Raymond W. Hayward Institute for Basic Standards, National Bureau of Standards, Washington, D. C. 20234 There are several difficulties that plague all existing relativistic equations of motion describing elementary fields having an intrinsic spin greater than one. While the free field equations can be shown to be explicitly covariant, the introduction of interactions gives rise to a phenomenon of noncausality. In the presence of interactions, the retarded solutions spread beyond the light cone and the influence travels faster than light. Furthermore, the solutions in certain simple potentials do not have a finite norm, violating the probabilistic requirements of quantum mechanics. This paper develops a relativistic theory that is free of the aforementioned difficulties. This Lagrangian theory describes fields and particles with arbitrary mass and charge and having any discrete spin, integer or half integer. Apart from gauge conditions there are no subsidiary conditions. A matrix formulation is used. The generators of the inhomogeneous Lorentz group for a field of any intrinsic spin and mass are defined in terms of Wigner operators of the group SU(2) and a metric operator. A maximal Abelian set of invariants is formed which defines two completely reducible representation bases of the inhomogeneous Lorentz group having distinct structures. A set of y matrices, obeying a Clifford algebra, is also defined in terms of the Wigner operators and the metric operator. State vectors having different structures and Lorentz transformation properties can be related to one another by operators involving they matrices. The equations of motion can be obtained from the Lagrangian by variational methods, and certain aspects of the canonical formalism can be used to quantize the fields. Invariance of the Lagrangian under infinitesimal displacements and rotations yield conservation laws and constants of the motion for pertinent physical observables. The metric of the Hilbert space of the states is uniquely defined for any spin field, assuring positive definite four momenta and charge. The Dirac formulation for the spin one-half field and the Maxwell-Lorentz formulation for the electromagnetic field are special cases of this theory. Key words: Causality; high spin fields; inhomogeneous Lorentz group; relativistic fields; wave equations. I. Introduction Elementary particles of finite mass and with spin greater than one half have assumed a role of increasing importance in physics in recent years. It is ironical, however, that no "true" dynamical theory of such particles has emerged that enjoys the privilege of existing as an entity like the Dirac theory of the spin one-half particle. There are a host of composite theories that are explicitly covariant but possess certain intrinsic difficulties in application. These theories make explicit or implicit use of the invariance properties of the inhomogeneous Lorentz group and often of the one-to-two homomorphism between the homogeneous Lorentz group and the group SL(2c). There have been proposed two classes of theories that yield first order equations of motion for the particles. The first of these is the Dirac-Fierz-Pauli type [1-3]1 involving the usual four-dimensional Dirac matrices obeying a Clifford algebra. A set of Dirac matrices is used for each constituent of the composite state vector and each set is orthogonal to all other sets. Specifically, for example, the formulation of Bargmann and Wigner [4] contains extraneous components that must be constrained or eliminated to yield the required number of degrees of freedom. The prescription for accomplishing this feat is not always unique, but depends rather on the choice of irreducible combinations that are to be considered on the basis of a preconceived model. Because of these auxiliary conditions the Bargmann-Wigner equations are somewhat intractable, and when interactions are present they become both formidable and questionable. In addition there are some difficulties in formulating a Lagrangian theory. The requirement that all differential equations of motion result from a variational principle involving an action integral requires the introduction of auxiliary fields into the Lagrangian. Only with the requirement that these auxiliary fields vanish in the absence of interactions is an explicit Lagrangian obtained. Fierz and Pauli [3] long ago recognized the difficulties arising from inconsistencies in this formulation. 1 Figures in brackets indicate the literature references at the end of this paper. equivalently described by the Klein-Gordon, Dirac and Proca equations, respectively. Free particles having spins greater than one half are described only by a set of coupled differential first order equations [5]. The number of equations may be reduced, however, by application of conditions for a lower spin field. For example, a spin 3/2 particle may be described by a set of three coupled equations involving a three-spinor. Conditions applicable to a spin-one field allow this three-spinor to be described in terms of a vector-spinor and an antisymmetrical tensor-spinor. This formalism proposed by Rarita and Schwinger [6] allows the set of three coupled equations to be reduced to two. A second class of theories are characterized as the Bhabba-type [7]. In these, attempts are made to avoid the auxiliary conditions by inclusion of only those irreducible composites that are necessary to maintain Lorentz covariance together with the requirement that all components of the state vector obey a second order wave equation involving the same mass. This procedure requires matrices of higher dimension than the Dirac matrices. These matrices obey a rather complicated algebra. For particles of spin 0 and 1 these matrices are of dimension five and ten, respectively, and obey the Duffin-KemmerPetiau algebra [8]. The Bhabba-type equations of motion may be readily obtained from a Lagrangian, however the representation of the Lagrangian is by no means unique. In fact, the Lagrangian used by Kemmer might be questionable from the point of view of having an excessive number of powers of the four-momenta needed to describe the fields. In any event, for spins greater than one there are profound difficulties in a theory with interactions. Because of the attractiveness of the possibility for the elimination of subsidiary conditions there is still lively attention [9] being given to the first order Bhabba-type theories. The problem remains to get a theory with interactions that describes a particle of unique mass and charge without subsidiary conditions and without extraneous components. Both classes suffer from an inherent nonuniqueness in the selection of irreducible combinations required to express the dynamics of a field of a particular spin. Theories that lead to second order equations of motion, apart from the familiar Proca [10] and Stuekelberg [11] formalisms for spin one fields, have received less attention. These are usually tensor formulations [12] and, like the first order theories, usually have supplementary conditions invoked to limit the number of degrees of freedom. These theories also become formidable when interactions are introduced. Velo and Zwanziger [13] have looked into the origin of some of the difficulties. Wave propagation is usually associated with hyperbolic systems of partial differential equations. Such equations allow an initial-value problem to be posed on a class of surfaces, called "space-like" with respect to the equations, and they possess solutions with wave fronts that travel along rays at finite velocities. The rays through any point form a ray cone that is entirely determined by the coefficients of the highest derivatives. Thus for hyperbolic systems when coupling occurs only in lower derivatives, the ray cone is the same in the interacting and free case. The free Klein-Gordon and Dirac equations are familiar examples of hyperbolic systems, and so, when they are coupled through lower order derivatives, the ray cone remains the light cone. On the other hand, for spins greater than one half, the free Lagrangian equations are not hyperbolic but constitute instead a degenerate system because they imply constraints. However, it may be shown that they are equivalent to a system of hyperbolic equations which describe the wave propagation, supplemented by constraints that are conserved in time. But, if any low or nonderivative coupling is added to the free higher-spin Lagrangian, the resulting equations do not remain equivalent to a hyperbolic system with the light cone as the ray cone supplemented by the same number of constraints. There is at present no known example of a satisfactory equation with interaction for spin greater than one. The case of spin one is marginal; some interactions appear to lead to satisfactory equations, but others are unacceptable. Similar doubts were expressed long ago by J. W. Weinberg [14]. Note that the requirements of special relativity are not automatically satisfied by equations that transform covariantly. This paper will develop a relativistic theory of higher-spin fields employing the variational methods of classical Lagrangian field theory. The chief aim is to present an unambiguous method for constructing a dynamical description of a field having any discrete spin, integer or half integer and including zero, while, at the same time, will have no supplementary conditions or inherent nonuniqueness except those resulting from internal symmetries arising from variation of the Lagrangian. One of the main considerations in elaborating relativistic quantum mechanics for particles comes from the fact that the law of conservation of the number of particles ceases, in general, to be true. To produce a complete theory we must encompass in a single scheme dynamical states specified, not only by the quantum state, but also by the number and the nature of the elementary particles of which they are composed. This requires that the theory, to be useful, must be quantizable. If it can be a Lagrangian type theory where the usual canonical methods apply, so much the better, for then the quantization methods may be straighforward. Another consideration in relativistic theories is concerned with positive definiteness of the proba bility density and the energy. In the early days of quantum mechanics, there was a belief that any effective theory describing a particle with spin should yield a first order equation of motion in such a way that the probability density should be positive definite at all times. Viewed in hindsight, the arguments for the superiority of first order equations over second order equations interpreted as single particle theories are not convincing. Of course, at the time the Dirac equation had the obvious advantage that it described most experimental facts involving spin one-half particles. A quantized theory removes these apparent differences between first and second order equations of motion, however neither is exempt from all difficulties, or even of contradictions. One of the characteristics of any quantized relativistic theory describing particles of spin one or greater is the appearance of a negative metric for some of the components of the Hilbert space [15]. This leads to a negative probability of a different sort than that previously discussed. Dirac suggested that, in a relativistic quantum theory, use should be made of an indefinite metric so that the negative probabilities could be eliminated in order to maintain the probabilistic interpretation of the formalism of quantum theory. A more fundamental reason for the indefinite metric may be seen in the fact that, disregarding the translations, the Lorentz group is a noncompact group, since there is no transformation that corresponds to the limiting velocity, c. Any finite representation of a non-compact group requires a space with an indefinite metric, although infinite representations with a definite metric may exist [16]. Although much of the contents of this paper may be based on arguments of group theory which could be used extensively to make the exposition more economical, much of this paper is written in terms of conventional matrix theory. The use of matrix theory in specific representations makes the physical arguments clearer and closer to the import of Dirac theory and also makes them more easily understood by people not well versed in spinor calculus and abstract algebra. In particular, we shall develop specific matrix representations of the inhomogeneous Lorentz group with the homogeneous subgroup employing the nonunitary four dimensional orthogonal group O(4) in a complex Minkowski space in which x = (x,it) where three of the six generators are regarded as specific representations of the three dimensional rotations about the spatial axes and the remaining three generators are specific representations of Lorentz transformations (boosts) along the three spatial axes. The use of O(4) in a Minkowski metric allows all matrix operators in spin space to be hermitian in our treatment2. The plan of the paper is as follows: In section 2 we establish and summarize the main features of the inhomogeneous Lorentz group. We introduce a set of invariants that supplement those customarily employed in order to obtain a set of commuting operators whose expectation values will completely characterize a physical state of a particular spin and mass. In section 3 we first develop certain matrix representations of hermitian operators in a subspace of the Hilbert space of the physical states in terms of Wigner operators of the rotation group SU(2) = 2 Alternatively, we could have used in our formalism the group 0(3,1) to describe the homogeneous Lorentz subgroup in a real space in which x" (t,x). It should be pointed out that the apparently trivial differences in sign in the real four-dimensional Euclidian group O(4) and the group 0(3,1) result in very important differences between the respective covering groups, SU (2) ✪ SU(2) which is homomorphic to real O(4) and SL(2c) which is homomorphic to 0(3,1). On the other hand, the group O(4) with the Minkowski metric has SL(2c) as its covering group, the same covering group as for O(3,1). There is no fundamental reason in special relativity, apart from personal taste, to choose 0(3,1) with a real metric over O(4) with a Minkowski metric. The latter is that which is used in the original Dirac-Pauli description of spin one-half fields. of y matrices obeying a Clifford algebra, and a set of spin operators S, that form a six parameter Lie algebra. We establish Lorentz transformation properties of the physical states with arbitrary spin using the generators of the inhomogeneous Lorentz group. In section 4 we establish a uniform Lagrangian formalism for classical fields of any spin and mass. From this Lagrangian formalism, the equations of motion are obtained and conservation laws established by means of variational methods and Noether's theorem. In section 5 we obtain the plane-wave solutions for fields having several specific spin values and determine the completeness and orthogonality properties of these solutions. The Hamiltonian formalism is developed and the conditions for quantization of these fields is established in the Heisenberg picture. In section 6 we subject the quantized fields to the discrete symmetries of space inversion, time reversal, and charge conjugation and find the transformation properties of these fields and bilinear combinations thereof. Furthermore we develop the commutation relations between the discrete symmetry operators and the generators of the inhomogeneous Lorentz group. 2. Lorentz Transformations Of the many important invariance principles of relativistic quantum mechanics, the most fundamental is that which arises from the group of transformations denoted as the inhomogeneous Lorentz group. In our canonical approach, it is necessary to consider the transformation properties of the Hilbert space of the physical states, as well as those of the coordinates, in order to discuss the symmetries of the Langrangian and the equations of motion obtained therefrom. The properties of the inhomogeneous group and its subgroups have been established and understood for many years since the classic paper of Wigner [17]. This understanding, however, has not led to a totally satisfactory dynamical theory for particles and fields of higher spin. This section is devoted to establishing and summarizing the main facts concerning the Lorentz group, and to introduce a supplement to the customary method of construction of invariants of this group in order to classify the representations employed in this presentation. The inhomogeneous Lorentz transformation, {a,A}, is a linear transformation of the coordinates conserving the norm of the intervals between different points of space-time. The new coordinates xa' are obtained from the old coordinates by the relation3 The quantity a represents the translation of the space-time coordinates, xa. The condition of invariance of the norm required that AasAay = 88, from which it follows that det A = ±1. = The set of transformations in which the translations are omitted (aa homogeneous Lorentz group, which in turn can be divided into four subsets: 0) is denoted as the full (1) The subset with det A = +1 and A441 is called the group of proper homogeneous Lorentz transformations. It is a six-parameter continuous group, containing the identity operator. = = (2) The subset with det A -1 and A44 ≥ 1 is called space inversion. 1 and A44 ≤ -1 is called time inversion. The latter three subsets are disjoint and not continuously connected. Those subsets having A44 ≥ 1 may be classified as orthochronous transformations, transforming a time-like vector into a time-like vector. = ... * We employ the following notation: All boldface letters, A, Q, p, x, J, etc., denote three vectors. The fourth components of the coordinates and momenta, x4 = it and p iE, are pure imaginaries. All Greek subscripts α, ẞ, μ, v, vary from 1 to 4 and all Roman subscripts i, j, k vary from 1 to 3, except when specifically indicated otherwise. Repeated indices are to be summed over. Scalar products of four vectors are written as p·x= P. The operator a/dx is often written as . The D'Alembertian operator a. Units are chosen so that ħ and c are equal to unity. = |