![\"Someone](\"https:\/\/imgs.xkcd.com\/comics\/duty_calls.png\")
<\/p>\n<\/p>\n
It’s getting late, but I can’t help myself. Reading too many wrong things about symmetry and physics on Twitter has forced me to do this. And, John Baez says I don’t explain things<\/a>. So, here’s what the relationship between symmetry and physics really is.<\/p>\n In the language of mathematicians, talking about “symmetries” means you are talking about groups (often Lie groups, or their infinitesimal versions, Lie algebras) and representations. The relation to physics is:<\/p>\n Classical mechanics (Hamiltonian form)<\/strong><\/p>\n In classical mechanics the state of a system with $n$ degrees of freedom is given by a point in phase space $P=\\mathbf R^{2n}$ with $n$ position coordinates $q_j$ and $n$ momentum coordinates $p_j$. Functions on this space are a Lie algebra, with Lie bracket the Poisson bracket Quantum mechanics<\/strong><\/p>\n Quantization of a classical system is something mathematically obvious: go from the above Lie algebra to a unitary representation of the Lie algebra. This takes elements of the Lie algebra (functions on $P$) to skew-adjoint operators on a Hilbert space, the space of quantum states. There’s a theorem (Stone-von Neumann) that says that (modulo technicalities) there’s only one way to do this, and it gives an irreducible unitary representation that works for polynomials up to degree two. For higher degree polynomials there will always be “operator ordering ambiguities”. The representation is given by For more details, I wrote a whole book about this<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":" It’s getting late, but I can’t help myself. Reading too many wrong things about symmetry and physics on Twitter has forced me to do this. And, John Baez says I don’t explain things. So, here’s what the relationship between symmetry … Continue reading
\n$\\{f,g\\}$. Dynamics is given by choosing a distinguished function, the Hamiltonian $h$. Then the value of any function on $P$ evolves in time according to
\n$$\\frac {df}{dt}=\\{f,h\\}$$
\nThe Hamiltonian $h$ generates the action of time translations. Applying the same formula, other functions generate the action of other groups (spatial translations, rotations, etc.). If your function satisfies $\\{f,h\\}=0$, it generates a “symmetry”, and doesn’t change with time (is a conserved quantity).<\/p>\n
\n$$1\\rightarrow -i\\mathbf 1,\\ \\ q_j\\rightarrow -iQ_j,\\ \\ p_j\\rightarrow -iP_j$$
\nThis is a representation because
\n$$\\{q_j,p_k\\}=\\delta_{jk}\\rightarrow [-iQ_j,-iP_k]=-i\\delta_{jk}\\mathbf 1$$
\nThe right-hand side is the Heisenberg commutation relations for $\\hbar=1$.<\/p>\n