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Linear operators and vector spaces form the backbone of the operator formulation of quantum mechanics. I want to ask are there operators in classical physics too? Are these operators defined on some vector space? Examples can also be helpful.

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    $\begingroup$ Don't get me wrong but your question sounds like: are there vectors in classical physics? $\endgroup$
    – Diracology
    Sep 20 '17 at 17:43
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    $\begingroup$ This post (v2) seems like a list question. $\endgroup$
    – Qmechanic
    Sep 20 '17 at 17:44
  • $\begingroup$ @Diracology Hmm...what about more non-trivial cases such as a classical field? Is there a vector space associated with a classical field? $\endgroup$ Sep 20 '17 at 17:51
  • $\begingroup$ @mithusengupta123 Of course! The classical field lives in an infinite dimensional vector space. $\endgroup$
    – Diracology
    Sep 20 '17 at 17:52
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    $\begingroup$ Rotation, differentiation, integration, cross product, and dot product are all linear operators. $\endgroup$ Sep 20 '17 at 18:03
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One can also make the distinction between linear operators and their representations on vector spaces. You do not necessarily need to find a basis to work with linear operators. For the most part, one can assume that there is some representation with vector spaces possible.

Space and time are vector spaces, space-time of special relativity is a vector space. However, since the latter has a non-trivial metric tensor, it is different from $\mathbb R^4$. General relativity adds curvature to this space, so one would rather call it a manifold. All the tangent spaces, however, are still usual vector spaces.

You have vector spaces as tangent spaces wherever you have some manifold. For some trivial manifolds, they are a vector space themselves. And there are manifolds everywhere in physics:

  • space and time, as well as space-time
  • gauge groups like U(1) for electromagnetism and then for all the quantum field theoretical forces like electroweak (SU(2)) and strong (SU(3)) force; string theory, super-symmetry and grand unified theories have even more complex groups
  • configuration space in Lagrangian mechanics
  • phase space in Hamiltonian mechanics

You can also make it more general with differential geometry where there are not only vectors but also co-vectors (differential forms). And classical mechanics and classical electromagnetism can be described beautifully using differential forms.

So in short: Vectors and their generalizations are everywhere in physics.

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