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UDC 530.145.1
PHYSICS
B. V. MEDVEDEV, A. D. SUKHANOV
ON THE \(S\)-MATRIX IN THE HEISENBERG REPRESENTATION
(Presented by Academician N. N. Bogolyubov, March 18, 1965)
1. With the development of axiomatic methods in quantum field theory, representations of the \(S\)-matrix in the form of functional expansions in the operators of asymptotic fields \(\varphi_{in}(x)\) (or \(\varphi_{out}(x)\)) have become widespread \((^{1,2})\). At the same time, in \((^3)\), outside the framework of perturbation theory, the possibility was substantiated of representing the \(S\)-matrix in the form
\[ S_{in}=T_W \exp\left\{ i\int_{-\infty}^{\infty} \mathcal{L}_{in}(z)\,dz \right\}, \tag{1} \]
where \(T_W\) is the Wick \(T\)-product \((^{3,4})\), defined by the expansion according to Wick’s theorem, and the operator \(\mathcal{L}_{in}(z)\) has the meaning of an effective interaction Lagrangian (with counterterms) \((^5)\), expressed through normal products of the operators \(\varphi_{in}(x)\) and their derivatives.
Of special interest is the derivation of the \(S\)-matrix directly in the Heisenberg representation, since the corresponding results (see, for example, \((^6)\)) have not received due circulation. Below we propose a consistent method for obtaining the \(S\)-matrix in the Heisenberg representation, based on the assumption of the existence of the so-called “half” \(S\)-matrix*.
2. As the starting point it is natural to choose the \(in\)-representation of the \(S\)-matrix in the form (1), and, by the very meaning of the introduction of the \(S\)-matrix, it carries out \((^6)\) a unitary transformation to a new — \(out\)-representation, for example, for the field operators
\[ \varphi_{out}(x)=S_{in}^{+}\varphi_{in}(x)S_{in}; \qquad \varphi_{in}(x)=S_{in}\varphi_{out}(x)S_{in}^{+}, \tag{2} \]
and similarly for the others. Since we construct the theory on the basis of a number of general axioms \((^{2,3})\), we shall obtain the “half” \(S\)-matrix \(S_{in}(\sigma,-\infty)\) not in the usual way, by solving the Tomonaga–Schwinger equation, but directly from the matrix \(S_{in}\), guided by the requirements: a) relativistic covariance; b) independence of the specific choice of the spacelike surface \(\sigma\); c) finiteness; d) unitarity:
\[ S_{in}^{+}(\sigma,-\infty)S_{in}(\sigma,-\infty) = S_{in}(\sigma,-\infty)S_{in}^{+}(\sigma,-\infty) = 1; \tag{3} \]
e) fulfillment of the group property:
\[ S_{in}=S_{in}(\infty,\sigma)S_{in}(\sigma,-\infty), \tag{4} \]
f) boundary conditions:
\[ \lim_{\sigma\to\infty} S_{in}(\sigma,-\infty)=S_{in}; \qquad \lim_{\sigma\to-\infty} S_{in}(\sigma,-\infty)=1. \tag{5} \]
The simplest possibility for obtaining a matrix \(S_{in}(\sigma,-\infty)\), satisfying the enumerated requirements, from \(S_{in}\) of the form (1) consists in the following:
* The results were reported at the Fifth All-Union Conference on the Theory of Elementary Particles (Uzhgorod, October 1963).
perform the functional substitution \(\mathcal L_{in}(z)\) by \(\theta(\sigma(x)-z^0)\mathcal L_{in}(z)\). However, as has already been shown in perturbation theory \((^{7,4})\), the half \(S\)-matrix obtained in this way will not satisfy requirements c), d), and e), if only \(\mathcal L_{in}(z)\) contains time derivatives of fields higher than first, i.e., in all renormalized theories (counterterms!). At the same time, in \((^4)\) it was shown that the indicated difficulties in the definition of \(S_{in}(\sigma,-\infty)\) can be avoided if one first passes, in the expression for \(S_{in}\) of the type (1), from the \(T_W\)-product to the \(T_D\)-product (Dyson \(T\)-product, defined by explicit chronological ordering \((^{3,4})\)) and simultaneously from \(\mathcal L_{in}(z)\) to \(H_{in}(z;\sigma)\), the interaction Hamiltonian \(H_{in}(z;\sigma)\) being obtainable, according to definite rules, from \(\mathcal L_{in}(z)\) \((^{4,8})\).
Since in the present case we do not have at our disposal a concrete expression for \(\mathcal L_{in}(z)\), relying on the analogy with perturbation theory \((^4)\), we shall suppose that the matrix \(S_{in}\) of the form (1) can be represented in the form*
\[ S_{in}=T_D\exp\left\{-i\int_{-\infty}^{\infty} H_{in}(z;\sigma)\,dz\right\}=S_{in}[H_{in}(z;\sigma)], \tag{6} \]
where, generally speaking, \(H_{in}(z;\sigma)\ne -\mathcal L_{in}(z)\).
Then the “half” \(S\)-matrix, if it exists at all, may be defined by the relation**
\[ S_{in}(\sigma,-\infty)=S_{in}[\theta(\sigma(x)-z^0)H_{in}(z;\sigma')] =T_D\exp\left\{-i\int_{-\infty}^{\sigma} H_{in}(z;\sigma')\,dz\right\}, \tag{7} \]
which, as is known from perturbation theory \((^4)\), in renormalizable theories can satisfy all requirements a)—e), in connection with which below, for simplicity, we shall put \(\sigma(x)=x^0=\mathrm{const}\).
- We shall now pass to the out-representation. By the general rule for transforming operators, one must have
\[ S_{out}=S_{in}^{+}S_{in}S_{in}, \tag{8} \]
and, since under the transformation (2) products are transformed in the same way, and consequently also polynomials and series of operators, it follows that
\[ S_{out}=S_{in}[H_{out}(z)], \tag{9} \]
where \(H_{out}(z)=S_{in}^{+}H_{in}(z)S_{in}\), while the notation in (9) means that the functional dependence (in the present case the \(T_D\)-product) of \(S_{out}\) on \(H_{out}(z)\) is exactly the same as that of \(S_{in}\) on \(H_{in}(z)\) in (6). However, the meaning of the operator \(S_{out}\) is not yet clear. If, moreover, we take into account in (8) the unitarity of \(S_{in}\), it turns out that
\[ S_{out}=S_{in}=S, \tag{10} \]
i.e., the value of \(S_{out}\) coincides with the value of \(S_{in}\).
At the same time,
\[ S_{out}(x^0,-\infty)=S^{+}S_{in}(x^0,-\infty)S =T_D\exp\left\{-i\int_{-\infty}^{x^0} H_{out}(z)\,dz\right\}. \tag{11} \]
Thus, the matrix \(S_{out}(x^0,-\infty)\) differs from the matrix \(S_{in}(x^0,-\infty)\), although its functional dependence on \(H_{out}(z)\) has remained
* The interaction Hamiltonian used by us should, more properly, be denoted \(H^{\mathrm v}_3(z;\sigma)\), since it is, generally speaking, distinct from the in-image of the interaction Hamiltonian \((^4)\). Our notation is intended to emphasize that this operator depends precisely on the fields \(\varphi_{in}(z)\), and not on \(\varphi_{out}(z)\) or on anything else. Further, since in our case the free operators are \(\varphi_{in}(x)\), the matrix \(S_{in}(\sigma,-\infty)\) has no relation to the evolution operator \(U(\sigma,-\infty)\), but coincides with the matrix \(V_{+}(\sigma)\) in the notation of \((^9)\).
** Such a definition, naturally, contains a factor of the form \(\exp[iF_{in}(\sigma)]\), where \(F_{in}(\sigma)\) is a Hermitian local operator which, from the point of view of the complete \(S\)-matrix, is immaterial.
same \((T_D\)-product). Similarly, for the other “half” of the \(S\)-matrix we have
\[ S_{out}(\infty,x^0)=S^+S_{in}(\infty,x^0)S=S_{in}^+(x^0,-\infty)S. \tag{12} \]
However, the group property (4) is still preserved in the out-representation, i.e.
\[ S_{out}(\infty,x^0)S_{out}(x^0,-\infty) =S^+S_{in}(\infty,x^0)SS^+S_{in}(x^0,-\infty)S=S. \tag{13} \]
Let us note that from (12) there follows a noteworthy formula, on which the subsequent transformations are based:
\[ S_{in}(x^0,-\infty)S_{out}(\infty,x^0)=S. \tag{14} \]
Thus, for the complete scattering matrix there are three expressions in terms of “halves,” namely (4), (13), and (14).
Let us now introduce, in addition to the in- and out-representations, a collection of \(j\)-representations \((j=1,\ldots,n)\), according to the formula
\[ A_j(z)=S_{in}^+(x_j^0,-\infty)\,A_{in}(z)\,S_{in}(x_j^0,-\infty), \tag{15} \]
where \(A_{in}(z)\) is an arbitrary in-operator. Then
\[ S_j(\infty,x_j^0)=S_{in}^+(x_j^0,-\infty)S;\qquad S_j(x_j^0,-\infty)=S_{in}(x_j^0,-\infty), \tag{16} \]
where the latter equality denotes not only the same functional dependence, but also coincidence of the operators.
Let us note that the matrix \(S_j=S_j(\infty,x_j^0)S_j(x_j^0,-\infty)\) is not equal to the matrix \(S\), but is related to it by the formula
\[ S_j=S_{in}^+(x_j^0,-\infty)SS_{in}(x_j^0,-\infty), \tag{17} \]
i.e. it has only the same functional dependence. At the same time, from the first equality in (16) we obtain
\[ S=S_{in}(x_j^0,-\infty)S_j(\infty,x_j^0). \tag{18} \]
Considering now simultaneously two such representations \(i\) and \(j\) (taking \(i>j\)), we obtain, with allowance for the fact that all transformations (15) form a group, that
\[ A_i(z)=S_j^+(x_i^0,x_j^0)A_j(z)S_j(x_i^0,x_j^0), \tag{19} \]
where
\[ S_j(x_i^0,x_j^0)=S_{in}^+(x_j^0,-\infty)S_{in}(x_i^0,x_j^0)S_{in}(x_j^0,-\infty). \tag{20} \]
Applying next the transformation (19) to the matrix \(S_j(\infty,x_j^0)\), we obtain the relation
\[ S_j(\infty,x_j^0)=S_j(x_i^0,x_j^0)S_i(\infty,x_i^0), \tag{21} \]
which, in combination with (18), gives for the complete \(S\)-matrix the expression
\[ S=S_{in}(x_j^0,-\infty)S_j(x_i^0,x_j^0)S_i(\infty,x_i^0). \tag{22} \]
Carrying out these arguments now for all \(j\) successively, beginning with the first up to \(n\), we arrive at
\[ S=S_{in}(x_1^0,-\infty)\prod_{j=1}^{n-1}S_j(x_{j+1}^0,x_j^0)S_n(\infty,x_n^0). \tag{23} \]
Writing (18) for \(j=n\) and comparing the expression obtained with (23), we also see that
\[ S_{in}(x_n^0,-\infty)=S_{in}(x_1^0,-\infty)\prod_{j=1}^{n-1}S_j(x_{j+1}^0,x_j^0). \tag{24} \]
To pass to the expressions of interest to us, it is necessary in (23) and (24) to carry out the limiting transition \(n \to \infty\) in such a way that, in doing so, \(x_1^0 \to -\infty\), \(\Delta_j = \max (x_{j+1}^0 - x_j^0) \to 0\), while \(x_n^0 \to \infty\) in (23) and \(x_n^0 \to x^0 = \mathrm{const}\) in (24). Let us first note that the expression \(S_{in}(x_{j+1}^0, x_j^0)\) entering into \(S_j(x_{j+1}^0, x_j^0)\), according to (20), with account of (7), in the limit \(\Delta_j \to 0\), if only terms of first order of smallness in \(\Delta_j\) are retained, gives
\[ \lim_{\Delta_j \to 0} S_{in}(x_{j+1}^0, x_j^0) = 1 - i \int_{x_j^0}^{x_{j+1}^0} H_{in}(z)\,dz . \tag{25} \]
Substituting now (25) into (20), and (20) into (23) and (24), we obtain for the \(S\)-matrix, in the indicated limit, an expression in the form of a Dyson antichronological exponential
\[ S = \widetilde{T}_D \exp\left\{ - i \int_{-\infty}^{\infty} \mathbf{H}_z(z)\,dz \right\} \tag{26} \]
of Hamiltonians taken, for each point \(z\), in its own special representation, determined by the “half” \(S\)-matrix with the separating surface passing through this point \(z\):
\[ H_{(z)}(z) = S_{in}^{+}(z^0, -\infty)\,H_{in}(z)\,S_{in}(z^0, -\infty). \tag{27} \]
But it is easy to see that such an infinite collection of representations forms precisely the Heisenberg representation
\[ H_z = S_{in}^{+}(z^0, -\infty)\,H_{in}(z)\,S_{in}(z^0, -\infty), \tag{28} \]
i.e. formula (26) means that
\[ S = S_{in} = S_{out} = S_{\Gamma} = T_D \exp\left\{ - i \int_{-\infty}^{\infty} \mathbf{H}(z)\,dz \right\}; \tag{29} \]
and analogously from (24) we obtain
\[ S_{in}(x^0, -\infty) = S_{\Gamma}(x^0, -\infty) = \widetilde{T}_D \exp\left\{ - i \int_{-\infty}^{x^0} \mathbf{H}(z)\,dz \right\}. \tag{30} \]
Thus, in the transition from the \(in\)-representation to the Heisenberg one, the functional dependence in the \(S\)-matrix changes; namely, chronological ordering passes into antichronological ordering.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Moscow Institute
of Radio Electronics and Mining Electromechanics
Received
5 III 1965
CITED LITERATURE
- H. Lehmann, K. Symanzik, W. Zimmermann, Nuovo Cim., 1, 205 (1955); 6, 319 (1957).
- N. N. Bogolyubov, B. V. Medvedev, M. K. Polivanov, Problems in the Theory of Dispersion Relations, Moscow, 1958; B. V. Medvedev, ZhETF, 40, 826 (1961).
- B. V. Medvedev, M. K. Polivanov, Lectures at the International Winter School of Theoretical Physics, Dubna, March 1964.
- A. D. Sukhanov, ZhETF, 47, 1303 (1964); 48, 538 (1956).
- N. N. Bogolyubov, D. V. Shirkov, Introduction to the Theory of Quantized Fields, 1957.
- C. N. Yang, D. Feldman, Phys. Rev., 79, 972 (1950).
- A. D. Sukhanov, ZhETF, 43, 1400 (1962).
- A. D. Sukhanov, ZhETF, 41, 1915 (1961).
- C. Schweber, Introduction to Relativistic Quantum Field Theory, IL, 1963.
- W. Weidlich, Nuovo Cim., 30, 803 (1963).