Poincaré group on quantum Klein-Gordon field (C*-algebraic scenario)

Question

on the same topic as this question, I have been trying to fool around with the free real K-G field in flat spacetime on the C*-algebraic scenario (Haag-Kastler axioms, Weyl quantization, etc).

Since I'm talking about the free (linear) Klein-Gordon field, the C*-algebra is taken as the CCR algebra generated by the unitary Weyl operators $W(f)$ (with $f$ a test function) which can be looked as the exponential of the field operators, $\exp\big( i\Phi(f)\big)$.

The action of the Poincare group usually is given for the field operators (Borchers algebra), with something as $\alpha_{(\Lambda,a)}\Phi(x) = \Phi(\Lambda x + a)$, as operator-valued distributions. Now I imagine that you can transport that to Weyl operators, something as $\alpha_{(\Lambda,a)} W(f) = \exp\big( i [\alpha_{(\Lambda,a)}\Phi](f)\big)$.

My questions are

1. Is the expression for $W(f)$ correct?
2. Does the action on the Weyl unitaries extend to a nice action on the CCR-algebra? By *-automorphisms? Is it inner or outer or what?
3. Where can I read about it? I could use a "for dummies" reference...

[EDIT: fixed the notation, as sugestion of user1504]

Answer 1

0) It's weird to denote the action by $Ad$; this is usually reserved for adjoint actions. I'm going to use $\rho$.

1) Your expression is correct. Note that $(\rho\Phi)(f)$ is defined to be $\Phi(\rho f)$. In the end, we're just translating and transforming the test functions.

2) It should. I'm not 100% sure. It really ought to be an inner automorphism, since one can construct generators for the Poincare algebra from the field operators. (See Peskin & Schroder Chap 2, the discussion of Noether's theorem.) But there might be annoying technicalities, stemming from your decision to use the Weyl operators, instead of the raw $\Phi(f)$ observables.

3) If I recall correctly, Baez's book Introduction to Algebraic & Constructive Field Quantum Field Theory covers this material in the language you seem to prefer.