The assumption of positive energy

The Fourier transform of $\Delta\varphi_{n}$ is

$$\Delta\varphi_{n}=\frac{1}{N}\displaystyle\sum_{k}\Delta\varphi_{k}e^{-2\pi ikn/N}$$ (78)

We assume that for any $\varphi_{n}$, we can express $\Delta\varphi_{k}$ as

$$\Delta\varphi_{k}=-if(k)\varphi_{k}$$ (79)

where $f(k)$ is some function of $k$. In the general case, the function $f(k)$ is not specified. Specific conditions, like the self-duality condition, must however be reflected in the form of $f$.
In order to ensure that theory be local on the lattice, $f(k)$ must be continuous, and because of the lattice periodicity, $f(k)$ must also be periodical. We reformulate $\Delta\varphi_{n}$ as

$$\Delta\varphi_{n}=\frac{-i}{N}\displaystyle\sum_{k}f(k)\varphi_{k}e^ {-2\pi ikn/N}$$ (80)

In the momentum space, the $f(k)$ corresponding to the generic "symmetric" discretization is

$$f(k)=\frac{1}{p}\displaystyle\sum_{p}\sin\frac{2\pi kp}{N}$$ (81)

The Hamiltonian corresponding to $(\ref{ping})$ can thus be written

$${\cal{H}} = \frac{a}{2}\displaystyle\sum_{n}\big[\dot{\varphi}_{n}^{2}+ (\frac{\Delta\varphi_{n}}{a})^{2}\big]=$$

$$= \frac{a}{2N}\displaystyle\sum_{k}\big[\dot{\varphi}_{k} \dot{\varphi}_{-k}-\frac{1}{a^{2}}\varphi_{k}\varphi_{-k}f(k)f(-k)\big]$$

In the self-dual case, the Hamiltonian takes the simple form

$${\cal{H}} = \frac{-1}{aN}\displaystyle\sum_{k}\dot{\varphi}_{k}\dot{\varphi}_{-k}f(k)f(-k)$$

In order to ascertain that ${\cal{H}}$ be positive, we assume that $f(-k)=-f(k)$. Hence, the self-dual Hamiltonian is

$${\cal{H}}= \frac{1}{aN}\displaystyle\sum_{k} \varphi_{k} \varphi_{-k}f^{2}(k),$$ (82)

and the energy depends on the momentum through $f^{2}(k)$. The self-duality condition as a condition on $f(k)$ is thus

$$f(-k)=-f(k)$$ (83)

where $f(k)$ is a real, continuous and periodic function.