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MATHEMATICAL PHYSICS
V. S. VLADIMIROV and V. F. NIKITIN
ON THE INTEGRAL REPRESENTATION OF JOST—LEHMANN—DYSON
(Presented by Academician N. N. Bogolyubov, 26 I 1961)
1. In the works \(^{1,2}\) of Jost, Lehmann, and Dyson an integral representation of the causal commutator was obtained. More precisely, the following theorem was proved. Let the generalized function \(f(x)\), \(x=(x_0,x_1,x_2,x_3)=(x_0,\bar{x})\), belong to the Sobolev–Schwartz space \(S^*\) and vanish at spacelike points: \(x^2=x_0^2-\bar{x}^2<0\). Suppose, further, that its Fourier transform \(\tilde f(p)\) vanishes in a domain \(G\), whose boundary consists of two spacelike surfaces*. Then \(\tilde f(p)\) can be represented in the form
\[ \tilde f(p)=\int \operatorname{sgn}(p_0-u_0)\,\delta\bigl[(p-u)^2-\lambda\bigr]\psi(u,\lambda)\,du\,d\lambda, \tag{1} \]
where the support of the (generalized) function \(\psi(u,\lambda)\) consists of those \(u\) and \(\lambda \ge 0\) for which the interior \((p-u)^2>\lambda\) of the two-sheeted hyperboloid \((p-u)^2=\lambda\) lies entirely outside the domain \(G\)**.
In this connection Dyson put forward the conjecture (see \(^{2}\), note 7) that for any open set \(G\) the support of the weight function \(\psi\) in the representation (1) consists of those \(u\) and \(\lambda \ge 0\) for which the hyperboloid \((p-u)^2=\lambda\) lies wholly outside \(G\). This conjecture was in fact established by them for such open sets \(G=\bigcup G_j\), for which each connected component \(G_j\) is bounded by two spacelike surfaces and any points of distinct connected components \(G_j\) are separated by spacelike intervals, i.e. \((x_1-x_2)^2<0\), if \(x_1\in G_{j_1}\), \(x_2\in G_{j_2}\), \(j_1\ne j_2\).
2. In works \(^{3,4}\), one of the authors (V. S. V.) established the validity of Dyson’s conjecture for a broader class of domains, namely for such domains \(G\) for which the least convex hull \(B_0(G)\) with respect to timelike curves is bounded by two spacelike surfaces***, and also for such open sets \(G=\bigcup G_j\) (\(G_j\) are the connected components of \(G\)) for which any points of distinct connected components of the set \(\bigcup B_0(G_j)\) are separated by spacelike intervals. This result is a consequence of the following theorem \(^{3}\): if \(f(x)\in S^*\), \(f(x)=0\) for \(x^2<0\), and \(\tilde f(p)=0\) for \(p\in G\), where \(G\) is an arbitrary domain, then \(\tilde f(p)=0\) for \(p\in B_0(G)\), where \(B_0(G)\) is the least convex hull of \(G\) with respect to timelike curves (the definition of \(B_0(G)\) is given in \(^{4}\)).
3. The aim of the present note is to clarify the validity of Dyson’s conjecture for domains and open sets \(G\) that do not satisfy
* For notation and definitions, see \(^{4}\).
** Such hyperboloids were called admissible by Dyson.
*** In \(^{3}\) this condition is omitted.
restrictions of item 2. It turns out that for them Dyson’s conjecture is then not always valid. For example, let us take as the domain \(G\) the half-space \(p_0<0\). If Dyson’s conjecture were true, then, by virtue of (1), \(\tilde f(p)\) would be equal to zero in the whole space \(R^4\). But a simple example shows that this is not so. Indeed, let
\[ \tilde f(p)=P(p)\lim_{\varepsilon\to +0}\frac{\varepsilon}{\pi} \int_{-\infty}^{\infty}\frac{d\sigma(t)}{(t-p_0)^2+\varepsilon^2}, \tag{2} \]
where \(P(p)\) is an arbitrary polynomial and \(\sigma(t)\ne \mathrm{const}\) is a function of bounded variation, constant for \(t<0\). The limit in (2) exists in the sense of \(S^*\). The function \(\tilde f(p)\) constructed by formula (2) is representable as the difference of the boundary values \(F(p_0\pm i0,\bar p)\) of the function
\[ F(\zeta)=P(\zeta)\frac{1}{2\pi i} \int_{-\infty}^{\infty}\frac{d\sigma(t)}{t-\zeta_0}, \qquad \zeta=(\zeta_0,\bar\zeta)=p+iq, \tag{3} \]
holomorphic in the domain bounded by the analytic hyperplane \(\zeta_0=\rho,\ \rho\geqslant 0\) (in the space \(C^4\)). Therefore the Fourier transform \(f(x)\) of the function \(\tilde f(p)\) vanishes for \(x^2<0\) (see \((^{3,4})\)).
- As a second example, let us consider the open set \(G\) consisting of two strips:
\[ G=[p:\ -n<p_0<-m\ \text{and}\ m<p_0<n,\ \bar p\text{ arbitrary}]. \tag{4} \]
If Dyson’s conjecture is accepted, then \(\tilde f(p)\) must be zero in the entire strip \(|p_0|<n\). But, taking in (2) \(\sigma(t)\) constant on the intervals \((-n,-m)\), \((m,n)\) and nonconstant on the interval \((-n,n)\), we obtain, as above, a counterexample.
- Denote by \(T\) the tubular domain in \(C^4\) of the form
\[ T=[\zeta=p+iq:\ q^2>0,\ \bar p\text{ arbitrary}]. \]
Let \(G\) be an arbitrary open set in \(R^4\); by \(\tilde G\) denote a complex neighborhood of \(G\) in \(C^4\) such that, if a ball of radius \(r\) belongs to \(G\), then the corresponding complex ball with the same center and radius \(\delta r\) belongs to \(\tilde G\) \((\delta=0,1)\).
Let \(f(x)=0\) for \(x^2<0\) and \(\tilde f(p)=0\) in the open set \(G\). Then, as is known \((^4)\), there exists a function \(F(\zeta)\), holomorphic in the domain \(T\cup\tilde G\), such that, in the sense of convergence in \(S^*\), the following limiting relation holds:
\[ \tilde f(p)=\lim_{\varepsilon\to +0} \left[F(p_0+i\varepsilon,\bar p)-F(p_0-i\varepsilon,\bar p)\right]. \tag{5} \]
This result is a consequence of the edge-of-the-wedge theorem \((^{5,6})\).
- In connection with the result stated in item 5, the problem arises of constructing the envelope of holomorphy \(H(T\cup\tilde G)\) of the domain \(T\cup\tilde G\). The integral representation (1) makes it possible to do this, at least for those \(G\) that satisfy the conditions of item 2 (see \((^4)\)). If these conditions are not fulfilled, however, then it is sometimes possible to construct or estimate \(H(T\cup\tilde G)\). Thus, in the case of the half-space \(p_0<0\) (see item 3), \(H(T\cup\tilde G)\) is the limit, as \(N\to+\infty\), of the sequence \(H(T\cup\tilde G_N)\), where \(G_N\) is the strip \(-N<p_0<0\). Since for a strip the envelope of holomorphy is known \((^{5,6})\):
\[ H(T\cup\tilde G_N)=\{\,|\bar q|<|\operatorname{Im}\sqrt{\zeta_0(\zeta_0+N)}|\,\}, \tag{6} \]
then, passing in (6) to the limit as \(N\to+\infty\), we find that the desired \(H(T\cup\tilde G)\) is the domain in \(C^4\) bounded by the hyperplane \(\zeta_0=\rho,\ \rho\geqslant 0\).
- Let \(G\) consist of the two strips (4). Using the definition of \(\widetilde G\) (item 5), it is not hard to show that \(T \cup \widetilde G\) contains a semitubular domain of the form
\[ [\zeta=(\zeta_0,\bar{\xi}):\zeta_0\in B,\ |\bar q|<V(\zeta_0),\ \bar p\text{ arbitrary}], \tag{7} \]
where \(B\) is the plane of the complex variable \(\zeta_0=p_0+iq_0\) with three cuts removed:
\[ -\infty<p_0\leq -n,\qquad -m\leq p_0\leq m,\qquad n\leq p_0<\infty,\qquad q_0=0; \tag{8} \]
\[ \begin{aligned} V(\zeta_0)=\sup_{\zeta_0\in B}\biggl\{& |q_0|,\, \frac{\delta}{\sqrt2}\left(\frac{n-m}{2}-\left|p_0-\frac{n+m}{2}\right|\right) \theta\!\left[ \frac{\delta}{\sqrt2}\left(\frac{n-m}{2}-\left|p_0-\frac{n+m}{2}\right|\right)-|q_0| \right],\\ &\frac{\delta}{\sqrt2}\left(\frac{n-m}{2}-\left|p_0+\frac{n+m}{2}\right|\right) \theta\!\left[ \frac{\delta}{\sqrt2}\left(\frac{n-m}{2}-\left|p_0+\frac{n+m}{2}\right|\right)-|q_0| \right]\biggr\}, \end{aligned} \tag{9} \]
where \(\theta(\xi)=1\) for \(\xi\geq 0\) and \(\theta(\xi)=0\) for \(\xi<0\).
Taking into account the fact that automorphisms of a domain are preserved also in its envelope of holomorphy (see \((^7)\), p. 260), and using Bremermann’s theorem \((^8)\), we conclude that the envelope of holomorphy of the domain (6) has the same form as the domain itself, with \(V(\zeta_0)\) necessarily replaced by the least superharmonic majorant \(V_0(\zeta_0)\) of the function \(V(\zeta_0)\).
- We shall show that the function
\[ V_1(\zeta_0)=\left|\operatorname{Im}\sqrt{\zeta_0^2-n^2}\right| -\frac{1}{\pi}\int_{-m}^{m}\frac{|q_0|\sqrt{\,n^2-t^2\,}\,dt}{(p_0-t)^2+q_0^2} \tag{10} \]
is a superharmonic majorant of the function \(V(\zeta_0)\) in the domain \(B\). Indeed, the function \(\operatorname{Im}\sqrt{\zeta_0^2-n^2}\) is harmonic and preserves its sign in \(B\). Therefore the function \(\left|\operatorname{Im}\sqrt{\zeta_0^2-n^2}\right|\) is also harmonic in \(B\). Noting that the modulus of a harmonic function is a subharmonic function \((^9)\), and taking into account the equality
\[ \int_{-m}^{m}\frac{|q_0|\sqrt{\,n^2-t^2\,}}{(p_0-t)^2+q_0^2}\,dt = \left|\operatorname{Im}\int_{-m}^{m}\frac{\sqrt{\,n^2-t^2\,}\,dt}{t-\zeta_0}\right|, \]
we conclude that the second term in (10), and consequently \(V_1(\zeta_0)\), are superharmonic functions. It remains to prove that \(V_1(\zeta_0)\) majorizes the function \(V(\zeta_0)\). In view of (9), for this it is sufficient to prove two inequalities:
\[ V_1(\zeta_0)-|q_0|\geq 0 \quad \text{for } \zeta_0\in B, \tag{11} \]
\[ V_2(\zeta_0)=V_1(\zeta_0)-\frac{\delta}{\sqrt2}\left(\frac{n-m}{2}-\left|p_0-\frac{n+m}{2}\right|\right)\geq 0 \tag{12} \]
in the triangles
\[ \begin{aligned} &m\leq p_0\leq \frac{n+m}{2},\qquad 0\leq q_0\leq \frac{\delta}{\sqrt2}(p_0-m);\\ &\frac{n+m}{2}\leq p_0\leq n,\qquad 0\leq q_0\leq \frac{\delta}{\sqrt2}(n-p_0). \end{aligned} \tag{13} \]
Since the function \(V_1(\zeta_0)-|q_0|\) is superharmonic in \(B\), in order to prove inequality (11) it is sufficient to prove that \(V_1(\zeta_0)=|q_0|\) on the boundary of the domain \(B\), i.e. on the cuts (8) and at infinity. But this is easily verified directly (on the cut \(|p_0|\leq m\) it is necessary to use the property of the \(\delta\)-shaped sequence; at the points \(\zeta_0=\pm m\) the function \(V_1(\zeta_0)\) has no limiting values). To prove inequality (12), note that the function \(V_2(\zeta_0)\) is harmonic
inside the triangles (13). Therefore it suffices to establish the inequality \(V_2(\zeta_0)\geq 0\) on the boundaries of these triangles:
\[ m\leq p_0\leq n,\quad q_0=0\quad (a); \qquad m\leq p_0\leq \frac{n+m}{2},\quad q_0=\frac{\delta}{\sqrt{2}}(p_0-m)\quad (b); \]
\[ \frac{n+m}{2}\leq p_0\leq n,\quad q_0=\frac{\delta}{\sqrt{2}}(n-p_0)\quad (c); \]
\[ 0\leq q_0\leq \frac{\delta}{2\sqrt{2}}(n-m),\quad p_0=\frac{n+m}{2}\quad (d). \]
On the line (a) inequality (12) reduces to
\[ \sqrt{n^2-p_0^2}\geq \frac{\delta}{\sqrt{2}}\left(\frac{n-m}{2}-\left|p_0-\frac{n+m}{2}\right|\right), \qquad m\leq p_0\leq n, \]
which is verified directly. On the lines (b), (c), and (d), inequalities (12) reduce respectively to the inequalities
\[ V_1\left(\frac{\sqrt{2}}{\delta}q_0+m+iq_0\right)\geq q_0,\qquad V_1\left(-\frac{\sqrt{2}}{\delta}q_0+n+iq_0\right)\geq q_0, \]
\[ V_1\left(\frac{n+m}{2}+iq_0\right)\geq q_0, \]
which are ensured by (11).
- We have not succeeded in showing that the constructed function \(V_1(\zeta_0)\) is the least superharmonic majorant of the function \(V(\zeta_0)\). Therefore one can only conclude that the envelope of holomorphy of the domain (7) is contained in the semitubular domain of holomorphy
\[ |\bar q|<V_1(\zeta_0). \tag{14} \]
Since the domain \(T\cup \widetilde G\) is contained in the domain (14), (14) majorizes the envelope of holomorphy \(H(T\cup \widetilde G)\), i.e.,
\[ H(T\cup \widetilde G)\subset \left\{|\bar q|<\left|\operatorname{Im}\sqrt{\zeta_0^2-n^2}\right| -\frac{1}{\pi}\int_{-m}^{m} \frac{|q_0|\sqrt{n^2-t^2}}{(p_0-t)^2+q_0^2}\,dt \right\}. \tag{15} \]
As a lower estimate one may take the sum of the domains of holomorphy of the separate strips \({}^{5,6}\):
\[ \text{either}\quad |\bar q|<\left|\operatorname{Im}\sqrt{(\zeta_0-n)(\zeta_0-m)}\right|, \qquad \text{or}\quad |\bar q|<\left|\operatorname{Im}\sqrt{(\zeta_0+n)(\zeta_0+m)}\right|. \]
Remark. From the inclusion (15) it follows, in accordance with the example of Sec. 4, that the points
\[ [\zeta:\ |p_0|\leq m,\ q=0,\ \bar p\ \text{arbitrary}] \]
do not belong to \(H(T\cup \widetilde G)\).
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
25 I 1961
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