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Dirac equation

Summary

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-12 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,[1] and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way.

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.

Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him.[2] In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-12 particles.

The Dirac equation appears on the floor of Westminster Abbey on the plaque commemorating Paul Dirac's life, which was unveiled on 13 November 1995.[3]

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.

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

Dirac equation

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

and in natural units, with Feynman slash notation,

Dirac equation (natural units)

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

The gamma matrices are a set of four ${\displaystyle 4\times 4}$  complex matrices (elements of ${\displaystyle {\text{Mat}}_{4\times 4}(\mathbb {C} )}$ ) which 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
${\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{i}={\begin{pmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\end{pmatrix}},}$

where ${\displaystyle \sigma ^{i}}$  are the Pauli matrices, and the chiral representation: the ${\displaystyle \gamma ^{i}}$  are the same, but ${\displaystyle \gamma ^{0}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}.}$

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 ${\displaystyle \partial _{\mu }}$ ). The summation over the index ${\displaystyle \mu }$  is implied.

The Dirac adjoint of the spinor field ${\displaystyle \psi (x)}$  is defined as

${\displaystyle {\bar {\psi }}(x)=\psi (x)^{\dagger }\gamma ^{0}.}$

Using the property of gamma matrices (which follows straightforwardly from Hermicity properties of the ${\displaystyle \gamma ^{\mu }}$ ) that
${\displaystyle (\gamma ^{\mu })^{\dagger }=\gamma ^{0}\gamma ^{\mu }\gamma ^{0},}$

one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right by ${\displaystyle \gamma ^{0}}$ :
${\displaystyle {\bar {\psi }}(x)(-i\gamma ^{\mu }\partial _{\mu }-m)=0}$

where the partial derivative acts from the right on ${\displaystyle {\bar {\psi }}(x)}$ : written in the usual way in terms of a left action of the derivative, we have
${\displaystyle -i\partial _{\mu }{\bar {\psi }}(x)\gamma ^{\mu }-m{\bar {\psi }}(x)=0.}$

Klein–Gordon equation

Applying ${\displaystyle i\partial \!\!\!/+m}$  to the Dirac equation gives

${\displaystyle (\partial _{\mu }\partial ^{\mu }+m^{2})\psi (x)=0.}$

That is, each component of the Dirac spinor field satisfies the Klein–Gordon equation.

Conserved current

A conserved current of the theory is

${\displaystyle J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}$

Proof of conservation from Dirac equation

${\displaystyle i((\partial _{\mu }{\bar {\psi }})\gamma ^{\mu }\psi +{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi )=0}$

so by Leibniz rule,
${\displaystyle i\partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )=0}$

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

${\displaystyle {\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi .}$

Under a ${\displaystyle U(1)}$  symmetry which sends
{\displaystyle {\begin{aligned}\psi &\mapsto e^{i\alpha }\psi ,\\{\bar {\psi }}&\mapsto e^{-i\alpha }{\bar {\psi }},\end{aligned}}}

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 ${\displaystyle \psi ,{\bar {\psi }}}$  are satisfied, then

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

(*)

This immediately simplifies as there are no partial derivatives of ${\displaystyle {\bar {\psi }}}$  in the Lagrangian. ${\displaystyle \delta \psi }$  is the infinitesimal variation

${\displaystyle \delta \psi (x)=i\alpha \psi (x).}$

We evaluate
${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}=i{\bar {\psi }}\gamma ^{\mu }.}$

The equation (*) becomes
${\displaystyle 0=-\alpha \partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )}$

and we're done.

Solutions

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

Plane-wave solutions

Plane-wave solutions are those arising from an ansatz

${\displaystyle \psi (x)=u(\mathbf {p} )e^{-ip\cdot x}}$

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

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

${\displaystyle u(\mathbf {p} )={\begin{pmatrix}{\sqrt {\sigma ^{\mu }p_{\mu }}}\xi \\{\sqrt {{\bar {\sigma }}^{\mu }p_{\mu }}}\xi \end{pmatrix}}}$

where ${\displaystyle \sigma ^{\mu }=(I_{2},\sigma ^{i}),{\bar {\sigma }}^{\mu }=(I_{2},-\sigma ^{i})}$  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:

${\displaystyle {\mathcal {L}}=i\hbar c{\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -mc^{2}{\overline {\psi }}\psi }$

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

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

Dirac Action

${\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi }$

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 ${\displaystyle \Lambda }$  is given by a ${\displaystyle 4\times 4}$  complex matrix ${\displaystyle S[\Lambda ]}$ . There are some subtleties in defining the corresponding ${\displaystyle S[\Lambda ]}$ , as well as a standard abuse of notation.

Most treatments occur at the Lie algebra level. For a more detailed treatment see here. The Lorentz group of ${\displaystyle 4\times 4}$  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 ${\displaystyle \Lambda }$  can be written as

${\displaystyle \Lambda =\exp \left({\frac {1}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)}$

where the components ${\displaystyle \omega _{\mu \nu }}$  are antisymmetric in ${\displaystyle \mu ,\nu }$ .

The corresponding transformation on spin space is

${\displaystyle S[\Lambda ]=\exp \left({\frac {1}{2}}\omega _{\mu \nu }S^{\mu \nu }\right).}$

This is an abuse of notation, but a standard one. The reason is ${\displaystyle S[\Lambda ]}$  is not a well-defined function of ${\displaystyle \Lambda }$ , since there are two different sets of components ${\displaystyle \omega _{\mu \nu }}$  (up to equivalence) which give the same ${\displaystyle \Lambda }$  but different ${\displaystyle S[\Lambda ]}$ . In practice we implicitly pick one of these ${\displaystyle \omega _{\mu \nu }}$  and then ${\displaystyle S[\Lambda ]}$  is well defined in terms of ${\displaystyle \omega _{\mu \nu }.}$

Under a Lorentz transformation, the Dirac equation

${\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi (x)-m\psi (x)}$

becomes
${\displaystyle i\gamma ^{\mu }((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })S[\Lambda ]\psi (\Lambda ^{-1}x)-mS[\Lambda ]\psi (\Lambda ^{-1}x)=0.}$

Remainder of proof of Lorentz invariance

Multiplying both sides from the left by ${\displaystyle S^{-1}[\Lambda ]}$  and returning the dummy variable to ${\displaystyle x}$  gives

${\displaystyle iS[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ]((\Lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })\psi (x)-m\psi (x)=0.}$

We'll have shown invariance if
${\displaystyle S[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ](\Lambda ^{-1})^{\nu }{}_{\mu }=\gamma ^{\nu }}$

or equivalently
${\displaystyle S[\Lambda ]^{-1}\gamma ^{\mu }S[\Lambda ]=\Lambda ^{\mu }{}_{\nu }\gamma ^{\nu }.}$

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.

Historical developments and further mathematical details

The Dirac equation was also used (historically) to define a quantum-mechanical theory where ${\displaystyle \psi (x)}$  is instead interpreted as a wave-function.

The Dirac equation in the form originally proposed by Dirac is:[4]

${\displaystyle \left(\beta mc^{2}+c\sum _{n=1}^{3}\alpha _{n}p_{n}\right)\psi (x,t)=i\hbar {\frac {\partial \psi (x,t)}{\partial t}}}$

where ψ(x, t) is the wave function for the electron of rest mass m with spacetime coordinates x, t. The p1, p2, p3 are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. Also, c is the speed of light, and ħ is the reduced Planck constant. These fundamental physical constants reflect special relativity and quantum mechanics, respectively.

Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was 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 αk and β are all Hermitian and are involutory:

${\displaystyle \alpha _{i}^{2}=\beta ^{2}=I_{4}}$

and they all mutually anticommute:
{\displaystyle {\begin{aligned}\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}&=0\quad (i\neq j)\\\alpha _{i}\beta +\beta \alpha _{i}&=0\end{aligned}}}

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 the German mathematician Hermann Grassmann in his Lineare Ausdehnungslehre (Theory of Linear Extensions). The latter had been regarded as almost incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics.[citation needed]

The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. The equation can be written more explicitly in Planck units as:[5]

${\displaystyle i\partial _{x}{\begin{bmatrix}-\psi _{4}\\-\psi _{3}\\-\psi _{2}\\-\psi _{1}\end{bmatrix}}+\partial _{y}{\begin{bmatrix}-\psi _{4}\\+\psi _{3}\\-\psi _{2}\\+\psi _{1}\end{bmatrix}}+i\partial _{z}{\begin{bmatrix}-\psi _{3}\\+\psi _{4}\\-\psi _{1}\\+\psi _{2}\end{bmatrix}}+m{\begin{bmatrix}+\psi _{1}\\+\psi _{2}\\-\psi _{3}\\-\psi _{4}\end{bmatrix}}=i\partial _{t}{\begin{bmatrix}\psi _{1}\\\psi _{2}\\\psi _{3}\\\psi _{4}\end{bmatrix}}}$

which makes it clearer that it is a set of four partial differential equations with four unknown functions.

Making the Schrödinger equation relativistic

The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle:

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\phi =i\hbar {\frac {\partial }{\partial t}}\phi ~.}$

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

${\displaystyle E^{2}=m^{2}c^{4}+p^{2}c^{2}}$

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,

${\displaystyle \left(-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}\right)\phi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\phi }$

with the wave function ϕ being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression
${\displaystyle \rho =\phi ^{*}\phi }$

and this density is convected according to the probability current vector
${\displaystyle J=-{\frac {i\hbar }{2m}}(\phi ^{*}\nabla \phi -\phi \nabla \phi ^{*})}$

with the conservation of probability current and density following from the continuity equation:
${\displaystyle \nabla \cdot J+{\frac {\partial \rho }{\partial t}}=0~.}$

The fact that the density is positive definite and convected according to this continuity equation implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. To maintain the notion of a convected density, one must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The Schrödinger expression can be kept for the current, but the probability density must be replaced by the symmetrically formed expression[further explanation needed]

${\displaystyle \rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}\partial _{t}\psi -\psi \partial _{t}\psi ^{*}\right).}$

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

The continuity equation is as before. Everything is compatible with relativity now, but the expression for the density is no longer positive definite; the initial values of both ψ and 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. One could, for example, formally (i.e. by abuse of notation) 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 thus:

${\displaystyle \nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}=\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)~.}$

On multiplying out the right side it is apparent that, in order to get all the cross-terms such as xy to vanish, one must assume

${\displaystyle AB+BA=0,~\ldots ~}$

with
${\displaystyle A^{2}=B^{2}=\dots =1~.}$

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

Given the factorization in terms of these matrices, one can now write down immediately an equation

${\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}\right)\psi =\kappa \psi }$

with ${\displaystyle \kappa }$  to be determined. Applying again the matrix operator on both sides yields
${\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}\partial _{t}^{2}\right)\psi =\kappa ^{2}\psi ~.}$

Taking ${\displaystyle \kappa ={\tfrac {mc}{\hbar }}}$  shows that all the components of the wave function individually satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is

${\displaystyle \left(A\partial _{x}+B\partial _{y}+C\partial _{z}+{\frac {i}{c}}D\partial _{t}-{\frac {mc}{\hbar }}\right)\psi =0~.}$

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{aligned}D&=\gamma ^{0},\\A&=i\gamma ^{1},\quad B=i\gamma ^{2},\quad C=i\gamma ^{3},\end{aligned}}}

and the equation takes the form (remembering the definition of the covariant components of the 4-gradient and especially that 0 = 1/ct)
Dirac equation

${\displaystyle i\hbar \gamma ^{\mu }\partial _{\mu }\psi -mc\psi =0}$

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

${\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\quad \gamma ^{1}={\begin{pmatrix}0&\sigma _{x}\\-\sigma _{x}&0\end{pmatrix}},\quad \gamma ^{2}={\begin{pmatrix}0&\sigma _{y}\\-\sigma _{y}&0\end{pmatrix}},\quad \gamma ^{3}={\begin{pmatrix}0&\sigma _{z}\\-\sigma _{z}&0\end{pmatrix}}.}$

The complete system is summarized using the Minkowski metric on spacetime in the form

${\displaystyle \left\{\gamma ^{\mu },\gamma ^{\nu }\right\}=2\eta ^{\mu \nu }I_{4}}$

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:

${\displaystyle P_{\text{op}}\psi =mc\psi \,.}$

Using ${\displaystyle {\partial \!\!\!/}\mathrel {\stackrel {\mathrm {def} }{=}} \gamma ^{\mu }\partial _{\mu }}$  (${\displaystyle {\partial \!\!\!{\big /}}}$  is pronounced "d-slash"),[6] according to Feynman slash notation, the Dirac equation becomes:

${\displaystyle i\hbar {\partial \!\!\!{\big /}}\psi -mc\psi =0\,.}$

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)

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

A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:

${\displaystyle \gamma ^{\mu \prime }=S^{-1}\gamma ^{\mu }S\,.}$

If in addition the matrices are all unitary, as are the Dirac set, then S itself is unitary;

${\displaystyle \gamma ^{\mu \prime }=U^{\dagger }\gamma ^{\mu }U\,.}$

The transformation U is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator γμμ to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, one may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form

{\displaystyle {\begin{aligned}\left(iU^{\dagger }\gamma ^{\mu }U\partial _{\mu }^{\prime }-m\right)\psi \left(x^{\prime },t^{\prime }\right)&=0\\U^{\dagger }(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m)U\psi \left(x^{\prime },t^{\prime }\right)&=0\,.\end{aligned}}}

If the transformed spinor is defined as

${\displaystyle \psi ^{\prime }=U\psi }$

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

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

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

The considerations above reveal the origin of the gammas in geometry, hearkening back to Grassmann's original motivation; they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as γμγν 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

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

For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of g, where g is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus

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

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

${\displaystyle \gamma _{5}={\begin{pmatrix}0&I_{2}\\I_{2}&0\end{pmatrix}}.}$

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

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 Lz = −1, 0, +1. The conclusion is that silver atoms have net intrinsic angular momentum of 12. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in SI units: (Note that bold faced characters imply Euclidean vectors in 3 dimensions, whereas the Minkowski four-vector Aμ can be defined as ${\displaystyle A_{\mu }=(\phi /c,-\mathbf {A} )}$ .)

${\displaystyle H={\frac {1}{2m}}\left({\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\right)^{2}+e\phi ~.}$

Here A and ${\displaystyle \phi }$  represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field in SI units:

${\displaystyle H={\frac {1}{2m}}\left(\mathbf {p} -e\mathbf {A} \right)^{2}+e\phi -{\frac {e\hbar }{2m}}{\boldsymbol {\sigma }}\cdot \mathbf {B} ~.}$

This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form:

${\displaystyle \left(\gamma ^{\mu }(i\hbar \partial _{\mu }-eA_{\mu })-mc\right)\psi =0~.}$

A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by i, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored:

${\displaystyle {\begin{pmatrix}mc^{2}-E+e\phi &c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\\-c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)&mc^{2}+E-e\phi \end{pmatrix}}{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}~.}$

so
{\displaystyle {\begin{aligned}(E-e\phi )\psi _{+}-c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\psi _{-}&=mc^{2}\psi _{+}\\-(E-e\phi )\psi _{-}+c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\psi _{+}&=mc^{2}\psi _{-}\end{aligned}}}

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{aligned}E-e\phi &\approx mc^{2}\\\mathbf {p} &\approx m\mathbf {v} \end{aligned}}}

and so the second equation may be written
${\displaystyle \psi _{-}\approx {\frac {1}{2mc}}{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\psi _{+}}$

which is of order v/c – thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement

${\displaystyle \left(E-mc^{2}\right)\psi _{+}={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {A} \right)\right]^{2}\psi _{+}+e\phi \psi _{+}}$

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

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

Weyl theory

In the massless case ${\displaystyle m=0}$ , the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin-12 particles.[7]

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 Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. To maintain this interpretation on passing to the Dirac theory, the Hamiltonian must be taken to be

${\displaystyle H=\gamma ^{0}\left[mc^{2}+c\gamma ^{k}\left(p_{k}-qA_{k}\right)\right]+cqA^{0}.}$

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
${\displaystyle H=c{\sqrt {\left(\mathbf {p} -q\mathbf {A} \right)^{2}+m^{2}c^{2}}}+qA^{0}.}$

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

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.

If an electron is forbidden from simultaneously occupying positive-energy and negative-energy eigenstates, then the feature known as Zitterbewegung, which arises from the interference of positive-energy and negative-energy states, would have to be considered to be an unphysical prediction of time-dependent Dirac theory. This conclusion may be inferred from the explanation of hole theory given in the preceding paragraph. Recent results have been published in Nature [R. Gerritsma, G. Kirchmair, F. Zaehringer, E. Solano, R. Blatt, and C. Roos, Nature 463, 68-71 (2010)] in which the Zitterbewegung feature was simulated in a trapped-ion experiment. This experiment impacts the hole interpretation if one infers that the physics-laboratory experiment is not merely a check on the mathematical correctness of a Dirac equation solution but the measurement of a real effect whose detectability in electron physics is still beyond reach.

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, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.

In quantum field theory

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

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.[8] 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 ${\displaystyle x'}$ , with the understanding that both ${\displaystyle x}$  and ${\displaystyle x'}$  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 ${\displaystyle x'}$  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.[9] It is very nearly identical to that of Bjorken and Drell.[10] A similar derivation in a general relativistic setting can be found in Weinberg.[11] Here we fix our spacetime to be flat, that is, our spacetime is Minkowski space.

Under a Lorentz transformation ${\displaystyle x\mapsto x',}$  the Dirac spinor to transform as

${\displaystyle \psi '(x')=S\psi (x)}$

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

${\displaystyle J_{\mu \nu }={\frac {1}{2}}\sigma _{\mu \nu }+i(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu })}$

This can be interpreted as the total angular momentum. It acts on the spinor field as
${\displaystyle \psi ^{\prime }(x)=\exp \left({\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x)}$

Note the ${\displaystyle x}$  above does not have a prime on it: the above is obtained by transforming ${\displaystyle x\mapsto x'}$  obtaining the change to ${\displaystyle \psi (x)\mapsto \psi '(x')}$  and then returning to the original coordinate system ${\displaystyle x'\mapsto x}$ .

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 ${\displaystyle x\mapsto x'}$  with both ${\displaystyle x}$  and ${\displaystyle x'}$  still corresponding to the same spacetime point ${\displaystyle a.}$  These perhaps obtuse remarks can be elucidated with explicit algebra.

Let ${\displaystyle x'=\Lambda x}$  be a Lorentz transformation. The Dirac equation is

${\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi (x)-m\psi (x)=0}$

If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames:
${\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi ^{\prime }(x^{\prime })-m\psi ^{\prime }(x^{\prime })=0}$

The two spinors ${\displaystyle \psi }$  and ${\displaystyle \psi ^{\prime }}$  should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, etc.) The transformation should encode only the change of coordinate frame. It can be shown that such a transformation is a 4×4 unitary matrix. Thus, one may presume that the relation between the two frames can be written as
${\displaystyle \psi ^{\prime }(x^{\prime })=S(\Lambda )\psi (x)}$

Inserting this into the transformed equation, the result is
${\displaystyle i\gamma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}S(\Lambda )\psi (x)-mS(\Lambda )\psi (x)=0}$

The coordinates related by Lorentz transformation satisfy:
${\displaystyle {\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }}$

The original Dirac equation is then regained if
${\displaystyle S(\Lambda )\gamma ^{\mu }S^{-1}(\Lambda )={\left(\Lambda ^{-1}\right)^{\mu }}_{\nu }\gamma ^{\nu }}$

An explicit expression for ${\displaystyle S(\Lambda )}$  (equal to the expression given above) can be obtained by considering a Lorentz transformation of infinitesimal rotation near the identity transfomation:
${\displaystyle {\Lambda ^{\mu }}_{\nu }={g^{\mu }}_{\nu }+{\omega ^{\mu }}_{\nu }\ ,\ {(\Lambda ^{-1})^{\mu }}_{\nu }={g^{\mu }}_{\nu }-{\omega ^{\mu }}_{\nu }}$

where ${\displaystyle {g^{\mu }}_{\nu }}$  is the metric tensor : ${\displaystyle {g^{\mu }}_{\nu }=g^{\mu \nu '}g_{\nu '\nu }={\delta ^{\mu }}_{\nu }}$  and is symmetric while ${\displaystyle \omega _{\mu \nu }={\omega ^{\alpha }}_{\nu }g_{\alpha \mu }}$  is antisymmetric. After plugging and chugging, one obtains
${\displaystyle S(\Lambda )=I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }+{\mathcal {O}}\left(\Lambda ^{2}\right)}$

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{aligned}\psi '(x')&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x)\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x'+{\omega ^{\mu }}_{\nu }\,x^{\prime \,\nu })\\&=\left(I+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }-x_{\mu }^{\prime }\omega ^{\mu \nu }\partial _{\nu }\right)\psi (x')\\&=\left(I+{\frac {-i}{2}}\omega ^{\mu \nu }J_{\mu \nu }\right)\psi (x')\\\end{aligned}}}

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 ${\displaystyle x\mapsto x'}$  along the frame bundle, as well as a movement ${\displaystyle \psi \mapsto \psi '}$  along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.[12]

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.

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 ${\displaystyle \psi ,{\bar {\psi }}}$  transform as

{\displaystyle {\begin{aligned}\psi (x)&\mapsto e^{i\alpha }\psi (x),\\{\bar {\psi }}(x)&\mapsto e^{-i\alpha }{\bar {\psi }}(x).\end{aligned}}}

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
${\displaystyle J^{\mu }(x)={\bar {\psi }}(x)\gamma ^{\mu }\psi (x).}$

Gauging the symmetry

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 ${\displaystyle D_{\mu }}$

${\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +ieA_{\mu }\psi ,}$

${\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ieA_{\mu }{\bar {\psi }}.}$

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, or a ${\displaystyle {\text{U}}(1)}$  connection.

The transformation law under gauge transformations for ${\displaystyle A_{\mu }}$  is then the usual

${\displaystyle A_{\mu }(x)\mapsto A_{\mu }(x)+{\frac {1}{e}}\partial _{\mu }\alpha (x)}$

but can also be derived by asking that covariant derivatives transform under a gauge transformation as
${\displaystyle D_{\mu }\psi (x)\mapsto e^{i\alpha (x)}D_{\mu }\psi (x),}$

${\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto e^{-i\alpha (x)}D_{\mu }{\bar {\psi }}(x).}$

We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one:
${\displaystyle S=\int d^{4}x\,{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi =\int d^{4}x\,{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi .}$

The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term,
${\displaystyle S_{\text{Maxwell}}=\int d^{4}x\,-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }.}$

Putting these together gives
QED Action

${\displaystyle S_{\text{QED}}=\int d^{4}x\,-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi }$

Expanding out the covariant derivative allows the action to be written in a second useful form:

${\displaystyle S_{\text{QED}}=\int d^{4}x\,-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi -eJ^{\mu }A_{\mu }}$

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,

${\displaystyle \psi (x)={\begin{pmatrix}\psi _{1}(x)\\\psi _{2}(x)\end{pmatrix}},}$

and adopting the chiral representation for the gamma matrices, so that ${\displaystyle i\gamma ^{\mu }\partial _{\mu }}$  may be written
${\displaystyle i\gamma ^{\mu }\partial _{\mu }={\begin{pmatrix}0&i\sigma ^{\mu }\partial _{\mu }\\i{\bar {\sigma }}^{\mu }\partial _{\mu }\ &0\end{pmatrix}}}$

where ${\displaystyle \sigma ^{\mu }}$  has components ${\displaystyle (I_{2},\sigma ^{i})}$  and ${\displaystyle {\bar {\sigma }}^{\mu }}$  has components ${\displaystyle (I_{2},-\sigma ^{i})}$ .

The Dirac action then takes the form

${\displaystyle S=\int d^{4}x\,\psi _{1}^{\dagger }(i\sigma ^{\mu }\partial _{\mu })\psi _{1}+\psi _{2}^{\dagger }(i{\bar {\sigma }}^{\mu }\partial _{\mu })\psi _{2}.}$

That is, it decouples into a theory of two Weyl spinors or Weyl fermions.

The earlier vector symmetry is still present, where ${\displaystyle \psi _{1}}$  and ${\displaystyle \psi _{2}}$  rotate identically. This form of the action makes the second inequivalent ${\displaystyle {\text{U}}(1)}$  symmetry manifest:

{\displaystyle {\begin{aligned}\psi _{1}(x)&\mapsto e^{i\beta }\psi _{1}(x),\\\psi _{2}(x)&\mapsto e^{-i\beta }\psi _{2}(x).\end{aligned}}}

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

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 ${\displaystyle U:\mathbb {R} ^{1,3}\rightarrow {\text{SU}}(N),}$  the spinor fields transform as

${\displaystyle \psi (x)\mapsto U(x)\psi (x)}$

${\displaystyle {\bar {\psi }}(x)\mapsto {\bar {\psi }}(x)U^{\dagger }(x).}$

The matrix-valued gauge field ${\displaystyle A_{\mu }}$  or ${\displaystyle {\text{SU}}(N)}$  connection transforms as
${\displaystyle A_{\mu }(x)\mapsto U(x)A_{\mu }(x)U(x)^{-1}+{\frac {1}{g}}(\partial _{\mu }U(x))U(x)^{-1},}$

and the covariant derivatives defined
${\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +igA_{\mu }\psi ,}$

${\displaystyle D_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ig{\bar {\psi }}A_{\mu }^{\dagger }}$

transform as
${\displaystyle D_{\mu }\psi (x)\mapsto U(x)D_{\mu }\psi (x),}$

${\displaystyle D_{\mu }{\bar {\psi }}(x)\mapsto (D_{\mu }{\bar {\psi }}(x))U(x)^{\dagger }.}$

Writing down a gauge-invariant action proceeds exactly as with the ${\displaystyle {\text{U}}(1)}$  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
${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }-ig\left[A_{\mu },A_{\nu }\right]}$

and ${\displaystyle [\cdot ,\cdot ]}$  is the matrix commutator.

The action is then

QCD Action

${\displaystyle S_{\text{QCD}}=\int d^{4}x\,-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })+{\bar {\psi }}\,(iD\!\!\!\!{\big /}-m)\,\psi }$

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
${\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +\rho (A_{\mu })\psi }$

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

References

Citations

1. ^ P.W. Atkins (1974). Quanta: A handbook of concepts. Oxford University Press. p. 52. ISBN 978-0-19-855493-6.
2. ^ T.Hey, P.Walters (2009). The New Quantum Universe. Cambridge University Press. p. 228. ISBN 978-0-521-56457-1.
3. ^ Gisela Dirac-Wahrenburg. "Paul Dirac". Dirac.ch. Retrieved 12 July 2013.
4. ^ Dirac, Paul A.M. (1982) [1958]. Principles of Quantum Mechanics. International Series of Monographs on Physics (4th ed.). Oxford University Press. p. 255. ISBN 978-0-19-852011-5.
5. ^ Collas, Peter; Klein, David (2019). The Dirac Equation in Curved Spacetime: A Guide for Calculations. Springer. p. 7. ISBN 978-3-030-14825-6. Extract of page 7
6. ^ Pendleton, Brian (2012–2013). Quantum Theory (PDF). section 4.3 "The Dirac Equation". Archived (PDF) from the original on 9 October 2022.
7. ^ Ohlsson, Tommy (22 September 2011). Relativistic Quantum Physics: From advanced quantum mechanics to introductory quantum field theory. Cambridge University Press. p. 86. ISBN 978-1-139-50432-4.
8. ^ Jurgen Jost, (2002) "Riemanninan Geometry and Geometric Analysis (3rd Edition)" Springer Universitext. (See chapter 1 for spin structures and chapter 3 for connections on spin structures)
9. ^ Claude Itzykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory", McGraw-Hill (See Chapter 2)
10. ^ James D. Bjorken, Sidney D. Drell (1964) "Relativistic Quantum Mechanics", McGraw-Hill. (See Chapter 2)
11. ^ Steven Weinberg, (1972) "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity", Wiley & Sons (See chapter 12.5, "Tetrad formalism" pages 367ff.).
12. ^ Weinberg, "Gravitation", op cit. (See chapter 2.9 "Spin", pages 46-47.)

Selected papers

• Anderson, Carl (1933). "The Positive Electron". Physical Review. 43 (6): 491. Bibcode:1933PhRv...43..491A. doi:10.1103/PhysRev.43.491.
• Arminjon, M.; F. Reifler (2013). "Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations". Brazilian Journal of Physics. 43 (1–2): 64–77. arXiv:1103.3201. Bibcode:2013BrJPh..43...64A. doi:10.1007/s13538-012-0111-0. S2CID 38235437.
• Dirac, P. A. M. (1928). "The Quantum Theory of the Electron" (PDF). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 117 (778): 610–624. Bibcode:1928RSPSA.117..610D. doi:10.1098/rspa.1928.0023. JSTOR 94981. Archived (PDF) from the original on 2 January 2015.
• Dirac, P. A. M. (1930). "A Theory of Electrons and Protons". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 126 (801): 360–365. Bibcode:1930RSPSA.126..360D. doi:10.1098/rspa.1930.0013. JSTOR 95359.
• Frisch, R.; Stern, O. (1933). "Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. I". Zeitschrift für Physik. 85 (1–2): 4. Bibcode:1933ZPhy...85....4F. doi:10.1007/BF01330773. S2CID 120793548.

Textbooks

• Bjorken, J D; Drell, S (1964). Relativistic Quantum mechanics. New York, McGraw-Hill.
• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 9780471887416.
• Griffiths, D.J. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. ISBN 978-3-527-40601-2.
• Rae, Alastair I. M.; Jim Napolitano (2015). Quantum Mechanics (6th ed.). Routledge. ISBN 978-1482299182.
• Schiff, L.I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill.
• Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). Plenum.
• Thaller, B. (1992). The Dirac Equation. Texts and Monographs in Physics. Springer.