The microlocal spectrum condition, initial value formulations, and background independence (Q2795551)

Let \((M,g)\) be a globally hyperbolic spacetime; (\(M=\mathbb{R}\times \Sigma\)), \(E\) a vector bundle over \(M\) equipped with a fiber metirc \(g_E\). In this paper, first a formally self-adjoint, normally hyperbolic differential operator \(P:\mathcal{E}(M,E)\to \mathcal{E}(M,E)\), \(\mathcal{E}(M,E)\) is the space of smooth sections of \(E\), where the principal symbol of \(P\) is given by the spacetime matric \(g\), is considered, and theNEWLINENEWLINE{Theorem 2.2} is proven. For any distributions \(u_0,u_1\) on \(\Sigma(=\{0\}\times\Sigma)\), there exists a unique proper distribution \(u\) such that \(Pu=0\) and \(u|\Sigma =u_0\),\(\nabla_nu|\Sigma=u_1\). Here, a proper solution means it is approximate by regular solutions by the weak topology.NEWLINENEWLINEBy using the causal propagator \(G\) of \(P\), the solution space of \(P\) has an aymplectic structure given by NEWLINENEWLINE\[NEWLINE\sigma^M(f,f')=\int_M g_E(G(f),f')dV_g .NEWLINE\]NEWLINE NEWLINE((2.44), an alternative expression of this symplectic structure defined on the space of initial date is also given as (2.45)). Let \(\mathcal{W}_P\) be the associated Weyl algebra, and \(\omega\) its quasi-free state. Then, \(\omega(W(G(f)))=\exp(-\frac{1}{2}\omega_2(f,f'))\), where the two-point function \(\omega_2\) is a distributional bi-solution of \(P\): \(\omega_2(Pf,f')=\omega_2(f,Pf')=0\). A quasi-free Hadamard state is characterized by NEWLINENEWLINE\[NEWLINE\mathrm{WF}(\omega_2)=\mathrm{WF}(K_G)\cap(C^\ast_+M\times C^\ast_-M), NEWLINE\]NEWLINE NEWLINEwhere \(C^\ast_\pm M\) are the future/past-directed co-light cone bundles of \(M\), \(K_g\) is the causal propagator kernel [\textit{M. J. Radzikowski}, Commun. Math. Phys. 179, No. 3, 529--553 (1996; Zbl 0858.53055)]. Explanations on wave front sets are given in A2 of the appendix.NEWLINENEWLINEA Hadamard state can be regarded as a replacement of the vacuum state of quantum field theory on Minkowski space and defines a Feynman propagator \(\Omega_F=i\omega_2-K_{G^-}\). Since \(\omega_2\) is a bisolution of \(P\), if \(\omega_2\) is proper, we may associate unique initial date \(\omega_{2,00}.\omega_{2,01},\omega_{2,10}, \omega_{2,11}\) in \(\mathcal{D}'(\Sigma\times \Sigma, E^\ast_\Sigma\boxtimes E^\ast_\Sigma)\) ((2.50), (2.51)). Then, the main theorem of this paper isNEWLINENEWLINE{Theorem 2.8}. Let \(\omega_2\) be the two-point function of a quasi-free Hadamard state \(\omega\) and proper. Then, its initial data satisfies NEWLINE\[NEWLINE\bigcup_{i,j=0}^1\mathrm{WF}(\omega_{2,ij})=N_\Delta\setminus\{0\},NEWLINE\]NEWLINE NEWLINEwhere \(N_\Delta\) is the conormal of the diagonal map \(\Delta:\Sigma\to \Sigma\times\Sigma\).NEWLINENEWLINEProof is done by using several properties of wave front sets explained in A2.NEWLINENEWLINEThe authors say that this is plausible from the point of view of canonical quantization on \(\Sigma\), because a proper solution \(u\) arises from \(u|_\Sigma\) by the causal propagator \(K_G\) of \(P\). Since \(u|_\Sigma\) is the restriction of \(u\), the knowledge of \(\mathrm{WF}(\omega_2)\) gives a two sided estimate on the wave front sets \(\mathrm{WF}(\omega_{2,ij})\). Furthermore, \(K_G\) propagates singularities along co-light cone bundle (appendix, Theorem A.22), the initial data for a Hadamard state contain enough singular directions to satisfy the microlocal spectrum condition.NEWLINENEWLINEAs an example, explicit computations of wave front sets arising from the Klein-Gordon equation are given and verify Theorem 2.8 as Example 2.9.NEWLINENEWLINEIn concluding Remarks (\S3), the authors say Theorem 2.8 tells us that the wave front sets of the initial data are already restricted in terms of the geometry of the Cauchy surface \(\Sigma\) only. There is no reference to the metirc (causal) structure of \(M\). Thus, we have a condition that is applicable to settings, where no spacetime metric is available, e.g., in loop quantum gravity. Researches in this direction, we refer to [the authors, ``Coherent states, quantum gravity and Born-Oppenheimer approximation, I,II,III'', \url{arXiv:1504.02169}, \url{arXiv:1504.02170}, \url{arXiv:1504.02171}].NEWLINENEWLINEIn the appendix, to make this paper self contained, distributions on manifolds (A1) and the wave front sets, tools from microlocal analysis (A2) are explained.