Consider a real scalar field $\Phi(x)$ in one spatial dimension which asymptotically goes to its vacuum values $\Phi_{+}$ or $\Phi_{-}$. Given the requirement of finiteness of energy, we deduce that no continuous transformation can change a field configuration to one with different asymptotic values. Thus the field values at $\pm\infty$ classify all possible field configurations into (four) homotopy sectors.

Now consider the configuration space. I am not so sure about this part, but to visualize this space, what I do is imagine an infinite-dimensional vector space with an orthogonal (or orthonormal, if necessary) basis (like Euclidean space, but infinite-dimensional), where every basis vector is labeled $\Phi_x$ and $\Phi_x=\Phi(x)$, with $x$ varying continuously from -$\infty$ to +$\infty$ .

Now, every point in this space corresponds to a field configuration and every path in this space is made up of series of such points.

My questions are:

- What is wrong with the image I have of the configuration space of a scalar field?

If my image is acceptable:

What do we mean by a path in this space, in more physical terms?

If I can draw a path from one vacuum to the other in this space, how is that possible, given the homotopy argument mentioned above? Does this imply that configuration space is somehow larger than merely the space of all configurations?

This post imported from StackExchange Physics at 2017-08-19 14:49 (UTC), posted by SE-user Optimus Prime