Considering the typical Lagrangian:

$$L=1/2(\partial_\mu\phi)(\partial^\mu\phi) - V(\phi)$$

I interpret the above (please correct me) as a theory consisting of a field which can move through spacetime with a certain potential.

However, this theory does not contain any forces (no gauge fields). Intuitively, a potential necessitates a force. I think back to examples of gravitational potential and electrical potential in class, where: $$F(x)=-\frac{dV(x)}{x}$$

How can there be a notion of potential without any forces?

Thanks

Consider the typical Lagrangian: $$L=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) - V(\phi).$$ I interpret the above (please correct me) as a theory consisting of a field which can move through spacetime with a certain potential.

However, this theory does not contain any forces (no gauge fields). Intuitively, a potential necessitates a force. I think back to examples of gravitational potential and electrical potential in class, where: $$F(x)=-\frac{\mathrm{d}V(x)}{\mathrm{d}x}.$$

How can there be a notion of potential without any forces?