Well the debate has been going on the science and physics OU Fora about the meaning or not of quantum physics as all such debates we do seem to going around in circles those of my fellow OU students reading this blog may find part 1 of the discussion here

http://learn.open.ac.uk/mod/forumng/discuss.php?d=1016527

and part 2 here

http://learn.open.ac.uk/mod/forumng/discuss.php?d=1029089

You will see that I have come close to losing my temper some times with a certain person, Still I'm not the only one who finds his attitude slightly infuriating which is reassuring.

Any way as promised here is a brief summary of the Pilot wave approach. This was intially thought of by De-Broglie and the basic idea is that each particle is accompanied by a Pilot wave. (Not that a particle is a wave or vice-versa), this Pilot wave guides the particles in situations such as the two slit experiment giving rise to the characteristic interference pattern on the screen, but a particle only passes through one slit at time. It was proposed at a conference at Solvay in 1927 but wasn't really taken up as most physicists were more impressed with the views of Bohr/Heisenberg and Schrodinger. The rest is history as they say, physicists got on with the business of using quantum physics to calculate the properties of molecules, atoms and solids etc and the pilot wave theory got quietly dropped.

That was until David Bohm rediscovered it and published two papers in the early 1950's these weren't taken seriously most notoriously because David Bohm claimed that the theory required the use of hidden variables to explain things like paticle states. So it was ignored until Bell came on the scene, as is well known he devised an experimental test which could distinguish between hidden variables and the standard predictions of quantum mechanics, it was shown that the predictions of quantum were vindicated and hidden variables were ruled out. You might have thought that would be the end of the story physicists could get on with the real business of developing the applciations of quantum mechanics to ever more and more complex problems. However there was a get out clause (there always is) In deriving his contradiction Bell made two assumptions

i) There were no hidden variables dictating spin components

ii) There were no non local interactions affecting the measurement of 1 particle a long distance away from another one.

As both were required it was perfectly possible for Bohmians to reject ii) and keep i). Most physicists were until quite recently prepared to accept i) and reject ii). However i) has its own problems if taken literally it implies that properties are created by an act of measurement against our notions of common sense.

Of course all that is a bit of an exaggeration, if on the statistical interpretation the wave function is simply a means to generate probabilities then measuring something does not create a property of a particle. All that happens as say when one throws a dice is that one of the possiblities is realised. But then that means that quantum mechanics is no more than an algorithm for correlating the mathematics of quantum mechanics with probabilities and doesn't really explain anything. Well I think I still hold that view, but nevertheless I'm slowly being persuaded, that there is more to the De-Broglie Pilot wave theory than I first would expect.

There has been quite an industry actually using the De-Broglie Bohm theory to perform calculations. Most impressive is it's explanation of the two slit experiment, It shows how the two slit experiment can be explained with each particle taking a definite trajectory and passing through a single slit, there is none of the usual problems associated with believing that an electron splits in two then magically reforms when it is detected. Neither is there any notion of wave packet collapse occuring as a result of measurement.

A really good introduction to the De-Broglie Bohm pilot wave theory is given here by these lectures by Mike Towler.

http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html

Best to start with the popular lecture

http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/towler_pilot_waves.pdf

I still think there are major problems with it which I will expound on in another post. Anyway I;ll leave you to enjoy the lectures and make your own minds up. The fact that something has moved from being a speculative tool to one where real calculations can be made has to be something.

On another note to do with real waves I was slightly disappointed with the results of my last TMA from MS324 in the low 80's. I missed a key point in one of the questions about the boundary conditions and in my favourite question number 2 I dropped a few marks because I missed out the first term in the series still a reasonable score but not as high as I hoped.

Bye for now

## Tuesday, 29 May 2012

## Sunday, 20 May 2012

### Interpretation of quantum mechanics part I (Again)

Hi been having a heated discussion on the OU science fora about the meaning or not of quantum mechanics

Those who follow this blog will know that I tend to be quite sceptical about any attempts to go beyond the current formalism as

a) No new physics will come out of it (Or none that we can distinuguish between experimentally)

b) Any attempt to see the wavefunction as somehow real leads to all sorts of problems

i) The idea of superposition being something physical until observed seems to imply that we create reality

by an act of measurement

ii) In the two slit experiment the idea that an electron or large particle somehow splits in two and then magically reforms (if taken literally) at the detector seems totally incredible (where does the energy come from etc) why bother with CERN if we can split electrons in two simply by passing them through slits.

iii) If a particle really is a wave how come the pattern only emerges after several impacts on the screen rather than all at once. It is only after a statisitically significant number of events have occurred that anything like a pattern interpreable as a wave function can occur. So that the 'wave aspects' are esssentially statistical the usual fuss about the pattern still occurring even though there is only one particle in the intererometer being irrelevant (or just as relevant as the throw of a single dice).

For these reasons I prefer the statistical interpretation of quantum mechanics, which says that the 'wavefunction' is essentially a probability amplitude whose modulus squared gives the probability of certain events happening. This implies that the wave function is not a property of a single system but more a mathematical device for generating probabilities, it differs from that of classical probability in the sense that to account for the quantum mechanical viewpoint we have to use complex numbers. I then went on to show how you could account for the sinusoidal dependence of the probabilites on the phase factor for a two state system. Also how it was quite striking how classical probabilty could be recast in the language of quantum mechanics specifically the Dirac formalism. For a recap see these two posts

http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/quantum-mechanics-of-two-state-systems.html

http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/mathematics-of-two-state-systems-2.html

I also gave a reference to a paper by Marcella which gave an explanation of how the typical form of the two slit interference pattern can be interpreted as a single particle build up of many events, where the particle does pass through a single slit, which acts as a measuring device the uncertainty in the particles position being due to not being able to know precisely the position of the particle and being responsible for the wave like appearance.

http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

In all fairness I should point out that a subsequent paper has been written criticising the above paper

http://arxiv.org/abs/1009.2408

And this was presented as a falsification of the Marcella paper, by one of the contributers to the forum. I beg to differ, all it shows is that Marcella has hidden some of his assumptions and that the use of free particle eigenstates is equivalent to a classical wave theory. It would be possible to adapt the Marcella paper to make it more accurate eg by the use of a superposition of free particle eigenstates (often referred to as a Gaussian wavepacket) and removing the assumption that it was equally likely for the particle to emerge from one of the slits. All this would be a distraction from the main point, namely that it is possible to give a particle like interpretation of the two slit experiment using the formalism of quantum mechanics. Indeed as that is what physically happens namely particles really do appear individually and only eventually is a wave like pattern revealed it would seem bizzare to attribute wave like properties to individual electrons, neutrons or buckyballs. Obviously collectively the wave like properties are manifest so wave particle duality is simply when considered as a single entity quantons (for want of a better word) behave like particles but when considered collectively they behave like waves.

The statistical interpretation to my mind is the bottom line, it makes the least number of metaphysical interpretations, one can avoid all the usual problems, real wave collapse, and so forth. It by definition is consistent with the formalism so physicists can get on with the real job, namely developing and applying the formalism to predict and understand the properties of solids, stars, quantum fluids, elementary particles, lasers etc. Or in a word physicists can 'shut up and calculate' and leave the 'interpretational stuff' to other people.

For more information on the statistical interpretation this web site gives a good introduction and overview

http://statintquant.net/siq/siq.html

However as a consequence of the dialogue on the OU science forum, I have become a bit more interested in the so called Pilot wave theory initiated by De-Broglie and subsequently developed by David Bohm. I'll discuss more about it in another blog, giving my reasons as to why out of all the myriad interpretations of quantum mechanics which seek to go beyond the statistics, this is the one I consider most promising. One of the striking things is that the motivations behind the pilot wave theory, seem very similar to the motivations behind the statistical ensemble theory. I will point out the comparisons between the two in a later post. Also there has recently been an experiment showing that classical systems can show wave particle duality.Something that hitherto has never happened before. I'll leave you to ponder about the significance of this experiment for now.

http://phys.org/news78650511.html

Those who follow this blog will know that I tend to be quite sceptical about any attempts to go beyond the current formalism as

a) No new physics will come out of it (Or none that we can distinuguish between experimentally)

b) Any attempt to see the wavefunction as somehow real leads to all sorts of problems

i) The idea of superposition being something physical until observed seems to imply that we create reality

by an act of measurement

ii) In the two slit experiment the idea that an electron or large particle somehow splits in two and then magically reforms (if taken literally) at the detector seems totally incredible (where does the energy come from etc) why bother with CERN if we can split electrons in two simply by passing them through slits.

iii) If a particle really is a wave how come the pattern only emerges after several impacts on the screen rather than all at once. It is only after a statisitically significant number of events have occurred that anything like a pattern interpreable as a wave function can occur. So that the 'wave aspects' are esssentially statistical the usual fuss about the pattern still occurring even though there is only one particle in the intererometer being irrelevant (or just as relevant as the throw of a single dice).

For these reasons I prefer the statistical interpretation of quantum mechanics, which says that the 'wavefunction' is essentially a probability amplitude whose modulus squared gives the probability of certain events happening. This implies that the wave function is not a property of a single system but more a mathematical device for generating probabilities, it differs from that of classical probability in the sense that to account for the quantum mechanical viewpoint we have to use complex numbers. I then went on to show how you could account for the sinusoidal dependence of the probabilites on the phase factor for a two state system. Also how it was quite striking how classical probabilty could be recast in the language of quantum mechanics specifically the Dirac formalism. For a recap see these two posts

http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/quantum-mechanics-of-two-state-systems.html

http://chrisfmathsphysicsmusic.blogspot.co.uk/2011/05/mathematics-of-two-state-systems-2.html

I also gave a reference to a paper by Marcella which gave an explanation of how the typical form of the two slit interference pattern can be interpreted as a single particle build up of many events, where the particle does pass through a single slit, which acts as a measuring device the uncertainty in the particles position being due to not being able to know precisely the position of the particle and being responsible for the wave like appearance.

http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

In all fairness I should point out that a subsequent paper has been written criticising the above paper

http://arxiv.org/abs/1009.2408

And this was presented as a falsification of the Marcella paper, by one of the contributers to the forum. I beg to differ, all it shows is that Marcella has hidden some of his assumptions and that the use of free particle eigenstates is equivalent to a classical wave theory. It would be possible to adapt the Marcella paper to make it more accurate eg by the use of a superposition of free particle eigenstates (often referred to as a Gaussian wavepacket) and removing the assumption that it was equally likely for the particle to emerge from one of the slits. All this would be a distraction from the main point, namely that it is possible to give a particle like interpretation of the two slit experiment using the formalism of quantum mechanics. Indeed as that is what physically happens namely particles really do appear individually and only eventually is a wave like pattern revealed it would seem bizzare to attribute wave like properties to individual electrons, neutrons or buckyballs. Obviously collectively the wave like properties are manifest so wave particle duality is simply when considered as a single entity quantons (for want of a better word) behave like particles but when considered collectively they behave like waves.

The statistical interpretation to my mind is the bottom line, it makes the least number of metaphysical interpretations, one can avoid all the usual problems, real wave collapse, and so forth. It by definition is consistent with the formalism so physicists can get on with the real job, namely developing and applying the formalism to predict and understand the properties of solids, stars, quantum fluids, elementary particles, lasers etc. Or in a word physicists can 'shut up and calculate' and leave the 'interpretational stuff' to other people.

For more information on the statistical interpretation this web site gives a good introduction and overview

http://statintquant.net/siq/siq.html

However as a consequence of the dialogue on the OU science forum, I have become a bit more interested in the so called Pilot wave theory initiated by De-Broglie and subsequently developed by David Bohm. I'll discuss more about it in another blog, giving my reasons as to why out of all the myriad interpretations of quantum mechanics which seek to go beyond the statistics, this is the one I consider most promising. One of the striking things is that the motivations behind the pilot wave theory, seem very similar to the motivations behind the statistical ensemble theory. I will point out the comparisons between the two in a later post. Also there has recently been an experiment showing that classical systems can show wave particle duality.Something that hitherto has never happened before. I'll leave you to ponder about the significance of this experiment for now.

http://phys.org/news78650511.html

## Sunday, 6 May 2012

### The Importance of Pedantry

Hi as a break from Topology and Waves I have been working on my long term ambition to understand the Big Bang line by line. I'm currently trying to understand the derivation of the Friedmann equations from General relativity. This exercise is somewhat tedious and heartbreaking to say the least. However reading some text books eg Lasenby I have noticed at best a misleading error in the calculations of the connection coefficients and this has led me to all sorts of confusion. A connection coefficient is a three indexed quanitity which can be calculated from the metric tensor g as follows

$$\Gamma^{i}_{jk}= \frac{1}{2}g^{il}(\partial_{k}g_{li}+\partial_{j}g_{lk}-\partial_{l}g_{jk})$$

Where the indices run from 0-3 0 corresponding to t and 1,2,3 x,y,z or whatever coodinate system is required. As there are potentially 128 different terms to caclulate one can see the temptation to take short cuts. However if we interchange the indices we get

$$\Gamma^{i}_{kj}= \frac{1}{2}g^{il}(\partial_{j}g_{li}+\partial_{k}g_{lj}-\partial_{l}g_{kj})$$

Which for cases where g is a diagonal metric ie the only non zero terms are

$$g_{00},g_{11},g_{22},g_{33}$$ or $$g_{ii}$$

as in the case of the Robertson Walker, the first equation reduces to

$$\Gamma^{i}_{jk}= \frac{1}{2}g^{ii}(\partial_{j}g_{ii}+\partial_{k}g_{ii}-\partial_{i}g_{kj})$$

and the second equation reduces to

$$\Gamma^{i}_{kj}= \frac{1}{2}g^{ii}(\partial_{k}g_{ii}+\partial_{j}g_{ii}-\partial_{i}g_{jk})$$

But as addition is commutative this is just the first equation with the first two terms interchanged and the last term in both equations will be zero for the off diagonal terms (ie when j does not equal k) . So for diagonal metrics then we must have

$$\Gamma^{i}_{jk} = \Gamma^{i}_{kj}$$

However this symmetry is often missed by standard textbook accounts (eg Lasenby et al page 377) and the claim is made that the only non zero off diagonal connection coefficients for the Robertson Walker metric are

$$\Gamma^{1}_{01}, \Gamma^{2}_{02},\Gamma^{2}_{12},\Gamma^{3}_{03},

\Gamma^{3}_{13},\Gamma^{3}_{23}$$

When they should also include the other off diagonal terms:

$$\Gamma^{1}_{10}, \Gamma^{2}_{20},\Gamma^{2}_{21},\Gamma^{3}_{30},

\Gamma^{3}_{31},\Gamma^{3}_{32}$$

The reason why this is important is because a key quantity in general relativity is the Ricci Tensor which can be calculated from the connection coeffients as

$$R_{ij}=\partial_{j}\Gamma^{k}_{ik}-\partial_{k}\Gamma^{k}_{ij}+

\Gamma^{l}_{ik}\Gamma^{k}_{l}{j}-\Gamma^{l}_{ij}\Gamma^{k}_{lk}$$

So if some gullible reader like myself neglects the other off diagonal terms then you will not get the correct answer for the Ricci Tensor and will spend many hours of frustration wondering why you can't get the answers quoted in the text books. Of the many books I have which give results for the connection coefficients for the Robertson Walker metric at least three namely

Narlikar " Introduction to Cosmology" page 106

http://www.amazon.co.uk/An-Introduction-Cosmology-J-Narlikar/dp/0521793769/ref=sr_1_1?ie=UTF8&qid=1336329189&sr=8-1

Lasenby "General Relativity An Introduction for physicists" page 377

http://www.amazon.co.uk/General-Relativity-An-Introduction-Physicists/dp/0521829518/ref=sr_1_1?s=books&ie=UTF8&qid=1336329247&sr=1-1

and

Collins, Martin and Squires "Particle physics and Cosmology" page 373

http://www.amazon.co.uk/Particle-Physics-Cosmology-P-Collins/dp/0471600881/ref=sr_1_2?s=books&ie=UTF8&qid=1336329321&sr=1-2

All neglect the other off diagonal connection coefficients for the Robertson Walker metric in their tables of connection coefficients.

All this goes to show is that one should not just naively take on trust results quoted in text books I can only agree with my colleague Duncan in his last post about the importance of pedantry when trying to understand maths or physics books

http://matrices-reloaded.blogspot.co.uk/2012/04/last-book.html

On another note I have decided not to embark on the composition course immediately as my finances couldn't really cope with it and also I want to spend more time on my extra curricular physics.

$$\Gamma^{i}_{jk}= \frac{1}{2}g^{il}(\partial_{k}g_{li}+\partial_{j}g_{lk}-\partial_{l}g_{jk})$$

Where the indices run from 0-3 0 corresponding to t and 1,2,3 x,y,z or whatever coodinate system is required. As there are potentially 128 different terms to caclulate one can see the temptation to take short cuts. However if we interchange the indices we get

$$\Gamma^{i}_{kj}= \frac{1}{2}g^{il}(\partial_{j}g_{li}+\partial_{k}g_{lj}-\partial_{l}g_{kj})$$

Which for cases where g is a diagonal metric ie the only non zero terms are

$$g_{00},g_{11},g_{22},g_{33}$$ or $$g_{ii}$$

as in the case of the Robertson Walker, the first equation reduces to

$$\Gamma^{i}_{jk}= \frac{1}{2}g^{ii}(\partial_{j}g_{ii}+\partial_{k}g_{ii}-\partial_{i}g_{kj})$$

and the second equation reduces to

$$\Gamma^{i}_{kj}= \frac{1}{2}g^{ii}(\partial_{k}g_{ii}+\partial_{j}g_{ii}-\partial_{i}g_{jk})$$

But as addition is commutative this is just the first equation with the first two terms interchanged and the last term in both equations will be zero for the off diagonal terms (ie when j does not equal k) . So for diagonal metrics then we must have

$$\Gamma^{i}_{jk} = \Gamma^{i}_{kj}$$

However this symmetry is often missed by standard textbook accounts (eg Lasenby et al page 377) and the claim is made that the only non zero off diagonal connection coefficients for the Robertson Walker metric are

$$\Gamma^{1}_{01}, \Gamma^{2}_{02},\Gamma^{2}_{12},\Gamma^{3}_{03},

\Gamma^{3}_{13},\Gamma^{3}_{23}$$

When they should also include the other off diagonal terms:

$$\Gamma^{1}_{10}, \Gamma^{2}_{20},\Gamma^{2}_{21},\Gamma^{3}_{30},

\Gamma^{3}_{31},\Gamma^{3}_{32}$$

The reason why this is important is because a key quantity in general relativity is the Ricci Tensor which can be calculated from the connection coeffients as

$$R_{ij}=\partial_{j}\Gamma^{k}_{ik}-\partial_{k}\Gamma^{k}_{ij}+

\Gamma^{l}_{ik}\Gamma^{k}_{l}{j}-\Gamma^{l}_{ij}\Gamma^{k}_{lk}$$

So if some gullible reader like myself neglects the other off diagonal terms then you will not get the correct answer for the Ricci Tensor and will spend many hours of frustration wondering why you can't get the answers quoted in the text books. Of the many books I have which give results for the connection coefficients for the Robertson Walker metric at least three namely

Narlikar " Introduction to Cosmology" page 106

http://www.amazon.co.uk/An-Introduction-Cosmology-J-Narlikar/dp/0521793769/ref=sr_1_1?ie=UTF8&qid=1336329189&sr=8-1

Lasenby "General Relativity An Introduction for physicists" page 377

http://www.amazon.co.uk/General-Relativity-An-Introduction-Physicists/dp/0521829518/ref=sr_1_1?s=books&ie=UTF8&qid=1336329247&sr=1-1

and

Collins, Martin and Squires "Particle physics and Cosmology" page 373

http://www.amazon.co.uk/Particle-Physics-Cosmology-P-Collins/dp/0471600881/ref=sr_1_2?s=books&ie=UTF8&qid=1336329321&sr=1-2

All neglect the other off diagonal connection coefficients for the Robertson Walker metric in their tables of connection coefficients.

All this goes to show is that one should not just naively take on trust results quoted in text books I can only agree with my colleague Duncan in his last post about the importance of pedantry when trying to understand maths or physics books

http://matrices-reloaded.blogspot.co.uk/2012/04/last-book.html

On another note I have decided not to embark on the composition course immediately as my finances couldn't really cope with it and also I want to spend more time on my extra curricular physics.

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