FEYNMAN PROPAGATOR AND LIMITING ABSORPTION PRINCIPLEON ASYMPTOTICALLY MINKOWSKI SPACETIMESKOUICHI TAIRA

Propagators are one of the fundamental objects in Quantum Field Theory (QFT). Mathematically, it is deﬁned by fundamental solutions to the Klein-Gordon equation such as thosegiven by the advanced/retarded propagators and the Feynman/anti-Feynman propagators. The
existence of the advanced/retarded propagators is classically known. These are constructed bysolving a Cauchy problem and deﬁned for globally hyperbolic spacetimes. On the other hand,
how the Feynman/anti-Feynman propagators are generalized is non-trivial and these existence
problem is global in nature and is more complicated. See the introductions of [1], [2], [3], [5], [6]and [7].
The massive Feynman propagator (m > 0) in exact Minkowski spacetime is given by(∂t2 − ∆y + m2 − i0)−1 .

(1)More precisely, its integral kernel (Green function) (∂t2 − ∆y + m2 − i0)−1 (x, x′ ) is given by∫′1ei(x−x )·ξ22−1′(∂t − ∆y + m − i0) (x, x ) = limdξ,ε→0, ε>0 (2π)n+1 Rn+1 −τ 2 + |η|2 + m2 − iεwhere we write ξ = (τ, η) ∈ R×Rn . This formula is deﬁned by using the Fourier analysis and thedistribution theory, so how Feynman propagators are deﬁned in more general curved spacetimes
is non-trivial from this formula.
Duistermaat-Hörmander [4] gave a precise deﬁnition of Feynman propagator in terms of thewavefront set and showed the existence of Feynman parametrices (inverses up to smoothingerrors) under general settings. Recently, actual inverses (Feynman propagators) have been
constructed on various spacetimes by using the scattering theory or the microlocal theory ([1],[2], [3], [5], [6], [7] and [11]). However, their constructions diﬀer from each other and relationshipbetween such Feynman propagators and the formula (1) had not been addressed. In this study,
following a program of [1], [3], we show that the (anti-)Feynman propagator constructed in [6]and [7] coincides with a limit of resolvent as (1) on curved spacetimes which are close to theMinkowski spacetime near spacetime inﬁnity.
Now we state our main theorem. Let g0 be the Minkowski metric on Rn+1= Rt × Rny andx−1
g0 be its dual metric:
g0 = −dx21 + dx22 + ... + dx21+n ,g0−1 = −∂x21 + ∂x22 + ... + ∂x21+n = (g0ij )ni,j=1 .

1We write ⟨x⟩ = (1 + |x|2 ) 2 and introduce the function space
S k (Rn+1 ) := {a ∈ C ∞ (Rn+1 ) | |∂xα a(x)| ≤ C⟨x⟩k−|α| },k ∈ R.

Assumption 1. A Lorentzian metric g on Rn+1 satisﬁes the following conditions: The inverse
matrix g −1 (x) = (g jk (x))nj,k=1 of g(x) satisﬁes g jk − g0jk ∈ S −µ (Rn+1 ) for some µ > 0.
Assumption 2 (Null non-trapping condition). All non-constant null geodesics escape to thespacetime inﬁnity.
Assumption 3. There exists a time function t̃ such that t̃ − t ∈ Cc∞ (Rn+1 ).
It is shown in [8], [9] and [11] that P := −□g is essentially self-adjoint on Cc∞ (Rn+1 ) under
Assumptions 1 and 2. We denote the unique self-adjoint extension by the same symbol P .
Moreover, we set L2,s (Rn+1 ) := ⟨x⟩−s L2 (Rn ).
12KOUICHI TAIRATheorem 1. [10] We suppose Assumptions 1 and 2.
(i) Let s > 21 and λ ∈ R \ {0}. Then the limits(P − λ ∓ i0)−1 :=lim (P − λ ∓ iε)−1ε→0, ε>0exist in B(L2,s (Rn+1 ), L2,−s (Rn+1 )).
(ii) In addition, we suppose Assumption 3, µ > 1 and λ = m2 > 0. The operator (P +m2 −i0)−1coincides with the (anti-)Feynman propagator deﬁned in [6] and [7]. In particular, (P +m2 −i0)−1
is actually a Feynman propagator in the deﬁnition of Duistermaat-Hörmander [4].
Remark 1. The convention of the Feynman/anti-Feynman propagators in [6] and [7] are different from ones in physics books. Here we follow the convention in physics books.
Remark 2. A weaker statement of (i) is also proved in [11] by a diﬀerent method but under avery short-range condition.
The proof of the part (i) is essentially due to the Mourre theory, which is commonly used in
the scattering theory. A main diﬃculty here is due to the lack of ellipticity of P = −□g .
To describe the key idea of the proof of the part (ii), we rewrite (1) as∫√i′
2(∂t2 − ∆y + m2 − i0)−1 f (t, y) = √e−i|t−t | −∆y +m f (t′ , y)dt
22 −∆y + m Rfor (t, y) ∈ R × Rn . If f ∈ Cc∞ (Rn+1 ), then we write√{−it −∆y +m2eg+ (t, y) t >> 1√(∂t2 − ∆y + m2 − i0)−1 f (t, y) =2eit −∆y +m g− (t, y) t << −1√
for some function g± and it satisﬁes (Dt ± −∆y + m2 )u = 0 for ±t >> 1. We regard this√n−1equation as an analogue of Sommerfeld’s radiation condition (∂r − i λ)u = o(|x|− 2 ) for theHelmholtz equation (−∆ − λ)u = 0 on Rn . This observation is more or less justiﬁed for moregeneral setting by using the microlocal analysis. Moreover, uniqueness of solutions under theradiation condition yields an identiﬁcation of Feynman propagators (proof of the part (ii)).
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Department of Mathematical Sciences, Ritsumeikan UniverLesity, 1-1-1 NojiHigashi, Kusatsu,
525-8577 Japan.
Email address: ktaira@fc.ritsumei.ac.jp