Full Text
UDC 517.433
MATHEMATICS
V. P. GACHOK, A. V. ZOLOTARYUK, Ya. M. YAKIMIV
ON THE REGULARIZATION OF SINGULAR OPERATOR EXPRESSIONS IN QUANTUM FIELD THEORY
(Presented by Academician N. N. Bogolyubov, 28 X 1969)
Up to now, the only illustration of the general postulates of quantum field theory has been the divergent series of perturbation theory \((^1)\). One of the approaches to constructing a nontrivial model using Fock space consists in violating the conditions of Haag’s theorem \((^{2,3})\). In this connection, regularization methods that violate Euclidean invariance acquire important significance. In this direction we propose a method for regularizing singular operator expressions as applied to \(\varphi^4\)-theory, as a development of an approach proposed earlier for generalizing annihilation and creation operators (see \((^4)\), § 1).
I. Let us introduce the basic notation and definitions. Let \(K(\mathbf p,\mathbf q)\) be a kernel satisfying the following requirements:
1) \(K(\mathbf p,\mathbf q)\) is a continuous function of the variables \(\mathbf p,\mathbf q \in R^3\);
2) \(K(\mathbf p,\mathbf q)=K(\mathbf q,\mathbf p)\) for all \(\mathbf p,\mathbf q\);
3)
\[
\iint f(\mathbf p)K(\mathbf p,\mathbf q)f(\mathbf q)\,d\mathbf p\,d\mathbf q \ge 0
\]
for all \(f(\mathbf p)\in L_2(R^3)\), and equality to zero implies \(f(\mathbf p)\equiv 0\);
4)
\[
\iint K(\mathbf p,\mathbf q)[\omega(\mathbf p)\omega(\mathbf q)]^{-1/2}\,d\mathbf p\,d\mathbf q<\infty,\quad
\omega(\mathbf p)=\sqrt{\mathbf p^2+m^2},\quad m>0.
\]
These properties are satisfied, for example, by the kernel
\[ K_s(\mathbf p,\mathbf q)=\frac{1}{(\pi s)^{3/2}} \exp\left[-\frac{(\mathbf p-\mathbf q)^2}{s}-s(\mathbf p^2+\mathbf q^2)\right], \quad 0<s<\infty, \]
which will be used essentially below.
We are now ready to introduce an analogue of the Fock space \(F_0\). Denote by \(F_{-}\) the Hilbert space of sequences
\(G=\{G_0,\ G_1(\mathbf p_1),\ldots,G_n(\mathbf p_1,\ldots,\mathbf p_n),\ldots\}\), whose scalar product has the form
\[ (F,G)_{-} = \sum_{n=0}^{\infty} \iint d\mathbf p_1\,d\mathbf p'_1\,K(\mathbf p_1,\mathbf p'_1)\cdots \iint d\mathbf p_n\,d\mathbf p'_n\,K(\mathbf p_n,\mathbf p'_n)\times \]
\[ \times F_n(\mathbf p_1,\ldots,\mathbf p_n)\, \overline{G(\mathbf p'_1,\ldots,\mathbf p'_n)}. \]
All functions \(G_n(\mathbf p_1,\ldots,\mathbf p_n)\), for arbitrary \(n=1,2,\ldots\), are assumed to be symmetric with respect to their arguments. Corresponding to the scalar product introduced,
\[ F_{-}=\oplus\sum_{n=0}^{\infty} F^{(n)}(R^{3n}). \]
Let us note the following two important properties of the spaces \(F_0\) and \(F_{-}\). First, the embedding \(F_0\to F_{-}\) is quasinuclear. Second, among the functions \(G_n(\mathbf p_1,\ldots,\mathbf p_n)\) there are also \(\delta\)-functions, i.e., there exists a function \(\delta_{\mathbf p_1,\ldots,\mathbf p_n}\) for which
\[ \left\|\delta_{\mathbf p_1,\ldots,\mathbf p_n}\right\|_{F_{-}^{(n)}}= K(\mathbf p_1,\mathbf p_1)\cdots K(\mathbf p_n,\mathbf p_n)>0. \]
II. In the space \(F_{-}\) we introduce analogues of the Fock annihilation and creation operators \(a^{-}(\mathbf{k})\), \(a^{+}(\mathbf{k})\) as follows:
\[ (A^{-}(F_{1})G)_{n-1}(\mathbf{k}_{1},\ldots,\mathbf{k}_{n-1}) = \]
\[ = \sqrt{n}\iint d\mathbf{k}_{n}\,d\mathbf{k}'_{n}K(\mathbf{k}_{n},\mathbf{k}'_{n})F_{1}(\mathbf{k}'_{n})G_{n}(\mathbf{k}_{1},\ldots,\mathbf{k}_{n}), \]
\[ (A^{+}(F_{1})G)_{n+1} = \frac{1}{\sqrt{n+1}} \sum_{j=1}^{n+1} G_{n}(\mathbf{k}_{1},\ldots,\widehat{\mathbf{k}}_{j},\ldots,\mathbf{k}_{n+1})F_{1}(\mathbf{k}_{j}), \]
where \(F_{1}(\mathbf{k}_{1})\) is any real function from \(F_{-}^{(1)}(R^{3})\). It is convenient to write these operators also in the following form
\[ (A^{-}(\mathbf{k})G)_{n-1}(\mathbf{k}_{1},\ldots,\mathbf{k}_{n-1}) = \sqrt{n}\int d\mathbf{k}_{n}K(\mathbf{k},\mathbf{k}_{n})G_{n}(\mathbf{k}_{1},\ldots,\mathbf{k}_{n}), \]
\[ (A^{+}(\mathbf{k})G)_{n+1}(\mathbf{k}_{1},\ldots,\mathbf{k}_{n+1}) = \frac{1}{\sqrt{n+1}} \sum_{j=1}^{n+1} G_{n}(\mathbf{k}_{1},\ldots,\widehat{\mathbf{k}}_{j},\ldots,\mathbf{k}_{n+1}) \delta(\mathbf{k}-\mathbf{k}_{j}), \]
where \(A^{+}(\mathbf{k})\) has the meaning of a generalized operator-valued function.
The following properties hold:
1) the operators \(A^{-}(F_{1})\) and \(A^{+}(F_{1})\) have a common dense domain of definition;
2) the operators \(A^{-}(F_{1})\) and \(A^{+}(F_{1})\) are mutually adjoint in the scalar product \((\cdot,\cdot)_{-}\), and \([A^{-}(F_{1})]^{*}=A^{+}(F_{1})\);
3) the commutation relations hold
\[ [A^{-}(F_{1}),A^{-}(G_{1})]=[A^{+}(F_{1}),A^{+}(G_{1})]=0, \]
\[ [A^{-}(F_{1}),A^{+}(G_{1})]=(F_{1},G_{1})_{F_{-}^{(1)}(R^{3})}. \]
III. Introduce the operators
\[ \varphi(x)=\frac{1}{(2\pi)^{3/2}}\int e^{-i\mathbf{k}x}\{A^{+}(\mathbf{k})+A^{-}(-\mathbf{k})\}\frac{d\mathbf{k}}{\sqrt{2\omega(\mathbf{k})}} \]
in the space \(F_{-}\), generated by the kernel \(K_{s}(p,q)\).
The operator \(\varphi(F_{1})\), where \(F_{1}\) is an arbitrary real function \(F_{1}(\mathbf{k})\in F_{-}(R^{3})\), is self-adjoint in \(F_{-}\).
Construct the operators
\[ H_{0}=\int \omega(\mathbf{k})A^{+}(\mathbf{k})A^{-}(\mathbf{k})\,d\mathbf{k}, \qquad H_{\mathrm{int}}=\int :\varphi^{4}(x):\,dx = \]
\[ = \sum_{j=0}^{4}\binom{4}{j} \idotsint \frac{\delta(\mathbf{k}_{1}+\cdots+\mathbf{k}_{4})}{\omega(\mathbf{k}_{1})\cdots\omega(\mathbf{k}_{4})} A^{+}(\mathbf{k}_{1})\cdots A^{+}(\mathbf{k}_{j}) A^{-}(-\mathbf{k}_{j+1})\cdots A^{-}(-\mathbf{k}_{4}) \,d\mathbf{k}_{1}\cdots d\mathbf{k}_{4}, \]
\[ H=H_{0}+\lambda H_{\mathrm{int}}, \qquad \lambda>0. \]
By obvious analogy, we shall call the expression \(H\) the regularized full Hamiltonian of \(\varphi^{4}\)-theory.
Theorem. The regularized full Hamiltonian \(H\) is a symmetric operator in the Hilbert space \(F_{-}\), generated by the kernel \(K_{s}(p,q)\).
The self-adjoint properties of the Hamiltonians \(H_{0}\), \(H_{\mathrm{int}}\), and \(H\) will be studied in a subsequent work. Here we restrict ourselves to one remark, which will make it possible to understand the proposed regularization of the annihilation and creation operators from the point of view of regularizing singular functions in quantum field theory.
IV. In our scheme the usual positive-frequency function \(D^{+}(x-y)\) is represented by the function
\[ D^{+}_{s}(x,y) = \frac{i}{(2\pi)^{3}} \iint e^{ipx+iqy}\theta(p^{0})\delta(p^{2}-m^{2})\theta(q^{0})\delta(q^{2}-m^{2})K_{s}(p,q)\,dp\,dq. \]
since \(D_s^+(x,y)\) is defined in terms of the operators \(\varphi(\mathbf{x},x^0)=\varphi(x)\) in the following way:
\[ \frac{1}{i}D_s^+(x,y)=(\psi_0,\varphi(x)\varphi(y)\psi_0)_{-}, \]
where \(\psi_0=\{1,0,0,\ldots\}\) is a cyclic vector in \(F_{-}\) and \(A^{-}(k)\psi_0=0\). Since the function \(D_s^+(x,y)\) has no singularities on the light cone and is smooth, and since \(K_s(\mathbf{p},\mathbf{q})\to\delta(\mathbf{p}-\mathbf{q})\), \(s\to0\) in the weak sense, the following is valid.
Lemma. The product
\[ \prod_{(u<v)}^{N} D^+(x_u-y_v) \]
can be defined as the weak limit of regular functions
\[ \prod_{(u<v)}^{N} D_s^+(x_u,y_v) \]
on the class of basic functions \(S(R^{3N})\).
V. As follows from the explicit form of \(D_s^+(x,y)\), the proposed regularization is essentially based on a violation of the translational invariance of the theory.
Nevertheless, the commutation relation
\[ [\varphi(\mathbf{x},0),\varphi(\mathbf{y},0)]=0 \]
holds in the Hilbert space \(F_{-}\), since
\[ [A^{-}(\mathbf{k}),A^{+}(\mathbf{k}')]=K_s(\mathbf{k},\mathbf{k}'). \]
Consequently, the constructed regularization approximates the canonical commutation relations by replacing \(\delta(\mathbf{k}-\mathbf{k}')\) with the \(\delta\)-like sequence \(K_s(\mathbf{k},\mathbf{k}')\).
The authors warmly thank N. N. Bogolyubov for valuable advice concerning the further development of the results of this work.
Institute of Theoretical Physics
Academy of Sciences of the Ukrainian SSR
Kiev
Received
22 X 1969
CITED LITERATURE
- N. N. Bogolyubov, D. V. Shirkov, Introduction to the Theory of Quantized Fields, Moscow, 1958.
- J. Glimm, International School of Physics, Varenna, 1968.
- A. Jaffe, ibid.
- V. P. Gachok, On Existence of \(G\)-ergodic Vector States, Kiev, Preprint ITP-68-9.