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Advances in Algebraic Quantum Field Theory pp 191–251Cite as

Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction

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Abstract

Within this chapter we introduce the overall idea of the algebraic formalism of QFT on a fixed globally hyperbolic spacetime in the framework of unital \(*\)-algebras. We point out some general features of CCR algebras, such as simplicity and the construction of symmetry-induced homomorphisms. For simplicity, we deal only with a real scalar quantum field. We discuss some known general results in curved spacetime like the existence of quasifree states enjoying symmetries induced from the background, pointing out the relevant original references. We introduce, in particular, the notion of a Hadamard quasifree algebraic quantum state, both in the geometric and microlocal formulation, and the associated notion of Wick polynomials.

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Notes

  1. 1.

    \(\pi '_w\) can equivalently be defined as \(\{A \in {\mathscr {B}}(\mathcal{H}) \,\,|\,\, A\phi (a)= \pi (a^*)^\dagger A\,,\quad \forall a \in \mathcal{A}\}\).

  2. 2.

    There do not exist one-parameter group of unital \(*\)-algebra anti-linear automorphisms, this is because \(\beta _t = \beta _{t/2}\circ \beta _{t/2}\) is linear both for \(\beta _{t/2}\) linear or anti-linear.

  3. 3.

    It holds that \([a(\psi ), a^\dagger (\xi )] = \langle K\psi |K\xi \rangle \), \([a(\psi ),a(\xi )] = 0 = [a^\dagger (\psi ), a^\dagger (\xi )]\) if \(\xi ,\psi \in \mathsf {Sol}\), and \(a(\xi )\), \(a(\psi )\) are defined on \(\mathcal{D}_\omega \), with \(a^\dagger (\psi )= a(\psi )^\dagger |_{\mathcal{D}_\omega }\).

  4. 4.

    Recall that this isomorphism was established in Proposition 5.2.12, based on the well-posedness properties of the Klein-Gordon equation. From now one, we will be making use of this isomorphism implicitly.

  5. 5.

    It should be evident that the given definition does not depend on the particular GNS representation chosen for each state \(\omega _i\).

  6. 6.

    Note that this result is stated incorrectly in Theorem 4.4.1 of [65], where the condition on the operator Q is incorrectly given as trace class instead of Hilbert-Schmidt. The correct condition is actually given in Equation (4.4.21) of [65] as the Hilbert-Schmidt property of the operator \(\mathcal {E}\) and the mistake appears in identifying the corresponding property of Q. We thank Rainer Verch and especially Ko Sanders for bringing this to our attention.

  7. 7.

    By Riesz lemma, it exists if an only if the map \(\mathcal{D}_{\omega _{\mathbb M}} \ni \varPsi ' \mapsto \langle \varPsi '| :\phi ^2:(f) \varPsi \rangle \) is continuous for every \(\varPsi \in \mathcal{D}_{\omega _{\mathbb M}}\).

  8. 8.

    Observe in particular that \(:\hat{\phi }(f) \hat{\phi }(g): - :\hat{\phi }(g) \hat{\phi }(f):\,= iE(f,g)1\!\!1- \omega _{\mathbb M2}(iE(f,g)1\!\!1)1\!\!1=0\).

  9. 9.

    The function \(z \mapsto I_1(\sqrt{z})/\sqrt{z}\), initially defined for \(Re(z)>0\), admits a unique analytic extension on the whole space \(\mathbb C\) and the formula actually refers to this extension.

  10. 10.

    Our convention for the Fourier transform is so that \(f(x) = \frac{1}{(2\pi )^m}\int e^{-ikx} \hat{f}(k)\, d^m k\). This convention agrees with those of [29, 52, 53], but has the opposite sign in the exponential with respect to [62]. This means that our wavefont sets need to be negated to be compared to those of [62]. Fortunately, in all cases where this is done, the wavefront sets happen to be negation symmetric.

  11. 11.

    Results in [52, 53] are stated for \(\xi =0\) in KG operator, however they are generally valid for \(m^2\) replaced by a given smooth function, as specified at the beginning of p. 533 in [52].

  12. 12.

    The gap is the content of the three lines immediately before th proof (ii) \(\mathbf{3} \Rightarrow \mathbf{2}\) on p. 547 of [52]: The reasoning presented there cannot exclude elements of the form either \((x_1,x_2, 0,p_2)\) or \((x_1,x_2,p_1, 0)\) from \(WF(\omega _2)\) outside \(\mathcal{N}\). The idea of our proof was suggested by N. Pinamonti to the authors.

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Acknowledgments

The authors are grateful to R. Brunetti, C. Dappiaggi, C. Fewster, T. Hack, N.Pinamonti, K. Sanders, A. Strohmaier, Y. Tanimoto and R. Verch for useful discussions and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Trento, via Sommarive 14, 38123, Povo (Trento), Italy

    Igor Khavkine & Valter Moretti

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  1. Igor Khavkine
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Correspondence to Valter Moretti .

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Editors and Affiliations

  1. University of Trento, Trento, Italy

    Romeo Brunetti

  2. University of Pavia, Pavia, Italy

    Claudio Dappiaggi

  3. University of Hamburg, Hamburg, Germany

    Klaus Fredenhagen

  4. University of Vienna, Vienna, Austria

    Jakob Yngvason

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Khavkine, I., Moretti, V. (2015). Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds) Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-21353-8_5

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