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Quaternions in QM

  1. Mar 6, 2009 #1
    Has anybody ever thought of using quaternions in QM? If so, why stop there? WHy not use octonions, etc. ? Just curious ...
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  3. Mar 6, 2009 #2


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    I just recently ran across a http://arxiv.org/abs/hep-th/0607121" [Broken] on that matter by Charles Schwartz (a Berkeley prof emeritus).

    It's mostly a novelty, I think. It doesn't really make sense to do so unless it simplifies things, which it doesn't appear to.
    Last edited by a moderator: May 4, 2017
  4. Mar 6, 2009 #3
    There is a famous theorem, which says that axioms of "quantum logic" can be satisfied only in Hilbert spaces over R (real numbers), C (complex numbers) and Q (quaternions)

    C. Piron, "Foundations of Quantum Physics", (W. A. Benjamin, Reading, 1976)

    Octonions are not in this list. Nevertheless, there were attempts to build octonionic QM. See, for example,


    and references there.

    Real and quaternionic QM were also investigated:

    E. C. G. Stueckelberg, "Quantum theory in real Hilbert space", Helv. Phys. Acta, 33 (1960),

    J. M. Jauch, "Projective representation of the Poncare group in a quaternionic Hilbert space",
    in Group theory and its applications, edited by E.M. Loebl, (Academic Press, New York, 1971).

    However, as far as I know, nothing exciting came out of that. The main problem with quaternions is that it is not possible to define the tensor product of two quaternionic Hilbert spaces. So, the description of multiparticle systems is questionable.
  5. Mar 6, 2009 #4


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    Why is that? (Is it explained in one of the references you mentioned? - I don't have Piron.)

  6. Mar 6, 2009 #5


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    Quaternions show up in physics whenever they make your life easy, which is relatively infrequently b/c of notation clutter. They show up from time to time when dealing with clifford algebras (so spinors and fermions), division algebras and so forth. I am unaware of a physical context where they absolutely must be used over anything else.
  7. Mar 6, 2009 #6

    George Jones

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    Quaternionic Quantum Mechanics and Quantum Fields by Stephen L. Adler,


    Adler gives his answer in section 2.7 Nonextendability to Octonionic Quantum Mechanics.

    It's been over ten years since I looked briefly at this book; I forget what's in it.
  8. Mar 6, 2009 #7


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  9. Mar 6, 2009 #8
    This is related to the non-commutativity of quaternions. I don't remember exact arguments now, and I don't have good references. The best thing would be to search the web for "quaternionic" "quantum mechanics" "tensor product". It yields a number of hits that look useful.
  10. Mar 7, 2009 #9
    Yes, definitely the noncommutivity of quaternions would pose a problem, not to mention defining probability densities, wavefuctions, etc... Although the noncommutativity could shine some light on operators, (but then, the Hilbert space problem would come up again) I guess it would be generally infeasible. Thinking farther, complex numbers are open to a more 'natural' interpretation than quaternions, octonions, etc. (see for exp. Nahin's interesting book "Imaginary Tale: The History of the square root of -1"), so maybe this could partly explain my problem
  11. Mar 12, 2009 #10


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    I realize this thread is a bit old, but I would like to draw your attention to something related to quaternions and that is Geometric Algebra. You might be interested in the book Geometric Algebra by Chris Doran and Anthony Lasenby. Geometric algebra actually is one of the Cliffird algebras and quaternions fall under that heading as well. Doran and Lasenby give the details. David Hestenes and others have done a lot of work, with some success, in applying Geometric Algebra to quantum mechanics.
    Last edited: Mar 12, 2009
  12. Apr 18, 2009 #11
    James Clerk Maxwell originally used quaternions in his work on Electromagnetism but there was a dispute/difference of opinion regarding their use that led him to use vectors later on. As Dr. David Hestenes points out in his many writings on Geometric Algebra and Geometric Calculus, quaternions have been popping up in quantum mechanics for many years but often disguised as matrices and spinors etc. so they were not recognized as quaternions.
    You will find that there are several physicists that are using quaternions in their work. For example, Dr. Mendel Sachs used quaternions in his research program as a common mathematical language in which to write both Quantum Mechanical and General Relativistic equations (See "Quantum Mechanics from General Relativity") https://www.amazon.com/exec/obidos/ASIN/9027722471/qid=926450603/sr=1-3/002-5908934-8425436.
    Another physicist that is using quaternions is Doug Sweetser (http://world.std.com/~sweetser/quaternions/qindex/qindex.html).

    The authors of "The Vector Calculus Gap" have a paper on Octonions and the Standard Model (http://fqxi.org/large-grants/awardee/details/2008/dray)

    I highly recommend approaching quaternions from the Geometric Algebra/Geometric Calculus side to avoid some of the problems cited by Dr. Hestenes with redundancy of formalisms in mathematical physics.
    Last edited by a moderator: May 4, 2017
  13. Apr 18, 2009 #12
    Thanks for the links!
    To me it seems EM should be with Quaternions and only if you write at as components, it become vectors. Don't quaternions have a shorter notation?
  14. Apr 19, 2009 #13
    Quaternions not having commutivity isn't as big as a problem as it sounds - neither do vectors and cross products. However, each "generation" of complex dimensions you add makes you lose additional mathematical properties. For example, octonions aren't associative. This property is one of the biggest reasons why generations beyond quaternions are relatively uncommon. You'd only use octonions (for example) if you actually want loss of associativity.
  15. Apr 23, 2009 #14
    Imo, i think that quaternions (or more accurately, bivector-valued tensors) provide such a powerful bridge between differential geometry and complex operator theory (see e.g. http://modelingnts.la.asu.edu/html/GCgravity.html [Broken]) is because the clifford bivector is the pseudo-complex analogue of the real vector cross product, an anticommutative lie bracket. so every bivector corresponds to a unique topology representable as a traceless matrix (the trace part being the scalar inner product term of a given spinor).

    As for multiparticle systems, ...errr...we're working on it. There's not really much support for field theories, partly because the fourier analysis on clifford algebras is still being researched =). but electrodynamics shows promising results, and i'm doing my part with a bit of interesting new Bayesian theory (to appear in IJTP).
    Last edited by a moderator: May 4, 2017
  16. Jun 29, 2010 #15
    Physicists that think that quaternions are of no importance for quantum mechanics and that instead Clifford algebras must be used have never considered the effect of a quaternion waltz ab/a. This is in general not equal to b. b is precessed! (a part of b is rotated). The combination of a unitary transformation and an observation already involves a quaternion waltz. Physicists that stick to complex QM will never notice this effect. If you analyse it then you will encounter the source of relativity!!!

    Ever heard of 2n-ons? See http://www.math.temple.edu/~wds/homepage/nce2.pdf [Broken]

    For Hilbert space on octonions see Horwitz : http://arxiv.org/abs/quant-ph/9602001

    If you specify a Hilbert space over the quaternions, then that should not withhold you of using higher 2n-ons as eigenvalues of operators. How else would you get all fields represented in the action S, which according to Dirac appears in the argument of a 'unitary' transform?
    Last edited by a moderator: May 4, 2017
  17. Jun 29, 2010 #16

    Ben Niehoff

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    The pairwise products of the Pauli matrices are basically the unit quaternions:

    [tex]\mathbf{i} = \sigma_2 \sigma_3[/tex]

    [tex]\mathbf{j} = \sigma_3 \sigma_1[/tex]

    [tex]\mathbf{k} = \sigma_1 \sigma_2[/tex]

    So technically speaking, quaternions show up in the theory of angular momentum; however, physicists prefer the matrix notation.
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