Abstract Generated abstract
The paper develops an operator representation of the relativistic scattering matrix in quantum field theory using an invariant one-dimensional parameter and the Fourier transform of the interaction Lagrangian. Starting from an integral equation for the reaction matrix, it introduces auxiliary quasiparticle state vectors and shift operators, whose algebra permits the equation to be rewritten and formally solved in operator form. The resulting expressions represent the scattering matrix as a vacuum expectation value involving the physical interaction Lagrangian and the quasiparticle operators, with an illustration for a cubic scalar interaction. The formulation is proposed as a tool for deriving relations among scattering matrix elements and for studying expansions in inverse powers of the coupling constant.
Full Text
PHYSICS
V. G. KADYSHEVSKY
ON A REPRESENTATION FOR THE SCATTERING MATRIX IN QUANTUM FIELD THEORY
(Presented by Academician N. N. Bogolyubov, 3 VIII 1964)
Let \(S = 1 + iR\) be the relativistic scattering matrix, specified in the interaction representation, and let \(\mathscr{L}(p)\) be the Fourier transform of the interaction Lagrangian \(\mathscr{L}(x)\) in the same representation:
\[ \widetilde{\mathscr{L}}(p)=\int e^{-ipx}\mathscr{L}(x)\,dx . \tag{1} \]
According to (1),
\[ R=R(\lambda\tau)\big|_{\tau=0}, \tag{2} \]
where \(\tau\) is a one-dimensional invariant parameter, \(\lambda\) is a 4-vector satisfying the conditions
\[ \lambda^2=1,\qquad \lambda_0>0, \]
and the matrix \(R(\lambda\tau)\) is determined from the equation
\[ R(\lambda\tau)=\widetilde{\mathscr{L}}(\lambda\tau)+\frac{1}{2\pi}\int_{-\infty}^{\infty} \widetilde{\mathscr{L}}(\lambda\tau-\lambda\tau')\, \frac{d\tau'}{\tau'-i\varepsilon}\,R(\lambda\tau'). \tag{3} \]
Instead of (3) one may consider the operator equation
\[ \hat R=\hat L+\frac{1}{2\pi}\hat L\,\frac{1}{\tau-i\varepsilon}\,\hat R, \tag{4} \]
if, by definition, one takes
\[ \tau'\delta(\tau'-\tau'')=\langle \tau'|\tau|\tau''\rangle, \tag{5a} \]
\[ \mathscr{L}(\lambda\tau'-\lambda\tau'')=\langle \tau'|\hat L|\tau''\rangle, \tag{5b} \]
\[ R(\lambda\tau)=\langle \tau|\hat R|0\rangle. \tag{5c} \]
(\(|\tau\rangle\) is the state vector of certain “quasiparticles” possessing 4-momentum equal to \(\lambda\tau\) \((^2)\).) It is not difficult to verify that the solution of equation (4) has the form
\[ \hat R=2\pi\,\frac{1}{2\pi-\hat L-i\varepsilon}\,\hat L. \tag{6} \]
In order that formula (7) be effective, it is necessary to find a concrete realization of the state vectors \(|\tau\rangle\) and of the operator \(\hat L\). To this end we introduce into consideration the operator \(a(\tau)\), assuming
\[ \langle \tau'|a(\tau)|\tau''\rangle=\delta(\tau'+\tau-\tau''). \tag{7} \]
It is easy to establish that
\[ a(\tau_1)a(\tau_2)\ldots a(\tau_n)=a(\tau_1+\tau_2+\ldots+\tau_n), \tag{8} \]
\[ a^+(\tau)=a(-\tau),\qquad [a(\tau),a^+(\tau')]-=0. \]
Moreover, with the aid of (5a) and (7), we find
\[ [a(\tau'),\tau]=\tau'a(\tau');\qquad [\tau,a^+(\tau')]=\tau'a^+(\tau'). \tag{9} \]
We shall now assume that
\[ |\tau\rangle=a^+(\tau)|0\rangle. \tag{10} \]
Obviously, with such a definition,
\[ \langle \tau'|\tau\rangle=\delta(\tau'-\tau),\qquad \tau|\tau'\rangle=\tau'|\tau'\rangle . \tag{11} \]
Indeed, according to (7), (8), and (9) we shall have
\[ \langle \tau'|\tau\rangle=\langle 0|a(\tau')a(-\tau)|0\rangle =\langle 0|a(\tau'-\tau)|0\rangle=\delta(\tau'-\tau); \]
\[ \tau|\tau'\rangle=\tau a^+(\tau')|0\rangle =a^+(\tau')\tau|0\rangle+\tau'a^+(\tau')|0\rangle =\tau'|\tau'\rangle, \]
if \(\tau|0\rangle=0\).
Taking into account definitions (5b) and (7), the operator \(L\) can be represented in the form
\[ \hat L=\int_{-\infty}^{\infty} a(\tau)\,\widetilde{\mathcal L}(-\lambda\tau)\,d\tau . \tag{12} \]
Thus, the description of quasiparticles in terms of \(a(\tau)\) and \(a^+(\tau)\) resembles the formalism of second quantization used to describe physical particles, while the operator \(\hat L\) plays the role of the Lagrangian of the interaction of physical particles with quasiparticles.
Substituting (12) into formula (6), we obtain the final expression for the operator \(\hat R\):
\[ \hat R =2\pi\tau\, \frac{1}{\displaystyle 2\pi\tau-\int_{-\infty}^{\infty} a(\tau)\widetilde{\mathcal L}(-\lambda\tau)\,d\tau-i\varepsilon} \int_{\infty-}^{\infty} a(\tau)\widetilde{\mathcal L}(-\lambda\tau)\,d\tau , \tag{13} \]
whence, on the basis of (2),
\[ S=1+2\pi i\langle 0|\tau \left[ 2\pi\tau-\int_{-\infty}^{\infty} a(\tau)\widetilde{\mathcal L}(-\lambda\tau)\,d\tau-i\varepsilon \right]^{-1} \int_{-\infty}^{\infty} a(\tau)\widetilde{\mathcal L}(-\lambda\tau)\,d\tau |0\rangle . \tag{14} \]
Let us emphasize that the Lagrangian \(\widetilde{\mathcal L}(-\lambda\tau)\) in (13) and (14) is an operator only in the state space of the physical particles. For example, in the case
\[ \mathcal L(x)=g:\varphi^3(x):, \]
by virtue of (1) and under the condition that
\[ \varphi(x)=\frac{1}{(2\pi)^{3/2}}\int \varphi(k)e^{ikx}\,dk, \]
we shall have
\[ \widetilde{\mathcal L}(-\lambda\tau) = \frac{g}{\sqrt{2\pi}} \int \delta^{(4)}(\lambda\tau+k_1+k_2+k_3): \varphi(k_1)\varphi(k_2)\varphi(k_3)\, dk_1dk_2dk_3 . \tag{15} \]
Relations (13) and (14) may prove useful for obtaining chains of equations connecting the matrix elements \(S\) with one another, and also in studies using an expansion in inverse powers of the coupling constant \(g\). In this case it is expedient to work precisely with the operator \(\hat R\), and to carry out the averaging over the “quasiparticle vacuum” only at the end of all calculations.
The author expresses his gratitude to B. M. Barbashov for a fruitful discussion.
Joint Institute for Nuclear Research
Received
27 VI 1964
References
- V. G. Kadyshevskii, ZhETF, 46, 654 (1964).
- V. G. Kadyshevskii, ZhETF, 46, 872 (1964).