What is a Spinor?

Recently Jean-Pierre Bourguignon recently gave the Inaugural Atiyah Lecture, with the title What is a Spinor? The title was a reference to a 2013 talk by Atiyah at the IHES with the same title. Bourguignon’s lecture is not yet online, but I realized there are lectures explaining what a spinor is that I can highly recommend: my own, in this semester’s course on the mathemematics of quantum mechanics. I’m closely following the textbook I wrote.

Teaching this course this past academic year has made me all too aware of things that are less than ideal about the book, and I unfortunately haven’t had time to get to work on making any significant improvements. Going through the material on spinors though, I’m pretty happy with how that part of the book turned out, think it provides a clear explanation of a beautiful and important story, one that is not readily available elsewhere.

One aspect of this that I emphasize is the remarkable parallel between

  • The usual story of canonical quantization, which is based on an antisymmetric bilinear form on phase space, giving an algebra of operators generated by $Q_j,P_j$ acting on the usual quantum state space.
  • Replacing antisymmetric by symmetric, you get the Clifford algebra, generated by $\gamma$-matrices, acting on the spinors.

For a table summarizing precisely this parallelism, see chapter 32 of the book.

For more video from my office, I recently had a long conversation with Reza Katebi, who has a Youtube channel of interviews called The Edge of Science.

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11 Responses to What is a Spinor?

  1. More Anonymous says:

    Does anyone know Micheal Atiyah’s reason for distaste of supersymmetry (mentioned around minute 36 of the video)? He seems to evade the question :/

  2. Bass ackwards, starting from something inherently bosonic to get to something fermionic. As pointed out by John Conway, each of the normed division algebras is a spinor space. Starting from there you get – among other things – Dirac neutrinos and a matter dominated universe.

    I know this is too self serving to pass muster, but sometimes these conversations using ideas that are fundamentally flawed are a tad irksome. Start with spinors.

  3. Peter Woit says:

    Geoffrey Dixon,
    The discussion of spinors I give in the book and lectures isn’t bosonic, it’s inherently fermionic. What I find remarkable and wanted to emphasize is the tight parallelism between the initially very different looking bosonic and fermionic versions of canonical quantization.

    What this is most useful for is getting a uniform way of dealing with spinors in any dimension, including infinite dimensions. Fundamental matter particles are described by a quantum field theory based on fermionic canonical quantization of the solutions of a Dirac equation. The solution space is infinite dimensional, fermionic quantization gives a state space that can be thought of as the spinors for this infinite dimensional space.

    The question of how to best think about the physical spin 1/2 degree of freedom and spinors in real 3 space or 4 space-time dimensions is at the other end of the problem from a uniform treatment of spinors valid in complex vector spaces of arbitrarily high dimensions. For real vector spaces you get a much more intricate algebraic story, bringing in the quaternions at least. The mysteries of division algebras are relevant there, as is my currently favorite idea about fundamental space-time geometry (twistors, where points in space-time are precisely spinors).

  4. Several years ago I encountered Atiyah in London. This was not long after he’d given a talk at Princeton extolling the algebra C ⊗ H ⊗ O as a basis for a unified HEP theory, with the octonion algebra, O, accounting for gravity. Those attending the talk were nonplussed, for they’d expected something else. Anyway, back in London, surrounded by a fawning throng, I told him his division algebra idea was wrong in its essentials. He was chagrined.

    Still, that algebra, like twistors, has the hope of forming a bottom up theoretical HEP foundation, one that might lead inexorably to unique and unavoidable explanations of many of the things that perplex us. As I am not an expert in QFT, this comment may be out of line, but it has always seemed to me that reliance on QFT to show the way is top down, relying on inconsistencies, infinities, and anomalies to exclude paths that are not viable – trial and error.

    As to twistors, in one of Penrose’s tomes he referenced me, along with a cluster of others who exploited division algebras beyond C. But it was not a positive reference. His twistors are intimately connected to C, and he felt that going beyond C would be like pouring molasses on caviar – unnecessary, and unwanted. Your willingness to consider pouring higher dimensional division algebras onto twistors I find intriguing. I have hopes it’ll be more like pouring maple syrup on pancakes.

  5. Peter Woit says:

    Geoffrey Dixon,
    Unfortunately, as far as I can tell, Atiyah’s enthusiasms in the last few years of his life were not based on anything very substantive, and I think that includes whatever he was thinking about octonions and gravity.

    Most of what has been achieved with twistors crucially uses holomorphicity which is based on complex numbers, so I can see why Penrose thinks those are crucial. Twistors naturally give you not space-time but complexified space-time, and if you want to work with fields on this, you need holomorphicity in order to get usual fields on usual space-time.

    One place where it is clear that quaternions play a role in twistor theory is in the Euclidean signature version, where the conformal group is $SL(2,\mathbf H)$. Penrose must know about this, but to get back to Minkowski signature you need holomorphicity, so complex not quaternionic analysis. The fact that twistor space can be taken to variously be $\mathbf C^4$ and $\mathbf H^2$ makes one suspicious that there’s an octonionic story buried there somewhere. But when I try and understand what people have tried to do along these lines, I remain baffled so far, unclear to me what’s going on.

  6. Chris Oakley says:

    Backtracking a little bit, physicists quite reasonably objected to relativity on the grounds that measurements of distance and time could not possibly depend on the observer’s motion and to quantum theory on the grounds something could not possibly be a particle and a wave at the same time. Electron spin, although it explained doublets in spectral lines nicely, was another idea that took a while to be accepted. I am guessing that one objection was that all objects in nature surely had to be invariant under a 360° rotation. So if spinors are real, spacetime is not as simple as we imagined. I would not know what to do with Penrose’s notion of spacetime being formed from composite spinors, but it does at least “explain” this one thing.

  7. Bertie says:

    The interview with Reza Katebi was wide-ranging and insightful.
    At 2 hrs long, I never expected to watch it through to the end, but am happy I did
    🙂

  8. Rollo Burgess says:

    Thanks v much for your video interview which is is very interesting.

    I am also thrilled finally to find a topic on this blog in which I possess some cognitive authority: per the comment in the video, in cricket a googly is a delivery which has an off spin but is delivered with a leg spin action, thus tricking the batsman into thinking the ball will break in the opposite direction on the bounce from what it in fact does. 🙂

  9. I says:

    Well, your book is quite good for an auto-didact. Trouble is, it is lacking in exercises. What other flaws do you think there are?

  10. Peter Woit says:

    I,
    Yes, there are some exercises, but too few. There are a bunch of places in the book where I’d now do things somewhat differently, or in a different order. One major flaw I think now is chapters 24-26, which spend too much time developing certain things that will get used later, but probably should have been relegated to an appendix, to be consulted as needed.

    In recently years I’ve spent a lot of time thinking about the Euclidean version of QM, and someday would like to incorporate some of that into the book.

  11. Art says:

    Look likes the Bourguignon video is now up.

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