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Does Clifford algebra solve both QFT and GR

  1. Feb 13, 2012 #1
    I noticed a few sources that seem to indicate that Clifford algebra may be used in both QFT and GR. I've seen where the Clifford algebra is a type of associative algebra that generalizes the real numbers, complex numbers, quaternions, and octonions, see Wikipedia on Clifford Algebra. And I've seen where the complex numbers, quaternions, and octernions can be used in the description for the U(1), SU(2), SU(3) symmetries of the Standard Model, see this article, and this book. However, I've also seen where the Clifford algebra can be used in an alternative description of differential geometry used in the formulation of GR, see this book for example.

    So my question is does this common algebra allow us to derive GR in terms of QFT or visa versa? Or if we could justify the use of the complex numbers, quaternions, octonions, and the Clifford algebra by some other means, could we derive both QFT and GR from that common justification of the algebra? What more information or constraints would be needed to do so?
  2. jcsd
  3. Feb 14, 2012 #2
    I'm trying to decide whether I should research the Clifford algebra and how it applies to physics. I'm looking to get a sense of how valid is the application of Clifford algebra in physics. Is there a consenses that Clifford algebra, quaternions, and octonions are indeed fundamental or at least have a valuable use?

    If the use of these algebras is valid in physics, then I have to wonder... Since the Clifford algebra is used in differential geometry, and quaterions and octonions are used in the SM, then can the quaternions and octonions be used to contrain the Clifford algebra in such a way in differential geometry as to product GR if the SM is assumed? Or is it more the case that the Clifford algebra is used on the metric whereas the quaternion and octonions are used on wavefunctions, which is a totally different animal? Any insight would be appreciated. Thanks.
  4. Feb 22, 2012 #3
  5. Mar 1, 2012 #4
    Yes, some people are using it as the basis of the SM. But I'm having trouble getting started with my studies of quaternions, octonions, and clifford algebra. My linear algebra and group theory is weak. Do you know of any easy introductions to quaternions, octonions, and clifford algebra?

    So when they use the quaternions and octonions in the SM, do they use the algebra to establish commutation relation between fields? How is the Clifford algebra used in GR? Is it used on connection fields perhaps? Does it establish commutation relations, or perhaps the Bianchi identity, what? Thanks.
  6. Mar 1, 2012 #5


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    Clifford algebras are particularly necessary for understanding spinor representations and spin groups. Depending on what you're interested in studying, they could be important (a good knowledge of the mathematical underpinnings of SM and QFT will require knowledge of Clifford algebras). I'd say that quaternions (and octonions) are less important: quaternions offer an alternative to the tensor formulation of GR, and are not something I'd concern myself with on an initial read through.

    I would certainly get up to speed on your linear algebra and group theory (in particular Lie groups and Lie algebras which are widespread in particle physics) before I'd worry about Clifford algebras and quaternions.
  7. Mar 1, 2012 #6
    The problem is that I seem to have been able to justify the use of quaternions and octonions from purely logical reasons. And now I wonder if this is a road into physics. So I feel like I need to go from these hypercomplex numbers to SM physics. You seem to indicate that you can go from the SM to hypercomplex numbers. But can one go from hypercomplex numbers to the SM?
  8. Mar 2, 2012 #7
    If with SM you mean standard model, then yes some parts of the Lagrangian are involving the Pauli matrices and you can consider that these matrices are a representation of the quaternions or of a Clifford algebra. See the theory of spinors by E. Cartan himself.
  9. Mar 2, 2012 #8
    I hope it is allowed to propose a (commercial) link here but if you need a reference either type the words on google or go to the following link (but it is not the only possible one): https://www.amazon.com/Theory-Spinors-Dover-Books-Mathematics/dp/0486640701

    You get everything you need to beginn with and if you are patient, at the very end of the book, a part of the answer to your initial question.
  10. Mar 3, 2012 #9
    Yes, the book is on its way. Thanks.

    But from what I can tell from what I've seen on the Net, clifford algebra is used either for geometry or for the SM in the form of quaternions and octonions. What I don't see is anyone connecting the two. Is this because the algebra operates on two different objects that can't be connected in some way?
    Last edited: Mar 3, 2012
  11. Mar 3, 2012 #10
    For example, is there a connection between spin foam networks of LQG and the spinors derived using clifford algebra?
  12. Mar 3, 2012 #11
    It is far far far over my head and I cannot help you or explain details.

    But I just guess that the answer is: some where yes.

    Good luck

  13. Mar 28, 2014 #12
    Last edited by a moderator: May 6, 2017
  14. Mar 29, 2014 #13
    What I'm reading so far is that the S0(1,3) symmetry of Lorentz invariance is a "double cover" for the SU(2) symmetry of the Standard Model. And I'm read other places that the spin of a particle is deeply connected to Lorentz invariance. Is this deep connection the double cover?

    What does double cover mean anyway? Is this covered in the book you mention? Thanks.
  15. Mar 29, 2014 #14
    Double Cover in Doran and Lasenby

    It is mentioned in D&L. In GA rotations (in any dimension) are preformed with rotors which are scalars plus bivectors, R, with the condition that R*rev(R) = 1 (rev(R) is the reverse of R and * the geometric product). Then to rotate a vector, x, in the plane that R defines you evaluate


    Since rev(-R) = -rev(R) if you replace R with -R you get the same result. This is the double cover. Note that R = exp((theta/2)*B) where B is a normalized bivector defining the plane of rotation. For the Lorentz transformation in Minkowski space if B is a spacelike plane if B**2 = -1 and

    R = cos(theta/2)+sin(theta/2)*B

    If the plane is timelike B**2 = 1 and

    R = cosh(theta/2)+sinh(theta/2)*B

    also note that rev(R) = exp(-(theta/2)*B). Multiple Lorentz transformation R1, R2 are evaluated with


    and the geometric product * is the group operation.
  16. Mar 29, 2014 #15


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    Hello friend, there's been lots of good advice here so far. While playing with Clifford algebra is great fun, there's a risk of both overestimating and underestimating its importance in theoretical physics, and it's good to understand its context. Clifford algebra is useful in physics because the antisymmetric Clifford product between Clifford bivector elements is isomorphic to the Lie bracket between corresponding elements of the spin Lie algebra. This means Clifford algebra is hugely important in fundamental physics, but not a replacement for all of differential geometry. One still needs to understand Lie groups and differential forms, and then have even more fun with Clifford valued differential forms, specifically for describing connections (gauge fields). Even more specifically, gravity is well described using a spin connection, which is a spin(1,3)-valued 1-form, which can also be written as a Cl2(1,3)-valued 1-form, which usually acts on a Cl1(1,3)-valued gravitational frame as well as on spinors, which are defined using matrix representations of the spin group which correspond to matrix representations of Clifford algebras. Clifford algebra is also useful in the Standard Model because u(1)xsu(2)xsu(3) embeds in spin(10) (the Lie algebra of the Spin(10) GUT) or equivalently Cl2(10), with the SM gauge fields all parts of a spin(10)-valued 1-form, and the SM fermion multiplets living in 16 dimensional spinor representation spaces. There are, of course, other things one can do with Clifford algebras, including how this structure unifies in spin(11,3) and E8, but that's my opinion on their importance in fundamental physics. Hope you find that helpful.
  17. Mar 29, 2014 #16
    Remind me again, U(1) is the symmetry of the Electromagnetic force and is represented by the dirac matrix? And SU(2) is the symmetry of the Weak force and is represented by the Pauli matix, and SU(3) is the symmetry of the Strong force and is represented by the what matix? Am I close?
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