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MATHEMATICAL PHYSICS
B. V. MEDVEDEV
THE AXIOMATIC METHOD AND PERTURBATION THEORY
(Presented by Academician N. N. Bogolyubov, 28 VI 1960)
- The usual approach to quantum field theory is the Hamiltonian formalism, which arose as a direct transfer to field theory of the path leading from classical mechanics to quantum mechanics. In this approach the theory is fixed by specifying the form of the Lagrangian; then, by variation, one obtains the equations of motion, which, after quantization according to known rules, are transformed into Heisenberg equations for operator field functions. In fact, by this path the theory can be formulated only within the framework of perturbation theory, since not only the solution of the equations but also their very writing down (“elimination of infinities”) can be carried out only in the form of series in powers of the coupling constant.
The difficulties of the Hamiltonian method led to the emergence of another approach—it is often (but not very successfully) called axiomatic—when certain general physical requirements are laid at the foundation of the theory, requirements that the solutions of the equations must satisfy, while the equations themselves are not explicitly formulated. Interest in this path of constructing the theory has especially increased in recent years in connection with the study of dispersion relations—the only exact result of quantum field theory.
The basic propositions of the axiomatic method can be formulated in various ways. Thus, one may include among the basic propositions the requirement that Heisenberg fields exist at each point and commute on any space-like hypersurface—this direction is being developed by Lehmann, Symanzik, and Zimmermann (see (¹,²) and numerous subsequent works). On the other hand, one may start from the program proposed in its time by Heisenberg (³) and restrict oneself to the consideration of the scattering matrix. The latter path was chosen by N. N. Bogolyubov, M. K. Polivanov, and the author (⁴) in connection with the theory of dispersion relations*.
In all variants of the axiomatic approach there arise natural questions concerning the compatibility of the introduced system of “axioms” and its sufficiency to determine the theory (and with what degree of arbitrariness?). To the first of these questions no definite answer can yet be given, since the existence of a noncontradictory scheme of quantum field theory in general has not been established. The purpose of the present note is to investigate the second question. Namely, we shall show that, within the framework of perturbation theory, the formal expansion of the scattering matrix in powers of the coupling follows from the basic propositions of the axiomatic approach, supplemented by assumptions about the transformation properties of the fields under consideration and the growth degrees of matrix elements, with the same degree of arbitrariness as in the usual theory.
- We shall proceed from the system of basic propositions formulated in (⁴), § 2. Fixing the transformation structure of the fields, we restrict—
* The system of basic propositions used in (⁴) arose from a system earlier proposed by N. N. Bogolyubov (⁵) within the framework of perturbation theory and the hypothesis of adiabatic switching on and off of the interaction.
we shall restrict ourselves, for simplicity, to the case of a single scalar field (the out-field \(\varphi(x)\)). The functional expansion (extended beyond the energy surface) of the scattering matrix in normal products of \(\varphi(x)\) we shall write in the form
\[ S=\sum_{\nu=0}^{\infty}\frac{(-i)^\nu}{\nu!}\int dx_1\ldots dx_\nu \Phi^\nu(x_1,\ldots,x_\nu):\varphi(x_1)\ldots \varphi(x_\nu):, \tag{1} \]
where \(\Phi^n(x_1,\ldots,x_n)\) are classical functions, symmetric with respect to their arguments; by \(n\)-fold variational differentiation we obtain for them the expressions
\[ \Phi^n(x_1,\ldots,x_n)=i^n\left\langle 0\left|\frac{\delta^{(n)}S}{\delta\varphi(x_1)\ldots \delta\varphi(x_n)}\right|0\right\rangle, \]
which it will be more convenient for us to rewrite, using the stability of the vacuum \(((4),\ \mathrm{I},\ (6))\), as
\[ \Phi^n(x_1,\ldots,x_n)=\left\langle 0\left|\frac{\delta^{(n)}S}{\delta\varphi(x_1)\ldots \delta\varphi(x_n)}S^+\right|0\right\rangle i^n. \tag{2} \]
By virtue of property \((4),\ \mathrm{II}\ (1)\), the \(\Phi^n(x_1,\ldots,x_n)\), with all arguments distinct, will be generalized functions integrable in one of the classes \(C(q,r)\). Consequently, for coincident arguments as well, their Fourier transforms \(\widetilde{\Phi}^n(p_1,\ldots,p_n)\), defined by the relation
\[ \int dx_1\ldots dx_n \exp\left(i\sum_{1}^{n} xp\right)\Phi^n(x_1,\ldots,x_n) =(2\pi)^4\delta(p_1+\ldots+p_n)\widetilde{\Phi}^n(p_1,\ldots,p_n) \tag{3} \]
(the \(\delta\)-function arises from taking translational invariance into account), would also be generalized functions, integrable in certain classes \(C(q',r')\), and therefore \(\widetilde{\Phi}(p)\) would be polynomially bounded as any one of the momenta tends to infinity. We shall impose on \(\widetilde{\Phi}(p)\) the weaker condition (satisfied in the usual theory) of polynomial boundedness under a uniform dilation of all momenta. Namely, we shall require that for every \(n\) there exist a finite growth index—the minimal integer \(\Omega(n)\)—such that under a dilation of all momenta
\[ p_1=\xi_1P,\ldots,p_n=\xi_nP,\qquad P\to\infty \tag{4} \]
the function \(\widetilde{\Phi}^n(\xi,P)\) grows more slowly than \(P^{\Omega(n)+\alpha}\) for any \(\alpha>0\).
In order to specify the theory concretely (which, in the usual approach, is done by specifying the interaction Lagrangian), one must specify the growth indices \(\Omega(n)\) for all \(n\). In particular, in a theory of the renormalizable type there must be no more than a finite number of functions \(\widetilde{\Phi}^n(p)\) with positive or zero index. It is also clear that the growth indices cannot be assigned completely arbitrarily; the question of admissible sets of growth indices requires special investigation.
- Let us now establish an infinite system of equations relating the functions \(\Phi^n(x)\) with different numbers of arguments. The basis for such a system will be the causality condition \(((4),\ \mathrm{II},\ (2))\):
\[ \frac{\delta}{\delta\varphi(x)} \left( \frac{\delta S}{\delta\varphi(y)}S^+ \right)=0 \quad \text{for } x\lesssim y. \tag{5} \]
By induction one can show that from it there follows the more general condition
\[ \frac{\delta}{\delta\varphi(x)} \left( \frac{\delta^{(n)}S}{\delta\varphi(y_1)\ldots\delta\varphi(y_n)}S^+ \right)=0 \quad \text{for } x\lesssim \text{all } \{y_1,\ldots,y_n\}. \tag{6} \]
from which one can prove by induction the operator identity
\[ \frac{\delta^{(n)} S}{\delta\varphi(x_1)\ldots\delta\varphi(x_n)}\,S^+ = \frac{\delta^s S}{\delta\varphi(x_{j_1})\ldots\delta\varphi(x_{j_s})}\,S^+ \frac{\delta^{(n-s)} S}{\delta\varphi(x_{j_{s+1}})\ldots\delta\varphi(x_{j_n})}\,S^+, \tag{7} \]
which holds whenever (for any \(1\leq s\leq n-1\))
\[ \{x_{j_1},\ldots,x_{j_s}\}\gtrless \{x_{j_{s+1}},\ldots,x_{j_n}\}. \tag{7a} \]
To pass here to the functions \(\Phi^n(x)\), let us take from (7) the vacuum average, expand the products of operators on the right-hand side in a complete system of functions* by means of the formula
\[ 1=\sum_{\nu=0}^{\infty}\frac{1}{\nu!}\int dk_1\ldots dk_\nu\, |k_1,\ldots,k_\nu\rangle\langle k_1,\ldots,k_\nu| \tag{8} \]
and transform the matrix elements that arise into vacuum averages, using property \((^4)\), II, (3). We then obtain
\[ \begin{aligned} \Phi^n(x_1,\ldots,x_n) &= \sum_{r=0}^{\infty}\sum_{\nu=0}^{\infty}\sum_{\mu=0}^{\infty} \frac{i^{r+\nu+\mu}}{r!\nu!\mu!} \int dz_1\ldots dz_r\,du'_1\ldots du'_\nu\,du_\nu\ldots du_\nu \\ &\quad\times \int dv_1\ldots dv_\mu\,dz'_1\ldots dz'_r\,dv'_1\ldots dv'_\mu\, \Phi^{s+r+\nu}(x_{j_1},\ldots,x_{j_s},z_1,\ldots,z_r,u_1,\ldots,u_\nu) \\ &\quad\times D^{(-)}(z_1-z'_1)\ldots D^{(-)}(z_r-z'_r)\, D^{(-)}(u_1-u'_1)\ldots D^{(-)}(u_\nu-u'_\nu) \\ &\quad\times \Phi^{*\nu+\mu}(u'_1,\ldots,u'_\nu,v_1,\ldots,v_\mu)\, D^{(-)}(v_1-v'_1)\ldots D^{(-)}(v_\mu-v'_\mu) \\ &\quad\times \Phi^{\,n-s+r+\mu}(z'_1,\ldots,z'_r,v'_1,\ldots,v'_\mu,\ldots,x_{j_{s+1}},\ldots,x_{j_n}), \tag{9} \end{aligned} \]
whenever condition (7a) is satisfied. We have arrived at an infinite system of equations which the functions \(\Phi^n(x)\) defining the scattering matrix must satisfy. One may think that, together with the unitarity condition \((^4)\), I, (5), which is written in terms of the functions \(\Phi\) in the form
\[ \begin{aligned} \Phi^n(x_1,\ldots,x_n)+(-1)^n\Phi^{*n}(x_1,\ldots,x_n) &= \delta_{n0} - \sum_{k=1}^{n-1}\sum_{s=0}^{\infty} \frac{(-1)^k(-1)^s}{s!} \\ &\quad\times P\!\left(\frac{x_1,\ldots,x_{n-k}}{x_{n-k+1},\ldots,x_n}\right) \int dz_1\ldots dz_s\,dz'_1\ldots dz'_s\, \Phi^{\,n-k+s}(x_1,\ldots,x_{n-k},z_1,\ldots,z_s) \\ &\quad\times D^{(-)}(z_1-z'_1)\ldots D^{(-)}(z_s-z'_s)\, \Phi^{*\,k+s}(z'_1,\ldots,z'_s,x_{n-k+1},\ldots,x_n), \tag{10} \end{aligned} \]
and with the assignment of growth indices, this system is sufficient for finding all \(\Phi^n(x)\). We shall show that this is so, at any rate, in perturbation theory—all the \(\Phi\)’s are found up to a finite number of constants.
- We shall assume that all the functions \(\Phi\) are expanded in series
\[ \Phi^n(x_1,\ldots,x_n)=\delta_{n0}+\sum_{m=1}^{\infty}\lambda^m\Phi_m^n(x_1,\ldots,x_n) \tag{11} \]
in powers of the parameter \(\lambda\), which accounts for the smallness of the interaction. Suppose that in this expansion the coefficients of \(\lambda^m\) satisfying (9), (10) have already been determined for all \(m<M\). We shall show that then one can always find functions \(\Phi_M^n(x_1,\ldots,x_n)\) satisfying (up to \(\lambda^M\)) equations (9) and conditions (10), and we shall establish the degree of the arbitrariness that thereby arises.
We shall seek the function \(\Phi_M^n(x_1,\ldots,x_n)\). If the set of its arguments can be divided into two groups satisfying (7a), then, extracting from (9) the terms of order \(M\), we obtain an expression of the sought function through functions with another number of arguments, but of lower order. Indeed, the expansion (11) of all \(\Phi^n\), except \(\Phi^0\), begins with the term of first
* We assume that in the theory under consideration there are no bound states and, consequently, that the states entering the sum in (8) form a complete system.
order (in the absence of interaction \(S=S^{+}=1\)). Therefore the terms of the \(M\)-th order on the right-hand side of (9) cannot contain functions \(\Phi\) of order higher than \(M-1\). Thus, the values \(\Phi_M^n\) for arguments admitting the partition (7a) are determined from the system (9) in terms of functions of lower orders that are already known, by assumption. It can be shown that the unitarity condition (10) is then satisfied automatically. It remains for us to determine only the values \(\Phi_M^n\) for arguments that do not admit the partition (7a).
But the set of arguments \(\{x_1,\ldots,x_n\}\) admits no partition (7a) only when all these arguments coincide. In other words, the system (9) determines \(\Phi_M^n\) from the prescribed \(\Phi_m^{n'}\), \(m<M\), up to a function different from zero only when all arguments coincide. Such a function must be a linear combination of \(\delta\)-functions and their derivatives. Therefore its Fourier transform will be a polynomial in \(p_1,\ldots,p_n\), symmetric in these variables. By virtue of the assumption on the degree of growth of the functions \(\widetilde{\Phi}^n(p)\), the degree of this polynomial cannot exceed the corresponding growth index \(\Omega(n)\), i.e., to specify it it is enough to fix a finite number of constants (the unitarity condition (10) will determine for each of these constants either the real or the imaginary part).
Since for a renormalizable theory one can assign at most a finite number of nonnegative growth indices (and for a negative growth index an arbitrary polynomial must be equal to zero), the total number of constants that must be specified in constructing the \(M\)-th approximation from the preceding ones will be finite. Finally, because in our assumptions the growth index does not depend on the order of approximation, one may, as usual in renormalization arguments, sum the constants that arise in each approximation, reducing the total number of constants needed to construct the theory to a finite number.
The last argument, connected with the hypothetical convergence of the series (11), can easily be avoided. Namely, instead of assigning the values of the constants that arise in each approximation, one may fix the values of the functions \(\widetilde{\Phi}^{\,n}(p)\) for those \(n\) to which nonnegative degrees of growth correspond, at as many points as there are coefficients in the corresponding polynomial, and then require that this normalization be fulfilled in every order of perturbation theory. This possibility will correspond to working with renormalized quantities in the usual approach.
For example, if one requires that for \(n=2\) the growth index be equal to 2, for \(n=3\) and \(n=4\) be equal to 0, and that the remaining growth indices be negative, then we arrive at the theory of a self-interacting scalar field having cubic and quartic interactions without derivatives. There will be four constants in the theory. Two of them (corresponding to \(n=2\)) we fix by requiring that there be no renormalizations of the mass and of the wave function. The other two (for \(n=3\) and \(n=4\)) will be specified if we fix the values of the “triple” and “quadruple” charges for certain fixed sets of momenta.
In conclusion I express my deep gratitude to N. N. Bogolyubov, to whom the initiative and idea of this investigation belong, and also to M. K. Polivanov for a valuable discussion.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Joint Institute for Nuclear Research
Received
6 VI 1960
CITED LITERATURE
- H. Lehmann, Nuovo Cim., 11, 342 (1954).
- H. Lehmann, K. Symanzik, W. Zimmermann, Nuovo Cim., 1, 205 (1955).
- W. Heisenberg, Zs. f. Phys., 120, 513, 673 (1943).
- N. N. Bogolyubov, B. V. Medvedev, M. K. Polivanov, Problems in the Theory of Dispersion Relations, Moscow, 1958.
- N. N. Bogolyubov, Izv. AN SSSR, ser. fiz., 19, 237 (1955).