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Keeping things at a "basic level", a field is a function from a base manifold (of dimension $D4) to some other space. Usually, the base space is the spacetime but it may be something different (a lattice, etc etc).

For example, it's simple to construct the classical action of a complex scalar field over the usual 3+1 spacetime. Imagine now that we want to let this field interact with a classical membrane, or a string-like object, or a point particle immersed in spacetime. How to write the action for such a theory in which a "low dimensional field" (zero-dimensional in the case of the particle) is immersed into spacetime? The situation is not conceptually different from when we write the action for a charged particle immersed in a background electromagnetic field. How to generalise the theory (the variational principle) in such a way as to have the coupled equations of motion for the particle and the field?

Qmechanic
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Quillo
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1 Answers1

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OP's scenario is often the case. Perhaps an example is in order. The E&M gauge fields $$A_0,A_1,A_2,A_3:[t_i,t_f]\times \mathbb{R}^3\to \mathbb{R}$$ (whose base manifold is spacetime) and $N$ (non-relativistic) point charges $q_1,\ldots, q_N$ with positions $${\bf r}_1, \ldots, {\bf r}_N: [t_i,t_f]\to \mathbb{R}^3$$ (whose base manifold is a worldline) are described by the action $$S[A_0,A_1,A_2,A_3,{\bf r}_1, \ldots, {\bf r}_N]~=~\int \! dt ~L,$$ with Lagrangian $$\begin{align} L~=~&\sum_{i=1}^N\left( \frac{m_i}{2}{\bf v}^2_i + q_i\{A_0({\bf r}_i) + {\bf v}_i\cdot {\bf A}({\bf r}_i)\} \right)\cr ~-~&\frac{1}{4} \int d^3 {\bf x}\sum_{\mu,\nu=0}^3F_{\mu\nu}(t,{\bf x})F^{\mu\nu}(t,{\bf x}) \end{align} $$ [in $(-,+,+,+)$ Minkowski sign convention].

Qmechanic
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