Physlib

Physlib.QFT.QED.AnomalyCancellation.VectorLike

Vector like charges

For the `n`-even case we define the property of a charge assignment being vector like.

3 declarations

theorem

n+n=2nn + n = 2n

For any natural number n∈Nn \in \mathbb{N}, the sum of nn with itself is equal to two times nn, that is, n+n=2nn + n = 2n.

theorem

n+1+n=2n+1n + 1 + n = 2n + 1

For any natural number nn, it holds that n+1+n=2n+1n + 1 + n = 2n + 1.

definition

Vector-like property for 2n2n charges: xi=−xn+ix_i = -x_{n+i}

For a pure U(1)U(1) gauge theory with an even number of fermions 2n2n, a charge assignment S=(q0,q1,…,q2n−1)∈Q2nS = (q_0, q_1, \dots, q_{2n-1}) \in \mathbb{Q}^{2n} is defined to be **vector-like** if, when the charges are sorted in non-decreasing order as x0≤x1≤⋯≤x2n−1x_0 \le x_1 \le \dots \le x_{2n-1}, they satisfy the condition xi=−xn+ix_i = -x_{n+i} for all i∈{0,1,…,n−1}i \in \{0, 1, \dots, n-1\}.