How Gravity Can Explain the Collapse of the Wavefunction

How Gravity Can Explain the Collapse of the Wavefunction Sabine Hossenfelder1

I present a simple argument for why a fundamental theory that unifies matter and gravity gives rise to what seems to be a collapse of the wavefunction. The resulting model is local, parameter-free and makes testable predictions.

I. INTRODUCTION

The measurement problem in quantum physics is why we do not observe the outcome of the Schrodinger evolution, but merely one eigenstate of the measurement observable with a probability that can be computed from the wavefunction.

While we can mathematically describe this process with the reduction (or ‘update’ or ‘collapse’) of the wavefunction, the collapse is then not local. This is difficult to reconcile with general relativity.

Why do we never observe macroscopic superpositions (of measurement eigenstates)?

This becomes: How can the process to arrive in a measurement eigenstate happen locally?

The most prominent models to convert the sudden collapse of the wave function into a gradual, physical process are:

• the GhiradiRimini-Weber (GRW) model
• Diosi’s model of stochastic gravitational collapse
• Penrose’s model of gravitationally induced collapse

These models are all non-local in the sense of violating Bell’s condition of local causality.

I propose a local collapse model.

Bell’s theorem says that any locally causal model that correctly describes observations needs to violate measurement independence.

Such theories are sometimes called ‘superdeterministic’.

To arrive at a local collapse model, we must use a superdeterministic approach.

I show that a local collapse arises in a superdeterministic setting from quantum gravity.

In the following ℏ = c = 1.

2. MODEL DEFINITION

A. Geometry and Matter

Matter, radiation, and geometry are ultimately the same.

The particles in the standard model are really geometrical in nature: stable, noise-free, subsystems.

This idea was pursued by John Wheeler with the geon approach. It survives in more modern formulations such as:

• Spinor Gravity
• geometric engineering in string theory
• braided spin networks

Geometry is a purely relational property that arises entirely from matter.

This has been pursued by:

• Causal Fermion Systems
• Shape Dynamics
• Geometric Unity

Classical gravity already has these features.

We can infer the mass and charge of a particle from its gravitational field.

There are known links between the solutions of Yang-Mills theory and those of gravity.

A difficulty here is to account for quantum properties like fractional spin.

Gravity carries a lot of the information from the particle sector already.

In the to-be-found underlying theory, geometry carries the same information as the particles because they are the same. Gravity is in this sense fundamentally different from the other interactions.

The electromagnetic interaction, for example, does not carry any information about the mass of the particles. Yet gravity carries information about the particles’ charges.

We may note in passing that this would solve the black hole information loss problem.

Concretely, I will take this idea to imply that we have a fundamental quantum theory in which particles and their geometry are one and the same quantum state. That is, the geometry is fully determined by the particles’ properties, and vice versa.

There are no extra degrees of freedom. I am here including gravitons as a type of particle, one for which the relation is particularly obvious.

To be even more concrete, let us call this fundamental quantum state |Ψ⟩. We then want to recover the familiar Hilbert space of quantum gravity H , that is a product of matter and geometric degrees of freedom

H = Hm ⊗ Hg . (1)

The geometric degrees of freedom do not necessarily have to be the metric. They could be other geometric properties, like the connection, loops, networks, or anything else that, in the fundamental theory, might replace space-time.

The assumption that I have made, that matter and geometry are ultimately the same, means we will not get the full Hilbert space H , but rather a subset of product states M := {|Ψ⟩ ⊗ U|Ψ⟩}. I have here introduced a unitary operation U that accounts for the possibly different identification of degrees of freedom. We correspondingly assume that the underlying theory has a total Hamiltonian Hˆ tot that acts on each B Off the Hamiltonian 2 factor separately and is of the form

…

where Hˆu is the unknown underlying Hamiltonian and 11 is the identity operator.

The reason for this reduced Hilbert-space, M, is that an entangled state between matter and geometry would have extra information in the phases that are neither in the matter nor in the geometry sector. But by assumption each of those sectors already carries the full information.

This seemingly innocent assumption causes an immediate problem, which is that in the best understood approaches to quantum gravity—perturbatively quantised and canonically quantised gravity—the Hamiltonian evolution generates entanglement between matter and geometry. We therefore need to reconcile these two approaches.

One might at this point say, well, this discrepancy just serves to show that we cannot assume a product state! But as I will argue below, this might be the reason why we do not have a physical description of the measurement process in quantum mechanics.