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Dirac equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way. It has become vital in the building of the Standard Model.

The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.

In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1⁄2 particles.

Dirac did not fully appreciate the importance of his results, however, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him. The equation has been deemed by some physicists to be the "real seed of modern physics". The equation has also been described as the "centerpiece of relativistic quantum mechanics", with it also stated that "the equation is perhaps the most important one in all of quantum mechanics".

The Dirac equation is inscribed upon a plaque on the floor of Westminster Abbey. Unveiled on 13 November 1995, the plaque commemorates Dirac's life.

History
The Dirac equation in the form originally proposed by Dirac is: $$\left(\beta mc^2 + c \sum_{n = 1}^{3}\alpha_n p_n\right) \psi (x,t) = i \hbar \frac{\partial\psi(x,t) }{\partial t} $$ where $ψ(x, t)$ is the wave function for an electron of rest mass $m$ with spacetime coordinates $x, t$. $p_{1}, p_{2}, p_{3}$ are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. $c$ is the speed of light, and $ħ$ is the reduced Planck constant. These fundamental physical constants reflect special relativity and quantum mechanics, respectively.

Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, thus allowing the atom to be treated in a manner consistent with relativity. He hoped that the corrections introduced this way might have a bearing on the problem of atomic spectra.

Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity—which were based on discretizing the angular momentum stored in the electron's possibly non-circular orbit of the atomic nucleus—had failed, and the new quantum mechanics of Heisenberg, Pauli, Jordan, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics.

The new elements in this equation are the four 4 × 4 matrices $α_{1}$, $α_{2}$, $α_{3}$ and $β$, and the four-component wave function $ψ$. There are four components in $ψ$ because the evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron.

The 4 × 4 matrices $α_{k}$ and $β$ are all Hermitian and are involutory: $$\alpha_i^2 = \beta^2 = I_4$$ and they all mutually anti-commute: $$\begin{align} \alpha_i\alpha_j + \alpha_j\alpha_i &= 0\quad(i \neq j) \\ \alpha_i\beta + \beta\alpha_i &= 0 \end{align}$$

These matrices and the form of the wave function have a deep mathematical significance. The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th-century work of German mathematician Hermann Grassmann in his Lineare Ausdehnungslehre (Theory of Linear Expansion).

The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. The equation can be written more explicitly in Planck units as: $$ i \partial_x \begin{bmatrix} +\psi_4 \\ +\psi_3 \\  -\psi_2 \\  -\psi_1 \end{bmatrix} + i \partial_y \begin{bmatrix} +\psi_4 \\ -\psi_3 \\ -\psi_2 \\ +\psi_1 \end{bmatrix} + i \partial_z \begin{bmatrix} +\psi_3 \\ -\psi_4 \\ -\psi_1 \\ +\psi_2 \end{bmatrix} - m           \begin{bmatrix} +\psi_1 \\  +\psi_2 \\ +\psi_3 \\ +\psi_4 \end{bmatrix} = i \partial_t \begin{bmatrix} -\psi_1 \\ -\psi_2 \\  +\psi_3 \\  +\psi_4 \end{bmatrix} $$ which makes it clearer that it is a set of four partial differential equations with four unknown functions.

Making the Schrödinger equation relativistic
The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle: $$-\frac{\hbar^2}{2m}\nabla^2\phi = i\hbar\frac{\partial}{\partial t}\phi ~.$$

The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the four-momentum, and they are related by the relativistically invariant relation $$E^2 = m^2c^4 + p^2c^2 $$

which says that the length of this four-vector is proportional to the rest mass $m$. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the Klein–Gordon equation describing the propagation of waves, constructed from relativistically invariant objects, $$\left(-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2\right)\phi = \frac{m^2c^2}{\hbar^2}\phi $$ with the wave function $$\phi$$ being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression $$\rho = \phi^*\phi $$ and this density is convected according to the probability current vector $$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*) $$ with the conservation of probability current and density following from the continuity equation: $$\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0~.$$

The fact that the density is positive definite and convected according to this continuity equation implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. To maintain the notion of a convected density, one must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The Schrödinger expression can be kept for the current, but the probability density must be replaced by the symmetrically formed expression $$\rho = \frac{i\hbar}{2mc^2} \left(\psi^*\partial_t\psi - \psi\partial_t\psi^* \right) .$$ which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression $$J^\mu = \frac{i\hbar}{2m} \left(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^* \right) .$$

The continuity equation is as before. Everything is compatible with relativity now, but the expression for the density is no longer positive definite; the initial values of both $ψ$ and $∂_{t}ψ$ may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus, one cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.

Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson or Higgs boson). Historically, Schrödinger himself arrived at this equation before the one that bears his name but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the charge density, which can be positive or negative, and not the probability density.

Dirac's coup
Dirac thus thought to try an equation that was first order in both space and time. He postulated an equation of the form $$E\psi = (\vec{\alpha} \cdot \vec{p} + \beta m) \psi$$ where the operators $$(\vec{\alpha}, \beta)$$ must be independent of $$(\vec{p}, t)$$ for linearity and independent of $$(\vec{x}, t)$$ for space-time homogeneity. These constraints implied additional dynamical variables that the $$(\vec{\alpha}, \beta)$$ operators will depend upon; from this requirement Dirac concluded that the operators would depend upon 4x4 matrices, related to the Pauli matrices.

One could, for example, formally (i.e. by abuse of notation) take the relativistic expression for the energy $$E = c \sqrt{p^2 + m^2c^2} ~,$$ replace $p$ by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.

As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator (see also half derivative) thus: $$\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)\left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)~.$$

On multiplying out the right side it is apparent that, in order to get all the cross-terms such as $∂_{x}∂_{y}$ to vanish, one must assume $$AB + BA = 0, ~ \ldots ~$$ with $$A^2 = B^2 = \dots = 1~.$$

Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if $A$, $B$, $C$ and $D$ are matrices, with the implication that the wave function has multiple components. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least 4 × 4 matrices to set up a system with the properties required — so the wave function had four components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.

Given the factorization in terms of these matrices, one can now write down immediately an equation $$\left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t\right)\psi = \kappa\psi $$ with $$\kappa$$ to be determined. Applying again the matrix operator on both sides yields $$\left(\nabla^2 - \frac{1}{c^2}\partial_t^2\right)\psi = \kappa^2\psi ~.$$

Taking $$\kappa = \tfrac{mc}{\hbar}$$ shows that all the components of the wave function individually satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is $$\left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t - \frac{mc}{\hbar}\right)\psi = 0 ~.$$

Setting $$A = i \beta \alpha_1 \,, \, B = i \beta \alpha_2 \, , \, C = i \beta \alpha_3 \, , \, D = \beta ~, $$ and because $$D^2 = \beta^2 = I_4 $$, the Dirac equation is produced as written above.

Covariant form and relativistic invariance
To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows: $$\begin{align} D &=  \gamma^0, \\ A &= i \gamma^1,\quad B = i \gamma^2,\quad C = i \gamma^3, \end{align}$$ and the equation takes the form (remembering the definition of the covariant components of the 4-gradient and especially that $∂_{0} = 1⁄c∂_{t}$)

where there is an implied summation over the values of the twice-repeated index $μ = 0, 1, 2, 3$, and $∂_{μ}$ is the 4-gradient. In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation is $$ \gamma^0 = \begin{pmatrix} I_2 &       0 \\         0 & -I_2 \end{pmatrix},\quad \gamma^1 = \begin{pmatrix}  0 & \sigma_x \\ -\sigma_x &    0 \end{pmatrix},\quad \gamma^2 = \begin{pmatrix}  0 & \sigma_y \\ -\sigma_y &    0 \end{pmatrix},\quad \gamma^3 = \begin{pmatrix}  0 & \sigma_z \\ -\sigma_z &    0 \end{pmatrix}. $$

The complete system is summarized using the Minkowski metric on spacetime in the form $$\left\{\gamma^\mu, \gamma^\nu\right\} = 2 \eta^{\mu\nu} I_4$$ where the bracket expression $$\{a, b\} = ab + ba$$ denotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-dimensional space with metric signature $(+ − − −)$. The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this geometric algebra represents an enormous stride forward in the development of quantum theory.

The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light: $$P_\text{op}\psi = mc\psi \,.$$

Using $${\partial\!\!\!/} \mathrel{\stackrel{\mathrm{def}}{=}} \gamma^\mu \partial_\mu$$ ($${\partial\!\!\!\big /}$$ is pronounced "d-slash"), according to Feynman slash notation, the Dirac equation becomes: $$i \hbar {\partial\!\!\!\big /} \psi - m c \psi = 0 \,.$$

In practice, physicists often use units of measure such that $ħ = c = 1$, known as natural units. The equation then takes the simple form

A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation: $$\gamma^{\mu\prime} = S^{-1} \gamma^\mu S \,.$$

If in addition the matrices are all unitary, as are the Dirac set, then $S$ itself is unitary; $$\gamma^{\mu\prime} = U^\dagger \gamma^\mu U \,.$$

The transformation $U$ is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator $γ^{μ}∂_{μ}$ to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, one may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form $$\begin{align} \left(iU^\dagger \gamma^\mu U\partial_\mu^\prime - m\right)\psi\left(x^\prime, t^\prime\right) &= 0 \\ U^\dagger(i\gamma^\mu\partial_\mu^\prime - m)U \psi\left(x^\prime, t^\prime\right) &= 0 \,. \end{align}$$

If the transformed spinor is defined as $$\psi^\prime = U\psi$$ then the transformed Dirac equation is produced in a way that demonstrates manifest relativistic invariance: $$\left(i\gamma^\mu\partial_\mu^\prime - m\right)\psi^\prime\left(x^\prime, t^\prime\right) = 0 \,.$$

Thus, settling on any unitary representation of the gammas is final, provided the spinor is transformed according to the unitary transformation that corresponds to the given Lorentz transformation.

The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function. The representation shown here is known as the standard representation – in it, the wave function's upper two components go over into Pauli's 2 spinor wave function in the limit of low energies and small velocities in comparison to light.

The considerations above reveal the origin of the gammas in geometry, hearkening back to Grassmann's original motivation; they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as $γ_{μ}γ_{ν}$ represent oriented surface elements, and so on. With this in mind, one can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is $$V = \frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta .$$

For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of $√g$, where $g$ is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus $$V = i \gamma^0\gamma^1\gamma^2\gamma^3 .$$

This matrix is given the special symbol $γ^{5}$, owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors. In the standard representation, it is $$\gamma_5 = \begin{pmatrix} 0 & I_{2} \\ I_{2} & 0 \end{pmatrix}.$$

This matrix will also be found to anticommute with the other four Dirac matrices: $$\gamma^5 \gamma^\mu + \gamma^\mu \gamma^5 = 0$$

It takes a leading role when questions of parity arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime.

Pauli theory
The necessity of introducing half-integer spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into $N$ parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two; the ground state therefore could not be integer, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with $L_{z} = −1, 0, +1$. The conclusion is that silver atoms have net intrinsic angular momentum of $1⁄2$. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in SI units: (Note that bold faced characters imply Euclidean vectors in 3 dimensions, whereas the Minkowski four-vector $A_{μ}$ can be defined as $$A_\mu = (\phi/c,-\mathbf A)$$.) $$H = \frac{1}{2m}\left( \boldsymbol{\sigma}\cdot\left(\mathbf{p} - e \mathbf{A}\right)\right)^2 + e\phi ~.$$

Here $A$ and $$\phi$$ represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field in SI units: $$H = \frac{1}{2m}\left(\mathbf{p} - e \mathbf{A}\right)^2 + e\phi - \frac{e\hbar}{2m} \boldsymbol{\sigma} \cdot \mathbf{B} ~.$$

This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form: $$\left(\gamma^\mu(i\hbar\partial_\mu - eA_\mu) - mc\right) \psi = 0 ~.$$

A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by $i$, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored: $$ \begin{pmatrix} mc^2 - E + e \phi & c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) \\ -c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) & mc^2 + E - e \phi \end{pmatrix} \begin{pmatrix} \psi_{+} \\ \psi_{-} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} ~. $$ so $$\begin{align} (E - e\phi) \psi_{+} - c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) \psi_{-} &= mc^2 \psi_{+} \\ c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) \psi_{+}-(E - e\phi) \psi_{-} &= mc^2 \psi_{-} \end{align}$$

Assuming the field is weak and the motion of the electron non-relativistic, the total energy of the electron is approximately equal to its rest energy, and the momentum going over to the classical value, $$\begin{align} E - e\phi &\approx mc^2 \\ \mathbf{p} &\approx m \mathbf{v} \end{align}$$ and so the second equation may be written $$\psi_- \approx \frac{1}{2mc} \boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) \psi_{+} $$

which is of order $v⁄c$ – thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement $$ \left(E - mc^2\right) \psi_{+} = \frac{1}{2m} \left[\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right)\right]^2 \psi_{+} + e\phi \psi_{+} $$

The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so one can recover Pauli's theory upon identifying his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus, the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious $i$ that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves.

It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an irreducible whole, and the components just neglected here to arrive at the Pauli theory will bring in new phenomena in the relativistic regime – antimatter and the idea of creation and annihilation of particles.

Weyl theory
In the massless case $$m = 0$$, the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin-$1/undefined$ particles.

The theory acquires a second $$\text{U}(1)$$ symmetry: see below.

Identification of observables
The critical physical question in a quantum theory is this: what are the physically observable quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. To maintain this interpretation on passing to the Dirac theory, the Hamiltonian must be taken to be $$H = \gamma^0 \left[mc^2 + c \gamma^k \left(p_k - q A_k\right) \right] + c q A^0.$$ where, as always, there is an implied summation over the twice-repeated index $k = 1, 2, 3$. This looks promising, because one can see by inspection the rest energy of the particle and, in the case of $A = 0$, the energy of a charge placed in an electric potential $cqA^{0}$. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is $$H = c\sqrt{\left(\mathbf{p} - q\mathbf{A}\right)^2 + m^2c^2} + qA^0.$$

Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and one must take great care to correctly identify what is observable in this theory. Much of the apparently paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables.

Hole theory
The negative $E$ solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, they cannot simply be ignored, for once the interaction between the electron and the electromagnetic field is included, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by emitting energy in the form of photons.

To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.

Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a positive energy because energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but Hermann Weyl pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932.

It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the net electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.

In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, and although it too is referred to as an "electron hole", it is distinct from a positron. The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.

In quantum field theory
In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation.

Mathematical formulation
In its modern formulation for field theory, the Dirac equation is written in terms of a Dirac spinor field $$\psi$$ taking values in a complex vector space described concretely as $$\mathbb{C}^4$$, defined on flat spacetime (Minkowski space) $$\mathbb{R}^{1,3}$$. Its expression also contains gamma matrices and a parameter $$m > 0$$ interpreted as the mass, as well as other physical constants. Dirac first obtained his equation through a factorization of Einstein's energy-momentum-mass equivalence relation assuming a scalar product of momentum vectors determined by the metric tensor and quantized the resulting relation by associating momenta to their respective operators.

In terms of a field $$\psi: \mathbb{R}^{1,3}\rightarrow \mathbb{C}^4$$, the Dirac equation is then

and in natural units, with Feynman slash notation,

The gamma matrices are a set of four $$4 \times 4$$ complex matrices (elements of $$\text{Mat}_{4\times 4}(\mathbb{C})$$) which satisfy the defining anti-commutation relations: $$\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}I_4$$ where $$\eta^{\mu\nu}$$ is the Minkowski metric element, and the indices $$\mu, \nu$$ run over 0,1,2 and 3. These matrices can be realized explicitly under a choice of representation. Two common choices are the Dirac representation $$ \gamma^0 = \begin{pmatrix} I_2 &       0 \\         0 & -I_2 \end{pmatrix},\quad \gamma^i = \begin{pmatrix}  0 & \sigma^i \\ -\sigma^i &    0 \end{pmatrix}, $$ where $$\sigma^i$$ are the Pauli matrices, and the chiral representation: the $$\gamma^i$$ are the same, but $$\gamma^0 = \begin{pmatrix} 0 & I_2 \\         I_2 & 0 \end{pmatrix}.$$

The slash notation is a compact notation for $$A\!\!\!/ := \gamma^\mu A_\mu$$ where $$A$$ is a four-vector (often it is the four-vector differential operator $$\partial_\mu$$). The summation over the index $$\mu$$ is implied.

Dirac adjoint and the adjoint equation
The Dirac adjoint of the spinor field $$\psi(x)$$ is defined as $$\bar\psi(x) = \psi(x)^\dagger \gamma^0.$$ Using the property of gamma matrices (which follows straightforwardly from Hermicity properties of the $$\gamma^\mu$$) that $$(\gamma^\mu)^\dagger = \gamma^0\gamma^\mu\gamma^0,$$ one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right by $$\gamma^0$$: $$\bar\psi (x)( - i\gamma^\mu \overleftarrow{\partial}_\mu - m) = 0$$ where the partial derivative $$\overleftarrow{\partial}_\mu$$ acts from the right on $$\bar\psi(x)$$: written in the usual way in terms of a left action of the derivative, we have $$- i\partial_\mu\bar\psi (x)\gamma^\mu - m\bar\psi (x) = 0.$$

Klein–Gordon equation
Applying $$i\partial\!\!\!/ + m$$ to the Dirac equation gives $$(\partial_\mu\partial^\mu + m^2)\psi(x) = 0.$$ That is, each component of the Dirac spinor field satisfies the Klein–Gordon equation.

Conserved current
A conserved current of the theory is $$J^\mu = \bar{\psi}\gamma^\mu\psi.$$ $$

Another approach to derive this expression is by variational methods, applying Noether's theorem for the global $$\text{U}(1)$$ symmetry to derive the conserved current $$J^\mu.$$

$$

Solutions
Since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. The fact that the energies of the solutions do not have a lower bound is unexpected.

Plane-wave solutions
Plane-wave solutions are those arising from an ansatz $$\psi(x) = u(\mathbf{p})e^{-i p \cdot x}$$ which models a particle with definite 4-momentum $$p = (E_\mathbf{p}, \mathbf{p})$$ where $E_\mathbf{p} = \sqrt{m^2 + |\mathbf{p}|^2}.$

For this ansatz, the Dirac equation becomes an equation for $$u(\mathbf{p})$$: $$\left(\gamma^\mu p_\mu - m\right) u(\mathbf{p}) = 0.$$ After picking a representation for the gamma matrices $$\gamma^\mu$$, solving this is a matter of solving a system of linear equations. It is a representation-free property of gamma matrices that the solution space is two-dimensional (see here).

For example, in the chiral representation for $$\gamma^\mu$$, the solution space is parametrised by a $$\mathbb{C}^2$$ vector $$\xi$$, with $$u(\mathbf{p}) = \begin{pmatrix} \sqrt{\sigma^\mu p_\mu}\xi \\ \sqrt{\bar\sigma^\mu p_\mu}\xi \end{pmatrix}$$ where $$\sigma^\mu = (I_2, \sigma^i), \bar\sigma^\mu = (I_2, -\sigma^i)$$ and $$\sqrt{\cdot}$$ is the Hermitian matrix square-root.

These plane-wave solutions provide a starting point for canonical quantization.

Lagrangian formulation
Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by: $$\mathcal{L} = i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi - mc^2\overline{\psi}\psi$$

If one varies this with respect to $$\psi$$ one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to $$\bar\psi$$ one gets the Dirac equation.

In natural units and with the slash notation, the action is then

For this action, the conserved current $$J^\mu$$ above arises as the conserved current corresponding to the global $$\text{U}(1)$$ symmetry through Noether's theorem for field theory. Gauging this field theory by changing the symmetry to a local, spacetime point dependent one gives gauge symmetry (really, gauge redundancy). The resultant theory is quantum electrodynamics or QED. See below for a more detailed discussion.

Lorentz invariance
The Dirac equation is invariant under Lorentz transformations, that is, under the action of the Lorentz group $$\text{SO}(1,3)$$ or strictly $$\text{SO}(1,3)^+$$, the component connected to the identity.

For a Dirac spinor viewed concretely as taking values in $$\mathbb{C}^4$$, the transformation under a Lorentz transformation $$\Lambda$$ is given by a $$4\times 4$$ complex matrix $$S[\Lambda]$$. There are some subtleties in defining the corresponding $$S[\Lambda]$$, as well as a standard abuse of notation.

Most treatments occur at the Lie algebra level. For a more detailed treatment see here. The Lorentz group of $$4 \times 4$$ real matrices acting on $$\mathbb{R}^{1,3}$$ is generated by a set of six matrices $$\{M^{\mu\nu}\}$$ with components $$(M^{\mu\nu})^\rho{}_\sigma = \eta^{\mu\rho}\delta^\nu{}_\sigma - \eta^{\nu\rho}\delta^\mu{}_\sigma.$$ When both the $$\rho,\sigma$$ indices are raised or lowered, these are simply the 'standard basis' of antisymmetric matrices.

These satisfy the Lorentz algebra commutation relations $$[M^{\mu\nu}, M^{\rho\sigma}] = M^{\mu\sigma}\eta^{\nu\rho} - M^{\nu\sigma}\eta^{\mu\rho} + M^{\nu\rho}\eta^{\mu\sigma} - M^{\mu\rho}\eta^{\nu\sigma}.$$ In the article on the Dirac algebra, it is also found that the spin generators $$S^{\mu\nu} = \frac{1}{4} [\gamma^\mu,\gamma^\nu]$$ satisfy the Lorentz algebra commutation relations.

A Lorentz transformation $$\Lambda$$ can be written as $$\Lambda = \exp\left(\frac{1}{2}\omega_{\mu\nu}M^{\mu\nu}\right)$$ where the components $$\omega_{\mu\nu}$$ are antisymmetric in $$\mu,\nu$$.

The corresponding transformation on spin space is $$S[\Lambda] = \exp\left(\frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}\right).$$ This is an abuse of notation, but a standard one. The reason is $$S[\Lambda]$$ is not a well-defined function of $$\Lambda$$, since there are two different sets of components $$\omega_{\mu\nu}$$ (up to equivalence) which give the same $$\Lambda$$ but different $$S[\Lambda]$$. In practice we implicitly pick one of these $$\omega_{\mu\nu}$$ and then $$S[\Lambda]$$ is well defined in terms of $$\omega_{\mu\nu}.$$

Under a Lorentz transformation, the Dirac equation $$i\gamma^\mu\partial_\mu \psi(x) - m \psi(x)=0$$ becomes $$i\gamma^\mu((\Lambda^{-1})_\mu{}^\nu\partial_\nu)S[\Lambda]\psi(\Lambda^{-1} x) - mS[\Lambda]\psi(\Lambda^{-1} x) = 0.$$

$$

Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents $$(\mathcal{J}^{\rho\sigma})^\mu$$. Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents $$T^{\mu\nu}$$, which can be identified as the stress-energy tensor of the theory. The Lorentz current $$(\mathcal{J}^{\rho\sigma})^\mu$$ can be written in terms of the stress-energy tensor in addition to a tensor representing internal angular momentum.

Further discussion of Lorentz covariance of the Dirac equation
The Dirac equation is Lorentz covariant. Articulating this helps illuminate not only the Dirac equation, but also the Majorana spinor and Elko spinor, which although closely related, have subtle and important differences.

Understanding Lorentz covariance is simplified by keeping in mind the geometric character of the process. Let $$a$$ be a single, fixed point in the spacetime manifold. Its location can be expressed in multiple coordinate systems. In the physics literature, these are written as $$x$$ and $$x'$$, with the understanding that both $$x$$ and $$x'$$ describe the same point $$a$$, but in different local frames of reference (a frame of reference over a small extended patch of spacetime). One can imagine $$a$$ as having a fiber of different coordinate frames above it. In geometric terms, one says that spacetime can be characterized as a fiber bundle, and specifically, the frame bundle. The difference between two points $$x$$ and $$x'$$ in the same fiber is a combination of rotations and Lorentz boosts. A choice of coordinate frame is a (local) section through that bundle.

Coupled to the frame bundle is a second bundle, the spinor bundle. A section through the spinor bundle is just the particle field (the Dirac spinor, in the present case). Different points in the spinor fiber correspond to the same physical object (the fermion) but expressed in different Lorentz frames. Clearly, the frame bundle and the spinor bundle must be tied together in a consistent fashion to get consistent results; formally, one says that the spinor bundle is the associated bundle; it is associated to a principal bundle, which in the present case is the frame bundle. Differences between points on the fiber correspond to the symmetries of the system. The spinor bundle has two distinct generators of its symmetries: the total angular momentum and the intrinsic angular momentum. Both correspond to Lorentz transformations, but in different ways.

The presentation here follows that of Itzykson and Zuber. It is very nearly identical to that of Bjorken and Drell. A similar derivation in a general relativistic setting can be found in Weinberg. Here we fix our spacetime to be flat, that is, our spacetime is Minkowski space.

Under a Lorentz transformation $$x \mapsto x',$$ the Dirac spinor to transform as $$\psi'(x') = S \psi(x)$$ It can be shown that an explicit expression for $$S$$ is given by $$S = \exp\left(\frac{-i}{4} \omega^{\mu\nu} \sigma_{\mu\nu}\right)$$ where $$\omega^{\mu\nu}$$ parameterizes the Lorentz transformation, and $$\sigma_{\mu\nu}$$ are the six 4×4 matrices satisfying: $$\sigma^{\mu\nu} = \frac{i}{2} [\gamma^\mu,\gamma^\nu]~.$$

This matrix can be interpreted as the intrinsic angular momentum of the Dirac field. That it deserves this interpretation arises by contrasting it to the generator $$J_{\mu\nu}$$ of Lorentz transformations, having the form $$J_{\mu\nu} = \frac{1}{2} \sigma_{\mu\nu} + i (x_\mu\partial_\nu - x_\nu\partial_\mu)$$ This can be interpreted as the total angular momentum. It acts on the spinor field as $$\psi^\prime(x) = \exp\left(\frac{-i}{2} \omega^{\mu\nu} J_{\mu\nu}\right) \psi(x)$$ Note the $$x$$ above does not have a prime on it: the above is obtained by transforming $$x \mapsto x'$$ obtaining the change to $$\psi(x)\mapsto \psi'(x')$$ and then returning to the original coordinate system $$x' \mapsto x$$.

The geometrical interpretation of the above is that the frame field is affine, having no preferred origin. The generator $$J_{\mu\nu}$$ generates the symmetries of this space: it provides a relabelling of a fixed point $$x~.$$ The generator $$\sigma_{\mu\nu}$$ generates a movement from one point in the fiber to another: a movement from $$x \mapsto x'$$ with both $$x$$ and $$x'$$ still corresponding to the same spacetime point $$a.$$ These perhaps obtuse remarks can be elucidated with explicit algebra.

Let $$x' = \Lambda x$$ be a Lorentz transformation. The Dirac equation is $$i\gamma^\mu \frac{\partial}{\partial x^\mu} \psi(x) -m\psi(x)=0$$ If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames: $$i\gamma^\mu \frac{\partial}{\partial x^{\prime\mu}} \psi^\prime(x^\prime) -m\psi^\prime(x^\prime)=0$$ The two spinors $$\psi$$ and $$\psi^\prime$$ should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, etc.) The transformation should encode only the change of coordinate frame. It can be shown that such a transformation is a 4×4 unitary matrix. Thus, one may presume that the relation between the two frames can be written as $$\psi^\prime(x^\prime) = S(\Lambda) \psi(x)$$ Inserting this into the transformed equation, the result is $$i\gamma^\mu \frac{\partial x^\nu}{\partial x^{\prime\mu}} \frac{\partial}{\partial x^{\nu}} S(\Lambda)\psi(x) -mS(\Lambda)\psi(x) = 0$$ The coordinates related by Lorentz transformation satisfy: $$\frac{\partial x^\nu}{\partial x^{\prime\mu}} = {\left(\Lambda^{-1}\right)^\nu}_\mu$$ The original Dirac equation is then regained if $$S(\Lambda) \gamma^\mu S^{-1}(\Lambda) = {\left(\Lambda^{-1}\right)^\mu}_\nu \gamma^\nu$$ An explicit expression for $$S(\Lambda)$$ (equal to the expression given above) can be obtained by considering a Lorentz transformation of infinitesimal rotation near the identity transformation: $${\Lambda^\mu}_\nu = {g^\mu}_\nu + {\omega^\mu}_\nu\ ,\ {(\Lambda^{-1})^\mu}_\nu = {g^\mu}_\nu - {\omega^\mu}_\nu$$ where $${g^\mu}_{\nu}$$ is the metric tensor : $${g^\mu}_{\nu}=g^{\mu\nu'}g_{\nu'\nu}={\delta^\mu}_{\nu}$$ and is symmetric while $$\omega_{\mu\nu}={\omega^{\alpha}}_{\nu} g_{\alpha\mu}$$ is antisymmetric. After plugging and chugging, one obtains $$S(\Lambda) = I + \frac{-i}{4} \omega^{\mu\nu} \sigma_{\mu\nu} + \mathcal{O}\left(\Lambda^2\right)$$ which is the (infinitesimal) form for $$S$$ above and yields the relation $$\sigma^{\mu\nu} = \frac{i}{2} [\gamma^\mu,\gamma^\nu]$$. To obtain the affine relabelling, write $$ \begin{align} \psi'(x') &= \left(I + \frac{-i}{4} \omega^{\mu\nu} \sigma_{\mu\nu} \right) \psi(x) \\ &= \left(I + \frac{-i}{4} \omega^{\mu\nu} \sigma_{\mu\nu} \right) \psi(x' + {\omega^\mu}_\nu \,x^{\prime\,\nu}) \\ &= \left(I + \frac{-i}{4} \omega^{\mu\nu} \sigma_{\mu\nu} - x^\prime_\mu \omega^{\mu\nu} \partial_\nu\right) \psi(x') \\ &= \left(I + \frac{-i}{2} \omega^{\mu\nu} J_{\mu\nu} \right) \psi(x') \\ \end{align}$$

After properly antisymmetrizing, one obtains the generator of symmetries $$J_{\mu\nu}$$ given earlier. Thus, both $$J_{\mu\nu}$$ and $$\sigma_{\mu\nu}$$ can be said to be the "generators of Lorentz transformations", but with a subtle distinction: the first corresponds to a relabelling of points on the affine frame bundle, which forces a translation along the fiber of the spinor on the spin bundle, while the second corresponds to translations along the fiber of the spin bundle (taken as a movement $$x \mapsto x'$$ along the frame bundle, as well as a movement $$\psi \mapsto \psi'$$ along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.

Other formulations
The Dirac equation can be formulated in a number of other ways.

Curved spacetime
This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the Dirac equation in curved spacetime.

The algebra of physical space
This article developed the Dirac equation using four-vectors and Schrödinger operators. The Dirac equation in the algebra of physical space uses a Clifford algebra over the real numbers, a type of geometric algebra.

Coupled Weyl Spinors
As mentioned above, the massless Dirac equation immediately reduces to the homogeneous Weyl equation. By using the chiral representation of the gamma matrices, the nonzero-mass equation can also be decomposed into a pair of coupled inhomogeneous Weyl equations acting on the first and last pairs of indices of the original four-component spinor, i.e. $$\psi = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}$$, where $$\psi_L$$ and $$\psi_R$$ are each two-component Weyl spinors. This is because the skew block form of the chiral gamma matrices means that they swap the $$\psi_L$$ and $$\psi_R$$ and apply the two-by-two Pauli matrices to each:

$$\gamma^\mu \begin{pmatrix}\psi_L \\ \psi_R \end{pmatrix} = \begin{pmatrix}\sigma^\mu \psi_R \\ \overline{\sigma}^\mu \psi_L \end{pmatrix}$$.

So the Dirac equation

$$ (i\gamma^\mu \partial_\mu - m)\begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} = 0 $$

becomes

$$ i\begin{pmatrix} \sigma^\mu \partial_\mu \psi_R \\ \overline{\sigma}^\mu \partial_\mu \psi_L \end{pmatrix} = m\begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} $$

which in turn is equivalent to a pair of inhomogeneous Weyl equations for massless left- and right-helicity spinors, where the coupling strength is proportional to the mass:

$$ i\sigma^\mu \partial_\mu \psi_R = m \psi_L $$

$$ i\overline{\sigma}^\mu \partial_\mu \psi_L = m \psi_R $$.

This has been proposed as an intuitive explanation of Zitterbewegung, as these massless components would propagate at the speed of light and move in opposite directions, since the helicity is the projection of the spin onto the direction of motion. Here the role of the "mass" $$m$$ is not to make the velocity less than the speed of light, but instead controls the average rate at which these reversals occur; specifically, the reversals can be modeled as a Poisson process.

U(1) symmetry
Natural units are used in this section. The coupling constant is labelled by convention with $$e$$: this parameter can also be viewed as modelling the electron charge.

Vector symmetry
The Dirac equation and action admits a $$\text{U}(1)$$ symmetry where the fields $$\psi, \bar\psi$$ transform as $$\begin{align} \psi(x) &\mapsto e^{i\alpha}\psi(x), \\ \bar\psi(x) &\mapsto e^{-i\alpha}\bar\psi(x). \end{align}$$ This is a global symmetry, known as the $$\text{U}(1)$$ vector symmetry (as opposed to the $$\text{U}(1)$$ axial symmetry: see below). By Noether's theorem there is a corresponding conserved current: this has been mentioned previously as $$J^\mu(x) = \bar\psi(x)\gamma^\mu\psi(x).$$

Gauging the symmetry
If we 'promote' the global symmetry, parametrised by the constant $$\alpha$$, to a local symmetry, parametrised by a function $$\alpha:\mathbb{R}^{1,3} \to \mathbb{R}$$, or equivalently $$e^{i\alpha}: \mathbb{R}^{1,3} \to \text{U}(1),$$ the Dirac equation is no longer invariant: there is a residual derivative of $$\alpha(x)$$.

The fix proceeds as in scalar electrodynamics: the partial derivative is promoted to a covariant derivative $$D_\mu$$ $$D_\mu \psi = \partial_\mu \psi + i e A_\mu\psi,$$ $$D_\mu \bar\psi = \partial_\mu \bar\psi - i e A_\mu\bar\psi.$$ The covariant derivative depends on the field being acted on. The newly introduced $$A_\mu$$ is the 4-vector potential from electrodynamics, but also can be viewed as a $$\text{U}(1)$$ gauge field, or a $$\text{U}(1)$$ connection.

The transformation law under gauge transformations for $$A_\mu$$ is then the usual $$A_\mu(x) \mapsto A_\mu(x) + \frac{1}{e}\partial_\mu\alpha(x)$$ but can also be derived by asking that covariant derivatives transform under a gauge transformation as $$D_\mu\psi(x) \mapsto e^{i\alpha(x)}D_\mu\psi(x),$$ $$D_\mu\bar\psi(x) \mapsto e^{-i\alpha(x)}D_\mu\bar\psi(x).$$ We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one: $$S = \int d^4x\,\bar\psi\,(iD\!\!\!\!\big / - m)\,\psi = \int d^4x\,\bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi.$$ The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term, $$S_{\text{Maxwell}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\right].$$ Putting these together gives

Expanding out the covariant derivative allows the action to be written in a second useful form: $$S_{\text{QED}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(i\partial\!\!\!\big / - m)\,\psi - eJ^\mu A_\mu\right]$$

Axial symmetry
Massless Dirac fermions, that is, fields $$\psi(x)$$ satisfying the Dirac equation with $$m = 0$$, admit a second, inequivalent $$\text{U}(1)$$ symmetry.

This is seen most easily by writing the four-component Dirac fermion $$\psi(x)$$ as a pair of two-component vector fields, $$\psi(x) = \begin{pmatrix} \psi_1(x)\\ \psi_2(x) \end{pmatrix}, $$ and adopting the chiral representation for the gamma matrices, so that $$i\gamma^\mu\partial_\mu$$ may be written $$i\gamma^\mu\partial_\mu = \begin{pmatrix} 0 & i\sigma^\mu \partial_\mu\\ i\bar\sigma^\mu \partial_\mu\ & 0 \end{pmatrix} $$ where $$\sigma^\mu$$ has components $$(I_2, \sigma^i)$$ and $$\bar\sigma^\mu$$ has components $$(I_2, -\sigma^i)$$.

The Dirac action then takes the form $$S = \int d^4x\, \psi_1^\dagger(i\sigma^\mu\partial_\mu)\psi_1 + \psi_2^\dagger(i\bar\sigma^\mu\partial_\mu) \psi_2.$$ That is, it decouples into a theory of two Weyl spinors or Weyl fermions.

The earlier vector symmetry is still present, where $$\psi_1$$ and $$\psi_2$$ rotate identically. This form of the action makes the second inequivalent $$\text{U}(1)$$ symmetry manifest: $$\begin{align} \psi_1(x) &\mapsto e^{i\beta} \psi_1(x), \\ \psi_2(x) &\mapsto e^{-i\beta}\psi_2(x). \end{align}$$ This can also be expressed at the level of the Dirac fermion as $$\psi(x) \mapsto \exp(i\beta\gamma^5) \psi(x)$$ where $$\exp$$ is the exponential map for matrices.

This isn't the only $$\text{U}(1)$$ symmetry possible, but it is conventional. Any 'linear combination' of the vector and axial symmetries is also a $$\text{U}(1)$$ symmetry.

Classically, the axial symmetry admits a well-formulated gauge theory. But at the quantum level, there is an anomaly, that is, an obstruction to gauging.

Extension to color symmetry
We can extend this discussion from an abelian $$\text{U}(1)$$ symmetry to a general non-abelian symmetry under a gauge group $$G$$, the group of color symmetries for a theory.

For concreteness, we fix $$G = \text{SU}(N)$$, the special unitary group of matrices acting on $$\mathbb{C}^N$$.

Before this section, $$\psi(x)$$ could be viewed as a spinor field on Minkowski space, in other words a function $$\psi: \mathbb{R}^{1,3}\mapsto \mathbb{C}^4$$, and its components in $$\mathbb{C}^4$$ are labelled by spin indices, conventionally Greek indices taken from the start of the alphabet $$\alpha,\beta,\gamma,\cdots$$.

Promoting the theory to a gauge theory, informally $$\psi$$ acquires a part transforming like $$\mathbb{C}^N$$, and these are labelled by color indices, conventionally Latin indices $$i,j,k,\cdots$$. In total, $$\psi(x)$$ has $$4N$$ components, given in indices by $$\psi^{i,\alpha}(x)$$. The 'spinor' labels only how the field transforms under spacetime transformations.

Formally, $$\psi(x)$$ is valued in a tensor product, that is, it is a function $$\psi:\mathbb{R}^{1,3} \to \mathbb{C}^4 \otimes \mathbb{C}^N.$$

Gauging proceeds similarly to the abelian $$\text{U}(1)$$ case, with a few differences. Under a gauge transformation $$U:\mathbb{R}^{1,3} \rightarrow \text{SU}(N),$$ the spinor fields transform as $$\psi(x) \mapsto U(x)\psi(x)$$ $$\bar\psi(x)\mapsto \bar\psi(x)U^\dagger(x).$$ The matrix-valued gauge field $$A_\mu$$ or $$\text{SU}(N)$$ connection transforms as $$A_\mu(x) \mapsto U(x)A_\mu(x)U(x)^{-1} + \frac{1}{g}(\partial_\mu U(x))U(x)^{-1},$$ and the covariant derivatives defined $$D_\mu\psi = \partial_\mu \psi + igA_\mu\psi,$$ $$D_\mu\bar\psi = \partial_\mu \bar\psi - ig\bar\psi A_\mu^\dagger$$ transform as $$D_\mu\psi(x) \mapsto U(x)D_\mu\psi(x),$$ $$D_\mu\bar\psi(x) \mapsto (D_\mu\bar\psi(x))U(x)^\dagger.$$

Writing down a gauge-invariant action proceeds exactly as with the $$\text{U}(1)$$ case, replacing the Maxwell Lagrangian with the Yang–Mills Lagrangian $$S_{\text{Y-M}} = \int d^4x \,-\frac{1}{4}\text{Tr}(F^{\mu\nu}F_{\mu\nu})$$ where the Yang–Mills field strength or curvature is defined here as $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ig\left[A_\mu,A_\nu\right]$$ and $$[\cdot,\cdot]$$ is the matrix commutator.

The action is then

Physical applications
For physical applications, the case $$N=3$$ describes the quark sector of the Standard Model which models strong interactions. Quarks are modelled as Dirac spinors; the gauge field is the gluon field. The case $$N=2$$ describes part of the electroweak sector of the Standard Model. Leptons such as electrons and neutrinos are the Dirac spinors; the gauge field is the $$W$$ gauge boson.

Generalisations
This expression can be generalised to arbitrary Lie group $$G$$ with connection $$A_\mu$$ and a representation $$(\rho, G, V)$$, where the colour part of $$\psi$$ is valued in $$V$$. Formally, the Dirac field is a function $$\psi:\mathbb{R}^{1,3} \to \mathbb{C}^4\otimes V.$$

Then $$\psi$$ transforms under a gauge transformation $$g:\mathbb{R}^{1,3} \to G$$ as $$\psi(x) \mapsto \rho(g(x))\psi(x)$$ and the covariant derivative is defined $$D_\mu\psi = \partial_\mu\psi + \rho(A_\mu)\psi$$ where here we view $$\rho$$ as a Lie algebra representation of the Lie algebra $$\mathfrak{g} = \text{L}(G)$$ associated to $$G$$.

This theory can be generalised to curved spacetime, but there are subtleties which arise in gauge theory on a general spacetime (or more generally still, a manifold) which, on flat spacetime, can be ignored. This is ultimately due to the contractibility of flat spacetime which allows us to view a gauge field and gauge transformations as defined globally on $$\mathbb{R}^{1,3}$$.

Articles on the Dirac equation

 * Dirac field
 * Dirac spinor
 * Gordon decomposition
 * Klein paradox
 * Nonlinear Dirac equation

Other equations

 * Breit equation
 * Dirac–Kähler equation
 * Klein–Gordon equation
 * Rarita–Schwinger equation
 * Two-body Dirac equations
 * Weyl equation
 * Majorana equation

Other topics

 * Fermionic field
 * Feynman checkerboard
 * Foldy–Wouthuysen transformation
 * Quantum electrodynamics
 * Quantum chromodynamics
 * ELKO theory

title: Dirac equation


text:
XYY syndrome

XYY syndrome, also known as Jacobs syndrome, is an aneuploid genetic condition in which a male has an extra Y chromosome. There are usually few symptoms. These may include being taller than average and an increased risk of learning disabilities. The person is generally otherwise normal, including typical rates of fertility.

The condition is generally not inherited but rather occurs as a result of a random event during sperm development. Diagnosis is by a chromosomal analysis, but most of those affected are not diagnosed within their lifetime. There are 47 chromosomes, instead of the usual 46, giving a 47,XYY karyotype.

Treatment may include speech therapy or extra help with schoolwork, however outcomes are generally positive. The condition occurs in about 1 in 1,000 male births. Many people with the condition are unaware that they have it. The condition was first described in 1961.

Physical traits
People with the 47,XYY karyotype have an increased growth rate from early childhood, with an average final height approximately 7 cm above expected final height. In Edinburgh, Scotland, eight 47,XYY boys born 1967–1972 and identified in a newborn screening programme had an average height of 188.1 cm at age 18—their fathers' average height was 174.1 cm, their mothers' average height was 162.8 cm. The increased gene dosage of three X/Y chromosome pseudoautosomal region (PAR1) SHOX genes has been postulated as a cause of the increased stature seen in all three sex chromosome trisomies: 47,XXX, 47,XXY, and 47,XYY. p. 2304: Table 561-1. Differential diagnosis of tall stature and overgrowth syndromes. Postnatal overgrowth leading to childhood tall stature—includes: Klinefelter syndrome (XXY), SHOX excess syndromes, XYY. Severe acne was noted in a very few early case reports, but dermatologists specializing in acne now doubt the existence of a relationship with 47,XYY.

Prenatal testosterone levels are normal in 47,XYY males. Most 47,XYY males have normal sexual development and have normal fertility.

Cognitive and behavioral traits
In contrast to the other common sex chromosome aneuploidies—47,XXX and 47,XXY (Klinefelter syndrome)—the average of the IQ scores of 47,XYY boys identified by newborn screening programs was not reduced compared to the general population. In a summary of six prospective studies of 47,XYY boys identified by newborn screening programmes, twenty-eight 47,XYY boys had an average 100.76 verbal IQ, 108.79 performance IQ, and 105.00 full-scale IQ. In a systematic review including two prospective studies of 47,XYY boys identified by newborn screening programs and one retrospective study of 47,XYY men identified by screening men over 184 cm in height, forty-two 47,XYY boys and men had an average 99.5 verbal IQ and 106.4 performance IQ.

In prospective studies of 47,XYY boys identified by newborn screening programs, the IQ scores of 47,XYY boys were usually slightly lower than those of their siblings. In Edinburgh, fifteen 47,XYY boys with siblings identified in a newborn screening program had an average 104.0 verbal IQ and 106.7 performance IQ, while their siblings had an average 112.9 verbal IQ and 114.6 performance IQ.

Approximately half of 47,XYY boys identified by newborn screening programs had learning difficulties—a higher proportion than found among siblings and above-average-IQ control groups. In Edinburgh, 54% of 47,XYY boys (7 of 13) identified in a newborn screening program received remedial reading teaching compared to 18% (4 of 22) in an above-average-IQ control group of 46,XY boys matched by their father's social class. In Boston, USA 55% of 47,XYY boys (6 of 11) identified in a newborn screening program had learning difficulties and received part-time resource room help compared to 11% (1 of 9) in an above-average-IQ control group of 46,XY boys with familial balanced autosomal chromosome translocations.

Developmental delays and behavioral problems are also found, but these characteristics vary widely among affected boys and men, are not unique to 47,XYY and are managed no differently from in 46,XY males. Aggression is not seen more frequently in 47,XYY males.

Patients with Jacobs syndrome have been shown to have a higher risk of developing certain diseases such as asthma, seizure problems, and tremors. Some 47,XYY patients have been found to have genitourinary malformations. These include cryptorchidism, hypoplastic scrotum, microphallus, and hypospadias. These men could be diagnosed with infertility as a result of oligospermia or sperm chromosomal abnormalities. According to certain psychological studies, people with XYY syndrome may have problems with impulse control and emotional regulation. Increased testosterone levels were found to be correlated with an increased risk of aggressive behavior in incarcerated males with 47,XYY syndrome. [[File:XYY syndrome.svg|thumb| Diagram showing XYY syndrome formation. MI and MII are the stages of meiosis, while the blue and pink circles are male (sperm) and female (ovum) cells, respectively, and the blue and pink bars are Y- and X-chromosomes, respectively.

The purple cell has 2 Y-chromosomes and 1 X-chromosome caused by the ovum fusing with a sperm cell with two Y-chromosomes, which was due to division problems in MII of the male. ]]47,XYY is not inherited; it usually occurs as a random event during the formation of sperm cells. An incident in chromosome separation during anaphase II (of meiosis II) called nondisjunction can result in sperm cells with an extra copy of the Y-chromosome. If one of these atypical sperm cells contributes to the genetic makeup of a child, the child will have an extra Y-chromosome in each of the body's cells.

In some cases, the extra Y-chromosome results from nondisjunction during mitosis in early embryonic development. This can produce 46,XY/47,XYY mosaics.

Diagnosis
47,XYY syndrome is not usually diagnosed until learning issues are present. The syndrome is diagnosed in an increasing number of children prenatally by amniocentesis and chorionic villus sampling in order to obtain a chromosome karyotype, where the abnormality can be observed.

It is estimated that only 15–20% of children with 47,XYY syndrome are ever diagnosed. Of these, approximately 30% are diagnosed prenatally. For the rest of those diagnosed after birth, around half are diagnosed during childhood or adolescence after developmental delays are observed. The rest are diagnosed after any of a variety of symptoms, including fertility problems (5%) have been seen.

Epidemiology
Around 1 in 1,000 boys are born with a 47,XYY karyotype. The incidence of 47,XYY is not known to be affected by the parents' ages.

1960s
In April 1956, Hereditas published the discovery by cytogeneticists Joe Hin Tjio and Albert Levan at Lund University in Sweden that the normal number of chromosomes in diploid human cells was 46—not 48, as had been believed for the preceding thirty years. In the wake of the establishment of the normal number of human chromosomes, 47,XYY was the last of the common sex chromosome aneuploidies to be discovered, two years after the discoveries of 47,XXY, 45,X and 47,XXX in 1959. Even the much less common 48,XXYY had been discovered in 1960, a year before 47,XYY.

Screening for those X chromosome aneuploidies was possible before the advent of human chromosome analysis by noting the presence or absence of "female" sex chromatin bodies (Barr bodies) in the nuclei of interphase cells in buccal smears, a technique developed a decade before the first reported sex chromosome aneuploidy. An analogous technique to screen for Y-chromosome aneuploidies by noting supernumerary "male" sex chromatin bodies was not developed until 1970, a decade after the first reported male sex chromosome aneuploidy.

The first published report of a man with a 47,XYY karyotype was by the American cytogeneticist Avery Sandberg and his colleagues at Roswell Park Comprehensive Cancer Center (then known as Roswell Park Memorial Institute) in Buffalo, New York in 1961. It was an incidental finding in a normal 44-year-old, 6 ft. [183 cm] tall man of average intelligence who was karyotyped because he had a daughter with Down syndrome. Only a dozen isolated 47,XYY cases were reported in the medical literature in the four years following the first report by Sandberg.

The XYY syndrome, if named after the discoverer, should rightly be termed Sandberg syndrome and not Jacobs syndrome although the British cytogeneticist Patricia Jacobs did indeed contribute meaningly to our knowledge of XYY. In December 1965 and March 1966, Nature and The Lancet published the first preliminary reports by Jacobs and her colleagues at the MRC Human Genetics Unit at Western General Hospital in Edinburgh of a chromosome survey of 315 male patients at State Hospital in Carstairs, Lanarkshire—Scotland's only special security hospital for developmentally disabled people —that found nine patients, ages 17 to 36, averaging almost 6 ft. in height (avg. 5'11", range: 5'7" to 6'2"), had a 47,XYY karyotype, and mischaracterized them as aggressive and violent criminals.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.
 * Over the next decade, almost all published XYY studies were on height-selected, institutionalized XYY males.

In January 1968 and March 1968, The Lancet and Science published the first U.S. reports of tall, institutionalized XYY males by Mary Telfer, a biochemist, and colleagues at the Elwyn Institute. Telfer found five tall, developmentally disabled XYY boys and men in hospitals and penal institutions in Pennsylvania, and since four of the five had at least moderate facial acne, reached the erroneous conclusion that acne was a defining characteristic of XYY males. After learning that convicted mass murderer Richard Speck had been karyotyped, Telfer not only incorrectly assumed the acne-scarred Speck was XYY, but reached the false conclusion that Speck was the archetypical XYY male—or "supermale" as Telfer referred to XYY males outside of peer-reviewed scientific journals. "Why do men commit crimes of violence? For some, the urge to violence may be inborn—traced to something called the Y chromosome... Once in every 500 male births, for example, the sex chromosome complement is XXY rather than XY, thus erring in the direction of femaleness. The resulting individual, called a Klinefelter male, is usually retarded, unusually tall and sterile. Erring in the other direction, however, is the XYY complement resulting in the 'supermale.' He is also unusually tall and somewhat retarded, but appears to be highly, perhaps too highly, sexually motivated... We were intrigued by Dr. Jacobs' contention that an extra Y chromosome results in tall stature, mild mental retardation, and severely disordered personality characterized by violent, aggressive behavior. We therefore planned to confirm and extend her studies.

Syndrome Status for the XYY The XXY male has long been thought to display a constellation of symptoms that makes him diagnosable; that is, he has achieved syndrome status. It would seem that the XYY male is fast achieving similar status. His symptoms, as we and other laboratories tend to think of them, are: extremely tall stature, long limbs and strikingly long arm span, facial acne, mild mental retardation, severe mental illness (including psychosis) and aggressive, antisocial behavior with a long history of arrests, frequently beginning at an early age. On reading newspaper accounts of Richard Speck, who murdered eight Chicago student nurses in 1966, we noted all these traits and therefore concluded that Speck was a likely candidate for the XYY disorder. Independently, a cytogenetic laboratory in Chicago confirmed this hunch, reinforcing our inclination to believe that the XYY syndrome is really coming of age. It seems quite possible that in the XYY male, exemplified by Speck, biologists are describing in genetic terms a certain type of defective criminal who has long been explicitly recognized by the forensic psychiatrist."

In April 1968, The New York Times—using Telfer as a main source—introduced the XYY genetic condition to the general public in a three-part series on consecutive days that began with a Sunday front-page story about the planned use of the condition as a mitigating factor in two murder trials in Paris and Melbourne —and falsely reported that Richard Speck was an XYY male and that the condition would be used in an appeal of his murder conviction.  Getty displayed a letter of Sept. 26, 1966, relating that photographic evidence of 18 cells from Speck's blood showed no chromosome abnormality. He also exhibited a letter of last July 3, indicating that 100 of 101 cells in a sample of Speck's blood studied after the original tests showed the normal 46 chromosomes. The other cell had 45, regarded by the Vanderbilt investigators as having no significance."
 * "Dr. Pergament said he and Dr. Sato, a research fellow, had absolutely no connection with the Speck case and never examined Speck. The report was also denied by Speck's attorney, Public Defender Gerald W. Getty. 'I never knew those doctors existed before I read about them in the paper,' Getty said. Getty did say that a chromosomal test was performed on Speck, before Speck's trial, by a geneticist from outside the Chicago area. He declined to identify the geneticist, and he said the results of the test never have been disclosed. 'It was agreed,' he said, 'that the results would not be disclosed unless I wished them disclosed. And I still don't.' In any case, Getty said, the results could not be used in an appeal—since they were not part of the trial evidence. If anything, he said, they could only be used in connection with a new trial."
 * "At the same time he made public reports from Vanderbilt University showing no abnormal makeup of Speck's chromosomes...
 * "Dr. Pergament said he and Dr. Sato, a research fellow, had absolutely no connection with the Speck case and never examined Speck. The report was also denied by Speck's attorney, Public Defender Gerald W. Getty. 'I never knew those doctors existed before I read about them in the paper,' Getty said. Getty did say that a chromosomal test was performed on Speck, before Speck's trial, by a geneticist from outside the Chicago area. He declined to identify the geneticist, and he said the results of the test never have been disclosed. 'It was agreed,' he said, 'that the results would not be disclosed unless I wished them disclosed. And I still don't.' In any case, Getty said, the results could not be used in an appeal—since they were not part of the trial evidence. If anything, he said, they could only be used in connection with a new trial."
 * "At the same time he made public reports from Vanderbilt University showing no abnormal makeup of Speck's chromosomes...
 * "At the same time he made public reports from Vanderbilt University showing no abnormal makeup of Speck's chromosomes...
 * —article by Vanderbilt University endocrinologist and geneticist Eric Engel, who performed two confidential chromosome analyses of Speck in September 1966 and June 1968. Based on mischaracterizations of XYY males as aggressive and violent criminals in the December 1965 and March 1966 preliminary reports by Jacobs, et al., Engel had made an unsolicited request in August 1966 to Speck's appointed defense attorney, Cook County Public Defender Gerald W. Getty, to confidentially karyotype Speck—which was repeated after false news reports in April 1968 that Speck was XYY. The series was echoed the following week by articles—again using Telfer as a main source—in Time and Newsweek, and six months later in The New York Times Magazine.

In December 1968, the Journal of Medical Genetics published the first XYY review article—by Willam Michael Court Brown (1918–1969), director of the MRC Human Genetics Unit—which reported that he had found no overrepresentation of XYY males in nationwide chromosome surveys of prisons and hospitals for developmentally disabled and mentally ill people in Scotland, and concluded that studies confined to institutionalized XYY males may be guilty of selection bias, and that long-term longitudinal prospective studies of newborn XYY boys were needed.

In May 1969, at the annual meeting of the American Psychiatric Association, Telfer and her Elwyn Institute colleagues reported that case studies of the institutionalized XYY and XXY males they had found convinced them that XYY males had been falsely stigmatized and that their behavior may not be significantly different from chromosomally normal 46,XY males.

In June 1969, the National Institute of Mental Health (NIMH) Center for Studies of Crime and Delinquency held a two-day XYY conference in Chevy Chase, Maryland. In December 1969, with a grant from the NIMH Center for Studies of Crime and Delinquency, cytogeneticist Digamber Borgaonkar at Johns Hopkins Hospital began a chromosome survey of (predominantly African-American) boys ages 8 to 18 in all Maryland institutions for delinquent, neglected, or mentally ill juveniles, which was suspended from February–May 1970 due to an American Civil Liberties Union (ACLU) lawsuit regarding the study's lack of informed consent.
 * reprinted in Katz (1972). Experimentation with human beings, pp. 342–343.
 * reprinted in Katz (1972). Experimentation with human beings, pp. 343–344.
 * reprinted in Katz (1972). Experimentation with human beings, p. 344.
 * reprinted in Katz (1972). Experimentation with human beings, p. 344.

In the late 1960s and early 1970s, screening of consecutive newborns for sex chromosome abnormalities was undertaken at seven centers worldwide: in Denver (Jan 1964–1974), Edinburgh (Apr 1967–Jun 1979), New Haven (Oct 1967–Sep 1968), Toronto (Oct 1967–Sep 1971), Aarhus (Oct 1969–Jan 1974, Oct 1980–Jan 1989), Winnipeg (Feb 1970–Sep 1973), and Boston (Apr 1970–Nov 1974). The Boston study, led by Harvard Medical School child psychiatrist Stanley Walzer at Children's Hospital, was unique among the seven newborn screening studies in that it only screened newborn boys (non-private-ward newborn boys at the Boston Hospital for Women) and was funded in part by grants from the NIMH Center for Studies of Crime and Delinquency.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.
 * The Edinburgh study was led by Shirley Ratcliffe who focused her career on it and published the results in 1999.

1970s
In December 1969, Lore Zech at the Karolinska Institute in Stockholm first reported intense fluorescence of the A T-rich distal half of the long arm of the Y chromosome in the nuclei of metaphase cells treated with quinacrine mustard. In April 1970, Peter Pearson and Martin Bobrow at the MRC Population Genetics Unit in Oxford and Canino Vosa at the University of Oxford reported fluorescent "male" sex chromatin bodies in the nuclei of interphase cells in buccal smears treated with quinacrine dihydrochloride, which could be used to screen for Y chromosome aneuploidies like 47,XYY.

In December 1970, at the annual meeting of the American Association for the Advancement of Science (AAAS), its retiring president, geneticist H. Bentley Glass, cheered by the legalization of abortion in New York, envisioned a future where pregnant women would be required by the government to abort XYY "sex deviants". Mischaracterization of the XYY genetic condition was quickly incorporated into high school biology textbooks and medical school psychiatry textbooks, where misinformation still persists decades later.

In 1973, child psychiatrist Herbert Schreier at Children's Hospital told Harvard Medical School microbiologist Jon Beckwith of Science for the People that he thought Walzer's Boston XYY study was unethical; Science for the People investigated the study and, regarding the study, filed an ethics complaint with Harvard Medical School in March 1974. In November 1974, Science for the People went public with their objections to the Boston XYY study in a press conference and a New Scientist article alleging inadequate informed consent, a lack of benefit (since no specific treatment was available) but substantial risk (by stigmatization with a false stereotype) to the subjects, and that the unblinded experimental design could not produce meaningful results regarding the subjects' behavior. In December 1974, the Harvard Standing Committee on Medical Research issued a report supporting the Boston XYY study, and in March 1975, the faculty voted 199–35 to allow continuation of the study. After April 1975, screening of newborns was discontinued—changes to informed consent procedures and pressure from additional advocacy groups, including the Children's Defense Fund, having led to the discontinuation of the last active U.S. newborn screening programs for sex chromosome abnormalities in Boston and Denver.

In a paper published in New Scientist on November 14, 1974 entitled "XYY syndrome: a dangerous myth" found no link was found between having an extra Y chromosome and violent behavior. According to the paper, adolescents found to have an extra Y chromosome in a Maryland institution were chemically sterilized to attempt to maintain "normal behavior," despite this the paper found no major behavioral differences between XY and XYY individuals.

In August 1976, Science published a retrospective cohort study by Educational Testing Service psychologist Herman Witkin and colleagues that screened the tallest 16% of men (over 184 cm (6'0") in height) born in Copenhagen from 1944 to 1947 for XXY and XYY karyotypes, and found an increased rate of minor criminal convictions for property crimes among sixteen XXY and twelve XYY men may be related to the lower intelligence of those with criminal convictions, but found no evidence that XXY or XYY men were inclined to be aggressive or violent.

1980s and later
The March of Dimes sponsored five international conferences in June 1974, November 1977, May 1981, June 1984, and June 1989 and published articles from the conferences in book form in 1979, 1982, 1986, and 1991 from seven longitudinal prospective cohort studies on the development of over 300 children and young adults with sex chromosome abnormalities identified in the screening of almost 200,000 consecutive births in hospitals in Denver, Edinburgh, New Haven, Toronto, Aarhus, Winnipeg, and Boston from 1964 to 1975. These seven studies—the only unbiased studies of unselected individuals with sex chromosome abnormalities—have replaced the older, biased studies of institutionalized individuals in understanding the development of individuals with sex chromosome abnormalities.

In May 1997, Nature Genetics published the discovery by Ercole Rao and colleagues of the X/Y chromosome pseudoautosomal region (PAR1) SHOX gene, haploinsufficiency of which leads to short stature in Turner syndrome (45,X). It was subsequently postulated that the increased gene dosage of three SHOX genes leads to tall stature in the sex chromosome trisomies 47,XXX, 47,XXY, and 47,XYY.

In July 1999, Psychological Medicine published a case-control study by Royal Edinburgh Hospital psychiatrist Michael Götz and colleagues that found an increased rate of criminal convictions among seventeen XYY men identified in the Edinburgh newborn screening study compared to an above-average-IQ control group of sixty XY men, which multiple logistic regression analysis indicated was mediated mainly through lowered intelligence.

In June 2002, the American Journal of Medical Genetics published results from a longitudinal prospective cohort Denver Family Development Study led by pediatrician and geneticist Arthur Robinson, which found that in fourteen prenatally diagnosed 47,XYY boys (from high socioeconomic status families), IQ scores available for six boys ranged from 100 to 147 with a mean of 120. For the eleven of fourteen boys with siblings, in nine instances their siblings were stronger academically, but in one case the subject was performing equal to, and in another case superior to, his siblings.

Society and culture
Some medical geneticists question whether the term "syndrome" is appropriate for this condition because many people with this karyotype appear normal.

In popular culture
In June 1970, The XYY Man was published—the first of seven Kenneth Royce spy novels whose fictional tall, intelligent, nonviolent XYY hero was a reformed expert cat burglar recruited by British intelligence for dangerous assignments—and later adapted into a thirteen-episode British summer television series broadcast in 1976 and 1977.

In other fictional television works, a January 1971 episode "By the Pricking of My Thumbs ..." of the British science fiction TV series Doomwatch featured an XYY boy expelled from school because his genetic condition led him to be falsely accused of nearly blinding another boy. A November 1993 episode of the American police procedural TV series Law & Order, "Born Bad", portrays a 14-year-old XYY sociopathic murderer. The May 2007 season finale episode, "Born To Kill", of the American police procedural TV series CSI: Miami depicts a 34-year-old XYY serial killer.

The false stereotype of XYY boys and men as violent criminals has also been used as a plot device in the horror films Il gatto a nove code in February 1971 (dubbed into English as The Cat o' Nine Tails in May 1971) and Alien 3 in May 1992. The main character of the 2005 film Neo Ned is a neo-nazi who has an extra Y chromosome.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: XYY syndrome


text:
Spoiler effect

In social choice theory and politics, the spoiler effect or Arrow's paradox refers to a situation where a losing (that is, irrelevant) candidate affects the results of an election. A voting system that is not affected by spoilers satisfies independence of irrelevant alternatives or independence of spoilers.

Arrow's impossibility theorem is a well-known theorem showing that all rank-based voting systems are vulnerable to the spoiler effect. However, the frequency and severity of spoiler effects depends on the voting method.

Plurality and ranked choice (plurality-loser) are highly sensitive to spoilers, and can manufacture spoiler effects even when doing so is not forced. Majority-rule methods are usually not affected by spoilers, which are limited to rare situations called cyclic ties.

Rated voting systems are not subject to Arrow's theorem; as a result, many satisfy independence of irrelevant alternatives (sometimes called spoilerproofness).

Motivation
Social choice theorists have long argued that voting methods should be spoiler-independent (at least so far as this is possible). The Marquis de Condorcet studied the same property going back to the 1780s.

Rational behavior
In decision theory, independence of irrelevant alternatives (IIA) is a fundamental principle of rationality, which says that which of two outcomes A or B is better, should not depend on how good another outcome (C) is. A famous joke by Sidney Morgenbesser illustrates this principle:"A man is deciding whether to order apple or blueberry pie before settling on apple. The waitress informs him that cherry pie is also an option, to which the man replies 'in that case, I'll have the blueberry.'"Social choice theorists argue it would be better to have a mechanism for making societal decisions that behaves rationally (or if this is not possible, one that is at least usually rational).

Manipulation by politicians
Voting systems that violate independence of irrelevant alternatives are susceptible to being manipulated by strategic nomination. Some systems are particularly infamous for their ease of manipulation, such as the Borda count, which lets any party "clone their way to victory" by running a large number of candidates. This famously forced de Borda to concede that "my system is meant only for honest men," and eventually led to its abandonment by the French Academy of Sciences.

Vote-splitting systems like choose-one and instant-runoff (ranked choice) voting have the opposite problem: because running many similar candidates at once makes it difficult for any of them to win the election, these systems tend to concentrate power in the hands of parties and political machines, which serve the role of clearing the field and signalling a single candidate that voters should focus their support on; in many cases, this leads plurality voting systems to behave like a de facto two-round system, where the top-two candidates are nominated by party primaries.

In some situations, a spoiler can extract concessions from other candidates by threatening to remain in the race unless they are bought off, typically with a promise of a high-ranking political position.

Fairness
Because a candidate's quality and popularity clearly do not depend on whether an unpopular candidate runs for office, it seems intuitively unfair or undemocratic for a voting system to behave as if it does. A voting system that is objectively fair to candidates and their supporters should not behave like a lottery; it should select the highest-quality candidate regardless of factors outside of a candidate's control (like whether or not another politician decides to run).

Arrow's theorem
Arrow's impossibility theorem is a major result in social choice theory, which proves that every ranked-choice voting system is vulnerable to spoiler effects.

However, rated voting systems are not affected by Arrow's theorem. Approval voting, range voting, and median voting all satisfy the IIA criterion: if we disqualify or add losing candidates, without changing ratings on votes, the score (and therefore winner) remains unchanged.

By electoral system
Different electoral systems have different levels of vulnerability to spoilers. As a rule of thumb, spoilers are extremely common with plurality voting, common in plurality-runoff methods, rare with paired counting (Condorcet), and impossible with rated voting.

Plurality-runoff methods like the two-round system and instant-runoff voting still suffer from vote-splitting in each round. As a result, they do not eliminate the spoiler effect. The elimination of weak spoilers in earlier rounds somewhat reduces their effects on the results compared to single-round plurality voting, but spoiled elections remain common, moreso than in other systems.

Modern tournament voting eliminates vote splitting effects completely, because every one-on-one matchup is evaluated independently. If there is a Condorcet winner, Condorcet methods are completely invulnerable to spoilers; in practice, somewhere between 90% and 99% of real-world elections have a Condorcet winner. Some systems like ranked pairs have even stronger spoilerproofing guarantees that are applicable to most situations without a Condorcet winner.

Cardinal voting methods can be fully immune to spoiler effects.

Plurality voting
Vote splitting most easily occurs in plurality voting. In the United States vote splitting commonly occurs in primary elections. The purpose of primary elections is to eliminate vote splitting among candidates in the same party before the general election. If primary elections or party nominations are not used to identify a single candidate from each party, the party that has more candidates is more likely to lose because of vote splitting among the candidates from the same party. In a two-party system, party primaries effectively turn plurality voting into a two-round system.

Vote splitting is the most common cause of spoiler effects in the commonly-used plurality vote and two-round runoff systems. In these systems, the presence of many ideologically similar candidates causes their vote total to be split between them, placing these candidates at a disadvantage. This is most visible in elections where a minor candidate draws votes away from a major candidate with similar politics, thereby causing a strong opponent of both to win.

Runoff systems
Spoilers also occur in the two-round system and instant-runoff voting at a substantially higher rate than for modern pairwise-counting or rated voting methods, though slightly less often than in plurality. As a result, instant-runoff voting still tends towards two-party rule.

In Burlington, Vermont's second IRV election, spoiler Kurt Wright knocked out Democrat Andy Montroll in the second round, leading to the election of Bob Kiss (despite the election results showing Montroll would have won a one-on-one election with Kiss). In Alaska's first-ever IRV election, Nick Begich was defeated in the first round by spoiler candidate Sarah Palin.

Tournament (Condorcet) voting
Spoiler effects rarely occur when using tournament solutions, because each candidate's total in a paired comparison does not involve any other candidates. Instead, methods can separately compare every pair of candidates and check who would win in a one-on-one election. This pairwise comparison means that spoilers can only occur in the rare situation known as a Condorcet cycle.

For each pair of candidates, there is a count for how many voters prefer the first candidate (in the pair) to the second candidate, and how many voters have the opposite preference. The resulting table of pairwise counts eliminates the step-by-step redistribution of votes, which causes vote splitting in other methods.

Rated voting
Rated voting methods ask voters to assign each candidate a score on a scale (usually from 0 to 10), instead of listing them from first to last. The best-known of these methods is score voting, which elects the candidate with the highest total number of points. Because voters rate candidates independently, changing one candidate's score does not affect those of other candidates, which is what allows rated methods to evade Arrow's theorem.

While true spoilers are not possible under score voting, voters who behave strategically in response to candidates can create pseudo-spoiler effects (which can be distinguished from true spoilers in that they are caused by voter behavior, rather than the voting system itself).

Weaker forms
Several weaker forms of independence of irrelevant alternatives (IIA) have been proposed as a way to compare ranked voting methods. Usually these procedures try to insulate the process from weak spoilers, ensuring that only a handful of candidates can change the outcome.

Local independence of irrelevant alternatives
Local independence from irrelevant alternatives (LIIA) is a weaker kind of independence that requires both of the following conditions:
 * 1) If the option that finished in last place is deleted from all the votes, the winner should not change.
 * 2) If the option that finished in first place is deleted from all the votes, the runner-up should win.

For every electoral method, it is possible to construct an order-of-finish that ranks candidates in terms of strength. This can be done by first finding the winner, then repeatedly deleting them and finding a new winner. This process is repeated to find which candidates rank 3rd, 4th, etc. As a result, LIIA can also be thought of as indicating independence from the weakest alternative, i.e. the alternative who would not win unless every other candidate dropped out.

Despite being a very weak form of spoiler-resistance (requiring that only the last-place finisher is unable to affect the outcome), LIIA is satisfied by only a few voting methods. These include Kemeny-Young and ranked pairs, but not Schulze or instant-runoff voting. Rated methods such as approval voting, range voting, and majority judgment also pass.

Condorcet independence criteria
Besides its interpretation in terms of majoritarianism, the Condorcet criterion can be interpreted as a kind of spoiler-resistance. In general, Condorcet methods are highly resistant to spoiler effects. Intuitively, this is because the only way to dislodge a beats-all champion is by beating them, so spoilers can only exist when there is no beats-all champion (which is rare). This property, of stability for Condorcet winners, is a major advantage of Condorcet methods.

Smith-independence is another kind of spoiler-resistance for Condorcet methods. This criterion says that a candidate should not affect the results of an election, unless they have a "reasonable claim" to the title of Condorcet winner (fall in the Smith set). Smith candidates are ones who can defeat every other candidate either directly or indirectly (by beating some candidate A who defeats B).

Independence of clones
Independence of clones is the most commonly-fulfilled spoiler-resistance criterion, and says that "cloning" a candidate—adding a new candidate identical to an existing one—should not affect the results. Two candidates are considered identical if they are ranked equally on every ballot. The criterion is satisfied by instant-runoff voting, all systems that satisfy independence of irrelevant alternatives (including cardinal systems), and most tournament solutions.

However, it is worth noting this criterion is extremely fragile, as even a single voter expressing a preference for one candidate over the other (or placing another candidate between them) can nullify a system's protection.

Borda count
In a Borda count, 5 voters rank 5 alternatives [A, B, C, D, E].

3 voters rank [A>B>C>D>E]. 1 voter ranks [C>D>E>B>A]. 1 voter ranks [E>C>D>B>A].

Borda count (a=0, b=1): C=13, A=12, B=11, D=8, E=6. C wins.

Now, the voter who ranks [C>D>E>B>A] instead ranks [C>B>E>D>A]; and the voter who ranks [E>C>D>B>A] instead ranks [E>C>B>D>A]. They change their preferences only over the pairs [B, D], [B, E] and [D, E].

The new Borda count: B=14, C=13, A=12, E=6, D=5. B wins.

The social choice has changed the ranking of [B, A] and [B, C]. The changes in the social choice ranking are dependent on irrelevant changes in the preference profile. In particular, B now wins instead of C, even though no voter changed their preference over [B, C].

Condorcet methods
A single example is enough to show that every Condorcet method must fail independence of irrelevant alternatives. Say that 3 candidates are in a Condorcet cycle. Label them Rock, Paper, and Scissors. In a one-on-one race, Rock loses to Paper, Paper to Scissors, etc. Without loss of generality, say that Rock wins the election with a certain method. Then, Scissors is a spoiler candidate for Paper: if Scissors were to drop out, Paper would win the only one-on-one race (Paper defeats Rock). The same reasoning applies regardless of the winner.

This example also shows why Condorcet elections are rarely (if ever) spoiled: spoilers can only happen if there is no Condorcet winner. Condorcet cycles are rare in large elections, and the median voter theorem shows cycles are impossible whenever candidates are arrayed on a left-right spectrum.

Plurality
Plurality voting is a degenerate form of ranked-choice voting, where the top-rated candidate receives a single point while all others receive none. The following example shows a plurality voting system with 7 voters ranking 3 alternatives (A, B, C).

 * 3 voters rank (A>B>C)
 * 2 voters rank (B>A>C)
 * 2 voters rank (C>B>A)

In an election, initially only A and B run: B wins with 4 votes to A's 3, but the entry of C into the race makes A the new winner.

The relative positions of A and B are reversed by the introduction of C, an "irrelevant" alternative.

title: Spoiler effect