Bell's Argument

https://www.youtube.com/watch?v=ZuvK-od647c

EPR

Hello everyone, apologies for the delay!

In this post I will be talking about EPR, and in particular the way in which it is structured as an argument, which is something I wanted to understand as clearly as possible. There are a few things I feel are problematic, so I wanted to flag them for discussion.

As I've understood it, the argument in EPR works like this...

Let's call P the proposition that "QM is incomplete" (or, more accurately, "the quantum-mechanical description of physical reality given by wave functions is not complete"), and let Q be the proposition "Incompatible quantities cannot have simultaneous realities". "Incompatible quantities" here refers to quantities described by non-commuting operators.

EPR gives the following necessary condition for completeness: If every element of the physical reality has a counterpart in the physical theory, then the physical theory is complete. Furthermore, the sufficient condition for "having a counterpart in the physical theory" is: if without disturbing a system we can predict with certainty the value of a physical quantity, then there exists an element of reality with a counterpart in the physical theory. Let's call this the Criterion of Reality (COR).

The argument itself goes like this.

i) P ∨ Q ...(Premise 1)

ii) ¬P → ¬Q ...(Premise 2)

iii) P ∨ ¬Q ...(Equivalence from premise 2)

iv) P ∨ (Q ∧ ¬Q) ...(Distribution of ∨ over ∧ from i and iii)

v) P ...(Conclusion)

Argument for premise 1: "Either QM is incomplete, or incompatible quantities cannot have simultaneous realities"

Suppose ¬Q, i.e., incompatible quantities could have simultaneous realities. So, in accordance with COR, without disturbing a system we can predict with certainty the value of two physical quantities (position and momentum). However, by the uncertainty principle, the wave function of the system cannot describe both simultaneously. So there at least one element of physical reality with no counterpart in the physical theory. Therefore the theory does not fulfil the necessary criterion for completeness. 

So either the theory is incomplete (P), or the supposition is false (so, Q).

So, P ∨ Q.

Argument for premise 2: "If QM is complete, then incompatible quantities could have simultaneous realities"

Suppose we have a case of quantum entanglement, where two systems I and II interact with each other in a way such that conservation of relative position and conservation of momentum hold. Let us also assume separability (when a measurement is made in I, there is some reality that pertains to II) and locality (no real change can take place in II as a consequence of a measurement in I).

Suppose now we make a position measurement in I. Because of the way the systems are entangled, we can make a prediction about the position in II, and since this prediction only depends on a measurement in I, according to locality, no real change takes place in II. Therefore, we can predict the position in II with certainty without disturbing the system II, and therefore, according to COR, there exists an element of reality corresponding to the position value in II (and the same can be imagined of momentum). We can then construct the two conditionals: (1) If a position measurement is made in I, then II has a real position value, and (2) if a momentum measurement is made in I, then II has a real momentum value. 

Now, what if we don't make any measurement in I at all? By locality, the state of II has the same reality even if there is no measurement made in I. So locality suggests that the sufficient condition for II to have a real value for position or momentum is that (1) or (2) hold, respectively. But a conditional always holds when its antecedent is false (i.e., if no measurement is made). So given these assumptions, (1) and (2) hold simultaneously, so II has real values for both momentum and position simultaneously, so these incompatible quantities have simultaneous realities (AKA, ¬Q).

As far as I’ve understood, this is the structural gist of EPR. The main problem for me is that it makes use of several fuzzily grounded assumptions and vaguely defined concepts. However, formulated as above, it seems to make sense. I think ultimately the problem lies in the assumption of locality, which the Q theorist could simply deny. 

The definition given for completeness is intuitively plausible, but one thing I think I’m still not getting is the COR. We can say there is a physical reality corresponding to and element of the physical theory if it can be predicted without disturbing the system. But isn’t the whole point that it does disturb the system? If the two are entangled in the relevant way, then a measurement on I disturbs the system I, and consequently disturbs II, doesn’t it? So if we can only get the position value of I by disturbing I, we can make a prediction on II but only by disturbing II, meaning that the value is not an element of the physical reality, which still satisfies the universal quantifier in the condition for completeness. So it seems to me, but I may very well have made an error here, that the COR presupposes the incompleteness of QM, and therefore cannot be legitimately used as a condition for the completeness of QM.

I’ll leave it here for now. All in all I found this topic quite challenging, so I welcome any corrections in my understanding of it!

The Metaphysics of Space-Time Substantivalism - Manifold Substantivalism

Hi everybody! Hope you all had a great break, I think it was my turn to do the blog post (if not then sorry for stealing the moment!)

As the Hoefer paper is quite long and covers a lot I’ve decided to focus on a particular area (manifold substantivalism vs determinism).

Before beginning I think it’s important to state that (by his own admission) Hoefer is not a substantivalist. I think it’s important to reiterate this as it does explain his attitude to some of the articles cited in his work.

As we have discussed in an earlier blog post, a committed substantivalist considers spacetime to be a real thing. Hoefer admits that this is a fairly natural position to take given the past and present success of Newtonian physics and GTR, which allow for spacetime existing as a separate entity with its own intrinsic properties. However, according to Hoefer there are too many different brands of substantivalism for its merits to be properly assessed against alternative theories. So the question follows, if spacetime is a real thing in GTR, as a substantivalist would say, then what part of a GTR universe represents spacetime.

This is probably a good time to through in some definitions to save everybody checking back to the lecture slides!

Manifold of events: A set of events with a smooth coordinate system

Metrical structure: Temporal and spatial distances between events

Matter fields: Where the stuff is. Distinct lumps of matter like galaxies can be represented by world lines

Manifold Substantivalism –

The manifold seems to closest resemble what a substantivalist wants spacetime to be. Earman and Norton split energy bearing structures (physical fields) as being the contents of spacetime which therefore leaves the manifold to be “the container”. It is not clear why such a seemingly arbitrary decision is permitted, other than it does seem to resonate with what is implied when substantivalists ‘spacetime’.

Problems emerge with this interpretation of spacetime in GTR due to Leibniz equivalence. Simple mathematical manipulation can lead to the creation of multiple models in which the material content and the metric field is distributed differently. Hence Manifold Substantivalism would seem to allow for many possible indistinguishable worlds. Leibniz equivalence means that any model produced is actually a representation of one possible world. This is a similar argument to that used in the Leibniz shifts and must be dismissed by any substantivalist true believers!

The hole argument and the breakdown of determinism –

I won’t go into detail regarding the setup of the hole transformation or about general covariance which allows it but essentially the fields within the boundaries of the hole have been manipulated (Einstein initial considered the region to be empty – hence why it’s called the hole). The two setups are indistinguishable because all the observable features are invariant (they remain the same regardless of how much you play with the coordinate system as the coordinate system is arbitrary and doesn’t affect physical laws) This leads to some serious conflict with determinism. As is clear from the diagram the metric and stress fields within the hole cannot be determined using information gathered from outside. It is therefore impossible to link a point within and without, which means you cannot relate a previous state of affairs to the future state of affairs regardless of how well you understand the system. There is nothing too controversial about this in my opinion, it is not a given that the universe is deterministic and there is no reason to be bound to this worldview. But it is clear that that determinism and this form of substantivalism are not compatible. Earman and Norton (also critical of manifold substantivalism) say that it is unacceptable to promote a substantivalist model which dismisses determinism is insufficient. This is because the substantivalist properties are less/not essential (I don’t like my terminology here but I’m trying to put it in simpler terms) so are not grounds to let go determinism, or at least not sufficient in isolation. Here I have to agree with Hoefer in that this doesn’t really seem to be a case against substantivalism but more that manifold substantivalism is perhaps not a complete enough representation of spacetime. I think this would be a good topic to start discussion on Monday so I’ll stop here!