Quantum Bayesian Networks

November 8, 2014

I’m a Born-Again Group-Pest

Filed under: Uncategorized — rrtucci @ 9:23 am

Trying to learn Newtonian mechanics without a priori or concurrently learning Calculus is not a good idea, in my opinion. (I mean here Calculus taught in a non-rigorous, applied way; I don’t mean rigorous Mathematical Analysis). Indeed, non-rigorous Calculus is extremely helpful and insight-giving in doing Newtonian mechanics. It’s not surprising that both disciplines were invented concurrently by the same guy, Isaac Newton.

Analogously, I think that trying to learn Quantum Mechanics (QM) without a priori or concurrently learning Group theory (GT) is a terrible idea. (Again, I mean here GT taught in a non-rigorous, applied way). Non-rigorous GT is extremely helpful and insight-giving in doing QM. GT and QM were not invented concurrently by the same physicist, but they could easily have been. Mathematicians like Schur, Frobenius, Young, Galois, etc., had already developed GT to a high degree before QM came along, but there is no doubt that the invention of QM has spurred the invention of many new tools in GT.

As one can see from the Wikipedia “Timeline of Quantum Mechanics”, most of the fundamentals of QM were discovered during the miraculous decade 1920 to 1930. Note, for example, that in
1923 – Louis de Broglie extends wave–particle duality to particles,
1930 – Dirac’s textbook “Principles of Quantum Mechanics” was published

Note also that Hermann Weyl (1885-1955) came out with his book “The Theory of Groups and Quantum Mechanics” (available as a Dover edition) in 1930 and Eugene Wigner (1902-1995) with his book “Group Theory and its Application to the Quantum Mechanics of Atomic Spectra” in 1931. So already during the miraculous decade, and certainly long thereafter, Weyl and Wigner were common household “group-pests” who never lost a chance to point out to anyone within earshot how closely related QM and GT are.

As you can see, the observation that a knowledge of GT greatly helps one to understand QM is an observation that is almost as old as QM itself, and this observation has been widely publicized by some of the patron saints of QM.

I learned QM from some books that I think are very good (for example, the 2 volume set by Cohen-Tannoudji, Diu and Laloe), but, unfortunately, those books don’t use GT explicitly. Then I learned a bit of GT from some books that I think are pretty lousy. Either because of the shortcomings of my first GT books or due to my own shortcomings, or both, it took me a long time to feel comfortable with GT. The tide first started to turn for me when I read the two volume set by Elliot and Dawber entitled “Symmetries in Physics”, two books which I find truly wonderful and excellent. Using the Dawber and Elliot books as my “base camp”, I was able to explore GT further and read other more recent and specialized GT texts. You might prefer different books than I do. The field of GT is so vast, I think no single book or author can cover everything in detail and satisfy the topic selection and stylistic tastes of everyone.

Nowadays, I feel much more comfortable with GT, and I feel compelled to analyze in GT language everything that I do in QM, and I mean everything, be that physical chemistry, AMO (atomic-molecular-optical) physics, condensed matter physics, nuclear physics, particle physics, or relativity. It’s amazing how widely applicable GT is in physics.

Of course, finding applications of GT to quantum computation and quantum Bayesian networks is high in my agenda. My latest patent, number 12 in this list, is about GT. Two previous blog posts of mine on GT are

2 Comments »

  1. For some pesky Group-Pest work see…

    http://www.pspchv.com/Abstract-PJMPA/PJMPA%20Vol.%205,%20Issues%201-2,%202014,%20Pages%201-53.pdf

    This is my (finally) published result from 1991 where I used group theory to construct a ray representation of Classical Mechanics.

    That representation turns out the be unique (thanks to an old result of Hermann Weyl in the book you mention).

    Group theory is also central to the quantum invariant prior that I introduced in my PhD thesis: “Quantum inference and the optimal determination of quantum states”

    http://www.amazon.com/Quantum-Inference-Optimal-Determination-States/dp/146816449X

    On another front, it is absolutely central to the (surprising) result that QM dynamics is actually a special case of CM dynamics.

    This is due to the group identity:

    U(n) = Sp(2n) intersect O(2n)

    See the paper:

    https://www.academia.edu/4832116/The_Schr%C3%B6dinger_equation_from_three_postulates

    Indeed, physicists have long had a “problem” with group theory in having failed to fully appreciate its impact.

    This is especially true in quantum computing.

    The other blind spot is that physicists now associate group theory predominantly with *linear* group action (i.e. linear transforms).

    The physics community, even today, are pretty ignorant of where the original ideas of Sophus Lie came to light: in continuous symmetries for general differential equations.

    Those methods apply equally well to non-linear equations along with the related work by Emmy Noether (as should be obvious given the relevance of her work to classical General Relativity).

    The problem today is that graduate school students are exposed to a limited menu of “results” which fall within the canon of quantum field theory but which are misshapen to fit the assumptions of that particular theory. As a result, the development of mathematical physics has been greatly stunted in the last three decades.

    These days it is mostly baby talk and the burbling of idiots reciting standard results and considering them important because they are technically involved. To the extent that they rely on linear methods and assumptions they are actually pretty unrepresentative.

    I am reminded of Einstein’s great jibe on QFT:

    “I see in this approach only an attempt to treat essentially non-linear phenomena using linear methods.”

    Too true…

    Precious little of the contemporary work in physics on unification or strong-field regimes (like superconducting qubits) pays even a passing acknowledgment of this issue.

    It is what I call “Puff-Dumpling Physics.”

    It keeps the kiddies happy but there is no meat to it.

    Baby food accompanied by Baby talk.

    Comment by Kingsley Jones (@savvyyabby) — November 10, 2014 @ 2:47 am

  2. “This is especially true in quantum computing.”
    Kingsley, I tend to agree with you there. I’m guilty of it myself and will try to mend my ways. I’m no Wigner or Weyl, but I’d love to find some simple but powerful applications of group theory to quantum computing.

    Comment by rrtucci — November 10, 2014 @ 6:08 am


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