3D-Space and the preferred basis cannot uniquely emerge from the quantum structure
33D-Space and the preferred basis cannot uniquely emerge from the quantum structure
Ovidiu Cristinel Stoica
Dept. of Theoretical Physics, NIPNE—HH, Bucharest, Romania.Email: [email protected], [email protected] (Dated: March 2, 2021)Is it possible that only the state vector exists, and the 3D-space, a preferred basis, a preferredfactorization of the Hilbert space, and everything else, emerge uniquely from the Hamiltonian andthe state vector?In this article no-go theorems are given, showing that whenever such a candidate preferred struc-ture exists and can distinguish among physically distinct states, infinitely many physically distinctstructures of the same kind exist. The idea of the proof is very simple: it is always possible to makea unitary transformation of the candidate structure into another one of the same kind, but withrespect to which the state of the system at a given time appears identical to its (physically distinct)state at any other time, or even to states from “alternative realities”.Therefore, such minimalist approaches lead to strange consequences like “passive” travel in timeand in alternative realities, realized simply by passive transformations of the Hilbert space.These theorems affect all minimalist theories in which the only fundamental structures are thestate vector and the Hamiltonian, whether they assume branching or state vector reduction, in par-ticular, the version of Everett’s Interpretation coined by Carroll and Singh “Mad-dog Everettian-ism”, various proposals based on decoherence, proposals that aim to describe everything by thequantum structure, and proposals that spacetime emerges from a purely quantum theory of gravity.
I. INTRODUCTION
The Quantum Mechanics (QM) of a closed system isdefined in terms of a
Hilbert space H , a Hamiltonianoperator (cid:98) H , and a state vector | ψ ( t ) (cid:105) ∈ H which dependson time, according to the Schr¨odinger equation i (cid:126) dd t | ψ ( t ) (cid:105) = (cid:98) H | ψ ( t ) (cid:105) . (1) Definition:
MQS . In the following, the triple ( H , (cid:98) H, | ψ (cid:105) ) (2) together with the Schr¨odinger equation (1) , will be called minimalist quantum structure ( MQS ) . The unitary symmetry of the Hilbert space in the
MQS seems to be broken only by the Hamiltonian operator (cid:98) H . But to connect the Hilbert space formalism withthe empirical observations, certain Hermitian operatorsneed to represent positions and momenta, a particularfactorization of the Hilbert space is required to representparticles, and, in general, a much richer structure thanthe MQS seems to be needed. Given that the postulatesof various formulations of QM are perfectly symmetricto the unitary symmetry, it makes sense to expect thatsuch formulations lead somehow to the rich structure thatdescribes our physical world. And indeed, it is often be-lieved that these structures can be uniquely recovered.This is sometimes expected to be true in particularin Everett’s Interpretation and its Many-Worlds variants(MWI) [23, 24, 50, 68, 72], but also in the
ConsistentHistories approaches [25, 26, 30, 41, 42, 44]. Presum-ably, decoherence [32, 33, 35, 52, 74–76, 78, 80] is thekey that solves the preferred basis problem and leads tothe emergence of the classical world. Therefore, claims that the preferred basis problem is solved became verycommon cf.
Wallace [71, 72], Tegmark [67], Brown andWallace [10], Zurek [76], Schlosshauer [51, 52], Saunders[49, 50] etc . Such claims were criticized, at least for MWI,by Kent [34] for seeming to require a preferred choice ofa basis to start with. Some authors stated clearly thatat least the configuration space and even the 3D-space,and a pre-existent factorization, are prerequisites of thetheory [69, 72].Therefore, the weak version of such programs assumesthe representation of the state vector | ψ (cid:105) as a wavefunc-tion ψ ( x ) = (cid:104) x | ψ (cid:105) on the configuration space, and maybea special role played by the 3D-space. If we include theconfiguration space along with the MQS , then, at least innonrelativistic QM, the factorization and the distancescan be decoded from the potential term of the Hamilto-nian, as explained e.g. in [1]. But there is a view that,if we take the unitary symmetry seriously, we should in-terpret ψ ( x ) = (cid:104) x | ψ (cid:105) as just a particular representationfavored only if we pick a preferred basis ( | x (cid:105) ) x ∈ R n of theHilbert space, while the right structure is the state vec-tor | ψ (cid:105) . Taking | ψ (cid:105) as a vector is often seen as makingmore sense, since a preferred representation of the Hilbertspace would be akin to the notion of an absolute refer-ence frame of space. And indeed this is often the statedposition in the discussions about a preferred basis, emer-gent space, or preferred factorization. The proofs givenin this article concern this strong version. A brief discus-sion of the weak version is contained in the last section,and another paper will give more details.When it is said that a preferred structure emerges, itis assumed that it satisfies very strict constraints, whichdefine what is understood by “preferred”. Otherwise,simple arguments can be used to show that there aremultiple choices of the factorization, exhibiting different a r X i v : . [ qu a n t - ph ] M a r physical interactions [21, 31], which can even be reducedto simply changing the phase of the other systems [59].Claims that a preferred position basis and a preferredfactorization emerge uniquely are not to be understoodas applying to all possible Hamiltonian operators. IljaSchmelzer gave simple counterexamples [53, 54]. Heused a Hamiltonian whose potential is a solution of the Korteweg-de Vries equation , depending on a parameter s ∈ R , and applied a result connecting the solutions givenby different values of s [37] to obtain different choices ofthe 1D-space for q . Schmelzer combined such Hamil-tonians to build Hamiltonians on larger Hilbert spacesand obtained physically distinct factorizations. But doesthis non-uniqueness hold in general, or it is an excep-tion based on a very special Hamiltonian? Could theHamiltonian from QM, which is different, be sophisti-cated enough to allow unique preferred structures?Apparently, Carroll and Singh showed that the Hamil-tonian is sufficient, more precisely, that its spectrum isenough to determine an “essentially unique” space struc-ture ([13], p. 99)a generic Hamiltonian will not be local withrespect to any decomposition, and for thespecial Hamiltonians that can be written in alocal form, the decomposition in which thatworks is essentially unique.In [13], p. 95, they wrote about the MQS thatEverything else–including space and fieldspropagating on it–is emergent from theseminimal elements.Carroll and Singh based their reconstruction of spaceon the results obtained by Cotler et al. [15] regarding theuniqueness of factorization of the Hilbert space, so thatthe interaction encoded in the Hamiltonian is “local” ina certain sense. But this uniqueness is in fact unitaryequivalence that preserves the form of the Hamiltonian,and we will see that it allows infinitely many physicallydistinct choices of the same kind.In this article we will give proofs that, whenever theHamiltonian leads to a tensor product decomposition ofthe Hilbert space, a 3D-space structure, or a preferredgeneralized basis, it leads to infinitely many physicallydistinct structures of the exact same type. This remainstrue even for constructions that also take the state vectorinto account.In Sec. § II we show that there are infinitely many phys-ically distinct 3D-spaces in nonrelativistic QM. This caseis used to illustrate the main idea of the proof, which willbe given in full generality in Sec. § III. The main theo-rem shows that, if a candidate preferred structure is ableto distinguish physically distinct states, then there aremore (in fact, infinitely many) physically distinct suchstructures. The idea of the proof is very simple (Fig. 1). a1) a2)b)
FIG. 1. Schematic representation of the proof of Theorem 2. a) The state vector, in green, changes in time from | ψ ( t ) (cid:105) to | ψ ( t ) (cid:105) = (cid:98) U t ,t | ψ ( t ) (cid:105) . The solid blue triangle representsthe candidate preferred structure S (cid:98) H . The dashed blue linesrepresent the relations between S (cid:98) H and | ψ ( t ) (cid:105) . The conditionthat S (cid:98) H distinguishes physically distinct states at differenttimes implies that these relations change as | ψ ( t ) (cid:105) changes. b) But unitary symmetry implies that at t there is anotherstructure S (cid:48) (cid:98) H , represented in red, of the exact same kind as S (cid:98) H , which is in the same relation with | ψ ( t ) (cid:105) as S (cid:98) H is in re-lation with | ψ ( t ) (cid:105) at the time t . It is obtained by a unitarytransformation (cid:98) S = (cid:98) U − t ,t as S (cid:48) (cid:98) H = (cid:98) S [ S (cid:98) H ]. Distinguishingnessimplies that the structures S (cid:98) H and S (cid:48) (cid:98) H are different. There-fore, there is no preferred structure. In Sec. § IV we prove that this remains valid even ifwe supplement the
MQS with projections corresponding tothe state vector reduction. So the problem is not specificto the approaches based on branching of the state vec-tor, like Everett’s, but also plagues all purely quantumreconstructions of QM.In Sec. § V we apply the main theorem from Sec. § IIIto prove the non-uniqueness of generalized “preferred”bases (in § V A), factorizations into subsystems (in § V B),3D-space structures (both as in the approach by Carrolland Singh, in § V C, and in general, in § V D), generalizedbases based on coherent states (in § V E), environmentaldecoherence (in § V F), and emergent macro classicality(in § V G).In Sec. § VI we show that the assumption that the
MQS is the only fundamental structure has strange conse-quences: the state vector representing the present stateequally represents all the past and future states and al-ternative realities.In Sec. § VII we discuss the possible options with whichthese results leave us.
II. NON-UNIQUENESS OF SPACE INNONRELATIVISTIC QUANTUM MECHANICS
The general proof that there is no way to uniquelyrecover the 3D-space or other preferred structures for the
MQS alone will be given in Section § III, but it is useful toillustrate first the idea with a more explicit proof, for thecase of nonrelativistic Quantum Mechanics (NRQM).The usual presentation of the Hamiltonian operatoris in a form that emphasizes the positions. Here is anexample for n particles in NRQM, (cid:98) H = − n − (cid:88) j =0 (cid:126) m j (cid:88) a =1 ∂ ∂x j + a + n − (cid:88) j =0 n − (cid:88) k =0 j (cid:54) = k V j,k ( d j,k ) (cid:98) I H , (3)where ( x , . . . , x n ) ∈ R n is a point in the configurationspace, m j is the mass of the particle j , and V j,k , thepotential of interaction between the particles j and k ,depends on the 3D distance between them d j,k = (cid:32) (cid:88) a =1 ( x j + a − x k + a ) (cid:33) . (4)In this representation, the state vector takes the form ofa wavefunction defined as ψ ( x , . . . , x n , t ) := (cid:104) x , . . . , x n | ψ ( t ) (cid:105) , (5)which belongs to the Hilbert space of complex square-integrable functions L (cid:0) R n (cid:1) .This expression of the Hamiltonian in terms of vari-ables ( x j ) j ∈{ ,..., n } representing the positions in the clas-sical configuration space is due to using a position basis.Let us now consider a unitary transformation (cid:98) S of theHilbert space H . For all j ∈ { , . . . , n } , given the posi-tion operators (cid:98) x j and the momentum operators (cid:98) p j = − i (cid:126) ∂∂x j , (6)let us define their transformations by (cid:98) S , (cid:98)(cid:101) x j := (cid:98) S (cid:98) x j (cid:98) S − (cid:98)(cid:101) p j := (cid:98) S (cid:98) p j (cid:98) S − = − i (cid:126) ∂∂ (cid:101) x j . (7)In general, under the transformation (cid:98) S , the form of theHamiltonian (3) changes in the new basis.Since the spectrum of each of the operators (cid:98) x j and (cid:98) p j is R and the transformation (cid:98) S is unitary, the spectrum ofeach of the operators (cid:98)(cid:101) x j and (cid:98)(cid:101) p j is R as well. In the newbasis (7), the state vector | ψ (cid:105) is no longer represented inthe position basis as in (5), but as a wavefunction (cid:101) ψ ( (cid:101) x , . . . , (cid:101) x n , t ) = (cid:104) (cid:101) x , . . . , (cid:101) x n | ψ ( t ) (cid:105) , (8) in the basis parametrized by the eigenvalues (cid:101) x j of (cid:98)(cid:101) x j | (cid:101) x , . . . , (cid:101) x n (cid:105) = (cid:98) S | x , . . . , x n (cid:105) . (9)The wavefunction (8) is also a square-integrable com-plex function from a space L (cid:0) R n (cid:1) , but in general thisspace is not the same as the one of the wavefunctions ofpositions. In particular, the parameters ( (cid:101) x j ) j are in gen-eral not coordinate transformations of parameters ( x j ) j .The dependence of the potential V j,k on the 3D dis-tance (4) suggests a few remarks. First, if there is nointeraction, the Hamiltonian reduces to the kinetic term (cid:98) T = − n − (cid:88) j =0 (cid:126) m j (cid:88) a =1 ∂ ∂x j + a , (10)and we cannot even recover the number of dimensions(unless each particle has a different mass m j ), since theonly contribution of the three dimensions appears in theexpression of the potentials, see e.g. [1]. We will assumethat there are sufficiently many interactions to allow therecovery of the 3D-space, provided that we know the po-sition configuration space. So we only need to focus onrecovering the position configuration space.Second, it suggests that not all reparametrizations de-fined by unitary transformations (7) recover the original3D-space, even when (cid:98) S commutes with (cid:98) H . Is it thenpossible to uniquely recover the 3D-space?The kinetic term (cid:98) T in (10) is a function of the momen-tum operators (cid:98) p j . Since they all commute, the transfor-mation (cid:98) S (cid:98) T (cid:98) S − of (cid:98) T is a function of the operators (cid:98)(cid:101) p j .Similarly, the potential part of the Hamiltonian (cid:98) V = n − (cid:88) j =0 n − (cid:88) k =0 j (cid:54) = k V j,k ( d j,k ) (cid:98) I H (11)is a function of positions, and since it acts by multipli-cation and the position operators commute with one an-other, the transformation (cid:98) S (cid:98) V (cid:98) S − of (cid:98) V is a function ofthe operators (cid:98)(cid:101) x j . Thus, (cid:98) S (cid:98) V (cid:98) S − acts on (cid:101) ψ ( (cid:101) x , . . . , (cid:101) x n , t )from (8) by multiplication with a function of the eigen-values (cid:101) x j , obtained by a change of variables. Remark . If the unitary transformation (cid:98) S commuteswith the Hamiltonian (cid:98) H , then the Hamiltonian has thesame form as (3), but expressed in terms of the vari-ables ( (cid:101) x j ) j instead of ( x j ) j . However, in general, theform of | ψ ( t ) (cid:105) will be different. Only if, in addition,we require that the transformation (cid:98) S leaves | ψ ( t ) (cid:105) un-changed ( i.e. | ψ ( t ) (cid:105) is an eigenvector of (cid:98) S ), the wave-function (cid:101) ψ ( (cid:101) x , . . . , (cid:101) x n , t ) is identical as a function to thewavefunction ψ ( x , . . . , x n , t ), only the variables differ-ing. But this does not mean that, when the 3D-spacecan be recovered from the MQS , the result is “essentiallyunique”. If for example the transformation (cid:98) S is inducedby a coordinate transformation of the 3D-space, then theform of the wavefunction changes, but the system is phys-ically the same. So it would be too strong to require that | ψ ( t ) (cid:105) is an eigenvector of (cid:98) S . Even if we require it, theremay be transformations (cid:98) S that leave | ψ ( t ) (cid:105) invariant, butlead to a parametrization ( (cid:101) x j ) j that cannot represent thesame 3D-space that ( x j ) j does. Theorem 1.
Any procedure to recover the D-space fromthe NRQM Hamiltonian leads to infinitely many physi-cally distinct solutions.Proof.
Suppose we found a candidate position basis, inwhich the wavefunction has the form ψ ( x , . . . , x n , t ).Let us see what other parametrizations that look like the3D-space can we find at the time t .For the reparametrization (7), we take as unitarytransformation (cid:98) S = (cid:98) U − t j ,t , (12)where (cid:98) U t j ,t := e − i (cid:126) (cid:98) H ( t j − t ) (13)is the unitary time evolution operator , i.e. | ψ ( t j ) (cid:105) = (cid:98) U t j ,t | ψ ( t ) (cid:105) . (14)The transformation (cid:98) S is not to be seen as a time trans-lation, but as a unitary symmetry transformation of H at t . Since [ (cid:98) H, (cid:98) U t j ,t ] = 0, we obtain another parametriza-tion in which the Hamiltonian operator has exactly thesame form.Then, in the new parametrization (cid:101) x j , the wavefunctionhas the form (8). But from equations (12) and (14) (cid:101) ψ ( (cid:101) x , . . . , (cid:101) x n , t ) (8) = (cid:104) (cid:101) x , . . . , (cid:101) x n | ψ ( t ) (cid:105) (9) = (cid:16) (cid:104) x , . . . , x n | (cid:98) S † (cid:17) | ψ ( t ) (cid:105) = (cid:104) x , . . . , x n | (cid:16) (cid:98) S † | ψ ( t ) (cid:105) (cid:17) (12) = (cid:104) x , . . . , x n | (cid:16) (cid:98) U t j ,t | ψ ( t ) (cid:105) (cid:17) (14) = (cid:104) x , . . . , x n | ψ ( t j ) (cid:105) = ψ ( x , . . . , x n , t j ) . (15)This means that in the configuration space of positionsobtained by using the unitary transformation (12), thewavefunction is identical to the one of the physically dis-tinct state at t j . Hence, we obtained another structurethat is similar to the original configuration space, butit is physically distinct. Since there are infinitely manymoments of time t j when the state is physically different,this means that there are infinitely many physically dis-tinct ways to choose the configuration space of positions.Hence, there are also infinitely many ways to choose the3D-space. Remark . Theorem 1 is based on the observation that aunitary symmetry commuting with the Hamiltonian al-low us to change the basis defining the 3D-space whileleaving the state vector | ψ (cid:105) untouched. Normally wewould require | ψ (cid:105) to transform as well, but we we wereallowed to consistently apply the transformation to theposition basis independently of | ψ (cid:105) because the 3D-spaceshould be independent on | ψ (cid:105) . In Section § III these re-sults are extended to general candidate preferred struc-tures, which may depend on | ψ (cid:105) as well. The key require-ment will be the ability to distinguish among physicallydistinct states at different times. III. NON-UNIQUENESS OF GENERALPREFERRED STRUCTURES
Let us extend the results from Sec. § II to general kindsof structures. Since we are dealing with different kindsof structures (generalized basis, tensor product structure,and emerging 3D-space structure), one should also definethe “kind” of each structure. The symmetries of the
MQS require us to define such structures as tensor objects overthe Hilbert space, and the kinds of the structures as thetypes of these tensor objects plus unitary invariant con-ditions that they are required to satisfy [47, 73]. Theconditions are needed to express what it means to be“preferred”.We denote the space of tensors of type ( r, s ) over H , i.e. the space of C - multilinear functions from (cid:78) r H ∗ ⊗ (cid:78) s H to C , where H ∗ is the dual of H , by T rs ( H ) := (cid:78) r H ⊗ (cid:78) s H ∗ . (16)The tensor algebra is T ( H ) := (cid:76) ∞ r =0 (cid:76) ∞ s =0 T rs ( H ) . (17)If (cid:98)(cid:98) A ∈ T rs ( H ) is a tensor, and (cid:98) S is a unitary transfor-mation of H , we denote by (cid:98) S [ (cid:98)(cid:98) A ] the tensor obtained byunitary transformation from the tensor (cid:98)(cid:98) A . In particular,scalars c ∈ T ( H ) ∼ = C are invariant constants (cid:98) S [ c ] = c .For | ψ (cid:105) ∈ T ( H ) = H , (cid:98) S [ | ψ (cid:105) ] = (cid:98) S | ψ (cid:105) , (cid:98) S [ (cid:104) ψ | ] = (cid:104) ψ | (cid:98) S † ,and for (cid:98) A ∈ T ( H ), (cid:98) S [ (cid:98) A ] = (cid:98) S (cid:98) A (cid:98) S † . For general tensorstransform each of the factor Hilbert spaces in eq. (16).We denote by Herm ( H ) ⊂ T ( H ) the space of Hermitianoperators on H .If A , X are sets, X A is a standard notation for the setof functions defined on A with values in X .While the proof of non-uniqueness is simple, we mustgo first through the definitions of the structures involved. Definition 1 (Tensor structures) . Let A be a set, θ : A → N be a function θ ( α ) = ( r α , s α ) , (18)and let T θ ( α ) ( H ) := T r α s α ( H ) (19)for all α ∈ A . Denote by T θ ( H ) := (cid:89) α ∈ A T θ ( α ) = (cid:89) α ∈ A T r α s α = { ( (cid:98)(cid:98) A α ) α ∈ A | ( ∀ α ∈ A ) (cid:98)(cid:98) A α ∈ T r α s α ( H ) } (20)the set of all structures consisting of tensors( (cid:98)(cid:98) A α ) α ∈ A , (21)where (cid:98)(cid:98) A α ∈ T θ ( α ) ( H ) for all α ∈ A . We call the elementsof T θ ( H ) tensor structures of type θ . Definition 2 (Invariant tensor functions) . Let A be aset and let θ ∈ ( N ) A . An invariant tensor function is afunction F : T θ ( H ) → T ( H ) (22)which is invariant under unitary symmetries, i.e. for anyunitary operator (cid:98) S on H and any ( (cid:98)(cid:98) A α ) α ∈ A ∈ T θ ( H ), F (cid:16) ( (cid:98) S [ (cid:98)(cid:98) A α ]) α (cid:17) = (cid:98) S (cid:2) F (cid:0) ( (cid:98)(cid:98) A α ) α (cid:1)(cid:3) . (23) Definition 3 (Kind) . A kind K = { C β } β ∈ B is a set ofinvariant tensor functions C β : T θ ( H ) × Herm( H ) × H → T ( H ) , (24)where A , B are two sets and θ ∈ ( N ) A is fixed. Thefactor Herm( H ) in (24) is needed to allow the functions C β to depend on the Hamiltonian.The kinds are required to be invariant because other-wise we will assume a symmetry breaking of the MQS . Definition 4 ( K -structure ) . Let A , B two sets and θ ∈ ( N ) A . A structure of kind K = { C β } β ∈ B or K -structure for the Hamiltonian (cid:98) H is defined as a function S (cid:98) H : H → T θ ( H ) , S (cid:98) H ( | ψ (cid:105) ) = (cid:16) (cid:98)(cid:98) A | ψ (cid:105) α (cid:17) α ∈ A , (25)so that for any β ∈ B and | ψ (cid:105) ∈ HC β (cid:16) ( (cid:98)(cid:98) A | ψ (cid:105) α ) α ∈ A , (cid:98) H, | ψ (cid:105) (cid:17) = 0 . (26)The set of functions K is called the kind of the struc-ture S (cid:98) H , and eq. (26) gives its defining conditions . Notethat some of the defining conditions may be independenton some of the tensors (cid:98)(cid:98) A | ψ (cid:105) α .Definitions 3 and 4 may seem too abstract. Often allof the tensors (cid:98)(cid:98) A | ψ (cid:105) α will be Hermitian operators (cid:98) A | ψ (cid:105) α .In this case, we will call the K -structure Hermitian K -structure . Hermitian K -structures will turn out to besufficient for most of the cases discussed in the article. Apossible reason why Hermitian operators are sufficient forthe relevant cases is that they correspond to observables.Let us give a simple example, so that the reader can havesomething concrete in mind when following the proofs. Example . A basis ( | α (cid:105) ) α ∈ A of H de-fines a K -structure S (cid:98) H ( | ψ (cid:105) ) = (cid:16) (cid:98) A α := | α (cid:105)(cid:104) α | (cid:17) α ∈ A , (27)where the kind K is given by the functions (cid:98) A α (cid:98) A α (cid:48) − (cid:98) A α δ αα (cid:48) ,I H − (cid:80) α ∈ A (cid:98) A α , tr (cid:98) A α − , (28)where α, α (cid:48) ∈ A . Hence, the defining conditions are (cid:98) A α (cid:98) A α (cid:48) − (cid:98) A α δ αα (cid:48) = (cid:98) ,I H − (cid:80) α ∈ A (cid:98) A α = (cid:98) , tr (cid:98) A α − α, α (cid:48) ∈ A . The first condition encodes the fact that (cid:98) A α are projectors on mutually orthogonal subspaces of H , the second one that they form a complete system, andthe third one that these subspaces are one-dimensional.In the preferred basis case one does not usually expectthe operators (cid:98) A α to depend on | ψ (cid:105) , but we may wantto consider cases when additional conditions make themdependent. In this case, we will write (cid:98) A | ψ (cid:105) α instead of (cid:98) A α .We will see that, even so, there are infinitely many phys-ically distinct bases with the same defining conditions.In Sec. § V we will see that Definition 4 covers as par-ticular cases tensor product structures, more general no-tions of emergent 3D-space or spacetime, and general no-tions of generalized bases.Let us state the two main conditions that we expect tobe satisfied by a procedure of constructing a K -structure.The first condition that we will require a K -structureto satisfy is to be time-distinguishing , i.e. to be able todistinguish among physically distinct states the systemcan have at different times. Definition 5 (Time-distinguishing structure) . A succes-sion of states is a set of physically distinct state vectors V = {| ψ ( t j ) (cid:105) ∈ H | t j ∈ T } connected by unitary evo-lution, where T ⊆ R has at least two elements. A ten-sor structure (cid:16) (cid:98)(cid:98) A | ψ (cid:105) α (cid:17) α ∈ A is said to be time-distinguishing for the succession of states V , if for any pair | ψ ( t j ) (cid:105) (cid:54) = | ψ ( t k ) (cid:105) ∈ V there is an invariant scalar function I : T θ ( H ) × H → C (30)able to distinguish | ψ ( t j ) (cid:105) and | ψ ( t k ) (cid:105) , (cid:16) I (cid:16) (cid:98)(cid:98) A | ψ ( t j ) (cid:105) α , | ψ ( t j ) (cid:105) (cid:17)(cid:17) α ∈ A (cid:54) = (cid:16) I (cid:16) (cid:98)(cid:98) A | ψ ( t k ) (cid:105) α , | ψ ( t k ) (cid:105) (cid:17)(cid:17) α ∈ A . (31)The inequality (31) is given for the set A rather thanindividually for each α in order to avoid “false positives”due to possible permutation symmetries of A allowed bythe defining conditions (26).Often the K -structure S (cid:98) H ( | ψ (cid:105) ) will consist of Hermi-tian operators (cid:16) (cid:98) A | ψ (cid:105) α (cid:17) α ∈ A , and the invariants used toprove that they are distinguishing will be their mean val-ues (cid:104) ψ | (cid:98) A | ψ (cid:105) α | ψ (cid:105) .We should expect our K -structure to distinguishamong a succession of possibly infinitely many states thata system can have at different times. This justifies thefollowing condition: Condition 1 (Time-Distinguishingness) . The K -structure should be time-distinguishing for a set of statevectors {| ψ ( t j ) (cid:105)| t j ∈ T } representing physically distinctstates, where T ⊆ R has at least two elements.Condition 1 captures the idea, used in the proof ofTheorem 1, that different physical states “look different”with respect to a candidate preferred structure. Thisis true for a candidate preferred space, basis, and ten-sor product structure. Without this condition, the newstructure would add nothing interesting to the MQS . Observation 1.
For Condition 1 only the tensor struc-ture (cid:16) (cid:98)(cid:98) A | ψ ( t ) (cid:105) α (cid:17) α ∈ A matters, not its kind K . The kind onlyspecifies conditions to be satisfied by the tensor structure,but these only affect the existence of K -structures, notwhether they are distinguishing or not.The other condition is that of uniqueness. Condition 2 (Uniqueness) . If a K -structure S (cid:98) H existsfor a MQS , it is the only K -structure for that MQS .If the K -structure is not ordered, uniqueness should beunderstood up to a permutation of the indices α allowedby the defining conditions (26). Remark . Let us detail what Condition 2 means. Sup-pose that at the time t there is a K -structure for( H , (cid:98) H, | ψ ( t ) (cid:105) ), S (cid:98) H ( | ψ ( t ) (cid:105) ) = (cid:16) (cid:98)(cid:98) A | ψ ( t ) (cid:105) α (cid:17) α ∈ A (32)for any | ψ ( t ) (cid:105) ∈ H . Let (cid:98) S be a unitary transformation of H which commutes with (cid:98) H . Then, (cid:16)(cid:99)(cid:99) A (cid:48) (cid:98) S | ψ ( t ) (cid:105) α (cid:17) α ∈ A is a K -structure for ( H , (cid:98) H, (cid:98) S | ψ ( t ) (cid:105) ), where for each α ∈ A (cid:99)(cid:99) A (cid:48) (cid:98) S | ψ ( t ) (cid:105) α := (cid:98) S [ (cid:98)(cid:98) A | ψ ( t ) (cid:105) α ] . (33)The Uniqueness Condition 2 becomes S (cid:98) H ( (cid:98) S | ψ ( t ) (cid:105) ) = (cid:16) (cid:98) S [ (cid:98)(cid:98) A | ψ ( t ) (cid:105) α ] (cid:17) α ∈ A , (34)or, equivalently, the invariance identity (cid:98)(cid:98) A (cid:98) S | ψ ( t ) (cid:105) α = (cid:98) S [ (cid:98)(cid:98) A | ψ ( t ) (cid:105) α ] . (35)We will now prove that there is a contradiction betweenthe Conditions 1 and 2. Theorem 2.
If a K -structure is time-distinguishing,then it is not unique.Proof. The proof is based on the same idea of the proofof Theorem 1, illustrated in Fig. 1.Let us assume that the K -structure is time-distinguishing for a temporal succession of states {| ψ ( t j ) (cid:105)| t j ∈ T } which includes the present state | ψ ( t ) (cid:105) .Then, for any t j ∈ T , the Time-Distinguishingness Con-dition 1 states that there is an invariant I such that I (cid:16) ( (cid:98) (cid:98) A | ψ ( t j ) (cid:105) α ) α , | ψ ( t j ) (cid:105) (cid:17) (cid:54) = I (cid:16) ( (cid:98) (cid:98) A | ψ ( t ) (cid:105) α ) α , | ψ ( t ) (cid:105) (cid:17) . (36)But | ψ ( t j ) (cid:105) has the form | ψ ( t j ) (cid:105) = (cid:98) U t j ,t | ψ ( t ) (cid:105) , (37)where (cid:98) U t j ,t := e − i (cid:126) (cid:98) H ( t j − t ) . (38)Since the invariant I satisfies (23), I (cid:16) ( (cid:98)(cid:98) A | ψ ( t ) (cid:105) α ) α , | ψ ( t ) (cid:105) (cid:17) = I (cid:16) ( (cid:98) U t j ,t [ (cid:98)(cid:98) A | ψ ( t ) (cid:105) α ]) α , (cid:98) U t j ,t | ψ ( t ) (cid:105) (cid:17) = I (cid:16) ( (cid:98) U t j ,t [ (cid:98)(cid:98) A | ψ ( t ) (cid:105) α ]) α , | ψ ( t j ) (cid:105) (cid:17) . (39)Returning to eq. (36) we get I (cid:16) ( (cid:98)(cid:98) A | ψ ( t j ) (cid:105) α ) α , | ψ ( t j ) (cid:105) (cid:17) (cid:54) = I (cid:16) ( (cid:98) U t j ,t [ (cid:98)(cid:98) A | ψ ( t ) (cid:105) α ]) α , | ψ ( t j ) (cid:105) (cid:17) , (40)hence (cid:18) (cid:98)(cid:98) A (cid:98) U tj,t | ψ ( t ) (cid:105) α (cid:19) α (cid:54) = (cid:16) (cid:98) U t j ,t [ (cid:98)(cid:98) A | ψ ( t ) (cid:105) α ] (cid:17) α . (41)Let us keep (41) in mind and take the unitary trans-formation from Remark 3 detailing uniqueness to be (cid:98) S = (cid:98) U t j ,t , (42)where t j ∈ T . Again, the transformation (cid:98) S is not tobe seen as a time translation, but as a unitary symmetrytransformation of H at t . Since [ (cid:98) H, (cid:98) U t j ,t ] = 0, eq. (35),required by Condition 2 should hold. But this is contra-dicted by eq. (41), required by Condition 1. Hence, theTime-Distinguishingness Condition 1 and the UniquenessCondition 2 cannot both be true. Objection 1.
Theorem 2 was derived by assuming theinstantaneous state of the system at t , represented by | ψ ( t ) (cid:105) . But maybe if we take into account the dynam-ics, i.e. the values of | ψ ( t ) (cid:105) in an interval ( t − ∆ t/ , t +∆ t/ Reply . Theorem 2 applies to all times in the interval, sothat if a K -structure S (cid:98) H exists for ( t − ∆ t/ , t + ∆ t/ t j there will be a K -structure S (cid:48) (cid:98) H whoserelation with | ψ ( t ) (cid:105) in the time interval ( t − ∆ t/ , t +∆ t/
2) is exactly the same relation of S (cid:98) H with | ψ ( t ) (cid:105) in thetime interval ( t j − ∆ t/ , t j +∆ t/ (cid:3) Remark . The proof of Theorem 2 can easily be ex-tended to structures that distinguish states that theHamiltonian cannot distinguish and are not connectedby unitary evolution. All we need to do is to use a uni-tary transformation (cid:98) S which maps such state vectors oneinto another, and there are infinitely many such trans-formations. If one insists that the Hamiltonian’s formis essential, then we can choose unitary transformationsgenerated by Hermitian operators that commute with (cid:98) H .If H = (cid:76) λ ∈ σ ( (cid:98) H ) H λ is the decomposition of the Hilbertspace in eigenspaces of the Hamiltonian, then the groupof unitary transformations that commute with (cid:98) H is theinfinite-dimensional groupU (cid:98) H = (cid:89) λ ∈ σ ( (cid:98) H ) U ( H λ ) . (43)Even if we factor out of this group those transforma-tions generated by momentum or angular momentum op-erators (which assume a preferred 3D-space) and gaugesymmetries, which make no physical difference, the re-maining group is still infinite-dimensional, and all of itselements lead to distinct structures that are not distin-guished by the Hamiltonian (cid:98) H . Remark . There are ways to construct structures thatdepend on the
MQS alone and are unique, but they all vi-olate the Time-Distinguishingness Condition 1. Such ex-amples include the trivial ones | ψ (cid:105) , (cid:104) ψ | , | ψ (cid:105)(cid:104) ψ | , (cid:98) H , (cid:98) H | ψ (cid:105) ,more general operators like f ( (cid:98) H ), where f ( x ) is a for-mal polynomial or formal power series, but also morecomplex constructions like the direct sum decompositionof the Hilbert space into eigenspaces of (cid:98) H , projectionsof | ψ (cid:105) on these eigenspaces etc . In general, any invari-ant tensor function ( cf. Definition 2) F ( (cid:98) H, | ψ (cid:105) ), where F : Herm( H ) × H → T ( H ), leads to a unique structure,but no such structure is time-distinguishing.Theorem 2 applies to approaches based on true statevector reduction too, as Corollary 1 will show. IV. IMPACT ON STANDARD QUANTUMMECHANICS
At first sight, due to the use of unitary transforma-tions equivalent to unitary time evolution in the proofof Theorem 2, only the hard-core Everettianism has thisproblem. But the argument also works if we allow thestate vector to be reduced during measurements, and notmerely to branch. The reason is that we used the unitary evolution only to find out unitary transformations of theHilbert space in the “present” time t leading to phys-ically distinct structures. The role of unitary evolutionwas to show that these structures are physically distinct.Even if we extend the MQS to include not only theHamiltonian and the state vector, but also the observ-ables and the resulting eigenvalues, or the correspondingprojectors, we cannot avoid the implications of Theo-rem 2. Let us see why. Let us first notice that, given afactorization of the Hilbert space into subsystem spaces H = H S ⊗ H E , any projector (cid:98) P S ∈ Herm( H S ) acts on theentire Hilbert space H as the projector (cid:98) P ∈ Herm( H ), (cid:98) P := (cid:98) P S ⊗ (cid:98) I E . This includes the case of multiple mea-surements taking place simultaneously on different sub-systems of S , even if they are entangled, because the pro-jector (cid:98) P S may be the tensor product of multiple projec-tors corresponding to subsystems of S . So it is sufficientto consider sequences { (cid:98) P t j } t j of projectors on H . Corollary 1.
Assuming reductions of the state vectorcannot avoid the conclusion of Theorem 2.More precisely, let Q = (cid:16) H , (cid:98) H, | ψ ( t ) (cid:105) , { (cid:98) P t j } t j ∈ T (cid:17) be astructure consisting of the Hilbert space H , the Hamilto-nian (cid:98) H , a set of projectors { (cid:98) P t j } t j ∈ T , and the state vector | ψ ( t ) (cid:105) , assumed to evolve according to the Schr¨odingerequation (1) on R \ T , where T ⊂ R is a discrete orderedsequence of moments of time. Assume that, for all t j ∈ T , | ψ ( t j ) (cid:105) = | ψ j (cid:105) / || ψ j (cid:105)| , where | ψ j (cid:105) = (cid:98) P t j lim (cid:15) (cid:38) | ψ ( t j − (cid:15) ) (cid:105) and the projectors (cid:98) P t j satisfy the condition that | ψ j (cid:105) (cid:54) = 0.Then, at any time t ∈ R , if a time-distinguishing K -structure exists for ( H , (cid:98) H, | ψ ( t ) (cid:105) ), it is not unique. Thisholds even if we include in our structure Q the Hermitianoperators (cid:98) A t j corresponding to the measurements, andthe eigenvalues corresponding to their outcomes. Proof.
Let us consider a K -structure at the time t . Picka moment of time t (cid:48) such that ( t, t (cid:48) ) ∩ T (cid:54) = ∅ . This meansthat at least a quantum measurement begins after thetime t and ends before the time t (cid:48) , so that the state vectoris projected between t and t (cid:48) . Then, the time translationargument from the proof of Theorem 2 leads to the con-clusion that at t there is another K -structure of the samekind, in relation to which the system “appears” to be asuperposition of a state at the time t (cid:48) which is a superpo-sition of all possible outcomes of the measurements thathappened between t and t (cid:48) . But this is a physically dis-tinct state too, because otherwise there would be no needto postulate the state vector reduction in the first place .So we have two physically distinct K -structures at thetime t . Since t (cid:48) can be chosen in infinitely many ways,there are infinitely many physically distinct K -structuresat the time t .We can try to extend our structure Q to include theHermitian operators (cid:98) A t j corresponding to the measure-ments and the eigenvalues λ t j corresponding to the out-come of each measurement, so that each projector (cid:98) P t j corresponds to the eigenspace of (cid:98) A t j with eigenvalue λ t j .But all the data about (cid:98) A t j and λ t j should be encodedin the state of the measuring device, so it should be en-coded in the state vector | ψ ( t ) (cid:105) . And since our structure Q already includes the state vector, this will not help.We can even impose frequencies for the projections, ac-cording to the Born rule, this also does not help, becausethe Born rule is invariant to unitary symmetries.Therefore, the assumption that the reduction is realcannot avoid the conclusion of Theorem 2.Corollary 1 applies to all versions of Standard QM thatclaim to provide a purely quantum universal descriptionof the world, by including the measuring device in thestate vector. In particular, it is not relevant whether wesee the state vector as ontic or epistemic, what matters isthat it is supposed to represent the complete informationabout the quantum system. We can even see it as anabstract structure that allows us to predicts the outcomesof measurements and their probabilities, we can take theeigenvalues as the only “real” entities of the theory, thisis not relevant. It also does not matter whether the onlystructure considered to represent something real is theinformation about the outcomes of measurements.On the other hand, views like Bohr’s, that the mea-suring apparatus is a classical system and the observedsystem is quantum, are protected from the consequencesof Theorem 2, precisely because the measuring deviceis considered to have classical properties. In particular,its components have known positions, so the problem ofthe preferred 3D-space does not occur for the measur-ing device. Moreover, by interacting with the observedparticle and finding it in a certain place, this knowledgeof the position basis is extended from the system of themeasuring device to that of the observed particle. Butthere is of course a price: the theory does not include aquantum description of the measuring device, hence it isnot universal. The problem of recovering the 3D-space isavoided by “contaminating” the quantum representationof the observed system with information that can onlybe known from the macro classical level. V. APPLICATIONS TO VARIOUS CANDIDATEPREFERRED STRUCTURES
In this Section we will see that no preferred generalizedbasis, no preferred tensor product structure, no preferredemergent 3D-space, not even a preferred macro classicallevel, can be defined from a
MQS ( H , (cid:98) H, | ψ (cid:105) ) alone. Toprove this, we reduce each of these structures to a Hermi-tian K -structure, and then we apply Theorem 2 to showthat for each of them there are infinitely many physicallydistinct possible choices. A. Non-uniqueness of the preferred basis
We will now prove the non-uniqueness of time-distinguishing generalized bases.
Definition 6.
Let ( H , (cid:98) H, | ψ (cid:105) ) be a MQS . Let A , B (cid:48) besets, where B (cid:48) may be the empty set, and let B := ( H × A ) ∪ { } ∪ B (cid:48) . (44)A generalized basis is a K -structure S (cid:98) H ( | ψ (cid:105) ) = ( (cid:98) E | ψ (cid:105) α ) α ∈ A , (45)with the following defining conditions.The first condition is that for any α ∈ A and any | ψ (cid:105) ∈ H , the operator (cid:98) E | ψ (cid:105) α is positive semi-definite , i.e. C ( | φ (cid:105) ,α ) (cid:16) ( (cid:98) E | ψ (cid:105) α , | ψ (cid:105) ) α (cid:17) := h ( (cid:104) φ | (cid:98) E | ψ (cid:105) α | φ (cid:105) ) − , (46)where h : R → { , } , h ( x ) = 1 iff x ≥ | ψ (cid:105) ∈ H , theoperators ( (cid:98) E | ψ (cid:105) α ) α ∈ A form a resolution of the identity , C (cid:16) ( (cid:98) E | ψ (cid:105) α ) α , | ψ (cid:105) (cid:17) := I H − (cid:88) α (cid:98) E | ψ (cid:105) α = (cid:98) . (47)We see from conditions (46) and (47) that S (cid:98) H ( | ψ (cid:105) ) isa positive operator-valued measure (POVM).The set B (cid:48) is reserved for possible additional conditions C β ∈ B (cid:48) (cid:16) ( (cid:98) E | ψ (cid:105) α , | ψ (cid:105) ) α (cid:17) = 0 (48)reflecting a possible dependence of (cid:98) E | ψ (cid:105) α on | ψ (cid:105) . Example . Particular cases of POVM are orthogonalbases (Example 1), projection-valued measures (PVM)( i.e. projectors that give an orthogonal direct sum de-composition of the Hilbert space H ), and overcompletebases. All these cases are obtained by adding new con-ditions to the conditions (46) and (47) from Definition6. If we add the conditions that all (cid:98) E | ψ (cid:105) α are projectors, (cid:16) (cid:98) E | ψ (cid:105) α (cid:17) − (cid:98) E | ψ (cid:105) α = (cid:98) , (49)and that all distinct (cid:98) E | ψ (cid:105) α and (cid:98) E | ψ (cid:105) α (cid:48) are orthogonal, (cid:98) E | ψ (cid:105) α (cid:98) E | ψ (cid:105) α (cid:48) = (cid:98) , (50)we obtain a PVM that gives the orthogonal direct sumdecomposition of H H = (cid:76) α (cid:98) E | ψ (cid:105) α H . (51)If, in addition, we impose the condition that the pro-jectors (cid:98) E | ψ (cid:105) α are one-dimensional,tr (cid:98) E | ψ (cid:105) α − , (52)we obtain the preferred basis from Example 1. Theorem 3.
If there exists a time-distinguishing gen-eralized basis of kind K , then there exist infinitely manyphysically distinct generalized bases of the same kind K .Proof. Follows immediately by applying Theorem 2 tothe K -structure from Definition 6. Remark . Here we considered the generalized basis ofthe universe. The notion of a preferred generalized ba-sis related to quantum measurements or subsystems ingeneral, or selected by environmental decoherence, is adifferent issue, to be discussed in Sec. § V F.
B. Non-uniqueness of the tensor product structure
The Hilbert space H given as such, even in the pres-ence of the Hamiltonian, does not exhibit a preferredtensor product structure. Such a structure is needed toaddress the preferred basis problem for subsystems, andto reconstruct the 3D-space from the MQS . Definition 7. A tensor product structure (TPS) of aHilbert space H is an equivalence class of unitary iso-morphisms of the form (cid:78) ε ∈ E H ε (cid:55)→ H , (53)where H ε are Hilbert spaces, and the equivalence rela-tion is generated by local unitary transformations of each H ε and permutations of the set E . The Hilbert spaces H ε represent subsystems, e.g. they can be one-particleHilbert spaces.It is evident that, in the absence of other conditions,there are infinitely many TPS. But one may hope thatwe can add reasonable conditions that will make a uniqueTPS emerge from the MQS . Theorem 2 forbids this.
Theorem 4.
If there exists a time-distinguishing TPS ofa given kind K , then there exist infinitely many physicallydistinct TPS of the same kind K .Proof. We will show that the TPS structure is a K -structure, even though we will prove non-uniqueness byusing other invariants than the ones associated to its ten-sor structures. Following [15], we characterize the TPSin terms of operators on each of the spaces H ε , extendedto H . Let us define the subspaces of Hermitian operators H ε := (cid:16)(cid:78) ε (cid:48) (cid:54) = ε (cid:98) I ε (cid:48) (cid:17) ⊗ Herm( H ε ) , (54)where the factor Herm( H ε ) is inserted in the appropriateposition to respect the order. For ε (cid:54) = ε (cid:48) ∈ E , if (cid:98) A ε ∈ H ε and (cid:98) B ε (cid:48) ∈ H ε (cid:48) , then they commute, so their product isHermitian ( (cid:98) A ε (cid:98) B ε (cid:48) ) † = (cid:98) B † ε (cid:48) (cid:98) A † ε = (cid:98) B ε (cid:48) (cid:98) A ε = (cid:98) A ε (cid:98) B ε (cid:48) . Anyoperator from Herm( H ) can be expressed as a real linearcombination of products of operators from various H ε .We will now make an extravagant choice for the set A needed to define the kind K TPS for the TPS: A TPS := (cid:70) ε ∈ E Herm( H ε ) . (55) We choose the tensors (cid:98)(cid:98) A α giving our K TPS -structureas in Definition 4 to be the Hermitian operators (cid:98) A α ε := (cid:16)(cid:78) ε (cid:48) (cid:54) = ε (cid:98) I ε (cid:48) (cid:17) ⊗ (cid:98) α ε , (56)where (cid:98) α ε ∈ Herm( H ε ), and the defining conditions to bethe commutativity of (cid:98) A α ε and (cid:98) A α (cid:48) ε (cid:48) for ε (cid:54) = ε (cid:48) .The K TPS -structures satisfy time-distinguishingness,but rather than using invariants of its operators, we willuse other invariants. The reason is that the set of Hermi-tian operators from (56) is too extravagant, in the sensethat the operators (cid:98) A α ε corresponding to a fixed ε ∈ E can be transformed into one another by unitary trans-formations of H ε . This would make it difficult to keeptrack of the indices α ε when comparing the mean val-ues (cid:104) ψ | (cid:98) A α ε | ψ (cid:105) between unitary transformations of theHilbert space H to prove time-distinguishingness.So we rather use as invariants of the TPS the spectraof the reduced density operators ρ ε ( t ) obtained from thedensity operator ρ ( t ) = | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | by tracing over thespaces H ε (cid:48) with ε (cid:48) (cid:54) = ε . They are sufficient to show thetime-distinguishingness of the K TPS -structures, becausein general, subject to the constraint tr ρ ε ( t ) = 1, thespectrum changes in time due to the interactions betweenthe subsystems. Hence, Theorem 2 can be applied, andit follows that the K TPS -structures are not unique.Although we constructed the invariants directly fromthe TPS, and not from the K TPS -structure, we neededto show that the TPS correspond to a kind K TPS , toapply Theorem 2, but also to define more specific TPSby adding new defining conditions, as we will do in § V C.
C. Locality from the spectrum does not implyuniqueness of emergent 3D-space
Now whatever candidate preferred TPS structure wemay have in mind, we can define it as a K -structure byadding new defining conditions that can be expressed asinvariant tensor equations of the form (26). In particu-lar, anticipating the attempt to reconstruct the 3D-spacefrom the MQS described in [13], we need to talk about theTPS reports from [15].Cotler et al. obtained remarkable results concerningthe TPS for which the interactions between subsystemsare “local”, in the sense that the interaction encoded inthe Hamiltonian takes place only between a small numberof subsystems [15]. They showed that, in the rare caseswhen such a local TPS exists, it is unique up to an equiv-alence of TPS ( cf.
Definition 7). We do not contest theirresult, but we will see that, no matter how restrictive isthe algorithm to obtain a local TPS from the spectrumof the Hamiltonian, there are infinitely many physicallydistinct TPS of the same kind.Let us see what we need to add to the kind K TPS toobtain the notion of local TPS from [15]. First, Cotler etal. expand the Hamiltonian (cid:98) H as a linear combination of0products of operators (cid:98) A α ε ∈ H ε , such that each term isa product of operators (cid:98) A α ε with distinct values for ε ∈ E .Then they impose the condition of locality for the TPS,which is that the TPS has to be such that the numberof factors in each term of this expansion is not greaterthan some small number d ∈ N . This condition, which wewill call d -locality , is invariant to unitary transformationsas well, although it is not an equation like (26), but aninequation. But we already encountered a defining con-dition given by an inequation in Definition 6 eq. (46), sowe know how to express it as a tensor equation. Thus,we obtain a kind for the d -locality TPS, let us denote itby K TPS-L(d) , where L stands for “local”, and d is thesmall number from the d -locality condition. Theorem 5.
If there exists a time-distinguishing K TPS-L(d) -structure, then there exist infinitely manyphysically distinct K TPS-L(d) -structures.Proof.
The additional defining conditions required tomake the TPS d -local can be expressed as tensor equa-tions, as needed in the proof. But rather than makingit unique, these additional conditions only make it moredifficult to find a K TPS-L(d) -structure, and indeed mostHamiltonians do not admit a local TPS [15]. But whenthey admit one, because the TPS is still able to distin-guish different states, Condition 1 is satisfied. Theorem2 implies that the K TPS-L(d) -structure is not unique.Whenever a K TPS-L(d) -structure exists, infinitely manyphysically distinct ones exist. This does not challenge theresults of Cotler et al. , it only shows that it cannot beused to recover a unique 3D-space from the Hamiltonian’sspectrum the way Carroll and Singh want [13].Carroll and Singh have an interesting idea to startfrom the local TPS and construct a space. For d = 2,the K TPS-L(2) -structure defines a graph, whose verticesare in the set E , and whose edges are the pairs ( ε, ε (cid:48) ), ε (cid:54) = ε (cid:48) , corresponding to the presence of an interac-tion between the subsystems H ε and H (cid:48) ε . Carroll andSingh interpret E as space, and the edges as definingits topology. The topology of the space E depends onthe spectrum of the Hamiltonian only. They also usedthe mutual information between two regions R, R (cid:48) ⊂ E , I ( R : R (cid:48) ) = S R + S R (cid:48) − S RR (cid:48) , where S R = − tr ρ R ln ρ R is the von Neumann entropy of | ψ (cid:105) in the region R , todefine distances between regions. They associate shorterdistances to larger mutual information. Their programis to develop not only the spacetime, but also emergentclassicality, gravitation from entanglement etc . [11–14].Their results are promising, but unfortunately, there isno way for a unique or essentially unique 3D-space, orany other preferred structure in their program, to emergefrom the Hamiltonian’s spectrum. The additional con-structions they make on top of the TPS are invariant,and depend on the uniqueness of the TPS itself, becausethe topology depends on the TPS and the Hamiltonian.But Theorem 5 shows that the 2-local TPS, required todefine the points of this candidate space, is not unique. Just like in the NRQM case from Theorem 1, there areinfinitely many physically distinct choices. Therefore, Corollary 2.
The emergent 3D-space of Carroll andSingh [13] can be chosen in infinitely many physicallydistinct ways.
Proof.
It is a consequence of Theorem 5.This example illustrates the fact, already known fromTheorem 2, that uniqueness may be violated even whenthe K -structure depends on | ψ (cid:105) .Another interesting idea to recover spacetime from aquantum theory was proposed by Giddings [20, 28]. Fol-lowing a profound analysis of how Local Quantum FieldTheory extends to gravity, Giddings notices that the gen-eral relativistic gauge (diffeomorphism) invariance con-flicts with usual tensor product decompositions, evenin the weak field limit. For this reason, he rejects theidea of using a commuting set of observable subalge-bras, and proposes instead a network of Hilbert subspaces( H ε ) ε ∈ E , where H ε (cid:44) → H for all ε ∈ E , and each H ε consists of state vectors in H that are indistingushableoutside a neighborhood U ε . For separated neighborhoods U ε and U ε (cid:48) , the condition H ε ⊗ H ε (cid:48) (cid:44) → H is also required.This approach is arguably more appropriate to define lo-cality in the presence of gravity, and defines a structurethat is coarser than the usual spacetime. However, if wewould want to start for the network of Hilbert subspaces( H ε ) ε ∈ E and recover the spacetime structure or a coarsegraining of it, non-uniqueness is again unavoidable. Sucha network can be expressed in terms of projectors ( (cid:98) E ε ) ε ∈ E corresponding to each of the subspaces in the network.Their incidence and inclusion relations, as well as thetensor product condition H ε ⊗ H ε (cid:48) (cid:44) → H , are all invari-ant to unitary transformations, just like orthogonality isinvariant in the case of an orthogonal decomposition ofthe Hilbert space H . Therefore, we can apply Theorem2 just like in the case of decompositions into subspacestreated in § V A, and non-uniqueness follows.
D. Non-uniqueness of general emergent 3D-space
We will now deal with generic kinds of emergent space or emergent spacetime structure (ESS) from a MQS whichmay be a purely quantum theory of gravity. Rather thancatching one fish at a time, let us be greedy and catchthem all at once. But we will do this in two steps, the firstbeing to prove that time-distinguishing exact emergentspace structures are not unique. By an “exact ESS” weunderstand and ESS in which space emerges exactly, andnot as an approximation of some other structure like agraph, spin network, causal set etc . The difference isirrelevant, but we take this route for pedagogical reasons.An ESS requires, of course, a TPS. In this sense Theo-rems 4 and 5 already show that the ESS cannot be uniquein general. Moreover, Theorem 2 already shows that notime-distinguishing K -structure can be unique, so again1the ESS cannot be unique in general. And there is alsothe non-relativistic case from Theorem 1. But, again forpedagogical reasons, let us do it explicitly for exact ESS.We start with the NRQM case from Theorem 1, andgeneralize it to QFT. The proof of Theorem 1 can bereinterpreted in terms of the K -structures of the generalTheorem 2 if we notice that the set A is the configurationspace R n and the K -structure is given by the projectors (cid:98) A q := | q (cid:105)(cid:104) q | , (57)where q ∈ A . Then, the invariants can be chosen to be( (cid:104) ψ | (cid:98) A q | ψ (cid:105) ) q ∈ R n = ( |(cid:104) q | ψ (cid:105)| ) q ∈ R n , which are invariantup to permutations of A corresponding to transforma-tions of the configuration space R n . This is the reasonwhy we took the whole set of invariants to be used totime-distinguish states, as explained in Definition 5. Asseen in the proof of Theorem 1, transformations of theconfiguration space are not sufficient to undo the differ-ences between distinct K -structures, and uniqueness isviolated.But this K -structure gives the configuration space, andwe want it to give the 3D-space. So we rather choose A = R and the operators (cid:98) A x := | x (cid:105)(cid:104) x | = n (cid:88) j =1 (cid:90) R n − | ˘ q j (cid:105)(cid:104) ˘ q j | d ˘ q j , (58)where x ∈ R , q j ∈ R for j ∈ { , . . . , n } and | ˘ q j (cid:105) := | q , . . . , q j − , x , q j +1 , . . . , q n (cid:105) (59)d ˘ q j := d q . . . d q j − d q j +1 . . . d q n . (60)The Hermitian operators (cid:98) A x defined in eq. (58) con-vey much less information than those from eq. (57), be-cause they reduce the entire configuration space to the3D-space. But the densities (cid:104) ψ ( t ) | (cid:98) A x | ψ ( t ) (cid:105) are still ableto distinguish between | ψ ( t j ) (cid:105) and | ψ ( t k ) (cid:105) at differenttimes t j and t k , despite the symmetries of the 3D-space,because matter is not uniformly distributed in space.We notice that we can deal with more types of particlesby defining operators like in (58) for each type, and wecan also deal with superpositions of different numbersof particles, because now we are no longer tied to theconfiguration space of a fixed number of particles. Wecan now move to the Fock space representation.Let P be the set of all the types of particles. We treatthem as scalar particles, and push the degrees of freedomdue the internal symmetries and the spin in P . We canmake abstraction of the fact that the various componentstransform differently under space isometries (and moregeneral Galilean or Poincar´e symmetries) according totheir spin, and also the gauge symmetries require themto transform according to the representation of the gaugegroups, because all these are encoded in the Hamiltonian.We define A := P × R . For each pair ( P, x ) ∈ A , let (cid:98) A ( P, x ) := (cid:98) N P ( x ) = (cid:98) a † P ( x ) (cid:98) a P ( x ) , (61) where the operator (cid:98) a † P ( x ) creates a particle of type P at the 3D-point x ∈ R , (cid:98) a † P ( x ) | (cid:105) = | x (cid:105) P , | (cid:105) being thevacuum state, and (cid:98) N P ( x ) the particle number operator at x for particles of type P .This is all we need to represent the 3D-space in QFT,since in the Fock space representation in QFT, everythingis the same as in eq. (61), except that P represents nowthe types of fields instead of the types of particles. We ofcourse represent the states by state vectors from the Fockspace obtained by acting with creation and annihilationoperators on the vacuum state | (cid:105) .We notice that the tensor structure from (57) consistsof commuting projectors adding up to the identity. Thetensor structure from (58) also consists of commutingprojectors adding up to the identity, being sums or inte-grals of commuting projectors. The same is true for (61),both for the bosonic and for the fermionic case, becauseproducts of pairs of anticommuting operators commutewith one another. We conclude that the kind of struc-ture that stands for exact ESS should be given in termsof commuting projectors that form a resolution of theidentity. Let us denote by K EESS the kind consistingof the conditions to be satisfied by a tensor structure( (cid:98) A ( P, x ) ) ( P, x ) ∈ A , A = P × R , in order for it to be of theform (61). We will not be specific about the exact defin-ing conditions, because they also depend on the symme-tries allowed by the Hamiltonian, and anyway they do notmatter for time-distinguishingness, as explained in Ob-servation 1. All that matters is that the conditions areinvariant. But we will require at least that they are pro-jectors, as in eq. (49), that they commute, as in eq. (50),and that they form a resolution of the identity, as in eq.(47). We call this exact emergent space kind , and denoteit by K EESS . We call a K EESS -structure ( (cid:98) A ( P, x ) ) ( P, x ) ∈ A exact emergent space structure . Theorem 6.
If there is a K EESS -structure, there areinfinitely many possible physically distinct emergent D-space structures of the same kind K EESS .Proof.
The mean value (cid:104) ψ ( t ) | (cid:98) N P ( x ) | ψ ( t ) (cid:105) of the num-ber of particles of each type at any point x in the 3D-space structure changes in time. Therefore, we expect (cid:104) ψ ( t ) | (cid:98) A ( P, x ) | ψ ( t ) (cid:105) to also change in time. Hence, the K EESS has to be time-distinguishing, and Theorem 2implies that there are infinitely many possible physicallydistinct 3D-space structures of the kind K EESS .We now have to generalize our result to structuresfrom which space is expected to emerge in some approx-imation. This is highly theory-dependent. For example,some approaches may be based on local algebras of oper-ators, and entanglement among them or inclusion maps,as already discussed in § V C. Other may be based onnodes in a spin network or in a causal set. Dependingon the particular theory, we may want to keep the con-ditions that the tensor structure consists of projectors,that they commute, and that they form a resolution ofthe identity. Or we may drop some of these conditions2and replace them with other conditions. But whateverwe will do, the unitary symmetry of the H space requiresuch a structure to be a tensor structure as in Definition1, and whatever conditions we impose to this structure,they have to be invariant to unitary symmetries, as inDefinition 3. So no matter what we will do, if the the-ory only assumes a MQS and if such a notion of emergentspace exists, Theorem 2 applies to it and infinitely manyphysically distinct structures satisfying the same defin-ing conditions exist. Whatever idea of emergent spacewe may have in mind, if it is supposed to emerge from a
MQS and no other structure that already breaks the uni-tary symmetry, it cannot be unique.
E. Non-uniqueness of branching into coherentstates
A candidate preferred basis in NRQM is the positionbasis | x (cid:105) , where x ∈ R n is a point in the classical con-figuration space. But in NRQM there is another sys-tem of states, named coherent states , that look classicaland evolve approximately classically on short time inter-vals. In the position basis ( | x (cid:105) ) x ∈ R n , (squeezed) coherentstates | p , q (cid:105) have the form (cid:104) x | q , p (cid:105) := (cid:18) iπ (cid:126) (cid:19) n e i (cid:126) < p , x − q > e − (cid:126) | x − q | (62)for all points in the classical phase space ( q , p ) ∈ R n ,where < ., . > is the Euclidean scalar product in R n .Coherent states were first used by Schr¨odinger [55],then by Klauder [36], and in quantum optics by Sudar-shan [65] and Glauber [29]. Coherent states form an over-complete system, satisfying eq. (47). By being highlypeaked at phase space points, coherent states approx-imate well classical states, and their dynamics is closeto the classical one for short time intervals. Therefore,they are good candidates for preferred generalized bases,and were indeed used as such to address the preferred(generalized) basis problem, e.g. in [25, 26, 32, 33, 41–43, 72, 79].But can we recover the generalized basis of coherentstates ( | q , p (cid:105) ) ( q , p ) ∈ R n from the MQS alone? For this, wewould also need to recover the position basis ( | x (cid:105) ) x ∈ R n .This means that we need to supplement the definingcondition of a generalized basis from Definition 6 withthe defining conditions for an emergent spacetime struc-ture as in § V D. In fact, once we have the position basis( | x (cid:105) ) x ∈ R n and the metric, we can define the momentumbasis as in eq. (6), and then the coherent states (62), andthey will automatically satisfy the defining conditions ofPOVM structures from Definition 6. But if the problemof finding a system of coherent states as in (62) reduces toand depends on finding the space structure, Theorem 6implies that there are infinitely many physically distinctsolutions. F. Non-uniqueness of the preferred basis of asubsystem
When one says that decoherence solves the preferredbasis problem, this may mean two things. First, is that itleads to a preferred generalized basis of the entire world,and we have seen in § V A and § V D that this cannot hap-pen if our only structure is the
MQS . The second thing onemay have in mind when mentioning a preferred basis isin reference to subsystems. This involves a factorizationof the Hilbert space as a tensor product H = H S ⊗ H E , (63)where H S is the Hilbert spaces of the subsystem and H E is that of the environment. The environment acts likea thermal bath, and monitors the subsystem, making itto appear in an approximately classical state. The re-duced density matrix of the subsystem, ρ S := tr E | ψ (cid:105)(cid:104) ψ | ,evolves under the repeated monitoring by the environ-ment until it takes an approximately diagonal form.Then we need a mechanism to choose one of the diag-onal entries, and in MWI this is the simplest one: allthe diagonal entries of the reduced density matrix areequally real, and they correspond to distinct branches ofthe state vector. The same mechanism is considered ableto solve the measurement problem, since in this case thebranches correspond to different outcomes of the mea-surement. A toy model example was proposed by Zurek,who used a special Hamiltonian to show this for a spin -particle, where the environment also consists of suchparticles, cf. [77] and [52], page 89.This article does not contest this explanation based ondecoherence. The question that interests us is whetherthis mechanism of emergence of a preferred basis for thesubsystem can happen when the only structure that weassume is the MQS . In particular, no preferred tensor prod-uct structure is assumed a priori .We have already seen that Theorem 4 implies thatthere are infinitely many physically distinct ways tochoose a TPS, in particular a factorization like in eq.(63). However, if we assume that the system and theenvironment are in separate states, i.e. | ψ (cid:105) = | ψ S (cid:105) ⊗ | ψ E (cid:105) (64)at a time t before decoherence leads to the diagonaliza-tion of ρ S , then the possible ways to factorize the Hilbertspace are limited.The first problem is that, even if we assume (64), ingeneral there are infinitely many ways to choose the fac-torization (63). Even for a system of two qubits there areinfinitely many ways consistent with (64). We can choosea basis ( | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) ) of the total Hilbert space H ∼ = C so that | (cid:105) = | ψ (cid:105) , so | ψ (cid:105) has the components (1 , , , | (cid:105) S , | (cid:105) S ) of H S ∼ = C and thebasis ( | (cid:105) E , | (cid:105) E ) of H E ∼ = C , assuming that | (cid:105) S = | ψ (cid:105) S and | (cid:105) E = | ψ (cid:105) E . But, since there are infinitely many3ways to construct the basis ( | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) ), there are in-finitely many ways to factorize H as in eq. (63). Evenif we impose the restriction that the Hamiltonian hasa certain form in the basis ( | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) ), unless theeigenvectors of the Hamiltonian consists of distinct val-ues, there are infinitely many ways to choose it so that | (cid:105) = | ψ (cid:105) . But in reality the Hamiltonian has highly de-generate eigenspaces, and in each of these eigenspaces itfails to impose any constraints.But the major problem is that it would make no senseto assume that the systems are separated at the time t .The reason is that the subsystem of interest already inter-acted with the environment, and, unless we assume thatit was projected, it is already entangled with the environ-ment. Therefore, if we want to keep all the branches, asin MWI or consistent histories interpretations, we have toassume that in general | ψ ( t ) (cid:105) is already entangled. Andthis prevents us to assume that we are in a branch inwhich the state is separable. We would need first to havea preferred generalized basis, and we know from Theo-rem 4 that this is not possible from the MQS alone. Thereare infinitely many physically distinct ways to choose thebranching to start with, and then any solution for sub-systems that is based on environmental-induced decoher-ence will depend on this choice. Therefore, the
MQS is notsufficient to find a preferred basis for subsystems.
G. Non-uniqueness of classicality
From the previous examples we can conclude that theclassical level of reality cannot emerge uniquely from the
MQS alone. For this to be possible, we would need thatthe 3D-space, and of course the factorization into subsys-tems like particles, emerge. But we have seen in § V D and § V B that this does not happen. Another way for classi-cality to emerge would be if there was a preferred basis orgeneralized basis, which would correspond to states thatare distinguishable at the macro level, but Theorem 3prevents this too, as seen in § V A. Therefore, classicalitycannot emerge uniquely from the
MQS . VI. “PARADOXES”: PASSIVE TRAVEL INTIME AND IN ALTERNATIVE REALITIES
The impossibility of emergence of the preferred struc-tures from the
MQS alone may seem a benign curiosity, butit has bizarre consequences which may be problematic.
Problem 1 (Time machine problem) . If at the time t there is a time-distinguishing K -structure S (cid:98) H ( | ψ ( t ) (cid:105) ),then there are infinitely many K -structures at t , withrespect to which | ψ ( t ) (cid:105) looks like | ψ ( t ) (cid:105) for any otherpast or future time t . This means that any system can“passively” travel in time by a simple unitary transfor-mation of the preferred choice of the 3D-space or of thepreferred basis, so that the system’s state looks with re-spect to the new “preferred” structure as if it is from a different time. Another form of this problem is that anystate vector supports not only the state of the system atthe present time, but it equally supports the state of thesystem at any other past or future time, and there is noway to choose which is the present. Proof.
This occurred during the proof of Theorem 2.
Problem 2 (Alternative reality problem) . As explainedin Remark 4, Theorem 2 can be extended to states thatare not connected by unitary evolution and are not dis-tinguished by the Hamiltonian. This means that thereare infinitely many equally valid choices of the 3D-space,of the TPS, or of the preferred basis, in which the sys-tem’s state looks as if it is from an alternative world (notrestricted to MWI worlds). Again, this leads to the inprinciple possibility of traveling in alternative realities,and also it means that the state vector equally supportsinfinitely many physically distinct alternative realities,and there is no way to tell which is the “most real”.
Proof.
In the proof of Theorem 2 we could use insteadof (cid:98) U t j ,t any unitary operator (cid:98) S that commutes withthe Hamiltonian, the unitary symmetry of the kind K would allow another K -structure for each of them when-ever there is at least a K -structure. Problem 1 shows ex-plicitly an infinite family of such K -structures, generatedby (cid:98) H itself, but there are infinitely many such families,because infinitely many generators of the unitary groupU( H ) commute with (cid:98) H . Remark 4 explains that even ifwe would factor out the symmetries of spacetime (whichrequire us anyway to know the 3D-space structure) andthe gauge symmetries, we still remain with an infinite-dimensional group of unitary transformations. Remark . Problems 1 and 2 assumed that only unitarytransformations that preserve the form of the Hamilto-nian are allowed. But is there any reason to imposethis restriction? If not, this would mean that passivetravel in worlds having different evolution equations, dueto the Hamiltonian having a different form (albeit thesame spectrum), is possible as well.This does not necessarily mean that one can ac-tually travel in time and in “alternative worlds” likethis in practice, but it at least means that, at anytime, there is a sense in which all past and futurestates, as well as “alternative worlds” which are notdue to any version of the Many-Worlds Interpretation,are “simultaneous” with the present state (in the caseof MWI and other branching-based interpretations, forevery branching structure there are infinitely many al-ternative branching structures). The proliferation ofsuch “basis-dependent worlds” is ensured by the time-distinguishingness of the candidate preferred 3D-space,TPS, or generalized basis, under the assumption that theonly fundamental structure is the
MQS , and everythingelse should be determined by this.On the other hand, Problems 1 and 2 cannot simplybe dismissed, they should be investigated to see if indeed4the observers in a basis-dependent world cannot changetheir perspective to access information from other basis-dependent worlds allowed by unitary symmetry. In addi-tion, for both the Second Law of Thermodynamics andfor the existence of decohering branching structures thatonly branches in the future and not in the past (as in[72]), the initial state of the universal wavefunction hadto be very special, but if the initial state itself dependson the choice of the candidate preferred structures, thenmost such choices would fail to be special enough.
VII. WHAT APPROACHES TO QUANTUMMECHANICS AVOID THE PROBLEMS?
How should we resolve these problems for theories likethe universal versions of Standard Quantum Mechanics,or like Everett’s? Some implications and available op-tions of the too symmetric structure of the Hilbert spacewere already discussed in the literature, see e.g. [54, 59].Let us see several ways to sufficiently break the unitarysymmetry so that it allows the emergence of 3D-spaceand the factorization into subsystems.
A. Bohr and Heisenberg
We already mentioned in Sec. § IV that Bohr’s inter-pretation, by distinguishing systems like the measuringdevice as classical, introduces preferred choices of thequantum observables that have classical correspondent.Unfortunately, the theory does not provide unified lawsfor both the quantum and the classical regimes, partic-ularly the measuring devices, and neither does Heisen-berg’s version. But it is possible to formulate StandardQM in a way that gives a unified description of the tworegimes and introduces the necessary symmetry break-ing, see e.g.
Stoica [63].
B. Embracing the symmetry
A possible response to the implications of Theorem 2may be to simply bite the bullet and embrace its con-sequences. For example, in MWI, we can pick a pre-ferred space and TPS but accept that there are infinitelymany possible ways to do this, as suggested by Saunders[48]. Everything will remain the same as in MWI, butthere will be “parallel many-worlds”. Each of the “basis-dependent many-worlds” are on equal footing with theothers. In one-world approaches with state vector reduc-tion we will also have multiple basis-dependent worlds,but different from the many-worlds, in the sense that theyare not branches, they are just one and the same worldviewed from a different frame. But if we accept all ofthe possible basis-dependent worlds allowed by unitaryevolution, we should also try to show that this positiondoes not have unintended consequences. In particular, how can one prove that passive travel in time and in par-allel worlds ( cf.
Sec. § VI) is not possible? And, giventhat unitary symmetry allows worlds that are outside ofthe history of our own world and are not simply otherbranches, how do we know that these worlds satisfy the past hypothesis [38], and that they do not lead to a pro-liferation of
Boltzmann brains [4, 57]? How can the free-dom of choice of the preferred basis ensures that the ini-tial state of the universal wavefunction is consistent withbranching in the future but not in the past [72]?
C. Enforcing a preferred structure
Can we simply use the wavefunction rather than theabstract state vector, or simply extend the
MQS struc-ture with a preferred TPS and a preferred basis for thepositions? Even if we extend the
MQS with an addi-tional structure S , being it the 3D-space, the configu-ration space, or a preferred basis, one may object thatthe unitary symmetry makes any such structure S in-distinguishable from any other one obtained by unitarysymmetry from S , making the theory unable to predictthe empirical observations, which clearly emphasize par-ticular observables as representing positions, momenta etc . But such an objection would only be fair if we wouldapply the same standard to Classical Mechanics. In Clas-sical Mechanics, we can make canonical transformationsof the phase space that lead to similar problems like thosediscussed here for the Hilbert space. Yet, this is not con-sidered to be a problem, because we take the theory asrepresenting real things, like particles and fields, propa-gating in space. We may therefore think that we can justdo the same and adopt the “weak” claim of MWI thatthe state vector is in fact a wavefunction.We can even represent the wavefunction as multipleclassical fields in the 3D-space, as shown by Stoica [62].The representation from [62] is fully equivalent to thewavefunction on the configuration space, it works forall interpretations, and can provide them the necessarystructure and a 3D-space ontology.Another possibility is Barbour’s proposal, in which theworlds are points in a certain configuration space, spe-cific to Barbour’s approach to quantum gravity based on time capsules and shape dynamics , but which can easilybe generalized into a general version of Everett’s Inter-pretation. The wavefunction encodes the probabilities ofthe configurations [5].But the things are not that simple with Everett’s In-terpretation and MWI in general. The reason is that,in order to give an “ontology” to the branches, the solu-tion is, at least for the moment, to interpret the physicalobjects as patterns in the wavefunction, and apply Den-nett’s criterion that “patterns are real things”as Wallacecalls it ([70] p. 93 and [72] p. 50). For Dennett’s notionof pattern see [18]. The key idea can be stated as “asimulation of a real pattern is an equally real pattern”.For a criticism by Maudlin of the usefulness of this crite-5rion as applied by Wallace see [39] p. 798. We will havemore to say about this in another paper, in the contextof the results presented here. We will see that the solu-tion depends on the ability of the preferred structure toguarantee experiences of the world in a way unavailableto a mere unitary transformation of that structure. Andthis depends on the theory of mind, since for example acomputationalist theory of mind allows the transformed(“simulated”) patterns obtained by unitary transforma-tions of “real” patterns to have the same experiences asthe “real” ones, because whatever computation is per-formed on the “real” patterns, it is identical to a com-putation on the transformed patterns. Therefore, sinceat least Wallace’s approach based on Dennett’s idea ofpattern, and in fact Everett’s original idea and subse-quent variations, are implicitly or explicitly committedto computational theories of mind, enforcing a preferredstructure leads us in the same place as the option of em-bracing the symmetry mentioned in § VII B. To have asingle generalized basis or a single underlying 3D-space,one may need to assume that not all patterns that looklike mental activity of observers actually support con-sciousness, and the preferred structures are correlatedto those that support it, a
Many Minds
Interpretation[2, 3, 74] where consciousness is not completely reducibleto computation.
D. Breaking the unitary symmetry of the laws
Maybe the problem is better solved in theories thatactively break the unitary symmetry of the very lawsof QM, by either modifying the dynamics or includingbeables living in the 3D-space.An example is the
Pilot-Wave Theory [8, 9, 16, 17,22] and variants like [40], which extend the
MQS with a3D-space and point-particles with definite positions inthe 3D-space, and whose dynamics breaks the unitarysymmetry of the Hilbert space.
Objective Collapse Theories [27] and variations like [19, 45, 46] also supplement the Hilbert space with 3Dpositions, and the wavefunction of each particle collapsesspontaneously into a highly peaked wavefunction well lo-calized in space. If the 3D-space is assumed not known,it may be possible to recover the configuration space one-collapse-at-a-time, and then the 3D-space emerges fromthe way interactions depend with the distance, as in [1].
E. Breaking the unitary symmetry of the space ofsolutions
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