Monday, April 22, 2019

Comments on “Quantum Gravity in the Lab” in Scientific American

The April issue of Scientific American has an article by Tim Folger titled “Quantum Gravity in the Lab”. It is mainly about the experiment by the Aspelmeyer group in Vienna, which I wrote about three years ago here. Folger also briefly mentions the two other proposals by Bose et al and Marletto and Vedral. (For context and references see my recent blogpost about the possibility to experimentally test quantum gravity, or this 2016 summary.)

I am happy that experimental tests for quantum gravity finally receive some media coverage. I worked on the theory-side of this topic for more than a decade, and I am still perplexed by how little attention the possibility to measure quantum gravitational effects gets, both in the media and in the community. Even a lot of physicists think this research area doesn’t exist.

I find this perplexing because if you think one cannot test quantum gravity, then developing a theory for it is not science, hence it should not be pursued in physics departments. So we have a situation in which a lot of physicists think it is fine to spend money on developing a theory they believe is not testable, or whose testability they believe is not worth studying. Excuse me for not being able to make sense of this. Freeman Dyson at least is consistent in that he both believes quantum gravity isn’t testable and isn’t worth the time.

In any case, the SciAm piece contains some bummers that I want to sort out in the hope to prevent them from spreading.

First, the “In Brief” box says:
“Physicists are hoping that by making extremely precise measurements of gravity in small-sale set-ups – experiments that will fit onto a tabletop in a laboratory – they can detect effects from the intersection of gravity and quantum theory.”
But not everything that is at the intersection of “gravity” and “quantum” is actually quantum gravity. A typical example is black hole evaporation. It is caused by quantum effects in a gravitational field, but the gravitational field itself does not need to have quantum properties for the effect to occur. Black hole evaporation, therefore, is not quantum gravitational in origin.

The same goes, eg, for this lovely experiment that measured the quantization of momenta of neutrons in the gravitational potential. Again this is clearly a quantum effect that arises from the gravitational interaction, so it is at “the intersection of gravity and quantum theory.” But since the quantum properties of gravity itself do not play a role for the neutron-experiment, it does not test quantum gravity.

A more helpful statement would therefore be “experiments that can probe the quantum behavior of gravity”. Or, since gravity is really caused by the curvature of space-time, one could also write “experiments that can probe the quantum properties of space and time.”

 Another misleading statement in “In Brief” box is:
“The experiments aim to show whether gravity becomes quantized – that is, divisible into discrete bits – on extremely tiny scales.”
This is either badly phrased, or wrong, or both. I don’t know what it means for gravity to be divisible into discrete bits. The best interpretation I can think of is that the sentence refers to gravitons, particles which should be associated with a quantized gravitational field. The experiments that the article describes, however, do not attempt to measure gravitons.

In the text we further find the sentence:
”A quantum space-time… would become coarse-grained, like a digital photograph that becomes pixelated when magnified.”
But we have no reason to think there is any pixelation going on for space-time. Moreover, the phrase “magnified” refers to distance scales. Quantum gravity, however, does not become relevant just below a certain distance. Indeed, if it became relevant below a certain distance, that would be in conflict with the symmetries of special relativity; this distance-dependence is hence a highly problematic conjecture.

What happens in the known approaches to quantum gravity instead is that quantum uncertainties increase when space-time curvature becomes large. It is not so much a pixelation that is going on, but an inevitable blurring that stems from the quantum fluctuations of space and time.

And then there is this quote from Lajos Diosi:
”[We] know for sure that there will be a total scrambling of the spacetime continuity if you go down to the Planck scale.”
No one knows “for sure” what happens at the Planck scale, so please don’t take this at face value.

Besides this, the article is a recommendable read.

If you want to get a sense of what other ways there are to experimentally test quantum gravity (or don’t have a subscription) read my 2017 Nautilus essay “What Quantum Gravity Needs is more Experiment.”

49 comments:

  1. Einstein spent the last 30 years of his life trying to unify gravitation and electromagnetism (as did Weyl and many others), a unification that was never considered testable. These two forces still have not been unified, but now the brightest minds in physics have moved on to what is surely a far more difficult task, the unification of quantum mechanics and gravity.

    Is there any real hope that quantum gravity will be testable, given that we don't yet have even a decent theory?

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    1. Bill,

      We know that the current theories that we have (quantum theories for the particles and a non-quantum theory for gravity) are incompatible. They simply cannot describe certain situations, one of which is the gravitational field of a particle in a quantum superposition of two locations.

      If you can measure this field, you will measure something that none of the existing theories predicts. It is widely believed that the results would be consistent with a limit that is known as "perturbatively quantized gravity". This may or may not be so, we will not know until we actually measure it. Be that as it may, yes it is a "real hope" because it is a good prediction: Something new has to happen here. It's a prediction that is good in the same way as the prediction that something new had to happen at the LHC (that "something new" being the Higgs).

      Delete
    2. Sabine,

      I do not understand what the measurement of the gravitational field of a particle in superposition is supposed to reveal.

      If the measurement is precise enough to determine the location of the particle then, well, you will find the particle in a certain place in accordance to Born's rule. I do not see how this is different from, say, measuring the electric field of an electron that is in a superposition.

      If, on the other hand the measurement is not precise enough, the particle will remain in superposition and no new information will be extracted.

      Delete
    3. It seems to me that for a state that is the superposition of two locations, in a regime where the gravitation field is small and varies slowly, the naive expectation would be to mesure the average of the Newtonian attraction from each position.

      I understand we don't know and we have no theory to predict the outcome of such an experiment, but if that is what is measured I'm not sure it will help progress much (beside ruling out exotic theories).

      Delete
    4. Andrei,

      The difference is that we have ample evidence the electromagnetic field is quantized, but so far we have no evidence that the gravitational field is quantized.

      Delete
    5. dlb,

      A quantum superposition is *not* just an average - that would be a non-quantum field.

      Delete
    6. Sabine,

      So, what would you expect in case the gravitational field is quantized vs it not being quantized?

      Delete
    7. Bee,

      "They simply cannot describe certain situations, one of which is the gravitational field of a particle in a quantum superposition of two locations. "

      how do candidate QG theories froms string theory to LQG to AS to CDT answer this question?

      Delete
    8. @Bill - Of course it was considered testable. In fact Einstein shot down Weyl's unification in its original form in spite of the fact that Weyl's solution was of extraordinary mathematical beauty. He pointed out that sharp spectral lines could not exist because the electron would have a mass dependent on its spacetime history. That's a testable phenomenon - broadening of spectral lines.

      -drl

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    9. @Sabine, does anyone really know if antimatter falls up or down? Has that experiment ever been done? I'd say you'd have to do that experiment first, if for no other reason than to get warmed up. If you think about it, it's a hard experiment to do.

      -drl

      Delete
    10. neo,

      They don't. I commented on this in my earlier blogpost. There are no theoretical predictions. As I said, theoreticians have not been very interested in actually making contact to the real world.

      Delete
    11. Andrei,

      Depends on exactly what you measure, of course. Ideally you would want to measure something like a Bell-inequality violation that demonstrates entanglement (of the gravitational field), which would be proof of a quantum phenomenon. The point that was made by Bose et al is that if you can create entanglement through the gravitational interaction, then that too would require gravity to be quantized already.

      Delete
    12. drl,

      I wrote about this here: Why doesn't antimatter antigravitate? And, yes, there are experiments.

      Delete
  2. Hi Sabine,

    Thanks for your continued effort to keep some sanity in the literature on this topic, I have learned a lot about this from this blog over the years, especially your review post linked above.

    I have to agree with your dislike of phrases like "detect effects from the intersection of gravity and quantum theory". Ugh, talk about obfuscating the point. People have been doing experiments like that since the COW days...

    Anyway, with apologies for suggesting my own paper, in case anyone wants a (slightly) more technical overview of some of this stuff, including the conservative take regarding the effective field theory treatment of perturbative GR: https://arxiv.org/abs/1807.11494

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    Replies
    1. Dan,

      Just scanned over [1] (nice review) and [2]. Here some questions.
      - “… gravitational decoherence here would be feasible - they have not yet been done.” - Are there any activities to install this kind of experiment?
      - "path bunching" is caused by the correlations in CWL above m_CWL. In CWL gravity is regarded to be quantized, since S_G is in the path integral [1](51) and thus is also able to generate entanglement.
      What prevents CWL that all particles including spacetime become entangled, exceed m_CWL and thus "path bunching" destroys all of QM and makes CWL entirely classical?
      - in [2] it says “… along an imaginary time contour around a cylinder of circumference 1/kT”. I know that this is a common technique used in Unruh-Hawking temperature and is true (“for sure” ;-). But do you know why the assumption, the identification of cyclic imaginary time with temperature holds?

      Delete
    2. Hi Reimond,

      "Are there any activities to install this kind of experiment?" -- yes, see the review [1] for a bunch of links, or also you can check out the excellent review by Bassi Ulbricht and Grosshardt https://arxiv.org/abs/1706.05677

      As to your path bunching question, if I understand what you are after, the statement would be that we only really worked out how the model works at low curvature, where gravitational nonlinearity is small. Ultimately if you are in a very high-curvature regime the theory breaks down for exactly the same reasons as perturbatively quantized GR. We didn't make an attempt to UV-complete the model...

      Delete
    3. Dan,

      I would never dare to ask for UV completion. What I meant was far more modest and would take place in flat spacetime. From [1] (figure 8) I got the impression that CWL mimics classical behavior in a double slit once the object exceeds m_CWL. This is good news.
      What I worried about was that if also the internal QM structure of the object would become classical, then this is bad news (for the object).
      Maybe CWL counters this by weighing the paths somehow differently...

      Delete
  3. Sabine,

    Is there an "obvious" result to the tabletop experiments that you and SciAm are talking about?

    My guess off-hand would be that decoherence kicks in really fast, and so you never really see quantum effects. But, that is just a knee-jerk reaction, not a careful analysis. (I've acquired the, no doubt lamentable, habit in recent years of answering any question about QM by just mumbling "decoherence.")

    All the best,

    Dave

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    Replies
    1. PhysicistDave,

      The "obvious" result is that from perturbatively quantized gravity. Sure, decoherence is the major obstacle to overcome here, but there is no fundamental limit that would prevent quantum gravitational events from being observable.

      Delete
    2. Sabine,

      Can you point us to a reference that goes through the analysis?

      And, since the gravitational fields are so weak, is there any chance of any result other then "perturbatively quantized gravity"?

      If the experimentalists can really overcome the problem of decoherence, I will be enormously impressed.

      Dave

      Delete
    3. PhysicistDave,

      Reference for what? Please check the references in this earlier blogpost and let me know in case you meant something else.

      Delete
  4. Although I think Freeman Dyson is indeed consequential, I wonder if you meant "consistent"?

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    Replies
    1. Hi Amos,

      I didn't realize that "consequential" has two different meanings. I think you are right and "consistent" is the better word. Thanks for pointing out,

      Sabine

      Delete
    2. The German word "konsequent" means "consistent" in the sense of taking seriously the consequences of one's thoughts or actions, in other words avoiding hypocrisy. In both languages, "consistent" ("konsistent") is more general, meaning contradiction free. Same same, but different.

      The English word "consequent" means "resulting from" and is not applied to a person.

      Delete
  5. Sabine said.... quantum uncertainties increase when space-time curvature becomes large.

    Yet another reason to wonder if the gravitational curvature of space-time is the real problem in unifying the physics of gravity and Quantum Mechanics.

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  6. As far as I understand, there is no clear theoretical prediction about what gravitational effects would a Schrodinger's cat-like experiment detect.
    * Would the (effective) gravitational force point to the cat the way it stood the moment the box got isolated? And then jump to its final location when the box is opened again?
    * Would it jump the moment the poison is administered?
    * Would it slowly drift from the alive to the dead cat position?
    * Would it not even show any gravitational effects at all, because gravity is an effective phenomenon emerging at the scales where decoherence takes over and any meaningful isolation is impossible?

    There are good (and bad) reasons to consider each of those possibilities.

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    Replies
    1. Sergei,

      There is no way to shield/isolate the gravitational field. An object placed inside a box, regardless of the structure of the box will produce a gravitational field outside the box. That means that the Schrodinger's cat experiment cannot actually be done. The observer placed outside the box has all information that exists inside the box (in a more convoluted way, true). He can know what the cat is doing at all times.

      So, by measuring accurately the gravitational field, the outside observer will "see" the cat just like in the absence of the box. No superposition will take place, nothing happens when the box is opened.

      Delete
    2. Andrei,

      What you say is wrong because the gravitational field is not a faithful copy of all quantum degrees of freedom. You can see this easily by noting that the source is bi-linear. (If it did, the information loss problem wouldn't exist.)

      Delete
  7. "Einstein spent the last 30 years of his life trying to unify gravitation and electromagnetism (as did Weyl and many others)"

    I recently came across a quote by Hermann Weyl; he figured that his attempts at unified field theory had failed because, when faced with a choice between the true and the beautiful, he always chose the beautiful. :-) (With respect to the other meaning of true and another type of beauty, Weyl was also the lover of Erwin Schrödinger's wife. Schrödinger, when negotiating moving to Dublin, made sure that he could bring both his wife and his mistress. Cue puns about superposition and possibly decoherence. Schrödinger---like Einstein a sceptic regarding the Copenhagen interpretation and other aspects of quantum theory---also spent his later years looking for unified field theories in the style of Einstein.)

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    Replies
    1. @Philip, of course Weyl was right! Only it wasn't the Lorentz spacetime "gauge", length standard, that was changing place to place, it was the phase of the wave function.

      -drl

      Delete
  8. I recall reading from one of Dysons books that he managed to get himself a meeting with Einstein and in an effort to find something to talk to him about he read his last few papers, decided they were rubbish, and in embarrassment didn’t go to the meeting and then, in presumably further embarrassment, avoided him when he saw him around the institute. Later, his friends scolded him for his rude behaviour saying, don’t you think he could have defended his ideas, and besides this is Einstein you’re talking about!

    I’m amazed no-one has had the nous until recently to suggest that nano-technology might bridge the gap between speculation and actual experiment in quantum gravity. From what you’ve written it seems like a natural match and also exciting news to think we are possibly only three orders away from testing quantum gravity.

    In some ways I’m unsurprised that possible experimental tests of QG are under the radar given that during my masters in theoretical physics we weren’t exposed to any. It made me feel uneasy as well as other students. For me, since all the way through high school physics theory was taught alongside experimental verifications that anchored theory to the real world (I can’t say for university physics as I chose to do maths but I imagine the same ought to be true there). I appreciate that at the point where we are at, theory has outrun experiment, but that to me makes it speculative theory, a hypothesis, to use an older term that seems to have been discarded but to me still seems appropriate. I also appreciate that it’s a course on theory, and there’s a lot to learn, but then again, so does every field.

    It’s a great post, as well as the other linked posts and as a commenter has already said, a real sanity check on the field. It needs one.















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  9. Hi Sabine,
    In your blog-post http://backreaction.blogspot.com/2007/06/bouncing-neutrons-in-gravitational.html, which you link to, concerning the quantization of momenta of neutrons in the gravitational field seem unable to properly load figures/images. That at least is what happens in my browser.

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    Replies
    1. Hi John,

      It's not my blogpost, but Stefan's. And the links refer to a university website that does no longer exist. So, I am sorry, but I can't fix that right now. I'll have to try and figure out if the images still exist.

      Delete
    2. Quite ok, I figured that something along those lines was the case.

      Delete
  10. I apologize. I tried to post this in an old thread you had linked.

    It is the first of two parts. The first part is excerpts from an op-ed from Saturday's Wall Street Journal. Note that the lead author is best known as the architect of Obamacare.

    "How the U.S. Surrendered to China on Scientific Research: Washington could show its seriousness about key technologies like AI, gene editing and quantum computing by making big federal investments" By Ezekiel Emanuel, Amy Gadsden and Scott Moore on April 19, 2019 at https://www.wsj.com/articles/how-the-u-s-surrendered-to-china-on-scientific-research-11555666200

    * * *.

    The U.S. intelligence community’s recent threat-assessment report identifies a number of research areas that will determine military and economic superiority in the coming decades, from artificial intelligence and gene editing to synthetic biology, 5G wireless systems and quantum computing. It warns that America’s “lead in science and technology fields has been significantly eroded.”

    As we see it, this decline stems from two related trends: steadily declining U.S. budgets for basic scientific research and a lopsided emphasis on the life sciences to the detriment of emerging technologies. ...

    Investment in science surged, peak[ed] in 1965, when 3.6% of the federal budget was devoted to basic and applied scientific research.

    ... some 46% of all federal civilian science money goes to just one agency: the National Institutes of Health. That is why the intelligence community can say that the U.S. remains dominant in biomedical science.

    Much of the federally funded research in ... computing, artificial intelligence, physics and more—passes through the National Science Foundation, which has been forced to get by on a shoestring. In 2017, the foundation’s total research funding was a mere $5.6 billion, some 0.1% of the federal budget and less than 20% of the budget of the National Institutes of Health. Worse, this amount has been declining for years as a fraction of the federal budget.
    ... Washington needs to invest more—much more—in both basic and applied scientific research. Ideally, the federal government would restore funding to the 1965 levels.

    * * *

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  11. The second part is one comment on the first part and a question to Dr. Hossenfelder.

    Comment: Percentage of the Federal Budget is a ridiculous metric. In 1965, half of the Federal Budget was for defense. Medicare, Medicaid and other Federal health spending was in its infancy. The Federal Budget is now largely medical care and pensions.

    Question. Is there any reason to think that additional money spent on physics will produce anything other than higher salaries for tenured professors?

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  12. Phillip wrote: The English word "consequent" means "resulting from" and is not applied to a person.

    I'll comment on this as a matter of interest, not as a nitpick. In English, "consequence" can be applied to a person. If Freeman Dyson is consequential, or a person of consequence, it means he's important. Conversely, if a person is of little or no consequence, they are not important.

    Gravity may be weak, but it is consequential. :-)

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  13. The statement What happens in the known approaches to quantum gravity instead is that quantum uncertainties increase when space-time curvature becomes large.  is interesting. This interplay between curvature and quantum uncertainty is something I have thought interesting. I just did a back of the envelope calculation to find

    ΔpΔx ≈ √(⟨p^2⟩ - ⟨p⟩^2)√(⟨x^2⟩ - ⟨x⟩^2) + R^i_{00j}(⟨p_ix^j⟩ – ⟨p_i⟩⟨x^j⟩)√(⟨x^2⟩ - ⟨x⟩^2).

    The Riemann curvature does act to increase the quantum uncertainty. I hope the unicode for the bra and ket show up right. This is a modification of the uncertianty principle. For relatively weak gravity fields R^i_{00j} ~ GMd/r^3c^2 for d the distance (or x) between the two bodies.

    I worked up calculations 10 years ago on how general relativity similarly amplified chaos. The Lyapunov exponent for chaotic drift in planetary orbits is increased the larger a role general relativity played. I had this idea that somehow aspects of quantum gravitation might be found in the orbital dynamics of a small satellite body executing relativistic motion in a system with two larger bodies. I treated the orbits of the large bodies with just Newtonian mechanics.

    Zurek demostrated how quantum fluctuations had changed the angular configuration of the Chronian moon Hyperion. In effect quantum fluctuations had acted to change very slightly the time data of the rotation of the body and this sent this kicked rotor onto a different path. It was the butterfly effect by quantum fluctuations.

    It seems plausible to put these together. We might by general relativistic physics get an amplification of quantum uncertainty. This would not be entirely quantum gravitation, but it would be a further coupling in of quantum and spacetime physics. Potentially some of this information could be garnered within the solar system. If not then maybe future improvements in the ephemeris of extra solar system dynamcis with more general relativistic physics might work.

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  14. The real problem with the idea of quantum gravity IMO is that GR has serious problems all by itself that are more or less ignored. Until those get fixed there doesn't seem to be much point in pressing forward. Classical electromagnetism of point particles of course has huge problems, and surprise, these are not fixed by quantum electrodynamics, and in some ways are made even worse. Dirac's approach to QED was to fix electrodynamics first, and THEN quantize it. He tried several approaches and was still thinking about it at the end of his life. That people regard QED as it stands as a success (and it is), simply says the problems are not so dire as to prevent any progress.

    -drl

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  15. Hello Sabine,

    How can anyone make those statements about pixelation and the scrambling of spacetime continuity in the face of https://arxiv.org/abs/1106.1068 and the more recent https://arxiv.org/abs/1710.08432? Am I completely misinterpreting these two papers? Or does actual science just not matter to scientists anymore? :-(

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    Replies
    1. Hi Jeff,

      As Peter says, these papers have little to do with "pixelation" and "scrambling." I know that the idea has sometimes been sold as "atoms of spacetime" but really if you look at the math, it has nothing to do with pixels.

      Delete
  16. Hello Jeff,

    You're misinterpreting these two papers. From the abstract:


    ... can be used to test the violation of the Lorentz invariance (LIV), predicted by some quantum gravity theories ... (boldface mine)

    So there are other theories involving pixellation of space-time that satisfy Lorentz invariance.

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  17. Fat Man wrote: Percentage of the Federal Budget is a ridiculous metric

    How about percentage of GDP? Do you have a good metric in mind, or a different approach altogether?

    Fat Man wrote: Is there any reason to think that additional money spent on physics will produce anything other than higher salaries for tenured professors?

    Any discussion about "more money for physics" is actually just a discussion about how nations decide to support and fund scientific research. One approach is a percentage of GDP.

    Whatever that percentage is, nations have to decide how to spend the money. It looks like you're suggesting that too much of that money is wasted on overpaid professors.

    Full professors at the most elite universities make between $150K and $200K. Otherwise, the average salary is between $70K and $100K. Professor salaries have not been increasing nearly as much as the cost of tuition.

    Theoretically, professors have two functions: They're supposed to do some productive research as well as teach and mentor.

    Has anyone analyzed and compared how nations spend money on scientific research, both theoretical and applied, and how this translates to results? For example, if someone says that the US "lags" behind China in a particular area, is that a result of China spending more money than the US? Or is it because China spends its research money more efficiently? Do Chinese researchers work a lot harder because they are hungry to become as powerful, rich and relevant as the US? Is it partly a numbers game? Does China have a lot more researchers and engineers simply because they have a lot more people?

    All other things being equal, if one nation has twice as many scientists and engineers as another nation, the long-term results are predictable.

    Not everyone has the ability or desire to be a scientist or engineer, but if we could somehow magically remove all obstacles, I wonder what percentage of a population would want to be scientists and engineers. Can a nation have too many good scientists and engineers?

    The US will spend enormous sums of money on a space race or a military arms race (actually, a space race is a proxy for a military arms race), but our willingness to spend as much on a science and engineering race seems more ambivalent.

    Back to your point about high salary tenured professors, and assuming you agree that government should spend a certain amount of public money on science research, what salary do you think is fair? Keep in mind that whatever you think the salary should be, it has to be good enough to motivate someone to get a PhD.

    The ostensible purpose of getting a PhD is to prepare someone for a research career. A $100K salary puts someone in the middle of middle class. Full professors at elite universities earn upper middle class salaries (enough to barely afford a small house in Cambridge or Palo Alto). You think that's too much?

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  18. Thank you Peter and Sabine!

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  19. Sabine,

    In order to distinguish between a dead cat and a living one there is no need to know "all quantum degrees of freedom". It is enough to determine the mass distribution so that you can conclude if the cat still moves arund, breathe, etc.

    So, by placing a torsion balance or something like that outside the box you can determine the cat's position and if it moves or not, and there is no box that is able to screen that information. So, in my opinion, position-superposition of macroscopic objects where the uncertainty in position is greater than the uncertainty implied be Heisenberg's principle, is not possible.

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    1. … can't you tell from the smell before you open the box … ;-)
      Contrary to Smolin I think there is nothing wrong with QM – all we need is a more realistic view how gravity is incorporated in the process.

      Delete
  20. Fat Man: Question. Is there any reason to think that additional money spent on physics will produce anything other than higher salaries for tenured professors?

    Sure.

    1) Additional money doesn't have to be spent on higher salaries, it could be spent on more salaries. It could also be spent funding more experiments, or better educated overseers to prevent proposal hype, or more facilities. It could be spent to fund more education and support for graduate students. None of those require increasing anybody's salary.

    Most people choosing the academic life have already made the choice to live on the salaries we get; those for whom giant salaries are important pursued other fields, like medicine, finance and business.

    2) There are unsolved problems in physics. More people, less hype, and funding of more experiments is more likely to produce new knowledge in physics than maintaining the status quo.

    3) We know there is something wrong with physics, there are unexplained phenomena and irreconcilable theories (GR and the SM) that each work in their realm but not together. We need new ideas to move forward. More minds are likely to produce more new ideas. But minds are housed in bodies that require money to remain physically and mentally healthy and happy enough to work eight hours a day.

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  21. If gravity exists at the quantum level; perhaps it becomes shared when in a superposition? And in that respect could validate a mechanism of entanglement? Spooky action at a distance.

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  22. Prof.Durgadas Datta published a few papers giving graviton a mass of 750 proton mass in emergent push fermion gravity mechanism. According to him massive graviton is a component of dark matter.

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