UDC 150.145

MATHEMATICAL PHYSICS

L. Sh. KHODJAEV

ON THE PROPERTIES OF THE DECOMPOSITION INTO BUNDLES OF THE CAUSAL \(S\)-MATRIX

(Presented by Academician N. N. Bogolyubov on 24 V 1966)

In the present paper we shall prove space-like properties of the decomposition into bundles of vacuum mean values of radiation operators, as well as of matrix elements of the \(S\)-matrix within the framework of N. N. Bogolyubov’s axiomatic causal \(S\)-matrix theory \((^{1})\). The properties of the decomposition into bundles of the \(S\)-matrix have recently been formulated \((^{2})\) and proved \((^{3})\) within the framework of Wightman’s axiomatic quantum field theory \((^{4})\). For simplicity we shall consider the self-action of one real scalar field \(\varphi(x)\).

Consider

\[ S=\sum_{N=0}^{\infty}\frac{(-i)^N}{N}\int (d^4x)_N\,h_N(x)_N:\varphi(x_1)\ldots\varphi(x_N):, \tag{1} \]

where the coefficient functions \(h_N(x)_N\) are \(C\)-numbers with the property \(h_N(x_{\alpha N})=h_N(x_N)\). In formula (1) the following notation has been introduced:

\[ (z)_N=(z_1,\ldots,z_N),\qquad (z_\alpha)_N=(z_{\alpha_1},\ldots,z_{\alpha_N}), \]

\[ \int (d^4x)_N=\int\cdots\int d^4x_1\ldots dx_N^4. \]

Following N. N. Bogolyubov \((^{1})\), we define the causality condition according to

\[ \iint d^4x\,d^4y f(x)g(y)\frac{\delta}{\delta\varphi(x)} \left( \frac{\delta S}{\delta\varphi(y)}S^{+} \right)=0 \tag{2} \]

for arbitrary \(f(x),\,g(y)\in S(R^4)\), satisfying the condition \(f(x)g(y)=0\) for time-like intervals \((x-y)^2\ge 0\) and \((x^0-y^0)\ge 0\).

Let us introduce into consideration the radiation-operator-valued generalized function

\[ H_N(f)=\int (d^4x)_N f(x)_N \frac{\delta^N S}{\delta\varphi(x_1)\ldots\delta\varphi(x_N)}S^{+}= \]

\[ =(-i)^N\int(d^4x)_N\theta(x_1^0-x_2^0)\ldots\theta(x_{N-1}^0-x_N^0) J(x_1)\ldots J(x_N)\sum_{p(1,\ldots,N)} f(x_{\alpha})_N, \tag{3} \]

defined in \(D_S\subset H\) for any \(f(x)_N\in S_0(R^{4N})\subset S(R^{4N})\), vanishing together with its derivatives of sufficiently high order when \(x_1=x_2=\cdots=x_N\). The summation in (2) is taken over all possible permutations \(\alpha_1,\ldots,\alpha_N\) of the numbers \(1,2,\ldots\). By \(J(x)=i\,\delta S/\delta\varphi(x)\) we denote the operator of the bosonic current. We shall assume that

\[ h_N(f)=i^N\langle 0|H_N(f)|0\rangle\in S_0'(R^{4N}) \tag{4} \]

for every \(f(x_1,\ldots,x_N)\in S_0(R^{4N})\).

Theorem 1. Let \(a\) be an arbitrary space-like vector, \(\lambda < 0\). If condition (2) is satisfied, the relation

\[ \lim_{\lambda\to\infty} h_{m+n+r+s}\bigl(F_{\lambda a}^{(m+n+r+s)}\bigr) = h_{m+n}\bigl(f^{(m+n)}\bigr)\, h_{r+s}\bigl(g^{(r+s)}\bigr) \tag{5} \]

holds for any

\[ F_{\lambda a}^{(m+n+r+s)} = F^{(m+n+r+s)}(x,\,y-\lambda a)_{m+n,r+s} = \]

\[ = f^{(m+n)}(x)_{m+n} g^{(r+s)}(y-\lambda a)_{r+s} \in S_0\bigl(R^{4(m+n+r+s)}\bigr), \tag{6} \]

where \(f^{(m+n)}(x)_{m+n},\, g^{(r+s)}(y)_{r+s} \in S_0\bigl(R^{4(m+n)}, R^{4(r+s)}\bigr)\).

One can indicate a relation analogous to (5) for matrix elements of the \(S\)-matrix.

Between the matrix elements
\(S_{m+r,n+s}(\mathbf q',\,\mathbf p';\,\mathbf q,\,\mathbf p)_{m,r;n,s}\in S'\bigl(R^{3(m+n+r+s)}\bigr)\)
and the vacuum averages of radiation operators
\(h_{m+n+r+s}(x',\,y';\,x,\,y)_{m,r;n,s}\in S'\bigl(R^{4(m+n+r+s)}\bigr)\)
there holds the relation

\[ S_{m+r,n+s}\left( \exp\left[-i\lambda\left(\sum_{j=1}^{r} a\cdot p_j-\sum_{k=1}^{s} a\cdot p_{k+r}\right)\right] \tilde f^{(m+n)}\tilde g^{(r+s)} \right) = h_{m+n+r+s}\bigl(f^{(m+n)}g_{\lambda a}^{(r+s)}\bigr), \tag{7} \]

where

\[ f^{(m+n)}(x)_{m+n} = \frac{1}{2\pi^{3(m+n)/2}} \int \frac{ \exp\left[i\left(\sum_{l=1}^{m} q_l x_l-\sum_{t=1}^{n} q_{m+t}x_{m+t}\right)\right] }{ \sqrt{2q_1^0\ldots 2q_{m+n}^0} } \tilde f^{(m+n)}(\mathbf q)_{m+n}\,(dq)_{m+n}, \]

\[ q_i^0=\sqrt{\mathbf q_i^2+m^2},\qquad i=1,\ldots,m+n, \tag{8} \]

where \(\tilde f^{(m+n)}(\mathbf q)_{m+n}\in S(\tilde R^{3(m+n)})\) and \(f^{(m+n)}\bigl(R^{4(m+n)}\bigr)\),

The function \(g^{(r+s)}(y)_{r+s}\in S\bigl(R^{3(r+s)}\bigr)\) has the same structure as \(f^{(m+n)}(x)_{m+n}\), and \(g_{\lambda a}^{(r+s)}=g^{(r+s)}(y-\lambda a)_{r+s}\), where \(a\) is an arbitrary space-like vector, \(\lambda>0\).

Theorem 2. Suppose the condition of Theorem 1 is fulfilled. Then

\[ \lim_{\lambda\to\infty} S_{m+r,n+s}\left( \exp\left[-i\lambda\left(\sum_{j=1}^{r} a\cdot p_j-\sum_{k=1}^{s} a\cdot p_{r+k}\right)\right] \tilde f^{(m+n)}\tilde g^{(r+s)} \right) = \]

\[ = S_{m+n}\bigl(\tilde f^{(m+n)}\bigr)\, S_{r+s}\bigl(\tilde g^{(r+s)}\bigr) \tag{9} \]

for any \(\tilde f^{(m+n)}(\mathbf q_{m+n})\in S_0(\tilde R^{3(m+n)})\) and \(\tilde g^{(r+s)}(\mathbf p_{r+s})\in S_0(\tilde R^{3(r+s)})\).

It is obvious that the proof of Theorem 2 follows directly from Theorem 1. We divide the proof of the theorem into the proofs of several lemmas.

Lemma 1. Suppose the causality condition (1) is fulfilled. Then

\[ H_{m+n+r+s}\bigl(F^{(m+n+r+s)}\bigr) = H_{m+n}\bigl(f^{(m+n)}\bigr)\, H_{r+s}\bigl(g^{(r+s)}\bigr) \tag{10} \]

for any function

\[ F^{(m+n+r+s)}(x,y)_{m+n,r+s}\in S_0\bigl(R^{4(m+n+r+s)}\bigr), \]

representable in the form

\[ F^{(m+n+r+s)}(x,y)_{n+m,r+s} = f^{(m+n)}(x)_{m+n}g^{(r+s)}(y)_{r+s}, \tag{11} \]

provided \(\{x\}_{n+m}\geq \{y\}_{r+s}\), where \(f^{(m+n)}(x)_{m+n}\in S_0\bigl(R^{4(m+n)}\bigr)\) and \(g^{(r+s)}(y)_{r+s}\in S_0\bigl(R^{4(r+s)}\bigr)\).

The proof of the lemma is based on the results of work (5).

Now, using this lemma and the completeness condition for the system of eigen-amplitudes of the 4-energy–momentum operator \(\hat P^\mu\), we obtain

\[ h_{m+n+r+s}\bigl(F_a^{(m+n+r+s)}\bigr) = h_{m+n}\bigl(f^{(m+n)}\bigr)\, h_{r+s}\bigl(g^{(r+s)}\bigr) + h_{m+n+r+s}^{T}\bigl(F_{\lambda a}^{(m+n+r+s)}\bigr), \tag{12} \]

where \(h_{m+n+r+s}^{T}\bigl(F_{\lambda a}^{(m+n+r+s)}\bigr)\), the truncated vacuum mean value of the radiation operators, is defined according to

\[ \begin{aligned} h_{m+n+r+s}^{T}\bigl(F_{\lambda a}^{(m+n+r+s)}\bigr) &= \iint (d^4x)_{m+n-1}(d^4y)_{r+s} \theta(x_1^0-x_2^0)\cdots \\ &\quad \cdots \theta(x_{m+n-1}^0-x_{m+n}^0) \theta(y_1^0-y_2^0)\cdots \theta(y_{r+s-1}^0-y_{r+s}) \\ &\quad \times \left[ \sum_{\nu=1}^{\infty}\frac{1}{\nu!} \int (dk)_\nu \langle 0|J(x_1)\cdots J(x_{m+n})|(k)_\nu\rangle \langle (k)_\nu|J(y_1)\cdots J(y_{r+s})|0\rangle \right] \\ &\quad \times \sum_{P(1,\ldots,m+n)} f^{(m+n)}(x_\alpha)_{m+n} \sum_{P(1,\ldots,r+s)} g^{(r+s)}(y_\beta-\lambda a)_{r+s} \end{aligned} \tag{13} \]

for arbitrary \(f^{(m+n)}(x)_{m+n}\in S_0(R^{4(m+n)})\) and \(g^{(r+s)}(y)_{r+s}\in S_0(R^{4(r+s)})\).

Lemma 2. The Fourier transform of \(h_{m+n+r+s}^{T}\bigl(F_{\lambda a}^{(m+n+r+s)}\bigr)\) belongs to the space \(S_0(\widetilde R^{3(m+n+r+s)})\).

Indeed, using the completeness properties of the amplitude of the 4-energy–momentum operator \(\hat P^\mu\), from (11) we obtain

\[ \begin{aligned} h_{m+n+r+s}^{T}\bigl(F_{\lambda a}^{(m+n+r+s)}\bigr) &= i^{m+n+r+s} \int\cdots\int I(\ldots,\mu_{\alpha3}^2,\ldots;\ldots,m_{ij}^2,\ldots;M^2) \\ &\quad \times \widetilde D_{0,\mu_{\alpha3}^2,m_{ij}^2,M^2}^{(-)} \bigl(e^{i\lambda ak}\widetilde f_2^{(m+n)}\widetilde g_2^{(r+s)}\bigr) \\ &\quad \times \prod_{\alpha\leq\beta\leq 1}^{m+n+1} d\mu_{\alpha\beta}^2 \prod_{i<j\leq 1}^{r+s+1} dm_{ij}^2\, dM^2, \end{aligned} \tag{14} \]

where

\[ \widetilde D_{0,\mu_{\alpha3}^2,m_{ij}^2,M^2}^{(-)} \in S'\bigl(R(\mu_{\alpha3}^2,m_{ij}^2,M^2)\bigr) \]

and is a Lorentz-invariant generalized function. Since the carrier of the generalized function
\(I(\ldots,\mu_{\alpha\beta}^2,\ldots;\ldots,m_{ij}^2,\ldots;M^2)\in S'(R)\) lies in the region

\[ \{\ldots,\mu_{\alpha\beta}^2,\ldots;\ldots,m_{ij}^2,\ldots;M^2\}>0, \tag{15} \]

the integral (14) exists, and therefore \(h_{m+n+r+s}^{T}\bigl(F_{\lambda a}^{(m+n+r+s)}\bigr)\) depends linearly and continuously on the functions
\(\widetilde f_2(q,k)_{m+n-1,0}\in S_0(\widetilde R^{3(m+n)})\) and
\(\widetilde g_2^{(r+s)}(p,k)_{r+s-1,0}\in S_0(\widetilde R^{3(r+s)})\).

Lemma 3. Let \(a\) be an arbitrary spacelike vector, \(\lambda>0\). Then

\[ \lim_{\lambda\to\infty} \lambda^N h_{m+n+r+s}^{T}\bigl(F_{\lambda a}^{(m+n+r+s)}\bigr) =0 \tag{16} \]

for any \(F^{(m+n+r+s)}\in S_0(R^{4(m+n+r+s)})\) of the form (5).

We note that the proof of this lemma reduces, on the basis of formula (13), to proving the relation

\[ \lim_{\lambda\to\infty} \lambda^N \widetilde D_{0,\mu_{\alpha\beta}^2,m_{ij}^2,M^2}^{(-)} \bigl(e^{i\lambda ak'}\widetilde f_2^{(m+n)}\widetilde g_2^{(r+s)}\bigr) =0 \tag{17} \]

for
\(\widetilde f_2^{(m+n)}(q,k)_{m+n-1,0}\in S_0(\widetilde R^{3(m+n)})\) and
\(\widetilde g_2^{(r+s)}(p,k)_{r+s-1}\in S_0(\widetilde R^{3(r+s)})\).

In (16) the reference frame \(d\equiv(0,a,0,0)\) is chosen, and \(k'\) is the component of the vector
\(k=(k^0,k^1,k^2,k^3)\). Araki \([6]\) proved that relation (17) holds in the case when
\(\widetilde D_{0,\mu_{\alpha\beta}^2,m_{ij}^2,M^2}^{(-)}\) is a Lorentz-invariant generalized function. Since in the present case
\(\widetilde D_{0,\mu_{\alpha3}^2,m_{ij}^2,M^2}^{(-)}\) is a Lorentz-invariant generalized function, Lemma 3 is thereby proved.

On the basis of the lemmas, Theorem 1 is completely proved and, consequently, Theorem 2 as well.

Now define the extension of the radiation operator \(H_N(f)\), given by formula (3) in \(S_0(R^{4N})\), to the whole space \(S(R^{4N})\) according to

\[ H_N^C(f)=\int (d^4x)_N\,\theta(x_1^0-x_2^0)\cdots \theta(x_{N-1}^0-x_N^0)J(x_1)\cdots J(x_N)\times \]

\[ \times \sum_{P(1,\ldots,N)} f(x_\alpha)_N+\int (d^4x)_N f(x)_N C_N^\Lambda(x)_N \tag{18} \]

for any function \(f(x)_N\in S(R^{4N})\), where \(C_N^\Lambda(x)_N\) is expressed as a sum of symmetrized \(T\)-products of chains of quasilocal operators \(\Lambda_n(x_1,\ldots,x_n)\) (5), possessing the properties of locality, Hermiticity, symmetry, and local commutativity.

Since

\[ h_N^C(f)=i^N\langle 0|H_N^C(f)|0\rangle \in S'(R^{4N}) \tag{19} \]

for any \(f\in S(R^{4N})\), the properties of the cluster decomposition of the vacuum mean values of the radiation operators with coinciding arguments will be defined according to

\[ \lim_{\lambda\to\infty}h_{m+n+r+s}^C(F_{\lambda a})=h_{m+n}^C(f)h_{r+s}^C(g) \tag{20} \]

for any function \(F(x,y)_{m+n,r+s}=f(x)_{m+n}g(y)_{r+s}\in S(R^{4(m+n+r+s)})\), where \(f(x)_{m+n}\in S(R^{4(m+n)})\) and \(g(y)_{r+s}\in S(R^{4(r+s)})\), with causally independent carriers \(\{x\}_{m+n}>\{y\}_{r+s}\).

Extending relation (7) to the whole space \(S(R^{4(m+n+r+s)})\) and using formula (20), we obtain the cluster-decomposition properties of the elements of the \(S\)-matrix, expressed in the form of formula (9).

Taking this opportunity, I express my deep gratitude to Academician N. N. Bogolyubov, I. Todorov, B. V. Medvedev, and A. V. Efremov for discussions and valuable remarks.

United Institute
for Nuclear Research

Received
25 IV 1966

CITED LITERATURE

\(^{1}\) N. N. Bogolyubov, B. V. Medvedev, M. K. Polivanov, Problems in the Theory of Dispersion Relations, 1958.
\(^{2}\) E. H. Wichmann, J. H. Crichton, Phys. Rev., 132, No. 5 (1963).
\(^{3}\) K. Hepp, Helv. Phys. Acta, 37, Fasc. 7/8 (1964).
\(^{4}\) A. S. Wightman, Phys. Rev., 101, 860 (1956).
\(^{5}\) B. V. Medvedev, ZhETF, 40, No. 3 (1961).
\(^{6}\) H. Araki, Ann. Phys., 11, 260 (1960).