u  | \, n \rangle$. There is a particularly useful way to rewrite the above relation in Eq.  for the case of the electromagnetic current operator $$J_\mu^{\text{em}} (x) = \ol Q(x) \, \cQ \, \gamma_\mu Q(x)
,$$ with $\cQ = \diag ( 2/3, - 1/3 )$. This form is $$\label{eq:protonEMkey}
\langle p \, | \, \ol u \, \gamma_\mu \, d  | \, n \rangle 
= 
\langle p \, | \,  J_\mu^{\text{em}}  | \, p \rangle
- 
\langle n \, | \,  J_\mu^{\text{em}}  | \, n \rangle 
.$$

Of course one is not limited to baryonic matrix elements. Below we shall largely highlight the calculation of nucleon electromagnetic form factors as an example of how to utilize twisted boundary conditions. Our results, however, apply to the arbitrary quark bilinear operators in Eq. , using the relation in Eq.  between the desired states.

Implementation on the Lattice {#s:extension}
=============================

The above operator relations can be utilized to study isovector form factors at continuous values of the momentum transfer. In this Sect., we describe how to utilize partially twisted boundary conditions for this purpose. Here we focus on the kinematical effects; while in Sect. \[s:example\], we take up the dynamical effects at 