Among the postulates of AQFT there is no dynamical evolution principle postulated, i.e. there is no analog to a postulated Heisenberg equation of motion. How does one define a constant of motion in algebraic qft?
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$\begingroup$ One usually assumes the Time-Slice axiom on top of the Haag-Kastler postulates in order to guarantee a well-posed IVP. See ncatlab.org/nlab/show/time+slice+axiom $\endgroup$– Phoenix87Commented Aug 20, 2016 at 10:02
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$\begingroup$ @Phoenix87 Is the classical field theory underlying an aqft assumed to be known before constructing the aqft, or is it derived from a given aqft as its classical limit? The world is quantum mechanical. Classical physics is just a limit. $\endgroup$– Andrea BeckerCommented Aug 20, 2016 at 10:11
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1$\begingroup$ In the Haag-Kastler framework there is no notion of field. If you know the underlying field theory then you can go through the Wightman construction and land on an AQFT. The focus is then on the net of local algebras and the quasi-local algebra as well, i.e. in the way the "interact" algebraically. $\endgroup$– Phoenix87Commented Aug 20, 2016 at 10:27
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$\begingroup$ @Phoenix87 So if one assumes the TSA axiom and starts with a classical field theory in which there are classical constants of motion, then in both Wightman and Haag-Kastler frameworks, i.e. in both axiomatic and algebraic qft, there are corresponding operators (corresponding to the classical ones) that do not depend on time? Is this a correct statement? $\endgroup$– Andrea BeckerCommented Aug 20, 2016 at 10:45
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$\begingroup$ I would be surprised if things are this simple. In general, for nicely behaved (usually free) theories, a classical first integral should remain such after you quantise the theory. Also, Wightman and Haag-Kastler should not be confused. I think that if you start with a field theory on a globally hyperbolic space-time, then the time-slice axiom should follow. Hence why one is led to make this extra assumption in the HK framework. $\endgroup$– Phoenix87Commented Aug 20, 2016 at 11:16
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