Follow @Aristeidis_K
In our research the Kochen-Specker Theorem (K-S) will be of significant importance. More generally, we will be trying to focus on the of realism in QM, but always, and as far as it is possible, with an eye on the most current version and developments in quantum theory.
The K-S theorem provides strong indications (if not evidence) against the possibility of interpreting QM in terms of a Hidden Variables (HV)theory. Note that we make no reference here on whether or not this excludes also any realistic interpretation of QM. Therefore, and considering the above the significance of the K-S theorem is that it forces us to review some of the most basic (philosophical/metaphysical) assumptions upon which we base our realistic interpretation of our physical theories.
The important notion under consideration here is that of "contextuality". Simply put, that that the value of a quantum mechanical observable is dependent on the measurement arrangement.
...to be continued
link for more on the K-S theorem
Monday 26 May 2008
Saturday 24 May 2008
Research Proposal Draft - Take 1
Follow @Aristeidis_K
First attempt at a basic introduction to the subject for the research proposal:
Statement Of Topic
Quantum Mechanics is an extremely successful branch of science. It has enabled us to explain the structures of atoms and the details of atomic spectra, radioactivity, and chemical bonding. Elaborations of the fundamental theory has led to satisfactory explanations of nuclear structures and relations, the electrical and thermal properties of solids, superconductivity, the creation and annihilation of elementary particles, the production of anti-matter, Bose-Einstein condensation, the stability of white dwarfs, and neutron stars, and much else. It has also made possible major practical developments such as the electron microscope, the laser, the transistor. Exceedingly delicate experiments have confirmed subtle quantum effects to an astonishing degree of accuracy. It has never been shown to contradict the results of 50 years of experimentation.
If all that is asked from a scientific theory is that it should correctly predict the results of experiments, quantum mechanics works perfectly, and, to paraphrase John Bell, "ordinary quantum mechanics (as far as we know) is just fine for all practical purposes (FAPP)".
However, the basic conceptual foundations of quantum mechanics, when closely examined, can lead, depending on their interpretation, to some puzzling paradoxes and strange, counterintuitive, and to some commentators, unacceptable features. For these reasons, the problem of the interpretation of Quantum Mechanics has produced some very lively debates, and continues to do so. The importance of the interpretation of Quantum Mechanics for the realism-antirealism can hardly be overstressed.
One major point of contention is the way quantum mechanical states are represented within the theory and the conclusions that are to be drawn from such a representation. Quantum Mechanics has the peculiar property that it only assigns probabilities that a physical state will be found at a specific state upon measurement; contrary to a classical theory which could, at least in principle, provide us with definite values for the various properties of the system in question. From a realist point of view, the conclusion to be drawn is that Quantum Mechanics does not provide us with a complete description of the physical system under investigation, something which could, arguably, be achieved through a Hidden Variables theory.
The significance of Bell's Theorem on the possibility of reproducing all of the predictions of quantum mechanics through a (local) hidden variables theory is well known. Bell test experiments to date overwhelmingly show that Bell inequalities are violated. These results provide empirical evidence against local hidden variable theories.
Furthermore, realistic interpretations of quantum mechanics seem to rely, either explicitly or implicitly, on the principles of Value Definiteness and Non Contextuality .
Value Definiteness: All observables defined for a QM system have definite values at all times.
Non Contextuality: If a QM system possesses a property (value of an observable), then it does so independently of any measurement context, i.e. independently of how that value is eventually measured.
At first glance, these two principles might even appear to form a "null" hypothesis for any realistic approach, especially if we are trying to extend our classical intuitions into the quantum domain. Both Value Realism and Non-COntextuality incorporate the basic idea of an independence of physical reality from its being measured. Or in other words, we can view this as a form of Cartesian Dualism and a separating of the mental world from the physical one.
Qauntum Mechanics however challenges in a serious way these basic notions through the Kochen-Specker (KS) theorem. The KS theorem establishes a contradiction between VD and NC. Implying that if we are to accept quantum mechanics, and the experimental results do force us into that direction, we have to renounce either VD or NC. As can be readily appreciated, for the realist, it is unclear how a plausible realist interpretation of QM can be achieved that renounced VD but not NC, or vice versa.
All the above problems seem to reinforce Arthur Fine's pronouncement that “realism is dead”. However, these problems should not deter the realist. After the same man that pronounced the death of realism has also mentioned that realism is a powerful tool in the development of science, and therefore it is not advisable that we abandon it. On the other hand if the realist to meet the challenges level at him from the latest scientific developments, and from quantum mechanics more specifically, he needs to undertake a critical re-evaluation of some of the notions that had hitherto formed the basis, at a foundational level, of any realistic interpretation of any physical theory.
In order to achieve this we need to examine some the notions that have been viewed as principle conditions of our conception of the world, including those that we have already mentioned (Value Definiteness, Non-contextuality) but also notions like separability. In addition, we need to go further back and examine critically what is the function of the various theories of truth, within the context of scientific research and discourse, be it correspondence theories of truth of a Tarskian type or minimalistic theories of truth (e.g. Pascal Engel).
What we are aiming at showing, is that there is a way the realist can meet the challenges that quantum mechanics presents. However, in order to achieve this, the realist will need to incorporate into his conception of the natural world the results of quantum mechanics, or to put it differently, to allow his world view to be instructed by the results of quantum mechanics. An integral part of this effort will be focused in evaluating the possible solutions in escaping the consequences of the Kochen-Specker theorem. In summary, some the of available solutions are:
• Denial of Value Definiteness
• Denial of Non-Contextuality, in the form of either a Casual or an Ontological Contextuality
Although most realistic approaches to the interpretation quantum mechanics are directed at showing that quantum theory is incomplete, by pointing out some its bizarre consequences that run counter to some of our fundamental intuitions about the nature of reality, there is an alternative path that is worth exploring. Taking the traditional approach and turning it on its head, we accept the results of quantum mechanics and use them to instruct our world view.
However, such debates can often leave the reader with a sense of frustration. Such views have been expressed by prominent commentators such as Arthur Fine, when he declared realism as dead and went on produce a colourful portayal of realism when he said that all they seek to add is "A desk-thumping, foot stamping shout of "REALLY"!!".
[http://www.drury.edu/ess/philsci/AFine.html
]
Comments such as these, however, may not need be taken as signaling the end realism as such.
Statement Of Topic
Quantum Mechanics is an extremely successful branch of science. It has enabled us to explain the structures of atoms and the details of atomic spectra, radioactivity, and chemical bonding. Elaborations of the fundamental theory has led to satisfactory explanations of nuclear structures and relations, the electrical and thermal properties of solids, superconductivity, the creation and annihilation of elementary particles, the production of anti-matter, Bose-Einstein condensation, the stability of white dwarfs, and neutron stars, and much else. It has also made possible major practical developments such as the electron microscope, the laser, the transistor. Exceedingly delicate experiments have confirmed subtle quantum effects to an astonishing degree of accuracy. It has never been shown to contradict the results of 50 years of experimentation.
If all that is asked from a scientific theory is that it should correctly predict the results of experiments, quantum mechanics works perfectly, and, to paraphrase John Bell, "ordinary quantum mechanics (as far as we know) is just fine for all practical purposes (FAPP)".
However, the basic conceptual foundations of quantum mechanics, when closely examined, can lead, depending on their interpretation, to some puzzling paradoxes and strange, counterintuitive, and to some commentators, unacceptable features. For these reasons, the problem of the interpretation of Quantum Mechanics has produced some very lively debates, and continues to do so. The importance of the interpretation of Quantum Mechanics for the realism-antirealism can hardly be overstressed.
One major point of contention is the way quantum mechanical states are represented within the theory and the conclusions that are to be drawn from such a representation. Quantum Mechanics has the peculiar property that it only assigns probabilities that a physical state will be found at a specific state upon measurement; contrary to a classical theory which could, at least in principle, provide us with definite values for the various properties of the system in question. From a realist point of view, the conclusion to be drawn is that Quantum Mechanics does not provide us with a complete description of the physical system under investigation, something which could, arguably, be achieved through a Hidden Variables theory.
The significance of Bell's Theorem on the possibility of reproducing all of the predictions of quantum mechanics through a (local) hidden variables theory is well known. Bell test experiments to date overwhelmingly show that Bell inequalities are violated. These results provide empirical evidence against local hidden variable theories.
Furthermore, realistic interpretations of quantum mechanics seem to rely, either explicitly or implicitly, on the principles of Value Definiteness and Non Contextuality .
Value Definiteness: All observables defined for a QM system have definite values at all times.
Non Contextuality: If a QM system possesses a property (value of an observable), then it does so independently of any measurement context, i.e. independently of how that value is eventually measured.
At first glance, these two principles might even appear to form a "null" hypothesis for any realistic approach, especially if we are trying to extend our classical intuitions into the quantum domain. Both Value Realism and Non-COntextuality incorporate the basic idea of an independence of physical reality from its being measured. Or in other words, we can view this as a form of Cartesian Dualism and a separating of the mental world from the physical one.
Qauntum Mechanics however challenges in a serious way these basic notions through the Kochen-Specker (KS) theorem. The KS theorem establishes a contradiction between VD and NC. Implying that if we are to accept quantum mechanics, and the experimental results do force us into that direction, we have to renounce either VD or NC. As can be readily appreciated, for the realist, it is unclear how a plausible realist interpretation of QM can be achieved that renounced VD but not NC, or vice versa.
All the above problems seem to reinforce Arthur Fine's pronouncement that “realism is dead”. However, these problems should not deter the realist. After the same man that pronounced the death of realism has also mentioned that realism is a powerful tool in the development of science, and therefore it is not advisable that we abandon it. On the other hand if the realist to meet the challenges level at him from the latest scientific developments, and from quantum mechanics more specifically, he needs to undertake a critical re-evaluation of some of the notions that had hitherto formed the basis, at a foundational level, of any realistic interpretation of any physical theory.
In order to achieve this we need to examine some the notions that have been viewed as principle conditions of our conception of the world, including those that we have already mentioned (Value Definiteness, Non-contextuality) but also notions like separability. In addition, we need to go further back and examine critically what is the function of the various theories of truth, within the context of scientific research and discourse, be it correspondence theories of truth of a Tarskian type or minimalistic theories of truth (e.g. Pascal Engel).
What we are aiming at showing, is that there is a way the realist can meet the challenges that quantum mechanics presents. However, in order to achieve this, the realist will need to incorporate into his conception of the natural world the results of quantum mechanics, or to put it differently, to allow his world view to be instructed by the results of quantum mechanics. An integral part of this effort will be focused in evaluating the possible solutions in escaping the consequences of the Kochen-Specker theorem. In summary, some the of available solutions are:
• Denial of Value Definiteness
• Denial of Non-Contextuality, in the form of either a Casual or an Ontological Contextuality
Although most realistic approaches to the interpretation quantum mechanics are directed at showing that quantum theory is incomplete, by pointing out some its bizarre consequences that run counter to some of our fundamental intuitions about the nature of reality, there is an alternative path that is worth exploring. Taking the traditional approach and turning it on its head, we accept the results of quantum mechanics and use them to instruct our world view.
However, such debates can often leave the reader with a sense of frustration. Such views have been expressed by prominent commentators such as Arthur Fine, when he declared realism as dead and went on produce a colourful portayal of realism when he said that all they seek to add is "A desk-thumping, foot stamping shout of "REALLY"!!".
[http://www.drury.edu/ess/philsci/AFine.html
]
Comments such as these, however, may not need be taken as signaling the end realism as such.
Tuesday 15 January 2008
"The Structure & Interpratation of Qunatum Mechanics" by R.I.G. Hughes - 2
Follow @Aristeidis_K
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.
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.
Thursday 10 January 2008
"The Structure & Interpratation of Qunatum Mechanics" by R.I.G. Hughes
Follow @Aristeidis_K
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
Follow @Aristeidis_K
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.
Subscribe to:
Posts (Atom)