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UDC 530.145
MATHEMATICAL PHYSICS
L. Sh. KHODZHAEV
ON THE REPRESENTATION OF STATES IN QUANTUM FIELD THEORY
(Presented by Academician N. N. Bogolyubov on 16 XII 1965)
For the investigation of certain general properties of the \(S\)-matrix for arbitrary spin it is necessary to consider the problem of describing a dynamical system consisting of several noninteracting quantized fields characterized by operator-valued generalized functions \(\hat u_1,\ldots,\hat u_n\).
We introduce into consideration the countably normed Hilbert space \((^1)\)
\[ D_S=\bigotimes_{n,l=0}^{\infty}D_S^{(n,l)}{}_{(R^4(n,l))} \tag{1} \]
of regular states \(|\Phi\rangle\), where \(D_S^{(n,l)}{}_{(R^4(n,l))}\) is the space of states of a bundle of noninteracting \(n\) particles and \(l\) antiparticles, and, for spins \(s_r,j_t=0,\tfrac12,1,\ldots,\ r=1,\ldots,n,\ t=1,\ldots,l\), the space \(D_S^{(n,l)}{}_{(R^4(n+l))}\) consists of elements of the space \(S(R^{4(n+l)})\) of basic functions \((^2)\), represented in the form
\[ \Phi_{(\sigma)n;\,(\nu)l}^{(n,l)}(p,q)_{n,l} = \bigotimes_{r=1}^{n}\mathscr D_{\sigma_r'\sigma_r}^{s_r}\bigl(l^{-1}(p_r)\bigr) \bigotimes_{t=1}^{l}\mathscr D_{\nu_t'\nu_t}^{j_t}\bigl(l^{-1}(q_t)\bigr) \hat\Phi_{(\sigma')n;\,(\nu')l}^{(n,l)}(p,a)_{n,l}, \tag{2} \]
where
\[ \hat\Phi_{(\sigma)n;\,(\nu)l}^{(n,l)}(p,a)_{n,l} = \left( \bigotimes_{r=1}^{n}U\bigl(l(p_r)\bigr) \bigotimes_{t=1}^{l}U\bigl(l(q_t)\bigr)\Phi \right)_{(\sigma,\nu)n,l}^{(n,l)}; \]
\(\sigma_r\) and \(\nu_t\) are the spin projections, respectively, of a particle of mass \(m_r\) and spin \(s_r\) and of an antiparticle of mass \(\mu_t\) and spin \(j_t\) on the third axis; \(p_r\) and \(q_t\) are their 4-energy-momenta with domain of definition
\[ \Omega_{n,l}=\bigotimes_{r=1}^{n}\Omega_{m_r}\bigotimes_{t=1}^{l}\Omega_{\mu_t}, \]
where \(\Omega_m\), \(p_r^2=m_r^2,\ p_{0r}>0,\ r=1,\ldots,n\), and \(\Omega_{\mu_t}\): \(q_t^2=\mu_t^2,\ q_{0t}>0,\ t=1,\ldots,l\); \(D_{\sigma'\sigma}(l(p))\) is a unitary irreducible representation of the \((2s+1)\)-dimensional spin space \((^3)\), \(l(p)\in SL(2,c)\) and \(l(p)\tilde p=p,\ \tilde p=(m,\vec 0)\), \((k)_f=(k_1,\ldots,k_f)\). We shall assume that \(\Phi^{(n,l)}=0\) for \(n,l>\tilde N\), where \(\tilde N\) is a sufficiently large number. A topology is introduced in \(D_S\) in a certain way.
The transformation property of the elements \(|\Phi\rangle\in D_S\) with respect to the spinor Poincaré group \(\widetilde P_+^\uparrow\) with elements \((a,A)\), where \(a\) denotes 4-translations and \(A\) is an arbitrary element of \(SL(2,c)\) corresponding to the element \(\Lambda\in L_+^\uparrow\), is defined according to
\[ U(a,A)|\Phi\rangle= \left\{ \exp\left[ ia\left(\sum_r p_r+\sum_t q_t\right) \right]\times \right. \]
\[ \left. \times \bigotimes_{r=1}^{n}\mathscr D_{\sigma_r'\sigma_r}^{s_r}(A) \bigotimes_{t=1}^{l}\mathscr D_{\nu_t'\nu_t}^{j_t}(A) \Phi_{(\sigma')n;\,(\nu')l}^{(n,l)}(A^{-1}p;\,A^{-1}q)_{n,l} \right\}. \tag{3} \]
The latter follows from the fact that
\[ \begin{aligned} (U(a,A)\widehat{\Phi})^{n,l}_{(\sigma)_n;(\nu)_l}(p;q)_{n,l} &=\exp\left[ ia\left(\sum_r p_r+\sum_t q_t\right)\right]\times\\ &\quad \times \bigotimes_{r=1}^{n}\mathfrak{D}^{s_r}_{\sigma_r'\sigma_r}(R(A)p_r) \bigotimes_{t=1}^{l}\mathfrak{D}^{j_t}_{\nu_t'\nu_t}(R(A,q_t)) \widehat{\Phi}^{(n,l)}_{(\sigma')_n;(\nu')_l}(A^{-1}p;A^{-1}q)_{n,l}, \end{aligned} \tag{4} \]
where \(R(A,p)=l^{-1}(A^{-1}p)Al(p)\in SU(2,c)\).
In \(D_S\) the scalar product is introduced
\[ \langle \Phi_1\mid \Phi_2\rangle = \sum_{n,l}^{\infty} \langle \Phi_1^{(n,l)}\mid \Phi_2^{(n,l)}\rangle<\infty, \tag{5} \]
where
\[ \langle \Phi_1^{(n,l)}\mid \Phi_2^{(n,l)}\rangle = \langle (U(a,A)\Phi)^{(n,l)}\mid (U(a,A)\Phi_2)^{(n,l)}\rangle . \tag{6} \]
Let \(H_{L_2}\) be the Hilbert space obtained by completing \(D_S\) with respect to the nondegenerate scalar product (5). Then the family of spaces \(D_S\subset H_{L_2}\subset D_S'\), where \(D_S'\) is the space of generalized states, forms a rigged physical Hilbert space \((^1,^4)\).
The space \(D^{(1,0)'}_{S(R^4)}(m,s)\) of one-particle generalized states
\[ \eta^{(1)}(\Phi^{(1)}) = \eta^{(1)}_{\sigma}(\Phi^{(1)}_{\sigma}) = \int \eta^{(1)}_{\sigma}(p)\Phi^{(1)}_{\sigma}(p)\frac{d^3p}{p}, \tag{7} \]
where \(\Phi^{(1)}_{\sigma}(p)\in D^{(1,0)}_{S(R^4)}(m,s)\), is the space of irreducible representations of the spinor Poincaré group \(\widetilde{P}_{+}\uparrow\). Generalized states are defined as common generalized eigenfunctions of the operators \(P_\mu', \vec S^{\,2'}, S_3'\) and form a canonical basis of the spinor space \((^5)\). L. Schwartz’s kernel theorem \((^2)\) makes it possible to define many-particle generalized states in the form
\[ \rho^{(n)}_{(\sigma)_n}(\Phi^{(n)}_{(\sigma)_n}) = \eta^{(n)}_{(\sigma)_n}(\Phi^{(1)}_{\sigma_1},\ldots,\Phi^{(1)}_{\sigma_n}) = \eta^{(1)}_{\sigma_1}(\Phi^{(1)}_{\sigma_1})\ldots \eta^{(1)}_{\sigma_n}(\Phi^n_{\sigma_n}), \tag{8} \]
where \(\Phi^{(n)}_{(\sigma)_n}(p)_n=\Phi^{(1)}_{\sigma_1}(p_1)\ldots\Phi^{(1)}_{\sigma_n}(p_n)\). Therefore the vectors of an arbitrary generalized state \(\mid\rho(\Phi)\rangle\in D_S'\) can be represented in the form
\[ \mid\rho(\Phi)\rangle = \sum_{n,l\geq 0}'\int_{\Omega_{nl}}\cdots\int \rho^{(n,l)}_{(\sigma)_n;(\nu)_l}(p,q)_{n,l} \Phi^{(n,l)}_{(\sigma)_n;(\nu)_l}(p,q)_{n,l} \prod_{r=1}^{n}\frac{d^3p_r}{p_{0r}} \prod_{t=1}^{l}\frac{d^3q_t}{q_{0t}} \tag{9} \]
and possess the transformation property
\[ U(a,A)\mid\rho(\Phi)\rangle = \{\rho^{(n,l)}_{(\sigma)_n;(\nu)_l}((U(a,A)\Phi)^{(n,l)}_{(\sigma)_n;(\nu)_l}(p,q)_{n,l})\}, \tag{10} \]
where \((U(a,A)\Phi)^{(n,l)}\) is defined according to (3).
In the space \(D_S'\) one can define weakly dense sets of vectors of generalized correlations of a special type. We shall present a scheme for constructing such a set.
To each function \(f(x)\in S(R^4)\) we assign the Hermitian operator
\[ A(f)=\int d^4x\, f(x)\,A(x)\ \text{in }D_S,\qquad \langle \Psi\mid A(f)\mid \Phi\rangle,\quad \text{where } \Psi,\Phi\in D_S. \tag{11} \]
and we shall consider the quantity \(A(f)\) as a linear continuous functional in \(S(R^4)\) and call it an operator-valued generalized function.
Let us now consider the field operator \(\chi(f)\) for the creation of a particle of mass \(m \ne 0\) and spin \(s\), defined as follows:
\[ \chi(\tilde f)=\chi_\sigma(\tilde f^\sigma)=\int \frac{d^3p}{p_0}\,\tilde f^\sigma(p)\chi_\sigma(p),\quad \text{where }\quad \tilde f^\sigma(p)=\int e^{-ipx} f^\sigma(x)\,d^4x; \tag{12} \]
\[ \chi(f)|\Phi\rangle= \left\{ \frac{1}{\sqrt n}\sum_{r=1}^{n}\sum_{\sigma_r=-s_r,\ldots,s_r} \tilde f^{\sigma_r}(-p_r)\times \right. \]
\[ \left. {}\times \Phi_{\sigma_1,\ldots,\sigma_{r-1},\,\sigma^{r+1},\ldots,\sigma_n;\,(\nu)_l}^{(n-1,l)} (p_1,\ldots,p_{r-1},p_{r+1},\ldots,p_n;(q)_l) \right\}; \tag{13} \]
\[ U(a,A)\chi_\sigma(\tilde f^\sigma)U^{-1}(a,A) = \chi_\sigma\!\left(e^{-iaAp}\mathcal D_{\sigma'\sigma}^{s}(R^{-1}(A,Ap))\tilde f^{\sigma'}(Ap)\right) \tag{14} \]
for any \(f^\sigma(x)\in S(R^4)\) and \(|\Phi\rangle\in D_S\).
We construct the corresponding covariant operator by setting
\[ a(\tilde f)=a_\alpha(\tilde f^\alpha) = \chi_\sigma\!\left(\mathcal D_{\alpha\sigma}^{s}(l(p))\tilde f^\alpha(p)\right) \tag{15} \]
with the transformation property
\[ U(a,A)a_\alpha(\tilde f^\alpha)U^{-1}(a,A) = a_\alpha\!\left(e^{-iaAp}\mathcal D_{\beta}^{\,s\alpha}(A^{-1})\tilde f^\beta(Ap)\right). \tag{16} \]
Now the free field can be characterized by means of the covariant operator-valued generalized function \(\varphi(f)\), defined in \(D_S\):
\[ \varphi(f)|\Phi\rangle = a(\tilde f(-p))|\Phi\rangle + \overset{*}{a}(\tilde f(p))|\Phi\rangle = \]
\[ = \left\{ \frac{1}{\sqrt n}\sum_{r=1}^{n}\sum_{\sigma_r=-s_r,\ldots,s_n} \mathcal D_{\alpha_r\sigma_r}^{s_r}(l(p_r))\tilde f^{\alpha_r}(-p_r)\times \right. \]
\[ \left. {}\times \Phi_{\sigma_1,\ldots,\sigma_{r-1},\,\sigma_{r+1},\ldots,\sigma_n;\,(\nu)_l}^{(n-1,l)} (p_1,\ldots,p_{r-1},p_{r+1},\ldots,p_n;(q)_l) + \right. \]
\[ \left. {}+ \frac{1}{\sqrt{n+1}}\int\frac{d^3p}{p_0}\, \mathcal D_{\sigma\alpha}^{s}(l^{-1}(p))\tilde f^\alpha(p) \Phi_{\sigma,\,(\sigma)_n;\,(\nu)_l}^{(n+1,l)} (p,(p)_n;(q)_l) \right\}. \tag{17} \]
It is clear that
\[ \varphi(f)D_S\subset D_S, \]
\[ U(a,A)\varphi_\alpha(f^\alpha)U^{-1}(a,A) = \varphi_\alpha\!\left(\mathcal D_{\beta}^{\,s\alpha}(A^{-1})f^\beta(A^{-1}(x-a))\right) \tag{18} \]
for \(f^\alpha(x)\in S(R^4)\).
Analogously, introducing the field operator in \(D_S\) according to
\[ \chi(f)=\chi^{\dot\beta}(f_{\dot\beta}) = b^{\dot\beta}(\tilde f_{\dot\beta}) + (-1)^{2s}\overset{*}{b}{}^{\dot\beta}(\tilde f_{\dot\beta}), \tag{19} \]
where
\[ b(\tilde f)=b^{\dot\beta}(\tilde f_{\dot\beta}) = \chi_\sigma\!\left(\overset{*}{\mathcal D}{}_{\dot\beta}^{\,s\sigma}(l(p))\tilde f_{\dot\beta}(p)\right), \tag{20} \]
\[ U(a,A)b^{\dot\beta}(\tilde f_{\dot\beta})U^{-1}(a,A) = b^{\dot\beta}\!\left(e^{-iaAp}\overset{*}{\mathcal D}{}_{\dot\beta}^{\,\dot\alpha}(A)\tilde f_{\dot\alpha}(Ap)\right). \tag{21} \]
The field operators \(\varphi_\alpha\) and \(\chi^{\dot\beta}\) satisfy the Dirac equation
\[ \varphi_\alpha(\Phi^\alpha)=\chi^{\dot\beta}(f_{\dot\beta}) \tag{22} \]
for arbitrary functions \(\Phi^\alpha(x)\) and \(f_{\dot\beta}(x)\) from \(S(R^4)\) satisfying the relations
\[ \begin{pmatrix} 0 & (-1)^{2s}\mathcal D^{s}(i\sigma\cdot\partial)_{\alpha\dot\beta}\\ (-1)^{2s}\mathcal D^{s}(i\tilde\sigma\cdot\partial)^{\dot\beta\alpha} & 0 \end{pmatrix} \begin{pmatrix} \Phi^\alpha(x)\\ f_{\dot\beta}(x) \end{pmatrix} = m^{2s} \begin{pmatrix} \Phi^\alpha(x)\\ f_{\dot\beta}(x) \end{pmatrix}, \tag{23} \]
where \(\sigma=(\sigma_0,\vec\sigma)\), \(\tilde\sigma=(\sigma_0,-\vec\sigma)\), \(\vec\sigma=(\sigma_1,\sigma_2,\sigma_3)\) are the Pauli matrices, and \(\sigma_0\) is the identity matrix.
Now suppose that there exists a set of nonproper field operators \(\varphi_{\alpha_1}^{(1)}, \varphi_{\alpha_2}^{(2)},\ldots\), defined in \(D_S\). Then, using nuclear
by theorem (? ), we can construct a set of generalized-state vectors everywhere dense in \(D_S\),
\[ \{c_{n,l}\Omega^{*(n,l)}(f_{n,l})|0\rangle\}, \tag{24} \]
where \(c_{n,l}\) are arbitrary complex numbers, \(|0\rangle\in D_S\) are vacuum states, and
\[ \Omega^{*(n,l)}_{(\alpha)n;(\beta)l}(f^{(\alpha)n;(\beta)l}_{n,l}) = a^{*(1)}_{\alpha_1}\ldots a^{*(n)}_{\alpha_n} c^{*(1)}_{\beta_1}\ldots c^{*(l)}_{\beta_l} (f^{(\alpha)n;(\beta)l}_{n,l}) \tag{25} \]
with the transformation property
\[ U(a,A)\Omega^{*(n)}_{(\alpha)n}(f^{(\alpha)n}_n)U^{-1}(a,A)= \]
\[ =\Omega^{*(n)}_{(\alpha)n}\left( \exp\left[ia\sum_r Ap_r\right] \bigotimes_{r=1}^n \mathscr{D}^{\,s_r}_{\beta_r\alpha_r}(A^{-1}) f^{(\beta)n}_n(Ap)_n \right) \tag{26} \]
for \(\widetilde f^{(\alpha)n}_m(p)_n\in S(R^{4n})\).
In the following work we shall study relativistically invariant matrix elements of the \(S\)-matrix, defined in the form:
\[ S(\Phi^{(\alpha)m;(\beta)r}_{M,n}\Psi^{(\alpha)n;(\beta)l}_{n,l}) = \langle 0| \Omega^{(m,r)}_{(\alpha)m;(\beta)r} (\Phi^{(\alpha)m;(\beta)r}_{m,r}) S\Omega^{*(n,l)}_{(\alpha)n;(\beta)l} (\Psi^{(\alpha)n;(\beta)l}_{n,l}) |0\rangle . \tag{27} \]
Here we shall regard the \(S\)-matrix as a unitary operator defined in the equipped physical Hilbert space \(D\subset H\subset D'\), which maps one generalized state (25) from \(D'\), corresponding to an incident wave, into some other generalized state from \(D'\), corresponding to an outgoing wave.
Taking this opportunity, I express my deep gratitude to N. N. Bogoliubov, Nguyen Van Hieu, and A. V. Efremov for their interest in the work and for valuable comments.
Joint Institute
for Nuclear Research
Received
7 XII 1965
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