Abstract
In the algebraic approach to quantum field theory over curved space-time the main object is a family of C -algebras (or even von Neumann algebras) A(O) indexed by open bounded sets O from the space-time manifold. The isometries g of this manifold are represented by *-automorphisms αg of the whole algebra A = C*(∪o A(0)). These objects A(O), αg satisfy some properties such as isotony, covariance, and locality (for details see [3], [7], [9], [10]). In the case that the space-time is the Minkowski manifold the structure of these algebras and automorphisms has been intensively studied mostly under additional assumptions (see e.g. [4], [5], [8]). The aim of the present paper is to extend some of these results to more general manifolds. For this goal a slight abstraction of the algebraic approach to quantum field theory is given. It has the sense to simplify and unify the essential ideas and to abstract from details which are unimportant for some questions. Thus it can be applied to more general situations.
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References
Baumgartel, H.: Einstein causal quantum fields on lattices with Lorentz invariance, in Topics in quantum field theory and spectral theory, Report R-Math-01/86, Karl-Weierstrass Institut fur Mathematik, Berlin, 1986.
Bratteli, O.; Robinson, D.W.: Operator algebras and quantum statistical mechanics I, Springer, New York, 1979.
Dimock, J.: Algebras of local observables on a manifold, Comm. Math. Phys. 77 (1980), 219–228.
Emch, G.: Algebraic methods in statistical mechanics and quantum field theory, Wiley Interscience, New York, 1972.
Haag, R.; Kastler, D.: An algebraic approach to quantum field theory, J. Math. Phys. 5(1964), 848–861.
Haag, R.; Kastler, D.; Kadison, R.V.: Nets of C*-algebras and classification of states, Comm. Math. Phys. 16(1970), 81–104.
Haag, R.; Namhofer, H.; Stein, H.: On quantum field theory in gravitational background, Comm. Math. Phys. 94(1984), 219–238.
Horuzii, S.S.: Introduction to algebraic quantum field theory (Russian), Nauka, Moscow, 1986.
Isham, C.: Quantum field theory in curved space-time: a general mathematical framework, in Differential geometric methods in mathematical physics. II, eds. K. Bleuler et al., Springer, New York, 1978.
Kay, B.: The double wedge algebra for quantum fields on Schwarzschild and Minkowski space times, Comm. Math. Phys. 100(1985), 57–81.
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© 1990 Birkhäuser Verlag Basel
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Wollenberg, M. (1990). On Causal Nets of Algebras. In: Helson, H., Sz.-Nagy, B., Vasilescu, FH., Arsene, G. (eds) Linear Operators in Function Spaces. Operator Theory: Advances and Applications, vol 43. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7250-8_26
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DOI: https://doi.org/10.1007/978-3-0348-7250-8_26
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-7252-2
Online ISBN: 978-3-0348-7250-8
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