Monday, 15 October 2018

group theory - Symmetry transformation on Quantum Field


I stumbled upon this point several times, the latest beeing this question: Connection between conserved charge and the generator of a symmetry


I want to understand, why Quantum fields transform under symmetry transformations as


$(g\cdot\phi)(y) = T_g^{-1}\phi(y) T_g = e^{-tX}\phi(y) e^{tX} = \big[ 1 + t[X,\cdot]+\mathcal{O}(t^2)\big]\phi(y)$



Two ideas from me:


1) Fields are treated as operators in the Heisenberg picture, and instead of transforming the states with $ T_g |x>$, the states stay as they are and all operators, including the quantum fields transform as $ T_g^{-1}O T_g$


2) There is a good reason why quantum fields live in $T_e$, i.e. the tangent space at the identity of the group (=the Lie algebra), onto which the group acts with the adjoint action: $Ad_g(x)= T_g^{-1} X T_g$ $\forall X \in T_e$


Any ideas or reading tips would be awesome!



Answer



Your idea 1) is the right idea: It's just the law of transformation of matrices generalized from the transformation of matrices:


If we apply a general linear transformation $U : V \to V$ on a vector space, the matrices/operators on it transform as $M \mapsto U^\dagger MU$


For unitary operators $U^\dagger = U^{-1}$, so the transformation law becomes $M \mapsto U^{-1}MU$. Since $T_g$ is the representation of the group upon our space of states, the quantum fields as operators transform according to this law.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...