Take the 2-minute tour ×  The textbooks I have available explain
that due to the infinite degrees of freedom of a field, the
relevant object in QFT is the lagrangian density. A lagrangian
is then obtained for the field by integrating over space.  I
find the justification for this procedure unclear. In classical
mechanics, the lagrangians of two particles may be added only if
the particules do not interract. Does it mean that the
lagrangian density concept is only valid for a free field? What
happens in the case of interracting particles?  share|improve
this question Even classically, you can add interaction terms.
For instance, if you're doing two coupled simple harmonic
oscillators (with one fixed to a wall), you can have the
Lagrangian $L = \frac{1}{2}m_{1}\dot x_{1}^{2} +
\frac{1}{2}m_{2}\dot x_{2}^{2} - \frac{1}{2}k_{0}
\left(x_{1}-x_{2}\right)^{2}+\frac{1}{2}k_{1}x_{1}^{2}$, which
has a clear interaction term. –  Jerry Schirmer Jan 26 '11 at
4:09 I understand that it is possible to add interaction terms