Interpretation

D. Interpretation

In this framework, a massive object that is in a superposition of two different locations is somewhat like a virtual particle-antiparticle pair in a Feynman diagram. It can exist temporarily but will not appear in outgoing states, just that here the “outgoing” states are detector eigenstates that must be, to good precision, product states of matter and metric.

The required precision is given by the accumulated Penrose phase.

However, the temporal and spatial extension of these intermediate superpositions much exceeds those of virtual particles because quantum gravitational effects are so small. They can exist over a duration given by approximately τ ∼ 1/(m|Φ12)| which, for elementary particles, is enormously large (we will get to estimates in the next section).

The teleological element of this construction is that the question of whether the state stays in a superposition or collapses depends on whether it will be measured in the future, i.e. whether it will go on to interact with a detector or not. Personally I do not read much into this mathematical property because I do not think of this model as fundamental. There is also no good reason for why virtual particles can’t become real other than that we know they are just a way to keep track of integrals, and real particles must be on-shell by construction. Asking how virtual particles “know” that they need to disappear again is just a meaningless question.

I think that the question of how a particle in a superposition of two locations “knows” it must recombine is equally meaningless. But the reader who feels uncomfortable with the futuredependence might want to imagine that indeed all possible time evolutions happen, each in its own universe, it’s just that the probability that we find ourselves in a universe with a catstate is vanishingly small, as we saw in the previous subsection. The comparison to virtual particles also helps to understand why, in this approach, we do not need to integrate the residual all the way to infinity. This is because for practical purposes we can chop the time evolution apart into disconnected diagrams at any sufficiently localised and near classical (think “real”) final state, that is the end of a measurement process.

E. Weak Measurements

Once we have the probabilistic formulation, we can deal with weak measurements. One of the standard definitions for weak measurements is loosely speaking a detector that only sometimes makes a detection. Concretely, let us consider a weak measurement that answers the question “Is the system in state |q⟩?” The detector can then be described by two operators Mˆ+ and Mˆ−, the former describing a positive, the latter a negative (no) detection. These operators can be defined as Mˆ+ := √ p |q⟩⟨q| , (43) Mˆ− := 11 −  1 − p 1 − p  |q⟩⟨q| , (44) with some detection probability p ≪ 1. That is, we reformulate the weak measurement as a collapse that happens only with a small probability [26]. There is a simple way to interpret such a device in the model presented here. A weak measurement detector is one that only sometimes maintains the residual ||R|| and sometimes washes it out. This can happen if the imprint of the state |q⟩ was not large enough for the detector to reliably amplify it to a macroscopic level. An example could be that frequent interactions with atoms in a gas can make states less, rather than more, distinguishable, essentially by adding noise. Whether a system actually acts as a detector, therefore, depends not only on its number of constituents, but on how they interact and just how quickly that increases ||R||. A weak measurement device sits at the threshold between detection and no detection. A second definition of weak measurement which is frequently used [27] is that the detector itself is described by a state |M⟩ that changes only mildly to a state |M′ ⟩ with large overlap, i.e. |⟨M|M′ ⟩| ∼ 1. In this case, the ‘weakness’ of the measurement lies in the difficulty of reading out the difference between the two states, which only sometimes works. The distinction between the two definitions of a weak measurement is therefore about what one calls the “measurement” and not the actual mechanism. I think that the imprint in |M′ ⟩ would better be referred to as a pre-measurement and the term measurement be reserved for a successful amplification.

In any case, both definitions mean that the record of the observable is only sometimes amplified to macroscopic scales, which can be described in the context of this model as a system that only sometimes accumulates a large Penrose phase. One must keep in mind though that the question of whether a device acts as a detector does not depend on whether ||R|| grows, but whether R dt||R|| reaches ∼ 1.

That is, the detection threshold is a residual that grows faster than 1/t for a sufficient duration.

F. Free Particles

In the previous subsections we assumed that the wavefunctions on each branch are gravitationally coherent states. By this I mean that the matter ⊗ gravity sector is coherent. The matter and gravity sectors do not have to be each coherent internally. Gravitationally coherent states will, by construction, not build up a residual.

Canonically coherent states in quantum mechanics are defined as eigenstates of the annihilation operators. They are often interpreted as the closest approximation to a classical state. Their main feature is, as the name suggests, that they have only one overall phase. However, to avoid an accumulation of the Penrose-phase it is sufficient that a state remains coherent, it need not necessarily be a canonically coherent state: any state that maintains only one overall phase under the canonical evolution will do. If one strictly defines a freely propagating particles as one that asymptotically goes to infinity, then any arbitrarily small residual would build up and eventually exceed the residual cost of going off the Schrodinger evolution at the previous in- ¨ teraction. Therefore, in the model presented here, asymptotic particle states in Feynman diagrams must be gravitationally coherent. Then again, we never observe asymptotic states, because to observe them they must go into a detector and not to infinity.