Nice lecture about Dirac.

If you prefer to read, this 1939 paper by Dirac is a characteristically lucid discussion of similar themes:

The Relation between Mathematics and Physics [PDF]

http://mcs.une.edu.au/~pmth213/PapersOfInterest/Paul%20Dirac...

This has the perfect qoutation for the well known fact of the Galileo->Lorentz group overthrow from an indisputable source:

"The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature. The restricted theory { 1905 } changed our ideas of space and time in a way that may be summarised by stating that the group of transformations to which the space-time continuum is subject must be changed from the Galilean group to the Lorentz group. The latter group is a much more beautiful thing than the former - in fact, the former would be called mathematically a degenerate special case of the latter { c->∞ }. The general theory of relativity { 1915 } involved another step of a rather similar character" { diffeomorphism group/category }. I came to think lately that much of the basic groups in physics, Lorentz and gauge, have all more or less rotatory features.

	▲	griffzhowl 4 days ago \| parent [-]
		I think the rotations in the Lorentz group just reflect the isotropy of space, which comes down to the quite natural idea that if you observe a physical system from a different direction it doesn't change its dynamics. The full symmetry group of special relativity is the Poincare group, which includes the spatial translations, reflecting homogeneity of space. The gauge groups are interesting in being extra symmetries beyond the spacetime ones, and yet they're closely related to spacetime symmetries, e.g. SU(2) being the double cover of the rotations SO(3). I also find it interesting that the groups that are physically basic, such as SU(2) being the one required to represent the phenomenon of spin, are also mathematically significant, in this case since SU(2) is the unique simply-connected group associated with the shared Lie algebra of SU(2) and SO(3). That shows some kind of deep connection between mathematics and physics. I'm just at the beginning stages of learning QFT and differential geometry so I don't have a feel for why that is or what it means at an intuitive level, and haven't seen any explanation for it. I think at the moment it's just a feature of our deepest experimentally-verified theory and so it would need a deeper theory to explain it.