File:Conjugate and Physical Momentum Densities of a Field 1.png

English: Relation between conjugate momentum density (${\displaystyle \pi }$) and physical momentum density (${\displaystyle {\mathcal {p}}}$) of a field. The conjugate momentum density is

${\displaystyle \pi ={\partial {\mathcal {L}} \over \partial {\dot {\phi }}}}$ where ${\displaystyle {\mathcal {L}}}$ is Lagrangian density and ${\displaystyle {\dot {\phi }}}$ is the time-derivative of the scalar field ${\displaystyle \phi }$; ${\displaystyle \pi }$ is proportional to ${\displaystyle {\dot {\phi }}}$. The height of the wave corresponds to the intensity of ${\displaystyle \phi }$, which depends on ${\displaystyle x}$, which in the figure is the horizontal dimension. The physical momentum density is interpreted as a traveling of a field wave left or right. ${\displaystyle \phi _{0}}$ is the field at a time t = 0 and ${\displaystyle \phi _{1}}$ is the field at a slightly later time t = 1.

Reference: Student Friendly Quantum Field Theory by Robert Klauber, page 23.