Reports of the Academy of Sciences of the USSR
1962. Volume 147, No. 3

PHYSICS

Yu. V. NOVOZHILOV

THE GROUP OF FOUR-DIMENSIONAL ISOSPIN AND HYPERCHARGE IN AXIOMATIC QUANTUM FIELD THEORY

(Presented by Academician V. A. Fock, 23 VI 1962)

Let us suppose that spatial parity is conserved. Let us consider what symmetry group then exists (besides the group of real Lorentz transformations) in axiomatic quantum field theory, based on the quantum-mechanical axioms (I), the axiom of relativistic invariance (II), the axioms of positivity of energy (III), and uniqueness of the vacuum (IV).

1. Since the Wightman functions (W_\varphi{x}), containing physical fields (\varphi_1,\ldots,\varphi_n), must transform under space-time rotations according to (M^+)-representations of the complex proper Lorentz group (L_+(C)) ((^1)), the operator of reflection of spatial coordinates (P_L) has the properties ((j,k=1,2,3))

[
[P_L,M_{jk}^{+}]=0,\qquad [P_L,M_{\mu\nu}^{-}]=0,\qquad
P_LM_{0j}^{+}=-M_{0j}^{+}P_L
\tag{1}
]

(the notation is the same as in ((^1))). In consequence of (1), under the reflection (P_L) the generator (M_{1j}^{+}+M_{0j}^{-}), which produces purely real transformations, goes over into the generator (M_{0j}^{-}-M_{0j}^{+}) for imaginary transformations of the group (L_+(C)). In view of the restrictions imposed by gradient invariance ((^1)), the reflection (P_L) is impossible in a gradient-invariant theory.

Let us introduce the reflection operator (P_T) in isospin space, with the properties

[
[P_T,M_{jk}^{-}]=0,\qquad [P_T,M_{\mu\nu}^{+}]=0,\qquad
P_TM_{0j}^{-}=-M_{0j}^{-}P_T .
\tag{2}
]

The operation (P=P_LP_T) (“generalized reflection”) preserves the reality of the coordinates and is compatible with gradient invariance.

Thus, in gradient-invariant theories only the generalized reflection has meaning; conservation of spatial parity (P_L) also implies conservation of isospin parity (P_T). In the absence of unphysical fields (A_\mu^0) and (B_\mu^0), one may consider (P_T) and (P_L) separately (although conservation of (P_T) is still a consequence of conservation of (P_L)). Since (P_LA_\mu^1P_L^{-1}=-A_\mu^1) and (PA_\mu^0P^{-1}=-A_\mu^0), it follows that (P_TA_\mu^1P^{-1}=A_\mu^1), i.e. (A_\mu^1) is an isopseudovector.

1. The group of internal symmetry must possess a Euclidean metric, since otherwise either probabilities will not be invariant with respect to transformations of the group, or, with invariant transition amplitudes, positivity of transition probabilities is lost.

Transformations of the complex group (L_+(C)) generated by (M_{\mu\nu}^{-}) are not connected with the space-time properties of physical fields: in application to (W_\varphi{z}), ([M_{\mu\nu}^{-},P_\lambda]=0). But the invariants of the (M_{\mu\nu}^{-})-transformations have a pseudo-Euclidean metric. Only some of the transformations, namely the subgroup of transformations with the operators (M_{jk}^{-}), (j,k=1,2,3), are

unitary. This subgroup is the group of rotations in the three-dimensional isospin space (T_j=e_{jik}M_{ik}^{-}).

If spatial parity is conserved, an isospin reflection (P_T) is added to the isospin rotations. The resulting symmetry group with conserved (T^2), (T_3), (P_T) is equivalent to the phenomenological symmetry group proposed by d’Espagnat and Prentki ((^2)). Since, however, the functions (W_\varphi{x}) are, in addition, invariant with respect to Lorentz-like rotations with (M_{0k}^{-}), the isospinor field (u_\alpha) must be described not by a two-component isospinor ((^2)), but by a bispinor. In the case of a complex (iso-)bispinor, as is known, we automatically obtain conservation of the number of isofermions.

If (\Gamma_\mu) is an analogue of the Dirac matrices in isospin space and (\Gamma_\mu\Gamma_\nu=i\Sigma_{\mu\nu}), then an (M_{jk}^{-})-transformation for the bispinor (u_\alpha) means the substitution (u\to\left(1+\frac{1}{2}\Sigma_{jk}\omega_{jk}\right)u), while the reflection (P_T) means the substitution (u\to i\Gamma_4u). Invariance with respect to reflections of the function (W_\varphi), containing (n) isofermions, as well as isoscalars and isopseudovectors and containing no antiparticles, is expressed by the formula (\Sigma\Gamma_4(m)W_\varphi=0), where (m=1,\ldots,n), which corresponds to conservation of the difference between the number of isofermions and anti-isofermions, i.e. to conservation of hypercharge (Y). Consequently, (\Gamma_4) characterizes the hypercharge (Y) for particles. If (\Gamma_4) and (\Sigma_{12}) are diagonal, then for (u_1) and (u_2) one has (Y=+1), and for (u_3) and (u_4) one has (Y=-1); (T_3=1/2) for (u_1,u_3); (T_3=-1/2) for (u_2,u_4); for antiparticles the signs of (T_3) and (Y) are opposite.

The expression for the hypercharge of quantized fields is
[
\hat Y_\varphi=-i\int\left[(\pi\Gamma_4\varphi)-(\pi^+\Gamma_4\varphi^+)\right]\,dV
]
for a scalar (\varphi),
[
\hat Y_\psi=\frac{i}{2}\int[\bar\psi,\gamma_4\Gamma_4\psi]\,dV,\qquad \bar\psi=\psi^+\gamma_4,
]
for fermions (\psi).

1. Let us determine the conditions under which transformations with (M_{0j}^{-}) can also characterize an “internal” symmetry. To this end let us consider the transition from the pseudo-Euclidean metric of the group (M_{\mu\nu}^{-}) to a Euclidean one; here we shall use Schwinger’s results on Euclidean quantum electrodynamics ((^3)).

Suppose that Hermitian fields (\psi_1,\ldots,\psi_n) belong to an internal group with a Euclidean metric. Denote the Hermitian operators of the isospin group of rotations in four-dimensional space by (T_{ab}); (a,b=1,2,3,4),
[
[T_{ab},T_{cd}]=i[\delta_{ac}T_{bd}-\delta_{ad}T_{bc}-\delta_{bc}T_{ad}+\delta_{bd}T_{ac}].
\tag{3}
]
The matrices (T_{ab}=-T_{ba}) are antisymmetric. Put, for (j,k=1,2,3), (M_{jk}^{-}=T_{jk}). Since the matrix (M_{0j}^{-}) is symmetric, the transition from (iM_{0j}^{-}) to (T_{4j}) also includes a change of symmetry.

For integral isospin (T) the reflection operator (P_T) is Hermitian and symmetric; therefore
[
T_{4k}=i\exp\frac{i\pi}{4}P_T\cdot M_{2k}^{-}\exp\left(-\frac{i\pi}{4}P_T\right)=-P_TM_{0k}^{-}.
\tag{4}
]
(The operator (M_{0k}^{-}) is anti-Hermitian with respect to states constructed with the aid of the fields (\psi).)

In the case of isospin (T=1/2), the Hermitian isospinor (\chi) in four-dimensional Euclidean space has 8 components; the Hermitian isospinor (u) in pseudo-Euclidean space (the “rotation” operators (M_{\mu\nu}^{-})) has only 4 components. Therefore the transition to the Euclidean “internal” metric is possible if the isospinor field in the pseudo-Euclidean metric is complex; the additional degree of freedom then corresponds to anti-isofermions.

For the isospinor (\chi),
[
T_{ab}=\frac{1}{4i}[\alpha_a,\alpha_b],
]
where (\alpha_a) are symmetric anticommuting (8\times 8) matrices; the isospin-reflection operator (P_T) is ...

(P_T = i\alpha_4). The relation of (\alpha_4) to the reflection operator (i\Gamma'4) for (u) ((\Gamma'_4) is (\Gamma_4) in the Majorana representation) is given by the formula (\alpha_4 = \lambda_2\Gamma'_4), where (\lambda_2) is a two-row antisymmetric Hermitian matrix acting in the space of the two pseudo-Euclidean isospinors (u_I) and (u) and describing the isofermion number.

The transition from (M^-{0j}) to (T) is effected by means of the transformation

[
T_{4j}=-i\exp\frac{i\pi}{4}\alpha_4\cdot M^-{0j}\exp\left(-\frac{i\pi}{4}\alpha_4\right)=-\alpha_4 M^-.
\tag{5}
]

Thus, the transition from the three-dimensional isospin group to the four-dimensional isospin group (T_{ab}) is possible only with conservation of isospin parity, which, according to Sec. 1, is equivalent to conservation of spatial parity.

Representations of the 4-isospin group are labeled by the eigenvalues of the operators ((T^\pm)^2, T^\pm_3)

[
T^\pm_3=\frac{1}{2}\,[T_{12}\pm T_{34}]\quad\text{and so on.}
\tag{6}
]

An essential feature of the group (3) obtained is that four-dimensional symmetry can be realized only in states with hypercharge equal to zero, since the parity (P_T) anticommutes with (T_{j4}):

[
P_TT_{j4}+T_{j4}P_T=0.
\tag{7}
]

1. The expression obtained earlier for the electric charge remains valid in the case of isobosons. In the case of isospinor representations, however, there are two possibilities: either the charge (Q) has another relation to the isospin (T), or isofermions do not exist.

The electric charge is determined by such a phase transformation (D_Q) when (for invariant (W_\varphi{x,A^1})) (A^1_\mu) does not change (being the “3” component of an isopseudovector, or, like (T_3), the “12” component of an antisymmetric tensor (1)). One of such transformations may be (\exp(iM^-{12}\alpha)=D^0_A). In the case of isofermions, however, there is an additional phase transformation (D_Y=1+iY\beta) with real (\beta), where (Y) is the hypercharge, which commutes with (T_3=\frac{1}{2}\SigmaY\right)) or else (P_T\exp i\pi T_3) for isofermions. In view of this we set, similarly to (2),}) and is an isoscalar. The field (A^1_\mu) will be invariant with respect to (D^0_A\cdot D_Y), in particular, (A^1_\mu) is invariant with respect to the transformation (\exp i\left(T_3+\frac{\pi}{2

[
Q=T_3+\frac{1}{2}Y.
\tag{8}
]

This formula is therefore not only a consequence of relativistic and gauge invariance, but also a consequence of the possibility of passing from the pseudo-Euclidean “internal” space to a Euclidean one.

1. Thus, with conservation of spatial parity (P_L), the axiomatic theory contains an additional group of four-dimensional Euclidean rotations (T_{ab}) with a “spatial” isospinor reflection, which determines the hypercharge.

In the absence of the electromagnetic field (and of vector mesons associated with gauge invariance), the symmetry properties may be “maximal” with respect to what can be given by the axioms of quantum field theory together with the condition of conservation of spatial parity; in this case states can be characterized by the quantum numbers of isospin (T^2=T^2_{12}+T^2_{23}+T^2_{31}, T_3), and by hypercharge (parity), but not by four-dimensional symmetry in the general case, since (T_{34}P_T=-P_TT_{34}).

Let us compare our results with the conclusions of phenomenological theories. The set of quantum numbers (T, T_3, Y) coincides with the set in the theory of d’Espagnat and Prentki ((^2)). In that theory, however, there is no hidden four-dimensional symmetry obtained by us, which manifests itself when the hypercharge is equal to zero (the quantum numbers (T^{\pm}, T_3^{\pm})).

The isospin group of the axiomatic theory also does not coincide with the group proposed by Salam and Polkinghorne ((^4)), where there is no reflection, the charge is identified with (T_{12}), and the three-dimensional isospin with (T^{+}).

The relations (3), (7), (8), which determine the symmetry in the axiomatic theory, were postulated in the phenomenological group of A. M. Baldin and A. A. Komar ((^5)).

Leningrad State University
named after A. A. Zhdanov

Received
14 VI 1962

CITED LITERATURE

(^1) Yu. V. Novozhilov, DAN, 147, No. 1 (1962).
(^2) B. d’Espagnat, J. Prentki, Nucl. Phys., 1, 33 (1956).
(^3) J. Schwinger, Phys. Rev., 115, 721 (1959).
(^4) A. Salam, J. C. Polkinghorne, Nuovo Cim., 11, 685 (1955).
(^5) A. M. Baldin, A. A. Komar, Preprint of the Joint Institute for Nuclear Research, D-976, 1962.