Dirac equation

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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. The equation is validated by its rigorous accounting of the observed fine structure of the hydrogen spectrum and 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 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.

The equation, in its natural units formulation, is also prominently displayed in the auditorium at the ‘Paul A.M. Dirac’ Lecture Hall at the Patrick M.S. Blackett Institute (formerly The San Domenico Monastery) of the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Sicily.

History

The Dirac equation in the form originally proposed by Dirac is:$${\displaystyle \left(\beta m{c}^2+c\sum _{n=1}^3{\alpha }_n{p}_n\right)\psi (x,t)=i\hslash \frac{\mathrm{\partial }\psi (x,t)}{\mathrm{\partial }t}}$$ where ψ(x, t) is the wave function for an electron of rest mass m with spacetime coordinates x, t. p₁, p₂, p₃ 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. α_n and β are 4 × 4 gamma matrices.

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 α₁, α₂, α₃ 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: $${\displaystyle {\alpha }_i^2={\beta }^2=I_4}$$ and they all mutually anti-commute: $${\displaystyle \begin{array}{rl}{\alpha }_i{\alpha }_j+{\alpha }_j{\alpha }_i& =0(i\ne j)\\ {\alpha }_i\beta +\beta {\alpha }_i& =0\end{array}}$$

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).

Making the Schrödinger equation relativistic

The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle: $${\displaystyle -\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\varphi =i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\varphi .}$$

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 $${\displaystyle E^2=m^2{c}^4+p^2{c}^2,}$$ 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, $${\displaystyle \left(-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}+{\mathrm{\nabla }}^2\right)\varphi =\frac{m^2{c}^2}{{\hslash }^2}\varphi ,}$$ with the wave function ${\displaystyle \varphi }$ being a relativistic scalar: a complex number that 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 $${\displaystyle \rho ={\varphi }^{\ast }\varphi }$$ and this density is convected according to the probability current vector $${\displaystyle J=-\frac{i\hslash }{2m}({\varphi }^{\ast }\mathrm{\nabla }\varphi -\varphi \mathrm{\nabla }{\varphi }^{\ast })}$$ with the conservation of probability current and density following from the continuity equation: $${\displaystyle \mathrm{\nabla }\cdot J+\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}=0.}$$

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 $${\displaystyle \rho =\frac{i\hslash }{2m{c}^2}\left({\psi }^{\ast }{\mathrm{\partial }}_t\psi -\psi {\mathrm{\partial }}_t{\psi }^{\ast }\right),}$$ which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression $${\displaystyle J^{\mu }=\frac{i\hslash }{2m}\left({\psi }^{\ast }{\mathrm{\partial }}^{\mu }\psi -\psi {\mathrm{\partial }}^{\mu }{\psi }^{\ast }\right).}$$

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 ∂_tψ 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 $${\displaystyle E\psi =(\overrightarrow{\alpha }\cdot \overrightarrow{p}+\beta m)\psi }$$ where the operators ${\displaystyle (\overrightarrow{\alpha },\beta )}$ must be independent of ${\displaystyle (\overrightarrow{p},t)}$ for linearity and independent of ${\displaystyle (\overrightarrow{x},t)}$ for space-time homogeneity. These constraints implied additional dynamical variables that the ${\displaystyle (\overrightarrow{\alpha },\beta )}$ operators will depend upon; from this requirement Dirac concluded that the operators would depend upon 4 × 4 matrices, related to the Pauli matrices.

One could, for example, formally (i.e. by abuse of notation, since it is not straightforward to take a functional square root of the sum of two differential operators) take the relativistic expression for the energy $${\displaystyle E=c\sqrt{p^2+m^2{c}^2},}$$ 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: $${\displaystyle {\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}=\left(A{\mathrm{\partial }}_x+B{\mathrm{\partial }}_y+C{\mathrm{\partial }}_z+\frac{i}{c}D{\mathrm{\partial }}_t\right)\left(A{\mathrm{\partial }}_x+B{\mathrm{\partial }}_y+C{\mathrm{\partial }}_z+\frac{i}{c}D{\mathrm{\partial }}_t\right).}$$

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 $${\displaystyle AB+BA=0,\dots }$$ with $${\displaystyle A^2=B^2=\cdots =1.}$$

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 $${\displaystyle \left(A{\mathrm{\partial }}_x+B{\mathrm{\partial }}_y+C{\mathrm{\partial }}_z+\frac{i}{c}D{\mathrm{\partial }}_t\right)\psi =\kappa \psi }$$ with ${\displaystyle \kappa }$ to be determined. Applying again the matrix operator on both sides yields $${\displaystyle \left({\mathrm{\nabla }}^2-\frac{1}{c^2}{\mathrm{\partial }}_t^2\right)\psi ={\kappa }^2\psi .}$$

Taking ${\displaystyle \kappa =\frac{mc}{\hslash }}$ 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 $${\displaystyle \left(A{\mathrm{\partial }}_x+B{\mathrm{\partial }}_y+C{\mathrm{\partial }}_z+\frac{i}{c}D{\mathrm{\partial }}_t-\frac{mc}{\hslash }\right)\psi =0.}$$

Setting $${\displaystyle A=i\beta {\alpha }_1,B=i\beta {\alpha }_2,C=i\beta {\alpha }_3,D=\beta ,}$$ and because ${\displaystyle D^2={\beta }^2=I_4}$, 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: $${\displaystyle \begin{array}{rl}D& ={\gamma }^0,\\ A& =i{\gamma }^1,B=i{\gamma }^2,C=i{\gamma }^3,\end{array}}$$ and the equation takes the form (remembering the definition of the covariant components of the 4-gradient and especially that ∂₀ = ⁠1/c⁠∂_t)

Dirac equation

${\displaystyle (i\hslash {\gamma }^{\mu }{\mathrm{\partial }}_{\mu }-mc)\psi =0}$

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 $${\displaystyle {\gamma }^0=\left(\begin{array}{cc}{I}_2& 0\\ 0& -I_2\end{array}\right),{\gamma }^1=\left(\begin{array}{cc}0& {\sigma }_x\\ -{\sigma }_x& 0\end{array}\right),{\gamma }^2=\left(\begin{array}{cc}0& {\sigma }_y\\ -{\sigma }_y& 0\end{array}\right),{\gamma }^3=\left(\begin{array}{cc}0& {\sigma }_z\\ -{\sigma }_z& 0\end{array}\right).}$$

The complete system is summarized using the Minkowski metric on spacetime in the form $${\displaystyle \left\{{\gamma }^{\mu },{\gamma }^{\nu }\right\}=2{\eta }^{\mu \nu }{I}_4}$$ where the bracket expression $${\displaystyle \{a,b\}=ab+ba}$$ 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: $${\displaystyle {\mathrm{P}}_{\mathsf{o}\mathsf{p}}\psi =mc\psi .}$$

Using ${\displaystyle \mathrm{\partial }/\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }}$ (${\displaystyle \mathrm{\partial }/}$ is pronounced "d-slash"), according to Feynman slash notation, the Dirac equation becomes: $${\displaystyle i\hslash \mathrm{\partial }/\psi -mc\psi =0.}$$

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

Dirac equation (natural units)

${\displaystyle (i\mathrm{\partial }/-m)\psi =0}$

A foundational 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 transform: $${\displaystyle {\gamma }^{\mu \mathrm{\prime }}=S^{-1}{\gamma }^{\mu }S.}$$

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

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 previously mentioned foundational 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 $${\displaystyle \begin{array}{rl}\left(i{U}^{†}{\gamma }^{\mu }U{\mathrm{\partial }}_{\mu }^{\mathrm{\prime }}-m\right)\psi \left(x^{\mathrm{\prime }},t^{\mathrm{\prime }}\right)& =0\\ U^{†}(i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }^{\mathrm{\prime }}-m)U\psi \left(x^{\mathrm{\prime }},t^{\mathrm{\prime }}\right)& =0.\end{array}}$$

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

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_μγ_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 $${\displaystyle V=\frac{1}{4!}{ϵ}_{\mu \nu \alpha \beta }{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\alpha }{\gamma }^{\beta }.}$$

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 $${\displaystyle V=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3.}$$

This matrix is given the special symbol γ⁵, 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 $${\displaystyle {\gamma }_5=\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right).}$$

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

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.

Comparison with related theories

Pauli theory

See also: Pauli equation and Lévy-Leblond equation

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 that 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 ⁠${\displaystyle A_{\mu }=\left(\varphi /c,-\mathbf{A}\right)}$⁠.) $${\displaystyle H=\frac{1}{2m}(\mathit{\sigma }\cdot (\mathbf{p}-e\mathbf{A}){)}^2+e\varphi .}$$

Here A and ${\displaystyle \varphi }$ 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: $${\displaystyle H=\frac{1}{2m}(\mathbf{p}-e\mathbf{A}{)}^2+e\varphi -\frac{e\hslash }{2m}\mathit{\sigma }\cdot \mathbf{B}.}$$

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: $${\displaystyle ({\gamma }^{\mu }(i\hslash {\mathrm{\partial }}_{\mu }-e{A}_{\mu })-mc)\psi =0.}$$

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: $${\displaystyle \left(\begin{array}{cc}m{c}^2-E+e\varphi & +c\mathit{\sigma }\cdot \left(\mathbf{p}-e\mathbf{A}\right)\\ -c\mathit{\sigma }\cdot \left(\mathbf{p}-e\mathbf{A}\right)& m{c}^2+E-e\varphi \end{array}\right)\left(\begin{array}{c}{\psi }_{+}\\ {\psi }_{-}\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).}$$ so $${\displaystyle \begin{array}{rl}(E-e\varphi ){\psi }_{+}-c\mathit{\sigma }\cdot \left(\mathbf{p}-e\mathbf{A}\right){\psi }_{-}& =m{c}^2{\psi }_{+}\\ c\mathit{\sigma }\cdot \left(\mathbf{p}-e\mathbf{A}\right){\psi }_{+}-\left(E-e\varphi \right){\psi }_{-}& =m{c}^2{\psi }_{-}\end{array}.}$$

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, $${\displaystyle \begin{array}{rl}E-e\varphi & \approx m{c}^2\\ \mathbf{p}& \approx m\mathbf{v}\end{array}}$$ and so the second equation may be written $${\displaystyle {\psi }_{-}\approx \frac{1}{2mc}\mathit{\sigma }\cdot (\mathbf{p}-e\mathbf{A}){\psi }_{+},}$$ which is of order ${\displaystyle \frac{v}{c}.}$ 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 $${\displaystyle (E-m{c}^2){\psi }_{+}=\frac{1}{2m}[\mathit{\sigma }\cdot (\mathbf{p}-e\mathbf{A}){]}^2{\psi }_{+}+e\varphi {\psi }_{+}}$$

The operator on the left represents the particle's total energy reduced by its rest energy, which is just its classical kinetic 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 ostensibly in the form of a diffusion equation, actually represents wave propagation.

It should be strongly emphasized that the entire Dirac spinor represents an irreducible whole. The separation, done here, of the Dirac spinor into large and small components depends on the low-energy approximation being valid. The components that were neglected above, to show that the Pauli theory can be recovered by a low-velocity approximation of Dirac's equation, are necessary to produce new phenomena observed in the relativistic regime – among them antimatter, and the creation and annihilation of particles.

Weyl theory

In the massless case ${\displaystyle m=0}$, the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin-1/2 particles.

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

Physical interpretation

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 self-adjoint 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 $${\displaystyle H={\gamma }^0\left[m{c}^2+c{\gamma }^k\left(p_k-q{A}_k\right)\right]+cq{A}^0.}$$ 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 cqA⁰. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is $${\displaystyle H=c\sqrt{{\left(\mathbf{p}-q\mathbf{A}\right)}^2+m^2{c}^2}+q{A}^0.}$$

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

See also: Fermionic field

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 ${\displaystyle \psi }$ taking values in a complex vector space described concretely as ${\displaystyle {\mathbb{C}}^4}$, defined on flat spacetime (Minkowski space) ${\displaystyle {\mathbb{R}}^{1,3}}$. Its expression also contains gamma matrices and a parameter ${\displaystyle m>0}$ 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 ${\displaystyle \psi :{\mathbb{R}}^{1,3}\to {\mathbb{C}}^4}$, the Dirac equation is then

$${\displaystyle (i\hslash {\gamma }^{\mu }{\mathrm{\partial }}_{\mu }-mc)\psi (x)=0}$$

and in natural units, with Feynman slash notation,

$${\displaystyle (i\mathrm{\partial }/-m)\psi (x)=0}$$

The gamma matrices are a set of four ${\displaystyle 4\times 4}$ complex matrices (elements of ${\displaystyle {\text{Mat}}_{4\times 4}(\mathbb{C})}$) that satisfy the defining anti-commutation relations: $${\displaystyle \{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }{I}_4}$$ where ${\displaystyle {\eta }^{\mu \nu }}$ is the Minkowski metric element, and the indices ${\displaystyle \mu ,\nu }$ 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 and the chiral representation. The Dirac representation is $${\displaystyle {\gamma }^0=\left(\begin{array}{cc}{I}_2& 0\\ 0& -I_2\end{array}\right),{\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right),}$$ where ${\displaystyle {\sigma }^i}$ are the Pauli matrices.

For the chiral representation the ${\displaystyle {\gamma }^i}$ are the same, but ${\displaystyle {\gamma }^0=\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right).}$

The slash notation is a compact notation for $${\displaystyle A/:={\gamma }^{\mu }{A}_{\mu }}$$ where ${\displaystyle A}$ is a four-vector (often it is the four-vector differential operator ${\displaystyle {\mathrm{\partial }}_{\mu }}$). The summation over the index ${\displaystyle \mu }$ is implied.

Alternatively the four coupled linear first-order partial differential equations for the four quantities that make up the wave function can be written as a vector. In Planck units this becomes:$${\displaystyle i{\mathrm{\partial }}_x\left[\begin{array}{c}+{\psi }_4\\ +{\psi }_3\\ -{\psi }_2\\ -{\psi }_1\end{array}\right]+{\mathrm{\partial }}_y\left[\begin{array}{c}+{\psi }_4\\ -{\psi }_3\\ -{\psi }_2\\ +{\psi }_1\end{array}\right]+i{\mathrm{\partial }}_z\left[\begin{array}{c}+{\psi }_3\\ -{\psi }_4\\ -{\psi }_1\\ +{\psi }_2\end{array}\right]-m\left[\begin{array}{c}+{\psi }_1\\ +{\psi }_2\\ +{\psi }_3\\ +{\psi }_4\end{array}\right]=i{\mathrm{\partial }}_t\left[\begin{array}{c}-{\psi }_1\\ -{\psi }_2\\ +{\psi }_3\\ +{\psi }_4\end{array}\right]}$$ which makes it clearer that it is a set of four partial differential equations with four unknown functions. (Note that the ⁠${\displaystyle {\mathrm{\partial }}_y}$⁠ term is not preceded by i because σ_y is imaginary.)

Dirac adjoint and the adjoint equation

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

Klein–Gordon equation

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

Conserved current

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

Proof of conservation from Dirac equation

Adding the Dirac and adjoint Dirac equations gives $${\displaystyle i(({\mathrm{\partial }}_{\mu }\overline{\psi }){\gamma }^{\mu }\psi +\overline{\psi }{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }\psi )=0}$$ so by Leibniz rule, $${\displaystyle i{\mathrm{\partial }}_{\mu }(\overline{\psi }{\gamma }^{\mu }\psi )=0}$$

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

Proof of conservation from Noether's theorem

Recall the Lagrangian is $${\displaystyle \mathcal{L}=\overline{\psi }(i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }-m)\psi .}$$ Under a ${\displaystyle U(1)}$ symmetry that sends $${\displaystyle \begin{array}{rl}\psi & \mapsto e^{i\alpha }\psi ,\\ \overline{\psi }& \mapsto e^{-i\alpha }\overline{\psi },\end{array}}$$ we find the Lagrangian is invariant.

Now considering the variation parameter ${\displaystyle \alpha }$ to be infinitesimal, we work at first order in ${\displaystyle \alpha }$ and ignore ${\displaystyle \mathcal{O}{\alpha }^2}$ terms. From the previous discussion we immediately see the explicit variation in the Lagrangian due to ${\displaystyle \alpha }$ is vanishing, that is under the variation, $${\displaystyle \mathcal{L}\mapsto \mathcal{L}+\delta \mathcal{L},}$$ where ${\displaystyle \delta \mathcal{L}=0}$.

As part of Noether's theorem, we find the implicit variation in the Lagrangian due to variation of fields. If the equation of motion for ${\displaystyle \psi ,\overline{\psi }}$ are satisfied, then

$${\displaystyle \delta \mathcal{L}={\mathrm{\partial }}_{\mu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\psi )}\delta \psi +\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\overline{\psi })}\delta \overline{\psi }\right)}$$

*

This immediately simplifies as there are no partial derivatives of ${\displaystyle \overline{\psi }}$ in the Lagrangian. ${\displaystyle \delta \psi }$ is the infinitesimal variation $${\displaystyle \delta \psi (x)=i\alpha \psi (x).}$$ We evaluate $${\displaystyle \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\psi )}=i\overline{\psi }{\gamma }^{\mu }.}$$ The equation (*) finally is $${\displaystyle 0=-\alpha {\mathrm{\partial }}_{\mu }(\overline{\psi }{\gamma }^{\mu }\psi )}$$

Solutions

Further information: Dirac spinor and § Hole theory

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 $${\displaystyle \psi (x)=u(\mathbf{p})e^{-ip\cdot x}}$$ which models a particle with definite 4-momentum ${\displaystyle p=(E_{\mathbf{p}},\mathbf{p})}$ where $E_{\mathbf{p}}=\sqrt{m^2+|\mathbf{p}{|}^2}.$

For this ansatz, the Dirac equation becomes an equation for ${\displaystyle u(\mathbf{p})}$: $${\displaystyle \left({\gamma }^{\mu }{p}_{\mu }-m\right)u(\mathbf{p})=0.}$$ After picking a representation for the gamma matrices ${\displaystyle {\gamma }^{\mu }}$, 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 ${\displaystyle {\gamma }^{\mu }}$, the solution space is parametrised by a ${\displaystyle {\mathbb{C}}^2}$ vector ${\displaystyle \xi }$, with $${\displaystyle u(\mathbf{p})=\left(\begin{array}{c}\sqrt{{\sigma }^{\mu }{p}_{\mu }}\xi \\ \sqrt{{\overline{\sigma }}^{\mu }{p}_{\mu }}\xi \end{array}\right)}$$ where ${\displaystyle {\sigma }^{\mu }=(I_2,{\sigma }^i),{\overline{\sigma }}^{\mu }=(I_2,-{\sigma }^i)}$ and ${\displaystyle \sqrt{\cdot }}$ 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: $${\displaystyle \mathcal{L}=i\hslash c\overline{\psi }{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }\psi -m{c}^2\overline{\psi }\psi }$$

If one varies this with respect to ${\displaystyle \psi }$ one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to ${\displaystyle \overline{\psi }}$ one gets the Dirac equation.

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

Dirac Action

${\displaystyle S=\int d^{4}x\overline{\psi }(i\mathrm{\partial }/-m)\psi }$

For this action, the conserved current ${\displaystyle J^{\mu }}$ above arises as the conserved current corresponding to the global ${\displaystyle \text{U}(1)}$ 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 ${\displaystyle \text{SO}(1,3)}$ or strictly ${\displaystyle \text{SO}(1,3{)}^{+}}$, the component connected to the identity.

For a Dirac spinor viewed concretely as taking values in ${\displaystyle {\mathbb{C}}^4}$, the transformation under a Lorentz transformation ${\displaystyle \mathrm{\Lambda }}$ is given by a ${\displaystyle 4\times 4}$ complex matrix ${\displaystyle S[\mathrm{\Lambda }]}$. There are some subtleties in defining the corresponding ${\displaystyle S[\mathrm{\Lambda }]}$, 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 ${\displaystyle 4\times 4}$ real matrices acting on ${\displaystyle {\mathbb{R}}^{1,3}}$ is generated by a set of six matrices ${\displaystyle \{M^{\mu \nu }\}}$ with components $${\displaystyle (M^{\mu \nu }{)}^{\rho }{}_{\sigma }={\eta }^{\mu \rho }{\delta }^{\nu }{}_{\sigma }-{\eta }^{\nu \rho }{\delta }^{\mu }{}_{\sigma }.}$$ When both the ${\displaystyle \rho ,\sigma }$ indices are raised or lowered, these are simply the 'standard basis' of antisymmetric matrices.

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

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

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

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

Remainder of proof of Lorentz invariance

Multiplying both sides from the left by ${\displaystyle S^{-1}[\mathrm{\Lambda }]}$ and returning the dummy variable to ${\displaystyle x}$ gives $${\displaystyle iS[\mathrm{\Lambda }{]}^{-1}{\gamma }^{\mu }S[\mathrm{\Lambda }](({\mathrm{\Lambda }}^{-1}{)}_{\mu }{}^{\nu }{\mathrm{\partial }}_{\nu })\psi (x)-m\psi (x)=0.}$$ We'll have shown invariance if $${\displaystyle S[\mathrm{\Lambda }{]}^{-1}{\gamma }^{\mu }S[\mathrm{\Lambda }]({\mathrm{\Lambda }}^{-1}{)}^{\nu }{}_{\mu }={\gamma }^{\nu }}$$ or equivalently $${\displaystyle S[\mathrm{\Lambda }{]}^{-1}{\gamma }^{\mu }S[\mathrm{\Lambda }]={\mathrm{\Lambda }}^{\mu }{}_{\nu }{\gamma }^{\nu }.}$$ This is most easily shown at the algebra level. Supposing the transformations are parametrised by infinitesimal components ${\displaystyle {\omega }_{\mu \nu }}$, then at first order in ${\displaystyle \omega }$, on the left-hand side we get $${\displaystyle \frac{1}{2}{\omega }_{\rho \sigma }(M^{\rho \sigma }{)}^{\mu }{}_{\nu }{\gamma }^{\nu }}$$ while on the right-hand side we get $${\displaystyle \left[\frac{1}{2}{\omega }_{\rho \sigma }{S}^{\rho \sigma },{\gamma }^{\mu }\right]=\frac{1}{2}{\omega }_{\rho \sigma }\left[S^{\rho \sigma },{\gamma }^{\mu }\right]}$$ It's a standard exercise to evaluate the commutator on the left-hand side. Writing ${\displaystyle M^{\rho \sigma }}$ in terms of components completes the proof.

Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents ${\displaystyle ({\mathcal{J}}^{\rho \sigma }{)}^{\mu }}$. Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents ${\displaystyle T^{\mu \nu }}$, which can be identified as the stress-energy tensor of the theory. The Lorentz current ${\displaystyle ({\mathcal{J}}^{\rho \sigma }{)}^{\mu }}$ 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 ${\displaystyle a}$ 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 ${\displaystyle x}$ and ${\displaystyle x^{\prime }}$, with the understanding that both ${\displaystyle x}$ and ${\displaystyle x^{\prime }}$ describe the same point ${\displaystyle a}$, but in different local frames of reference (a frame of reference over a small extended patch of spacetime). One can imagine ${\displaystyle a}$ 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 ${\displaystyle x}$ and ${\displaystyle x^{\prime }}$ 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 ${\displaystyle x\mapsto x^{\prime },}$ the Dirac spinor to transform as $${\displaystyle {\psi }^{\prime }(x^{\prime })=S\psi (x)}$$ It can be shown that an explicit expression for ${\displaystyle S}$ is given by $${\displaystyle S=\exp \left(\frac{-i}{4}{\omega }^{\mu \nu }{\sigma }_{\mu \nu }\right)}$$ where ${\displaystyle {\omega }^{\mu \nu }}$ parameterizes the Lorentz transformation, and ${\displaystyle {\sigma }_{\mu \nu }}$ are the six 4×4 matrices satisfying: $${\displaystyle {\sigma }^{\mu \nu }=\frac{i}{2}[{\gamma }^{\mu },{\gamma }^{\nu }].}$$

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 ${\displaystyle J_{\mu \nu }}$ of Lorentz transformations, having the form $${\displaystyle J_{\mu \nu }=\frac{1}{2}{\sigma }_{\mu \nu }+i(x_{\mu }{\mathrm{\partial }}_{\nu }-x_{\nu }{\mathrm{\partial }}_{\mu })}$$ This can be interpreted as the total angular momentum. It acts on the spinor field as $${\displaystyle {\psi }^{\mathrm{\prime }}(x)=\exp \left(\frac{-i}{2}{\omega }^{\mu \nu }{J}_{\mu \nu }\right)\psi (x)}$$ Note the ${\displaystyle x}$ above does not have a prime on it: the above is obtained by transforming ${\displaystyle x\mapsto x^{\prime }}$ obtaining the change to ${\displaystyle \psi (x)\mapsto {\psi }^{\prime }(x^{\prime })}$ and then returning to the original coordinate system ${\displaystyle x^{\prime }\mapsto x}$.

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

Let ${\displaystyle x^{\prime }=\mathrm{\Lambda }x}$ be a Lorentz transformation. The Dirac equation is $${\displaystyle i{\gamma }^{\mu }\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mu }}\psi (x)-m\psi (x)=0}$$ If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames: $${\displaystyle i{\gamma }^{\mu }\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }\mu }}{\psi }^{\mathrm{\prime }}(x^{\mathrm{\prime }})-m{\psi }^{\mathrm{\prime }}(x^{\mathrm{\prime }})=0}$$ The two spinors ${\displaystyle \psi }$ and ${\displaystyle {\psi }^{\mathrm{\prime }}}$ 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 $${\displaystyle {\psi }^{\mathrm{\prime }}(x^{\mathrm{\prime }})=S(\mathrm{\Lambda })\psi (x)}$$ Inserting this into the transformed equation, the result is $${\displaystyle i{\gamma }^{\mu }\frac{\mathrm{\partial }{x}^{\nu }}{\mathrm{\partial }{x}^{\mathrm{\prime }\mu }}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\nu }}S(\mathrm{\Lambda })\psi (x)-mS(\mathrm{\Lambda })\psi (x)=0}$$ The coordinates related by Lorentz transformation satisfy: $${\displaystyle \frac{\mathrm{\partial }{x}^{\nu }}{\mathrm{\partial }{x}^{\mathrm{\prime }\mu }}={{\left({\mathrm{\Lambda }}^{-1}\right)}^{\nu }}_{\mu }}$$ The original Dirac equation is then regained if $${\displaystyle S(\mathrm{\Lambda }){\gamma }^{\mu }{S}^{-1}(\mathrm{\Lambda })={{\left({\mathrm{\Lambda }}^{-1}\right)}^{\mu }}_{\nu }{\gamma }^{\nu }}$$ An explicit expression for ${\displaystyle S(\mathrm{\Lambda })}$ (equal to the expression given above) can be obtained by considering a Lorentz transformation of infinitesimal rotation near the identity transformation: $${\displaystyle {{\mathrm{\Lambda }}^{\mu }}_{\nu }={g^{\mu }}_{\nu }+{{\omega }^{\mu }}_{\nu },{({\mathrm{\Lambda }}^{-1}{)}^{\mu }}_{\nu }={g^{\mu }}_{\nu }-{{\omega }^{\mu }}_{\nu }}$$ where ${\displaystyle {g^{\mu }}_{\nu }}$ is the metric tensor : ${\displaystyle {g^{\mu }}_{\nu }=g^{\mu {\nu }^{\prime }}{g}_{{\nu }^{\prime }\nu }={{\delta }^{\mu }}_{\nu }}$ and is symmetric while ${\displaystyle {\omega }_{\mu \nu }={{\omega }^{\alpha }}_{\nu }{g}_{\alpha \mu }}$ is antisymmetric. After plugging and chugging, one obtains $${\displaystyle S(\mathrm{\Lambda })=I+\frac{-i}{4}{\omega }^{\mu \nu }{\sigma }_{\mu \nu }+\mathcal{O}\left({\mathrm{\Lambda }}^2\right),}$$ which is the (infinitesimal) form for ${\displaystyle S}$ above and yields the relation ${\displaystyle {\sigma }^{\mu \nu }=\frac{i}{2}[{\gamma }^{\mu },{\gamma }^{\nu }]}$ . To obtain the affine relabelling, write $${\displaystyle \begin{array}{rl}{\psi }^{\prime }(x^{\prime })& =\left(I+\frac{-i}{4}{\omega }^{\mu \nu }{\sigma }_{\mu \nu }\right)\psi (x)\\ & =\left(I+\frac{-i}{4}{\omega }^{\mu \nu }{\sigma }_{\mu \nu }\right)\psi (x^{\prime }+{{\omega }^{\mu }}_{\nu }{x}^{\mathrm{\prime }\nu })\\ & =\left(I+\frac{-i}{4}{\omega }^{\mu \nu }{\sigma }_{\mu \nu }-x_{\mu }^{\mathrm{\prime }}{\omega }^{\mu \nu }{\mathrm{\partial }}_{\nu }\right)\psi (x^{\prime })\\ & =\left(I+\frac{-i}{2}{\omega }^{\mu \nu }{J}_{\mu \nu }\right)\psi (x^{\prime })\end{array}}$$

After properly antisymmetrizing, one obtains the generator of symmetries ${\displaystyle J_{\mu \nu }}$ given earlier. Thus, both ${\displaystyle J_{\mu \nu }}$ and ${\displaystyle {\sigma }_{\mu \nu }}$ 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 ${\displaystyle x\mapsto x^{\prime }}$ along the frame bundle, as well as a movement ${\displaystyle \psi \mapsto {\psi }^{\prime }}$ 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. ${\displaystyle \psi =\left(\begin{array}{c}{\psi }_L\\ {\psi }_R\end{array}\right)}$, where ${\displaystyle {\psi }_L}$ and ${\displaystyle {\psi }_R}$ are each two-component Weyl spinors. This is because the skew block form of the chiral gamma matrices means that they swap the ${\displaystyle {\psi }_L}$ and ${\displaystyle {\psi }_R}$ and apply the two-by-two Pauli matrices to each: $${\displaystyle {\gamma }^{\mu }\left(\begin{array}{c}{\psi }_L\\ {\psi }_R\end{array}\right)=\left(\begin{array}{c}{\sigma }^{\mu }{\psi }_R\\ {\overline{\sigma }}^{\mu }{\psi }_L\end{array}\right).}$$

So the Dirac equation $${\displaystyle (i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }-m)\left(\begin{array}{c}{\psi }_L\\ {\psi }_R\end{array}\right)=0}$$ becomes $${\displaystyle i\left(\begin{array}{c}{\sigma }^{\mu }{\mathrm{\partial }}_{\mu }{\psi }_R\\ {\overline{\sigma }}^{\mu }{\mathrm{\partial }}_{\mu }{\psi }_L\end{array}\right)=m\left(\begin{array}{c}{\psi }_L\\ {\psi }_R\end{array}\right),}$$ 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: $${\displaystyle i{\sigma }^{\mu }{\mathrm{\partial }}_{\mu }{\psi }_R=m{\psi }_L}$$ $${\displaystyle i{\overline{\sigma }}^{\mu }{\mathrm{\partial }}_{\mu }{\psi }_L=m{\psi }_R.}$$^{[clarification needed]}

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" ${\displaystyle m}$ 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 ${\displaystyle e}$: this parameter can also be viewed as modelling the electron charge.

Vector symmetry

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

Gauging the symmetry

See also: Quantum electrodynamics

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

The fix proceeds as in scalar electrodynamics: the partial derivative is promoted to a covariant derivative ${\displaystyle D_{\mu }}$ $${\displaystyle D_{\mu }\psi ={\mathrm{\partial }}_{\mu }\psi +ie{A}_{\mu }\psi ,}$$ $${\displaystyle D_{\mu }\overline{\psi }={\mathrm{\partial }}_{\mu }\overline{\psi }-ie{A}_{\mu }\overline{\psi }.}$$ The covariant derivative depends on the field being acted on. The newly introduced ${\displaystyle A_{\mu }}$ is the 4-vector potential from electrodynamics, but also can be viewed as a ${\displaystyle \text{U}(1)}$ gauge field (which, mathematically, is defined as a ${\displaystyle \text{U}(1)}$ connection).

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

QED Action

${\displaystyle S_{\text{QED}}=\int d^{4}x\left[-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }+\overline{\psi }(iD/-m)\psi \right]}$

Expanding out the covariant derivative allows the action to be written in a second useful form: $${\displaystyle S_{\text{QED}}=\int d^{4}x\left[-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }+\overline{\psi }(i\mathrm{\partial }/-m)\psi -e{J}^{\mu }{A}_{\mu }\right]}$$

Axial symmetry

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

This is seen most easily by writing the four-component Dirac fermion ${\displaystyle \psi (x)}$ as a pair of two-component vector fields, $${\displaystyle \psi (x)=\left(\begin{array}{c}{\psi }_1(x)\\ {\psi }_2(x)\end{array}\right),}$$ and adopting the chiral representation for the gamma matrices, so that ${\displaystyle i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }}$ may be written $${\displaystyle i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }=\left(\begin{array}{cc}0& i{\sigma }^{\mu }{\mathrm{\partial }}_{\mu }\\ i{\overline{\sigma }}^{\mu }{\mathrm{\partial }}_{\mu }& 0\end{array}\right)}$$ where ${\displaystyle {\sigma }^{\mu }}$ has components ${\displaystyle (I_2,{\sigma }^i)}$ and ${\displaystyle {\overline{\sigma }}^{\mu }}$ has components ${\displaystyle (I_2,-{\sigma }^i)}$.

The Dirac action then takes the form $${\displaystyle S=\int d^{4}x{\psi }_1^{†}(i{\sigma }^{\mu }{\mathrm{\partial }}_{\mu }){\psi }_1+{\psi }_2^{†}(i{\overline{\sigma }}^{\mu }{\mathrm{\partial }}_{\mu }){\psi }_2.}$$ That is, it decouples into a theory of two Weyl spinors or Weyl fermions.

The earlier vector symmetry is still present, where ${\displaystyle {\psi }_1}$ and ${\displaystyle {\psi }_2}$ rotate identically. This form of the action makes the second inequivalent ${\displaystyle \text{U}(1)}$ symmetry manifest: $${\displaystyle \begin{array}{rl}{\psi }_1(x)& \mapsto e^{i\beta }{\psi }_1(x),\\ {\psi }_2(x)& \mapsto e^{-i\beta }{\psi }_2(x).\end{array}}$$ This can also be expressed at the level of the Dirac fermion as $${\displaystyle \psi (x)\mapsto \exp (i\beta {\gamma }^5)\psi (x)}$$ where ${\displaystyle \exp }$ is the exponential map for matrices.

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

See also: quantum chromodynamics

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

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

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

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

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

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

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

The action is then

QCD Action

${\displaystyle S_{\text{QCD}}=\int d^{4}x\left[-\frac{1}{4}\text{Tr}(F^{\mu \nu }{F}_{\mu \nu })+\overline{\psi }(iD/-m)\psi \right]}$

Physical applications

For physical applications, the case ${\displaystyle N=3}$ 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 ${\displaystyle N=2}$ 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 ${\displaystyle W}$ gauge boson.

Generalisations

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

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

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

Emic and etic

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Emic_and_etic

In anthropology, folkloristics, linguistics, and the social and behavioral sciences, emic (/ˈiːmɪk/) and etic (/ˈɛtɪk/) refer to two kinds of field research done and viewpoints obtained.

The emic approach is an insider's perspective, which looks at the beliefs, values, and practices of a particular culture from the perspective of the people who live within that culture. This approach aims to understand the cultural meaning and significance of a particular behavior or practice, as it is understood by the people who engage in it.

The etic approach, on the other hand, is an outsider's perspective, which looks at a culture from the perspective of an outside observer or researcher. This approach tends to focus on the observable behaviors and practices of a culture, and aims to understand them in terms of their functional or evolutionary significance. The etic approach often involves the use of standardized measures and frameworks to compare different cultures and may involve the use of concepts and theories from other disciplines, such as psychology or sociology.

The emic and etic approaches each have their own strengths and limitations, and each can be useful in understanding different aspects of culture and behavior. Some anthropologists argue that a combination of both approaches is necessary for a complete understanding of a culture, while others argue that one approach may be more appropriate depending on the specific research question being addressed.

Definitions

"The emic approach investigates how local people think...". How they perceive and categorize the world, their rules for behavior, what has meaning for them, and how they imagine and explain things. "The etic (scientist-oriented) approach shifts the focus from local observations, categories, explanations, and interpretations to those of the anthropologist. The etic approach realizes that members of a culture often are too involved in what they are doing... to interpret their cultures impartially. When using the etic approach, the ethnographer emphasizes what he or she considers important."

Although emics and etics are sometimes regarded as inherently in conflict and one can be preferred to the exclusion of the other, the complementarity of emic and etic approaches to anthropological research has been widely recognized, especially in the areas of interest concerning the characteristics of human nature as well as the form and function of human social systems.

...Emic knowledge and interpretations are those existing within a culture, that are 'determined by local custom, meaning, and belief' (Ager and Loughry, 2004: n.p.) and best described by a 'native' of the culture. Etic knowledge refers to generalizations about human behavior that are considered universally true, and commonly links cultural practices to factors of interest to the researcher, such as economic or ecological conditions, that cultural insiders may not consider very relevant (Morris et al., 1999).

Emic and etic approaches of understanding behavior and personality fall under the study of cultural anthropology, which states that people are shaped by their cultures and their subcultures, and we must account for this in the study of personality. One way is looking at things through an emic approach. This approach "is culture specific because it focuses on a single culture and it is understood on its own terms." As explained below, the term "emic" originated from the specific linguistic term "phonemic", from phoneme, which is a language-specific way of abstracting speech sounds.

• An emic account is a description of behavior or a belief in terms meaningful (consciously or unconsciously) to the actor; that is, an emic account comes from a person within the culture. Almost anything from within a culture can provide an emic account.
• An etic account is a description of a behavior or belief by a social analyst or scientific observer (a student or scholar of anthropology or sociology, for example), in terms that can be applied across cultures; that is, an etic account attempts to be 'culturally neutral', limiting any ethnocentric, political or cultural bias or alienation by the observer.

When these two approaches are combined, the "richest" view of a culture or society can be understood. On its own, an emic approach would struggle with applying overarching values to a single culture. The etic approach is helpful in enabling researchers to see more than one aspect of one culture, and in applying observations to cultures around the world.

History

The terms were coined in 1954 by linguist Kenneth Pike, who argued that the tools developed for describing linguistic behaviors could be adapted to the description of any human social behavior. As Pike noted, social scientists have long debated whether their knowledge is objective or subjective. Pike's innovation was to turn away from an epistemological debate and turn instead to a methodological solution. Emic and etic are derived from the linguistic terms phonemic and phonetic, respectively, where a phone is a distinct speech sound or gesture (such distinction being referred to as phonetic), regardless of whether the exact sound is critical to the meanings of words, whereas a phoneme is a speech sound in a given language that, if swapped with another phoneme, could change one word to another. The possibility of a truly objective description was discounted by Pike himself in his original work; he proposed the emic–etic dichotomy in anthropology as a way around philosophic issues about the very nature of objectivity.

The terms were also championed by anthropologists Ward Goodenough and Marvin Harris with slightly different connotations from those used by Pike. Goodenough was primarily interested in understanding the culturally specific meaning of specific beliefs and practices; Harris was primarily interested in explaining human behavior.

Pike, Harris, and others have argued that cultural "insiders" and "outsiders" are equally capable of producing emic and etic accounts of their culture. Some researchers use "etic" to refer to outsider accounts, and "emic" to refer to insider accounts.

Margaret Mead was an anthropologist who studied the patterns of adolescence in Samoa. She discovered that the difficulties and the transitions that adolescents faced are culturally influenced. The hormones that are released during puberty can be defined using an etic framework, because adolescents globally have the same hormones being secreted. However, Mead concluded that how adolescents respond to these hormones is greatly influenced by their cultural norms. Through her studies, Mead found that simple classifications about behaviors and personality could not be used because peoples’ cultures influenced their behaviors in such a radical way. Her studies helped create an emic approach of understanding behaviors and personality. Her research deduced that culture has a significant impact in shaping an individual's personality.

Carl Jung, a Swiss psychoanalyst, is a researcher who took an emic approach in his studies. Jung studied mythology, religion, ancient rituals, and dreams, leading him to believe that there are archetypes that can be identified and used to categorize people's behaviors. Archetypes are universal structures of the collective unconscious that refer to the inherent way people are predisposed to perceive and process information. The main archetypes that Jung studied were the persona (how people choose to present themselves to the world), the anima and animus (part of people experiencing the world in viewing the opposite sex, that guides how they select their romantic partner), and the shadow (dark side of personalities because people have a concept of evil; well-adjusted people must integrate both good and bad parts of themselves). Jung looked at the role of the mother and deduced that all people have mothers and see their mothers in a similar way; they offer nurture and comfort. His studies also suggest that "infants have evolved to suck milk from the breast, it is also the case that all children have inborn tendencies to react in certain ways." This way of looking at the mother is an emic way of applying a concept cross-culturally and universally.

Importance as regards personality

Emic and etic approaches are important to understanding personality because problems can arise "when concepts, measures, and methods are carelessly transferred to other cultures in attempts to make cross-cultural generalizations about personality." It is hard to apply certain generalizations of behavior to people who are so diverse and culturally different. One example of this is the F-scale (Macleod). The F-scale, which was created by Theodor Adorno, is used to measure authoritarian personality, which can, in turn, be used to predict prejudiced behaviors. This test, when applied to Americans, accurately depicts prejudices towards black individuals. However, when a study was conducted in South Africa using the F-Scale (Pettigrew and Friedman), results did not predict any prejudices towards black individuals. This study used emic approaches of study by conducting interviews with the locals and etic approaches by giving participants generalized personality tests.

Schrödinger's cat

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat

Schrödinger's cat: a cat, a flask of poison, and a radioactive source connected to a Geiger counter are placed in a sealed box. As illustrated, the quantum description uses a superposition of an alive cat and one that has died.

In quantum mechanics, Schrödinger's cat is a thought experiment concerning quantum superposition. In the thought experiment, a hypothetical cat in a closed box may be considered to be simultaneously both alive and dead while it is unobserved, as a result of its fate being linked to a random subatomic event that may or may not occur. This experiment, viewed this way, is described as a paradox. This thought experiment was devised by physicist Erwin Schrödinger in 1935 in a discussion with Albert Einstein to illustrate what Schrödinger saw as the problems of the Copenhagen interpretation of quantum mechanics.

In Schrödinger's original formulation, a cat, a flask of poison, and a radioactive source are placed in a sealed box. If an internal radiation monitor such as a Geiger counter detects radioactivity (a single atom decaying), the flask is shattered, releasing the poison, which kills the cat. If no decaying atom triggers the monitor, the cat remains alive. The Copenhagen interpretation implies that the cat is therefore simultaneously alive and dead. Yet, when one looks in the box, one sees the cat either alive or dead, not both alive and dead. This poses the question of when exactly quantum superposition ends and reality resolves into one possibility or the other.

Although originally a critique on the Copenhagen interpretation, Schrödinger's seemingly paradoxical thought experiment became part of the foundation of quantum mechanics. It is often featured in theoretical discussions of the interpretations of quantum mechanics, particularly in situations involving the measurement problem. As a result, Schrödinger's cat has had enduring appeal in popular culture. The experiment is not intended to be actually performed on a cat, but rather as an easily understandable illustration of the behavior of atoms. Experiments at the atomic scale have been carried out, showing that very small objects may exist as superpositions, but superposing an object as large as a cat would pose considerable technical difficulties.

Fundamentally, the Schrödinger's cat experiment asks how long quantum superpositions last and when (or whether) they collapse. Different interpretations of the mathematics of quantum mechanics have been proposed that give different explanations for this process.

Origin and motivation

Unsolved problem in physics

 

How does the quantum description of reality, which includes elements such as the superposition of states and wavefunction collapse or quantum decoherence, give rise to the reality we perceive? Another way of stating this question regards the measurement problem: What constitutes a "measurement" that apparently causes the wave function to collapse into a definite state?

 

More unsolved problems in physics

Schrödinger intended his thought experiment as a discussion of the EPR article—named after its authors Einstein, Podolsky, and Rosen—in 1935. The EPR article highlighted the counterintuitive nature of quantum superpositions, in which a quantum system for two particles does not separate even when the particles are detected far from their last point of contact. The EPR paper concludes with a claim that this lack of separability meant that quantum mechanics as a theory of reality was incomplete.

Schrödinger and Einstein exchanged letters about Einstein's EPR article, in the course of which Einstein pointed out that the state of an unstable keg of gunpowder will, after a while, contain a superposition of both exploded and unexploded states.

To further illustrate, Schrödinger described how one could, in principle, create a superposition in a large-scale system by making it dependent on a quantum particle that was in a superposition. He proposed a scenario with a cat in a closed steel chamber, wherein the cat's life or death depended on the state of a radioactive atom, whether it had decayed and emitted radiation or not. According to Schrödinger, the Copenhagen interpretation implies that the cat remains both alive and dead until the state has been observed. Schrödinger did not wish to promote the idea of dead-and-live cats as a serious possibility; on the contrary, he intended the example to illustrate the absurdity of the existing view of quantum mechanics, thus employing reductio ad absurdum.

Since Schrödinger's time, various interpretations of the mathematics of quantum mechanics have been advanced by physicists, some of which regard the "alive and dead" cat superposition as quite real, others do not. Intended as a critique of the Copenhagen interpretation (the prevailing orthodoxy in 1935), the Schrödinger's cat thought experiment remains a touchstone for modern interpretations of quantum mechanics and can be used to illustrate and compare their strengths and weaknesses.

Thought experiment

A life-size cat figure in the garden of Huttenstrasse 9, Zurich, where Erwin Schrödinger lived from 1921 to 1926. Depending on the light conditions, the figure appears to be either a live cat or a dead one.

Schrödinger wrote:

One can contrive even completely burlesque [farcical] cases. A cat is put in a steel chamber along with the following infernal device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny amount of radioactive substance, so tiny that in the course of an hour one of the atoms will perhaps decay, but also, with equal probability, that none of them will; if it does happen, the counter tube will discharge and through a relay release a hammer that will shatter a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would tell oneself that the cat is still alive if no atom has decayed in the meantime. Even a single atomic decay would have poisoned it. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or spread out in equal parts.

It is typical of these cases that an indeterminacy originally restricted to the atomic domain turns into a sensually observable [macroscopic] indeterminacy, which can then be resolved by direct observation. This prevents us from so naïvely accepting a "blurred model" as representative of reality. Per se, it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.

Schrödinger developed his famous thought experiment in correspondence with Einstein. He suggested this 'quite ridiculous case' to illustrate his conclusion that the wave function cannot represent reality. The wave function description of the complete cat system implies that the reality of the cat mixes the living and dead cat. Einstein was impressed by the ability of the thought experiment to highlight these issues. In a letter to Schrödinger dated 1950, he wrote:

You are the only contemporary physicist, besides Laue, who sees that one cannot get around the assumption of reality, if only one is honest. Most of them simply do not see what sort of risky game they are playing with reality—reality as something independent of what is experimentally established. Their interpretation is, however, refuted most elegantly by your system of radioactive atom + amplifier + charge of gun powder + cat in a box, in which the psi-function of the system contains both the cat alive and blown to bits. Nobody really doubts that the presence or absence of the cat is something independent of the act of observation.

Note that the charge of gunpowder is not mentioned in Schrödinger's setup, which uses a Geiger counter as an amplifier and hydrocyanic poison instead of gunpowder. The gunpowder had been mentioned in Einstein's original suggestion to Schrödinger 15 years before, and Einstein carried it forward to the present discussion.

Analysis

In modern terms Schrödinger's hypothetical cat experiment describes the measurement problem: quantum theory describes the cat system as a combination of two possible outcomes but only one outcome is ever observed. The experiment poses the question, "when does a quantum system stop existing as a superposition of states and become one or the other?" (More technically, when does the actual quantum state stop being a non-trivial linear combination of states, each of which resembles different classical states, and instead begin to have a unique classical description?) Standard microscopic quantum mechanics describes multiple possible outcomes of experiments but only one outcome is observed. The thought experiment illustrates this apparent paradox. Our intuition says that the cat cannot be in more than one state simultaneously—yet the quantum mechanical description of the thought experiment requires such a condition.

Interpretations

Since Schrödinger's time, other interpretations of quantum mechanics have been proposed that give different answers to the questions posed by Schrödinger's cat of how long superpositions last and when (or whether) they collapse.

Copenhagen interpretation

Main article: Copenhagen interpretation

A commonly held interpretation of quantum mechanics is the Copenhagen interpretation. In the Copenhagen interpretation, a measurement results in only one state of a superposition. This thought experiment makes apparent the fact that this interpretation simply provides no explanation for the state of the cat while the box is closed. The wavefunction description of the system consists of a superposition of the states "decayed nucleus/dead cat" and "undecayed nucleus/living cat". Only when the box is opened and observed can we make a statement about the cat.

Role of consciousness

Main article: Consciousness causes collapse

In 1932, John von Neumann described in his book Mathematical Foundations of Quantum Mechanics a pattern where the radioactive source is observed by a device, which itself is observed by another device and so on. It makes no difference in the predictions of quantum theory where along this chain of causal effects the superposition collapses. This potentially infinite chain could be broken if the last device is replaced by a conscious observer. This solved the problem because it was claimed that an individual's consciousness cannot be multiple. Eugene Wigner asserted that an observer is necessary for a collapse to one or the other (e.g., either a live cat or a dead cat) of the terms on the right-hand side of a wave function. Wigner discussed the interpretation in a thought experiment known as Wigner's friend.

Wigner supposed that a friend opened the box and observed the cat without telling anyone. From Wigner's conscious perspective, the friend is now part of the wave function and has seen a live cat and seen a dead cat. To a third person's conscious perspective, Wigner himself becomes part of the wave function once Wigner learns the outcome from the friend. This could be extended indefinitely.

A resolution of the paradox is that the triggering of the Geiger counter counts as a measurement of the state of the radioactive substance. Because a measurement has already occurred deciding the state of the cat, the subsequent observation by a human records only what has already occurred. Analysis of an actual experiment by Roger Carpenter and A. J. Anderson found that measurement alone (for example by a Geiger counter) is sufficient to collapse a quantum wave function before any human knows of the result. The apparatus indicates one of two colors depending on the outcome. The human observer sees which color is indicated, but they don't consciously know which outcome the color represents. A second human, the one who set up the apparatus, is told of the color and becomes conscious of the outcome, and the box is opened to check if the outcome matches. However, it is disputed whether merely observing the color counts as a conscious observation of the outcome.

Bohr's interpretation

Analysis of the work of Niels Bohr, one of the main scientists associated with the Copenhagen interpretation, suggests he viewed the state of the cat before the box is opened as indeterminate. The superposition itself had no physical meaning to Bohr: Schrödinger's cat would be either dead or alive long before the box is opened but the cat and box form a inseparable combination. Bohr saw no role for a human observer. Bohr emphasized the classical nature of measurement results. An "irreversible" or effectively irreversible process imparts the classical behavior of "observation" or "measurement".

Many-worlds interpretation

Main article: Many-worlds interpretation

The quantum-mechanical "Schrödinger's cat" paradox according to the many-worlds interpretation. In this interpretation, every event is a branch point. The cat is both alive and dead—regardless of whether the box is opened—but the "alive" and "dead" cats are in different branches of the universe that are equally real but cannot interact with each other.

In 1957, Hugh Everett formulated the many-worlds interpretation of quantum mechanics, which does not single out observation as a special process. In the many-worlds interpretation, both alive and dead states of the cat persist after the box is opened, but are decoherent from each other. In other words, when the box is opened, the observer and the possibly-dead cat split into an observer looking at a box with a dead cat and an observer looking at a box with a live cat. But since the dead and alive states are decoherent, there is no communication or interaction between them.

When opening the box, the observer becomes entangled with the cat, so "observer states" corresponding to the cat's being alive and dead are formed; each observer state is entangled, or linked, with the cat so that the observation of the cat's state and the cat's state correspond with each other. Quantum decoherence ensures that the different outcomes have no interaction with each other. Decoherence is generally considered to prevent simultaneous observation of multiple states.

A variant of the Schrödinger's cat experiment, known as the quantum suicide machine, has been proposed by cosmologist Max Tegmark. It examines the Schrödinger's cat experiment from the point of view of the cat, and argues that by using this approach, one may be able to distinguish between the Copenhagen interpretation and many-worlds.

Ensemble interpretation

The ensemble interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble. The state vector would not apply to individual cat experiments, but only to the statistics of many similarly prepared cat experiments. Proponents of this interpretation state that this makes the Schrödinger's cat paradox a trivial matter, or a non-issue.

This interpretation serves to discard the idea that a single physical system in quantum mechanics has a mathematical description that corresponds to it in any way.

Relational interpretation

Main article: Relational quantum mechanics

The relational interpretation makes no fundamental distinction between the human experimenter, the cat, and the apparatus or between animate and inanimate systems; all are quantum systems governed by the same rules of wavefunction evolution, and all may be considered "observers". But the relational interpretation allows that different observers can give different accounts of the same series of events, depending on the information they have about the system. The cat can be considered an observer of the apparatus; meanwhile, the experimenter can be considered another observer of the system in the box (the cat plus the apparatus). Before the box is opened, the cat, by nature of its being alive or dead, has information about the state of the apparatus (the atom has either decayed or not decayed); but the experimenter does not have information about the state of the box contents. In this way, the two observers simultaneously have different accounts of the situation: To the cat, the wavefunction of the apparatus has appeared to "collapse"; to the experimenter, the contents of the box appear to be in superposition. Not until the box is opened, and both observers have the same information about what happened, do both system states appear to "collapse" into the same definite result, a cat that is either alive or dead.

Transactional interpretation

In the transactional interpretation the apparatus emits an advanced wave backward in time, which combined with the wave that the source emits forward in time, forms a standing wave. The waves are seen as physically real, and the apparatus is considered an "observer". In the transactional interpretation, the collapse of the wavefunction is "atemporal" and occurs along the whole transaction between the source and the apparatus. The cat is never in superposition. Rather the cat is only in one state at any particular time, regardless of when the human experimenter looks in the box. The transactional interpretation resolves this quantum paradox.

Objective collapse theories

According to objective collapse theories, superpositions are destroyed spontaneously (irrespective of external observation) when some objective physical threshold (of time, mass, temperature, irreversibility, etc.) is reached. Thus, the cat would be expected to have settled into a definite state long before the box is opened. This could loosely be phrased as "the cat observes itself" or "the environment observes the cat".

Objective collapse theories require a modification of standard quantum mechanics to allow superpositions to be destroyed by the process of time evolution. These theories could ideally be tested by creating mesoscopic superposition states in the experiment. For instance, energy cat states has been proposed as a precise detector of the quantum gravity related energy decoherence models.

Applications and tests

The experiment as described is a purely theoretical one, and the machine proposed is not known to have been constructed. However, successful experiments involving similar principles, e.g. superpositions of relatively large (by the standards of quantum physics) objects have been performed. These experiments do not show that a cat-sized object can be superposed, but the known upper limit on "cat states" has been pushed upwards by them. In many cases the state is short-lived, even when cooled to near absolute zero.

• A "cat state" has been achieved with photons.
• A beryllium ion has been trapped in a superposed state.
• An experiment involving a superconducting quantum interference device ("SQUID") has been linked to the theme of the thought experiment: "The superposition state does not correspond to a billion electrons flowing one way and a billion others flowing the other way. Superconducting electrons move en masse. All the superconducting electrons in the SQUID flow both ways around the loop at once when they are in the Schrödinger's cat state."
• A piezoelectric "tuning fork" has been constructed, which can be placed into a superposition of vibrating and non vibrating states. The resonator comprises about 10 trillion atoms.
• An experiment involving a flu virus has been proposed.
• An experiment involving a bacterium and an electromechanical oscillator has been proposed.

In quantum computing the phrase "cat state" sometimes refers to the GHZ state, wherein several qubits are in an equal superposition of all being 0 and all being 1; e.g.,

${\displaystyle |\psi ⟩=\frac{1}{\sqrt{2}}(|00\dots 0⟩+|11\dots 1⟩).}$

According to at least one proposal, it may be possible to determine the state of the cat before observing it.

Extensions

In August 2020, physicists presented studies involving interpretations of quantum mechanics that are related to the Schrödinger's cat and Wigner's friend paradoxes, resulting in conclusions that challenge seemingly established assumptions about reality.