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Advances in Algebraic Quantum Field Theory pp 125–189Cite as

Algebraic Quantum Field Theory in Curved Spacetimes

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Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

This chapter sets out the framework of algebraic quantum field theory in curved spacetimes, based on the idea of local covariance. In this framework, a quantum field theory is modelled by a functor from a category of spacetimes to a category of (\(C^*\))-algebras obeying supplementary conditions. Among other things: (a) the key idea of relative Cauchy evolution is described in detail, and related to the stress-energy tensor; (b) a systematic ‘rigidity argument’ is used to generalise results from flat to curved spacetimes; (c) a detailed discussion of the issue of selection of physical states is given, linking notions of stability at microscopic, mesoscopic and macroscopic scales; (d) the notion of subtheories and global gauge transformations are formalised; (e) it is shown that the general framework excludes the possibility of there being a single preferred state in each spacetime, if the choice of states is local and covariant. Many of the ideas are illustrated by the example of the free Klein–Gordon theory, which is given a new ‘universal definition’.

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Notes

  1. 1.

    We adopt signature convention \(+-\cdots -\) for the metric and in general adopt the conventions used in Chaps. 3, 5 and 6.

  2. 2.

    The algebra \({\mathscr {A}}({\varvec{M}})\) is simple (and not the zero algebra!), so \({\mathscr {A}}(\psi )\) either has trivial kernel or full kernel; the latter case is excluded because \({\mathscr {A}}(\psi )\mathbf {1}_{{\mathscr {A}}({\varvec{M}})}=\mathbf {1}_{{\mathscr {A}}({\varvec{N}})}\ne 0\).

  3. 3.

    An initial object in a category \({{{\mathbf {\mathsf{{C}}}}}}\) is an object I with the property that there is, to each object C of \({{{\mathbf {\mathsf{{C}}}}}}\), exactly one morphism from I to C.

  4. 4.

    Alternatively, and perhaps more in the spirit of a categorical description, one might say that the morphism \({\mathscr {A}}(\iota _{{\varvec{M}};O})\), regarded as defining a subobject of \({\mathscr {A}}({\varvec{M}})\), should be the focus here [69].

  5. 5.

    In a general categorical setting, one would employ the categorical union of the \({\mathscr {A}}^{\text {kin}}({\varvec{M}};O_i)\).

  6. 6.

    Some authors, notably Penrose [123] and Geroch [83], define the Cauchy development with timelike curves of various types. We follow [8, 90, 119, 161]. Many authors only define the Cauchy development for achronal sets. The fact that \(D_{\varvec{M}}(O)\) is open is most easily seen using limit curves cf. [8, Prop. 3.31] or [90, Lem. 6.2.1].

  7. 7.

    The orientation need not be changed when the metric changes; recall that \({\mathfrak {o}}\) is a component of the nonzero smooth n-forms, and not e.g., the volume form.

  8. 8.

    It is natural to write \({{{\mathbf {\mathsf{{T}}}}}}_{\varvec{M}}(\pounds _X g)=-2(\nabla \cdot {{{\mathbf {\mathsf{{T}}}}}}_{\varvec{M}})(X)\), regarding the divergence in a weak sense.

  9. 9.

    Decompose \(f=f_0 + P_{\varvec{M}}f_1\), where \(f_0\) is supported in the image of \({\mathcal {M}}^+\), and \(f_1\in {\mathscr {D}}({\varvec{M}})\). As \(\varPhi _{\varvec{M}}(f)=\varPhi _{\varvec{M}}(f_0)\), we may apply (4.19) to \(f_0\) and then use the fact that \(E_{\varvec{M}}f_0=E_{\varvec{M}}f\).

  10. 10.

    This result differs by a sign from that in [24] (and repeated e.g., in [69, 70]). The source of the difference arises on p. 61 of [24], where the action of an ‘advanced’ Green function is taken to have support in the causal future of the source. The sign error does not affect the results of [69, 70].

  11. 11.

    By default, all \(*\)-algebras here are unital, i.e. they have a unit element for the algebra product.

  12. 12.

    Note that \({\mathscr {S}}\), or its opposite covariant functor \({\mathscr {S}}^{\text {op}}:{{{\mathbf {\mathsf{{Loc}}}}}}\rightarrow {{{\mathbf {\mathsf{{Stsp}}}}}}^{\text {op}}\), contains all the information in \({\mathscr {A}}\), and could be used by itself to specify the theory in full—this is done e.g., in [48, 54].

  13. 13.

    Our convention on Fourier transforms of compactly supported distributions is (in Minkowski space) \(\hat{u}(k)=u(e_k)\), where \(e_k(x)=e^{ik_\mu x^\mu }\); this is extended to manifolds using coordinate charts.

  14. 14.

    Recall that \({\mathfrak {t}}\), \({\mathfrak {o}}\) and \({\mathfrak w}\) are all regarded as connected components of certain sets of nowhere zero forms; by \({\mathfrak {t}}\wedge {\mathfrak w}\) we denote the set of all possible exterior products from within \({\mathfrak {t}}\) and \({\mathfrak w}\).

  15. 15.

    Here, we use the stability of (multi)-diamonds under \({{{\mathbf {\mathsf{{Loc}}}}}}\) morphisms  [25, Lem. 2.8].

  16. 16.

    The same arguments could be applied to more general smearings of the field-strength.

  17. 17.

    Of course, each \(\varPhi _i\) is contained in the local algebra for regions containing the 2-surfaces \(S_i\).

  18. 18.

    The preorder is not a partial order, because \((S_1,T_1)\prec (S_2,T_2)\prec (S_1,T_1)\) implies \(D_{\varvec{M}}(S_1)=D_{\varvec{M}}(S_2)\) and \(D_{\varvec{M}}(T_1)=D_{\varvec{M}}(T_2)\), but not necessarily \(S_1=S_2\) and \(T_1=T_2\).

  19. 19.

    That is, we require \({\mathcal {R}}_S\varOmega _\omega \) to be dense in \(\mathscr {H}_\omega \) and \({\mathcal {R}}_T\ni A\mapsto A\varOmega _\omega \in \mathscr {H}_\omega \) to be injective.

  20. 20.

    In Hilbert-space representations, \({{{\mathbf {\mathsf{{T}}}}}}_{\varvec{M}}[h]\) is an unbounded operator, one also has to consider the domain of algebra elements A for which the commutator can be formed, or in which precise mathematical sense the commutator is to be understood.

  21. 21.

    In fact, the result on the Hadamard property of ground states and KMS-states on ultrastatic spacetimes holds for more general types of quantized linear fields, and more generally also on static (not necessarily ultrastatic) spacetimes.

  22. 22.

    While the results in [68] have only been established for compact \(\varSigma \), the results could be extended to noncompact \(\varSigma \) upon making suitable integrability assumptions on \(c_{{\varvec{M}}}(f,\mathbf{x})\) with respect to \(\mathbf{x} \in \varSigma \).

  23. 23.

    The reader might wonder why (4.43) is not adopted for \({\text {Fld}}({\mathscr {D}},{\mathscr {A}})\) in place of (4.42). The reason is that \(\varPhi ^*\varPhi \) is a positive element of \({\text {Fld}}({\mathscr {D}},{\mathscr {A}})\) in the sense that \((\varPhi ^*\varPhi )_{\varvec{M}}(f) = \varPhi _{\varvec{M}}(f)^*\varPhi _{\varvec{M}}(f)\) is a positive element in \({\mathscr {A}}({\varvec{M}})\) for every \(f\in {\mathscr {D}}({\varvec{M}})\), while \(\varPhi ^\star \varPhi \) need not be positive in this way. Order structure and functional calculus for abstract fields is discussed in [52].

  24. 24.

    Somewhat more technically, \({\mathscr {D}}_\rho ({\varvec{M}},\sigma )\) may be regarded as the space of compactly supported sections of the bundle \({\varvec{M}}\ltimes _\rho V_\rho \) associated to \(S{\varvec{M}}\) and \(\rho \).

  25. 25.

    Terminology here is parallel to [156] but one could equally make a case for labelling the type by the dual (also known as contragredient) representation \(\rho ^*\), which was our convention in Sect. 4.9.1.

  26. 26.

    An interesting variant shows what happens if negative-normed states are allowed [91].

  27. 27.

    Here, the trivial spin structure \(\sigma _0(A) = R_{\varLambda (A)}e\) is intended, where \(e=(\partial /\partial x^\mu )_{\mu =0,\ldots ,3}\) is the orthonormal frame on Minkowski space associated with standard inertial coordinates \(x^\mu \).

  28. 28.

    DHR work in the Hilbert space representation of the Poincaré invariant vacuum state, and require that global gauge transformations should leave the vacuum vector invariant. This also has an analogue in the present setting [54].

  29. 29.

    Suppose \(({\mathscr {A}},\varPhi )\) and \(({\mathscr {B}},\varPsi )\) both satisfy Definition 4.10.4. Then there are naturals \(\eta :{\mathscr {A}}\mathop {\rightarrow }\limits ^{\cdot }{\mathscr {B}}\) and \(\zeta :{\mathscr {B}}\mathop {\rightarrow }\limits ^{\cdot }{\mathscr {A}}\) such that \(\varPsi =\eta \cdot \varPhi \) and \(\varPhi =\zeta \cdot \varPsi \). Hence also \(\varPhi =(\zeta \circ \eta )\cdot \varPhi \). But by the universal property yet again, the only natural \(\xi :{\mathscr {A}}\mathop {\rightarrow }\limits ^{\cdot }{\mathscr {A}}\) such that \(\varPhi =\xi \cdot \varPhi \) is the identity, \(\xi _{\varvec{M}}=\text {id}_{{\mathscr {A}}({\varvec{M}})}\) for all \({\varvec{M}}\in {{{\mathbf {\mathsf{{Loc}}}}}}\). Hence \(\zeta \circ \eta =\text {id}_{{\mathscr {A}}}\) and by similar reasoning applied to \(({\mathscr {B}},\varPsi )\), we also have \(\eta \circ \zeta = \text {id}_{{\mathscr {B}}}\). Hence \(\eta \) is an equivalence.

  30. 30.

    In theories based on a classical Lagrangian one usually proceeds simply to use the ‘same’ Lagrangian (modulo some subtleties [53]) but this option is not open in a general AQFT context.

  31. 31.

    Recall that if \({\varvec{M}}\) has compact Cauchy surfaces then so does \({\varvec{N}}\).

  32. 32.

    This assumes that \({\mathscr {A}}\) is not isomorphic to \({\mathscr {A}}\otimes {\mathscr {A}}\). A convenient way of ruling out such isomorphisms is to check that \({\mathscr {A}}\) and \({\mathscr {A}}\otimes {\mathscr {A}}\) have nonisomorphic gauge groups.

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Acknowledgments

We are grateful to Francis Wingham for comments on the text.

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  1. Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK

    Christopher J. Fewster

  2. Institut für Theoretische Physik, Universität Leipzig, Brüderstr. 16, 04103, Leipzig, Germany

    Rainer Verch

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    Claudio Dappiaggi

  3. University of Hamburg, Hamburg, Germany

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Fewster, C.J., Verch, R. (2015). Algebraic Quantum Field Theory in Curved Spacetimes. 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_4

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