It is possible to write the Standard Model Lagrangian in a simple way by looking at the matrix representation of the 10 dimensional Clifford Algebra, which is connected to the 10d rotation group SO(10), that contains the 5d special unitary group SU(5), which in turn contains the SU(3) x SU(2) x U(1) of the Standard Model. The matrix representation of the 10d Clifford Algebra is made of matrices with dimension 210/2×210/2=32×32, which is enough for a 32 component spinor that contains all four-spinors of the 8 particles in one generation.
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Rotation Group SO(n)
For any real anti-symmetric matrix ω with ω⊺=−ω the matrix exponential eω is a rotation matrix that leaves invariant the scalar product between two vectors since:
exp(ω)⊺exp(ω)=exp(ω⊺+ω)=1The n-dimensional rotation group consisting of all O=eω⇒O⊺O=1 is called SO(n) and is generated by the 2n(n−1) independent components of a n-dimensional anti-symmetric matrix ω.
Unitary Group U(n)
The argument about the rotation group can be generalized to complex anti-hermitian ω with ω⊺=−ω, which when exponentiated leave invariant the complex scalar product:
exp(ω)⊺exp(ω)=exp(ω⊺+ω)=1The n-dimensional unitary group consisting of all U=eω⇒U⊺U=1 is called U(n) and generated by the 2n(n−1) real and 2n(n+1) imaginary independent components of an anti-hermitian matrix. The subgroup of U(n) where det(U)=1 (traceless ω) is called SU(n).
U(n) in SO(2n)
Any complex number a+bi can be represented as 2×2 matrix:
a+bi↦a(1001)+b(0−110)This can be used to write any complex matrix M and vector v as the corresponding real tensor products:
v∈Cn↦Re(v)⊗(10)+Im(v)⊗(01)∈R2nM∈Cn×n↦Re(M)⊗(1001)+Im(M)⊗(0−110)∈R2n×2nFor anti-hermitian, complex M, the new matrix becomes anti-symmetric and real, and thus the n2 dimensional U(n) is embedded in the n(2n−1) dimensional SO(2n).
Clifford Algebra
The Clifford algebra consists of products of γi that obey the following anti-commutation relation:
21{γi,γj}=δijThis means that for any i=j:
γiγi=1γiγj=−γjγiFrom the anti-commutation relation one can show that the γi rotate like vectors when sandwiched between two exponentials of ωijγiγj:
e−ωijγiγj/2γαeωijγiγj/2=eαiωγi=OαiγiThis allows one to translate rotations in real space ωij to rotations in the spinor space ωijγiγj/2.
Note: Einstein summation is used where any index occurs twice.
Clifford Algebra: Matrix representation
In three dimensions the γi can be represented as the Pauli matrices:
σx=(0110)σy=(0i−i0)σz=(100−1)In n dimensions the complete Clifford algebra is of dimension 2n, so its matrix representation must consist of at least 2n/2×2n/2 matrices.
One higher-dimensional representation of the γi 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⊗σySO(10) Unification
If one wants to bring all 8 fermions of one generation into a single spinor, then 2n/2=!8⋅4=32 requires at least n=10 dimensions. But 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):
Dμ=∂μ−λAαβ(x)γαγβThere are now 210⋅9=45 different gauge fields Aαβ(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):
- Two particles transforming under SU(2) x U(1): the left-handed electron and neutrino
- Six particles transforming under SU(3) x SU(2) x U(1): the left-handed up and down quarks
- Six particles transforming under SU(3) x U(1): the right-handed up and down quarks
- One particle transforming under U(1): the right-handed electron
- One particle transforming under nothing: the right-handed neutrino
The interpretation of the spinor components depends on the choice of the γi matrix representation, but the eigenvalues of the resulting gamma bivectors are independent of the choice of representation.
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