47.1 The Dirac equation in Minkowski space

47.1 The Dirac equation in Minkowski space

Concept links · terms present in this machine draft; source roles are unverified: vector space · eigenvalue · irreducible representation · complexification

Recall from section 34.4 that for any real vector space ${\mathbf{R}}^{r+s}$ with an inner product of signature $(r,s)$ we can use the Cliford algebra $\mathrm{Cliff}(r,s)$ to define a first-order diferential operator, the Dirac operator $\partial /.$ For the Minkowski space case of signature $(3,1)$ , the Cliford algebra $\mathrm{Cliff}(3,1)$ is generated by elements ${\gamma }_0$, ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ satisfying

$${\gamma }_0^2=-1,{\gamma }_1^2={\gamma }_2^2={\gamma }_3^2=+1,{\gamma }_j{\gamma }_k+{\gamma }_k{\gamma }_j=0\mathrm{for}j\ne k$$

Clif(3, 1) is isomorphic to the algebra $\mathcal{M}(4,\mathbf{R})$ of 4 by 4 real matrices. Several conventional identifications of the generators ${\gamma }_j$ with 4 by 4 complex matrices satisfying the relations of the algebra were described in chapter 41. Each of these gives an identification of $\mathrm{Cliff}(3,1)$ with a specific subset of the complex matrices $M(4,\mathbf{C})$ and of the complexified Cliford algebra $\mathrm{Cliff}(3,1)\otimes \mathbf{C}$ with $M(4,\mathbf{C})$ itself. The Dirac operator in Minkowski space is thus

$$\partial ={\gamma }_0\frac{\partial }{\partial x_0}+{\gamma }_1\frac{\partial }{\partial x_1}+{\gamma }_2\frac{\partial }{\partial x_2}+{\gamma }_3\frac{\partial }{\partial x_3}={\gamma }_0\frac{\partial }{\partial x_0}+\mathbf{\gamma }\cdot \nabla$$

and it will act on four-component functions $\psi (t,\mathbf{x})=\psi (x)$ on Minkowski space. These functions take values in the four dimensional vector space that the Cliford algebra elements act on (which can be ${\mathbf{R}}^4$ if using real matrices, ${\mathbf{C}}^4$ in the complex case).

We have seen in chapter 42 that $-P_0^2+P_1^2+P_2^2+P_3^2$ is a Casimir operator for the Poincar´e group. Acting on four-component wavefunctions $\psi (x)$ , the Dirac operator provides a square root of (minus) this Casimir operator since

$${\partial }^2=-\frac{{\partial }^2}{\partial x_0^2}+\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\frac{{\partial }^2}{\partial x_3^2}=-(-P_0^2+P_1^2+P_2^2+P_3^2)$$

For irreducible representations of the Poincar´e group the Casimir operator acts as a scalar (0 for massless particles, $-m^2$ for particles of mass m). Using the Dirac operator we can rewrite this condition as

$$(-\frac{{\partial }^2}{\partial x_0^2}+\Delta -m^2)\psi =(/\partial /+m)(/\partial /-m)\psi =0\text{\hspace{1em}(47.1)}$$

This motivates the following definition of a new wave equation:

Definition (Dirac equation)

The Dirac equation is the diferential equation

$$(\partial /-m)\psi (x)=0\text{\hspace{1em}(47.2)}$$

for four-component functions on Minkowski space.

Using equation 40.3, for the Minkowski space Fourier transform, the Dirac equation in energy-momentum space is

$$(i/p-m)\tilde{\psi }(p)=(i(-{\gamma }_0{p}_0+{\gamma }_1{p}_1+{\gamma }_2{p}_2+{\gamma }_3{p}_3)-m)\tilde{\psi }(p)=0\text{\hspace{1em}(47.3)}$$

Note that solutions to this Dirac equation are also solutions to equation 47.1, but in a sense only half of them. The Dirac equation is first-order in time, so solutions are determined by the initial value data

$$\psi (\mathbf{x})=\psi (0,\mathbf{x})$$

of $\psi$ at a fixed time, while equation 47.1 is second-order, with solutions determined by specifying both $\psi$ and its time derivative.

The Dirac equation

$$({\gamma }_0\frac{\partial }{\partial x_0}+\gamma \cdot \nabla -m)\psi (x)=0$$

can be written in the form of a Schr¨odinger equation as

$$i\frac{\partial }{\partial t}\psi (t,\mathbf{x})=H_D\psi (t,\mathbf{x})\text{\hspace{1em}(47.4)}$$

with Hamiltonian

$$H_D=i{\gamma }_0(\gamma \cdot \nabla -m)\text{\hspace{1em}(47.5)}$$

Fourier transforming, in momentum space the energy eigenvalue equation is

$$-{\gamma }_0(\mathbf{\gamma }\cdot \mathbf{p}+im)\tilde{\psi }(\mathbf{p})=E\tilde{\psi }(\mathbf{p})$$

The square of the left-hand side of this equation is

$$\begin{array}{rl}(-{\gamma }_0(\mathbf{\gamma }\cdot \mathbf{p}+im){)}^2& ={\gamma }_0(\mathbf{\gamma }\cdot \mathbf{p}+im){\gamma }_0(\mathbf{\gamma }\cdot \mathbf{p}+im)\\ & =(\mathbf{\gamma }\cdot \mathbf{p}-im)(\mathbf{\gamma }\cdot \mathbf{p}+im)\\ & =(\mathbf{\gamma }\cdot \mathbf{p}{)}^2+m^2=|\mathbf{p}{|}^2+m^2\end{array}$$

This shows that solutions to the Dirac equation have the expected relativistic energy-momentum relation

$$E=\pm {\omega }_{\mathbf{p}}=\pm \sqrt{|\mathbf{p}{|}^2+m^2}$$

For each $\mathbf{p},$ there will be a two dimensional space of solutions ${\tilde{\psi }}_{+}(\mathbf{p})$ to

$$-{\gamma }_0(\gamma \cdot \mathbf{p}+im){\tilde{\psi }}_{+}(\mathbf{p})={\omega }_{\mathbf{p}}{\tilde{\psi }}_{+}(\mathbf{p})\text{\hspace{1em}(47.6)}$$

(the positive energy solutions), and a two dimensional space of solutions ${\tilde{\psi }}_{-}(\mathbf{p})$ to

$$-{\gamma }_0(\gamma \cdot \mathbf{p}+im){\tilde{\psi }}_{-}(\mathbf{p})=-{\omega }_{\mathbf{p}}{\tilde{\psi }}_{-}(\mathbf{p})\text{\hspace{1em}(47.7)}$$

(the negative energy solutions). Solutions to the Dirac equation can be identified with either

• Four-component functions $\psi (\mathbf{x})$ , initial value data at a time $t=0$

• Four-component functions $\tilde{\psi }(\mathbf{p})$ , Fourier transforms of the initial value data. These can be decomposed as

$$\tilde{\psi }(\mathbf{p})={\tilde{\psi }}_{+}(\mathbf{p})+{\tilde{\psi }}_{-}(\mathbf{p})\text{\hspace{1em}(47.8)}$$

into positive (solutions of 47.6) and negative (solutions of 47.7) energy components.

The four dimensional Fourier transform of a solution is of the form

$$\begin{array}{rl}\tilde{\psi }(p)& =\frac{1}{(2\pi {)}^2}{\int }_{{\mathbf{R}}^4}{e}^{-i(-p_0{x}_0+\mathbf{p}\cdot \mathbf{x})}\psi (x)d^{4}x\\ & =\theta (p_0)\delta (-p_0^2+|\mathbf{p}{|}^2+m^2){\tilde{\psi }}_{+}(\mathbf{p})+\theta (-p_0)\delta (-p_0^2+|\mathbf{p}{|}^2+m^2){\tilde{\psi }}_{-}(\mathbf{p})\end{array}$$

The Poincar´e group acts on solutions to the Dirac equation by

$$\psi (x)\to u(a,\Lambda )\psi (x)=S(\Lambda )\psi ({\Lambda }^{-1}\cdot (x-a))\text{\hspace{1em}(47.9)}$$

or, in terms of Fourier transforms, by

$$\tilde{\psi }(p)\to \tilde{u}(a,\Lambda )\tilde{\psi }(p)=e^{-i(-p_0{a}_0+\mathbf{p}\cdot \mathbf{a})}S(\Lambda )\tilde{\psi }({\Lambda }^{-1}\cdot p)\text{\hspace{1em}(47.10)}$$

Here $\Lambda$ is in $Spin(3,1)$ , the double cover of the Lorentz group, and $\Lambda \cdot x$ means the action of $Spin(3,1)$ on Minkowski space vectors. $S(\Lambda )$ is the spin representation, realized explicitly as 4 by 4 matrices by exponentiating quadratic combinations of the Cliford algebra generators (using a chosen identification of the ${\gamma }_j$ with 4 by 4 matrices). Spinor fields $\psi$ can be interpreted as elements of the tensor product of the spinor representation space $({\mathbf{R}}^4$ or ${\mathbf{C}}^4)$ and functions on Minkowski space. Then equation 47.9 means that $S(\Lambda )$ acts on the spinor factor, and the action on functions is the one induced from the Poincar´e action on Minkowski space.

Recall that (equation 29.4) conjugation by $S(\Lambda )$ takes vectors $v$ to their Lorentz transform $v^{\prime }=\Lambda \cdot v.$ in the sense that

$$S(\Lambda {)}^{-1}\psi S(\Lambda )={\psi }^{\prime }$$

so

$$/pS(\Lambda )\tilde{\psi }({\Lambda }^{-1}\cdot p)=S(\Lambda )/p^{\prime }{S}^{-1}(\Lambda )S(\Lambda )\tilde{\psi }(p^{\prime })=S(\Lambda )/p^{\prime }\tilde{\psi }(p^{\prime })$$

where $p^{\prime }={\Lambda }^{-1}\cdot p$ . As a result the action 47.10 takes solutions of the Dirac equation 47.3 to solutions, since

$$\begin{array}{rl}(i/p-m)\tilde{u}(a,\Lambda )\tilde{\psi }(p)& =(i/p-m)e^{-i(-p_0{a}_0+\mathbf{p}\cdot \mathbf{a})}S(\Lambda )\tilde{\psi }({\Lambda }^{-1}\cdot p)\\ & =e^{-i(-p_0{a}_0+\mathbf{p}\cdot \mathbf{a})}S(\Lambda )(i/p^{\prime }-m)\tilde{\psi }(p^{\prime })=0\end{array}$$

Chapter contents · Next section: 47.2 Majorana spinors and the Majorana field · 中文版本

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