So finished the first TMA01 for the linear statistical modelling course M346
This course is dull as dishwater but a necessary evil for me to complete my maths and statistics degree
The first question covered the usual stuff t tests, various distributions and chi squared and F tests and a bit of hypothesis testing.
All quite mind numbingly tedious you look up the formulae in Unit 1 and apply them
Question 2 covered how to use Gen Stat to test for normality and how to use the tools to calculate various probabilities
Question 3 covered linear regression the first a straightforward application of linear regression to a data set and the other requiring a transformation of the variables. You are taught how to interpret the linear regression output from GenStat
Really am going to die of boredom from this course but wont give up as I need the qualification
Not even sure what this is doing in a maths course as there are no derivations or proofs in the course
Addendum 16th DEC
Got the tma back on Monday 4 marks short of distinction
lost marks on the transformation question as I didn't think of transforming the x variable and made a silly mistake in the linear regression question as I used the prediction of mean instead of future prediction. Still my tutor has provided a really useful summary of the correct answers
Probably heading for grade 2 if I don't die of boredom before I aim to do 1 questions from the TMA's per week, Might even finish the 2nd TMA over the Xmas holidays
Thursday, 3 December 2015
Monday, 30 November 2015
Grade 6 result
So got the percentage mark for my Grade 6 theory result it was a respectable pass 72% but way short of a merit so glad to have passed but slightly disappointed the mark wasn't higher. I will know in 2 weeks just what the breakdown in marks was and round about the end of January ABRSM will publish the 2014 papers along with the model answers. So I will be able to learn from the experience
Grade 7 next depending on how it goes I will try and do it in June but at least by November of Next year
Grade 7 next depending on how it goes I will try and do it in June but at least by November of Next year
Wednesday, 4 November 2015
Grade 6 Music Theory Debrief
Well yesterday I braved the gloomy Fog of Edinburgh and went to the darkest frontier of Sighthill to sit my grade 6 Music theory exam. I think it went reasonably well although there were one or two tricky bits. The exam consists of 5 sections and I had prepared myself by working through quite a few past papers upto 2014 over the past year or so. Also with the help of Victoria Williams my music theory website and Udemy videos
http://www.mymusictheory.com/
https://www.udemy.com/u/vickywilliams/
A hint for those doing Udemy courses if you invest in one then practically every week they will have course offers between £8 and £15 pounds. For grade 6 I did the Grade 6 melody course and the Figured bass course. Definitely worth investing in if you are contemplating doing the Grade 6 exam
So to the exam itself there are 5 questions
1 Given a melody can you work out an appropriate chord sequence or alternatively given the start of a melody with figured bass can you complete it.
I had got into the habit of doing the first question. To start with it is usual to start a melody with either an anacrusis leading onto the tonic chord or just usually the tonic chord so the first chord is either the tonic (Ia or ia) or if an anacrusis is used (Va-1a)
Usually the piece ends with a perfect cadence (Va-Ia) if the melody ends with a descending scale figure ending on the tonic of the piece then it is possible to use the cadential 6-4 formula (Ic-Va-Ia)
So that is usually 3 or 4 chords that you can write down almost automatically.
The rest of the chords are worked out by first writing each possible chord that would fit (at this level there are only 3 that would work) and seeing if a logical sequence can be fitted in
For example if in the middle of the piece 3 notes form a descending scale passage leading to the tonic then a good fit is a sequence in the opposite direction ascending upwards
Suppose we have (in the key of C major) the notes E D C then in the bass we would have C D E and the chords would be Ia -viiob - Ib or vice versa if the melody was ascending
Then it is usually possible to find a sequence which follows part of the circle of 5th's progression namely
I-IV-vii-iii-vi-ii-V-I
So to some extent this question is a bit like a Sudoku puzzle you have certain clues and you have to fill in the rest of the grid. There are certain rules which one must follow, Never have the bass line or any of the other lines move in parallel fifths or parallel octaves with each other, So if the soprano line is G and you have chosen to use the tonic chord C if the next note in the soprano line is A do not make the note in the bass line D (a ii chord) One way to avoid this as far as possible is to ensure for the most part that the bass line moves in contrary motion to the soprano line.
Also never double the third of a major triad and a few other rules all explained brilliantly by Victoria Williams in her notes, sticking to these and learning a few basic chord progressions, usually gets you through. For this question you don't have to fill in the actual chords but just give the letters. One thing I find helpful in these exercises is to fill in the bass line making sure it doesn't give rise to parallel fifths or octaves with the Soprano line. If I were doing a full harmonisation I would always write the bass line first. Then fill in the other notes. One thing that is slightly odd about the ABRSM exam is that you can only label the chords by their letters or fill in the notes, So if you do leave the bass line in you will get marked down, So you end up rubbing out the bass line. A bit pointless in my opinion
The actual question worked out OK I think but I was a bit repetitive using far more Tonic and Dominant chords than I would normally expect to and also repeating the Ia-viib-Ib chord progression twice, I'll wait and see what the examiner thinks,
The second question was figured bass a topic which can be quite confusing but again explained brilliantly by Victoria in her notes and video. This is even more like a Sudoku puzzle in that the figures on the bass line tell the arranger or accompanist the type of chord to use, So example if there are no figures underneath the chord then the chord is in root position, If the note has a 6 below it is in first inversion and finally if it has 6-4 then it is in second inversion
Thus if the bass note were C with no figures under it would mean a chord of C major in root position C - E -G. If a figure of 6 were below it would be an A minor chord (A-C-E) in first inversion i.e with the note C as the bass note. Finally if 6-4 were the figures under the note then it would be the F major chord (FAC) in second inversion. Also there may be an accidental included. So the first thing to do is to work out which notes the figures are telling you to use. After this is done write out a soprano line which is interesting again trying for contrary motion with the bass line as much as possible and again making sure that the third of a major triad is not used.
Then comes a slightly trickier part filling in the Alto and tenor parts making sure that there are no parallel fifths or octaves between all 4 of the other parts. The only way to fully check this is to systematically work out the intervals between all the other parts. In the context of exam pressure this is really quite a tedious exercise to say the least. I generally trust to judgement and intuition hoping that provided I have no parallel fifths or octaves between the Bass and Soprano lines no others will creep in. Generally speaking provided the alto and tenor parts just move to the nearest available note in the next chord then things should work out, But I have in my practice been known to let the odd parallel fifth creep in. I suspect the same would happen here. In order to make the Soprano part a bit more interesting then it is possible to fill in gaps with quavers so that suppose in the soprano part you had a descent from D to B then make the D a quaver and put in another quaver before the B.
The question was slightly odd in that there were quite a few quavers in the bass line needing a different chord. However provided I got the correct notes and didn't have to many parallel fifths or octaves then I should be OK (fingers crossed anyway)
The third question is to complete a melody of which the first few bars are given again Victoria explains how to go about doing this very well in her Grade 6 video. This often involves a modulation in the last few bars. If you plan out in advance the key points then it helps give a structure. Generally speaking a melody will have 8 - 10 bars and will move from the tonic chord to an imperfect cadence V on the 4th bar and then back to the tonic of either the same key if there is no modulation or to the new key. So again the first note of the 4th bar will be a note which fits the Vth of the given key and the last note will be the tonic of whatever key you have ended up with.
The examiner is looking for you to take what ever is given and use it again with some possible variations in rhythm. I generally try and sequence the second bar in the third bar by raising it up a tone and ensuring a movement by step to the dominant note in the first half of the 4th bar. Then using a logical sequence such as the circle of fifths to get to the modulation section usually with an inversion of the movement in the first phrase so if the first phrase has an ascending passage I will have a descending passage in the second phrase, If there is a large upwards movement then will have downwards leap and so forth finally if there is a modulation ending with a pivot chord from the old key to the new key and then V7-I in the new key. Where in the penultimate bar or half bar the notes of the new key are emphasised.
The question was a bit odd in that it involved a rapid scale like figure occupying 4 hemi-semi quavers and also a modulation from a minor key to the dominant of that minor key. Also despite being in 6/8 time the harmony of the second bar was static just oscillating about 1 note. The modulation caused me a bit of trouble but I managed to scrape something together hopefully it will be enough to get me through,
The last two questions are usually a bit more straightforward than the first 3 and I usually do them first to get them out of the way. They involve asking questions about 2 extracts of music the first one being a piece for piano or with piano and another instrument or voice. and the second a short orchestral extract.
In these questions you are generally asked to identify various chords and intervals the meaning of certain terms such as Etwas Geschwind (Somewhat agitated, which I was before the exam), Also can you spot certain things such as a triad in first inversion and so forth.
The orchestral score question usually involves asking you transpose the music written out for horns in F or clarinets in Bb to how they would actually sound. If a horn in F plays what is notated as C it actually sounds like F so you would transpose down a 5th,(7 semitones), Similarly for clarinets in Bb a notated C sounds like Bb you transpose down a 2nd (2 semitones). So the easiest way to do this is to write out the notes of the chromatic scale of C major and underneath write out the notes of F major underneath it for horns in F or Bb for clarinets. Then write out explicitly the notes for each part as given and then under them the corresponding transposed notes. This can be a bit time consuming, also and I never got the hang of this, if the original note has a flat attached do you transpose with the original note or with a sharp instead. Technically the notes would sound the same so Db is equivalent to C# but an interval with either C# or Db would have a different name. Thus Db - A is a 5th but C# - A is a 6th. As the interval questions on the orchestral score often involves one of the transposing instruments. It's not clear whether or not after you have made the correct transposition that you will get the correct interval
Anyway that was that I think I got most of the last two questions correct and hopefully will have done OK on the first 3. We'll just have to see. I await the examiners report with interest.
Next year provided everything went reasonably well I hope to do grade 7. For now upto Xmas I need to focus on my statistics course.
If you took the grade 6 exam or anyother grade theory exam I hope it went well for you as well
As a final comment it may seem to the average person that this approach takes the creativity out of music. However just as physicists or mathematicians can be creative within the context of a given framework (i,e the laws of physics or axioms of mathematics) so in music the laws of harmony provide a framework whereby some one can be creative. Of course if you are a Mozart or Beethoven then you can use the framework to construct quite different types of pieces but the idea that is often floated that composers don't use the laws of harmony but just write whatever comes into their head is quite a misconception. They will usually have an idea of a melody either by picking notes out on a piano or something that they hear in their head, But those fragments will be developed into complete pieces only by following a given structure. I find it quite remarkable that provided you follow the basic laws of harmony and form you can write down a competent piece of music and that this craft can be taught.
http://www.mymusictheory.com/
https://www.udemy.com/u/vickywilliams/
A hint for those doing Udemy courses if you invest in one then practically every week they will have course offers between £8 and £15 pounds. For grade 6 I did the Grade 6 melody course and the Figured bass course. Definitely worth investing in if you are contemplating doing the Grade 6 exam
So to the exam itself there are 5 questions
1 Given a melody can you work out an appropriate chord sequence or alternatively given the start of a melody with figured bass can you complete it.
I had got into the habit of doing the first question. To start with it is usual to start a melody with either an anacrusis leading onto the tonic chord or just usually the tonic chord so the first chord is either the tonic (Ia or ia) or if an anacrusis is used (Va-1a)
Usually the piece ends with a perfect cadence (Va-Ia) if the melody ends with a descending scale figure ending on the tonic of the piece then it is possible to use the cadential 6-4 formula (Ic-Va-Ia)
So that is usually 3 or 4 chords that you can write down almost automatically.
The rest of the chords are worked out by first writing each possible chord that would fit (at this level there are only 3 that would work) and seeing if a logical sequence can be fitted in
For example if in the middle of the piece 3 notes form a descending scale passage leading to the tonic then a good fit is a sequence in the opposite direction ascending upwards
Suppose we have (in the key of C major) the notes E D C then in the bass we would have C D E and the chords would be Ia -viiob - Ib or vice versa if the melody was ascending
Then it is usually possible to find a sequence which follows part of the circle of 5th's progression namely
I-IV-vii-iii-vi-ii-V-I
So to some extent this question is a bit like a Sudoku puzzle you have certain clues and you have to fill in the rest of the grid. There are certain rules which one must follow, Never have the bass line or any of the other lines move in parallel fifths or parallel octaves with each other, So if the soprano line is G and you have chosen to use the tonic chord C if the next note in the soprano line is A do not make the note in the bass line D (a ii chord) One way to avoid this as far as possible is to ensure for the most part that the bass line moves in contrary motion to the soprano line.
Also never double the third of a major triad and a few other rules all explained brilliantly by Victoria Williams in her notes, sticking to these and learning a few basic chord progressions, usually gets you through. For this question you don't have to fill in the actual chords but just give the letters. One thing I find helpful in these exercises is to fill in the bass line making sure it doesn't give rise to parallel fifths or octaves with the Soprano line. If I were doing a full harmonisation I would always write the bass line first. Then fill in the other notes. One thing that is slightly odd about the ABRSM exam is that you can only label the chords by their letters or fill in the notes, So if you do leave the bass line in you will get marked down, So you end up rubbing out the bass line. A bit pointless in my opinion
The actual question worked out OK I think but I was a bit repetitive using far more Tonic and Dominant chords than I would normally expect to and also repeating the Ia-viib-Ib chord progression twice, I'll wait and see what the examiner thinks,
The second question was figured bass a topic which can be quite confusing but again explained brilliantly by Victoria in her notes and video. This is even more like a Sudoku puzzle in that the figures on the bass line tell the arranger or accompanist the type of chord to use, So example if there are no figures underneath the chord then the chord is in root position, If the note has a 6 below it is in first inversion and finally if it has 6-4 then it is in second inversion
Thus if the bass note were C with no figures under it would mean a chord of C major in root position C - E -G. If a figure of 6 were below it would be an A minor chord (A-C-E) in first inversion i.e with the note C as the bass note. Finally if 6-4 were the figures under the note then it would be the F major chord (FAC) in second inversion. Also there may be an accidental included. So the first thing to do is to work out which notes the figures are telling you to use. After this is done write out a soprano line which is interesting again trying for contrary motion with the bass line as much as possible and again making sure that the third of a major triad is not used.
Then comes a slightly trickier part filling in the Alto and tenor parts making sure that there are no parallel fifths or octaves between all 4 of the other parts. The only way to fully check this is to systematically work out the intervals between all the other parts. In the context of exam pressure this is really quite a tedious exercise to say the least. I generally trust to judgement and intuition hoping that provided I have no parallel fifths or octaves between the Bass and Soprano lines no others will creep in. Generally speaking provided the alto and tenor parts just move to the nearest available note in the next chord then things should work out, But I have in my practice been known to let the odd parallel fifth creep in. I suspect the same would happen here. In order to make the Soprano part a bit more interesting then it is possible to fill in gaps with quavers so that suppose in the soprano part you had a descent from D to B then make the D a quaver and put in another quaver before the B.
The question was slightly odd in that there were quite a few quavers in the bass line needing a different chord. However provided I got the correct notes and didn't have to many parallel fifths or octaves then I should be OK (fingers crossed anyway)
The third question is to complete a melody of which the first few bars are given again Victoria explains how to go about doing this very well in her Grade 6 video. This often involves a modulation in the last few bars. If you plan out in advance the key points then it helps give a structure. Generally speaking a melody will have 8 - 10 bars and will move from the tonic chord to an imperfect cadence V on the 4th bar and then back to the tonic of either the same key if there is no modulation or to the new key. So again the first note of the 4th bar will be a note which fits the Vth of the given key and the last note will be the tonic of whatever key you have ended up with.
The examiner is looking for you to take what ever is given and use it again with some possible variations in rhythm. I generally try and sequence the second bar in the third bar by raising it up a tone and ensuring a movement by step to the dominant note in the first half of the 4th bar. Then using a logical sequence such as the circle of fifths to get to the modulation section usually with an inversion of the movement in the first phrase so if the first phrase has an ascending passage I will have a descending passage in the second phrase, If there is a large upwards movement then will have downwards leap and so forth finally if there is a modulation ending with a pivot chord from the old key to the new key and then V7-I in the new key. Where in the penultimate bar or half bar the notes of the new key are emphasised.
The question was a bit odd in that it involved a rapid scale like figure occupying 4 hemi-semi quavers and also a modulation from a minor key to the dominant of that minor key. Also despite being in 6/8 time the harmony of the second bar was static just oscillating about 1 note. The modulation caused me a bit of trouble but I managed to scrape something together hopefully it will be enough to get me through,
The last two questions are usually a bit more straightforward than the first 3 and I usually do them first to get them out of the way. They involve asking questions about 2 extracts of music the first one being a piece for piano or with piano and another instrument or voice. and the second a short orchestral extract.
In these questions you are generally asked to identify various chords and intervals the meaning of certain terms such as Etwas Geschwind (Somewhat agitated, which I was before the exam), Also can you spot certain things such as a triad in first inversion and so forth.
The orchestral score question usually involves asking you transpose the music written out for horns in F or clarinets in Bb to how they would actually sound. If a horn in F plays what is notated as C it actually sounds like F so you would transpose down a 5th,(7 semitones), Similarly for clarinets in Bb a notated C sounds like Bb you transpose down a 2nd (2 semitones). So the easiest way to do this is to write out the notes of the chromatic scale of C major and underneath write out the notes of F major underneath it for horns in F or Bb for clarinets. Then write out explicitly the notes for each part as given and then under them the corresponding transposed notes. This can be a bit time consuming, also and I never got the hang of this, if the original note has a flat attached do you transpose with the original note or with a sharp instead. Technically the notes would sound the same so Db is equivalent to C# but an interval with either C# or Db would have a different name. Thus Db - A is a 5th but C# - A is a 6th. As the interval questions on the orchestral score often involves one of the transposing instruments. It's not clear whether or not after you have made the correct transposition that you will get the correct interval
Anyway that was that I think I got most of the last two questions correct and hopefully will have done OK on the first 3. We'll just have to see. I await the examiners report with interest.
Next year provided everything went reasonably well I hope to do grade 7. For now upto Xmas I need to focus on my statistics course.
If you took the grade 6 exam or anyother grade theory exam I hope it went well for you as well
As a final comment it may seem to the average person that this approach takes the creativity out of music. However just as physicists or mathematicians can be creative within the context of a given framework (i,e the laws of physics or axioms of mathematics) so in music the laws of harmony provide a framework whereby some one can be creative. Of course if you are a Mozart or Beethoven then you can use the framework to construct quite different types of pieces but the idea that is often floated that composers don't use the laws of harmony but just write whatever comes into their head is quite a misconception. They will usually have an idea of a melody either by picking notes out on a piano or something that they hear in their head, But those fragments will be developed into complete pieces only by following a given structure. I find it quite remarkable that provided you follow the basic laws of harmony and form you can write down a competent piece of music and that this craft can be taught.
Wednesday, 7 October 2015
A bit more on Bell
Some readers of the last post on Bell may claim that I have missed the point. What my last account fails to do they would say is take into account the effect of the polarisation settings and the fact that they can be changed during the time of flight. So that as the usual story goes if the polarisation of the left detector is changed then the polarisation of the photon (or electron) will change in accordance with that polarisation but because the other photon must have opposite spin then it will change its polarisation accordingly.
I think from a statistical point of view this is misguided. A polariser acts as a filter and for a given polarisation angle only a fraction of the particles in the beam will be able to pass the polariser in a accordance with Malus's Law the probability of passing through the filter being proportional to the square of the cosine of the angle between the polariser setting and the photons polarisation angle. On emergence the photons that pass through the filter will have the polarisation of the polariser.
However some photons will not pass through. The polariser has destroyed any correlation there might have been with an EPR pair. Similarly at the other end a different number of photons will also pass through. The number of photons that pass through both ends being dependent on the polariser settings All this can be calculated from the basic laws of quantum physics. But given that polarisers will destroy any prior correlation. I can't see why it is inconsistent with the idea that before 1 or both of the photons hit the polarisers they were emitted from the source with perfect anti-correlation.
Clearly if one thinks the Bell State applies to single pairs alone then one would have to invoke some spooky action at a distance, But from a statistical point of view all we have is the probability of the photons electrons or whatever emerging through the polarisers. One has to imagine the pairs being randomly emitted from the source with all sorts of polarisation angles, each pair being anti-correlated especially as the Bell state is rotationally invariant. However it is not clear, if one is just concerned with the overall transmission probability of photons through the polarisers which is all one can measure, that the photon that passes through the left polariser is one that is paired with the photon that passes through the right hand polariser. Because the polariser obviously destroys any correlation. Imagine if for example there was only 1 polariser say the left side then clearly the polarisation of the other photon would not change.
Given that in order to verify the statistics many events have to occur then there is no way of ever detecting a single pair of photons.
On that basis then I still feel justified in denying the need for any form of super-luminal signalling between a single pair of photons.
I think from a statistical point of view this is misguided. A polariser acts as a filter and for a given polarisation angle only a fraction of the particles in the beam will be able to pass the polariser in a accordance with Malus's Law the probability of passing through the filter being proportional to the square of the cosine of the angle between the polariser setting and the photons polarisation angle. On emergence the photons that pass through the filter will have the polarisation of the polariser.
However some photons will not pass through. The polariser has destroyed any correlation there might have been with an EPR pair. Similarly at the other end a different number of photons will also pass through. The number of photons that pass through both ends being dependent on the polariser settings All this can be calculated from the basic laws of quantum physics. But given that polarisers will destroy any prior correlation. I can't see why it is inconsistent with the idea that before 1 or both of the photons hit the polarisers they were emitted from the source with perfect anti-correlation.
Clearly if one thinks the Bell State applies to single pairs alone then one would have to invoke some spooky action at a distance, But from a statistical point of view all we have is the probability of the photons electrons or whatever emerging through the polarisers. One has to imagine the pairs being randomly emitted from the source with all sorts of polarisation angles, each pair being anti-correlated especially as the Bell state is rotationally invariant. However it is not clear, if one is just concerned with the overall transmission probability of photons through the polarisers which is all one can measure, that the photon that passes through the left polariser is one that is paired with the photon that passes through the right hand polariser. Because the polariser obviously destroys any correlation. Imagine if for example there was only 1 polariser say the left side then clearly the polarisation of the other photon would not change.
Given that in order to verify the statistics many events have to occur then there is no way of ever detecting a single pair of photons.
On that basis then I still feel justified in denying the need for any form of super-luminal signalling between a single pair of photons.
Monday, 5 October 2015
Long term plans
So my (current) long term plans are as follows
2015 - 2016 M346 Linear Statistical Modelling
2016 - 2017 M347 Mathematical Statistics and possibly M140
So hopefully I will then have a BSc in Maths and Statistics
Then I shall build on what I have already to convert this to a Maths and Economics degree
For which I will need the 2 economics courses
2017 -2018 DD209 Running the economy
2018-2019 DD309 Doing Economics People Markets and Policy
and also the Maths Methods course MST224 which I would do alongside DD209 probably
2015 - 2016 M346 Linear Statistical Modelling
2016 - 2017 M347 Mathematical Statistics and possibly M140
So hopefully I will then have a BSc in Maths and Statistics
Then I shall build on what I have already to convert this to a Maths and Economics degree
For which I will need the 2 economics courses
2017 -2018 DD209 Running the economy
2018-2019 DD309 Doing Economics People Markets and Policy
and also the Maths Methods course MST224 which I would do alongside DD209 probably
Then having got the economics courses under my belt I could go on to do the Politics Philosophy and Economics degree by doing the 2nd level politics course,
2019-2020 DD211 Understanding politics: ideas and institutions in the modern world
Then a third level politics course and finally the third level philosophy course. So that would complete 3 Open university degrees within the next 7 years or so.
Concurrently I would be continuing with my music studies Grade 6 theory beckons in a months time
then Grade 7 and
Grade 8. Also by the middle of next year I would hope (money permitting to embark on the Open College of Arts composition course with the aim of writing an extended piece of music within the next 5 years or so.
http://www.oca.ac.uk/courses/music-courses/
Grade 8. Also by the middle of next year I would hope (money permitting to embark on the Open College of Arts composition course with the aim of writing an extended piece of music within the next 5 years or so.
http://www.oca.ac.uk/courses/music-courses/
So that should keep me busy for a while.
Saturday, 26 September 2015
The Bell State a defence of naivety
In this post I wish to defend a naive interpretation of the Bell State against some of the more wilder claims made for it in the literature
The Bell state (for those who have been asleep for the past 40 years) applies to a pair of particles emitted from a common source in opposite directions. The Bell State describes the joint spin of the two particles which because of the conservation of angular momentum must be zero. But until measurement occurs we do not know the spin of a given particle so the probability amplitude takes the form
$$|B> = \frac{1}{\sqrt{2}}(|u1>|d2>-|u2> |d1>)$$
where u represents spin up and d represents spin down and the naive interpretation I want to defend is that this simply represents the conservation of angular momentum, so that once the spin of 1 particle is known by measurement, then the other spin is automatically known and must be equal and opposite. All the Bell state represents is the 50 % probability on measurement that the particle a person observes at one detector will either be measured with spin up or spin down. Repeated measurements will confirm the fact that at a particular detector the spin will be measured up or down 50% of the time and at the other detector the spin will be equal and opposite. It is well known that the Bell state is rotationally invariant so if one the particles are emitted at an angle $$\theta$$ with respect to the polarisation axis between the particles, the other particle will be emitted at an angle $$-\theta$$ with respect to the polarisation axis . Applying a bit of Pauli spin algebra to the Bell state is well known to give the correct correlation function as was spectacularly confirmed by Aspect in the early 1980's
So why all the fuss, apart from the intrinsic skill in being able to measure the relative spin states of two particles separated a long distance away, from the naive perspective I am offering this is just another routine success of quantum mechanics.
As many people will know, the fuss has arisen because it is claimed that this experiment encapsulates the essence of the debate that Bohr and Einstein had in the early 1930's about the interpretation of quantum mechanics and which culminated in the so called EPR paradox.
To cut a long story short the claim is made that because we don't know the spin state of a given particle before measurement, The Bell state represents a physical state of superposition so that the particles spin state is undetermined until measurement which then collapses the Bell state into one of the two possibilities. So that it is claimed the other particle instantly knows what spin take to take as soon as the first particles spin state has been determined. As the particles can in principle be millions of miles away, it is claimed that this means that some signal faster than the speed of light has been sent to the other particle. Thus there would appear to be a prima facia contradiction between special relativity and quantum mechanics.
Furthermore in his paper
It is claimed that Bell showed that the 2 assumptions
a) that there were hidden variables that predetermined the spins of the particles
b) the assumption that the two detectors were independent of each other
Led to a contradiction with the predictions of quantum mechanics.
The failure of assumption a) would imply that quantum measurements cause reality to happen giving rise to the 'California Interpretation' and the failure of assumption b) gives rise to the idea that a measurement of the spin of a particle at one detector affects the other. This is the so called 'Spooky action at a distance' On this reading (which is the one that seems to pervade the popular literature) The Aspect experiment would imply that quantum mechanics does either involve spooky action at a distance (non-locality) or a form of non realism in which measurement determines properties of particles. What is worse in the popular literature, both claims are made about quantum mechanics when it is clear that only one of the two need be made. So that ever since the Aspect experiment quantum mechanics is seen in many circles as being both non-realist and non-local.
I want to challenge not the assumptions that Bell made but the interpretation of their significance as they have become known in the popular literature.
First it should be stressed that Bell himself did not think quantum mechanics in itself showed non-locality or spooky action at a distance. If one reads the conclusion ( section VI) Bell states explicitly in the first sentence.
In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover the signal must propagate instantaneously, so that such a theory could not be Lorentz invariant.
So what Bell thought he had proven was that hidden variable theories had to have some form of non-locality or spooky action at a distance not quantum mechanics. Indeed if one goes through the actual calculation of the correlation function measured in the Aspect experiment using quantum mechanics no mention is made of any form of signalling at all. All one does is apply a bit of Pauli Algebra to the Bell state and out pops the result almost by magic. It seems odd to claim non-local signalling to explain an experimental result when the correct quantum mechanical prediction makes no mention of it.
Lets consider the situation in a bit more detail
Quantum mechanics is essentially a statistical theory, the solution of Schrodinger's equation for a given problem gives rise to a probability amplitude the modulus squared of which gives rise to a probability density function via the Born rule. The solution to Schrodinger's equation is not a real wave existing in (3N+1)*S configuration space where N is the number of particles and S is the product of all the possible spin states of the particles involved. But effectively the square root of a probability density function and the (3N+1)*S configuration space is simply the probability sample space not a real physical space.
That being so the Bell state is nothing physical it is (as is any quantum superposition) a superposition of possible outcomes with coefficients representing the probability of one of the possible outcomes. When a measurement is made the so called collapse of the probability state vector is not a physical process, but a realization of one of the possibilities, just as say I can assign a probability state vector to a coin or a dice or any situation which results in a number of possible outcomes N
That being so the Bell state is nothing physical it is (as is any quantum superposition) a superposition of possible outcomes with coefficients representing the probability of one of the possible outcomes. When a measurement is made the so called collapse of the probability state vector is not a physical process, but a realization of one of the possibilities, just as say I can assign a probability state vector to a coin or a dice or any situation which results in a number of possible outcomes N
This is simply
$$ |Final> =\frac{1}{\sqrt{N}}{a_1|1a_>+.........a_n|N>} $$
where the a's represent the square roots of the probability of one of the outcomes being realized.
But the system, is not before measurement physically in some hybrid state of all the possibilities. All one can say is that one of the possibilities will be realized on measurement, not that measurement causes the system to jump from it's limbo state into one of the possible outcomes. The probability amplitude can be seen as akin to a Bayesian prior representing our best guess as to what the possible outcomes will be. Given that one can by definition only ever see a quantum system in one state speculation as to what happens before the act of measurement occurs is totally meaningless.
Furthermore if as the standard story has it the spin states of the particles are only determined at measurement then that would make a nonsense of the conservation of angular momentum. It would imply that the conservation of angular momentum only applies at the point of measurement. On the naive view I'm defending the relative orientations of the particles spin states is fixed at the point of emission but we wont know what spin state we will measure at a given detector. So whilst a given individual particles spin state is not determined, the relative orientation with respect to the other is, If that is not so then the conservation of angular momentum is meaningless and we would not be able to predict the outcome of experiments such as those at CERN. I hope then to have dispelled the myth that the Aspect experiment or any other such experiment involves spooky action at a distance.
Ok so what does entanglement or non-locality actually mean? Again there is a fairly straightforward interpretation. If the joint probability state vector of a system of many objects is entangled that means that it cannot be written as the product of the individual objects probability state vectors. This is well known in statistics, If the probability density function of two or more variables cannot be written as the product of the probability density functions of the individual variables, then that is is an indication that the two variables are not independent of each other. For the Bell state we know that the individual particle spin states are not independent of each other, as they are equal and opposite because of the law of conservation of angular momentum. Again there is no need to invoke spooky action at a distance and I have to say Bell has misled a whole generation of physicists by invoking this in his conclusion.
Is my interpretation consistent with the quantum formalism undoubtedly yes as I am using the Bell state to make the correct predictions as measured in the Aspect experiment. The question has to be why invoke spooky action at a distance or collapse of the wave packet when a simple statistical interpretation in accordance with the Born interpretation is all one needs to successfully predict the outcome of quantum mechanical experiments.
As a final point if Bell has really disproved the existence of hidden variables then we cannot go beyond the statistics to find out "what is really going on". The hope of going beyond the statistical predictions of quantum mechanics to predict the results of individual measurements is an illusion and "Nature really does play dice."
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Friday, 18 September 2015
More quantum madness
Yet another episode in the quantum madness game
This time the claim is the rather somewhat ludicrous claim that a microbe before it's position is measured is in two places at once.
This really is such gibberish that it is hardly worth bothering with. But at the heart of it lies a misunderstanding of the nature of quantum superposition
The claim is that a quantum superposition is real so that particles really can be in two places at once until measured in which case the 'position wavfunction' collapses into one of the two positions.
Of course the more pragmatic point of view would see quantum superposition as a superposition of possibilities with a given probability of the object being in one position after measurement.
The latter interpretation is much more sensible and still consistent with the formalism of quantum mechanics. I really do despair why such a view isn't propagated more widely.
I don't understand the need to see quantum mechanics as something mysterious or weird has taken hold on so many people.
One can only sympathise with David Hume when he says regarding the tendency of rationalist philosophers to overstate their case
"Though the chain of arguments which conduct us to it were ever so logical, there must arise a strong suspicion …that it has carried us quite beyond the reach of our faculties when it leads us to conclusions, so extraordinary, and so remote from common life and experience, We are got into fairy land.‘ (An Enquiry Concerning Human Undestanding VII ) .
This time the claim is the rather somewhat ludicrous claim that a microbe before it's position is measured is in two places at once.
This really is such gibberish that it is hardly worth bothering with. But at the heart of it lies a misunderstanding of the nature of quantum superposition
The claim is that a quantum superposition is real so that particles really can be in two places at once until measured in which case the 'position wavfunction' collapses into one of the two positions.
Of course the more pragmatic point of view would see quantum superposition as a superposition of possibilities with a given probability of the object being in one position after measurement.
The latter interpretation is much more sensible and still consistent with the formalism of quantum mechanics. I really do despair why such a view isn't propagated more widely.
I don't understand the need to see quantum mechanics as something mysterious or weird has taken hold on so many people.
One can only sympathise with David Hume when he says regarding the tendency of rationalist philosophers to overstate their case
"Though the chain of arguments which conduct us to it were ever so logical, there must arise a strong suspicion …that it has carried us quite beyond the reach of our faculties when it leads us to conclusions, so extraordinary, and so remote from common life and experience, We are got into fairy land.‘ (An Enquiry Concerning Human Undestanding VII ) .
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