Abstract
Free quantum field theories on curved backgrounds are discussed via three explicit examples: the real scalar field, the Dirac field and the Proca field. The first step consists of outlining the main properties of globally hyperbolic spacetimes, that is the class of manifolds on which the classical dynamics of all physically relevant free fields can be written in terms of a Cauchy problem. The set of all smooth solutions of the latter encompasses the dynamically allowed configurations which are used to identify via a suitable pairing a collection of classical observables. As a last step we use such collection to construct a \(*\)-algebra which encodes the information on the dynamics and the canonical commutation or anti-commutation relations depending on the Bosonic or Fermionic nature of the underlying field.
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Notes
- 1.
Notice that, in this chapter, we employ the following convention for the tensor components: Latin indices, \(a,b,c,\ldots \), are used for abstract tensor indices, Greek ones, \(\mu ,\nu ,\ldots \) for coordinates, while i, j, k are used for spatial components or coordinates.
- 2.
We are grateful to Zhirayr Avetisyan for pointing us out Theorem 3.68 in [7].
- 3.
A covariant derivative \(\nabla \) on F is P-compatible if there exists a section \(A\in \varGamma (\mathrm {End}(F))\) such that \(\Box _\nabla +A=P\).
- 4.
A partition of unity such as the one described exists on account of Theorem 3.1.4. In fact, after splitting the globally hyperbolic spacetime \({\varvec{M}}\) in the Cartesian product of \({\mathbb R}\) and a spacelike Cauchy surface \(\varSigma \) and for any choice of \(t_\pm \in {\mathbb R}\) with \(t_-<t_+\), one can introduce a partition of unity \(\{\chi _+,\chi _-\}\) on \({\mathbb R}\) such that \(\chi _\pm (t)=1\) for \(\pm t \ge \pm t_\pm \). Pulling this partition of unity back to \({\mathcal M}\) along the projection on the time factor \(t:{\mathcal M}\rightarrow {\mathbb R}\), one obtains a partition of unity on \({\mathcal M}\) of the sought type.
- 5.
In Remark 3.3.3, we shall show that, in the case of the real scalar field, \(N=P(C^\infty _0({\mathcal M}))\). More generally, using the same argument, one can prove an analogous result for any field whose dynamics is ruled by a Green hyperbolic operator.
- 6.
Even though the term “symplectic structure” is mathematically correct, it would be more appropriate to refer to this as a constant Poisson structure. Yet, we shall adhere to the common nomenclature of quantum field theory on curved spacetimes.
- 7.
The function in the right-hand-side of the equation which defines L is the extension by zero to the whole spacetime of the function appearing in the left-hand-side.
- 8.
The volume form \(d\varSigma \) on \(\varSigma \) is defined out of the structure induced on \(\varSigma \) itself as a submanifold of the globally hyperbolic spacetime \({\varvec{M}}\). More explicitly, on \(\varSigma \) we take the Riemannian metric \(g\vert _\varSigma \) and the orientation specified by the orientation and time-orientation of \({\varvec{M}}\). Then \(d\varSigma \) is the natural volume form defined out of these data.
- 9.
Note that the Dirac representation T is usually regarded as a unitary representation of \(\mathrm {SL}(2,{\mathbb C})\) on \({\mathbb C}^4\), yet \(\mathrm {Spin}(1,3)\) is isomorphic to \(\mathrm {SL}(2,{\mathbb C})\) as a Lie group.
- 10.
The sections on the right-hand-side in the definitions of \(L_\mathrm {s}\) and of \(L_\mathrm {c}\) are the extensions by zero to the whole spacetime of the sections which appear on the left-hand-side.
- 11.
As usual, the component of the direct sum corresponding to the degree \(k=0\) is simply \({\mathbb C}\).
- 12.
The differential form on the right-hand-side of the equation which defines L is the extension by zero to the whole spacetime of the differential form which appears on the left-hand-side.
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Acknowledgments
The work of M.B. has been supported by a grant of the University of Pavia, which is gratefully acknowledged. C.D. is grateful to Zhirayr Avetisyan for the useful discussions.
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Benini, M., Dappiaggi, C. (2015). Models of Free Quantum Field Theories on Curved Backgrounds. 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_3
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