3D-Space and the preferred basis cannot uniquely emerge from the quantum structure

Abstract

Hilbert-Space Fundamentalism (HSF) states that the only fundamental structures are the quantum state vector and the Hamiltonian, and from them everything else emerge uniquely, including the 3D-space, a preferred basis, and a preferred factorization of the Hilbert space. In this article it is shown that whenever such a structure emerges from the Hamiltonian and the state vector alone, if it is physically relevant, it is not unique. Moreover, HSF leads to strange effects like "passive" travel in time and in alternative realities, realized simply by passive transformations of the Hilbert space. The results from this article affect all theories that adhere to HSF, whether they assume branching or state vector reduction (in particular the version of Everett's Interpretation coined by Carroll and Singh "Mad-dog Everettianism"), various proposals based on decoherence, proposals that aim to describe everything by the quantum structure alone, and proposals that spacetime emerges from a purely quantum theory of gravity.

Figures (1)

• Figure 1: Schematic representation of the idea of the proof in the vector space of Hermitian operators (see Theorem \ref{['thm:nogo']}). a) The candidate preferred structures $\textcolor{blueStruct}{\mathcal{S}_{\widehat{H}}^{|\psi\rangle}}$ and $\textcolor{blueStruct}{\mathcal{S}_{\widehat{H}}^{|\psi'\rangle}}$ are symbolized as solid blue triangles. The vertices represent the preferred observables of the structure, and the edges relations between the observables. The dashed blue lines symbolize their relations with the corresponding states, represented by $|\psi\rangle\langle\psi|$ and $|\psi'\rangle\langle\psi'|=\widehat{S}|\psi\rangle\langle\psi|\widehat{S}^{-1}$, where $[\widehat{S},\widehat{H}]=0$. For the candidate $3$D-space and other candidate preferred structures expected to emerge, for any state vector $|\psi\rangle$ there are state vectors $|\psi'\rangle\neq|\psi\rangle$ for which these relations are different. Without this, the structure is not physically relevant. An obvious example is obtained by taking $\widehat{S}$ to be a time evolution operator, because we expect that $|\psi\rangle$ changes in time with respect to space, the preferred bases, and the preferred factorization. In addition to the time evolution ones, other choices of $\widehat{S}$ give different families of unitary transformations. b) Since the structure $\textcolor{blueStruct}{\mathcal{S}_{\widehat{H}}^{|\psi\rangle}}$ is invariant to any unitary transformation $\widehat{S}$ that commutes with $\widehat{H}$, $\widehat{S}^{-1}$ transforms $\textcolor{blueStruct}{\mathcal{S}_{\widehat{H}}^{\widehat{S}|\psi\rangle}}$ into another structure for $|\psi\rangle$, $\textcolor{redStruct}{\mathcal{S}_{\widehat{H}}^{'|\psi\rangle}}=\widehat{S}^{-1}\textcolor{blueStruct}{\mathcal{S}_{\widehat{H}}^{\widehat{S}|\psi\rangle}}$, which is of the same kind as $\textcolor{blueStruct}{\mathcal{S}_{\widehat{H}}^{|\psi\rangle}}$. Since the relations between $\textcolor{redStruct}{\mathcal{S}_{\widehat{H}}^{'|\psi\rangle}}$ and $|\psi\rangle$ are different from those between $\textcolor{blueStruct}{\mathcal{S}_{\widehat{H}}^{|\psi\rangle}}$ and $|\psi\rangle$, we conclude that $\textcolor{blueStruct}{\mathcal{S}_{\widehat{H}}^{|\psi\rangle}}$ is not unique. Theorem \ref{['thm:nogo']} is more general, since it also covers uniqueness up to a physical equivalence.

Theorems & Definitions (73)

• Definition MQS
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Proposition 1
• proof
• Theorem 1