The collapse of the wave function, and two sorts of time

Suppose a star emits a photon. Its wave function spreads over space, perhaps over light years.

On a far planet, a poet is looking at the night sky: if they  see that photon they will be inspired to write a great poem. Meanwhile at the other edge of the galaxy there is a pig, also, for its own porcine reasons, gazing skywards, and the photon wave function also passes through its retina. The pig may see the photon or the poet may see the photon. (Or, of course, it may be seen by neither.)  But it absolutely cannot be seen by both of them. If the photon materialises in an eyeball of one, it cannot do so in an eyeball of the other.  The wave function collapses, instantaneously and simultaneously across all space.

This rings alarm bells. The Theory of Relativity tells us loud and clear that ‘simultaneity’ is a dirty word, but it really does apply here. The arrival of the photon wave function at the pig and the poet may be simultaneous in the picture above, but there will be reference frames in which it arrives at the pig before the poet, and frames where the poet comes before the pig. Whatever frame you’re working in, the wave function collapse is simultaneous everywhere in that frame.

It’s the truly spooky bit of quantum mechanics that really nobody understands. An object – be it a photon, an electron, an atom, or a cat – has a wave function, and we can study the behaviour of that wave function by solving complicated differential equations. Then a measurement is made of some property, and the wave function changes randomly but instantly into a state where that property is defined. If the wave function is ‘real’ it defies relativity. If it is not ‘real’ then what is?

Most physicists solve this puzzle by ignoring it – the so-called ‘shut up and calculate’ school. I’m not going to explain the puzzle here – nobody can. What I do want to do is show that it is not really new, but linked to an older one.

Let’s introduce a logical formalism for discussing the collapse of the wave function in an ordered and well-defined way.  This was done by John von Neumann in his “Mathematical Foundations of Quantum Mechanics”, (Springer 1932, English translation by R T Beyer, Princeton, 1955) and what follows is his development, with modernised notation.

We need to introduce a neat concept called the density matrix.  This combines the two sorts of uncertainty that we have to deal with: quantum uncertainty and the established uncertainty of statistics and thermodynamics.

First, from quantum mechanics we take the idea of a basis set of states |i>, which are eigenstates of some measurement operator  (so Â |i> = ai|i>). In what follows I’ll use as an example the simple case where there are just two states describing the spin of an electron as up |↑> or down |↓>;  they could also be the states |x> of delta functions at particular positions, or the pure sine wave states |p> that have definite wavelength and thus definite momentum.

Secondly, from Statistical Mechanics we take the idea of a large ensemble of N states |ν>, which are the actual states of many systems in equilibrium with one another.  The  |ν> states are (in general) not the same as the |i> states but they can be written in terms of them: |ν> = Σi <i|ν>  |i>, because the |i> states are a basis set. <i|ν>  is a number, the i-component of the ν state in the ensemble.

Right, that’s the apparatus in place. The density matrix is just the average over the ensemble of the Cartesian product of the components

ρij=(1/N) Σν <i|ν> <ν|j>

What can you do with it? Well the diagonal elements ρii correspond to the average over the ensemble of | <ν|i> |2, which, according to the Born interpretation, is the probability of finding state |ν> in state |i> if you measure it with  . It tells us the probability of getting the result ai where the probability includes both the quantum uncertainty and the statistical uncertainty from the ensemble.

Density matrix for an up/down mixture

Let’s take an example.  If we take an ensemble of many electrons of which half are spin up and half are spin down then the matrix has 1/2  on each diagonal element and the off-diagonals are zero, because each    |ν> is either up or down, so the product of  <ν|> and <ν|↓> is always zero.  The diagonal tells us that if you pick an electron at random from the ensemble, there’s a 50% chance each for it being spin up or spin down.

Now let’s get a bit less trivial. We’ll take a sample of spin-down electrons, all the same this time,  and then rotate them 90 degrees about the y axis, so they point in the +x direction.

Rotating a spinor from down to sideways
Density matrix for electrons pointing sideways, along the x axis

These states give the density matrix which has the same diagonal elements as the last one, but non-zero off-diagonal elements. The diagonal elements tell us that there is, again, a 50:50 chance of detecting the electron in an up or down state, though this time it’s because of quantum uncertainty.

If the diagonal elements give the probabilities, you might wonder whether there’s any point to the off-diagonal elements. But they do play a part. Suppose that, for both examples, we rotate the spins buy another 90 degrees before we measure them.  Under a rotation R the density matrix becomes  ρ’=RρR. A bit of  matrix arithmetic shows the matrix for the first example is unchanged, on the second example, the sideways states, is

which is obvious, with hindsight. The second rotation converts the spin direction from the x to the z axis,  so they will always be in the spin up state if you measure them, whereas for the first example the result is still 50:50 unpredictable.

So the density matrix encompasses quantum uncertainties, which may become certain if you ask a different question, and statistical uncertainties which cannot.  Diagonal elements give probabilities and off-diagonal elements contain information on the degree of quantum coherence. If you want to know more about it, try Feynman’s textbook  “Statistical Mechanics: a set of lectures” (CRC press, 1998).

As time goes by the states will evolve, and the evolution of the density matrix has the apparently simple form ρ’=e-iHt/ℏ ρ eiHt/ℏ. I say ‘apparently simple’ because H, which is being exponentiated, is a matrix. But this is standard quantum mechanics and the techniques exist to handle it. The |i> wave functions  oscillate at their characteristic frequencies.

But the matrix can also describe the effect of a measurement of the quantity corresponding to the operator Â. The measurement asks of each state |ν > which of the |i> basis states it belongs to: if it is in one of those states it stays in it, if it is in a superposition then it will drop into one of the|i>, the probability for each being   |<i|ν >|2.  So the density matrix becomes ρ’ij= δij ρij. The diagonal elements are preserved, and all the off-diagonal elements vanish, as each member of the ensemble is now in a definite basis state.

To say ‘a measurement zeroes all the off-diagonal terms of the density matrix’ expresses the ‘collapse of the wave function’ correctly and completely, but in a smooth and non-sensational way.

These two equations,  ρ’=e-iHt/ℏ ρ eiHt/ℏ and ρ’ij= δij ρij, both describe how a system changes with time. They are different because the ‘time’ is different.

It is often (unkindly) said that there are as many philosophies of time as there are philosophers of time, but if you ignore the weirdest ones and brush over the details there are basically two.

In first concept, the time of Parmenides, Plato, Newton, Einstein and Hawking,  sometimes called ‘Being Time‘,  is the 4th dimension, analogous to the dimensions of space.  Events take place in a 4 dimensional space and relativity describes, in great and successful detail, the metric of that space.  Which is fine. But the ‘block universe’ this describes has no sense of direction, no sense of time passing. There is no difference between ‘earlier’ and ‘later’. A space-time diagram completely describes events and the world-lines of objects, including ourselves, but contains nothing to say that you and I are at a particular ‘now’ point on our world-lines, and are making our way along them.  This was encapsulated in the very moving letter that Einstein wrote to the widow of his great friend Michele Besso:

“For those of us who believe in physics, the distinction between past, present and future is only a stubbornly persistent illusion.”

The second concept of time, due to Heraclitus, Aristotle, Leibniz and Heidegger, sometimes called ‘Becoming Time‘, is an ordering relation between events. If event A causes event B, then A comes before B, and B comes after A. We write A→B.  (Actually A→B is shorthand for ‘if a choice is made as to what happens at A, that can affect what happens at B’. If I shoot an arrow (A) it hits the target (B); if I refrain then it does not. If I shoot an arrow A and, due to my incompetence, it misses the target B, we can still say A→B because it might have done.) This is a transitive relation, so if A→B and B→C then A→C , and we can establish an order for all possible events. (Relativity says there are some events in which neither A→B nor B→A, but this can be handled.)  Which is fine. the sequence has a sense of direction, and the past and future are clearly different.  But there is no metric. Events are ordered like competitors in a race where only the final places are given – we know that A, B and C came first, second, and third, but not their individual timings.

Being-time is like a clock with a continuous movement. The hands sweep round smoothly – but the ‘clockwise’ direction is an arbitrary convention. Becoming-time is like a tear-off calendar: the present event is visible on the top, future events are in the stack beneath, past events are in the wastebasket.

So the dual nature of time is a longstanding and unsolved puzzle. We’re not going to solve it here. But we can note that the two ways in which the density matrix changes,  ρ’=e-iHt/ℏ ρ eiHt/ℏ and ρ’ij= δij ρij, correspond to the two different sorts of time. The wave function develops in being-time; measurements are made (and the wave function ‘collapses’)  in becoming-time.  The collapse of the wave function is not a new puzzle produced by quantum mechanics, just a new form of an old puzzle that philosophers have argued about since the time of the ancient Greeks.

 

The Monty Hall Puzzle

There are many probability paradoxes, but the Monty Hall Puzzle is much the greatest of these, provoking more head scratching and bafflement than any other.

It is easy to state. Monty Hall hosted a TV quiz show “Let’s Make a Deal”, in which a contestant has to choose one of 3 doors: behind one of these is a sports car, whereas the other two both contain a goat. (Some discussions of the puzzle – and there are many – speak of ‘a large prize or smaller prizes’, but they can be dismissed as non-canonical; the goats are essential.) There is no other information, so the contestant has a 1 in 3 chance of guessing correctly. Let’s say, without loss of generality, that they pick door 1.

But Monty doesn’t open it straight away. Instead he opens one of the other 2 doors – let’s say it’s door 3 – and shows that it contains a goat. He then offers the contestant a chance to switch their choice from the original door 1 to door 2.

Should the contestant switch? Or stick? Or does it make no difference?

That’s the question. I suggest you think about it before reading on. What would you do?  Bear in mind that the pressure is on, you are in a spotlight with loud music building up tension, and Monty is insistent for an answer. Putting the contestant under pressure makes good television.

Several arguments are put forward – often vehemently

  1. You should switch: the odds were 1/3 that door 1 was the winner, and 2/3 that it was one of the other doors. You now know the car isn’t behind door 3, so all that 2/3 collapses onto door 2. Switching doubles your chance from 1/3  to 2/3.
  2. There’s no point in switching: all you actually know, discarding the theatricality, is that the car is either behind door 1 or door 2, so the odds are equal.
  3. But you should switch! Suppose there were 100 doors rather than 3. You choose one, and Monty opens 98 others, revealing 98 goats, leaving just one of the non-chosen doors unopened.  You’d surely want to switch to that door he’s so carefully avoided opening.

Thought about it?  OK, the answer is that there is no answer. You don’t yet have enough information to make the decision, as you need to know Monty’s strategy. Maybe he wants you to lose, and only offers you the chance to switch because you’ve chosen the winning door. Or maybe he’s kind and is offering because you’ve chosen the wrong door. (There’s a pragmatic let-out which says that if you don’t know whether to switch or stick you might as well switch, as it can’t do any harm – we can close that bolthole by supposing that Monty will charge you a small amount, $10 or so, to change your mind.)

OK, let’s suppose we know the rules and they are

  1. Monty always opens another door.
  2. He always opens a door with a goat and offers the chance to switch. If both non-chosen doors contain goats he chooses either at random.

Now we have enough information. We can analyse this using frequentist probability, which is what we learnt at school.

 Suppose we did this 1800 times ( a nice large number with lots of useful factors). Then the car would be behind each door 600 times. Alright, not exactly 600 because of the randomness of the process, but the law of large numbers ensures it will be close.

A door is then chosen – this is also random so in each of the 3 x 600 cases door 1 will be chosen in only 3 x 200 times. The other cases can now be discarded as we know they didn’t happen. 

For the 200 cases where the car is behind door 1,  Monty will open door 2 and door 3 100 times each. We know he didn’t open door 2, so only 100 cases survive. But all 200 cases with the car behind door 2 survive, as for them he is sure open door 3. When the car is behind door 3 he is never going to open it. So of the original 1800 instances, door 1 is chosen and door 3 is opened in 300 cases, of which 200 involve a winning door 2 and only 100 have door 1 as the winner. Within this sample the odds are 2:1 in favour of door 2. You should switch!

You can also show show this using Bayes’ theorem. Maybe I’ll write about Bayes’ theorem another time.  For the moment, let’s just accept that when you have data, prior probabilities are multiplied by the likelihood of getting that data, subject to overall normalisation.

The initial probability is 1/3 for each door.

P1=P2=P3=1/3

The ‘data’ is that Monty chose to open door 3.  If the winner is door  2, he will certainly open door 3. If it is door 3, he will not open it. If it is door 1, there is a 50% chance of picking door 3 (and 50% for door 2). So the likelihoods are   1/ 2 , 1 and 0 respectively, and after normalisation

P1‘ = 1/ 3         P2‘ = 2/ 3         P3‘=0

So switch! It doubles your chances. 

If you think that’s all obvious and are feeling pretty smug, let’s try a slightly different version of the rules:

  1. Monty always opens another door.
  2. He does this at random. If it reveals a car, he says ‘Tough.” If it contains a goat, he offers a switch.

The frequentist analysis is similar: starting with 1800 cases, if door 1 is chosen then that leaves 600, with 200 for each door being the winner. Now he opens doors 2 and 3 with equal probability, whatever the winning door may be. If it’s door 1, 100 survive as before. If it’s door 2, this time only 100 survive, and in the other hundred he opens door 2 to show a car. For door 3 there are no survivors as he either reveals a goat behind door 2 or a car behind door 3, neither of which has happened. So in this scenario there are 200 survivors, 100 each for doors 1 and 2. The odds are even and there is no point in switching.

Using Bayes’ theorem gives (of course) the same result. The prior probabilities are still all  1/ 3.  The likelihood for Monty to pick door 3 and reveal a goat is  1/ 2 for both door 1 and door 2 concealing a car, and zero for door 3.  Normalising

P1‘ = 1/ 2         P2‘ = 1/ 2         P3‘=0

and theres no point in switching.

So a slight change in the rules switches the result. The arguments 1 to 3 are all suspect.  Even the 3rd argument (which I personally find pretty convincing) is not valid for the second set of rules. If Monty opens 98 doors at random to reveal 98 goats this does not make it any more likely that the 99th possibility is the winner.

If you don’t believe that – or any of the other results – then the only cure is to write a simulation program in the language of your choice. This will only take a few lines, and seeing the results will convince you where mathematical logic can’t.

So the moral is to be very wary of common sense and “intuition” when dealing with probabilities, and to trust only in the results of the calculations. Thank you, Monty!

 

 

 

 

The Legacy of Stephen Hawking

Few great scientists are famous to the general public, but Stephen Hawking achieved rock star status, thanks to his best selling ‘Brief History of Time’, and to the epic saga of his struggle not merely to survive but to master the disease that struck him – his inspirational refusal to give in to the unfair twists of fate.   But the man or woman in the street would claim no knowledge of his scientific achievements: what it was that made his reputation in the scientific community, enabling him to write and speak to the public with such authority.

Although there was a lot of solid first rate work that gave him a reputation in the specialised community of cosmologists, the big item was undoubtedly his development of the theory of Hawking Radiation (as we now call it).  This was a complete paradigm shift, revolutionising our ideas of black holes, but based on a simple and understandable concept. 

Black holes – at the time they were just an oddity predicted by equations, today they are matter of fact items of galactic astronomy – occur when a body is so massive that its escape velocity is more than the velocity of light, so nothing can escape its gravitational pull.  Such black hole will accumulate matter as it falls into it, never to re-emerge: it will just get more and more massive forever, or until the end of the universe.  

Hawking took this description from General Relativity, and linked it to the picture of the vacuum that comes from particle physics. At the quantum level the vacuum is not empty but abuzz with virtual particle-antiparticle pairs (predominantly electrons and positrons) being created out of nothing and then re-merging into nothing. Hawking pointed out that in a very strong gravitational field, such as you get just outside black holes, one of these particles could fall down into the gravity well, giving the other enough energy to become real. The surface of a black hole will radiate electrons and positrons (“Hawking Radiation”), losing energy and mass as it does so. It will not exist forever, but will evaporate.  Our whole concept of black holes changed: they are not just gobblers of everything, but dynamic objects that can be formed and also destroyed.

As well as its own intrinsic importance, his proposal showed how cosmology and particle physics could be fruitfully combined. Previously the former, which involves physics at the largest scale, had been very separate from the smallest-scale physics of the latter.  Theoretical physicists might specialise in General Relativity or Quantum Field Theory, but very few worked on both, and never at the same time.  Today that’s all changed, and our attempts to understand the earliest stages of the big bang, and the most fundamental laws of nature, are framed using combinations of microscopic and megascopic physics.

So he wasn’t just the author of ‘that book by that wheelchair guy’, as Homer Simpson put it, or the character that enabled Eddie Redmayne to win his Oscar. He revolutionised our ideas about black holes and their role in the universe, and he showed the way to a combination of the physics of the very small and the very large which is being carried forward today. That is his gift to us and future generations.

   

Beginning of a blog

This is actually my third blog.

The first was at the general election in 2010, when I was standing as the Lib Dem candidate in Macclesfield. It was early days for social media campaigns and look pretty primitive now. It includes a passionate Liberal Democrat pledge to abolish student fees – subsequent events made me feel pretty sick about that. I didn’t quit the party, but it was close.

The second charted events in the Huddersfield Accelerator Institute, and it gave way of recording a lot of the things we did. I got some of the students to write some of the posts, which was good.  It’s not clear if anyone ever read it, and all the university blogs seem to have dropped out of sight in a recent web reorganisation.

So this, my third, is a retirement project. (Well, it’s better than taking up golf or buying a motorbike.) As any researcher will tell you, the urge to find out about things is closely linked to a ‘mission to explain’ need to share them with other people.  Before retirement, teaching students gave me that outlet.  This is a substitute.  When I’ve understood (by hard struggle or by blinding insight) something in physics, or statistics, or whatever, this will give me a platform to tell others about it.

Of course there’s no guarantee that anyone will read it – there is so much published on the web. But I can only try, and see what happens.