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Lemmas 1 and 2 (Theoretical and Mathematical Physics, Vol. 211, No. 3, pp. 792–793 and 795–796) must be stated as follows.
Let \(d\in\mathcal S(\mathbb{R}^4;\mathbb{C})^*\) , and let
with the kernel
corresponding to the Wick product (at the same space–time point \(x\) )
of the integral kernel operators \(\Xi_{l_k,m_k}(\kappa_{l_k,m_k}^{n_k}(x))\) .
Let the integral kernel \(d\ast\kappa_{l,m}\) be equal to
where \(\xi_{i_k}\in E_{i_k}\) , \(\phi\in\mathscr{E}\) and \(\mathscr{E}=\mathcal S(\mathbb{R}^4;\mathbb{}C)\) or \(\mathscr{E}=\mathcal S^{00}(\mathbb{R}^4;\mathbb{C})\) .
Then
-
1.
If the convolution \(d_n\ast d_{n-1}\ast\cdots\ast d_1\ast\kappa_{l,m}\) exists, then it is continuous, i.e.,
$$d_n\ast d_{n-1}\ast\cdots\ast d_1\ast\kappa_{l,m}\in\mathscr{L}(E_{i_1}\otimes\cdots\otimes E_{i_{l+m}},\mathscr{E}^*),$$provided
$$\kappa_{l,m}^{}=(\kappa_{l_1,m_1}^{n_1})\dot{\otimes}\cdots\dot{\otimes}(\kappa_{l_M,m_M}^{n_M}),\qquad l+m=M,$$and each \(d_i\) is equal to the product of pairings or to the retarded or advanced part of the causal combinations of products of pairings and \(M>1\) , which we encounter as higher-order contributions to interacting fields in spinor QED.
-
2.
Let, moreover, in the case \(M=1\) , \(\kappa_{l_1,m_1}^{n_1}=\kappa_{0,1},\kappa_{1,0}\) be equal to the kernel of a free field with a mass \(m_{i_1}\) . If further among the distributions \(d_n, d_{n-1},\ldots, d_1\) there are no (retarded or advanced parts of the) commutation functions of a free field of the mass \(m_2=m_{i_1}\) , then the convolutions
$$d_n\ast\cdots\ast d_1\ast\kappa_{0,1}(\xi),\quad d_n\ast\cdots\ast d_1\ast\kappa_{1,0}(\xi),\qquad \xi\in E,$$are well-defined and
$$d_n\ast\cdots\ast d_1\ast\kappa_{0,1},\;d_n\ast\cdots\ast d_1\ast\kappa_{1,0}\in\mathscr{L}(E_{i_1},\mathscr{E}^*).$$ -
3.
If \(\kappa_{l_1,m_1}^{n_1}=\kappa_{0,1},\kappa_{1,0}\) is the kernel of a free field with the mass not equal to the mass of the free field whose commutation function (or its retarded or advanced part) is equal to \(d\) , then the convolutions
$$d\ast\kappa_{0,1},\;d\ast\kappa_{1,0}\in\mathscr{L}(E_{i_1}^*,\mathscr{E}^*)=\mathscr{L}(\mathscr{E}, E_{i_1})\subset\mathscr{L}(E_{i_1},\mathscr{E}^*)$$are well-defined.
-
4.
If \(\kappa_{l_1,m_1}^{n_1}=\kappa_{0,1},\kappa_{1,0}\) is the kernel of a free field with the mass equal to the mass of the free field whose commutation function (or its retarded or advanced part) is equal to \(d\) , then the convolutions \(d\ast\kappa_{0,1}\) and \(d\ast\kappa_{1,0}\) are not well-defined.
The following statements hold.
-
1.
Let \(d_i\) be equal to the product of pairings or to the retarded or advanced part of the causal combinations of products of pairings and \(M>1\) , which we encounter as the kernels of higher-order contributions to interacting fields in spinor QED and with the “natural ” splitting of the causal distributions in the computation of the scattering operator. Assume that the convolution \(d_n\ast d_{n-1}\ast\cdots\ast d_1\ast\kappa_{l,m}\) exists. Then the operator
$$\begin{aligned} \, d_n&{}\ast\cdots\ast d_1\ast\Xi_{l,m}(\kappa_{l,m})(x)= \\ &=\int_{[\mathbb{R}^4]^{\times n}} d_n(x-y_n) d_{n-1}(y_n-y_{n-1})\ldots d_1(y_2-y_1)\Xi_{l,m}(\kappa_{l,m}(y_1))\,d^4y_1\ldots d^4y_n= \\ &=\Xi_{l,m}\biggl(\,\int_{[\mathbb{R}^4]^{\times n}} d_n(x-y_n)d_n(y_{n-1}-y_{n-2})\ldots d_1(y_2-y_1)\kappa_{l,m}(y_1)\, d^4y_1\ldots d^4y_n\biggr)= \\ &=\Xi_{l,m}(d_n\ast\cdots\ast d_1\ast\kappa_{lm}(x)) \end{aligned}$$defines an integral kernel operator
$$\Xi_{l,m}(d_n\ast\cdots\ast d_1\ast\kappa_{lm})\in\mathscr{L}((\boldsymbol{E})\otimes\mathscr{E}, (\boldsymbol{E})^*) \cong\mathscr{L}(\mathscr{E},\mathscr{L}((\boldsymbol{E}),(\boldsymbol{E})^*))$$with the vector-valued kernel
$$d_n\ast\cdots\ast d_1\ast\kappa_{lm}\in\mathscr{L}(\mathscr{E},(E_{i_1}\otimes\cdots\otimes E_{i_{l+m}})^*) \cong\mathscr{L}(E_{i_1}\otimes\cdots\otimes E_{i_{l+m}},\mathscr{E}^*).$$ -
2.
Let, moreover, for the higher-order contributions, in the case \(M=1\) , \(\kappa_{l_1,m_1}^{n_1}=\kappa_{0,1},\kappa_{1,0}\) be equal to the kernel of a free field with a mass \(m_{i_1}\) . If further among the distributions \(d_n, d_{n-1},\ldots, d_1\) there are no (retarded or advanced parts of the) commutation functions of a free field of the mass \(m_2=m_{i_1}\) , then
$$\begin{aligned} \, d_n\ast&{}\ldots\ast d_1\ast\Xi_{\,0,1}(\kappa_{0,1})(x)= \\ &=\int_{[\mathbb{R}^4]^{\times n}} d_n(x-y_n) d_{n-1}(y_n-y_{n-1})\ldots d_1(y_2-y_1)\Xi_{\,0,1}(\kappa_{0,1}(y))\,d^4y_1\ldots d^4y_n= \\ &=\Xi_{\,0,1}\biggl(\,\int_{[\mathbb{R}^4]^{\times n}} d_n(x-y_n) d_{n-1}(y_n-y_{n-1})\ldots d_1(y_2-y_1)\kappa_{0,1}(y)\,d^4y_1\ldots d^4y_n\biggr)= \\ &=\Xi_{\,0,1}(d_n\ast\cdots\ast d_1\ast\kappa_{0,1}(x)) \end{aligned}$$defines an integral kernel operator
$$\Xi_{\,0,1}(d_n\ast\cdots\ast d_1\ast\kappa_{lm})\in\mathscr{L}((\boldsymbol{E})\otimes\mathscr{E}, (\boldsymbol{E})^*) \cong\mathscr{L}(\mathscr{E},\mathscr{L}((\boldsymbol{E}),(\boldsymbol{E})^*))$$with the vector-valued kernel
$$d_n\ast\cdots\ast d_1\ast\kappa_{0,1}\in\mathscr{L}(E_{i_1},\mathscr{E}^*);$$and similarly for the kernel \(\kappa_{1,0}\) .
-
3.
If, moreover, in the case \(M=1\) , \(\kappa_{l_1,m_1}^{n_1}=\kappa_{0,1},\kappa_{1,0}\) is the kernel of a free field with the mass not equal to the mass of the free field whose commutation function (or its retarded or advanced part) is equal to \(d\) , then the integral kernel operators
$$d\ast\Xi_{\,0,1}(\kappa_{0,1})=\Xi_{\,0,1}(d\ast\kappa_{0,1}),\qquad d\ast\Xi_{1,0}(\kappa_{1,0})=\Xi_{\,0,1}(d\ast\kappa_{1,0})$$are well-defined and belong to
$$\mathscr{L}(\mathscr{E},E_{i_1})=\mathscr{L}(E_{i_1}^*,\mathscr{E}^*)\subset\mathscr{L}(E_{i_1},\mathscr{E}^*).$$ -
4.
If \(\kappa_{l_1,m_1}^{n_1}=\kappa_{0,1},\kappa_{1,0}\) is the kernel of a free field with the mass equal to the mass of the free field whose commutation function (or its retarded or advanced part) is equal to \(d\) , then the integral kernel operators
$$d\ast\Xi_{\,0,1}(\kappa_{0,1})=\Xi_{\,0,1}(d\ast\kappa_{0,1}),\qquad d\ast\Xi_{1,0}(\kappa_{1,0})=\Xi_{\,0,1}(d\ast\kappa_{1,0})$$are not well-defined.
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 212, pp. 489–491 https://doi.org/10.4213/tmf10345.
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Wawrzycki, J. Erratum to: Bogoliubov’s causal perturbative QED and white noise. Interacting fields. Theor Math Phys 212, 1312–1314 (2022). https://doi.org/10.1134/S0040577922090124
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DOI: https://doi.org/10.1134/S0040577922090124