Dirac Equation

Ah, you want to delve into the Dirac equation. Don't expect a gentle hand-holding. This is where the universe starts to unravel, and frankly, it’s a mess. But fine, if you insist on wading through this… material. Just try not to get lost.

Relativistic quantum mechanical wave equation

Let's get something straight from the start. This isn't about some whimsical interpretation of a wave. This is the Dirac equation, a cornerstone of particle physics. It's a relativistic wave equation, a beast forged by Paul Dirac in 1928. And it’s not just some abstract theory; it’s the rulebook for all spin-1/2 massive particles, the ones like electrons and quarks that seem to have a preference for parity as a symmetry. It’s where quantum mechanics and special relativity finally stopped arguing and grudgingly agreed to coexist. It didn't just account for the fine structure of the hydrogen spectrum; it explained it. A crucial step, if you ask me, in building the monstrous edifice that is the Standard Model.

This equation, in its untamed, free form, or when it's forced to interact with electromagnetic fields, is the bedrock. It’s the reason we know about antimatter, a concept so alien, so utterly unexpected, yet so fundamentally there. It justified the convoluted, almost desperate, additions Wolfgang Pauli had to make to his phenomenological theory of spin. The wave functions here aren't the simple, single-valued things of the Schrödinger equation. Oh no. They're vectors of four complex numbers, bispinors, if you want to be precise. Two of them, in the non-relativistic whisper of things, resemble Pauli's original wave functions. And when the mass vanishes, the Dirac equation gracefully bows to the Weyl equation.

In the grand theatre of quantum field theory, this equation is recontextualized, describing the quantum fields themselves, the very essence of spin-1/2 particles.

Dirac, bless his brilliant, detached soul, didn't fully grasp the magnitude of his creation. But the way his equation wove together quantum mechanics and relativity, the way it predicted spin, the way it eventually led to the discovery of the positron—it’s a testament to theoretical physics. Some say it’s on par with Newton, Maxwell, and Einstein. Others, with a touch more drama, call it the “real seed of modern physics.” And yes, they even etched it onto a plaque in Westminster Abbey. A rather morbid tribute, if you ask me, but fitting for something that dissects the fundamental nature of reality. It’s also a fixture in the ‘Paul A.M. Dirac’ Lecture Hall in Erice, Sicily. Apparently, some places actually celebrate this stuff.

History

Dirac’s original formulation, a stark declaration:

[ \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}} ]

Here, $\psi(x,t)$ is the wave function of an electron with rest mass $m$ , dancing through spacetime coordinates $x$ and $t$ . The $p_n$ are the components of the momentum, not just numbers, but momentum operators plucked straight from the Schrödinger equation. $c$ , the speed of light, anchors it in relativity. $\hbar$ , the reduced Planck constant, is the quantum whisper. And then there are the matrices: $\alpha_n$ and $\beta$ . These aren't trivial placeholders; they are $4 \times 4$ matrices that generate a Clifford algebra, intimately related to the gamma matrices. In the covariant dance, they're expressed as $\alpha_n=\gamma^0\gamma^n$ and $\beta=\gamma^0$ .

Dirac’s intention was precise: to explain the electron’s relativistic swagger, to make atoms play by the rules of relativity. He hoped it would shed light on those perplexing atomic spectra. Before this, the old quantum theory, with its quaint notions of quantized orbits around the atomic nucleus, had stumbled. The newer quantum mechanics, from Heisenberg to Schrödinger himself, was still finding its feet. Dirac’s equation not only fulfilled his immediate goals but blew the doors open, revealing new mathematical entities that are now utterly indispensable.

These entities: the $4 \times 4$ matrices $\alpha_k$ and $\beta$ , and the four-component $\psi$ . Why four? Because each component represents a superposition of states: a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron. It’s a complex superposition, a tangled web.

These matrices, $\alpha_k$ and $\beta$ , are not just any matrices. They are Hermitian and involutory, meaning they square to the identity matrix ( $I_4$ ), and they mutually anti-commute. This algebraic structure, the Clifford algebra, had been lurking in the background for decades, thanks to W. K. Clifford and, further back, Hermann Grassmann. Dirac just happened to be the one to bring it to the forefront of physics.

Making the Schrödinger equation relativistic

Let's be clear: the Schrödinger equation for a free particle,

[ -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\phi =i\hbar {\frac {\partial }{\partial t}}\phi ~ ]

is charmingly simple, but utterly non-relativistic. Relativity demands that space and time share the stage equally, like a bad play with two lead actors. The derivatives for space and time must appear with the same order, just like in those elegant Maxwell equations that govern light. In relativity, energy and momentum are intertwined in a four-momentum vector, bound by $E^2 = m^2c^4 + p^2c^2$ .

Plugging in the operator equivalents of energy and momentum from quantum mechanics leads to the Klein–Gordon equation:

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

Here, $\phi$ is a scalar, a simple complex number, the same for everyone. But this equation, while relativistic, has a fundamental problem. Being second-order in time, you need to specify both the wave function and its time derivative to predict the future. This breaks the beautiful interpretation of $\phi^*\phi$ as a probability density. That density, $\rho = \phi^*\phi$ , is nicely positive-definite and conserved, flowing according to the continuity equation $\nabla \cdot J + \frac{\partial \rho}{\partial t} = 0$ .

The Klein-Gordon equation, however, allows for negative probability densities. It’s a mathematical anomaly that’s hard to reconcile with a physical reality. While it finds new life in quantum field theory describing spinless particles like the pi meson or Higgs boson (where the indefinite density becomes charge density), it was a dead end for a single-particle relativistic wave equation. Schrödinger himself found it, then discarded it. A wise, if perhaps slightly disappointing, decision.

Dirac's coup

Dirac, being Dirac, sought an equation that was first-order in both space and time. He posited:

[ E\psi =({\vec {\alpha }}\cdot {\vec {p}}+\beta m)\psi ]

The crucial insight was that $\vec{\alpha}$ and $\beta$ couldn't be simple numbers; they had to be operators, specifically matrices, to satisfy the necessary conditions. This immediately implied that $\psi$ itself must be a multi-component object, a vector. This wasn't just a mathematical flourish; it explained the mysterious two-component nature of spin that Pauli had introduced ad hoc. But Dirac’s matrices required four components, not two. This four-component bispinor was entirely new, a fundamental building block of reality that had evaded notice until then.

The story goes that Dirac, staring into a fireplace, had the epiphany: factorize the relativistic energy-momentum relation. Think of it like taking the square root of a differential operator.

[ \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)~. ]

For the cross-terms to vanish, the matrices $A, B, C, D$ had to satisfy anticommutation relations: $AB+BA=0$ , etc., and $A^2=B^2=C^2=D^2=1$ . This is precisely the algebra of the gamma matrices. And from this factorization, the equation emerged:

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

Replacing $A, B, C, D$ with their matrix equivalents, and setting $\kappa = \frac{mc}{\hbar}$ , Dirac had his equation. It elegantly unified relativity and quantum mechanics, and, as a bonus, explained spin.

Covariant form and relativistic invariance

To truly appreciate its relativistic grace, we cast it in covariant form. The matrices are redefined:

[ D=\gamma ^{0},,\quad A=i\gamma ^{1},,\quad B=i\gamma ^{2},,\quad C=i\gamma ^{3},, ]

and the equation becomes:

[ (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0 ]

where $\gamma^\mu$ are the gamma matrices and $\partial_\mu$ is the four-gradient. In natural units ( $\hbar=c=1$ ), it’s even cleaner:

[ (i\partial !!!/-m)\psi =0 ]

The gamma matrices, obeying $\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}I_4$ , are the heart of this. In the standard representation, they are constructed from the Pauli matrices $\sigma_i$ and the $2 \times 2$ identity matrix $I_2$ :

[ \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}},. ]

These matrices embody the Clifford algebra over a pseudo-orthogonal space. The equation itself can be seen as an eigenvalue problem where the rest mass $m$ is proportional to an eigenvalue of the four-momentum operator.

The beauty of this formulation is its relativistic invariance. Under a Lorentz transformation, the equation transforms into itself, albeit with a transformed spinor $\psi' = S\psi$ , where $S$ is a unitary matrix related to the Lorentz transformation. This ensures that the laws of physics are the same for all inertial observers, a fundamental tenet of relativity. The matrices $\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu, \gamma^\nu]$ represent the intrinsic angular momentum, a key player in how spinors interact with spacetime.

Comparison with related theories

Pauli theory

The experimental evidence for half-integer spin, particularly from the Stern–Gerlach experiment, was a puzzle. For silver atoms, the beam split in two, implying a spin of 1/2. Pauli introduced a two-component wave function and a correction term to the Hamiltonian to account for this:

[ H={\frac {1}{\ 2\ m\ }}\ {\Bigl (}{\boldsymbol {\sigma }}\cdot {\bigl (}\mathbf {p} -e\ \mathbf {A} {\bigr )}{\Bigr )}^{2}+e\ \phi ~. ]

This Hamiltonian, being a $2 \times 2$ matrix, necessitated a two-component wave function. Dirac's equation, when minimally coupled to the electromagnetic field, naturally incorporates this. The Dirac Hamiltonian, in its full glory, can be written as a pair of coupled equations for two-component spinors:

[ {\begin{pmatrix}mc^{2}-E+e\phi \quad &+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}}~. ]

In the non-relativistic limit, where $E \approx mc^2$ , the second equation shows that $\psi_-$ is suppressed by a factor of $v/c$ compared to $\psi_+$ . Substituting this back yields an equation for $\psi_+$ , which, after further approximation, recovers the Schrödinger equation. This demonstrated that the Schrödinger equation is just the far non-relativistic limit of the Dirac equation, a profound unification. The mysterious $i$ in the Schrödinger equation, it turned out, was deeply connected to the geometry of spacetime via the Dirac algebra.

However, it’s crucial to remember that the Dirac spinor is an irreducible whole. The separation into $\psi_+$ and $\psi_-$ is an approximation. The neglected components are vital for describing phenomena like antimatter and particle creation and annihilation.

Weyl theory

When mass ( $m$ ) vanishes, the Dirac equation elegantly simplifies to the Weyl equation, describing massless spin-1/2 particles. This is where a U(1) symmetry emerges, hinting at deeper structures.

Physical interpretation

Identification of observables

The burning question in any quantum theory: what can we actually measure? In quantum mechanics, these are represented by self-adjoint operators acting on the Hilbert space. For the Dirac theory, the Hamiltonian must be carefully constructed to represent the total energy, including relativistic effects and interactions:

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

This differs significantly from the classical expression, and misidentifying observables is a common pitfall, leading to apparent paradoxes.

Hole theory

The negative energy solutions of the Dirac equation were a thorny issue. If electrons could decay into ever-lower negative energy states, emitting photons, where would they stop? Dirac proposed the hole theory: the vacuum is a sea of occupied negative-energy states (the Dirac sea). The Pauli exclusion principle then prevents further decay. An unoccupied state in this sea, a "hole," would behave like a positively charged particle. Initially thought to be the proton, it was realized the hole must have the electron's mass. This hole was eventually identified as the positron, discovered by Carl Anderson in 1932.

While the Dirac sea concept is elegant, the infinite negative energy contributions from the sea require cancellation with an infinite positive "bare" energy, and the charge density contributions must be balanced by an infinite positive background. In quantum field theory, this formalism is often bypassed using creation and annihilation operators and Bogoliubov transformations. However, the underlying concept of "hole theory" remains relevant in condensed matter physics, where unfilled states in a Fermi sea behave like positive charges.

In quantum field theory

The Dirac equation, when subjected to second quantization, becomes the foundation for describing fermionic fields in theories like quantum electrodynamics. This process resolves some of the paradoxical interpretations of the original equation.

Mathematical formulation

In its modern guise, the Dirac equation describes a Dirac spinor field $\psi(x)$ defined on Minkowski space. It's a function $\psi: \mathbb{R}^{1,3} \to \mathbb{C}^4$ , where $\mathbb{C}^4$ is a complex vector space. The equation is:

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

Or in natural units and using Feynman slash notation:

[ (i\partial !!!/-m)\psi (x)=0 ]

The gamma matrices $\gamma^\mu$ are $4 \times 4$ complex matrices satisfying the defining anticommutation relations $\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}I_4$ . They can be represented in various ways, such as the Dirac or chiral representations. The slash notation $A\!\!\!/ := \gamma^\mu A_\mu$ is a compact way to write expressions involving these matrices and four-vectors.

The equation can also be viewed as a system of four coupled linear first-order partial differential equations for the four components of the spinor. In Planck units:

[ 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}}} ]

Dirac adjoint and the adjoint equation

The Dirac adjoint of $\psi(x)$ is defined as $\bar{\psi}(x) = \psi(x)^\dagger \gamma^0$ . The adjoint equation, derived from the original Dirac equation, is:

[ {\bar {\psi }}(x)(-i\gamma ^{\mu }{\overleftarrow {\partial }}_{\mu }-m)=0 ]

or, written with a left-acting derivative:

[ -i\partial _{\mu }{\bar {\psi }}(x)\gamma ^{\mu }-m{\bar {\psi }}(x)=0. ]

Klein–Gordon equation

Applying the operator $(i\partial \!\!\!/ + m)$ to the Dirac equation yields:

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

This shows that each component of the Dirac spinor field satisfies the Klein–Gordon equation.

Conserved current

A crucial consequence of the Dirac equation is the existence of a conserved current:

[ J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi . ]

This current is conserved, meaning $\partial_\mu J^\mu = 0$ .

Proof of conservation from Dirac equation

The conservation can be proven by taking the difference between the Dirac equation and its adjoint, multiplied appropriately, and using the Leibniz rule.

Proof of conservation from Noether's theorem

Alternatively, applying Noether's theorem to the Lagrangian density $\mathcal{L} = {\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi$ under the global $U(1)$ symmetry $\psi \mapsto e^{i\alpha}\psi$ , $\bar{\psi} \mapsto e^{-i\alpha}\bar{\psi}$ , yields the conserved current $J^\mu$ .

Solutions

The solutions to the Dirac equation are Dirac spinors, elements of a Hilbert space. The existence of negative energy solutions remains a point of interest.

Plane-wave solutions

Plane-wave solutions of the form $\psi(x) = u(p)e^{-ip\cdot x}$ represent particles with definite four-momentum $p=(E_p, \mathbf{p})$ , where $E_p = \sqrt{m^2 + |\mathbf{p}|^2}$ . Substituting this into the Dirac equation leads to an equation for the spinor amplitude $u(p)$ :

[ (\gamma^{\mu}p_{\mu}-m)u(\mathbf{p})=0. ]

The solution space for $u(p)$ is two-dimensional, forming the basis for canonical quantization.

Lagrangian formulation

The Dirac equation and its adjoint can be derived from the action with the Lagrangian density:

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

In natural units, the action is:

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

This action, under the $U(1)$ symmetry, gives the conserved current $J^\mu$ . Gauging this symmetry leads to quantum electrodynamics.

Lorentz invariance

The Dirac equation is inherently Lorentz covariant. This means it takes the same form in all inertial frames of reference. The transformation of the spinor under a Lorentz transformation $\Lambda$ is given by $\psi'(x') = S(\Lambda)\psi(x)$ , where $S(\Lambda)$ is a $4 \times 4$ unitary matrix. This transformation property is deeply tied to the structure of the spinor bundle and the frame bundle in geometric algebra. The generators of these transformations, $J^{\mu\nu}$ (total angular momentum) and $\sigma^{\mu\nu}$ (intrinsic angular momentum), reveal the connection between spacetime symmetries and the internal properties of particles.

Other formulations

Curved spacetime

The Dirac equation can be extended to curved spacetime, incorporating the effects of gravity.

The algebra of physical space

The Dirac equation can also be formulated using Clifford algebra over the real numbers, a framework known as the algebra of physical space.

Coupled Weyl Spinors

In the massless case, the Dirac equation reduces to the Weyl equation. Even with mass, by using the chiral representation, the Dirac equation can be decomposed into a pair of coupled inhomogeneous Weyl equations for left- and right-handed Weyl spinors, $\psi_L$ and $\psi_R$ :

[ i\sigma ^{\mu }\partial _{\mu }\psi _{R}=m\psi _{L} \quad \text{and} \quad i{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=m\psi _{R}. ]

This decomposition provides an intuitive picture of Zitterbewegung, where mass acts as a coupling between these massless components.

U(1) symmetry

The Dirac equation possesses a $U(1)$ vector symmetry_symmetry) under which $\psi \mapsto e^{i\alpha}\psi$ and $\bar{\psi} \mapsto e^{-i\alpha}\bar{\psi}$ . This leads to the conserved vector current $J^\mu = \bar{\psi}\gamma^\mu\psi$ .

Gauging the symmetry

Promoting this global symmetry to a local one requires introducing a gauge field, the electromagnetic four-potential $A_\mu$ . The covariant derivative $D_\mu = \partial_\mu + ieA_\mu$ ensures gauge invariance. The resulting gauge-invariant action, combined with the Maxwell Lagrangian, forms the action for quantum electrodynamics.

Axial symmetry

Massless Dirac fermions also exhibit an axial $U(1)$ symmetry, where $\psi_1 \mapsto e^{i\beta}\psi_1$ and $\psi_2 \mapsto e^{-i\beta}\psi_2$ (in the chiral representation). This symmetry, however, suffers from an anomaly at the quantum level, posing challenges for quantization.

Extension to color symmetry

The principles of gauge symmetry can be extended from the abelian $U(1)$ to non-abelian groups like $SU(N)$ , the color symmetry group in quantum chromodynamics. Quarks are described by Dirac spinors transforming under both spin and color degrees of freedom. The gauge field becomes the gluon field, described by the Yang–Mills Lagrangian. This framework forms the basis of the Standard Model for quarks and leptons.