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Yu. V. NOVOZHILOV
GRADIENT INVARIANCE AND THE AXIOMATIC APPROACH IN QUANTUM FIELD THEORY
(Presented by Academician V. A. Fock, 11 VI 1962)
1. Let \(\varphi_1, \ldots, \varphi_n\) be Hermitian operators describing quantum fields (both with integral and with half-integral spin), for which the quantum-mechanical axioms (I), the axiom of relativistic invariance (II), the axioms of positivity of energy (III) and uniqueness of the vacuum (IV) are satisfied. The electromagnetic field \(A_\mu\) is not included among the \(\varphi_j\). Let us add the condition that the theory possesses gradient invariance, i.e. invariance under the substitution
\[ A_\mu \to A_\mu + \partial f/\partial x_\mu,\qquad \varphi_j \to \varphi_j + i e q_j f\varphi_i, \tag{1} \]
where \(e q_j\) is the electric charge; \(q_j\) is a Hermitian antisymmetric matrix; the infinitesimally small operator \(f\) is an arbitrary function of \(x\).
Let us consider the question: is it possible, starting from axioms (I)—(IV), to obtain as a result of the transformation (1) an electric-charge operator different from zero? In other words: does the possibility of the existence of charged particles and the rule for constructing the charge operator follow solely from axioms (I)—(IV) and gradient invariance?
2. Let us turn to the Wightman functions \(W_\varphi\) \((^1)\), or the vacuum expectation values of products of field operators \(W_\varphi\{x\}=\langle 0|\varphi_1(x_1)\ldots \varphi_n(x_n)|0\rangle\), the complete set of which determines the state of the quantum field. The functions \(W_\varphi\{x\}\) are translationally invariant; let us choose the differences in the form \(\xi_j=x_{j+1}-x_j\); according to axioms (I), (III), and (IV), \(W_\varphi\{x\}\) is the boundary value of a function \(W_\varphi\{z\}\), \(z=x-iy\), analytic in the domain \(T_n\), consisting of points \(\zeta_1=z_2-z_1,\ldots,\zeta_{n-1}\), \(\zeta=\xi-i\eta\), \(\eta_0^2-\vec{\eta}^{\,2}>0\), \(\eta_0>0\). Taking axiom (II) into account, the domain of analyticity can be extended to the domain \(T'_n\), consisting of points \(\Lambda_c\zeta_1\ldots\Lambda_c\zeta_{n-1}\), where \(\Lambda_c\) is a complex proper Lorentz transformation which is the analytic continuation of a real proper Lorentz transformation, \(\det\Lambda_c=+1\); the functions \(W_\varphi\{z\}\) are invariant with respect to the transformations \(\Lambda_c\) \((^2)\).
3. Let us introduce the electromagnetic field \(A_\mu\). This field can be relativistically invariantly decomposed into two parts \(A_\mu^1+A_\mu^0\), where \(A_\mu^1\) describes a field with spin 1, and \(A_\mu^0\) a field with spin 0. The field \(A_\mu^1\) is not changed under the transformation (1); the field \(A_\mu^0\), which is not a dynamical variable and does not obey equations of motion, takes on the entire gradient addition \(\partial f/\partial x_\mu\).
The operators \(A_\mu^0\) act in a space \(\mathfrak{H}\) different from the Hilbert space of physical states. The arbitrariness of \(A_\mu^0\) and their nonphysical character are described, as is known, by introducing states \(|m\rangle\) with negative energy or an indefinite metric in the space \(\mathfrak{H}\) with matrix elements \(\langle m'|A_\mu^0|m\rangle\ne0\). In view of this, the vacuum expectations \(W_\varphi\{x,A^0\}\), containing the fields \(A_\mu^0\), cannot be continued into the complex domain \(T_n\) with respect to the coordinates \(\xi_A\) entering into \(\langle m'|A_\mu^0(\xi_A)|m\rangle\).
After continuation into the complex domain with respect to the other variables, the function \(W_\varphi\{z,z^*,A^0\}\) will depend on \(z^*\) by means of \(\xi_A=\zeta_A+\zeta_A^*\); \(\zeta_j=z_{j+1}-z_j\).
The question posed by us can now be formulated as follows: the function \(W_\varphi\{x,A\}\) is the sum of two parts \(W_\varphi\{x,A^0\}+W_\varphi\{x,A^1\}\), where \(W_\varphi\{x,A^1\}\) is the boundary value of the function \(W_\varphi\{z,A^1\}\), analytic in the domain \(T'_n\), while the function \(W_\varphi\{z,z^*,A^0\}\) depends not only on \(z\), but also on \(z^*\) by means of \(\xi_A=\zeta_A+\zeta_A^*\). Is this circumstance essential in order that one may assert invariance of the theory with respect to phase transformations leading to the appearance of the charge \(q_j\)?
Indeed, if \(f\) does not depend on the coordinates, then \(f\) may be regarded as a parameter, and the invariance of the theory with respect to the transformations \(1+ieQf\) leads to conservation of charge:
\[ W_{\varphi+\delta\varphi}\{x,A\}=W_\varphi\{x,A\};\qquad \sum q_j W_\varphi\{x,A\}=0. \tag{2} \]
- The analyticity of \(W_\varphi\{z\}\) as a function of the coordinates, by virtue of the translational invariance of the theory, may be expressed by the formula
\[ -i\frac{\partial}{\partial z_{j\mu}^*}\,W_\varphi\{z\}\equiv P_\mu^{-}(j)W_\varphi\{z\}=0,\qquad j=1,2,\ldots,n. \tag{3} \]
The functions \(W_\varphi\{z,z^*,A^0\}\) depend both on \(z\) and on \(z^*\); therefore one must also consider transformations that do not preserve the domains of analyticity. The generators of such transformations do not commute with \(P_\mu^{-}\); the generators of the analytic part of the complex Lorentz group (see item 2), however, commute with \(P_\mu^{-}\), but not with \(P_\mu^{+}=-i\,\partial/\partial z_\mu\). Thus, starting from the existence of \(A^0\), we have arrived at the conclusion that it is necessary to introduce the complex Lorentz group \(L_+(C)\).
Let us denote by \(M_{\mu\nu}\) that generator of the inhomogeneous group \(L_+(C)\) which, together with \(P_\lambda\), corresponds to the real subgroup: \([P_\lambda,M_{\mu\nu}]=i[g_{\lambda\mu}P_\nu-g_{\lambda\nu}P_\mu]\). The commutation relations for the inhomogeneous group \(L_+(C)\) split into two independent parts, containing the operators \(P_\mu^\pm\) and \(M_{\mu\nu}^{\pm}=\frac{1}{2}[M_{\mu\nu}\pm M'_{\mu\nu}]\):
\[ [P_\lambda^{\pm},M_{\mu\nu}^{\pm}] =i\,[g_{\lambda\mu}P_\nu^{\pm}-g_{\lambda\nu}P_\mu^{\pm}], \]
\[ [M_{\mu\nu}^{\pm},M_{\lambda\sigma}^{\pm}] =i\,[g_{\mu\lambda}M_{\nu\sigma}^{\pm} -g_{\nu\sigma}M_{\mu\lambda}^{\pm} -g_{\mu\sigma}M_{\nu\lambda}^{\pm} +g_{\nu\sigma}M_{\mu\lambda}^{\pm}], \tag{4} \]
where either the upper or the lower indices must be taken. The operators \(P_\lambda^{+}, M_{\mu\nu}^{+}\) commute with the operators \(P_\sigma^{-}, M_{\gamma\tau}^{-}\). The commutation relations (4) have the same form as in the case of the inhomogeneous real Lorentz group (the inhomogeneous group \(L_+(C)\), from the point of view of interest to us, is described in \((3)\)).
It follows from relations (4) that, when applied to the functions \(W_\varphi\{z\}\), analytic with respect to the variables \(z\) (condition (3)), the operators \(M_{\mu\nu}^{-}\) do not depend on the space-time rotations and displacements, which in this case are determined exclusively by the operators
\[ M_{\mu\nu}^{+}\ \text{and}\ P_\lambda^{+}:\qquad P_\lambda^{-}\simeq 0,\ [P_\lambda,M_{\mu\nu}^{-}]\simeq 0, \]
\[ [P_\lambda,M_{\mu\nu}^{+}]\simeq ig_{\lambda\mu}P_\nu-ig_{\lambda\nu}P_\mu. \tag{5} \]
The sign \(\simeq\) in (5) means that the equality has meaning for functions satisfying condition (3). In this case the generators \(M_{\mu\nu}^{-}\) determine an additional linear transformation of the components \(W_\varphi\{z\}\). The necessity of considering these generators is a consequence of the existence of the nonphysical field \(A_\mu^0\).
- Let us list the properties of the vacuum functions, using the group \(L_+(C)\). Let \(W_\varphi(x)\) refer only to physical fields and not contain \(A^0\). By
axiom (II), \(W_\varphi\{x\}\) is invariant with respect to real proper Lorentz transformations \(L_+\); the spectral axioms (III), (IV) make it possible to analytically continue \(W_\varphi\{x\}\) into the domain \(T_n\), with \(W_\varphi\{x\}\) being the boundary value of \(W_\varphi\{z\}\); since the function \(W_\varphi\{z\}\) is Lorentz-invariant, by virtue of the analyticity of the coefficients of the transformation \(L_+\) it follows that \(W_\varphi\{z\}\) is invariant with respect to transformations with \(M_{\mu\nu}^{+}\) and \(P_\lambda^{+}\):
\[ \sum_j M_{\mu\nu}^{+}(j)\,W_\varphi\{z\}=0;\qquad \sum_j P_\lambda^{+}(j)\,W_\varphi\{z\}=0, \tag{6} \]
thereby the domain of analyticity is enlarged to \(T'_n\), and among the points of \(T'_n\) there are also real points (Jost points).
In other words, under space-time transformations the vacuum expectations transform according to \(M^{+}\)-representations of the complex group \(L_+(C)\), i.e., physical fields in vacuum expectations behave as \(M^{+}\)-spinors, \(M^{+}\)-vectors, etc.
Let us turn to the possible transformations \(W_\varphi\{z,A\}\). Since the coordinates \(x_A\) in \(W_\varphi\{z,A^0\}\), arising from the field \(A_\mu^0\), can only be real, \(x_A=z_A+z_A^*\), this function can be invariant only with respect to transformations of the real subgroup \(L_+(C)\), with operators \(P_\lambda, M_{\mu\nu}\); \(A_\sigma^0\) is a 4-vector in this subgroup (an “\(M\)-vector”):
\[ D_A=1+i\omega^{\mu\nu}M_{\mu\nu}. \tag{7} \]
The application of \(D_A\) to \(W_\varphi\) gives, by virtue of (6),
\[ D_A W_\varphi\{z\} = \left\{1+i\sum_j M_{\mu\nu}\omega^{\mu\nu}\right\}W_\varphi\{z\}=0, \tag{8} \]
i.e., invariance with respect to the transformation \(1+iM_{\mu\nu}\omega^{\mu\nu}\), containing only \(M_{\mu\nu}\); the generators \(M_{jk}\), \(j,k=1,2,3\), produce in this case a phase transformation.
- Gradient invariance of the theory means, in particular, that \(W_\varphi\{x,A^1\}\) must have the same transformation properties as \(W_\varphi\{x,A^0\}\) (for real coordinates). Choosing as \(\xi_1,\ldots\) the Jost points \(\rho_1,\ldots\), we obtain that invariance with respect to a transformation of type (7) is preserved for \(W_\varphi\{z,A^1\}\) also in the domain \(T_n\). Thus the field \(A_\mu^1\) must be simultaneously an \(M\)-vector and an \(M^{+}\)-vector.
In the inhomogeneous Lorentz group with generators \(M_{\mu\nu}, P_\lambda\), the invariants \(W^2=W_\mu W^\mu\), where \(W_\mu=\frac12\varepsilon^{\mu\lambda\nu\sigma}M_{\lambda\nu}P_\sigma\), and \(W^\lambda n_\lambda\), where the 4-vector \(n_\lambda\) is spacelike, are equal to zero for the field \(A_\mu^0\), since the spin of \(A_\mu^0\) is zero. Assuming the transformation properties of \(A_\mu^0\) and \(A_\mu^1\) to be identical (with respect to the real group \(P_\lambda, M_{\mu\nu}\)), we find that for \(A_\mu^1\) also \(W^2=0\) and \(W^\lambda n_\lambda=0\). But the spin properties of the field \(A_\mu^1\) are determined by the group \(M_{\mu\nu}^{+}, P_\lambda\) with invariants \(W_\lambda^{+}n^\lambda\), \((W^{+})^2\), where \(W^{+\mu}=\frac12\varepsilon^{\mu\lambda\nu\sigma}M_{\lambda\nu}^{+}P_\sigma\), and here \(W_\lambda^{+}n^\lambda\) is not equal to zero. Hence it follows that \(M_{\mu\nu}^{-}\) cannot vanish for \(A_\mu^1\) for all \(\mu,\nu\), i.e. \(A_\mu^1\) is not a scalar with respect to transformations with \(M_{\mu\nu}^{-}\).
The exact form of the transformation \(D_A^0\) for \(A_\mu^1\) as an \(M\)-vector and an \(M^{+}\)-vector may also depend on the number of independent components of \(A_\mu^1\); the minimal symmetry corresponds to the fields \(A_{\mu\, in,\,out}^1\) (two independent components). In this case one can choose such a coordinate system where there are only \(A_1^1\) and \(A_2^1\), and then only the generators \(M_{12}, M_{03}\) will enter into \(D_A^0\). But the quantities \(M_{\lambda\nu}^{-}\) as applied to \(W_\varphi\{z,A^1\}\) are invariant with respect to the \(M^{+}\)-group, i.e., with respect to the Lorentz group for the \(M^{+}\)-vector \(A_\mu^1\). Consequently,
\[ D_A^0=1+i\omega^{\mu\nu}M_{\mu\nu}^{+}+iM_{12}^{-}\alpha+M_{03}^{-}\beta \tag{9} \]
with real \(\alpha,\ \beta\) and Hermitian \(M_{\overline{12}}^{-},\ M_{\overline{03}}^{-}\); moreover, according to (8), \(M_{\overline{12}}^{-}\) describes a phase transformation. Since, according to (2), the electric charge is associated precisely with such a transformation, the charge operator is
\[ Q=M_{\overline{12}}^{-}+\text{const.} \tag{10} \]
Thus, the charge operator is the operator of rotation about the “3” axis in the space \(T\), which we shall call isospin space. Since under a phase transformation the field \(A'_{\mu}\) does not change, the field \(A^1_{\mu}\) is the projection of the pseudovector onto the “3” axis in isospin space: \(A^1_{\mu}\sim M_{\overline{12}}^{1}\).
For the interacting field \(A^1_{\mu}\) the number of independent components is increased by one, associated with the Coulomb interaction. Taking \(A^1_3=0\) in some coordinate system, we obtain invariance with respect to the transformation \(1+iM_{\overline{12}}\alpha+M_{\overline{01}}\beta_1+M_{\overline{02}}\beta_2\) with real \(\alpha,\beta_1,\beta_2\) and Hermitian \(M_{\overline{12}},\ M_{\overline{01}},\ M_{\overline{02}}\). As before, only \(M_{\overline{12}}\) leads to a phase transformation in accordance with condition (9) for \(A^1_{\mu}\) as an \(M^{-}\)- and \(M^{+}\)-vector.
7. Suppose that the mass of particles associated with the vector field \(B_{\mu}\) is nonzero, but the theory is gradient invariant. The electromagnetic field is not considered.
Put \(B_{\mu}=B^1_{\mu}+B^0_{\mu}\), where \(B^1_{\mu}\) belongs to a physical field with spin 1, while the operator \(B^0_{\mu}\) is not a dynamical variable and corresponds to a field with spin 0. Following the same path as in the case of \(A_{\mu}\), we arrive at the conclusion that the class of possible transformations \(D_B\) coincides with \(D_A\); formula (9) also remains unchanged, since the field \(B^1_{\mu}\) must simultaneously be an \(M^{+}\)-vector and an \(M\)-vector and, owing to \(W^2=0\) for \(B^0_{\mu}\), cannot be a scalar with respect to \(M_{\mu\nu}\)-transformations. Suppose that in the case of the field \(B^1_{\mu}\) there are three independent components in isospace \(B^1_{\mu 1},\ B^2_{\mu 2},\ B^3_{\mu 3}\), which (by analogy with \(A^1_{\mu}\)) transform respectively as \(M_{\overline{23}}^{-},\ M_{\overline{13}}^{-},\ M_{\overline{12}}^{-}\). Then \(D^0_B\) will contain quantities \(M_{\overline{jk}}^{-}\), \(j,k=1,2,3\), invariant (with respect to transformations of the subgroup \(M^{+}\)):
\[ D_B^0=1+iM_{\mu\nu}^{+}\omega^{\mu\nu}+iM_{\overline{jk}}^{-}\alpha_{jk}; \tag{11} \]
where \(\alpha_{jk}\) are real, and \(M_{\overline{jk}}^{-}\) are Hermitian. The additional group \(M_{\overline{jk}}^{-}\) is the group of rotations in three-dimensional isospin space with invariants \(T^2,\ T_3\). The fields interacting with \(B_{\mu}\) can be classified according to the values of isospin \(T\) and the projection \(T_3=M_{\overline{12}}^{-}\). For \(T_3=Q\) only the values \(T=0,1\) are possible.
The field \(B_{\mu}\) possesses isospin (and charge), whereas the electromagnetic field carries no charge. The fields \(B_{\mu in,out}\) can be only isovector fields, which follows from \(W^2=0\) and \((W^{+})^2=0\) for these fields.
In conclusion, I express my sincere gratitude to Acad. V. A. Fock for interesting and useful discussions.
Leningrad State University
named after A. A. Zhdanov
Received
5 VI 1962
CITED LITERATURE
- A. S. Wightman, Phys. Rev., 101, 860 (1956).
- S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, N. Y., 1961.
- Yu. V. Novozhilov, Vestn. LGU, No. 4, 5 (1962).