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Chapter 1 is mostly a review of the theory of vector spaces. As such it nothing extremely interesting or new and therefore I have little to comment here.
However the last section of the chapter is concerned with giving a kind of definition of Hilbert Spaces, which I found to be more confusing than helpful. Therefore, a few comments on this section lest I forget what (I think) I already know.
A Hilbert Space, or rather the mathematical concept of a Hilbert Space generalises the notion of an Euclidian Space, and vector algebra, from a 2 or 3 dimensional space, and vector to an infinite one. In formal terms, a Hilbert Space is an inner product space which is complete.
To put it in simpler terms, an inner product space is an abstract vector space where angles and distances can be measured. By 'complete' we mean that if a vector series in a space approaches a limit, that limit will exist in the space as well.
A few examples to clarify the concept of completeness:
The space Q of rational numbers, with the standard metric given by the absolute value, is not complete. Consider for instance the sequence defined by x1 = 1 and xn+1 = xn/2 + 1/xn. This is a complete sequence of rational numbers, but it does not converge towards any rational limit: Such a limit x of the sequence would have the property that x² = 2, but no rational numbers have that property. But considered as a sequence of real numbers R it converges towards the irrational number , the square root of two.
The open interval (0,1), again with the absolute value metric, is not complete either. The sequence (1/2, 1/3, 1/4, 1/5, ...) is complete, but does not have a limit in the space. However the closed interval [0,1] is complete; the sequence above has the limit 0 in this interval.
It is worth mentioning that intuitively a complete a complete set is one that had "no holes" in it, or that no points are missing from it, whether that happens inside the set or at the boundaries.
Tuesday 15 January 2008
Thursday 10 January 2008
"The Structure & Interpratation of Qunatum Mechanics" by R.I.G. Hughes
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Of most book that I have read that attempt to introduce some of the peculiarities of QM, I have found the one by Hughes the most instructive and mostly for thisd reason I decided to include some comments here.
Hughes uses the Stern-Gerlach experiment as a means for demonstrating some of the peculiarities or ‘weirdness’ of quantum theory.
Very briefly, and omitting a considerable deal of detail, a collimated beam of electrons enters a non-uniform magnetic field produced by specially arranged magnets. What is observed is that the beam splits into 2 parts of equal intensity, one travelling upwards and the other downwards.
Using the ‘spin’ of the electrons we can account for this behaviour in classical terms. Half of the electrons in the beam posses spin -1/2 in the vertical direction, and the other half +1/2 (for simplicity Plank’s constant is set to 1). The ones with he negative-valued spin feel a net magnetic force downwards and the positive-valued ones a net force upwards. This is verified by simply blocking off one of the beams (up or down) and placing another similar set of magnets. As we would expect no further splitting is observed.[Hughes refers to this set-up as a ‘Experiment VV’, meaning that we force the beam to be split in the Vertical direction twice].
The problems start when we alter this set-up. By rotating the second apparatus by 90∘we force the incoming beam to be split into two horizontal components, spin-left and spin-right. [This set up would be ‘Experiment VH’ - H for Horizontal splitting]. Let’s assume that we have blocked the spin-up component from the first apparatus, and the spin-left from the second. It follows that the resulting beam has a horizontal spin-right component only, and a vertical spin-down component only. Therefore, if we passed this beam though yet another apparatus we would expect to see no further splitting.
This, however, is not the case. Placing a third apparatus set in the vertical direction, the emergent beam will be split into two. Therefore, there must be something faulty in this account we offered.
Hughes lists 4 separate (unwarranted) assumptions that might have found their way into this account.
1. When we assign a numerical value to a physical constant (in this case a specific component of spin), we can think of this quantity as a property of the system.
2. We can assign a value for each physical quantity to a system at any given time. For example we can say that the electron has both spin-up and spin-left.
3. The apparatus sorts out the atoms according to the values of one particular quantity, in other words according to the properties they posses.
4. As the apparatus does so, the other properties of the system remain unchanged.
Obviously, here something has to go. Here I think Hughes describes in less technical terms the notions of Value Realism (assumption 1), Value Definiteness (assumption 2), and Non-contextuality (assumption 3&4). Which assumption or assumptions have to go, modified, replaced, etc., is something I will postpone commenting on for two reasons. Firstly, because it will be more productive and clear if done after some more material is covered. Secondly and more importantly, I think it will depend on one’s ‘allegiances’ and research programme. Possibly someone supporting a more traditional scientific realism will not want to abandon the above assumption... or at least not without a serious fight. Someone on the opposite camp, an instrumentalist, would probably have no problems throwing them out of the window altogether.
Hughes uses the Stern-Gerlach experiment as a means for demonstrating some of the peculiarities or ‘weirdness’ of quantum theory.
Very briefly, and omitting a considerable deal of detail, a collimated beam of electrons enters a non-uniform magnetic field produced by specially arranged magnets. What is observed is that the beam splits into 2 parts of equal intensity, one travelling upwards and the other downwards.
Using the ‘spin’ of the electrons we can account for this behaviour in classical terms. Half of the electrons in the beam posses spin -1/2 in the vertical direction, and the other half +1/2 (for simplicity Plank’s constant is set to 1). The ones with he negative-valued spin feel a net magnetic force downwards and the positive-valued ones a net force upwards. This is verified by simply blocking off one of the beams (up or down) and placing another similar set of magnets. As we would expect no further splitting is observed.[Hughes refers to this set-up as a ‘Experiment VV’, meaning that we force the beam to be split in the Vertical direction twice].
The problems start when we alter this set-up. By rotating the second apparatus by 90∘we force the incoming beam to be split into two horizontal components, spin-left and spin-right. [This set up would be ‘Experiment VH’ - H for Horizontal splitting]. Let’s assume that we have blocked the spin-up component from the first apparatus, and the spin-left from the second. It follows that the resulting beam has a horizontal spin-right component only, and a vertical spin-down component only. Therefore, if we passed this beam though yet another apparatus we would expect to see no further splitting.
This, however, is not the case. Placing a third apparatus set in the vertical direction, the emergent beam will be split into two. Therefore, there must be something faulty in this account we offered.
Hughes lists 4 separate (unwarranted) assumptions that might have found their way into this account.
1. When we assign a numerical value to a physical constant (in this case a specific component of spin), we can think of this quantity as a property of the system.
2. We can assign a value for each physical quantity to a system at any given time. For example we can say that the electron has both spin-up and spin-left.
3. The apparatus sorts out the atoms according to the values of one particular quantity, in other words according to the properties they posses.
4. As the apparatus does so, the other properties of the system remain unchanged.
Obviously, here something has to go. Here I think Hughes describes in less technical terms the notions of Value Realism (assumption 1), Value Definiteness (assumption 2), and Non-contextuality (assumption 3&4). Which assumption or assumptions have to go, modified, replaced, etc., is something I will postpone commenting on for two reasons. Firstly, because it will be more productive and clear if done after some more material is covered. Secondly and more importantly, I think it will depend on one’s ‘allegiances’ and research programme. Possibly someone supporting a more traditional scientific realism will not want to abandon the above assumption... or at least not without a serious fight. Someone on the opposite camp, an instrumentalist, would probably have no problems throwing them out of the window altogether.
Sunday 6 January 2008
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This website is designed to present some of the preliminary work, or rather readings, concerning a potential PhD project in the Philosophy of Science.
Unfortunately, the timing for this is far from perfect. I had to join the Greek Army for my compulsory national service this February (12 months service). Meaning that not only the time I get fro studying/research is limited and fragmented, but I do not have the necessary resource with on whenever I need them. This blog is some form of a solution to this, since it saves me from having to carry lots of notes around and it is accessible from (almost) anywhere.
The first milestone is the submission of the research proposal around end the of May - beginning of June. The preliminary title for the PhD is "Theories of Truth and the Problem of Realism in Quantum Mechanics". Some of the key concepts will include the context dependency, the Kochen-Specker Theory, and the realisation or actualisation of physical measurable quantities, as well as the assigning of truth-values with reference to a quantum mechanical context.
Particular emphasis will be placed on correspondence theories of truth and their critique with reference to the context dependency in quantum mechanics.
Unfortunately, the timing for this is far from perfect. I had to join the Greek Army for my compulsory national service this February (12 months service). Meaning that not only the time I get fro studying/research is limited and fragmented, but I do not have the necessary resource with on whenever I need them. This blog is some form of a solution to this, since it saves me from having to carry lots of notes around and it is accessible from (almost) anywhere.
The first milestone is the submission of the research proposal around end the of May - beginning of June. The preliminary title for the PhD is "Theories of Truth and the Problem of Realism in Quantum Mechanics". Some of the key concepts will include the context dependency, the Kochen-Specker Theory, and the realisation or actualisation of physical measurable quantities, as well as the assigning of truth-values with reference to a quantum mechanical context.
Particular emphasis will be placed on correspondence theories of truth and their critique with reference to the context dependency in quantum mechanics.
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