of the wave packet’s effects are lacking. It is our objective to study the deviations of one-particle states from simple plane waves and their consequences in many-body quantum systems. We mainly study the systems of relativistic invariance, where space and time are treated equally.

The minimum wave packet is an idealistic wave packet that satisfies the minimum uncertainty relation between the variances of coordinates and momenta. Although they have been studied often,[@Schiff; @Schiff2] it is instructive to review the properties of the minimum wave packet here for later convenience.

From the canonical commutation relation, Eq. $(\ref{commutation-relation})$, uncertainties in the position and momentum satisfy $$\begin{aligned}
\delta x \delta p \geq \frac{\hbar}{2}.
\label{uncertainty-product}\end{aligned}$$ In Gaussian wave packet, the coherent state of one variable $x$, $$\begin{aligned}
& &\langle x |P_0,X_0\rangle =N_1e^{i{ P_0 \over \hbar} (x-X_0)-{1 \over 2{\gamma}}(x-X_0)^2},\\
& &N_1^2=(\pi {\gamma})^{-{1 \over 2}},~{P_0 \over \hbar}=k_0, \end{aligned}$$ where $X_0$ and $P_0$ are the expectation values of $x$ and $p$, the equality of Eq. $(\ref{uncertainty-product})$ is satisfied. The variances of $x$ and $p$ are, in fact, $$\begin{aligned}
& &( \delta x)^2=  \langle| x^2| \rangle-\langle |x |\rangle^2= {1 \over 2}{\gamma}^2,    \\
& & ( \delta p)^2= \langle| p^2| \rangle-\langle |p|\rangle^2  ={{\hbar}^2 \over {2 {\gamma}^2}}\label{parameter-sigma}
. \end{aligned}$$ The 