Why Is Quantum Mechanics So Difficult?
Table of Contents
Quantum Mechanics Key points:
- Quantum mechanics (QM) is often perceived as difficult, especially by non-physicists.
- The difficulty lies in the conceptual foundation of QM, as it doesn’t connect well with classical understanding.
- Unlike other areas of physics, there’s no direct continuity between classical concepts and QM.
- While the conceptual understanding is challenging, the mathematical formulation of QM is familiar and follows from existing knowledge.
- Mathematical formalism is crucial in QM, as it provides a bridge between classical and quantum worlds.
- QM’s mathematical formalism is the foundation on which our understanding is built.
- Disagreements often arise in the interpretation of QM, but the source, mathematical formalism, remains consistent.
- The mathematical aspect of QM is a means of conveying ideas and principles accurately, akin to musical notes conveying music.
- In QM, mathematics is a form of communication that accurately describes our universe.
- Quantum mechanics doesn’t fully make “sense” without embracing its mathematical continuity.
QM’s formalism
Strangely enough, QM’s formalism isn’t any more difficult than other areas of physics. The mathematics of the “standard” QM isn’t any worse than, let’s say, electromagnetism. Yet, to many people, especially non-physicists, QM presents a very daunting effort to understand.
I strongly believe that it all comes down to how we understand things and how we expand our knowledge. Typically, when we teach students new things, what we do is build upon their existing understanding. We hope that a student already has a foundation of knowledge in certain areas, such as basic mathematics, etc. so that we can use that to teach them about forces, motion, energy, and other fun stuff in intro physics. Then, after they understand the basic ideas, we show them the same thing, but with more complications added to it.
The same thing occurs when we try to help a student doing a homework problem. We always try to ask what the student knows already, such as the basic principle being tested in that question. Does the student know where to start? What about the most general form of the equation that is relevant to the problem? Once we know a starting point, we then build on that to tackle that problem.
The common thread in both cases is that there exists a STARTING point as a reference foundation on which, other “new” stuff is built upon. We learn new and unknown subjects based on what we have already understood. This is something crucial to keep in mind because, in the study of QM, this part is missing! I am certain that for most non-physicists, this is the most common reason why QM is puzzling, and why quacks and other people who are trying to use QM in other areas such as “metaphysics” or mysticism, are using it in a completely hilarious fashion.
The Disconnect
There is a complete disconnect between our “existing” understanding of the universe based on classical understanding and QM. There is nothing about our understanding of classical mechanics that we can build on to understand QM. We use identical words such as particle, wave, spin, energy, position, momentum, etc… but in QM, they attain a very different nature. You can’t explain these using existing classical concepts. The line between these two is not continuous, at least, not as of now. How does one use the classical idea of a “spin” to explain a spin 1/2 particle in which one only regains the identical symmetry upon two complete revolutions? We simply have to accept that we use the same word but to ONLY mean that it produces a magnetic moment. It has nothing to do with anything that’s spinning classically. We can’t build the understanding of the QM spin using the existing classical spin that we have already understood.
Now interestingly enough, the MATHEMATICAL FORMULATION of QM is quite familiar! The time-dependent Schrodinger equation has the same structure as a standard wave equation. We call the energy operator the Hamiltonian not for nothing since it looks very familiar with the Hamiltonian approach to classical mechanics. The matrix formulation isn’t anything new. What this means is that while the conceptual foundation of QM is completely disconnected from our traditional conceptual understanding, the mathematical formulation of QM completely follows from our existing understanding! Mathematically, there is no discontinuity. We build the formalism of QM based on our existing understanding!
Mathematical formulation
This is why, in previous threads in PF, I disagree that we should teach students the concepts of QM FIRST, rather than the mathematical formulation straightaway. There is nothing to “build on” in terms of conceptual understanding. We end up telling the students what they are out of thin air. The postulates of QM did not come out of our classical understanding of our world. Instead, mathematical formalism is the only thing that saves us from dangling in mid-air. It is the only thing on which our existing understanding can be built.
What this implies is that, if one lacks the understanding of the mathematical formalism of QM, one hasn’t understood QM at all! One ends up with all these weird, unexplained, unfamiliar, and frankly, rather strange ideas on how the world works. These conceptual descriptions of QM may even appear “mystical”. It is not surprising that such connections are being made between QM and various forms of mysticism. One lacks any connection with the existing reality that one has understood. So somehow, since QM can do this, it seems as if it’s a license to simply invent stuff..
The mathematical formalism of QM is what defines the QM description. The “conceptual description” is secondary, and is only present because we desire some physical description based on what we already have classically. It is why people can disagree on the interpretation of QM, yet they all agree on the source, the mathematical formalism of QM.
QM as musical notes
This, however, does not mean that QM is nothing more than “just mathematics”. This is no more true than saying the musical notes on a sheet of paper are just scribbles. The notes are not the important object. Rather, it is the sound that it represents that’s the main point. The musical notes are simply a means to convey that point clearly and unambiguously. Similarly, the mathematics that is inherent in QM and in all of physics, is a means to convey an idea or principle. It is a form of communication, and so far it is the ONLY form of communication accurate and unambiguous enough to describe our universe. It reflects completely our understanding of phenomena. So a mathematical formulation isn’t “just math”.
You cannot use your existing understanding of the universe to try to understand the various concepts of QM. There is a discontinuity between the two. It is only via the mathematical continuity of the description can there is a smooth transition to build upon. Without this, QM will not make “sense”.
PhD Physics
Accelerator physics, photocathodes, field-enhancement. tunneling spectroscopy, superconductivity
From my own learning experience I'm with the first paper. For me it was quite difficult to "unlearn" the Bohr-Sommerfeld model, which is mathematically appealing and to a certain extent intuitive although it's totally inconsistent in itself since if the electron was moving in elliptic orbits around the nucleus (or even more accurately both moving around their center of mass), the atom should radiate and be instable; in the BS-model it's simply stated that there are "allowed orbits", where this doesn't happen, but it's still inconsistent with classical physics upon which the model rests.
Considering the amount of answers, the question: Why is quantum so difficult, seems pretty difficult itself.If one focuses on teaching of qf, the idea that one should or should not start with the old quantum theory is not so evident.In 1992 Fischler & Lichtfelt (http://dx.doi.org/10.1080/0950069980200905) concluded that teaching the Bohr-model was a bad idea.In 2008 McKagan (http://dx.doi.org/10.1103/PhysRevSTPER.4.010103) concludes that the Bohr-model can be used, if it is done right (and mcKagan tells how).
[QUOTE="Greg Bernhardt, post: 4821250, member: 1"]Author: ZapperZhttps://www.physicsforums.com/insights/quantum-mechanics-difficult/“well, Quantum Physics is hard to grasp because it lacks the sound base; the atom model is incorrect; you don't know why the so called electrons don't crash into the nucleus, since it continually radiates energy; you don't know what happen in the double split experiment. Particles don't exist! The Great Unified Field exists, with different emanations, but always the same force!! You should begin to learn the correct model of atom; its 99.9 empy space is a nonsense!
"same structure as a standard wave equation": hmm… really? It always looked more like the difussion equation.
[QUOTE="vanhees71, post: 5554468, member: 260864"]That's science, not philosophy!”Science and philosophy are not mutually exclusive. If a scientific method can answer a deep question interesting also to philosophers, then it's also philosophy. Science is defined by a certain objective method, but philosophy is not defined by negation of that method. Philosophy is defined by the type of questions it asks. The intersection between scientific method and philosophic questions is not zero.
[QUOTE="atyy, post: 5553933, member: 123698"]Why should we even care about a local hidden-variable theory? That is philosophy, since hidden variables are motivated by reality. If you don't like philosophy, Bell's inequality is not about hidden variables.”I don't understand this statement. To test a theory Bell thought about a class of alternative theories, namely a deterministic local theories, derived a consequence (Bell's inequality) which is violated by QT. Thus you can test it with experiments in the lab (nowadays there are many of them, starting with the pioneering work by Aspect). That's science, not philosophy!
[QUOTE="Demystifier, post: 5553913, member: 61953"]The right question is this. Without using a philosophic question as a motivation, can you explain why Bell inequalities are important and interesting?”I consider the question of how to understand the indeterminism of quantum theory for both part of physics and philosophy. It's a very fundamental question whether nature is deterministic or not and thus it's part of philosophy as well as the natural sciences. The merit of Bell's work, in my opinion, was to make it a clearly answerable question of the natural sciences.
[QUOTE="lavinia, post: 5554005, member: 243745"]classical phase space e.g. a smooth manifold with some non-trivial topology, I would love to know how to take linear combinations of points on the manifold.”Classical phase space has very often a trivial topology, so that it is ##C^{3n}## in a very natural coordinatization. Taking linear combinations is straightforward.On the other hand, thinking about superpositions rather than states is not needed in most of quantum theory, as it is not needed in most of classical theory. The real actors are the density operators resp. density functions, which encode the states once the problems get somewhat realistic (i.e., include the dissipative effect of the environment resp. friction). One cannot do this with superpositions.
[QUOTE="A. Neumaier, post: 5553113, member: 293806"]No. Using vectors, matrices and functions is the natural way of describing any (mathematical or physical) system with a large number of degrees of freedom. For example, nonlinear manifolds are represented in terms of vectors when doing actual computations.The classical phase space for a particle in an external field is also a vector space ##R^6## (or ##C^3## if you combine position and momentum to a complex position ##z=q+ikappa p## with a suitable constant ##kappa##). And, unlike in the quantum case, one can form linear combinations of classical states.Thus the problem with quantum mechnaics cannot lie in the use of vectors and their linear combinations. In the quantum case you just have many more states than classically, which is no surprise since it describes systems form a more microscopic (i.e., much more detailed) point of view.What one must get used to is not the superpositions but the meaning attached to a pure quantum state, since this meaning has no classical analogue.However, for mixed states (and almost all states in Nature are mixed when properly modelled), quantum mechanics is very similar to classical mechanics in all respects, as you could see from my book. (Note that the math in my book is no more difficult than the math you know already, but the intuition conveyed with it is quite different from what you can get from a textbook.)Thus the difficulty is not intrinsic to quantum mechanics. It is created artificially by following the historically earlier road of Schroedinger rather than the later statistical road of von Neumann.”I frankly think you are misinterpreting what I said. That this is more linguistic than anything else. But if not I suggest that you get in touch with Leonard Susskind and explain to him what he has been getting wrong all these years.Superposition to me is linear combination in a complex vector space. Then I guess if you want to be super accurate you then have to normalize. In classical phase space e.g. a smooth manifold with some non-tricvial topology, I would love to know how to take linear combinations of points on the manifold.
[QUOTE="vanhees71, post: 5553906, member: 260864"]But Bell's inequality is a perfect example of the opposite! By giving a clear definition of what's meant by a deterministic local hidden-variable theory he derived is famous inequality which is violated in quantum theory, and thus it became a question of science which could be empirically tested. So it's completely (and in my opinion only) understandable from science, and it's not even too complicated. It can be explained in QM1 easily, as are the experiments like the Aspect experiment with polarization-entangled photons. There's no philosophy.Of course, Bell's work was strongly motivated by philosophical issues and all the hype about the EPR paper, but the breakthrough was that this work brought these vague philosophical questions into the realm of objectively testable observational facts about nature!”Why should we even care about a local hidden-variable theory? That is philosophy, since hidden variables are motivated by reality. If you don't like philosophy, Bell's inequality is not about hidden variables.
[QUOTE="Demystifier, post: 5553908, member: 61953"]That's quite an unusual definition of philosophy.”I wouldn't try and force the mathematical rigor of a "definition" on more of a working description. But it is based on defining science as testable in repeatable experiments. Historically, the experiments testing the "less philosophical" parts came much earlier.My point is that an awful lot of experimental results can be explained by the aspects of QM that were historically used for decades to describe experimental results without the aspects that have been more debated on the philosophical side but not as important historically in the early decades of QM.You seem to be wanting something more fundamental, I am offering something more practical and pedagogical – what to emphasize in undergrad QM courses.I've published a lot of papers in atomic physics and QM without much use of the philosophical side, and so have a lot of atomic, molecular, and optical physicists.
[QUOTE="vanhees71, post: 5553906, member: 260864"]But Bell's inequality is a perfect example of the opposite! By giving a clear definition of what's meant by a deterministic local hidden-variable theory he derived is famous inequality which is violated in quantum theory, and thus it became a question of science which could be empirically tested. So it's completely (and in my opinion only) understandable from science, and it's not even too complicated. It can be explained in QM1 easily, as are the experiments like the Aspect experiment with polarization-entangled photons. There's no philosophy.Of course, Bell's work was strongly motivated by philosophical issues and all the hype about the EPR paper, but the breakthrough was that this work brought these vague philosophical questions into the realm of objectively testable observational facts about nature!”The right question is this. Without using a philosophic question as a motivation, can you explain why Bell inequalities are important and interesting?
[QUOTE="Dr. Courtney, post: 5553905, member: 117790"]I prefer to describe the "philosophical bits" by exclusion rather than enumeration. Is it a part of QM that is essential for understanding the periodic table or for computing the spectra of atoms or molecules?”That's quite an unusual definition of philosophy.
[QUOTE="Demystifier, post: 5553892, member: 61953"]Well, you are certainly right that quantum optics can be done without philosophy. Nevertheless, if you think about typical quantum-optics experiments such as those that involve violation of Bell inequalities, weak measurements, or delayed choice quantum erasers, philosophic questions occur more naturally than in other branches of quantum physics. Yes, you can resist thinking about philosophical aspects of such experiments if you have a strong character, but temptation is quite strong. Some experimentalists in that field even call it – experimental metaphysics.”But Bell's inequality is a perfect example of the opposite! By giving a clear definition of what's meant by a deterministic local hidden-variable theory he derived is famous inequality which is violated in quantum theory, and thus it became a question of science which could be empirically tested. So it's completely (and in my opinion only) understandable from science, and it's not even too complicated. It can be explained in QM1 easily, as are the experiments like the Aspect experiment with polarization-entangled photons. There's no philosophy. Of course, Bell's work was strongly motivated by philosophical issues and all the hype about the EPR paper, but the breakthrough was that this work brought these vague philosophical questions into the realm of objectively testable observational facts about nature!
[QUOTE="Demystifier, post: 5553862, member: 61953"]Perhaps by philosophical bits you don't mean the same thing as most of us do? Or perhaps you are working in a field such as quantum optics where quantum bits are more important than in most other branches of quantum physics? To check this out, can you name a few philosophical bits which you have in mind?”I prefer to describe the "philosophical bits" by exclusion rather than enumeration. Is it a part of QM that is essential for understanding the periodic table or for computing the spectra of atoms or molecules? If not, I tend to regard it more on the philosophical side that likely need not receive much emphasis in the first two semesters of undergrad QM. I don't mind a section in the book or some brief classroom discussions to set the stage for what may be learned in more detail later, but I would regard it as out of balance if more than a few percent of the points in a 2 semester undergrad sequence depended on the philosophical material.
[QUOTE="vanhees71, post: 5553712, member: 260864"]Can you explain the meaning of these enigmatic statements? Also in condensed matter physics the highest symmetry is reached at (asymptotic) high energies, where matter becomes an ideal gas of elementary particles (of quite probably yet unknown fundamental degrees of freedom), but that cannot be what the condensed-matter physicist wanted to express.”Probably best to hear him explain it himself, rather than my garbling it. He says it right at the start: http://online.kitp.ucsb.edu/online/adscmt_m09/xu/.
[QUOTE="vanhees71, post: 5553884, member: 260864"]Hm, where do you need philosophy for quantum optics? For me the fascinating thing about quantum optics is that you just do the very fundamental experiments discussed as thought experiments only in older textbooks. There's no need for philosophy at all but quite basic manipulations of bras and kets to predict the outcome of measurements.”Well, you are certainly right that quantum optics can be done without philosophy. Nevertheless, if you think about typical quantum-optics experiments such as those that involve violation of Bell inequalities, weak measurements, or delayed choice quantum erasers, philosophic questions occur more naturally than in other branches of quantum physics.
[QUOTE="Demystifier, post: 5553862, member: 61953"]Perhaps by philosophical bits you don't mean the same thing as most of us do? Or perhaps you are working in a field such as quantum optics where quantum bits are more important than in most other branches of quantum physics? To check this out, can you name a few philosophical bits which you have in mind?”Hm, where do you need philosophy for quantum optics? For me the fascinating thing about quantum optics is that you just do the very fundamental experiments discussed as thought experiments only in older textbooks. There's no need for philosophy at all but quite basic manipulations of bras and kets to predict the outcome of measurements.
I took 11 courses (grad and undergrad) in QM and QFT (to include solid state, nuclear, and particle physics) using several different texts and different approaches (historical, Dirac notation, Schrodinger eqn, etc.). How you decide to teach quantum physics depends on what problems you want to solve. If you want atomic and molecular energy levels, you don't need to worry about the measurement problem or violations of the Bell inequality, for example. As I said I in post #22 of this thread, I teach QM based on foundations of physics, so I only teach the weird stuff. I have since added the following two problems to my course https://www.physicsforums.com/insights/weak-values-part-1-asking-photons/ (based on experiment published in Phys. Rev. Lett. in 2013) and https://www.physicsforums.com/insights/weak-values-part-2-quantum-cheshire-cat-experiment/ (based on experiment published in Nature Comm in 2014). These analyses would be worthless in a course on chemical physics, for example.
[QUOTE="Dr. Courtney, post: 5553837, member: 117790"]At some point, original research (PhD level and beyond) in QM likely requires wrestling with the philosophical bits”Perhaps by philosophical bits you don't mean the same thing as most of us do? Or perhaps you are working in a field such as quantum optics where quantum bits are more important than in most other branches of quantum physics? To check this out, can you name a few philosophical bits which you have in mind?
[QUOTE="vanhees71, post: 5553656, member: 260864"]That's a very healthy approach. Too much philosophy hinders the understanding of science ;-)).”I certainly agree at the level of late high school and early undergraduate, which is what the original Insight article seemed to be discussing (the absence of a starting point). I proffered my experience that a great high school chemistry course that focused on the periodic table provided a pretty good starting point. At some point, original research (PhD level and beyond) in QM likely requires wrestling with the philosophical bits, but an awful lot of the applications of QM (intro through a lot of PhD and beyond atomic and molecular physics) can be accomplished in a satisfactory manner without wrestling with the philosophy of it.A lot of confusion arises because the philosophical bits get introduced too early, that is before there is a sound foundation of the parts that are more immediately experimentally testable and that serve as the necessary basis for the periodic table and atomic physics. For me, a good two semester undergrad course in QM is likely the necessary starting point to really understand the philosophical aspects.
[QUOTE="Demystifier, post: 5553696, member: 61953"]How can unitary group representations help to understand the measurement problem?”I didn't claim it would. But as you probably know from the discussions here, I don't think that there is a measurement problem in quantum mechanics. At least not one deeper than the corresponding classical measurement problem, which is usually taken to be absent. Everyone who measures something knows how to apply the theory to match experiments, and that's all needed.
[QUOTE="Demystifier, post: 5553713, member: 61953"]Interesting, you are the first person I know disappointed with the Zee's QFT in a Nutshell.But that's OK. I guess I am one or rare persons who does not like Weinberg's QFT2. (Even though I do like his QFT1.)”I was particularly disappointed about QFT in a Nutshell. It's just too superficial. Just not mentioning the subtleties properly doesn't mean they are not there! Weinberg QT of Fields is, in my opinion, the best book on relativistic QFT for experts. It's not so good to start with. My favorite intro textbook for QFT is M. Schwartz, Quantum Field Theory and the Standard Model.
[QUOTE="dextercioby, post: 5553722, member: 1064"]I think you are not addressing the purpose of the book by comparing it to his former 2. This is a book about applied mathematics, not a book about physics, thus you need to judge it from a different angle, i.e. not how well and full of insightful facts it teaches physics, but how well it teaches mathematics to (present/future to be) physicists. Therefore, we have some questions to ask:1. Is this book necessary in the context of the available literature on this subject?2. Is this book too abstract ?3. Does it contain new facts/discoveries from mathematics and physics compared to, let's say the much older books by Barut & Raczka and Cornwell ?4. Is the exposition clear enough, or is itsimply a tough reading”My answers:1. No.2. No.3. I haven't find any.4. Perhaps it's no so tough, but it's boring. What I was hoping for is to see group theory from a new angle. I hoped that his book might change the way I think about groups and representations. That didn't happen, and that's why I was disappointed.
[QUOTE="Demystifier, post: 5553700, member: 61953"]Speaking of symmetries, groups and representations, I was very disappointed with the new book by Zeehttps://www.amazon.com/Group-Theory-Nutshell-Physicists-Zee/dp/0691162697In his previous books (Quantum Field Theory and Gravity) I have found a lot of new deep original insights, but that didn't happen with his last book on Group Theory. Perhaps it tells more about me than about the book, but I would certainly like to see what others think.”I think you are not addressing the purpose of the book by comparing it to his former 2. This is a book about applied mathematics, not a book about physics, thus you need to judge it from a different angle, i.e. not how well and full of insightful facts it teaches physics, but how well it teaches mathematics to (present/future to be) physicists. Therefore, we have some questions to ask:1. Is this book necessary in the context of the available literature on this subject?2. Is this book too abstract ?3. Does it contain new facts/discoveries from mathematics and physics compared to, let's say the much older books by Barut & Raczka and Cornwell ?4. Is the exposition clear enough, or is itsimply a tough reading
[QUOTE="vanhees71, post: 5553712, member: 260864"]Can you explain the meaning of these enigmatic statements? Also in condensed matter physics the highest symmetry is reached at (asymptotic) high energies, where matter becomes an ideal gas of elementary particles (of quite probably yet unknown fundamental degrees of freedom), but that cannot be what the condensed-matter physicist wanted to express.”I would put it this way. Both communities use symmetry to understand the most interesting part of their branch of physics. In high-energy physics the most interesting question is what happens at the highest energies, for this is were new particles are expected to appear. By contrast, in condensed matter the high-energy level (atoms) is well understood and quite boring, while the most interesting phenomena, including new quasi-particles, happen at low energies (large scales).In general, of course, old symmetries may disappear and new ones appear in both directions in the energy-scale, in both particle physics and cond mat physics. But it has to do with psychology in the two communities, with what is considered "interesting" and "important".
[QUOTE="vanhees71, post: 5553709, member: 260864"]I was disappointed by all books by Zee”Interesting, you are the first person I know disappointed with the Zee's QFT in a Nutshell.But that's OK. I guess I am one or rare persons who does not like Weinberg's QFT2. (Even though I do like his QFT1.)
[QUOTE="atyy, post: 5553699, member: 123698"]I have heard one condensed matter physicist (Cenke Xu) explain it this way:In HEP, the higher the energy, the more the symmetry.In condensed matter, the lower the energy, the more the symmetry.”Can you explain the meaning of these enigmatic statements? Also in condensed matter physics the highest symmetry is reached at (asymptotic) high energies, where matter becomes an ideal gas of elementary particles (of quite probably yet unknown fundamental degrees of freedom), but that cannot be what the condensed-matter physicist wanted to express.
[QUOTE="Demystifier, post: 5553700, member: 61953"]Speaking of symmetries, groups and representations, I was very disappointed with the new book by Zeehttps://www.amazon.com/Group-Theory-Nutshell-Physicists-Zee/dp/0691162697In his previous books (Quantum Field Theory and Gravity) I have found a lot of new deep original insights, but that didn't happen with his last book on Group Theory. Perhaps it tells more about me than about the book, but I would certainly like to see what others think.”Hm, I was disappointed by all books by Zee, I've had a look at. I've not looked at the newest one yet. So I can't say, whether I like it or not.
[QUOTE="atyy, post: 5553699, member: 123698"]I have heard one condensed matter physicist (Cenke Xu) explain it this way:In HEP, the higher the energy, the more the symmetry.In condensed matter, the lower the energy, the more the symmetry.”Exactly! :smile:Now vanhees71 will find important counterexamples, but that will not change the fact that the above statement greatly summarizes the general spirit in the two communities.
[QUOTE="Demystifier, post: 5553683, member: 61953"]In their daily work, mathematicians don't start from axioms any more than physicists start from experimental facts. And you certainly know that physicists, in their daily work, often do not really start from experimental facts.”The great thing about Bourbaki is that they started from experimental facts, just like quantum mechanics! They said, well, we know what it means by two symbols on the page being the "same", even though it all probability, two different "ψ"s are almost certainly not the same down to the last atom.
Speaking of symmetries, groups and representations, I was very disappointed with the new book by Zeehttps://www.amazon.com/Group-Theory-Nutshell-Physicists-Zee/dp/0691162697In his previous books (Quantum Field Theory and Gravity) I have found a lot of new deep original insights, but that didn't happen with his last book on Group Theory. Perhaps it tells more about me than about the book, but I would certainly like to see what others think.
[QUOTE="Demystifier, post: 5553687, member: 61953"]Indeed, you just confirmed my claim that high-energy spirit of QFT differs from condensed-matter spirit of QFT. Condensed-matter physicists rarely make such a strong emphasis on symmetry. They use symmetry in practice, but they rarely base their intuition on it.”I have heard one condensed matter physicist (Cenke Xu) explain it this way:In HEP, the higher the energy, the more the symmetry.In condensed matter, the lower the energy, the more the symmetry.
[QUOTE="A. Neumaier, post: 5553691, member: 293806"]Whereas the older I get the more I realize that properly understanding quantum mechanics means properly understanding unitary group representations. Whether spin and Stern-Gerlach experiments, or the Schroedinger equation, or the harmonic oscillator, or the spectrum of the hydrogen atom, or multiparticle scattering, or Hartree-Fock theory, or coupled cluster expansions, or coherent states, or the fractional Hall effect, or equilibrium statistical mechanics, or dissipative quantum mechanics, or free (condensed matter or relativistic) quantum fields, or QED and the standard model, or exactly solvable models, or nonperturbative quantum field theory, or conformal field theory, unitary group representations always give the best structural insights into what really matters. It is a great organizational principle.”How can unitary group representations help to understand the measurement problem?
[QUOTE="vanhees71, post: 5553690, member: 260864"]Well, I don't know condensed-matter physics as well as high-energy particle/nuclear physics, but I don't think that symmetries are less important in condensed-matter physics than in HEP. Also I think symmetries is a common thing for all subtopics of physics, and one should not specialize too early. As parts of the general theory course (and not lecture aiming at specialization) QM1 and QM 2 should provide the theoretical methodology for a broad range of "users", including experimental physicists. So, I think, that no matter in which field of research you specialize in your research, symmetries seems to be a good basis to understand theoretical physics (not only QT but also classical physics).”Well, every theoretical physicist needs symmetry at the intuitive level, e.g. to develop instinct of using spherical coordinates whenever the spherical symmetry is obvious. But I don't think that use of advanced group theory is always necessary.
[QUOTE="Demystifier, post: 5553687, member: 61953"]When I was young (late high school and early college days), I thought that one of the keys for understanding the deepest secrets of nature is to understand the meaning of symbols such as SU(2) and SU(3). Needless to say, I don't think that anymore.”Whereas the older I get the more I realize that properly understanding quantum mechanics means properly understanding unitary group representations. Whether spin and Stern-Gerlach experiments, or the Schroedinger equation, or the harmonic oscillator, or the spectrum of the hydrogen atom, or multiparticle scattering, or Hartree-Fock theory, or coupled cluster expansions, or coherent states, or the fractional Hall effect, or equilibrium statistical mechanics, or dissipative quantum mechanics, or free (condensed matter or relativistic) quantum fields, or QED and the standard model, or exactly solvable models, or nonperturbative quantum field theory, or conformal field theory, unitary group representations always give the best structural insights into what really matters. It is a great organizational principle.
Well, I don't know condensed-matter physics as well as high-energy particle/nuclear physics, but I don't think that symmetries are less important in condensed-matter physics than in HEP. Also I think symmetries is a common thing for all subtopics of physics, and one should not specialize too early. As parts of the general theory course (and not lecture aiming at specialization) QM1 and QM 2 should provide the theoretical methodology for a broad range of "users", including experimental physicists. So, I think, that no matter in which field of research you specialize in your research, symmetries seems to be a good basis to understand theoretical physics (not only QT but also classical physics).
[QUOTE="vanhees71, post: 5552094, member: 260864"]I've once taught QM 2, where I emphasized the role of symmetries, which is in my opinion the right intuition for modern physics anyway.”Only if, by "modern" physics, you mean high-energy physics. But most physicists would not agree with such a definition of modern physics. Indeed, you just confirmed my claim that high-energy spirit of QFT differs from condensed-matter spirit of QFT. Condensed-matter physicists rarely make such a strong emphasis on symmetry. They use symmetry in practice, but they rarely base their intuition on it.When I was young (late high school and early college days), I thought that one of the keys for understanding the deepest secrets of nature is to understand the meaning of symbols such as SU(2) and SU(3). Needless to say, I don't think that anymore.
[QUOTE="vanhees71, post: 5552095, member: 260864"]It should, however, be clear that in physics the axioms are not the starting point but the goal of theory building. In math you start with the axioms, because it defines your "universe". In physics the "universe" is imposed on us by nature, and we have to figure out the "axioms" for describing it. It's amazing enought, how far you come with this axiomatic approach to physics, although admittedly the most successful theory we have has no rigid foundation in math, relativistic QFT and the Standard Model of HEP.”In their daily work, mathematicians don't start from axioms any more than physicists start from experimental facts. And you certainly know that physicists, in their daily work, often do not really start from experimental facts.
[QUOTE="vanhees71, post: 5553656, member: 260864"]Too much philosophy hinders the understanding of science”Too much abstract formalism (Bourbaki) also hinders the understanding of science.Too much equations without verbal explanations also hinders the understanding of science.Too much verbal explanations without equations also hinders the understanding of science.Too much of numerical computation also hinders the understanding of science.Too much of general equations without putting numbers in also hinders the understanding of science.Too much theory without experiments also hinders the understanding of science.Too much experiments without theory also hinders the understanding of science.Even too much science without looking at it from the outside (meta-science) hinders the understanding of science.The point is to find a healthy dose of all that. And to realize that the correct dose depends on the individual.
[QUOTE="vanhees71, post: 5553656, member: 260864"]That's a very healthy approach. Too much philosophy hinders the understanding of science ;-)).”I think this comment comes from a misunderstanding. Let me explain my point with an example:There are physicists who work on lattice QCD simulations and other people who work on the problem of confinement. Some of the results from those simulations may be useful for developing models for confinement. But imagine there is a particular result that is still controversial among lattice QCD people. Should confinement people accept or reject that result? Should they participate in a serious discussion about that result although they don't know as much as lattice QCD people about the subject?I think the answers to the above questions are clear. The confinement people just wait until the lattice QCD people come to a consensus and until then they just ignore that result. Of course, they may accept it or deny it or try to contribute to the discussion but because of the simple fact that they don't know enough about the subject, they may get confused or have uninformed ideas.The field of "Foundations of QM" is just another field of research in physics with its own community of experts. The fact that people not in this community may get confused by the subject or have uninformed ideas doesn't mean that the subject is flawed, it just means that people should focus on what they know!
[QUOTE="Dr. Courtney, post: 5553236, member: 117790"]I had a great high school chemistry course which served as the underpinnings for QM when I got there in college. My teacher was an old school former nun who made us memorize the periodic table and the orbitals for many of the atoms. We understood how the orbitals corresponded to the columns with the "filled shells" business and all of that.Somewhere between my 2nd semester general physics course and my 3rd semester modern physics course, I realized that QM held the promise of explaining the "why" of the periodic table as well as explaining an awful lot about atomic and molecular spectra, which fascinated me since I was in elementary school.The "electron cloud" and the probabilistic bit never bothered me. I learned to focus on the parts that could be measured in experiments and learned not to worry about bits which did not make predictions.”That's a very healthy approach. Too much philosophy hinders the understanding of science ;-)).
[QUOTE="lavinia, post: 5553097, member: 243745"]Yeah I know that and I am not sure why you are correcting this since it is a given on the first day of a Quantum Mechanics course that you have to normalize . Yeah normalize so you really have a projective space. But this is all obvious. The important thing is the difference in the nature of state space and that is intimately a consequence of linear combination. I suppose if you only want physically equivalent sates you would form the complex projective space of 2 planes in the Hilbert space.”Sure, the superposition principle is one very important feature of quantum theory, i.e., (except when there are superselection rules) any Hilbert space vector can be representant of a pure state (which indeed is only determined up to a multiplicative constant and thus is in fact rerpesented by the entire ray in Hilbert space or, equivalently by the corresponding projection operator of any normalized representant of that ray) and thus also the superpositions of any such vectors.On the other hand it is very important to keep in mind that the true representants of the states are the statistical operators (or in case of pure states rays) since otherwise you'd have a hard time to define non-relativistic quantum theory: it's not the unitary reprsentations of the Galileo group but the ray representations, and thus you have more freedom, i.e., you can represent any central extension of the covering group, and thus you can introduce the mass as a non-trivial central charge of the Galilei algebra and use SU(2) to represent rotations. The former is the only way to make a physically sensible dynamics possible and the latter enables to describe half-integer spins, without which the description of the matter around us wouldn't be possible either.
I had a great high school chemistry course which served as the underpinnings for QM when I got there in college. My teacher was an old school former nun who made us memorize the periodic table and the orbitals for many of the atoms. We understood how the orbitals corresponded to the columns with the "filled shells" business and all of that.Somewhere between my 2nd semester general physics course and my 3rd semester modern physics course, I realized that QM held the promise of explaining the "why" of the periodic table as well as explaining an awful lot about atomic and molecular spectra, which fascinated me since I was in elementary school.The "electron cloud" and the probabilistic bit never bothered me. I learned to focus on the parts that could be measured in experiments and learned not to worry about bits which did not make predictions.
[QUOTE="lavinia, post: 5553097, member: 243745"]The important thing is the difference in the nature of state space and that is intimately a consequence of linear combination.”No. Using vectors, matrices and functions is the natural way of describing any (mathematical or physical) system with a large number of degrees of freedom. For example, nonlinear manifolds are represented in terms of vectors when doing actual computations.The classical phase space for a particle in an external field is also a vector space ##R^6## (or ##C^3## if you combine position and momentum to a complex position ##z=q+ikappa p## with a suitable constant ##kappa##). And, unlike in the quantum case, one can form linear combinations of classical states.Thus the problem with quantum mechnaics cannot lie in the use of vectors and their linear combinations. In the quantum case you just have many more states than classically, which is no surprise since it describes systems form a more microscopic (i.e., much more detailed) point of view.What one must get used to is not the superpositions but the meaning attached to a pure quantum state, since this meaning has no classical analogue.However, for mixed states (and almost all states in Nature are mixed when properly modelled), quantum mechanics is very similar to classical mechanics in all respects, as you could see from my book. (Note that the math in my book is no more difficult than the math you know already, but the intuition conveyed with it is quite different from what you can get from a textbook.)Thus the difficulty is not intrinsic to quantum mechnaics but created artificially by following the historically earlier road of Schroedinger rather than the later statistical road of von Neumann.
[QUOTE="A. Neumaier, post: 5552395, member: 293806"]Linear combinations of state vectors give new state vectors, which is as it should be in a vector space, no problem for the intuition.But state vectors are not states – physical states are normalized state vectors determined only up to a phase – i.e., rays in the Hilbert space, or points in the projective space.It makes no sense to take linear combinations of states. Thus what is a difficulty for your intuition is based on a misunderstanding.”Yeah I know that and I am not sure why you are correcting this since it is a given on the first day of a Quantum Mechanics course that you have to normalize . Yeah normalize so you really have a projective space. But this is all obvious. The important thing is the difference in the nature of state space and that is intimately a consequence of linear combination. I suppose if you only want physically equivalent sates you would form the complex projective space of 2 planes in the Hilbert space.
[QUOTE="Stephen Tashi, post: 5552346, member: 186655"]or is calling different things the same thing the essential idea of symmetry?”Not quite. It is the essential idea of equivalence relations. Without calling different things the same we cannot form a single concept….
[QUOTE="lavinia, post: 5552350, member: 243745"]that linear combinations of states to get new states is difficult to intuit.”Linear combinations of state vectors give new state vectors, which is as it should be in a vector space, no problem for the intuition. But state vectors are not states – physical states are normalized state vectors determined only up to a phase – i.e., rays in the Hilbert space, or points in the projective space. It makes no sense to take linear combinations of states. Thus what is a difficulty for your intuition is based on a misunderstanding.
[QUOTE="A. Neumaier, post: 5552084, member: 293806"]No, it is a complex projective space formed by the rays in the Hilbert space. Geometrically it is again a symplectic manifold (though often infinite-dimensional). Note that rays cannot be superimposed, only state vectors.You might like my online book Classical and Quantum Mechanics via Lie algebras, where I present quantum mechanics in a mathematician-friendly way.”Thanks but the math isn't what worries me. Leonard Susskind and Richard Feynmann teach it explicitly as a complex vector space. That is what I am learning from. Your book sounds too mathematical for a first trip around the block.My point was that linear combinations of states to get new states is difficult to intuit.
[QUOTE="bhobba, post: 5552237, member: 366323"]So much so I believe, and this is simply conjecture on my part, some truly great and strikingly simple symmetry, lies at the heart of all physics. “Perhaps an even more fundamental concept than symmetry is that we agree to call different things the same – or is calling different things the same thing the essential idea of symmetry?The idea of repeating the "same" experiment is, at face value, self-contradictory because if experiment #2 was precisely a repeat of experiment #1 then it wouldn't be a different experiment, it would just be another label for experiment #1. So when an experiment is repeated it only certain aspects of it are repeated. The "unessential" aspects of the experiment tend to be ignored, but if they didn't exist then we wouldn't have a repeated experiment. Any particular unessential aspect of an experiment (e.g. what color t-shirt the lab technician wore) is not critical, but it is critical that there be some unessential aspect that distinguishes two repeated experiments. The concept of physical probability involves the convention that we will define "an event" in a way that actually denotes a collection of different events. The mathematical model of repeated independent trials as some sort of tensoring together of copies of the same sample space doesn't quite capture the requirement that a "repetition" of an experiment requires that something be different when an experiment is repeated.
[QUOTE="vanhees71, post: 5552094, member: 260864"]I have the opposite impression. Modern mathematics textbooks are very often in an awful "Bourbaki style". It's all well ordered in terms of axioms, definitions, lemmas, and theorems with proofs, but no intuition whatsoever. I think in math the intuition comes from the applications, and physics is of course full of applications of interesting math issues (from analytical geometry via calculus to group theory and topology nearly everything interesting in math finds applications in physics).the absolute measure of entropy, etc. are solved either).”o0)o0)o0)o0)o0)o0)o0)o0)o0)o0)How true that is.But it requires its own thread, not in the QM sub-forum but in one of the math ones.It should look at the history of Rigged Hilbert spaces and its now central role in White Noise Theory that has far reaching applications to QM.[QUOTE="vanhees71, post: 5552094, member: 260864"]I've once taught QM 2, where I emphasized the role of symmetries, which is in my opinion the right intuition for modern physics anyway.”Again – :smile::smile::smile::smile::smile::smile::smile::smile::smile::smile::smile::smile::smile:To start this journey, and its the most important insight of 20th century physics IMHO, see Landau who in his terse no BS style gets to the heart of the matter:https://www.amazon.com/Mechanics-Third-Course-Theoretical-Physics/dp/0750628960I could explain it, but IMHO its important the OP, and anyone reading this, needs to nut it out for themselves, which is the only way for true understanding. Its best done in the classical mechanics section. Once that is understood that then rest falls rather quickly into place via Noerther's beautiful theorem:https://www.amazon.com/Noethers-Wonderful-Theorem-Dwight-Neuenschwander/dp/0801896940It left Einstein basically – well his own words are besthttp://cwp.library.ucla.edu/articles/noether.asg/noether.htmlhttps://www.washingtonpost.com/news/comic-riffs/wp/2015/03/23/emmy-noether-google-doodle-why-einstein-called-her-a-creative-mathematical-genius/http://www.math.wichita.edu/history/women/noether.html'Einstein also wrote in a letter to Professor David Hilbert that Emmy Noether display "penetrating mathematical thinking."Its the type of thing that makes me laugh about philosophy. I never ever see philosophy articles about it, they get into all sorts of irrelevant tangents that skirt around the real issue, but it's central, vital, and one of the truly great insights of modern physics. So much so I believe, and this is simply conjecture on my part, some truly great and strikingly simple symmetry, lies at the heart of all physics. But unraveling that first requires figuring out what it is a symmetry in. In classical physics we know its symmetries in the Lagrangian which depends on QM. In QFT we know its symmetries again in Field Equations, but they can also be expressed in terms of Lagrangians. That however is not explainable in terms of QM because QFT is what explains QM. To me this is the deep deep mystery who solution will illuminate the fundamentalist law at the foundation of all of physics ie the TOE. Just an opinion of course – only time and further research will solve it.ThanksBill
[QUOTE="vanhees71, post: 5552095, member: 260864"]In math you start with the axioms, because it defines your "universe".”Only in a textbook treatment, just like in theoretical physics. In a historical context, real numbers were used 1000 years before their axiomatization, matrices and quadratic forms were used 150 years before the axiomatic concept of a vecor space appeared, transformation groups were studied over 100 years before the notion of an abstract group was coined, probability 150 years before Kolmogorov's axioms, etc.. The appropriate axiomatic setting of a subject matter arises only after enough experience has accumulated, both in mathematics and in theoretical physics.The goal of a theory, both in mathematics and physics, is to fully account for the corresponding part of the domain of discourse the theory covers, and to do so in an efficient way that is easy to teach and provides quickly all relevant tools and results. The axioms are a device serving the concise introduction of the concepts to be used later, again both in mathematics and in theoretical physics. Except that the mathematician's axioms cover the full conceptual content while physical axioms usually have an exemplary nature and are not sufficient to give a solid basi of the theory. Born's rule (which applies only to certain highly idealized experiments) is the best known example of such a caricature axiom, supposedly defining the formal meaning of a measurement.
[QUOTE="A. Neumaier, post: 5552089, member: 293806"]Because they allow a concise description of models of real physical systems. Just like in mathematics axioms for group theory or for vector spaces are assumed because they allow a concise description of symmetries and linear algebra. In both cases, the experience with the concrete stuff comes first and gives later rise to a formal, abstract edifice in which the experience gained is transformed into powerful tools that allow to tackle more complex problems than what one can do when staying on the concrete level.”It should, however, be clear that in physics the axioms are not the starting point but the goal of theory building. In math you start with the axioms, because it defines your "universe". In physics the "universe" is imposed on us by nature, and we have to figure out the "axioms" for describing it. It's amazing enought, how far you come with this axiomatic approach to physics, although admittedly the most successful theory we have has no rigid foundation in math, relativistic QFT and the Standard Model of HEP.
[QUOTE="lavinia, post: 5551954, member: 243745"]I am just starting out in quantum mechanics. The approach has been to talk about Hermitian operators as obervables acting on a complex vector space of states.Mathematically this is simple stuff, linear algebra. The Shroedinger equation makes it a little more difficult as does the use of symmetries to find degeneracies. But net net this is easy mathematics. Basic Quantum Mechanics is not difficult because of the mathematics. Classical phase space is just a manifold of points. This is a straight forward generalization of our visual picture of the space around us. Quantum Mechanical phase space is a complex vector space, and points in the phase space can be linearly combined to make new states. I think for anybody this is difficult to intuit. Also the whole deal with eigen states as the results of observations. We end up with a mathematical formalism that is interpreted in terms of weird experiments. That is hard. In Mathematics unintuitive ideas come to life through examples and applications to other areas of mathematics. That is how I want to learn Quantum Mechanics. In this context the historical approach has got to be enlightening. I don't agree that it should not be taught. I want to know it.Another conceptual problem with Quantum Mechanics is that it is taught deductively from a set of Axioms (as is Special Relativity). These Axioms can seem like pure formalism and the exposition of the subject a syllogism. That can not be a good way to teach. That is not the way mathematics is taught nor is it the way mathematical ideas are discovered. This again urges the historical approach at least to gain insight into why these Axioms are necessary and how they were thought of. I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.”I have the opposite impression. Modern mathematics textbooks are very often in an awful "Bourbaki style". It's all well ordered in terms of axioms, definitions, lemmas, and theorems with proofs, but no intuition whatsoever. I think in math the intuition comes from the applications, and physics is of course full of applications of interesting math issues (from analytical geometry via calculus to group theory and topology nearly everything interesting in math finds applications in physics).Concerning the historic approach, I'm a bit unsure. From my own experience, I don't like it at all. In high school we learnt "old quantum mechanics" first. This is phsics of a transition era from classical ways of thought, which were proven wrong by observations at the time (e.g., the very familiar stability of matter is a complete enigma to classical physics as soon as you know that atoms consist of a positive charged particle (now called atomic nucleus) surrounded by negatively chargend electrons, as was revealed by the famous gold-foil experiment by Rutherford. The collisions of ##alpha## particles (He nuclei) were perfectly described by classical scattering theory in the Coulomb field (that's why the corresponding cross section is named after Rutherford). So far so good, but then you had to understand the bound state making up an atom itself, and this caused a lot of trouble since on the one hand Bohr had just to solve the classical motion of a charged particle running around the nucleus under the influence of the mutual electrostatic interaction and assuming the quantization of the action, which was an ingeneous ad-hoc extrapolation of Planck's treatment of black-body radiation. Lateron the model was refined by Sommerfeld. However it was contradicting classical electromagnetics according to which there should be some bremsstrahlung and in a very short time the electrons should crash into the nucleus, i.e., the atom should be unstable. Instead of the bremsstrahlun, however, what's observed are clean pretty narrow spectral lines whose frequency was given by the distance between the energy levels known from the Bohr-Sommerfeld model, which is totally ununderstandable from classical physics. Last but not least the chemists knew that hydrogen atoms are spherical and not little flat disks.Learning the old quantum mechanics first, cements very wrong pictures in the students's mind whic h have to be unlearnt and corrected when you want them to understand modern quantum theory. As you rightly realized the problem is indeed not the math. Taking the wave-mechanics approach for a long time of the QM 1 lecture you deal just with a scalar field. Compared to electromagnetics, where you deal with a lot of scalar and vector fields which are coupled via Maxwell's equations, it's a piece of cake to deal with a single pseudoparabolic pde, known as the Schrödinger equation. The problem is that you have to build the entire intuition behind this math via this math itself. There is no correct intuitive picture from our classical experience, which ironically becomes a pretty tough question to answer, why it is right (the answer is decoherence, and the course-graining concept of quantum statistical physics, and after you swallowed this many loose ends of classical statistics like the Ehrenfest paradoxon the question of the absolute measure of entropy, etc. are solved either). On the other hand, you are right in saying that learning about the historical development of theories can help a lot to gain intuition, and particularly for QM 1 I'd not know how to teach it without a (however brief) historical introduction about how the idea to have the wave-mechanics approach or, in my opinion even better, the representation free Hilbert space formulation a la Dirac. I've never taught QM 1, but I guess, I'd give a brief introduction about the historical development, starting with Planck's black-body law, then talking about "wave-particle dualism" (sparing out however, the completely misleading picture of photons being "particles" in any classical sense, and the photoelectric effect does NOT prove the necessity of photons, i.e., the necessity of quantizing the em. field, which is anyway not discussed in QM 1) and finally getting via the wave-mechanics heuristics a la Schrödinger as quickly as possible over to Dirac's very neat mathematical heuristics to the Hilbert-space ##C^*##-algebra approach, where first the commutation relations can be reduced to the Heisenberg algebra of position and momentum operators, i.e., a canonical quantization approach. I've once taught QM 2, where I emphasized the role of symmetries, which is in my opinion the right intuition for modern physics anyway. The ##C^*## algebra of non-relativistic QM is then very clearly deduced from the ray-representation theory of the Galileo group, which then helps very much to understand, by the notion of mass is so different in Galilean physics (central charge of the Galileo algebra) compared to Minkowskian physics (Casimir operator of the proper orthochronous Poincare group). Given the evaluations of this lecture by the students, this approach seems to be not too bad since they liked it, although they argued that one better keeps the discussion of the Galilean part shorter and rather go farther in the relativstic part, whic h they found much more interesting. Of course, I did not teach the wrong wave-mechanics interpretation of relativsitic QT a la Bjorken-Drell vol. I. This is really too much sacrifice to the historical approach, since once the Dirac formalism is motivated in QM 1, there is no reason, not to start relativistic QT right away as a local quantum field theory.I think the best way in a good balance between the historical and the deductive approach to teach theoretical physics can be found in Weinberg's textbooks, where he usually has a historical introduction but then develops the theory itself in the deductive way emphasizing the mathematical structure and its meaning in context of the physics. I can only recommend to read his "Lectures on Quantum Mechanics" and of course "Quantum Theory of Fields".
[QUOTE="lavinia, post: 5551954, member: 243745"]why these axioms are assumed.”Because they allow a concise description of models of real physical systems. Just like in mathematics axioms for group theory or for vector spaces are assumed because they allow a concise description of symmetries and linear algebra. In both cases, the experience with the concrete stuff comes first and gives later rise to a formal, abstract edifice in which the experience gained is transformed into powerful tools that allow to tackle more complex problems than what one can do when staying on the concrete level.
[QUOTE="lavinia, post: 5551954, member: 243745"]Quantum Mechanical phase space is a complex vector space”No, it is a complex projective space formed by the rays in the Hilbert space. Geometrically it is again a symplectic manifold (though often infinite-dimensional). Note that rays cannot be superimposed, only state vectors.You might like my online book Classical and Quantum Mechanics via Lie algebras, where I present quantum mechanics in a mathematician-friendly way.
[QUOTE="lavinia, post: 5551954, member: 243745"]I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.”I don't think a theory deals with physical probability can be deduced from the standard (Kolmogorov) axioms for probability. Those axioms lack assumptions having to do with a "probable" event becoming an "actual event". I see nothing in the Kolmogorov axioms that even asserts it is possible to take a random sample – i.e. to have a "realization" of random variable. It seems to me that in order to axiomatize physical probability, one would have to introduce the notion of time into the axioms, so that an event could have a probability at one time and then become (or fail to become) actual at a later time.
[QUOTE="lavinia, post: 5551954, member: 243745"]I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.”Physical theories are not axiomatic like mathematics is.They are models which fit with observations.If the model predicts something that can be tested, it's a good theory.
I am just starting out in quantum mechanics. The approach has been to talk about Hermitian operators as obervables acting on a complex vector space of states.Mathematically this is simple stuff, linear algebra. The Shroedinger equation makes it a little more difficult as does the use of symmetries to find degeneracies. But net net this is easy mathematics. Basic Quantum Mechanics is not difficult because of the mathematics. Classical phase space is just a manifold of points. This is a straight forward generalization of our visual picture of the space around us. Quantum Mechanical phase space is a complex vector space, and points in the phase space can be linearly combined to make new states. I think for anybody this is difficult to intuit. Also the whole deal with eigen states as the results of observations. We end up with a mathematical formalism that is interpreted in terms of weird experiments. That is hard. In Mathematics unintuitive ideas come to life through examples and applications to other areas of mathematics. That is how I want to learn Quantum Mechanics. In this context the historical approach has got to be enlightening. I don't agree that it should not be taught. I want to know it.Another conceptual problem with Quantum Mechanics is that it is taught deductively from a set of Axioms (as is Special Relativity). These Axioms can seem like pure formalism and the exposition of the subject a syllogism. That can not be a good way to teach. That is not the way mathematics is taught nor is it the way mathematical ideas are discovered. This again urges the historical approach at least to gain insight into why these Axioms are necessary and how they were thought of. I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.
Discussing the PF Insight : My experience with my 2 undergraduate QM courses has been rather strange in that, to me, the math part was actually the most challenging part. I did not understand most of it, I believe. For instance in the 1st course we spent some classes to reach the spectral theorem and how it relates to operators in QM. There were many homework problems involving pure math proofs. It seemed to me like a deep linear algebra course (I had already took a linear algebra course but QM really digged much deeper. For example I had never dealt with unbounded operators and some theorems that were valid for finite dimension vector spaces were suddenly not valid for infinite dimensional ones. The ones that appear in QM!)In my second course what really got me ENTIRELY lost was the spherical tensors part. I did not grasp a single bit out of it. Whilst on the "physics" part of QM that challenged my intuition most, or got me think "wow… that's freaking weird" was :1) That in a central potential, the wavefunction of the electron is not restriced to a plane. That's entirely different than in classical mechanics. 2)The wavefunction of the ground level of the SHO for 1 particle, implies that there's a greater probability to find the particle at the center rather than on the tails of the quadratic potential. That is opposite to classical mechanics.There are a few others, dealing for example with perturbations that demolished my intuition again and again. But it was not what made the course hard.
[QUOTE="vanhees71, post: 5551555, member: 260864"]So it's tricky to ask, what's invented and what's discovered in theoretical physics, but it's fortunately completely irrelevant to the progress of science either :-)).”Well, it's not completely irrelevant. If one says it's invented and not discovered, then it's close to saying that it's mathematics and not physics, which can influence someone's decision on whether such research should be financed or not, which may have a high impact on progress of science.
Hm, isn't this purely semantic again. In a sense SUSY seems to be triggered by the famous Coleman-Mandula no-go theorem, and some physicists tend to dislike no-go theorems and event more general mathematical structures to circumvent it. In this sense the introduction of a ##mathbb{Z}_2## graded algebra into the symmetry structure of Lagrangians is an invention (interesting would be to know, whether mathematicians had thought of such a struture before; I guess so since if you have Grassmann numbers, it's not far to come to the idea to formulate such structures). It stays of course invented even if LHC confirms SUSY, because that's the way it was found historically. The only point is, how likely it is that it is really found. It's very rare that from such a pure invention without being triggered by some necessity from observations something turns out to be finally true, although the prediction of the Higgs boson is close to something like it. It has been introduced to get a consistent gauge theory with massive gauge bosons to describe the weak interactions within a renormalizable QFT, and nobody could find another way to do this (except for the Abelian case, which however is not describing QFD in accordance with experiments). The idea to use the apparent spontaneous breaking of a local gauge theory was, however, in some sense a mathematical invention (by condensed-matter physicists by the way, mostly Andersen, in the context of superconductivity) like SUSY. On the other hand the application to the ew. interaction was very much triggered by phenomenology (even in form of Fermi's theory of ##beta## decay and its variants after parity violation was discovered and in this context the correct (V-A)-realization of this breaking was found based on these observations).So it's tricky to ask, what's invented and what's discovered in theoretical physics, but it's fortunately completely irrelevant to the progress of science either :-)).
[QUOTE="vanhees71, post: 5551499, member: 260864"]Well, I was talking about discovering new theories, not work on established ones, which of course makes the most of what's done by theoretical physicists in their daily work.”Is SUSY discovered or invented? If it is invented, would you be still claiming so if it happened that LHC confirmed SUSY in experiments?
[QUOTE="A. Neumaier, post: 5551497, member: 293806"]If I look into a typical issue of ''Nuclear Physics B'' or ''Classical and Quantum Gravity'' I find lots of theory obtained primarily by thinking about other theory, not about observed phenomena.”Excellent point!
Well, I was talking about discovering new theories, not work on established ones, which of course makes the most of what's done by theoretical physicists in their daily work.
[QUOTE="vanhees71, post: 5551491, member: 260864"]Well, but mathematical structures are pure inventions of the human mind, only restricted for being consistent within the usual logics.”Well, physical structures are also pure inventions of the human mind, only restricted for being consistent with experiment within their domain of validity. Whether they are interesting is another question. As in mathematics, it is only the interesting, most widely applicable ones that are preserved and taught; the remaining ones, though consistent with experiment within their domain of validity, fall into oblivion.[QUOTE="vanhees71, post: 5551491, member: 260864"]while physical theories are usually discovered by thinking about observed phenomena.”In each discipline, the relevant results are usually discovered by thinking about its subject matter. If I look into a typical issue of ''Nuclear Physics B'' or ''Classical and Quantum Gravity'' I find lots of theory obtained primarily by thinking about other theory, not about observed phenomena.
[QUOTE="A. Neumaier, post: 5551481, member: 293806"]The mathematical structures you mention are also discoveries and not inventions. You can invent a set of axioms by listing some properties and give the structures that have these properties a name – but nobody would be interested in them. Even if new it is dead wood, not more interesting than listing new truths of the form a+b=c where a,b are huge numerically given numbers whose sums were never computed before by anyone.On the other hand, one discovers that certain very special structures (like groups, rings, fields, vector spaces, calculus,…) have interesting properties that help to organize prior knowledge in a more powerful way and thereby give rise to progress, including new and interesting quests – just as with progress in physics.”Well, but mathematical strucures are pure inventions of the human mind, only restricted for being consistent within the usual logics. Whether they are interesting is another question, while physical theories (usually formulated using well-established mathematical structures or inventing new ones as needed, which are then often happily adopted by the mathematicians like functional analysis (Dirac's ##delta## function made rigorous) or fiber bundles (setting gauge theories on a rigorous mathematical footing)), are usually discovered by thinking about observed phenomena.
[QUOTE="vanhees71, post: 5551470, member: 260864"]So to discover a successful model or even a paradigm shifting general theory (which is a very rare thing in the history of science; in my opinion there are only two: … the advent of the field-point of view … and quantum theory”You forgot the biggest paradigm shift, that physics should be formulated in terms of equations (Newton).Of course, there are also many "smaller" paradigm shifts. For example, that QFT does not need to be renormalizable in order to make sense, that there may be more than 3 spacial dimensions, … All they require a bit of philosophy.
[QUOTE="vanhees71, post: 5551470, member: 260864"]physical theories are discoveries and not inventions (as are mathematical structures like groups, rings, fields, vector spaces, calculus,…).”The mathematical structures you mention are also discoveries and not inventions. You can invent a set of axioms by listing some properties and give the structures that have these properties a name – but nobody would be interested in them. Even if new it is dead wood, not more interesting than listing new truths of the form a+b=c where a,b are huge numerically given numbers whose sums were never computed before by anyone.On the other hand, one discovers that certain very special structures (like groups, rings, fields, vector spaces, calculus,…) have interesting properties that help to organize prior knowledge in a more powerful way and thereby give rise to progress, including new and interesting quests – just as with progress in physics.
[QUOTE="stevendaryl, post: 5551459, member: 372855"]Falsifiable predictions isn't the goal of the scientist, but is a measure of his progress.”I would like to add that it is not the only measure of progress. For instance, reformulation of the theory such that calculations become simpler (e.g. matrix mechanics vs wave mechanics) is also a progress. Or a reformulation of the theory such that abstract mathematical objects can more easily be visualized (the same example) is also a progress. Or a reformulation of the theory such that it becomes more intuitive (which is a matter of interpretation) is also a progress.Of course, different individuals do not need to agree which method of calculation is simpler, which mathematical object can more easily be visualized, or which interpretation is more intuitive. That's why different methods of calculation, different mathematical formulations, and different interpretations simultaneously exist. It is subjective and depends on someone's taste, but it does mean that it is not a part of science.
[QUOTE="Demystifier, post: 5551451, member: 61953"]Yes, Einstein is a good example. But do you think that any big step in science, any change of paradigm, is possible without some philosophy as a heuristic line of thought for theory building? Newton, Darwin, Einstein, Bohr, could any of them make their main achievements without any philosophy?”In the model building itself, I think it's an important aspect, but I don't know of any example in the history of physics that progress has been made without solid foundation on empirical facts. That's why I think physical theories are discoveries and not inventions (as are mathematical structures like groups, rings, fields, vector spaces, calculus,…). Take "modern physics". It's almost entirely based on observations contradicting previous theories. The key point from the 19th century was the discovery (!) of the modern (and still valid) form of the theory of the electromagnetic interactions by Faraday (experiment) and Maxwell (theory). Note that Maxwell's heuristics is entirely wrong, while his theory has had a solid foundation in Faraday's experiments and thus still holds in modern form today. Maxwell himself later abandoned the wrong mechanical heuristics by the way. The discovery of quantum theory is also triggered in major flaws of the classical picture. Here also the theory of light (i.e., Maxwell electrodynamics) plays some role (photoelectric effect) but even more importantly the flaw of classical thermodynamics and statistical physics in explaining the stability of macroscopic matter. Last but not least it was Planck's derivation of his previously empirically found formula for the black-body spectrum based on highly accurate measurements by Kurlbaum et al at the Reichsanstalt (BTW with the main motivation to define a reproducible and reliable measure for the luminosity of all kinds of lightning, among them the then pretty new electric light bulbs :-)).Of course there is also other input which I'd summarize as "intuition" of the physicists going into model building. E.g., the "old quantum theory", particularly concerning the Bohr-Sommerfeld model of atomic structure and spectral lines (again electromagnetism!) was considered (even by Bohr and Sommerfeld themselves) as a pretty poor picture with all its ad-hoc assumptions just to tweak the theory to all the accurate measurements of atomic spectra, the wrong counting of multiplicity of spectral lines, the socalled anomalous Zeeman effect etc. Here the main motivation to develop a more consistent theory was mostly on esthetical grounds, i.e., to find a more convincing or if you wish "more beautiful" theory. Best known is Heisenberg's heuristics of his famous Helgoland paper: He wanted to include only "observable quantities" into his theory and invented what Born has recognized as matrix calculus and worked it out together with Heisenberg and Jordan to matrix mechanics. Schrödinger's heuristics was very mathematical and strongly influenced by de Broglie's thesis, which itself can be seen also as an attempt to unify the disturbing heuristics of old quantum theory to a new basic principle, "wave-particle dualism", which is of course also flawed, as the development of modern QT shows. Schrödinger based his "wave mechanics" entirely on the analogy between ray optics and wave optics (the former being the leading-order eikonal approximation of the latter) on the one hand and point particle mechanics in terms of the Hamilton-Jacoby partial differential equations, taking the Hamilton-Jacobi PDE as the eikonal approximation of his "matter waves". Of course, he had a completely wrong intuition about the meaning of his waves, which he never got completely convinced until the end of his life.So to discover a successful model or even a paradigm shifting general theory (which is a very rare thing in the history of science; in my opinion there are only two: the first is the advent of the field-point of view, based on observations by Faraday (finally leading to the relativistic space-time model; in full form achieved by Einstein with his General Relativity Theory) and quantum theory, which was initially mainly enforced on the usually rather conservative physicists by overwhelming empirical evidence) there is a very strong need for a solid foundation in empirical evidence, usually consisting of high accuracy measurements of clearly defined experimentally setup situations, but as well some intuition of ingenious physicists, including a lot of (often even wrong) heuristics. New models and theories in turn lead to new experimental setups as ingenious as the theories. For the longest time scientific progress is made in testing and investigating the predictions of models at higher and higher accuracy and ever new ways (again a lot of intuition is needed to invent the proper technology to perform the pertinent measurements), not so often in paradigm-shifting "revolutions" in theory. That's why I think Kuhn's famous work on these revolutions is flawed in the sense that it describes very rare special events in the history of sciences not the usual slow progress eventually leading to them. An example are gravitational waves. Predicted, with many doubts, by Einstein in 1916 almost precisely 100 years later they could be observed for the first time with an instrument taking the collaborative effort of many physicists around the world for decades. Who knows, what will come out of this completely new "window" (which can be understood in a quite literal sense) to observing the universe!
[QUOTE="stevendaryl, post: 5551459, member: 372855"]Falsifiable predictions isn't the goal of the scientist, but is a measure of his progress.”Exactly!I have one falsifiable prediction: The collision of two pink elephants in liquid helium produces 7 red frogs. Nobody so far has made that experiment, but it can be done with present technology. Is that science? :biggrin:
[QUOTE="Demystifier, post: 5551451, member: 61953"]Yes, Einstein is a good example. But do you think any big step any science, any change of paradigm, is possible without some philosophy as a heuristic line of thought for theory building? Newton, Darwin, Einstein, Bohr, could any of them make their main achievements without any philosophy?”This is the question of what science is about. Starting with Popper (or earlier, with the logical positivists), many people think that science is about making better and better predictions. I actually don't think that is the goal of science. I think the goal of science is understanding the universe, and that the focus on predictions is an attempt to keep science grounded, so that it doesn't drift too far into pure speculation. Falsifiable predictions isn't the goal of the scientist, but is a measure of his progress.When Einstein worked on Special Relativity, his goal wasn't to make new predictions, but to reconcile Newtonian mechanics with electrodynamics (which turned out to require a modification of the first). When he worked on General Relativity, his goal wasn't to make new predictions, but to reconcile Special Relativity with Newtonian gravity (which turned out to require a modification of both). In both cases, there were falsifiable predictions, but that wasn't the goal. To me, it's like exams in school. They are an important check on a student's understanding, but it's a mistake to think that doing well on exams is the purpose of learning.
[QUOTE="vanhees71, post: 5551431, member: 260864"]It can also provide heuristical lines of thought in model/theory building, as shows the example of Einstein.”Yes, Einstein is a good example. But do you think that any big step in science, any change of paradigm, is possible without some philosophy as a heuristic line of thought for theory building? Newton, Darwin, Einstein, Bohr, could any of them make their main achievements without any philosophy?
[QUOTE="Demystifier, post: 5551368, member: 61953"]So, in your opinion, could Bohmian mechanics have some value, even if it is not science? After all, it has some non-trivial mathematical structure. In addition, similarly to ethics if you wish, it offers some meaning of QM for those human physicists who, for some personal reasons, need some meaning in physics for internal motivation. (After all, if physics does not have any meaning for you, then why do you do it?)Or let me put it this way. Even if BM is not science, it is certainly a non-trivial intellectual discipline. So how should we classify it? Philosophy? Philosophy of science? Isn't philosophy of science a part of science as much as it is a part of philosophy?”Well, in the case of BM I don't see any value. Maybe it's interesting as mathematics. Physics and the natural sciences do not provide "meaning" to anything. It is a method to learn in an objective and quantitative way about how nature is (or let's say more carefully about the objectively by human beings knowable part). Philosophy of science (or metaphysics) is part of philosophy and not science. It's providing amazingly little to the progress of science itself. It can, however, be valuable to subsume science into a bigger view of human knowledge and to systematially analyze its role in the progress of culture in general. It can also provide heuristical lines of thought in model/theory building, as shows the example of Einstein. His case is paradigmatic for the good and the bad of the use of philosophy of science in science itself: On the one hand Einstein's strong believe in a deterministic world view for sure helped him to very clearly resolve some of the outstanding puzzles of physics of his time starting from the incompatibility of Maxwell electrodynamics with the Newtonian space-time picture (ingeniously solving it by adapting the space-time picture and mechanics to the findings included in Maxwell's equations rather than the other way around as was the approach by the other physicists of his time, including Hertz, Lorentz, and Poincare) as well as early quantum theory (which however he found dissatisfying from the very beginning). On the other hand, this strong philosophical (if not quasi-religious) believes also hindered Einstein to participate in the development of modern quantum (field) theory for the last 20-30 years of his life (most probably more to the disadvantage for science than for himself).