Niklas Buschmann

SO(10) Grand Unified Theory

Rotation Group SO(n)

For any real anti-symmetric matrix ω \omega with ω⊺=−ω \omega^\intercal=-\omega the matrix exponential eω e^\omega is a rotation matrix that leaves invariant the scalar product between two vectors since:

exp⁡(ω)⊺exp⁡(ω)=exp⁡(ω⊺+ω)=1\exp(\omega)^\intercal \exp(\omega) = \exp(\omega^\intercal+\omega) = \bm{1}

The n-dimensional rotation group consisting of all O=eω⇒O⊺O=1 O=e^\omega \Rightarrow O^\intercal O = \bm{1} is called SO(n) and is generated by the n(n−1)2 \frac{n(n-1)}{2} independent components of a n-dimensional anti-symmetric matrix ω \omega .

Unitary Group U(n)

The argument about the rotation group can be generalized to complex anti-hermitian ω \omega with ω‾⊺=−ω \overline{\omega}^\intercal=-\omega , which when exponentiated leave invariant the complex scalar product:

exp⁡(ω)‾⊺exp⁡(ω)=exp⁡(ω‾⊺+ω)=1\overline{\exp(\omega)}^\intercal \exp(\omega) = \exp(\overline{\omega}^\intercal+\omega) = \bm{1}

The n-dimensional unitary group consisting of all U=eω⇒U‾⊺U=1 U=e^\omega \Rightarrow \overline{U}^\intercal U = \bm{1} is called U(n) and generated by the n(n−1)2 \frac{n(n-1)}{2} real and n(n+1)2 \frac{n(n+1)}{2} imaginary independent components of an anti-hermitian matrix. The subgroup of U(n) where det⁡(U)=1 \det(U)=1 (traceless ω \omega ) is called SU(n).

U(n) in SO(2n)

Any complex number a+bi a + bi can be represented as 2×2 2 \times 2 matrix:

a+bi↦a(1001)+b(01−10)a+bi \hspace{0.5em} \mapsto \hspace{0.5em} a\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix} + b\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}

This can be used to write any complex matrix M M and vector v v as the corresponding real tensor products:

v∈Cn↦Re(v)⊗(10)+Im(v)⊗(01)∈R2nM∈Cn×n↦Re(M)⊗(1001)+Im(M)⊗(01−10)∈R2n×2nv \in \mathbb{C}^n \hspace{0.5em} \mapsto \hspace{0.5em} \mathfrak{Re}(v) \otimes \begin{pmatrix}1 \\ 0\end{pmatrix} + \mathfrak{Im}(v) \otimes \begin{pmatrix}0 \\ 1\end{pmatrix} \in \mathbb{R}^{2n} \\[1ex] M \in \mathbb{C}^{n \times n} \hspace{0.5em} \mapsto \hspace{0.5em} \mathfrak{Re}(M) \otimes \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix} + \mathfrak{Im}(M) \otimes \begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix} \in \mathbb{R}^{2n \times 2n}

For anti-hermitian, complex M M , the new matrix becomes anti-symmetric and real, and thus the n2 n^2 dimensional U(n) is embedded in the n(2n−1) n(2n-1) dimensional SO(2n).

Clifford Algebra

The Clifford algebra consists of products of γi \bm{\gamma}_i that obey the following anti-commutation relation:

12{γi,γj}=δij\frac{1}{2}\{\bm{\gamma}_i,\bm{\gamma}_j\} = \delta_{ij}

This means that for any i≠j i \neq j :

γiγi=1γiγj=−γjγi\bm{\gamma}_i\bm{\gamma}_i = \bm{1} \qquad \bm{\gamma}_i\bm{\gamma}_j = -\bm{\gamma}_j\bm{\gamma}_i

From the anti-commutation relation one can show that the γi \bm{\gamma}_i rotate like vectors when sandwiched between two exponentials of ωijγiγj \omega_{ij}\bm{\gamma}_i\bm{\gamma}_j :

e−ωijγiγj/2γαeωijγiγj/2=eαiωγi=Oαiγie^{-\omega_{ij}\bm{\gamma}_i\bm{\gamma}_j/2} \bm{\gamma}_\alpha e^{\omega_{ij}\bm{\gamma}_i\bm{\gamma}_j/2}=e^\omega_{\alpha i}\bm{\gamma}_i=O_{\alpha i}\bm{\gamma}_i

This allows one to translate rotations in real space ωij \omega_{ij} to rotations in the spinor space ωijγiγj/2 \omega_{ij}\bm{\gamma}_i\bm{\gamma}_j/2 .

Note: Einstein summation is used where any index occurs twice.

Clifford Algebra: Matrix representation

In three dimensions the γi \bm{\gamma}_i can be represented as the Pauli matrices:

σx=(0110)σy=(0−ii0)σz=(100−1)\sigma_x = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix} \quad \sigma_y = \begin{pmatrix}0 & -i\\ i & 0\end{pmatrix} \quad \sigma_z = \begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}

In n n dimensions the complete Clifford algebra is of dimension 2n 2^n , so its matrix representation must consist of at least 2n/2×2n/2 2^{n/2} \times 2^{n/2} matrices.

One higher-dimensional representation of the γi \bm{\gamma}_i can be found using the Weyl-Brauer construction, where one builds gamma matrices according to the following scheme, where each new row adds two new dimensions:

γ1=σx⊗1⊗1⊗1⊗1γ2=σy⊗1⊗1⊗1⊗1γ3=σz⊗σx⊗1⊗1⊗1γ4=σz⊗σy⊗1⊗1⊗1γ5=σz⊗σz⊗σx⊗1⊗1γ6=σz⊗σz⊗σy⊗1⊗1γ7=σz⊗σz⊗σz⊗σx⊗1γ8=σz⊗σz⊗σz⊗σy⊗1γ9=σz⊗σz⊗σz⊗σz⊗σxγ10=σz⊗σz⊗σz⊗σz⊗σy\bm{\gamma}_1=\sigma_x \otimes 1 \otimes 1 \otimes 1 \otimes 1 \qquad \bm{\gamma}_2=\sigma_y \otimes 1 \otimes 1 \otimes 1 \otimes 1 \\ \bm{\gamma}_3=\sigma_z \otimes \sigma_x \otimes 1 \otimes 1 \otimes 1 \qquad \bm{\gamma}_4=\sigma_z \otimes \sigma_y \otimes 1 \otimes 1 \otimes 1 \\ \bm{\gamma}_5=\sigma_z \otimes \sigma_z \otimes \sigma_x \otimes 1 \otimes 1 \qquad \bm{\gamma}_6=\sigma_z \otimes \sigma_z \otimes \sigma_y \otimes 1 \otimes 1 \\ \bm{\gamma}_7=\sigma_z \otimes \sigma_z \otimes \sigma_z \otimes \sigma_x \otimes 1 \qquad \bm{\gamma}_8=\sigma_z \otimes \sigma_z \otimes \sigma_z \otimes \sigma_y \otimes 1 \\ \bm{\gamma}_9=\sigma_z \otimes \sigma_z \otimes \sigma_z \otimes \sigma_z \otimes \sigma_x \qquad \bm{\gamma}_{10}=\sigma_z \otimes \sigma_z \otimes \sigma_z \otimes \sigma_z \otimes \sigma_y

SO(10) Unification

If one wants to bring all 8 fermions of one generation into a single spinor, then 2n/2=!8⋅4=32 2^{n/2} \overset{!}{=} 8 \cdot 4 = 32 requires at least n=10 n = 10 dimensions. But n=10 n = 10 is not only the smallest dimension to contain all fermions of one generation in one spinor, but is also connected to the group SO(10) that contains SU(5) which is the smallest group to contain all twelve SU(3) x SU(2) x U(1) generators of the Standard Model.

The general form of the SO(10) gauge covariant derivative is (with coupling strength λ \lambda ):

Dμ=∂μ−λAαβ(x)γαγβD_\mu = \partial_\mu - \lambda A_{\alpha\beta}(x)\bm{\gamma}_\alpha\bm{\gamma}_\beta

There are now 10⋅92=45 \frac{10 \cdot 9}{2} = 45 different gauge fields Aαβ(x) A_{\alpha\beta}(x) . Looking only at how the 12 known gauge fields / their generators from SU(5) from the Standard Model - translated to SO(10) - act on the 32-component spinor gives exactly the Standard Model (including the correct hypercharges):

The interpretation of the spinor components depends on the choice of the γi \bm{\gamma}_i matrix representation, but the eigenvalues of the resulting gamma bivectors are independent of the choice of representation.

Appendix

12 of the 24 SU(5) generators

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Corresponding ωijγiγj/4 \omega_{ij}\bm{\gamma}_i\bm{\gamma}_j/4

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& 0 & 0 & 0 & 0 & \frac{i}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{2 i}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{4 i}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{3} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{i}{3} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{2 i}{3} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 i & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & i & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]

Python code

from sympy import *
from sympy.physics.matrices import msigma
from sympy.physics.quantum import TensorProduct

SU5 = [Matrix([[0,1,0,0,0], [1,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0]]), Matrix([[0,-I,0,0,0], [I,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0]]), Matrix([[1,0,0,0,0], [0,-1,0,0,0], [0,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0]]), Matrix([[0,0,1,0,0], [0,0,0,0,0], [1,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0]]), Matrix([[0,0,-I,0,0], [0,0,0,0,0], [I,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0]]), Matrix([[0,0,0,0,0], [0,0,1,0,0], [0,1,0,0,0], [0,0,0,0,0], [0,0,0,0,0]]), Matrix([[0,0,0,0,0], [0,0,-I,0,0], [0,I,0,0,0], [0,0,0,0,0], [0,0,0,0,0]]), Matrix([[1,0,0,0,0], [0,1,0,0,0], [0,0,-2,0,0], [0,0,0,0,0], [0,0,0,0,0]])/sqrt(3), Matrix([[0,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0], [0,0,0,0,1], [0,0,0,1,0]]), Matrix([[0,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0], [0,0,0,0,-I], [0,0,0,I,0]]), Matrix([[0,0,0,0,0], [0,0,0,0,0], [0,0,0,0,0], [0,0,0,1,0], [0,0,0,0,-1]]), Matrix([[-2,0,0,0,0], [0,-2,0,0,0], [0,0,-2,0,0], [0,0,0,3,0], [0,0,0,0,3]])/3]

def gamma(i, n = 10):
    return TensorProduct(*[msigma(3) for j in range(i // 2)] + [msigma(1) if i % 2 == 0 else msigma(2)] + [eye(2) for j in range(n // 2 - (i // 2 + 1))])

for element in SU5:
    SO10 = TensorProduct(Matrix(re(element*I)), eye(2)) + TensorProduct(Matrix(im(element*I)), msigma(3)*msigma(1))
    generator = sum([SO10[x,y]*gamma(x)*gamma(y)/4 for x in range(10) for y in range(10)], zeros(32))
    pprint(element)
    pprint(generator)