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“Notes on Symmetry.” In K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections. Cambridge University Press (2003), pp. 393-412.
[preprint]
Abstract. These notes discuss some aspects of the sort of symmetry considerations that arise in philosophy of physics. They describe and provide illustration of: (i) one common sort of symmetry argument; and (ii) a construction that allows one to eliminate symmetries from a given structure.
Note. The discussion of Section 4.1 of the paper is very misleading: from knowledge of the differential equations alone one can of course determine the physically relevant symmetries.
But there is a reasonable point behind this misleading discussion.
Consider the following two observations (i) Perfectly generally, a symmetry of a structure is a map from the structure to itself that preserves the relations between the elements of the structures. (ii) The Lorentz (or Galilei) group is a symmetry of a physical theory if under the obvious action of this group on the space of kinematic possibilities, the group maps solutions of equations of motion to solutions of equations of motion.
These are both sound. But taken together they provide some temptation to think that we can characterize a symmetry of a theory as a (smooth) map on the space of kinematic possibilities that maps solutions to solutions. This characterization is in fact not uncommon among philosophers. But the resulting notion of a symmetry is useless, since under it, e.g., any two solutions of the n-body problem are related by a symmetry of Newton’s theory.
What has gone wrong? Clearly we need to think of the relevant structure encoding the physics of a theory as more than just a (smooth) space of solutions sitting inside of a space of kinematic possibilities.
There are two ways we can go here, each of which is completely standard.
The first, emphasized in the paper, is that we can take the space of solutions to be equipped with a symplectic form and a Hamiltonian. Then the physical symmetries that we know and love can be characterized as maps from the space of solutions to itself that preserve these structures.
The second option is to look more carefully at the structure implicit in the differential equations of the theory. This gives us more than just a smooth structure on the spaces of kinematic and dynamic possibilities, it also allows us to single out some maps from the space of kinematic possibilities to itself as arising from transformations of the dependent and independent variables of the theory that are local in the latter. The symmetries of the theory can then be taken to be maps from the space of kinematic possibilities to itself that (i) arise in this way and (ii) map solutions to solutions (the obvious action of a group of spacetime symmetries on the space of kinematic possibilities of a field theory always satisfies (i)).
This second approach is developed in detail in, e.g., Olver, Applications of Lie Groups to Differential Equations. It is sketched in Section 2 of Giulini, “Algebraic and geometric structures of Special Relativity” (where it is observed that if one drops the locality condition then Maxwell’s theory counts as Galilei-covariant).
N.B. Each of the two approaches just described can lead to surprising results: it is not plausible, for instance, that being related by a symmetry of one or the other sort is sufficient for two solutions of a theory to be physically equivalent. See my "Symmetry and Equivalence" for further discussion.
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