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UDC 530.145
MATHEMATICAL PHYSICS
L. Sh. Khodjaev
GENERALIZED BOGOLIUBOV FUNCTIONS AND ELEMENTS OF THE CAUSAL \(S\)-MATRIX
(Presented by Academician N. N. Bogoliubov on 16 II 1968)
In all investigations carried out within the framework of the axiomatic \(S\)-matrix approach to quantum field theory by N. N. Bogoliubov, B. V. Medvedev, M. K. Polivanov \((^{1,2})\) and others \((^{3})\), the principal quantities determining the theory are taken to be the class of so-called chronological operators
\[ \begin{aligned} T_n^c(x_1,\ldots,x_n) &= T'\bigl(J(x_1)\ldots J(x_n)\bigr) + \\ &\quad + \sum_{\substack{2\le m\le n-1\\ \left(\sum_{j=1}^{m}\nu_j=n,\ \nu_j\ge 1\right)}} \frac{(-i)^{\,n-m}}{m!}\, P(x_1,\ldots,x_{\nu_1}\mid x_{\nu_1+1},\ldots,x_{\nu_1+\nu_2}\mid \ldots x_n)\times \\ &\quad \times T\bigl(\Lambda_{\nu_1}(x_1,\ldots,x_{\nu_1}) \Lambda_{\nu_2}(x_{\nu_1+1},\ldots,x_{\nu_1+\nu_2})\ldots \Lambda_{\nu_m}(\ldots x_n)\bigr) + \\ &\quad + i^{\,n-1}\Lambda_n(x_1,\ldots,x_n),\qquad n=1,2,\ldots; \end{aligned} \tag{1} \]
\[ \Lambda_1(x_1)=J(x_1)= i\,\frac{\delta S}{\delta\varphi(x)}\,S^{+}; \tag{2} \]
\[ \Lambda_\nu(x_1,\ldots,x_\nu),\qquad \nu=2,3,\ldots, \]
are arbitrary timelike operators possessing the properties of Poincaré invariance, locality, symmetry, and local commutativity. In addition to these linear properties they must satisfy the equation of motion (see \((^{1})\)).
However, from the mathematical point of view the most convenient quantities for constructing the theory are not the chronological operators \(T_n^c(x_1,\ldots,x_n)\), \(n=1,2,\ldots\), defined according to (1), but the infinite set of their vacuum averages, i.e.
\[ B_n(x_1,\ldots,x_n)=\langle 0|T_n^c(x_1,\ldots,x_n)|\rangle,\qquad n=1,2,\ldots, \tag{3} \]
which serve as coefficient functions in the functional expansion of the \(S\)-matrix in asymptotic fields, i.e.
\[ S=\sum_{n=0}^{\infty}\frac{(-i)^n}{n!}\int\ldots\int \left(\prod_{j=1}^{n} d^4x_j\right) B_n(x_1,\ldots,x_n):\varphi_{\mathrm{out}}(x_1)\ldots\varphi_{\mathrm{out}}(x_n): . \tag{4} \]
The content of the BMP axioms \((^{4,5})\) can be completely translated into the language of the vacuum averages \(B_n(x_1,\ldots,x_n)\), \(n=1,2,\ldots\), defined according to (3), characterized by linear and nonlinear relations.
A. Linear properties.
1°. Functional structures. The vacuum averages \(B_n(x_1,\ldots,x_n)\), \(n=1,2,\ldots\), are generalized functions: a) belonging to the space \(S'(R^{4n})\); b) extendable to the space \(S_+^{\mathrm{KG}}(R^{4n})\) of sufficiently smooth solutions of the Klein–Gordon (KG) equation with positive energies, i.e. to the space \(S_+^{\mathrm{KG}}(R^{4n})\) of functions \(f(x_1,\ldots,x_n)\) representable in the form \(f(x_1,\ldots,x_n)=f_1(x_1)\ldots\)
\(\ldots f_n(x_n)\), where
\[ f_j(x_j)=\frac{1}{(2\pi)^{3/2}}\int d^4p_j\,\theta(p_j^0)\,\delta(p_j^2-\mu^2)\,\widetilde f_j(p_j)\,e^{ip_jx_j}\in S_+^{\mathrm{KT}}(R^4), \tag{5} \]
\[ \widetilde f_j(p_j)\in S(\overline{\Omega}_\mu^+),\qquad j=1,2,\ldots,n. \]
By \(S(\overline{\Omega}_\mu^+)\) is denoted the space of test functions with
\(\operatorname{supp}\widetilde f_j(p_j)\in\overline{\Omega}_\mu^+\), where
\[ \overline{\Omega}_\mu^+=\{p_j\in \overline{R}:\ (p_j^0)^2-(\mathbf p_j)^2=\mu^2,\quad p_j^0>0\}, \]
\(j=1,2,\ldots,n\).
Condition b), in other words, means the existence of generalized functions
\(\widehat B_n(x_1,\ldots,x_n)\in S_+^{\prime \mathrm{KT}}(R^{4n})\), \(n=1,2,\ldots\), such that
\[ B_n(f)=\widehat B_n(f)\quad \text{for any } f\in S_+^{\mathrm{KT}}(R^{4n}). \tag{6} \]
Property b) requires the extendability (preextendability) of the generalized functions
\(B_n(x_1,\ldots,x_n)\in S'(R^{4n})\), \(n=1,2,\ldots\), to the mass surface, i.e. the existence of generalized \(M\)-functions defined by
\[ M_n(p_1,\ldots,p_n)=\prod_{j=1}^n\theta(p_j^0)\,\delta(p_j^2-\mu^2)\,\widetilde B_n(p_1,\ldots,p_n)\in S'(\overline{\Omega}_n^+), \tag{7} \]
where
\[ \widetilde B_n(p_1,\ldots,p_n)= \frac{1}{(2\pi)^{2n}}\int\cdots\int\left(\prod_{j=1}^n d^4x_j\right) \exp\left[i\sum_{j=1}^n p_jx_j\right]B_n(x_1,\ldots,x_n). \tag{8} \]
By \(S(\overline{\Omega}_n^+)\) is denoted the space of test functions
\(\widetilde f(p_1,\ldots,p_n)\) with
\(\operatorname{supp}\widetilde f(p_1,\ldots,p_n)\in\overline{\Omega}_n^+\), where
\[ \overline{\Omega}_n^+(p_1,\ldots,p_n)=\bigotimes_{j=1}^n \overline{\Omega}_\mu^+(p_j). \]
The existence of the generalized functions \(M_n(x_1,\ldots,x_n)\), \(n=1,2,\ldots\), defined according to (7), follows from the existence of the generalized functions
\(\widehat B_n(x_1,\ldots,x_n)\in S_+^{\prime \mathrm{KT}}(R^{4n})\) and the definition
\[ M_n(\widetilde f_1,\ldots,\widetilde f_n) =\int\cdots\int\left(\prod_{j=1}^n d^4p_j\,\widetilde f_j(p_j)\right) M_n(p_1,\ldots,p_n) = \]
\[ =(2\pi)^{-n/2}\widehat B_n(f_1,\ldots,f_n) \tag{9} \]
for arbitrary \(\widetilde f_j(p_j)\in S(\overline{\Omega}_\mu^+)\), where
\(f_j(x_j)\in S_+^{\mathrm{KT}}(R^4)\), \(j=1,2,\ldots,n\), are determined according to (5).
\(2^\circ.\) Symmetry property.
\[ B_n(F)=0, \tag{10} \]
where
\[ F(x_1,\ldots,x_n)=f(x_1,\ldots,x_n)-f(x_{\alpha_1},\ldots,x_{\alpha_n}) \tag{11} \]
for any \(f(x_1,\ldots,x_n)\in S(R^{4n})\).
\(3^\circ.\) Poincaré invariance
\[ B_n(f)=B_n(f_{(a,\Lambda)}), \tag{12} \]
where
\[ f_{(a,\Lambda)}(x_1,\ldots,x_n)=f\bigl(\Lambda^{-1}(x_1-a),\ldots,\Lambda^{-1}(x_n-a)\bigr) \tag{13} \]
for any \(f(x_1,\ldots,x_n)\in S(R^{4n})\) and \((a,\Lambda)\in P_+^\uparrow\), where \(a\) is a 4-translation vector, and \(\Lambda\) is an arbitrary element of the proper Lorentz group \(L_+^\uparrow\).
B. Nonlinear properties.
4°. Generalized unitarity relations
\[ i\sum_{s=0}^{n} P\left(\frac{f_1,\ldots,f_s}{f_{s+1},\ldots,f_n}\right) \sum_{l=0}^{\infty}\frac{1}{l!}\sum_{a_1\ldots a_l} B_{s+l}(f_1,\ldots,f_s,h_{a_1}^{*},\ldots,h_{a_l}^{*})\times \]
\[ \times B_{n-s+l}^{*}(f_{s+1},\ldots,f_n,h_{a_1},\ldots,h_{a_l}) =\delta_{n0},\quad n=1,2,\ldots, \tag{14} \]
\[ B_0=B_0^{*}=1 \]
for arbitrary \(f_j(x_j)\in S(R^4)\), \(j=1,\ldots,n\), \(\{h_\alpha(z)\}\) is a complete set of normalized functions of the space \(S_+^{\mathrm{kr}}(R^{4n})\), representable in the form
\[ h_{a_j}(z_j)=\frac{1}{(2\pi)^{3/2}}\int d^4p_j\,\theta(p_j^0)\delta(p_j^2-\mu^2)e^{ip_jz_j}\tilde h_{a_j}(p_j), \tag{15} \]
where \(\tilde h_{a_j}(p_j)\in S(\bar\Omega_\mu^+)\), and \((h_\alpha,h_\beta)<\infty\),
\[ (h_\alpha,h_\beta)=i\int d^3x\, f_\alpha(x)\overset{\leftrightarrow}{\partial}_0 f_\beta(x). \tag{16} \]
The Lorentz-invariant scalar product is
\[ \sum_a h_a(x)h_a(x')=-iD^{(-)}(x'-x). \tag{17} \]
By \(P\left(\frac{f_1,\ldots,f_s}{f_{s+1},\ldots,f_n}\right)\) is denoted the symmetrization operator, meaning \(n!/s!(n-s)!\) partitions of the set of \(n\) functions into two groups of \(s\) and \((n-s)\) functions in each.
5°. Generalized causality relations
\[ (-i)^n B_{n+1}(f_0,\ldots,f_n) +(i)^n\sum_{s=0}^{n-1}P\left(\frac{f_1,\ldots,f_s}{f_{s+1},\ldots,f_n}\right)\times \]
\[ \times\sum_{l=0}^{\infty}\frac{1}{l!}\sum_{a_1\ldots a_l} B_{s+l+1}(f_0,\ldots,f_s,h_{a_1}^{*},\ldots,h_{a_l}^{*})\times \]
\[ \times B_{n-s+l}^{*}(h_{a_1},\ldots,h_{a_l},f_{s+1},\ldots,f_n)=0, \tag{18} \]
\[ B_0=B_0^{*}=1 \]
for arbitrary functions \(f_j(x_j)\in S(R^4)\) with causally independent supports, i.e. satisfying the condition
\[ f_0(x_0)f_j(x_j)=0 \tag{19} \]
when
\[ (x_0^0-x_j^0)\leq 0\quad \text{and}\quad (x_0-x_j)^2\geq 0,\quad j=1,2,\ldots,n, \tag{20} \]
for at least one function \(f_j(x_j)\in S(R^4)\), \(j=1,2,\ldots,n\), and for arbitrary functions \(h_1(z_1),\ldots,h_l(z)\in S(\bar\Omega_\mu^+)\), defined according to (15)—(17).
Restoration theorem. From an infinite set of numerical generalized \(B\)-functions satisfying conditions 1°—5°, one can construct the \(S\)-matrix according to
\[ S=\sum_{n=0}^{\infty}\frac{(-i)^n}{n!}\int\ldots\int \left(\prod_{j=1}^{n}d^4x_j\right) B_n(x_1,\ldots,x_n):\varphi_{\mathrm{out}}(x_1)\ldots\varphi_{\mathrm{out}}(x_n):, \tag{21} \]
which will:
a) commute with the unitary representation of the proper Poincaré group \(P_+^{\uparrow}\), i.e.
\[ [U(a,\Lambda),S]=0, \tag{22} \]
where \(a\) is a vector of 4-translations, and \(\Lambda\) is an arbitrary proper transformation of the Lorentz group \(L_+^{\uparrow}\),
b) unitary, i.e.
\[
SS^{+}=S^{+}S=I;
\tag{23}
\]
c) satisfy the causality condition in the Bogoliubov form.
The generalized \(B\)-functions defined according to (3), and their linear and nonlinear properties \(1^\circ\)—\(5^\circ\), may henceforth be adopted as the basic mathematical conditions defining the axiomatic \(S\)-matrix BMP approach in quantum field theory, free of the original BMP formalism.
Now the elements of the \(S\)-matrix can be represented through generalized \(B\)-functions as follows:
\[
S_{mn}(\mathbf p_1,\ldots,\mathbf p_m,-\mathbf q_1,\ldots,-\mathbf q_n)=
\]
\[
=\frac{(-i)^{m+n}}{(2\pi)^{\frac32(m+n)}}\int\cdots\int
\frac{\left(\prod_{r=1}^{n} d^4x_r\right)\left(\prod_{s=1}^{m} d^4y_s\right)}
{\prod_{r=1}^{n}(2q_r^0)^{1/2}\prod_{s=1}^{m}(2p_s^0)^{1/2}}
\times
\]
\[
\times \exp\left[-i\left(\sum_{r=1}^{n}q_rx_r-\sum_{s=1}^{m}p_sy_s\right)\right]
B_{m+n}(x_1,\ldots,x_n,y_1,\ldots,y_m),
\]
\[
m,n=0,1,2,\ldots,
\tag{24}
\]
where
\[
q_r^0=\sqrt{\mathbf q_r^2+\mu^2},\quad r=1,2,\ldots,n,\qquad
p_s^0=\sqrt{\mathbf p_s^2+\mu^2},\quad s=1,2,\ldots,m,
\]
\[
\sum_{r=1}^{n}q_r=\sum_{s=1}^{m}p_s
\]
for all noncoincident momenta \(p_1,\ldots,p_m\) and \(q_1,\ldots,q_n\).
From the functional structure of the generalized \(B\)-functions it follows that
\[
S_{mn}(p_1,\ldots,p_m,-q_1,\ldots,-q_n)\in S'(\mathbb R^{3(m+n)}).
\]
Taking this opportunity, I express my sincere gratitude to N. N. Bogoliubov, A. N. Tavkhelidze, B. V. Medvedev, A. D. Sukhanov, A. V. Efremov, and G. V. Efimov for their interest and valuable comments.
Joint Institute
for Nuclear Research
Received
27 II 1967
REFERENCES
- B. V. Medvedev, ZhETF, 40, 926 (1961).
- B. V. Medvedev, M. K. Polivanov, International Winter School of Theoretical Physics at the Joint Institute for Nuclear Research, 1, 1964.
- A. D. Sukhanov, ZhETF, 51, 1195 (1966).
- N. N. Bogoliubov, B. V. Medvedev, M. K. Polivanov, Problems of the Theory of Dispersion Relations, Moscow, 1958.
- L. Sh. Khodzhaev, Preprint R5-3179, Joint Institute for Nuclear Research, Dubna, 1967.
- L. Sh. Khodzhaev, DAN, 182, No. 5 (1968).