seems that to not accept that a cat could be in a superposition, even if we can Accepting the Everett interpretation never demonstrate it, however, is a failure raises an important question: if the macro- of induction—a rejection of an empirically- scopic world can be in a superposition of demonstrated trend. multiple states, what differentiates them? While the Everett interpretation ended up Stephen Hawking has the answer: “in order implying the existence of multiple worlds, to determine where one is in space-time one this was never Everett’s starting point. The has to measure the metric and this act of “multiple worlds” of the Everett interpre- measurement places one in one of the various tation were not added to traditional quan- different branches of the wave function in tum mechanics as new postulates, but rather the Wheeler-Everett interpretation of quan- fell out from the act of taking away the ex- tum mechanics.”[17] When we perform an isting ad-hoc postulates of the Copenhagen observation on a system whose state is in a interpretation—a consequence of taking the superposition of eigenfunctions, a version of wave function seriously as a fundamental us sees each different, possible eigenfunction. physical entity. In Everett’s own words, “The The different worlds are defined by the aim is not to deny or contradict the conven- different eigenfunctions that are observed. tional formulation of quantum theory, which We can show this, as Everett did, just has demonstrated its usefulness in an over- by acknowledging the existence of universal, 5 joint system-observer wave functions.[7][8] are as a random choice from that weighted Before measuring the state of a system in a distribution. Formally, we can prove that superposition, the observer and the system under the Everett interpretation, if an ob- are independent—we can get their joint wave server interacts with many systems each in function simply by multiplying together their a superposition of multiple states, the dis- individual wave functions. After measure- tribution of states they see will follow the ment, however, the two become entangled— Born rule.[7][8][18][11][19][14] A portion of that is, the state of the observer becomes de- Everett’s proof of this fact is included in sec- pendent on the state of the system that was tion III C. observed. The result is that for each eigenfunction in the system’s superposition, the observer’s wave function evolves differently. Thus, we can no longer express their joint wave function as the product of their individual wave functions. Instead, we are forced to express the joint wave function as a sum of different components, one for each possible eigenfunction of the system that could be observed. These different components are the different “worlds” of the Everett interpretation, with the only difference between them being which eigenfunction of the system was observed. We will formalize this reasoning in section III A. We are still left with the question, however, of why we experience a particular probability of seeing some states over others, if every state that can be observed is observed. Informally, we can think of the different worlds—the different possible observations— as being “weighted” by their squared amplitudes, and which one of the versions of us we 6

III. THE MATHEMATICS OF THE are working under the Everett interpretation, EVERETT INTERPRETATION we will let ourselves define a joint system- observer wave function Ψ with initial config- Previously, we asserted that universally- uration applied wave mechanics was sufficient, with- X out ad-hoc postulates such as wave func- Ψ0 = ψϕ = ψ aiϕi i tion collapse, to imply all the oddities of Then, our goal is to understand what hap- the Copenhagen interpretation. We will now pens to Ψ when O repeatedly observes S. prove that assertion. In this section, as Thus, we will define Ψn to represent the state per the Everett interpretation, we will ac- of Ψ after n ∈ N independent observations of cept that basic wave mechanics is obeyed S are performed by O. for all physical systems, including those con- Consider the simple case where ϕ = ϕ0 taining observers. From that assumption, and thus we are in initial state Ψ0 = ψϕ0. we will show that the apparent phenomena In this case, by our previous definition of ψi of wave function collapse, random measure- and requirement that ϕi remain unchanged, ment, and the Born Rule follow. The proofs we can write the state after the observation given below are adopted from Everett’s orig- as Ψ1 = ψ0ϕ0. Since quantum mechanics is inal paper.[7][8] linear, and the eigenfunctions ϕi are orthogo- nal, it must be that this same process occurs A. The Apparent Collapse of The Wave for each ϕi. Function Thus, by the principle of superposition, we can write Ψ in its general form as Suppose we have a system S with eigen- 1 P functions {ϕi} and initial state ϕ = i aiϕi. X Ψ1 = aiψiϕi Consider an observer O with initial state ψ. i

Let ψi,j,... be the state of O after observ- For the next observation, each ψi will once

ing eigenfunctions ϕi, ϕj,... of S. Since we again see the same ϕi, since it has not would like to demonstrate how repeated mea- changed state. As previously deﬁned, we use

surements see a collapsed wave function, we the notation ψi,i to denote the state of O af-

will assume that repeated measurement is ter observing S in state ϕi twice. Thus, we

possible, and thus that the states ϕi of S can write Ψ2 as

remain unchanged after observation. As we X Ψ2 = aiψi,iϕi i 7

and more generally, we can write Ψn as the joint wave function, however, that is not the case—in fact, the entire superposition X Ψn = aiψi,i,...,iϕi i still exists. What has changed is only that where i is repeated n times in i, i, . . . , i. ψ, the state of O, is no longer independent of Thus, once a measurement of S has been that superposition, and has instead become performed, every subsequent measurement entangled with it. will see the same eigenfunction, even though all eigenfunctions continue to exist. We can B. The Apparent Randomness of see this from the fact that the same i is re- Measurement peated in each state ψi,i,...,i of O. In this way, Suppose we now have many such systems we see how, despite the fact that the original P S, which we will denote Sn where n ∈ N. wave function ϕ = i aiϕi for S is in a super- Consider O from before, but with the modifi- position of many eigenfunctions, once a mea- cation that instead of repeatedly observing a surement has been performed, each subse- single S, O observes different S in each mea- quent measurement will always see the same n surement, such that Ψ is the joint system- eigenfunction. n observer wave function after measuring the Note that there is no longer a single, in- nth S . dependent state ψ of O. Instead, there n As before, we will define the initial joint are many ψi,i,...,i, one for each eigenfunction. wave function Ψ as What does that mean? It means that for ev- 0 ery eigenfunction ϕi of S, there is a corre- X Ψ0 = ψ ai1,i2,...,in i1,i2,...,in sponding state ψi,i,...,i of O wherein O sees ϕ (x )ϕ (x ) ··· ϕ (x ) that eigenfunction. Thus, one is required to i1 1 i2 2 in n accept that there are many observers O , with i where we are summing over all possible com- corresponding state ψ , each one seeing a i,i,...,i binations of eigenfunctions for the differ- different eigenfunction ϕi. This is the ori- ent systems Sn with arbitrary coefficients gin of the Everett interpretation’s “multiple ai1,i2,...,in for each combination. worlds.” Then, as before, we can use the principle From the perspective of each Oi in this of superposition to find Ψ1 as scenario it will appear as if ϕ has “collapsed” P X from a complex superposition i aiϕi into a Ψ1 = ψi1 ai1,i2,...,in i1,i2,...,in single eigenfunction ϕi. As we can see from ϕi1 (x1)ϕi2 (x2) ··· ϕin (xn) 8 since the first measurement will see the state the equation

ϕi1 of S1. More generally, we can write Ψn as q ∗ P (ai) = P ai ai = P (|ai|) X Ψn = ψi1,i2,...,in ai1,i2,...,in i1,i2,...,in Furthermore, by the linearity of quantum me- ϕi1 (x1)ϕi2 (x2) ··· ϕin (xn) chanics, we will impose the condition on P such that for aϕ deﬁned as following the same principle, as each mea- X surement of an Sn will see the corresponding aϕ = aiϕi i state ϕin . P must satisfy the equation Thus, when subsequent measurements of identical systems Sn are performed, the re- X P (a) = P (ai) sulting sequence of eigenfunctions observed i by O in each ψ appear random (according to Together, these two conditions fully spec- what distribution we will show in the next ify what function P must be. Assuming ϕ is P ∗ subsection), since there is no structure to the normalized, such that i ϕi ϕi = 1, it must be that sequences i1, i2, . . . , in. This appearance of ∗ X ∗ randomness is true even though the entire a a = ai ai i process is completely deterministic. If, al- or equivalently ternatively, O was to return to a previously- s X 2 |a| = |ai| measured Sn, we would get a repeat of the i ﬁrst analysis, wherein O would always see the such that same state as was previously measured.   s X 2 P (|a|) = P  |ai|  i C. The Born Probability Rule which, using the phase invariance condition As before, consider a system S in state that P (|a|) = P (a), gives P a ϕ . To be able to talk about a proba-   i i i s X 2 P (a) = P  |ai|  bility for an observer O to see state ϕi, we i need some function P (ai) that will serve as a Then, from the linearity condition, we have measure of that probability. X Since we know that quantum mechanics is P (a) = P (ai) invariant up to an overall phase, we will im- i pose the condition on P that it must satisfy 9

which, by the phase invariance condition, is the wave function as the “probability.” Nev- equivalent to ertheless, for nearly every reasonable prob-

ability theory that exists, such proofs have X q 2 P (a) = P |ai| i been provided. Everett provided a proof Putting it all together, we get based on the standard frequentist definition   of probability[7][8], David Deutsch (Oxford s X 2 X q 2 P (a) = P  |ai|  = P |ai| theoretical physicist) has provided a proof i i based on game theory[18], and David Wal- √ then, defining a new function g(x) = P ( x), lace (USC theoretical physicist) has provided yields a proof based on decision theory[11]. For any ! X 2 X 2 reasonable definition of probability, wave me- g |ai| = g |ai| i i chanics is able to show that the above mea- which implies that g must be a linear function sure satisfies it in the limit without any ad- such that for some constant c ditional postulates.[19][14][20]

g(x) = cx

Therefore, since P (x) = g(x2),

P (x) = cx2 which, imposing the phase invariance condition, becomes

P (x) = c |x|2 which, where c is normalized to 1, is the Born rule. The fact that this measure is a probability, beyond that it is the only measure that can be, is deserving of further proof. The concept of probability is notoriously hard to deﬁne, however, and without a deﬁnition of probability, it is just as meaningful to call P something as arbitrary as the “stallion” of 10

IV. ARGUMENTS FOR AND every demonstration performed of superpo- AGAINST THE EVERETT sition at larger and larger length scales— INTERPRETATION such as for Carbon 60 as was previously mentioned[15]—is a test of the Everett inter- “Despite the unrivaled empirical pretation. Arguably, it is the Copenhagen success of quantum theory, the very interpretation which is unfalsifiable, since suggestion that it may be literally true it makes no claim about where the bound- as a description of nature is still ary lies at which wave function collapse oc- greeted with cynicism, curs, and thus proponents can respond to the incomprehension, and even anger.”[21] evidence of larger superpositions simply by changing their theory and moving their pro- David Deutsch, 1996 posed boundary up. Another method of falsification regards A. Falsifiability and Empiricism the interaction between the Everett interpretation and quantum gravity. The Ev- Perhaps the most common criticism of erett interpretation makes a definitive predic- the Everett interpretation is the claim that tion that gravity must be quantized. Were it is not falsifiable, and thus falls outside gravity not quantized—not wrapped up in the realm of empirical science.[22] In fact, the wave function like all the other forces— this claim is simply not true—many differ- and instead simply a background metric ent methods for testing the Everett interpre- for the entire wave function, we would be tation have been proposed, and, a great deal able to detect the gravitational impact of of empirical data regarding the Everett inter- the other states we were in a superposition pretation is already available. with.[10][23] In 1957, Richard Feynman, who One such method we have already dis- would later come to explicitly support the cussed: the Everett interpretation removes Everett interpretation[16] as well as become the Copenhagen interpretation’s postulate a Nobel laureate, presented an early version that the wave function must collapse at a par- of the above argument as a reason to believe ticular length scale. Were it ever to be con- in quantum gravity, arguing, “There is a bare clusively demonstrated that superposition possibility (which I shouldn’t mention!) that was impossible past some point, the Everett quantum mechanics fails and becomes clas- interpretation would be disproved. Thus, sical again when the amplification gets far 11 enough [but] if you believe in quantum me- Earth, we would not accept it as a new phys- chanics up to any level then you have to be- ical controversy—we would say that, unless lieve in gravitational quantization.”[24] there is incredibly strong proof otherwise, we Another proposal concerns differing prob- should by default assume that the same laws abilities of finding ourselves in the universe of physics apply everywhere. The Everett in- we are in depending on whether the Everett terpretation is just that default—it is only interpretation holds or not. If the Everett by historical accident that it happened to be interpretation is false, and the universe only discovered after the Copenhagen interpreta- has a single state, there is only one state for tion. Thus, to the extent that one has con- us to find ourselves in, and thus we would fidence in the universal applicability of the expect to find ourselves in an approximately principles of quantum mechanics, one should random universe. On the other hand, if the have equal confidence in the Everett inter- Everett interpretation is true, and there are pretation, since it is a logical consequence. many different states that the universe is in, It is in fact all the more impressive—and we could find ourselves in any of them, and tantamount to its importance to quantum thus we would expect to find ourselves in one mechanics—that the Everett interpretation which was more disposed than average to- manages to achieve falsifiability and empiri- wards the existence of life. Approximate cal- cal support despite its primary virtue of sim- culations of the relative probability of the ob- ply saying that quantum mechanics be ap- served universe based on the Hartle-Hawking plied universally. boundary condition strongly support the Ev- erett interpretation.[10] B. Simplicity Finally, as we made a point of being clear about in section II, the Everett interpretation Another common objection to the Everett is simply a consequence of taking the wave interpretation is that it “postulates too many function seriously as a physical entity. Thus, universes,” which Sean Carroll, a Caltech cos- it is somewhat unfair to ask the Everett in- mologist and supporter of the Everett inter- terpretation to achieve falsifiability indepen- pretation, calls “the basic silly objection.”[25] dently of the theory—quantum mechanics— At this point, it should be very clear why this which implies it.[22] If a new theory were pro- objection is silly: the Everett interpretation posed that said quantum mechanics stopped postulates no such thing—the existence of working outside of the future light cone of “many universes” is an implication, not a pos- 12

2 tulate, of the theory. Opponents of the Ev- have a set of programs {Ti} which encode for erett interpretation, however, have accused empirically indistinguishable physical theo- it of a lack of simplicity on the grounds that ries, the probability P of the theory described adding in all those additional universes is un- by a given program Ti with length in bits (0s necessary added complexity, and since by the and 1s) |Ti| is given by principle of Occam’s razor the simplest expla- −|Ti| P(Ti) ∼ 2 nation is probably correct, the Everett interpretation can be rejected.[26] up to a constant normalization factor calcu- In fact, Occam’s razor is an incredibly lated across all the {Ti} to make the probabil- strong argument in favor of the Everett inter- ities sum to 1.[27] We can see how this makes pretation. To explain this, we will first need intuitive sense, since if we are predicting an to formalize what we mean by Occam’s razor, arbitrary system, and thus have no informa- which will require some measure of theoret- tion about the correctness of a program im- ical computer science. Specifically, we will plementing a theory other than its length in make use of Solomonoff’s theory of induc- bits, we are forced to assign equal probability tive inference: the best, most general frame- to each of the two options for each bit, 0 and work we have for comparing the probabil- 1, and thus each additional bit adds a factor ity of empirically indistinguishable physical 1 of 2 to the total probability of the program. 1 theories.[27][28][29] To use Solomonoff’s for- Furthermore, we can see how Solomonoff in- malism, only one assumption is required of duction serves as a formalization of Occam’s us: under some encoding scheme, competing razor, since it gives us a way of calculating theories of the universe can be modeled as how much to discount longer, more complex programs. This assumption does not imply theories in favor of shorter, simpler ones. that the universe must be computable, only Now, we will attempt to apply this for- that it can be computably described, which malism to assign probabilities to competing all physical theories capable of being writ- interpretations of quantum mechanics, which ten down must abide by. From this assump- we will represent as elements of the set {Ti}. tion, and the axioms of probability theory, Let W be the shortest program which com- Solomonoff induction can be derived.[27] putes the wave equation. Since the wave Solomonoff induction tells us that, if we equation is a component of all quantum the-

1 In some of these sources, the equivalent formalism 2 To be precise, these should be universal Turing ma- of Kolmogorov complexity is used instead. chine programs. 13

ories, it must be that |W | ≤ |Ti|. Thus, the very modest estimate, considering the small- smallest that any Ti could possibly be is |W |, est Chess program ever written is 3896 bits such that any Ti of length |W | is at least twice long.[30] Then, the relative probability is at as probable as a Ti of any other length. The most

Everett interpretation is such a Ti, since it re- (C) P = 2−|O|−|L| < 2−|L| ≈ 2−800 ≈ 2 · 10−241 quires nothing else beyond wave mechanics, P(E) and follows directly from it. Therefore, from which is about the probability of picking four the perspective of Solomonoﬀ induction, the random atoms in the universe and getting the Everett interpretation is provably optimal in same one each time, and is thus so small as terms of program length, and thus also in to be trivially dismissible. terms of probability.

To get a sense of the magnitude of these C. The Arrow of Time eﬀects, we will attempt to approximate how much less probable the Copenhagen interpre- Another objection to the Everett in- tation is than the Everett interpretation. We terpretation is that it is time-symmetric. will represent the Copenhagen interpretation Since the Everett interpretation is just the C as made of three parts: W , wave mechan- wave equation, its time symmetry follows ics; O, a machine which determines when to from the fact that the Schrodinger equation collapse the wave function; and L, classical is time-reversal invariant, or more techni- mechanics. Then, where the Everett inter- cally, charge-parity-time-reversal (CPT) in- pretation E is just W , we can write their rel- variant. The Copenhagen interpretation, ative probabilities as however, is not, since wave function collapse

(C) 2−|W |−|O|−|L| is a fundamentally irreversible event.[31] P = = 2−|O|−|L| P(E) 2−|W | In fact, CPT symmetry is not the only How large are O and L? As a quick Fermi natural property that wave function col- estimate for L, we will take Newton’s three lapse lacks that the Schrodinger equation laws of motion, Einstein’s general relativistic has—wave function collapse breaks linear- field equation, and Maxwell’s four equations ity, unitarity, differentiability, locality, and of electromagnetism as the principles of clas- determinism.[13][12][16][32] The Everett in- sical mechanics, for a total of 8 fundamen- terpretation, by virtue of consisting of noth- tal equations. Assume the minimal imple- ing but the Schrodinger equation, preserves mentation for each one averages 100 bits—a all of these properties. This is an argu- 14 ment in favor of the Everett interpretation, tions. since there are strong theoretical and empirical reasons to believe that such symmetries V. OTHER INTERPRETATIONS OF are properties of the universe.[33][34][35][5] QUANTUM MECHANICS Nevertheless, as mentioned above, it has been argued that the Copenhagen interpre- “The mathematical formalism of the tation’s breaking of CPT symmetry is ac- quantum theory is capable of yielding tually a point in its favor, since it suppos- its own interpretation.”[9] edly explains the arrow of time, the idea that Bryce DeWitt, 1970 time does not behave symmetrically in our everyday experience.[31] Unfortunately for the Copenhagen interpretation, wave func- A. Decoherence tion collapse does not actually imply any of It is sometimes proposed that wave me- the desired thermodynamic properties of the chanics alone is sufficient to explain the ap- arrow of time.[31] Furthermore, under the parent phenomenon of wave function col- Everett interpretation, the arrow of time can lapse without the need for the Everett in- be explained using the standard thermody- terpretation’s multiple worlds. The justifi- namic explanation that the universe started cation for this assertion is usually based on in a very low-entropy state.[36] the idea of decoherence. Decoherence is the In fact, accepting the Everett interpreta- mathematical result, following from the wave tion gets rid of the need for the current state equation, that tightly-interacting superpo- of the universe to be dependent on subtle ini- sitions tend to evolve into non-interacting tial variations in that low-entropy state.[36] superpositions.[37][38] Importantly, decoher- Instead, the current state of the universe is ence does not destroy the superposition—it simply one of the many different components merely “diagonalizes” it, which is to say, it re- of the wave function that evolved determinis- moves the interference terms.[37] After deco- tically from that initial state. Thus, the Ev- herence, one is always still left with a super- erett interpretation is even simpler—from a position of multiple states.[39][40] The only Solomonoff perspective—than was shown in way to remove the resulting superposition section IV B, since it forgoes the need for its is to assume wave function collapse, which program to specify a complex initial condi- every statistical theory claiming to do away tion for the universe with many subtle varia- 15 with multiple worlds has been shown to im- which are all reasons to favor the Everett plicitly assume.[41][19] There is no escaping interpretation over any single-world theory. the logic presented in section III A—if one ac- Specifically, calculations of the probability of cepts the universal applicability of the wave the current state of the universe support the function, one must accept the multiple worlds Everett interpretation[10], as does the fact it implies. that the Everett interpretation allows for the That is not to say that decoherence initial state of the universe to be simpler[36]. is not an incredibly valuable, useful concept for the interpretation of quantum me- B. Consistent Histories chanics, however. In the Everett interpretation, decoherence serves the very im- The consistent histories interpretation of portant role of ensuring that macroscopic quantum mechanics, owing primarily to superpositions—the multiple worlds of the Robert Griffiths, eschews probabilities over Everett interpretation—are non-interacting, “measurement” in favor of probabilities over and that each one thus behaves approxi- “histories,” which are defined as arbitrary mately classically.[41][40] Thus, the simplest sequences of events.[43] Consistent histories decoherence-based interpretation of quantum provides a way of formalizing what classi- mechanics is in fact the Everett interpreta- cal probabilistic questions make sense in a tion. From the Stanford Encyclopedia of Phi- quantum domain and which do not—that losophy, “Decoherence as such does not pro- is, which are consistent. Its explanation for vide a solution to the measurement problem, why this consistency always appears at large at least not unless it is combined with an ap- length scales is based on the idea of decoher- propriate interpretation of the theory [and it ence, as discussed above.[43][44] In this con- has been suggested that] decoherence is most text, consistent histories is a very useful tool naturally understood in terms of Everett-like for reasoning about probabilities in the con- interpretations.”[39] The discoverer of deco- text of quantum mechanics, and for providing herence himself, German theoretical physicist yet another proof of the natural origin of the Heinz-Dieter Zeh, is an ardent proponent of Born rule. the Everett interpretation.[42][36] Proponents of consistent histories claim Furthermore, we have given general argu- that it does not imply the multiple worlds of ments in favor of the existence of the multiple the Everett interpretation.[43] However, since worlds implied by the Everett interpretation, the theory is based on decoherence, there 16 are always multiple different consistent his- not made, then the result is macroscopic su- tories, which cannot be removed via any nat- perposition, which is to say, the Everett in- ural history selection criterion.[45][44] Thus, terpretation. This formulation of consistent just as the wave equation implies the Ev- histories without any extra postulates has erett interpretation, so too does consistent been called the theory of “the universal path histories. To see this, we will consider the integral,” exactly mirroring Everett’s theory fact that consistent histories works because of the universal wave function.[46] The the- of Feynmann’s observation that the ampli- ory of the universal wave function—the Ev- tude of any given final state can be calculated erett interpretation—is to the theory of the as the sum of the amplitudes along all the universal path integral as wave mechanics is possible paths to that state.[44][46] Impor- to the sum-over-paths approach, which is to tantly, we know that two different histories— say that they are both equivalent formalisms for example, the different branches of a Mach- with the same implications. Zender interferometer—can diverge and then later merge back together and interfere with C. Pilot Wave Theory each other. Thus, it is not in general possible to describe the state of the universe as a The pilot wave interpretation, otherwise single history, since other, parallel histories known as the de Broglie-Bohm interpre- can interfere and change how that state will tation, postulates that the wave function, later evolve. A history is great for describing rather than being physically real, is a back- how a state came to be, but not very use- ground which “guides” otherwise classical ful for describing how it might evolve in the particles.[47] As we saw with the Copenhagen future. For that, including the other paral- interpretation, the obvious question to ask lel histories—the full superposition—is nec- of the pilot wave interpretation is whether essary. its extra postulate—in this case adding in Once one accepts that the existence of classical particles—is necessary or useful in multiple histories is necessary on a micro- any way. The answer to this question is scopic level, their existence on a macroscopic a definitive no. Heinz-Dieter Zeh says of level follows—excluding them would require the pilot wave interpretation, “Bohm’s pi- an extra postulate, which would make consis- lot wave theory is successful only because tent histories equivalent to the Copenhagen it keeps Schrodinger’s (exact) wave mechan- interpretation. If such an extra postulate is ics unchanged, while the rest of it is obser- 17 vationally meaningless and solely based on of the required initial condition[36]. Thus, classical prejudice.”[42] As we have previously the pilot wave interpretation, despite being shown in section III, wave mechanics is capa- strictly more complicated than the Everett ble of solving all supposed problems of mea- interpretation—both in terms of its extra surement without the need for any additional postulate and the concerns above—produces postulates. While it is true that pilot wave exactly no additional explanatory power. theory solves all these problems as well, it Therefore, we can safely dismiss the pilot does so not by virtue of its classical add-ons, wave interpretation on the grounds of the but simply by virtue of including the entirety same simplicity argument used against the of wave mechanics.[42][48] Copenhagen interpretation in section IV B. Furthermore, since pilot wave theory has no collapse postulate, it does not even get rid VI. CONCLUSION of the existence of multiple words. If the universe computes the entirety of the wave func- Harvard theoretical physicist Sidney Cole- tion, including all of its multiple worlds, then man uses the following parable from Wittgen- all of the observers in those worlds should ex- stein as an analogy for the interpretation of perience physical reality by the act of being quantum mechanics: “‘Tell me,’ Wittgen- computed—it is not at all clear how the clas- stein asked a friend, ‘why do people always sical particles could have physical reality and say, it was natural for man to assume that the rest of the wave function not.[21][42] In the sun went round the Earth rather than the words of David Deutsch, “pilot-wave the- that the Earth was rotating?’ His friend ories are parallel-universes theories in a state replied, ‘Well, obviously because it just looks of chronic denial. This is no coincidence. as though the Sun is going round the Earth.’ Pilot-wave theories assume that the quantum Wittgenstein replied, ‘Well, what would it formalism describes reality. The multiplicity have looked like if it had looked as though of reality is a direct consequence of any such the Earth was rotating?”’[49] Of course, the theory.”[21] answer is it would have looked exactly as it However, since the extra classical particles actually does! To our fallible human intu- only exist in one of these worlds, the pilot ition, it seems as if we are seeing the sun wave interpretation also does not resolve the rotating around the Earth, despite the fact problem of the low likelihood of the observed that what we are actually seeing is a helio- state of the universe[10] or the complexity centric solar system. Similarly, it seems as if 18 we are seeing the wave function randomly col- There is a tendency of many physicists lapsing around us, despite the fact that this to describe the Everett interpretation sim- phenomenon is entirely explained just from ply as one possible answer to the mea- the wave equation, which we already know surement problem. It should hopefully be empirically is a law of nature. clear at this point why that view should It is perhaps unfortunate that the Ev- be rejected—the Everett interpretation is erett interpretation ended up implying the not simply yet another solution to the mea- existence of multiple worlds, since this fact surement problem, but rather a straightfor- has led to many incorrectly viewing the Ev- ward conclusion of quantum mechanics it- erett interpretation as a fanciful theory of self that shows that the measurement problem alternative realities, rather than the best, should never have been a problem in the first simplest theory we have as of yet for ex- place. Without the Everett interpretation, plaining measurement in quantum mechan- one is forced to needlessly introduce complex, ics. The Everett interpretation’s greatest symmetry-breaking, empirically-unjustifiable virtue is the fact that it is barely even an postulates—either wave function collapse or interpretation of quantum mechanics, hold- pilot wave theory—just to explain what was ing as its most fundamental principle that already explicable under basic wave mechan- the wave equation can interpret itself. In ics. The Everett interpretation is not just the words of David Wallace: “If I were to another possible way of interpreting quan- pick one theme as central to the tangled de- tum mechanics, but a necessary component velopment of the Everett interpretation of of any quantum theory that wishes to explain quantum mechanics, it would probably be: the phenomenon of measurement in a natu- the formalism is to be left alone. What ral way. In the words of John Wheeler, Ev- distinguished Everett’s original paper both erett’s thesis advisor, “No escape seems possi- from the Dirac-von Neumann collapse-of-the- ble from [Everett’s] relative state formulation wavefunction orthodoxy and from contempo- if one wants to have a complete mathemati- rary rivals such as the de Broglie-Bohm the- cal model for the quantum mechanics that is ory was its insistence that unitary quantum internal to an isolated system. Apart from mechanics need not be supplemented in any Everett’s concept of relative states, no self- way (whether by hidden variables, by new consistent system of ideas [fully explains the dynamical processes, or whatever).”[11] universe].”[6] 19

[1] W. Heisenberg, The actual content of quan- [11] D. Wallace, Quantum probability from sub- tum theoretical kinematics and mechanics, jective likelihood: Improving on deutsch’s Zeitschrift für Physik (1927). proof of the probability rule, Studies in His- [2] The solvay conference, probably the most tory and Philosophy of Science (2007). intelligent picture ever taken, 1927. [12] S. Saunders, J. Barrett, A. Kent, and [3] A. Einstein, B. Podolsky, and N. Rosen, D. Wallace, Many Worlds?: Everett, Quan- Can quantum-mechanical description of tum Theory, & Reality (Oxford University physical reality be considered complete?, Press, 2010). Physical Review (1935). [13] D. Wallace, The Emergent Multiverse (Ox- [4] D. M. Greenberger, Bell’s theorem without ford University Press, 2014). inequalities, American Journal of Physics [14] D. Wallace, Epistemology quantized: cir- (1990). cumstances in which we should come to [5] J. Townsend, Quantum Physics: A Funda- believe in the everett interpretation, The mental Approach to Modern Physics (Uni- British Journal for the Philosophy of Sci- versity Science Books, 2010). ence (2006). [6] J. A. Wheeler, Assessment of everett’s “rel- [15] M. Arndt, O. Nairz, J. Vos-Andreae, ative state” formulation of quantum theory, C. Keller, G. van der Zouw, and Reviews of Modern Physics (1957). A. Zeilinger, Wave-particle duality of [7] H. Everett, The theory of the universal c60 molecules, Nature (1999). wave function, Princeton University Press [16] M. C. Price, The everett faq (1995). (1957). [17] S. S. Hawking, Black holes and thermody- [8] H. Everett, “relative state” formulation of namics, Physical Review D (1975). quantum mechanics, Reviews of Modern [18] D. Deutsch, Quantum theory of probabil- Physics (1957). ity and decisions, Proceedings of the Royal [9] B. S. DeWitt, Quantum mechanics and re- Society of London (1999). ality, Physics Today (1970). [19] D. Wallace, Everettian rationality: defend- [10] A. Barrau, Testing the everett interpreta- ing deutsch’s approach to probability in the tion of quantum mechanics with cosmology everett interpretation, Studies in History (2015). and Philosophy of Science (2003). 20

[20] C. Clark, A theoretical introduction to wave [32] E. Yudkowsky, The quantum physics se- mechanics (2010). quence: Collapse postulates (2008). [21] D. Deutsch, Comment on lockwood, The [33] J. Ellis and J. S. Hagelin, Search for vi- British Journal for the Philosophy of Sci- olations of quantum mechanics, Nuclear ence (1996). Physics (1984). [22] S. Carroll, The wrong objections to the [34] J. Ellis, J. L. Lopez, N. E. Mavromatos, many-worlds interpretation of quantum me- and D. V. Nanopoulos, Precision tests of chanics (2015). cpt symmetry and quantum mechanics in [23] J. B. Hartle, Spacetime quantum mechanics the neutral kaon system, Physical Review and the quantum mechanics of spacetime D (1996). (2014). [35] M. Agrawal, Linearity in quantum mechan- [24] H. D. Zeh, Feynman’s interpretation of ics (2003). quantum theory, The European Physical [36] H. D. Zeh, Measurement in bohm’s versus Journal (2011). everett’s quantum theory, Foundations of [25] S. Carroll, Why the many-worlds formula- Physics (1988). tion of quantum mechanics is probably cor- [37] W. H. Zurek, Decoherence and the transi- rect (2014). tion from quantum to classical—revisited, [26] A. I. M. Rae, Everett and the born rule, Los Alamos Science (2002). Studies in History and Philosophy of Sci- [38] M. Schlosshauer, Decoherence, the meaus- ence (2009). rement problem, and interpretations of [27] R. J. Solomonoﬀ, A preliminary report on a quantum mechanics (2005). general theory of inductive inference (1960). [39] G. Bacciagaluppi, The Role of Decoherence [28] A. N. Soklakov, Occam’s razor as a formal in Quantum Mechanics (Stanford Encyclo- basis for a physical theory (2001). pedia of Philosophy, 2012). [29] A. Altair, An intuitive explanation of [40] D. Wallace, Everett and structure, Stud- solomonoﬀ induction (2012). ies in History and Philosophy of Science [30] L. Kelion, Coder creates smallest chess (2003). game for computers (2015). [41] H. D. Zeh, On the interpretation of mea- [31] M. Bitbol, The concept of measurement and surement in quantum theory, Foundations time symmetry in quantum mechanics, Phi- of Physics (1970). losophy of Science (1988). [42] H. D. Zeh, Why bohm’s quantum theory?, 21

Foundations of Physics Letters (1999). (2015). [43] R. B. Griﬃths, Consistent histories and the [47] D. J. Bohm and B. J. Hiley, The de broglie interpretation of quantum mechanics, Jour- pilot wave theory and the further develop- nal of Statistical Physics (1984). ment of new insights arising out of it, Foun- [44] M. Gell-Mann and J. B. Hartle, Quantum dations of Physics (1982). mechanics in the light of quantum cosmol- [48] H. R. Brown and D. Wallace, Solving the ogy, Int. Symp. Foundations of Quantum measurement problem: De broglie-bohm Mechanics (1989). loses out to everett, Foundations of Physics [45] P. Wallden, Contrary inferences in consis- (2005). tent histories and a set selection criterion [49] S. Coleman, Quantum mechanics in your (2014). face (1994). [46] S. Lloyd and O. Dreyer, The universal path integral, Quantum Information Processing