Dear Professor Braun,

We read your essay with lot of interest, and enjoyed it, especially for the novelty of the idea.

We have a question in the context of non relativistic quantum mechanics and the process of quantum measurement. Do you adhere to the Copenhagen interpretation (this is the impression we get from reading your essay)? Many physicists including us regard this interpretation as problematic, because of the artificially assumed quantum - classical divide between system and apparatus. Let us consider the following three popular alternatives to Copenhagen:

Many worlds interpretation: no collapse of the wave-function, branching of alternatives [origin of Born probability rule obscure]

Bohmian mechanics as an equivalent mathematical reformulation of quantum mechanics - probabilities arise because of the so-called typicality assumption of initial condirions [We could not understand your remark "The lack of a consistent hidden variable description of the microscopic world entails a jump in cardinality of the set of possible physical theories from א2 to א3! "....to our understanding non-local hidden variable theories (generally referred to as Bohmian mechanics nowadays) are consistent at the non-relativistic level].

Phenomenologically Modified QM : GRW / Continuos spontaneous Localisation (CSL): Born rule arises from the stochastic nature of the modification.

While all these three variants have their own limitations and may turn out to be wrong eventually, it is our impression they do better than the Copenhagen interpretation.

We were wondering if the cardinality issue gets modified (looking more like classical) f you consider many-worlds / Bohemian / CSL, or is there no change compared to standard QM? We would be very interested to know what the answer is.

Thanks and regards,

Anshu, Tejinder

Dear Tejinder,

Thank you very much for your kind remarks and interest.

I did not make an assumption about the interpretation of quantum mechanics, and I don't think it enters into the cardinality calculation. The reason is that I consider possible QM theories as mappings from Hamiltonians to time evolution laws (plus possibly representations of operators for relevant physical quantities), which I think should be uncontroversial between different interpretations. The status of the wave function does not enter anywhere.

This being said, I did have in an earlier version a remark about Bohmian mechanics referring exactly to the point about the hidden variables that you mention, but had to cut it due to the length restriction. So I profit from your question to post part of that remark:

"Bohmian mechanics does not help in reducing the complexity of the set of

quantum theories: One still has to solve Sch\"odinger's equation in order to calculate the guiding field, and so the whole Hilbert-space setup of quantum mechanics (which we hold responsable for the jump from $\aleph_2$ to $\aleph_3$) is not avoided by Bohmian mechanics, even if the interpretation of the wave function is different."

I think the same conclusion holds for GRW-type theories. There, the mentioned mapping from the hamiltonian to the time evolution is different from standard QM, of course, (i.e. they are not just a different interpretation of QM) but it is just one mapping in the whole set considered already. So the cardinality of the set of QM theories is not changed.

Many worlds is more tricky. I think one would have to specify whether one wants to include the entire plethora of branches with their observables, Hilbert spaces and so on. If so, I have the gut feeling that the cardinality of the set of possible QM theories might be affected, but haven't thought about it. Definitely, I only considered QM in one universe.

Does this answer your questions? Thanks once more for bringing up these interesting points!

Best regards,

Daniel

    Dear Daniel,

    Many thanks for your answer. I tend to agree with you. Many worlds is likely to be different, because no collapse takes place, and probabilities are likely only an apparent effect.

    Best regards,

    Tejinder

    25 days later

    Dear Daniel,

    Thank you for an interesting and ambitious essay. You raise original questions about calculating the cardinality of the ensemble of all physical quantities and physical theories. I found the following sentence particularly interesting:

    "We can safely claim that no useful physical quantity is known that involves powers larger than say one hundred."

    Indeed, of all the SI derived units (newton, joule, watt, etc.), only 4 involve a third (or minus third) power, and only one, the farad, involves a fourth power : F = 1 kg^в€'1 x m^в€'2 x s^4 x A^2. I had never really given a second thought about the issue... very interesting! NASA engineers that worked on the Hubble space telescope found it useful to define the jounce as the fourth derivative of the position, so its units are m/s^-4, but once again, we don't get higher than fourth power!

    Later on in your essay, I think you made an interesting distinction between physical laws of "fundamental type" and of "secondary type" (that involve weird and arbitrary functions of dimensionless ratios).

    Your essay deals in an original and interesting way with the question of the cardinality of possible physical laws compared to the cardinality of all possible mathematical laws, and it makes a worthwhile contribution to the ideas that have been put forward in this contest. Strangely enough, it has been a bit forgotten so far in the competition, and I hope bumping it higher will make it more noticeable. Good luck!

    Marc

      Dear Marc,

      thank you very much for your kind remarks, and bumping up the rating of my essay. Given the small number of ratings so far, this had an immediate positive effect :-) [as an aside, initially the essay was in the top 5 for several days, till a community member shot it down by apparently giving it only 1 point, whereupon it ended up in the middle range and slipped into oblivion. I wish he/she who did that rather offered some public criticism, whereupon we could have a constructive discussion...]

      Anyhow, yes, I had looked up tables of known physical quantities, and also came to the conclusion that a power of order 3 or 4 is the largest one appearing. I put an upper bound of 100 to be on the safe side. I was not aware of the jounce - glad to learn about it!

      And yes, the apparent preference of Nature for laws of the "fundamental type" is strange. I wonder, however, whether there is a human bias in this in the sense that we favor simple laws, and are certainly also more capable of extracting them from experimental data than more complicated ones. Evolution trained us to make sense out of complex environments in a most efficient way, and we certainly have a bias towards "simple" laws which we experience as more beautiful than complicated ones.

      Recently we discovered a really ugly one: a formula for the Fisher information relevant for doing measurements with photon-added states ((A1) in here or (35) in there for the arxiv version). The formula goes over a whole page, contains up to 11th powers, and one can only make sense out of it by plotting it as function of some variables while keeping others fixed, or looking at limiting cases and so on. We were debating whether to publish it at all.

      Also, thanks to Jens Eisert and co-workers, we know that extracting dynamical equations from experimental data is NP-hard, but certainly it is easier to find simple patterns than complicated ones. I would doubt for example, that our mentioned formula going over one page could have been found empirically by analysing measurement data. In a certain sense it is rather a mathematical quantity (already it is dimensionless) and was derived mathematically (using Mathematica :-)

      So there is certainly a bias towards simle formulae in both our human appreciation, and in the possibility of finding them. But is that all?

      Maybe with machine learning becoming more and more powerful we'll have to get used to find ever more complex laws, in areas where we did not even expect any new laws nor physical quantities to be found...

      Or another competition could be created for finding the most (irreducibly) complicated formula connecting some (to be newly defined?) physical quantities. Of course, this runs completely against the whole philosophy of physics and our trying to make sense out of things...

      Thanks once more for your interest, and good luck in the contest,

      Daniel

      Dear Daniel,

      That's quite an equation in your paper! If I understand correctly, it has been obtained by expanding some parameters to the tenth order, so it could be even "worse" if it was expanded further?

      I agree with your that FQXi should organize a mock contest on the subject of the most ugly fundamental physical equation! Maybe next April 1st? :)

      Marc

      Dear Marc,

      no, that equation is supposed to be exact (incidentally, I just learned that it might contain some sign error, so I have to recheck it and possible write an Erratum. But other than that it should be exact). It results from an exact Gaussian integration of a high-order polynomial.

      And yes, such a mock contest on April 1st (with the ugly equation to be found still required to be non-trivial and correct) might change a little bit the perspective of us beauty-hungry physicists :-)

      Best regards,

      Daniel

      Dear Daniel,

      I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

      All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

      Joe Fisher

      7 days later

      Hello Daniel,

      I greatly enjoyed your excellent essay. As luck would have it; I use the term 'theory of theories' in my own essay, but in a rather different way. Phil Gibbs put forward a theory of theories in his 'Cyclotron Notebooks' referring to the idea that the reality we observe may not be the product of a singular mathematical pattern or model (a GUT or TOE), but rather that the universe results from a kind of path integral whose output is shaped by the full range of applicable theories - as a weighted average. In this way; we would not have to choose between loops and strings, for example, because the laws of nature employ the regularity of the Maths in both cases at once.

      This notion has shaped my thinking for quite some time, but I had not thought before now that one might be able to determine the cardinality of the full array of possible theories. So you give me a lot to think about! I have boosted your score a bit, but I hope others push it higher, because you deserve a higher standing than you presently enjoy.

      All the Best,

      Jonathan

      Hello Jonathan,

      thank you very much for your kind words and pushing my score. I read the paper by Lloyd and Dreyer on the universal path integral but frankly couldn't make much sense out of it. So now I am curious to read your article, and the paper by Phil Gibbs that you mention.

      I think it is an intriguing idea that laws of physics themselves might be the result of some interference, but is there even in principle a way to test this?

      All the best,

      Daniel

        Thanks for the thoughtful reply Daniel..

        Sorry someone else pushed your ranking back down. As regards to my essay's thesis in relation to an idea in your essay; exact determinations of various quantities may have a built-in difficulty, and this can present problems with mathematization. Iterating the squaring function on the complex plane yield a bounding surface at r = 1 - any value smaller converges to (0,0i) and seed values further from the origin all diverge or are repelling points, while a value whose distance is 1 from the origin stays on the boundary forever.

        But this requires infinite precision, or the preservation of unitarity as a fundamental value. If there is any noise on our calculation, or a truncation due to limited precision, this imprecision will drive the successive iterands to 0 or infinity. This is sort of like what Hawking has lately said about the fluctuation of an event horizon for a black hole being like weather forecasting. A built-in uncertainty, indeterminacy, or measurement imprecision, can all have the same effect - to drive a system to a particular result.

        If we use z^2 z instead of just z^2, things become much more interesting, but perhaps a little jitter is part of the equation. This gives me much food for thought.

        Regards,

        Jonathan

        I want to thank you Daniel..

        I had not seen the Lloyd and Dreyer paper before your mention above (or had filed it away and forgot), but now I have downloaded it and it looks very interesting.

        All the Best,

        Jonathan