Skip to main content

Hardy's Paradox

I want to discuss one further argument against local realism from Lucien Hardy in another well known set of publications, that were submitted in 1992 and 1993. In two papers titled "Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories" and "Nonlocality for two particles without inequalities for almost all entangled states". These papers present what is more commonly known as "Hardy's paradox".

These must be the nail in the coffin for any reasonable person who wants to still defend local realism. The experiment shows that the predictions of quantum mechanics differ from that of any local realist and local hidden variable theory.

The experiment involves using a stream of positions and electrons both sent through separate beam splitters on the left and the right. The wave function of each particle "splits" along the w path and the v path. These are then rejoined at the second beam splitter in phase so that if there's no interaction between the electrons and the positrons then an interference pattern is detected at c (constructive interference). 

However if you set up the experiment so that the elections on the right interfere with the positrons on the left, the electrons and positrons will annihilate producing photons

So that only particles in the v path reach the second beam splitter. These waves are not coherent with anything so they meet at detectors c and d with equal probability. We can therefore assume that any detection at either d detector means that there was an anihilation event between the two beams at point P. Whenever a detection is made at d the corresponding particle is at u so that

However if we analyse this experiment from the perspective of a local hidden variable theory, wherein particles are independent there can be no detection at d because they would have to travel though the u path where here is an annihilation event. Local hidden variables have a problem then, because these experiments have been performed and there is a detection at d with a Pr(d) =1/16. 

There's a Pr(e+) and Pr(e-) entering both w beams of 1/2 each, so 1/4 and then a further 1/4 probability of each reaching either d detector. So both taken together give a probability of 1/16.

Some more context is probably necessary, if one thinks of quantum theory as being local and as maintaining realism then there is a frame of reference in which a detection can be made at the d detector for the positron, while the corresponding particle in the electron beam has not reached the beam splitter. Similarly you can can have a frame of reference in the opposite direction in which the reverse is also true, where the electron has reached the detector but the positron has not yet reached the beam splitter. Both of these claims cannot be correct, if you compare both frames of reference the particles must have come through either w path and anihilated. So we've arrived at Hardy's paradox.

One possibility is that the particles detection at d is not independent of the second particle in u and there was some influence that occurred faster-than-light or another is that perhaps the measuring device does not register a property the particle had before detection and so there was no definite measurement outcome. The experiment appears to confirm the long held belief that any realist or hidden variable theory must deny locality in order to be consistent with the predictions of quantum mechanics. 


Popular posts from this blog

Margaret Thatcher's Legacy for Britain

The following is an adaptation of my thoughts at UCL's Conservative Society some months ago concerning the issue of the Conservative Parties vote of no confidence that lead to the resignation of Margaret Thatcher, her legacy for Britain and why she's so undeserving hated by the hard left.

When one enters parliament through members lobby there are four prime ministers commemorated and immortalized in statue form. The first of these figures, David Lloyd George seeded the beginnings of the welfare state, the second Winston Churchill served his tenure protecting us from physical annihilation during the Second World War, the third, Clement Attlee nationalized the health service and sought to drive Britain down the road of socialism and the fourth, the late Baroness Thatcher brought great economic revolution at the end of the Cold War.

It's been said of British politics that these last two figures though diametrically opposed were the only elections that ever really mattered. B…

Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble paramet…

'Don't boo Labour, vote Conservative!' #ImWithHer

"My pitch is very simple, I'm Theresa May and I believe I'm the best person to be Prime Minister"

In an election one doesn't always get the option of voting for their primary candidate, for me that's been the case here. Originally I had supported Michael Gove and then Andrea Leadsom for leadership of the Conservative party but on June 8th we're expected to choose between Theresa May and Jeremy Corbyn. Whatever you think of the two personally, the choice of who offers better governance couldn't be clearer.

The most notable part of Jeremy Corbyn's "leadership" has been his support for the outrageous and policies of the far left. He supports the unilateral disarmament of British nuclear weapons, while supporting the right of Iran to have its own unrestricted nuclear program. He's had an industrial policy to nationalize the mining of coal but not to burn coal, and supports self-determination for the people of Palestine but not for the p…