# Is this integral always equal to 1?

This is my Hamiltonian. $$\psi_{\alpha}$$ is a bosonic field. $$H_{\alpha}=\int \mathrm{d} \mathbf{r} \psi_{\alpha}^{\dagger}(\mathbf{r})\left(-\frac{\nabla^{2}}{2 m}\right) \psi_{\alpha}(\mathbf{r})+\frac{V_{0}}{2} \int \mathrm{d} \mathbf{r} \psi_{\alpha}^{\dagger}(\mathbf{r}) \psi_{\alpha}^{\dagger}(\mathbf{r}) \psi_{\alpha}(\mathbf{r}) \psi_{\alpha}(\mathbf{r})$$

I'm interested in the potential term. I write it in bras and kets, and I get this:

$$\frac{V_{0}}{2}\left\langle\psi_{\alpha}\left|\left\langle\psi_{\alpha} | \psi_{\alpha}\right\rangle\right| \psi_{\alpha}\right\rangle$$

(Actually, I'm not 100% sure about this step, since I ignored the fact that there is only one integral, and for my step to work, I would have thought I would need a double integral).

Is $$\left\langle\psi_{\alpha}\left|\left\langle\psi_{\alpha} | \psi_{\alpha}\right\rangle\right| \psi_{\alpha}\right\rangle = 1$$? I know it would be normal quantum mechanics, but maybe that's not necessarily true in QFT with bosonic fields?

Any help would be much appreciated

It might be $$1$$ for a specific function, but it's not true in general, because the integrand is $$|\psi_\alpha|^4$$, so you would need to have that $$\psi_\alpha$$ is an unit vector both in $$L^2$$ and in $$L^4$$. Also: $$\left\langle\psi_{\alpha}\left|\left\langle\psi_{\alpha} | \psi_{\alpha}\right\rangle\right| \psi_{\alpha}\right\rangle=\int \mathrm{d}r \bar{\psi}_\alpha(r)\left(\int\mathrm{d}r' \bar{\psi}_\alpha(r')\psi_\alpha(r')\right)\psi_\alpha(r)$$ So it does not equal to your original integral.
• @user45757 Your braket expression is indeed $1$. I just pointed out why does it not equal to your original expression. Jun 16, 2020 at 10:10
• @user45757 the integrand in your "potential integral" equals to $|\psi_\alpha|^4$, so your integral is the 4th power of the $L^4$ norm of $\psi_\alpha$. It can happen that both the $L^2$ norm and the $L^4$ norm is one, for example in 1D, if $\psi_\alpha(x)=1$ on $[0,1]$ and zero otherwise, but usually the two norms are different. Is it better now? Jun 16, 2020 at 10:19