# Axiomatic QFT: Time-slice Axiom vs Transformation Properties

I am studying Wightman axioms and Haag–Kastler axioms for QFT from Haag's book "Local Quantum Physics". In both axiomatic frameworks, he introduces the "Time-slice Axiom" (axiom G) as

"There should be a dynamical law which allows one to compute fields at an arbitrary time in terms of fields in small time slice $\mathcal{O}=\{ x:|x^0-t|<\epsilon \}$."

I don't understand what is the difference between this axiom and the axiom concerning the transformation properties (axiom D) of the Wightman fields or local algebras. Doesn't the unitary operator $U(a,\mathbb{1})$ introduced in axiom D work as the dynamical law mentioned in axiom G?

If you know some old post or some book where I can find this answer and I didn't see please notify me.

• They are very similar, but the only difference is that the Haag-Kastler axioms do not require a notion of a field as in the Wightman formulation. All you need is a consistent construction of net of algebras. Does this help? – user106422 Feb 16 '16 at 16:53
• Thanks kid for your response, but my question goes in another way. I'm not asking the differences between both axiomatic frameworks. I don't understand the difference between the "Time-slice axiom" and the "Transformation properties axiom". I mistakenly think that the second one includes the first one. That should be wrong, but I don't know why. – Diego Feb 16 '16 at 19:26
• The time-slice axiom requires that given an initial condition, your dynamics is entirely determined. Maybe the best way of understanding the time-slice axiom is by stating it as the following: given an inclusion of space-times $f: X \rightarrow Y$ such that $\Sigma_Y \subset f(X)$, $A(f): A (X) \rightarrow A(Y)$ is an isomorphism, where $\Sigma$ is denoting a Cauchy surface (an initial condition). – user40276 Feb 17 '16 at 7:12