48.2 Fundamental fermions

48.2 Fundamental fermions

The Standard Model includes both left-handed and right-handed Weyl spinor fields, each coming in multiple copies and transforming under the $U(1)\times SU(2)\times$ $SU(3)$ group according to a very specific choice of representations. They are described by the following terms in the Hamiltonian

$$h={\int }_{{\mathbf{R}}^3}(\sum _a({\Psi }_{L,a}^{†}\sigma \cdot (\nabla -i{\mathbf{A}}_L){\Psi }_{L,a})-\sum _b({\Psi }_{R,b}^{†}\sigma \cdot (\nabla -i{\mathbf{A}}_R){\Psi }_{R,b}))d^3\mathbf{x}$$

Here the left-handed fermions take values in three copies (called “genera-${\mathrm{tions}}^{,,})$ of the representation

$$(-1\otimes 2\otimes 1)\oplus (\frac{1}{3}\otimes 2\otimes 3)$$

while the right-handed fermions use three copies of

$$(-2\otimes 1\otimes 1)\oplus (-\frac{2}{3}\otimes 1\otimes 3)\oplus (\frac{4}{3}\otimes 1\otimes 3)$$

with the first term in each tensor product giving the representation of $U(1)$ (the fractions indicate a cover is needed), the second the representation of $SU(2)$ , the third the representation of $SU(3)$ . 1 is the trivial representation, 2 the defining representation for $SU(2)$ , 3 that for $SU(3)$

The ${\mathbf{A}}_L$ and ${\mathbf{A}}_R$ are vector potential fields acting by the representations given above, scaled by a constant $g_j$ corresponding to the appropriate term in the gauge group.

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来源版本：2025-10-20

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