.$$

\[beans\] The index $k=2 \pi n/L$ where $n \in \mathbb{Z}^3$, and the word cutoff refers to the restriction to all values of $k$ with $|k| < \Lambda$.

The vectors $\varepsilon_{\lambda}(k)$ are the polarization vectors and are normalized in such a way that $$\varepsilon_{i}(k)\cdot \varepsilon_{j}(k) = \delta_{i,j}\ ,\
\varepsilon_{i}(k)\cdot k = 0 \ .$$

The operators $a_{\lambda}(k)$ and $a_{\lambda}^{*}(k)$ satisfy the commutation relation $$\bigl[a_{\lambda}(k), a_{\lambda^{\prime}}^{*}(k^{\prime})\bigr]= 
\delta_{\lambda,\lambda^{\prime}} \delta(k,k^{\prime}) \ ,$$ while all others commute with each other.

The energy of the radiation field can now be conveniently written as $$H_f = \sum_ {|k| < \Lambda} \sum_{ \lambda =1,2}|k| a_{\lambda}^{*}(k)
a_{\lambda}(k) \ .$$

These operators act on the Hilbert space generated by the polynomials in $a_{\lambda}^{*}(k)$ acting on the vacuum $|0 \rangle$.

The self energy of (one or more) particles is the [**ground state energy**]{} of the Hamiltonian $$H = {\rm kinetic \ energy} \ + \ H_f \ .$$ where, as usual, the ground state energy of $H$ is defined to be $$E_0 = \inf_{\Psi} \frac{ \langle \Psi, H\ \Psi \rangle }{\langle \Psi, \Psi \rangle}\
.$$

Typically, in the inquiry about the self–energy problem, i.e., the problem of computing the self–energy for fixed, albeit small, $\alpha$ and for large $\Lambda$, one 