SUSY and Quantum Mechanics

You may have read somewhere today that Columbia professor strips down to underwear in bizarre lesson to help baffled students learn quantum mechanics (first-hand sources here and here). That wasn’t me, but I have been talking to my class for the last couple weeks about quantizing fermionic variables, and some simple quantum mechanical examples of supersymmetry. Notes on the supersymmetry stuff are here, earlier notes on the fermionic version of quantization and what it has to do with Clifford algebras and spinors are on this page.

These notes are not quite finished, mainly because I’ve been trying to sort through the hairy issue of sign conventions that comes up when you start dealing with a Hermitian inner product on anti-commuting variables, something you need to do to get unitary representations. There’s a detailed treatise on the subject by Deligne and Freed, who are very smart and sensible, but I’d like to understand this better. The choices they make end up leading to odd self-adjoint operators having eigenvalues proportional to a square root of i, which is consistent, but not exactly intuitively clear. The best source for finding details of the mathematics used in SUSY is probably the IAS volume that Deligne/Freed is part of. The first part includes valiant efforts by Bernstein, Deligne, Freed, and Morgan to get the mathematics right, including the signs (they say “Writing this has been an absolute cauchemar de signes!”). One sign they get wrong is a typo on page 91 (equation 4.4.5).

The parallel stories of bosonic and fermionic oscillators are among the deepest things in theoretical physics, and involve just spectacularly intricate and deep mathematical ideas (symplectic geometry, rotation and spin groups, Heisenberg groups, the metaplectic representation, Clifford algebras and Weyl algebras, spinors, etc., etc…). I hope the course notes I’ve been writing give a little insight into this and the way Lie groups, Lie algebras, and their representations are involved. Generically, “supersymmetry” refers to generalizing the notion of a Lie algebra to include odd generators, and thus get a “super” Lie algebra, sometimes acting in an interesting way that mixes even and odd variables. In the notes I describe two very simple examples, showing how one gets a “square root” of the Hamiltonian operator.

There are all sorts of interesting structures one can get by looking for supersymmetrical versions of QFT, and the IAS volume describes a lot of them. One wonderful example is the N=2 susy gauge theory that gives a TQFT with observables four-manifold invariants. This is an unphysical theory, but tantalizingly close to physical theories. It involves a “twisting” mixing the space-time and internal symmetries which might be the sort of thing needed to avoid the problem of the kind of “superpartners” that is deadly for SUSY extensions of the standard model.

Perhaps the most compelling example though is the way the fact that the Dirac operator is a square root of the Laplacian can be thought of as an example of SUSY. This is one of the deepest ideas in mathematics, something whose implications I suspect we still don’t completely understand.

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18 Responses to SUSY and Quantum Mechanics

  1. P says:

    Hi Peter,

    Nice post – I’m curious to look at the notes.

    I’m confused by one of your paragraphs, though. You’re right to point out the importance of SUSY gauge theories in studying interesting mathematical quantities – including the Donaldson-Thomas invariants you presumably had in mind, but also many other quantities.

    I was wondering if you could elaborate on

    “It involves a “twisting” mixing the space-time and internal symmetries which might be the sort of thing needed to avoid the problem of the kind of “superpartners” that is deadly for SUSY extensions of the standard model.”

    As I see it, one can argue (as you do, but I disagree with) that the hierarchy problem is not a problem and that SUSY is not well motivated. One can also argue that we won’t see superpartners at the LHC. Was this the kind of “problem” you meant? There is no theoretical problem with superpartners in BSM models, and in particular the soft masses could be at a high scale and they could completely decouple from low energy physics. There is no theoretical problem, so I’m just very curious what you had in mind!

    Moreover, whatever the problem is, how you do expect the topological twist to help with the problem?


  2. Peter woit says:

    The basic problem with SUSY extensions of the SM is that in them SUSY doesn’t mix any degrees of freedom you know about. Instead it takes each degree of freedom you know about and mixes it with a new unobserved one (the superpartner) that you have to postulate, and then start jumping through hoops in order to make unobservable. This new “symmetry” one has introduced is trivial on the world as we know it.

    The reason I mention the Dirac operator and the N=2 TQFT case is that in these cases there is a supersymmetry, but it does something more interesting than taking bosons to fermions with spin changing by 1/2. In the Dirac operator case, the Z2 grading is that of even/odd spinors: the SUSY operator (Dirac operator itself) takes left-handed spinors to right-handed spinors and vice versa (this is just QM though, not QFT). In the N=2 TQFT case (Witten’s first 1988 TQFT) the twisting basically turns this into a theory of differential forms, and the SUSY is like a deRham differential taking even to odd forms and vice-versa. Because of this though, this theory violates spin statistics and is non-physical.

    I’m certainly not claiming I have an intelligible idea of how to construct a physical QFT that avoids the standard usage of SUSY, just pointing out that there are examples of theories where it does something different than the usual problematic thing.

  3. Bob Jones says:

    “You’re right to point out the importance of SUSY gauge theories in studying interesting mathematical quantities – including the Donaldson-Thomas invariants you presumably had in mind”

    Did you mean to say Donaldson invariants? Donaldson-Thomas theory is really a topic in enumerative algebraic geometry rather than topology.

  4. P says:

    Hi Peter,

    Very good! Thanks for explaining what you mean. I agree with you that this is something very important to emphasize to the outside community: SUSY is a theoretical framework that may or may not have relevance for particle physics, but like any truly deep theoretical structure it has implications for other fields. The mathematical impact of SUSY gauge theories and string is immense: from uncovering mirror symmetry, to the Donaldson-Thomas invariants you mentioned, to recent work on 3d-3d correspondence, 2d-4d correspondence, exact partition functions, and their relevance for three-manifolds, GW and GV invariants, etc. It’s really amazing that QFT has anything to say about mathematics that mathematicians didn’t already known, and this is certainly true of some SUSY QFT’s. Also, thanks for pointing out the relevance of the deRham complex – I haven’t looked at that paper in awhile.

    On the other issue: since you don’t like strings or BSM SUSY for the hierarchy problem, it surely seems like these new degrees of freedom are pointless. But for those of us that do, mixing known degrees of freedom with unknown degrees of freedom is the whole point. e.g. this is what allows for the non-renormalization of the Higgs mass (or any scalar mass) above the scale of supersymmetry breaking. Once one is building SUSY models BSM, the values of the soft masses are model dependent. Some models are killed; others are not. This isn’t jumping through hoops, it’s just the normal death that any set of models goes through as experimental bounds get better. Perhaps people start studying new models over time based on bounds, but that model was always there in theory space, regardless of whether people studied it! 🙂


  5. Peter Woit says:

    P and Bob Jones,

    I was referring to Donaldson, not Donaldson-Thomas. Specifically Witten’s 1988 paper
    “Topological quantum field theory”,
    this paper takes N=2 susy on a four-manifold and uses the twisting trick to get a TQFT with Donaldson invariants as observables. Many mathematicians were not so impressed by this because it didn’t seem to tell them anything new about Donaldson invariants. They had to eat their hats a few year later when the Seiberg-Witten paper about this same theory came along.

    While mirror symmetry and QFT arguments telling one new things about algebraic geometry are all well and good, what’s remarkable is that this is a simple 4d QFT with gauge fields, fermions, scalars, looking very close to the kind of theory that describes the real world. It doesn’t deserve to be forgotten…

  6. P says:

    Certainly SUSY QFT has importance consequences for enumerative algebraic geometry, also. I’m forgot the exact context of SW theory and four-manifolds, though. It’s D and not DT?

    “While mirror symmetry and QFT arguments telling one new things about algebraic geometry are all well and good, what’s remarkable is that this is a simple 4d QFT with gauge fields, fermions, scalars, looking very close to the kind of theory that describes the real world. It doesn’t deserve to be forgotten…”

    True, but remember that though N=2 theories can come close to being realistic in a certain sense that you just mentioned, the absence of chiral matter is fatal. In this sense N=1 theories have a shot, and N=2 theories do not.

    Also, I read a little too quickly and though you were talking about the magnificence of the standard model for a second – another model that also doesn’t deserve to be forgotten 🙂

  7. Lowell says:

    Gee, there is a long discussion in my book “Quantum Field Theory” Sec. 2.4 about
    Fermonic variables, Grassmann numbers, and such.

    Is there anything wrong with this exposition?

    Lowell Brown

  8. Peter Woit says:


    Thanks for writing. Your book is on my shelf, and I’ve been consulting it, maybe more so this semester as I cover some QFT.

    You give a very good discussion of coherent states for fermions and the fermionic path integral, but in my course I’m not covering this (for lack of time), sticking to the Hamiltonian formalism (which you don’t emphasize). What I’m trying to explain is the fermionic analog of Poisson brackets, quantization, and how this quantization gives you unitary representations of the symmetry groups of the situation. In the fermionic case things the part that’s not straightforward is unitarity, you have ghosts to worry about.

  9. D.R. Bocancea says:


    “This new “symmetry” one has introduced is trivial on the world as we know it.”
    There is a name, or a paper , for this ..?


  10. Peter Woit says:

    D.R. Bocancea,

    In SUSY extensions of the standard model, the SUSY symmetry generator act by taking known state to unknown superpartners. Restrict to the states we know, this is the zero operator, so the states are invariant.

  11. chris says:


    no, that is perfectly fine, but you might once again go over your proof of a massless particle having half-integer spin 🙂

  12. Shantanu says:

    Peter, something OT.
    but could (or someone elsE)you help me decipher this press release
    how is higgs boson related to fate of the universe?

  13. Nick M. says:

    Hi Shantanu,

    You might want to check out this recently published New Scientist article, written by supersymmetry evangelist Lisa Grossman :), that talks precisely about this issue with many helpful links to other news articles (most notably the BBC News and NBC News) as well as having links to other original technical papers and letters. For example, see this paper published in “Physics Letters B” from this past August, which talks about the relationship between the Higgs boson and top quark masses as it relates to vacuum stability, and this letter in “Letters to Nature” from August of 1982, which talks about vacuum stability in a more general sense.

    — Nick M.

  14. Peter Woit says:


    This is based upon assuming we completely understand the effective Higgs potential at currently accessible energies and then extrapolating up to GUT energies, assuming no new relevant physics. The fact that the Higgs mass is such that the quartic term in the Higgs is getting very small at such high energies is intriguing, but I wouldn’t describe any such metastability result as a reliable calculation.

  15. M Uppal says:

    Dr. Woit, may I ask that you please teach this class again next year? At the very least you’ll have one student enrolled 🙂

  16. P says:

    Peter and Shantanu,

    It’s important to differentiate between an unstable vacuum and a metastable vacuum. The story about extrapolating up to GUT scale and the quartic coupling becoming zero would cause our vacuum to become unstable. This is in contrast with the possibility that there is another vacuum and we tunnel to it out of a metastable vacuum. Talk of bubble universes and the like (as here) is the later case.

    In any case, Peter is absolutely right that this calculation makes huge assumptions not only about the Higgs potential and couplings, but also about the absence of other scalar fields. The latter completely changes the story, and from everything we know about cosmology and particle physics (let alone semi-speculative top-down approaches), the Higgs is almost certainly not the only scalar field in nature.


  17. Peter Woit says:

    M. Uppal,

    I won’t be teaching this next year, but anyone reading through the notes who has any questions is encouraged to come by my office, I’m usually there….

  18. Nick M. says:

    p>Thanks, Peter, for the clarifying comment. One often gets the feeling, when reading through the calculations that are presented in these papers, that a plethora of assumptions, both explicit and tacit, are required in order for the conclusions to hold up. Indeed, from the BBC News article that I have a link to in my comment above, the caveat is “If we use all the physics we know now, and you do this straightforward calculation – it’s bad news.” If I have any further questions or comments regarding the vacuum stability issue, I will direct them to the thread that you recently just started on this topic, i.e., Higgs News.

    — Nick M.

    P.S.: I, too, along with M. Uppal, wish you were going to offer the QM class (the one that you just taught) next semester. Unfortunately, it would be a bit of a morning walk from the Los Angeles area, where I currently live, to Columbia University in N.Y..

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