Editing Dirac equation (section)

=== Lorentz invariance ===

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

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

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

These satisfy the Lorentz algebra commutation relations
<math display="block">[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}.</math>
In the article on the [[Dirac algebra]], it is also found that the spin generators 
<math display="block">S^{\mu\nu} = \frac{1}{4} [\gamma^\mu,\gamma^\nu]</math>
satisfy the Lorentz algebra commutation relations.

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

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

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

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

Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents <math>(\mathcal{J}^{\rho\sigma})^\mu</math>. Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents <math>T^{\mu\nu}</math>, which can be identified as the stress-energy tensor of the theory. The Lorentz current <math>(\mathcal{J}^{\rho\sigma})^\mu</math> 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.<ref>Jurgen Jost, (2002) "Riemannian Geometry and Geometric Analysis (3rd Edition)" Springer Universitext. ''(See chapter 1 for spin structures and chapter 3 for connections on spin structures)''</ref> Let <math>a</math> be a single, fixed point in the [[spacetime]] [[manifold]]. Its location can be expressed in multiple [[Atlas (topology)|coordinate systems]]. In the physics literature, these are written as <math>x</math> and <math>x'</math>, with the understanding that both <math>x</math> and <math>x'</math> describe ''the same'' point <math>a</math>, but in different [[local reference frame|local frames of reference]] (a [[frame of reference]] over a small extended patch of spacetime). 
One can imagine <math>a</math> as having a [[fiber (mathematics)|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 <math>x</math> and <math>x'</math> in the same fiber is a combination of [[rotation]]s and [[Lorentz boost]]s. A choice of coordinate frame is a (local) [[section (fiber bundle)|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 [[generator (mathematics)|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.<ref name="iz">Claude Itzykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory", McGraw-Hill ''(See Chapter 2)''</ref> It is very nearly identical to that of Bjorken and Drell.<ref>James D. Bjorken, Sidney D. Drell (1964) "Relativistic Quantum Mechanics", McGraw-Hill. ''(See Chapter 2)''</ref> A similar derivation in a [[general relativistic]] setting can be found in Weinberg.<ref name="weinberg">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.)''.</ref> Here we fix our spacetime to be flat, that is, our spacetime is Minkowski space.

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

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 <math>J_{\mu\nu}</math> of [[Lorentz transformation]]s, having the form
<math display="block">J_{\mu\nu} = \frac{1}{2} \sigma_{\mu\nu} + i (x_\mu\partial_\nu - x_\nu\partial_\mu)</math>
This can be interpreted as the [[total angular momentum]]. It acts on the spinor field as
<math display="block">\psi^\prime(x) = \exp\left(\frac{-i}{2} \omega^{\mu\nu} J_{\mu\nu}\right) \psi(x)</math>
Note the <math>x</math> above does ''not'' have a prime on it: the above is obtained by transforming <math>x \mapsto x'</math> obtaining the change to <math>\psi(x)\mapsto \psi'(x')</math> and then returning to the original coordinate system <math>x' \mapsto x</math>.

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

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

After properly antisymmetrizing, one obtains the generator of symmetries <math>J_{\mu\nu}</math> given earlier.  Thus, both <math>J_{\mu\nu}</math> and <math>\sigma_{\mu\nu}</math> 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 <math>x \mapsto x'</math> along the frame bundle, as well as a movement <math>\psi \mapsto \psi'</math> along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.<ref>Weinberg, "Gravitation", ''op cit.'' ''(See chapter 2.9 "Spin", pages 46-47.)''</ref>