### Re: [Fis] Probability Amplitudes and philosophical basis of QM

Dear Lars-Göran and colleagues, I think that probability amplitudes should have a single meaning as well as any concept that our mind can invent. Heisenberg said in his Lectures 1955-1956 ”Physics and Philosophy”: “Descartes realizes that what we know about our mind is more certain than what we know about the outer world”. According to this correct notion no theory about the outer world can have interpretations and any concept should have a single meaning. Theories of physics as theories about the outer world should not have interpretations. But it is well known that interpretations of quantum mechanics are charaterized by huge diversity. Therefore quantum mechanics (QM) can not be considered as a physical theory. Schrodinger was right that QM is failed theory (see the Section “THE MAKESHIFT OF WAVE MECHANICS’ in the book Schrodinger E. Science and Humanism. Physics in Our Time. Cambridge: University Press, 1952). Schrodinger craved to interpret his wave function as a real wave. He tried to replace particles with wavepackets. But wavepackets diffuse. This diffuseness contradicts numerous observations. Most physicists have accepted the interpretation of Schrodinger’s wave function as probability amplitudes proposed by Born because the probability of observation should change at observation. Most physicists believe for the present that it was good solution of the problem to say that wavepacket can be localized at observation. But they ignore that it is localized first of all in the mind of the observer. The probability of observation describes the state of the mind of the observer. It is quite obvious that our knowledge changes when we observe anything. This change of our knowledge is discontinuous. In the Section ”The Copenhagen Interpretation of Quantum Theory” of his Lectures 1955-1956 Heisenberg stated ”that certainly our knowledge can change suddenly and that this fact justifies the use of the term ’quantum jump’”. Just therefore QM presupposes the instantaneous and non-local change at observation called Dirac jump, wave function collapse, or ’quantum jump’ from the ’possible’ to the ’actual’. This discontinuous change of the mind of the observer takes place at any observation at which an object (for example a physical system) has an influence on subject (the mind of the observer). Schrodinger noted in the Section “THE ALLEGED BREAK-DOWN OF THE BARRIER BETWEEN SUBJECT AND OBJECT” of his book “Science and Humanism” that ”…the mere contention that every observation depends on both the subject and the object, which are inextricably interwoven this contention is hardly new, it is almost as old as science itself” . But ”… in the present order of ideas the direct physical, causal influence between the two is regarded as mutual. It is said that there is also an unavoidable and uncontrollable impression from the side of the subject onto the object. This aspect is new…”. Heisenberg had in mind just this new aspect when he said in his Lectures 1955-1956 that “we have to criticise from the development of physics in our time” the Cartesian division. Quantum mechanics is vague and has numerous interpretation just because of the denial of the Cartesian division. Heisenberg said fairly: The mechanics of Newton and all the other parts of classical physics constructed after its model started from the assumption that one can describe the world without speaking about God or ourselves… If one follows the great difficulty which even eminent scientists like Einstein had in understanding and accepting the Copenhagen interpretation of quantum theory, one can trace the roots of this difficulty to the Cartesian partition. This partition has penetrated deeply into the human mind during the three centuries following Descartes and it will take a long time for it to be replaced by a really different attitude toward the problem of reality. But he was not right when he stated that a scientific theory could be possible without the acceptance of the Cartesian polarity between the 'res cogitans' and the 'res extensa'. No science can describe a mutual causal influence between the 'res cogitans' and the 'res extensa' (or between the subject and the object). Quantum mechanics presumes such mutual influence. Therefore it can not be considered as a scientific theory. Einstein wrote to Schrodinger as far back as 1928: The soothing philosophy-or religion?-of Heisenberg-Bohr is so cleverly concocted that it offers the believers a soft resting pillow from which they are not easily chased away. The diversity of opinions about quantum mechanics witnesses that Einstein's words turned out prophetic: the dissent can be about a religion but our true comprehension must be unified. At least we must believe that it is possible. Otherwise no science could be possible. The belief in quantum mechanics is a consequence of the illusion inherent to most contemporary scientists that what we

### Re: [Fis] Probability Amplitudes

Dear Hans, I would like just to point that 99,99% of people working in quantum theory would say that the complex amplitude of quantum probability is the main its intrinsic property, so if you try to exclude amplitudes from the model you can in principle do this and this is well known long ago in so called quantum tomographic approach of Vladimir Manko, but in this way quantum theory loses its simplicity and clarity, yours, andrei Andrei Khrennikov, Professor of Applied Mathematics, International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science Linnaeus University, Växjö-Kalmar, Sweden From: fis-boun...@listas.unizar.es [fis-boun...@listas.unizar.es] on behalf of Hans von Baeyer [henrikrit...@gmail.com] Sent: Wednesday, January 22, 2014 12:21 AM To: fis@listas.unizar.es Subject: [Fis] Probability Amplitudes Dear Dino and friends, thanks for bringing up the issue of probability amplitudes. Since they are technical tools of physics, and since I didn't want to go too far afield, I did not mention them in my lecture. The closest I came was the wavefunction, which, indeed, is a probability amplitude. In order to make contact with real, measurable quantities, it must be multiplied by its complex conjugate. This recipe is called the Born rule, and it is an ad hoc addition to the quantum theory. It lacks any motivation except that it works. In keeping with Einstein's advice (which he himself often flouted) to try to keep unmeasurable concepts out of our description of nature, physicists have realized long ago that it must be possible to recast quantum mechanics entirely in terms of probabilities, not even mentioning probability amplitudes or wavefunctions. The question is only: How complicated would the resulting formalism be? (To make a weak analogy, it must be possible to recast arithmetic in the language of Roman numerals, but the result would surely look much messier than what we learn in grade school.) Hitherto, nobody had come up with an elegant solution to this problem. To their happy surprise, QBists have made progress toward a quantum theory without probability amplitudes. Of course they have to pay a price. Instead of unmeasurable concepts they introduce, for any experiment, a very special set of standard probabilities (NOT AMPLITUDES) which are measurable, but not actually measured. When they re-write the Born rule in terms of these, they find that it looks almost, but not quite, like a fundamental axiom of probability theory called Unitarity. Unitarity decrees that for any experiment the sum of the probabilities for all possible outcomes must be one. (For a coin, the probabilities of heads and tails are both 1/2. Unitarity states 1/2 + 1/2 = 1.) This unexpected outcome of QBism suggests a deep connection between the Born rule and Unitarity. Since Unitarity is a logical concept unrelated to quantum phenomena, this gives QBists the hope that they will eventually succeed in explaining the significacne of the Born rule, and banishing probability amplitudes from quantum mechanics, leaving only (Bayesian) probabilities. So, I'm afraid dear Dino, that the current attitude of QBists is that probability amplitudes are LESS fundamental than probabilities, not MORE. But the story is far from finished! Hans ___ fis mailing list fis@listas.unizar.es https://webmail.unizar.es/cgi-bin/mailman/listinfo/fis

### Re: [Fis] Probability Amplitudes

Dear Joseph, you are going toward quantum probability theory where probabilities are determined by vectors; moreover, the vectors belong to complex Hilbert space, i.e., roughly speaking each probability has not only the direction but even the phase, andrei Andrei Khrennikov, Professor of Applied Mathematics, International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science Linnaeus University, Växjö-Kalmar, Sweden From: fis-boun...@listas.unizar.es [fis-boun...@listas.unizar.es] on behalf of Joseph Brenner [joe.bren...@bluewin.ch] Sent: Wednesday, January 22, 2014 8:54 AM To: Dino Buzzetti; Hans von Baeyer; fis Subject: Re: [Fis] Probability Amplitudes Dear Hans and Dino, This is a direct question to both of you, to which I have not found a clear answer: are value and amplitude the only parameters that have been assigned to probability? In my theory, the changing value of actuality and potentiality of specific antagonistic process elements are probability-like in not including 0 and 1, as I have said. Can, in addition, probabilities have some vector-like properties, that is, include a /direction/? This concept would be moving toward (and past) Dino and away from Hans . . . Your comments and those of others would be welcome. Best wishes, Joseph - Original Message - From: Dino Buzzettimailto:dino.buzze...@gmail.com To: Hans von Baeyermailto:henrikrit...@gmail.com ; fismailto:fis@listas.unizar.es Sent: Wednesday, January 22, 2014 3:53 AM Subject: Re: [Fis] Probability Amplitudes Dear Hans, Thank you for your explanation about probability amplitudes, that clarifies a lot. My only worry was about the *epistemological* implications of quantum mechanics in its standard formulation, that in my opinion point to a paradigm shift, which is felt not only in this domain, but in all fields where *emergent* phenomena are accounted for—a process that I thought was hinted to by Wheeler's famous words It from Bit, that I remember reading for the first time precisely in your book on information. That's the ground for expressing my worry that reverting to classical probability theory might entail a drawback to this decisive epistemological turn. But I might misunderstand the whole story, that is certainly not over yet :-) -dino On 22 January 2014 00:21, Hans von Baeyer henrikrit...@gmail.commailto:henrikrit...@gmail.com wrote: Dear Dino and friends, thanks for bringing up the issue of probability amplitudes. Since they are technical tools of physics, and since I didn't want to go too far afield, I did not mention them in my lecture. The closest I came was the wavefunction, which, indeed, is a probability amplitude. In order to make contact with real, measurable quantities, it must be multiplied by its complex conjugate. This recipe is called the Born rule, and it is an ad hoc addition to the quantum theory. It lacks any motivation except that it works. In keeping with Einstein's advice (which he himself often flouted) to try to keep unmeasurable concepts out of our description of nature, physicists have realized long ago that it must be possible to recast quantum mechanics entirely in terms of probabilities, not even mentioning probability amplitudes or wavefunctions. The question is only: How complicated would the resulting formalism be? (To make a weak analogy, it must be possible to recast arithmetic in the language of Roman numerals, but the result would surely look much messier than what we learn in grade school.) Hitherto, nobody had come up with an elegant solution to this problem. To their happy surprise, QBists have made progress toward a quantum theory without probability amplitudes. Of course they have to pay a price. Instead of unmeasurable concepts they introduce, for any experiment, a very special set of standard probabilities (NOT AMPLITUDES) which are measurable, but not actually measured. When they re-write the Born rule in terms of these, they find that it looks almost, but not quite, like a fundamental axiom of probability theory called Unitarity. Unitarity decrees that for any experiment the sum of the probabilities for all possible outcomes must be one. (For a coin, the probabilities of heads and tails are both 1/2. Unitarity states 1/2 + 1/2 = 1.) This unexpected outcome of QBism suggests a deep connection between the Born rule and Unitarity. Since Unitarity is a logical concept unrelated to quantum phenomena, this gives QBists the hope that they will eventually succeed in explaining the significacne of the Born rule, and banishing probability amplitudes from quantum mechanics, leaving only (Bayesian) probabilities. So, I'm afraid dear Dino, that the current attitude of QBists is that probability amplitudes are LESS fundamental than probabilities, not MORE. But the story is far from finished! Hans

### Re: [Fis] Probability Amplitudes

Dear Andrei, Hans and all I agree with Andrei. And why make quantum theory more complex than it is? One may use all kinds of mathematical tools in a scientific theory, and the more these tools simplify calculations the better. I see no reason to avoid using amplitudes or matrices in quantum theory. Using a mathematical concept for making calculations doesn't entail that I accept that that concept represent a physical property. To Hans: Where exactly did Einstein wrote that one should avoid unmeasurable concepts in the description of Nature? I can't remember having read that. The issue is how we should interpret quantum theory, in particular the wave function, i.e., probability amplitudes; are they just mathematical tools, or do they describe real physical features of quantum systems? I believe the latter alternative is true and so did Schrödinger. But there are formidable difficulties to give a realistic interpretation of wave functions, and Schrödinger didn't succeed. But I think the difficulties can be overcome and I have published my views about these things (Lars-Göran Johansson: Interpreting Quantum Mechanics. A realist view in Schrödinger's vein, Ashgate, Aldershot 2007). Lars-Göran 22 jan 2014 kl. 10:59 skrev Andrei Khrennikov andrei.khrenni...@lnu.semailto:andrei.khrenni...@lnu.se: Dear Hans, I would like just to point that 99,99% of people working in quantum theory would say that the complex amplitude of quantum probability is the main its intrinsic property, so if you try to exclude amplitudes from the model you can in principle do this and this is well known long ago in so called quantum tomographic approach of Vladimir Manko, but in this way quantum theory loses its simplicity and clarity, yours, andrei Andrei Khrennikov, Professor of Applied Mathematics, International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science Linnaeus University, Växjö-Kalmar, Sweden From: fis-boun...@listas.unizar.esmailto:fis-boun...@listas.unizar.es [fis-boun...@listas.unizar.esmailto:fis-boun...@listas.unizar.es] on behalf of Hans von Baeyer [henrikrit...@gmail.commailto:henrikrit...@gmail.com] Sent: Wednesday, January 22, 2014 12:21 AM To: fis@listas.unizar.esmailto:fis@listas.unizar.es Subject: [Fis] Probability Amplitudes Dear Dino and friends, thanks for bringing up the issue of probability amplitudes. Since they are technical tools of physics, and since I didn't want to go too far afield, I did not mention them in my lecture. The closest I came was the wavefunction, which, indeed, is a probability amplitude. In order to make contact with real, measurable quantities, it must be multiplied by its complex conjugate. This recipe is called the Born rule, and it is an ad hoc addition to the quantum theory. It lacks any motivation except that it works. In keeping with Einstein's advice (which he himself often flouted) to try to keep unmeasurable concepts out of our description of nature, physicists have realized long ago that it must be possible to recast quantum mechanics entirely in terms of probabilities, not even mentioning probability amplitudes or wavefunctions. The question is only: How complicated would the resulting formalism be? (To make a weak analogy, it must be possible to recast arithmetic in the language of Roman numerals, but the result would surely look much messier than what we learn in grade school.) Hitherto, nobody had come up with an elegant solution to this problem. To their happy surprise, QBists have made progress toward a quantum theory without probability amplitudes. Of course they have to pay a price. Instead of unmeasurable concepts they introduce, for any experiment, a very special set of standard probabilities (NOT AMPLITUDES) which are measurable, but not actually measured. When they re-write the Born rule in terms of these, they find that it looks almost, but not quite, like a fundamental axiom of probability theory called Unitarity. Unitarity decrees that for any experiment the sum of the probabilities for all possible outcomes must be one. (For a coin, the probabilities of heads and tails are both 1/2. Unitarity states 1/2 + 1/2 = 1.) This unexpected outcome of QBism suggests a deep connection between the Born rule and Unitarity. Since Unitarity is a logical concept unrelated to quantum phenomena, this gives QBists the hope that they will eventually succeed in explaining the significacne of the Born rule, and banishing probability amplitudes from quantum mechanics, leaving only (Bayesian) probabilities. So, I'm afraid dear Dino, that the current attitude of QBists is that probability amplitudes are LESS fundamental than probabilities, not MORE. But the story is far from finished! Hans ___ fis mailing list fis@listas.unizar.esmailto:fis@listas.unizar.es

### Re: [Fis] Probability Amplitudes in Macroscopic Processes

Let me clarify one point: by saying that probability amplitudes represent real physical features I reject the instrumentalist idea that they are mere calculational devices. But of course, the probability amplitude is no observable. But there is no need to claim that only observables have any physical significance. Robert Chen has, in a couple of papers argued that the square of real part of the wave function could be interpreted as the system's kinetic energy, whereas the square of the imaginary part represents the potential energy of the system. It is as far as I can see a possible and reasonable interpretation. Lars-Göran 22 jan 2014 kl. 15:14 skrev Joseph Brenner joe.bren...@bluewin.chmailto:joe.bren...@bluewin.ch: Dear Lars-Göran, Andrei and Hans, As you (I hope) have seen, I am trying to see how the evolution of macroscopic processes can be described in terms of changing probabilities, and I am encouraged to believe this is possible. If you allow the extension from QM, all of the following would seem to allow this (I am not concerned about whether QM itself becomes more or less complex): 1. Andrei confirms that the probability (in LIR, degree of potentiality or actuality) of a phenomenon can have a direction. 2. Lars-Göran says that probability amplitudes can represent real physical features. 3. Even though /a contrario/, Hans wrote: In order to make contact with real, measurable quantities, it (the probability amplitude) must be multiplied by its complex conjugate. This recipe is called the Born rule, and it is an ad hoc addition to the quantum theory. It lacks any motivation except that it works. In my Logic in Reality, since there is a reciprocal relation between actuality and potentiality, each should be the complex conjugate of the other. I have no problem in the two summing to 1 if the values of 0 or 1 are excluded for either of them. This non-quantum aspect of reality could provide the missing motivation for the recipe in quantum theory ;-) I am certainly looking for a measurable (or estimatable) quantity of the actuality and potentiality of interactive processes that is not a standard probability of outcomes, but of changing macroscopic states. This is of course an 'underdeveloped' concept, but I am encouraged to believe that this idea of another set of very special probabilities is neither totally wrong nor totally trivial. Many thanks, Joseph - Original Message - From: Lars-Göran Johanssonmailto:lars-goran.johans...@filosofi.uu.se To: fis@listas.unizar.esmailto:fis@listas.unizar.es Sent: Wednesday, January 22, 2014 12:45 PM Subject: Re: [Fis] Probability Amplitudes Dear Andrei, Hans and all I agree with Andrei. And why make quantum theory more complex than it is? One may use all kinds of mathematical tools in a scientific theory, and the more these tools simplify calculations the better. I see no reason to avoid using amplitudes or matrices in quantum theory. Using a mathematical concept for making calculations doesn't entail that I accept that that concept represent a physical property. To Hans: Where exactly did Einstein wrote that one should avoid unmeasurable concepts in the description of Nature? I can't remember having read that. The issue is how we should interpret quantum theory, in particular the wave function, i.e., probability amplitudes; are they just mathematical tools, or do they describe real physical features of quantum systems? I believe the latter alternative is true and so did Schrödinger. But there are formidable difficulties to give a realistic interpretation of wave functions, and Schrödinger didn't succeed. But I think the difficulties can be overcome and I have published my views about these things (Lars-Göran Johansson: Interpreting Quantum Mechanics. A realist view in Schrödinger's vein, Ashgate, Aldershot 2007). Lars-Göran 22 jan 2014 kl. 10:59 skrev Andrei Khrennikov andrei.khrenni...@lnu.semailto:andrei.khrenni...@lnu.se: Dear Hans, I would like just to point that 99,99% of people working in quantum theory would say that the complex amplitude of quantum probability is the main its intrinsic property, so if you try to exclude amplitudes from the model you can in principle do this and this is well known long ago in so called quantum tomographic approach of Vladimir Manko, but in this way quantum theory loses its simplicity and clarity, yours, andrei Andrei Khrennikov, Professor of Applied Mathematics, International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science Linnaeus University, Växjö-Kalmar, Sweden From: fis-boun...@listas.unizar.esmailto:fis-boun...@listas.unizar.es [fis-boun...@listas.unizar.esmailto:fis-boun...@listas.unizar.es] on behalf of Hans von Baeyer [henrikrit...@gmail.commailto:henrikrit...@gmail.com] Sent: Wednesday, January 22, 2014 12:21 AM To: fis@listas.unizar.esmailto:fis

### Re: [Fis] Probability Amplitudes

Dear Hans, Thank you for your explanation about probability amplitudes, that clarifies a lot. My only worry was about the *epistemological* implications of quantum mechanics in its standard formulation, that in my opinion point to a paradigm shift, which is felt not only in this domain, but in all fields where *emergent* phenomena are accounted for—a process that I thought was hinted to by Wheeler's famous words It from Bit, that I remember reading for the first time precisely in your book on information. That's the ground for expressing my worry that reverting to classical probability theory might entail a drawback to this decisive epistemological turn. But I might misunderstand the whole story, that is certainly not over yet :-) -dino On 22 January 2014 00:21, Hans von Baeyer henrikrit...@gmail.com wrote: Dear Dino and friends, thanks for bringing up the issue of probability amplitudes. Since they are technical tools of physics, and since I didn't want to go too far afield, I did not mention them in my lecture. The closest I came was the wavefunction, which, indeed, is a probability amplitude. In order to make contact with real, measurable quantities, it must be multiplied by its complex conjugate. This recipe is called the Born rule, and it is an ad hoc addition to the quantum theory. It lacks any motivation except that it works. In keeping with Einstein's advice (which he himself often flouted) to try to keep unmeasurable concepts out of our description of nature, physicists have realized long ago that it must be possible to recast quantum mechanics entirely in terms of probabilities, not even mentioning probability amplitudes or wavefunctions. The question is only: How complicated would the resulting formalism be? (To make a weak analogy, it must be possible to recast arithmetic in the language of Roman numerals, but the result would surely look much messier than what we learn in grade school.) Hitherto, nobody had come up with an elegant solution to this problem. To their happy surprise, QBists have made progress toward a quantum theory without probability amplitudes. Of course they have to pay a price. Instead of unmeasurable concepts they introduce, for any experiment, a very special set of standard probabilities (NOT AMPLITUDES) which are measurable, but not actually measured. When they re-write the Born rule in terms of these, they find that it looks almost, but not quite, like a fundamental axiom of probability theory called Unitarity. Unitarity decrees that for any experiment the sum of the probabilities for all possible outcomes must be one. (For a coin, the probabilities of heads and tails are both 1/2. Unitarity states 1/2 + 1/2 = 1.) This unexpected outcome of QBism suggests a deep connection between the Born rule and Unitarity. Since Unitarity is a logical concept unrelated to quantum phenomena, this gives QBists the hope that they will eventually succeed in explaining the significacne of the Born rule, and banishing probability amplitudes from quantum mechanics, leaving only (Bayesian) probabilities. So, I'm afraid dear Dino, that the current attitude of QBists is that probability amplitudes are LESS fundamental than probabilities, not MORE. But the story is far from finished! Hans ___ fis mailing list fis@listas.unizar.es https://webmail.unizar.es/cgi-bin/mailman/listinfo/fis ___ fis mailing list fis@listas.unizar.es https://webmail.unizar.es/cgi-bin/mailman/listinfo/fis

### Re: [Fis] Probability Amplitudes

Dear Hans and Dino, This is a direct question to both of you, to which I have not found a clear answer: are value and amplitude the only parameters that have been assigned to probability? In my theory, the changing value of actuality and potentiality of specific antagonistic process elements are probability-like in not including 0 and 1, as I have said. Can, in addition, probabilities have some vector-like properties, that is, include a /direction/? This concept would be moving toward (and past) Dino and away from Hans . . . Your comments and those of others would be welcome. Best wishes, Joseph - Original Message - From: Dino Buzzetti To: Hans von Baeyer ; fis Sent: Wednesday, January 22, 2014 3:53 AM Subject: Re: [Fis] Probability Amplitudes Dear Hans, Thank you for your explanation about probability amplitudes, that clarifies a lot. My only worry was about the *epistemological* implications of quantum mechanics in its standard formulation, that in my opinion point to a paradigm shift, which is felt not only in this domain, but in all fields where *emergent* phenomena are accounted for—a process that I thought was hinted to by Wheeler's famous words It from Bit, that I remember reading for the first time precisely in your book on information. That's the ground for expressing my worry that reverting to classical probability theory might entail a drawback to this decisive epistemological turn. But I might misunderstand the whole story, that is certainly not over yet :-) -dino On 22 January 2014 00:21, Hans von Baeyer henrikrit...@gmail.com wrote: Dear Dino and friends, thanks for bringing up the issue of probability amplitudes. Since they are technical tools of physics, and since I didn't want to go too far afield, I did not mention them in my lecture. The closest I came was the wavefunction, which, indeed, is a probability amplitude. In order to make contact with real, measurable quantities, it must be multiplied by its complex conjugate. This recipe is called the Born rule, and it is an ad hoc addition to the quantum theory. It lacks any motivation except that it works. In keeping with Einstein's advice (which he himself often flouted) to try to keep unmeasurable concepts out of our description of nature, physicists have realized long ago that it must be possible to recast quantum mechanics entirely in terms of probabilities, not even mentioning probability amplitudes or wavefunctions. The question is only: How complicated would the resulting formalism be? (To make a weak analogy, it must be possible to recast arithmetic in the language of Roman numerals, but the result would surely look much messier than what we learn in grade school.) Hitherto, nobody had come up with an elegant solution to this problem. To their happy surprise, QBists have made progress toward a quantum theory without probability amplitudes. Of course they have to pay a price. Instead of unmeasurable concepts they introduce, for any experiment, a very special set of standard probabilities (NOT AMPLITUDES) which are measurable, but not actually measured. When they re-write the Born rule in terms of these, they find that it looks almost, but not quite, like a fundamental axiom of probability theory called Unitarity. Unitarity decrees that for any experiment the sum of the probabilities for all possible outcomes must be one. (For a coin, the probabilities of heads and tails are both 1/2. Unitarity states 1/2 + 1/2 = 1.) This unexpected outcome of QBism suggests a deep connection between the Born rule and Unitarity. Since Unitarity is a logical concept unrelated to quantum phenomena, this gives QBists the hope that they will eventually succeed in explaining the significacne of the Born rule, and banishing probability amplitudes from quantum mechanics, leaving only (Bayesian) probabilities. So, I'm afraid dear Dino, that the current attitude of QBists is that probability amplitudes are LESS fundamental than probabilities, not MORE. But the story is far from finished! Hans ___ fis mailing list fis@listas.unizar.es https://webmail.unizar.es/cgi-bin/mailman/listinfo/fis -- ___ fis mailing list fis@listas.unizar.es https://webmail.unizar.es/cgi-bin/mailman/listinfo/fis ___ fis mailing list fis@listas.unizar.es https://webmail.unizar.es/cgi-bin/mailman/listinfo/fis