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Does the "field" in physics same as the "field" in mathematics? Which one those is more abstract in Nature?

Abu sayed
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1 Answers1

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The terminology addressing the meaning of field, in physics, tends to change according to the area and the context one is dealing with.

In general a field is a map $p\mapsto T(p)$ that assigns a quantity to each point in the domain of definition of the theory, in order to define the states of the theory itself and their evolution.

  1. In point particle classical mechanics a field is a map $$ t\in \mathbb{R}\to (\mathbf{r}(t),\mathbf{v}(t))\in\mathbb{R^6} $$ giving the position and the velocity of the particle at any time $t$. More generally, for a system of classical particles a field is a map $$ s\in \mathbb{R}\to \gamma(s)\in T^*Q $$ with $T^*Q$ being the phase space as cotangent bundle of the configuration space.

  2. In quantum mechanics a field is a map $$ t\in \mathbb{R}\to |\psi(t)\rangle\in\mathcal{H} $$ that assigns an element of a Hilbert space to any point $t$ in time.

  3. In classical electromagnetism a field is a map $$ (x,t)\in \mathbb{R^4}\to A^{\mu}(x,t)\in\mathbb{R^4} $$ that assigns a vector potential to every point in space and time $(x,t)$. More generally, for gauge theories, a field is a map $$ m\in \mathcal{M}\to T(m)\in\chi(\mathcal{M}) $$ that assigns a tensor to any point $m$ of a smooth manifold $\mathcal{M}$. Likewise for quantum gauge theories, where the above classical gauge fields are integrated in the path integral to give rise to correlation functions.

  4. In thermodynamics a field is a map $$ (X_1,\ldots,X_N)\in\mathbb{R^N}\to S(X_1,\ldots,X_N)\in\mathbb{R} $$ that assigns the entropy given a set of extensive variables $(X_1,\ldots,X_N)$

and many more examples can be provided.

gented
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