Zero-rest-mass field equations and the spinor form of Maxwell, Weyl, and Dirac fields

Intuition Beginner

A massless particle moves at the speed of light, and there is no frame in which it sits still. That single fact reshapes how its field is described. A particle with rest mass, like an electron, can be brought to rest and turned slowly in your hands; you can ask which way its spin points in any direction. A massless particle never stops, so its spin can only be read along its direction of travel. The two readings you ever get are "spinning the way a right-handed screw advances" or the left-handed mirror image. This single, sign-carrying number is called the helicity.

Spinors are the natural language for this story. A spinor is the squarest-possible root of a direction in spacetime: where a velocity arrow needs four numbers, a spinor needs two complex ones, and a velocity along the light cone is built by squaring a spinor. Light-cone directions and spinors fit together perfectly, which is why massless fields look simplest when written with spinors.

The punchline of this unit is that one compact spinor equation, with a tower of indices whose length records the spin, contains the source-free Maxwell equations, the equation for a massless spin-half particle, and (with a small twist) the Dirac equation, all at once.

Visual Beginner

Alt text: At the centre sits a single equation reading "spinor divergence of a symmetric spinor with 2s indices equals zero." Three arrows point outward to boxes. The spin-one arrow reaches Maxwell, a light ray with a dial reading plus or minus one for the photon's two helicities. The spin-one-half arrow reaches Weyl, a single light ray with a dial reading plus or minus one half. The third arrow reaches Dirac, drawn as two linked spinor symbols joined by a mass term, showing that a massive particle needs two coupled fields. The figure shows how varying the index count s reproduces three field theories, each labelled by helicity.

Worked example Beginner

Take the electromagnetic field. In the usual telling it is six numbers: a three-component electric field and a three-component magnetic field, packaged into the antisymmetric field tensor with components $F_{ab}$. Counting carefully, an antisymmetric pair of four-valued indices has six independent slots, matching the six numbers.

Now repackage. The six real numbers can be rewritten as three complex numbers, namely the electric field plus $i$ times the magnetic field. These three complex numbers are exactly the independent components of a symmetric spinor with two lower indices, written ${\varphi }_{AB}$, which also has three slots because a symmetric pair of two-valued indices has three. The dictionary is a clean match: six real becomes three complex becomes the spinor ${\varphi }_{AB}$.

What this buys us: the four Maxwell equations, in empty space, collapse into the single statement that a certain spinor derivative of ${\varphi }_{AB}$ is zero. The two source-free Maxwell equations that say the divergence and the curl behave correctly are repackaged into one tidy spinor equation. The same trick, with a shorter or longer tower of indices, handles spin one half and spin two.

Check your understanding Beginner

Formal definition Intermediate+

We work on Minkowski space (and later a curved Lorentzian $4$-manifold) equipped with the two-spinor calculus of 03.02.41: a complex rank-two spin space $\mathbb{S}$ with symplectic form ${\epsilon }_{AB}$, its complex conjugate $\bar{\mathbb{S}}$ with primed indices, and the soldering isomorphism $V^a\leftrightarrow V^{A{A}^{\prime }}$ between complexified tangent vectors and elements of $\mathbb{S}\otimes \bar{\mathbb{S}}$. Indices are raised and lowered with ${\epsilon }_{AB}$, ${\epsilon }^{AB}$ (and their primed versions) by the conventions ${\xi }^A={\epsilon }^{AB}{\xi }_B$, ${\xi }_B={\xi }^A{\epsilon }_{AB}$. The spinor covariant derivative ${\nabla }_{A{A}^{\prime }}$ is the soldered form of the Levi-Civita connection, annihilating ${\epsilon }_{AB}$.

Definition (zero-rest-mass field). A zero-rest-mass free field of spin $s=\frac{n}{2}$ ($n$ a positive integer) is a totally symmetric spinor field $$ \phi_{AB\cdots L} = \phi_{(AB\cdots L)}, \qquad n = 2s \ \text{indices}, $$ satisfying the zero-rest-mass field equation $$ \nabla^{AA'}\phi_{AB\cdots L} = 0 . $$ The derivative is contracted against the first index; by total symmetry the choice of contracted index is immaterial. The conjugate field ${\bar{\varphi }}_{A^{\prime }{B}^{\prime }\cdots L^{\prime }}$ describes the opposite helicity. The number of unprimed versus primed indices is the helicity grading: ${\varphi }_{AB\cdots L}$ with $n$ unprimed indices has helicity $+s$, and ${\bar{\varphi }}_{A^{\prime }\cdots L^{\prime }}$ has helicity $-s$.

The Maxwell spinor ($s=1$). A real antisymmetric two-form $F_{ab}=F_{[ab]}$ decomposes, via the dictionary of 03.02.41, as $$ F_{ab} = \phi_{AB},\varepsilon_{A'B'} + \bar\phi_{A'B'},\varepsilon_{AB}, \qquad \phi_{AB} = \tfrac12 F_{ab},\varepsilon^{A'B'} = \phi_{(AB)}, $$ where ${\varphi }_{AB}$ is the Maxwell spinor. The first term is the anti-self-dual part of $F$ and the second its self-dual part (the complex conjugate); the split of a real two-form into ${\varphi }_{AB}{\epsilon }_{A^{\prime }{B}^{\prime }}$ plus complex conjugate is the (anti-)self-dual decomposition. The source-free Maxwell equations ${\nabla }^a{F}_{ab}=0$ and ${\nabla }_{[a}{F}_{bc]}=0$ together read, in spinor form, ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}=0$ — the $s=1$ case of the master equation.

The Weyl (neutrino) field ($s=\frac{1}{2}$). A single unprimed spinor field ${\varphi }_A$ obeying $$ \nabla^{AA'}\phi_A = 0 $$ is the Weyl field, the massless spin-half field of one definite helicity. This is the chiral, two-component reduction of the Dirac field at zero mass 12.11.01.

The Dirac field as a coupled pair. A Dirac $4$-spinor splits into a pair $({\varphi }_A,{\chi }^{A^{\prime }})$ — an unprimed and a primed two-spinor, the two chiral halves 12.11.01. With mass parameter $\mu =m/\sqrt{2}$ (in natural units) the Dirac equation becomes the coupled first-order pair $$ \nabla^{AA'}\phi_A = \mu,\chi^{A'}, \qquad \nabla_{AA'}\chi^{A'} = \mu,\phi_A . $$ Setting $\mu =0$ decouples the pair into two independent Weyl equations, recovering the $s=\frac{1}{2}$ massless case for each chirality.

Counterexamples to common slips

• Symmetry is not optional. A solution of ${\nabla }^{A{A}^{\prime }}{\psi }_{AB\cdots L}=0$ with $\psi$ not totally symmetric is not a spin-$s$ zero-rest-mass field; its antisymmetric parts carry lower-spin content and obey different equations. The symmetric projection is part of the definition.
• Helicity is the index imbalance, not the index count. A field with equal numbers of primed and unprimed indices (a tensor field) has helicity zero in this grading; helicity is the difference of unprimed and primed counts, halved. Doubling all indices does not double the helicity if you add them symmetrically on both index types.
• The massive case needs two fields. One cannot write the massive Dirac equation as a single symmetric spinor equation. Mass couples the two chiralities, so a primed partner is mandatory; a lone ${\varphi }_A$ with a mass term on the right is inconsistent unless a ${\chi }^{A^{\prime }}$ is supplied to receive it.

Key theorem with proof Intermediate+

Theorem (Maxwell recovery and helicity). Let ${\varphi }_{AB}={\varphi }_{(AB)}$ be a Maxwell spinor on Minkowski space and let $F_{ab}={\varphi }_{AB}{\epsilon }_{A^{\prime }{B}^{\prime }}+{\bar{\varphi }}_{A^{\prime }{B}^{\prime }}{\epsilon }_{AB}$ be the associated real two-form. Then the single spinor equation ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}=0$ is equivalent to the full set of source-free Maxwell equations ${\nabla }^a{F}_{ab}=0$ and ${\nabla }_{[a}{F}_{bc]}=0$. Moreover the positive- and negative-helicity photon states correspond to ${\varphi }_{AB}$ and ${\bar{\varphi }}_{A^{\prime }{B}^{\prime }}$ respectively.

Proof. Apply ${\nabla }^{A{A}^{\prime }}$ to the decomposition of $F_{ab}$. In two-spinor form the divergence and the exterior-derivative parts of ${\nabla }^a{F}_{ab}$ are governed by the symmetric and antisymmetric parts of ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}$ in the index $A$. Compute ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}$ and split it on the free unprimed index $B$ and free primed index $A^{\prime }$. The contraction ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}$ is a spinor with one lower unprimed index $B$ and one upper primed index $A^{\prime }$, hence corresponds to a covector $G_b{}^{\prime }$, and the real and imaginary parts of the equation $G=0$ are precisely the divergence equation ${\nabla }^a{F}_{ab}=0$ and the Bianchi/curl equation ${\nabla }_{[a}{F}_{bc]}=0$. Concretely, writing ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}$ and using $F_{ab}={\varphi }_{AB}{\epsilon }_{A^{\prime }{B}^{\prime }}+\text{c.c.}$ together with ${\nabla }_{A{A}^{\prime }}{\epsilon }_{BC}=0$, the four-vector ${\nabla }^a{F}_{ab}$ equals ${\nabla }^A{}_{B^{\prime }}{\varphi }_{AB}\cdot (\text{soldering})+\text{c.c.}$, which vanishes for all $b$ iff ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}=0$; the homogeneous Maxwell pair is the conjugate statement, automatically included because $F$ is real. The two-to-one correspondence between $F_{ab}$ and the pair $({\varphi }_{AB},{\bar{\varphi }}_{A^{\prime }{B}^{\prime }})$ shows the field carries exactly the two photon polarisations, the self-dual part $\bar{\varphi }$ and the anti-self-dual part $\varphi$, which are the helicity $-1$ and $+1$ states. $■$

This is the prototype: a tensor field equation has been turned into a contracted-derivative condition on a symmetric spinor, and the (anti-)self-dual split that is awkward in tensor language is the plain real-and-imaginary split of ${\varphi }_{AB}$.

Bridge. This recovery builds toward the unified spin-$s$ picture and appears again in the linearised gravity case ${\varphi }_{ABCD}$, where the same contracted-derivative equation governs the Weyl spinor of 03.02.18. The foundational reason the spinor form is cleaner is that the irreducible pieces of a Lorentz tensor are exactly the totally symmetric spinors, so the (anti-)self-dual decomposition is dual to the split of ${\varphi }_{AB}$ into itself and its conjugate; this is exactly the pattern by which the Maxwell field, the Weyl neutrino, and the linearised graviton become a single equation graded by index count. Putting these together, the central insight is that helicity is read directly off the imbalance between primed and unprimed indices, and the zero-rest-mass equation generalises the source-free Maxwell system to every half-integer spin.

Exercises Intermediate+

Advanced results Master

The full spin-$s$ tower and conserved currents. For each $s=\frac{n}{2}$ the equation ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB\cdots L}=0$ on Minkowski space admits a positive-frequency solution space carrying the irreducible massless helicity-$\pm s$ representation of the Poincaré group ^{[Penrose-Rindler §5.7]}, matching the Wigner classification 07.07.06. The field carries a conserved, symmetric stress-energy-type current quadratic in $\varphi$; for $s=1$ this is the Maxwell stress tensor, and for $s=2$ the linearised Bel-Robinson tensor of 03.02.18. The contour-integral solution via the Penrose transform produces exactly these helicity-$\pm s$ fields as cohomology classes on twistor space 03.07.14, so the present equation is the spacetime shadow of that complex-analytic construction.

Maxwell as $s=1$ and linearised gravity as $s=2$. The Maxwell spinor ${\varphi }_{AB}$ and its equation reproduce source-free electromagnetism, with the principal null directions of ${\varphi }_{AB}$ (the spinors ${\alpha }_A$ with ${\varphi }_{AB}{\alpha }^A{\alpha }^B=0$) marking the special null directions of the field. For $s=2$, the totally symmetric ${\varphi }_{ABCD}$ is the linearised Weyl curvature spinor, and ${\nabla }^{A{A}^{\prime }}{\varphi }_{ABCD}=0$ is the linearised vacuum Bianchi identity; its four principal null spinors and their coincidence pattern are the Petrov classification of 03.02.18. The peeling behaviour at null infinity — the $r^{-k}$ falloff sorted by Petrov type — is a statement about this spinor field along outgoing null geodesics.

The Buchdahl consistency condition. In a general curved Lorentzian background the equation ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB\cdots L}=0$ for $s>1$ is overdetermined: cross-differentiating and applying the spinor Ricci identities yields an algebraic constraint coupling $\varphi$ to the conformal curvature spinor ${\Psi }_{ABCD}$. For $s>1$ this Buchdahl condition ${\Psi }_{(AB}{}^{EF}{\varphi }_{CD\cdots )EF\cdots }=0$ generically forces $\varphi$ to vanish unless the background is algebraically special or conformally flat ^{[Buchdahl 1958]}. Thus higher-spin massless fields do not propagate freely on a generic curved metric — a genuine obstruction with no analogue for $s\le 1$, where Maxwell and Weyl theory remain consistent on any background.

The massive case and the Dirac pair. Mass is the off-diagonal coupling ${\nabla }^{A{A}^{\prime }}{\varphi }_A=\mu {\chi }^{A^{\prime }}$, ${\nabla }_{A{A}^{\prime }}{\chi }^{A^{\prime }}=\mu {\varphi }_A$. Eliminating ${\chi }^{A^{\prime }}$ gives the spinor wave equation $(\square +2{\mu }^2){\varphi }_A=0$ up to a curvature term, recovering the Klein-Gordon mass shell $m^2=2{\mu }^2$. The same coupling pattern generalises to higher-spin massive fields as towers of symmetric spinors linked by mass terms, the spinor incarnation of the Bargmann-Wigner equations.

Synthesis. The zero-rest-mass equation is the foundational reason a single object unifies three classical field theories: this is exactly the statement that the irreducible content of a massless relativistic field is a totally symmetric spinor, graded by helicity. Putting these together, Maxwell ($s=1$) generalises to the linearised graviton ($s=2$) and reduces to the Weyl neutrino ($s=\frac{1}{2}$), while the massive Dirac field is dual to a coupled primed-unprimed pair whose decoupling at zero mass returns the Weyl case. The central insight is that helicity is the primed-versus-unprimed index imbalance, the (anti-)self-dual split is the real-imaginary split of the field spinor, and the Buchdahl obstruction shows precisely where this clean picture breaks under curvature for $s>1$ — the bridge from flat-space field theory to curved-space relativity, and the spacetime end of the twistor correspondence of 03.07.14.

Full proof set Master

Proposition (component count and helicity content). The totally symmetric spinor ${\varphi }_{AB\cdots L}$ with $n=2s$ indices has $n+1$ independent complex components, and the solution space of ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB\cdots L}=0$ carries helicity $+s$ (its conjugate, helicity $-s$).

Proof. A symmetric spinor in $n$ two-valued indices is determined by how many of its indices equal $0$ (the rest equal $1$), giving the values $0,1,\dots ,n$, hence $n+1$ complex components. For the helicity, decompose a plane-wave solution ${\varphi }_{AB\cdots L}={\pi }_A{\pi }_B\cdots {\pi }_L{e}^{ik\cdot x}$ with $k_a={\pi }_A{\bar{\pi }}_{A^{\prime }}$ a null covector (this is the general null-datum ansatz). The field equation ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB\cdots L}=0$ forces $k^{A{A}^{\prime }}{\pi }_A=0$ on the wave, i.e. the propagation direction is the repeated principal spinor; the response of $\varphi$ to a rotation about $k$ multiplies it by $e^{is\theta }$, the defining transformation of helicity $+s$. The conjugate field transforms by $e^{-is\theta }$. $■$

Proposition (Maxwell equivalence). For real $F_{ab}={\varphi }_{AB}{\epsilon }_{A^{\prime }{B}^{\prime }}+{\bar{\varphi }}_{A^{\prime }{B}^{\prime }}{\epsilon }_{AB}$, the system $\{{\nabla }^a{F}_{ab}=0,{\nabla }_{[a}{F}_{bc]}=0\}$ holds iff ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}=0$.

Proof. Using ${\nabla }_{A{A}^{\prime }}{\epsilon }_{BC}=0$, the four-vector ${\nabla }^a{F}_{ab}$ in spinor form is ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}\bar{\delta }+\text{c.c.}$, where the soldering identifies the free indices $(B,A^{\prime })$ with the covector index $b$. Its vanishing is the condition ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}=0$. The homogeneous pair ${\nabla }_{[a}{F}_{bc]}=0$ is the complex conjugate of the same spinor equation, automatically holding because $F$ is real and ${\varphi }_{AB}$ is symmetric: the antisymmetric part of ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}$ on the indices $A,B$ vanishes by symmetry of $\varphi$, leaving only the symmetric part, whose reality content is the inhomogeneous equation and whose imaginary content is the Bianchi equation. Hence the single complex spinor equation encodes both Maxwell pairs. $■$

Proposition (Buchdahl obstruction for $s>1$). On a curved background, ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB\cdots L}=0$ with $n=2s\ge 3$ implies the algebraic identity ${\Psi }_{(AB}{}^{EF}{\varphi }_{C\cdots L)EF}=0$, where ${\Psi }_{ABCD}$ is the Weyl conformal spinor.

Proof. Apply ${\nabla }_{C^{\prime }}{}^C$ to the field equation and antisymmetrise the two unprimed derivative indices to form the spinor curvature operator ${\square }_{AB}={\nabla }_{A^{\prime }(A}{\nabla }_{B)}{}^{A^{\prime }}$. The spinor Ricci identities give ${\square }_{AB}$ acting on a spinor index as a contraction with ${\Psi }_{ABCD}$ (the conformally invariant part) plus a Ricci-spinor term that cancels in vacuum or by the trace structure. Feeding the field equation in to remove the symmetric-derivative part leaves the purely algebraic curvature contraction; total symmetry of $\varphi$ projects it onto ${\Psi }_{(AB}{}^{EF}{\varphi }_{C\cdots L)EF}=0$. For $n\ge 3$ this is a nonempty constraint, the Buchdahl condition ^{[Buchdahl 1958]}; for $n\le 2$ the symmetrisation kills it and no obstruction arises. $■$

The detailed spinor Ricci identities and the full Bargmann-Wigner generalisation to massive higher-spin fields are stated above without proof — see Penrose-Rindler ^{[Penrose-Rindler §5.8]} and Wald ^{[Wald §13.2]}.

Connections Master

• The two-spinor calculus and the spinor-tensor dictionary 03.02.41. That unit (co-produced in this wave) builds the $\epsilon$-spinor, the soldering form $V^a\leftrightarrow V^{A{A}^{\prime }}$, and the decomposition of an antisymmetric two-form $F_{ab}={\varphi }_{AB}{\epsilon }_{A^{\prime }{B}^{\prime }}+\text{c.c.}$ that this unit uses to write Maxwell as ${\varphi }_{AB}$. The spinor covariant derivative ${\nabla }_{A{A}^{\prime }}$ appearing in every field equation here is the soldered Levi-Civita connection defined there.

• The Newman-Penrose spin-coefficient formalism 03.02.43. That unit (co-produced in this wave) turns the abstract ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB\cdots L}=0$ into component equations along a null tetrad, expressing the Maxwell scalars ${\varphi }_0,{\varphi }_1,{\varphi }_2$ and the Weyl scalars ${\Psi }_0\dots {\Psi }_4$ and their NP-equation dynamics; the present equation is the covariant statement that the NP formalism makes computational.

• The Dirac equation, relativistic spin 12.11.01. This unit recasts the four-component Dirac equation built there as the coupled two-spinor pair ${\nabla }^{A{A}^{\prime }}{\varphi }_A=\mu {\chi }^{A^{\prime }}$, ${\nabla }_{A{A}^{\prime }}{\chi }^{A^{\prime }}=\mu {\varphi }_A$, and identifies the massless chiral halves with the $s=\frac{1}{2}$ Weyl field. The chiral projectors there become the split into primed and unprimed spinors here.

• Free Maxwell and the Proca field 12.05.06. The source-free Maxwell system established there is exactly the $s=1$ case ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB}=0$; the photon's two helicities are the Maxwell spinor and its conjugate, and the electric-magnetic duality rotation is the phase freedom of ${\varphi }_{AB}$.

• The Petrov classification and the Weyl spinor 03.02.18. The $s=2$ field ${\varphi }_{ABCD}$ is the linearised Weyl spinor classified there; its four principal null directions and their coincidences are the Petrov types, and the peeling theorem describes this spinor field at null infinity.

Historical & philosophical context Master

The spinor form of the massless field equations grew out of the van der Waerden two-spinor calculus of the late 1920s and reached its mature, geometrically explicit form in the work of Roger Penrose and Wolfgang Rindler, whose Spinors and Space-Time, Vol. 1 (Cambridge University Press, 1984) devotes its fifth chapter to the zero-rest-mass fields and presents the equation ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB\cdots L}=0$ as a single object subsuming Maxwell, Weyl, and (in coupled form) Dirac ^{[Penrose-Rindler §5.7]}. The viewpoint that helicity is the fundamental label of a massless field — and that this label is read directly off the imbalance of primed and unprimed spinor indices — became the conceptual seed of the twistor programme, in which these fields are repackaged as cohomology on a complex projective space.

The obstruction to propagating higher-spin massless fields on a curved background was identified by H. A. Buchdahl, whose 1958 paper in Il Nuovo Cimento showed that the wave equations for particles of spin greater than one are inconsistent in a general gravitational field unless an algebraic curvature condition holds ^{[Buchdahl 1958]}. This result anticipated the modern no-go theorems for interacting higher-spin fields and sharpened the special status of spins $\frac{1}{2}$, $1$, and $2$ — the spins realised by the known fundamental fields. Wald's General Relativity (University of Chicago Press, 1984) and Stewart's Advanced General Relativity (Cambridge University Press, 1990) gave the now-standard textbook treatments tying the spinor field equations to the tetrad and Newman-Penrose machinery ^{[Wald §13.2]}.

Bibliography Master

@book{PenroseRindler1984,
  author    = {Penrose, Roger and Rindler, Wolfgang},
  title     = {Spinors and Space-Time, Volume 1: Two-Spinor Calculus and Relativistic Fields},
  publisher = {Cambridge University Press},
  year      = {1984},
  note      = {Chapter 5, especially \S\S5.7--5.8 (zero-rest-mass fields, Buchdahl condition)}
}

@article{Buchdahl1958,
  author  = {Buchdahl, H. A.},
  title   = {On the compatibility of relativistic wave equations for particles of higher spin in the presence of a gravitational field},
  journal = {Il Nuovo Cimento},
  volume  = {10},
  pages   = {96--103},
  year    = {1958}
}

@book{Wald1984,
  author    = {Wald, Robert M.},
  title     = {General Relativity},
  publisher = {University of Chicago Press},
  year      = {1984},
  note      = {Chapter 13 (spinors), \S13.2 (massless fields)}
}

@book{Stewart1990,
  author    = {Stewart, John},
  title     = {Advanced General Relativity},
  publisher = {Cambridge University Press},
  year      = {1990},
  note      = {Chapter 2 (spinors and the spinor form of the field equations)}
}

@book{ODonnell2003,
  author    = {O'Donnell, Peter},
  title     = {Introduction to 2-Spinors in General Relativity},
  publisher = {World Scientific},
  year      = {2003}
}

@article{NewmanPenrose1962,
  author  = {Newman, Ezra and Penrose, Roger},
  title   = {An approach to gravitational radiation by a method of spin coefficients},
  journal = {Journal of Mathematical Physics},
  volume  = {3},
  pages   = {566--578},
  year    = {1962}
}