Is the Poincare action on the Klein-Gordon quantum field strongly continuous?

Question

I am interested in checking continuity property of the Poincare group action on the Klein-Gordon quantum field theory defined over the Minkowski spacetime. Maybe the simplest example of QFT out there.

The setup I am working on is the C*-algebraic one, following Haag's local quantum physics (qft = local net of operator algebras). Let me explain briefly the setup. Take $\mathbf{R} = \mathbf{R}^{1+d}$ the Minkowski spacetime, $P = \partial_t^2 - \Delta + m^2$ the Klein-Gordon differential operator and $G = G^+ - G^-$ the advanced minus retarded propagator. This generates a sympletic vector space $V = C^\infty_c(\mathbf{M}) / P C^\infty_c(\mathbf{M})$ with sympletic form $\sigma([f],[g]) = \int_\mathbf{M} (f Gg) dx$. Now the local net of algebras is the CCR-algebra generated by the Weyl unitaries $W(f)$ for $[f]\in V$, with the rule $W(f)W(g)=e^{-i\sigma(f,g)}W(f+g)$, restricting the support of the test functions to the domain of your local algebra.

We have two translation actions on the test functions, which we can call $\alpha$ and $\beta$, such that $\alpha_a f(x) = f(x+a)$ and $\beta_a f(x) = f(x-a)$. Now, we can define the translation action on the C*-algebras by

$\alpha_aW(f) = W(\beta_af)$.

My question: Is this the right action for the theory? Is this strongly continuous? I think it is not, since for the CCR algebra, it is known that $\|W(f)-W(g)\| = 2$ whenever $[f] \ne [g]$. However, Haag's book he writes "it is possible and warranted to choose the algebras so that the action of the translation automorphisms on the elements is continuous in the norm topology". What is the problem here?

This post imported from StackExchange MathOverflow at 2014-08-31 09:11 (UCT), posted by SE-user Yul Otani

Comments

Your argument only shows that the action is not norm continous, and this means that the generators of the group action are unbounded operators.

This post imported from StackExchange MathOverflow at 2014-08-31 09:11 (UCT), posted by SE-user jjcale

Answer 1

Well, you have to represent that Weyl C*-algebra in a unitary representation. Referring to the Fock one built upon the Minkowski vacuum state you have that $$\hat{W}(f) = e^{ia(Gf) + ia^\dagger(Gf)} = e^{i\hat{\phi}(f)}\:.$$

The action of Poincaré group you consider is unitarily implementable: $$W(\beta_a(f)) = U_a W(f) U_a^\dagger$$

where $SO(3,1)_+ \ni a \mapsto U_a$ is a strongly continuous unitary representation of Poincaré group. I think you are confonding strong continuity of the *-authomorphism representation  $SO(3,1)_+ \ni a \mapsto s_a$, $s_a(W(f)):= W(\beta_a(f))$ in the Weyl C*-algebra and strongly continuity  of the unitary representation. The former is false for the reason you mention. The latter is true. It means that, for every vector $\psi$ in the Hilbert space of the representation, the map $SO(3,1)_+ \ni a \mapsto U_a \psi$ is continuous.