Quantum operations in algebraic QFT. (English) Zbl 07818742
Summary: Conformal Quantum Field Theories (CFT) in 1 or \(1+1\) spacetime dimensions (respectively called chiral and full CFTs) admit several “axiomatic” (mathematically rigorous and model-independent) formulations. In this note, we deal with the von Neumann algebraic formulation due to R. Haag and D. Kastler [J. Math. Phys. 5, 848–861 (1964; Zbl 0139.46003)], mainly restricted to the chiral CFT setting [18]. Irrespectively of the chosen formulation, one can ask the question \(\mathrm{s}\): given a theory \(\mathcal{A}\), how many and which are the possible extensions \(\mathcal{B} \supset \mathcal{A}\) or subtheories \(\mathcal{B} \subset \mathcal{A}\)? How to construct and classify them, and study their properties? Extensions are typically described in the language of algebra objects in the braided tensor category of representations of \(\mathcal{A}\), while subtheories require different ideas. In this paper, we review recent structural results on the study of subtheories in the von Neumann algebraic formulation (conformal subnets) of a given chiral CFT (conformal net), [M. Bischoff, Rev. Math. Phys. 29, No. 1, Article ID 1750002, 53 p. (2017; Zbl 1364.81213); J. Funct. Anal. 281, No. 1, Article ID 109004, 78 p. (2021; Zbl 1481.46066); Ann. Henri Poincaré 23, No. 8, 2979–3020 (2022; Zbl 1502.46050); Rev. Math. Phys. 35, No. 4, Article ID 2350007, 30 p. (2023; Zbl 1524.81078). Furthermore, building on [M. Bischoff et al., Rev. Math. Phys. 35, No. 4, Article ID 2350007, 30 p. (2023; Zbl 1524.81078)], we provide a “quantum Galois theory” for conformal nets analogous to the one for Vertex Operator Algebras (VOA) [C. Dong and G. Mason, Duke Math. J. 86, No. 2, 305–321 (1997; Zbl 0890.17031); J. Algebra 214, No. 1, 92–102 (1999; Zbl 0929.17031)]. We also outline the case of \(3+1\) dimensional Algebraic Quantum Field Theories (AQFT). The aforementioned results make use of families of (extreme) vacuum state preserving unital completely positive maps acting on the net of von Neumann algebras, hereafter called quantum operations. These are natural generalizations of the ordinary vacuum preserving gauge automorphisms, hence they play the role of “generalized global gauge symmetries”. Quantum operations suffice to describe all possible conformal subnets of a given conformal net with the same central charge.
MSC:
| 81T05 | Axiomatic quantum field theory; operator algebras |
| 46L37 | Subfactors and their classification |
| 94A40 | Channel models (including quantum) in information and communication theory |
| 20N20 | Hypergroups |
| 83C25 | Approximation procedures, weak fields in general relativity and gravitational theory |
| 83C22 | Einstein-Maxwell equations |
Keywords:
conformal subnet; algebraic quantum field theory; quantum operation; unital completely positive map; hypergroupCitations:
Zbl 0139.46003; Zbl 1364.81213; Zbl 1481.46066; Zbl 1502.46050; Zbl 1524.81078; Zbl 0890.17031; Zbl 0929.17031References:
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