Chapter 5 Free quantum Dirac field

This chapter studies the canonical quantization of the free Dirac field, which leads to fermionic spin \(\frac {1}{2}\) particles and their antiparticles.

5.1 Clifford algebra and spinor representation

Different types of relativistic fields are characterized by their transformation behavior under proper and orthochronous Poincaré transformations \(x^\prime = \Lambda x+b\). For instance, we have seen in Example 2.4 that a scalar field \(\Phi (x)\) transforms under active Poincaré transformations according to

\begin{flalign} T\,:\, \Phi (x)\longmapsto T\Phi (x) = \Phi \big (\Lambda ^{-1}(x-b)\big ) \end{flalign} and, more generally, a \((p,q)\)-tensor field \(A^{\mu _1\cdots \mu _p}_{\nu _1\cdots \nu _q}(x)\) transforms according to

\begin{flalign} T\,:\, A^{\mu _1\cdots \mu _p}_{\nu _1\cdots \nu _q}(x)\longmapsto (TA)^{\mu _1\cdots \mu _p}_{\nu _1\cdots \nu _q}(x) = \Lambda ^{\mu _1}_{~~\rho _1}\cdots \Lambda ^{\mu _p}_{~~\rho _p}\Lambda _{\nu _1}^{~~\sigma _1}\cdots \Lambda _{\nu _q}^{~~\sigma _q}\, A^{\rho _1\cdots \rho _p}_{\sigma _1\cdots \sigma _q}\big (\Lambda ^{-1}(x-b)\big )\quad . \end{flalign} These are particular cases of the general transformation law

\begin{flalign} \label {eqn:generalLorentztransformation} T\,:\, \phi _a(x)\longmapsto (T\phi )_a(x) = D[\Lambda ]_a^{~~c}\,\phi _c\big (\Lambda ^{-1}(x-b)\big ) \end{flalign} for a multicomponent field \(\phi _a(x)\), where \(D\) is a representation of the group \(\mathsf {SO}_0(d-1,1)\) of proper and orthochronous Lorentz transformations, i.e. \(D[\Lambda \,\Lambda ^\prime ]_{a}^{~~c} = D[\Lambda ]_{a}^{~~b}\,D[\Lambda ^\prime ]_{b}^{~~c}\) is multiplicative and it preserves the identity element \(D[1]_{a}^{~~b} = \delta _a^b\).

You might now ask the question: Do \((p,q)\)-tensor fields cover all possible cases of the general transformation law (5.3)? Mathematically, this amounts to classifying the representations \(D\) of the proper and orthochronous Lorentz group \(\mathsf {SO}_0(d{-}1,1)\), which is a problem that has been solved. Due to time constraints, we shall not discuss this classification in detail and I refer you to e.g. Maggiore (Chapter 2) for the case of \(d=4\) dimensions, which is the most relevant one for physics. The key observation is that, in addition to the \((p,q)\)-tensor representations of \(\mathsf {SO}_0(d{-}1,1)\), there exist spinor representations of the closely related spin group \(\mathsf {Spin}(d{-}1,1)\), which is a double cover of \(\mathsf {SO}_0(d{-}1,1)\). These are the relativistic analogs of the half-integer spin representations of the \(3\)-dimensional rotation group \(\mathsf {SO}(3)\), or more precisely of its double cover \(\mathsf {Spin}(3)\cong \mathsf {SU}(2)\), which you have previously seen in quantum mechanics.

To get our hands on such spinor representations, it is useful to work, as illustrated in Example 2.10, with infinitesimal Lorentz transformations. Let us recall and further expand on this point. The infinitesimal Lorentz transformations are characterized by real \(d\times d\)-matrices \(\omega ^{\mu }_{~~\nu }\), whose index lowering is antisymmetric \(\omega _{\mu \nu } = -\omega _{\nu \mu }\). The passage to finite Lorentz transformations is via matrix exponentiation \(\Lambda = e^{\omega }\). It is useful to factorize

\begin{flalign} \label {tmp:omegagenerators} \omega = \frac {\ii }{2}\,\omega _{\mu \nu }\,J^{\nu \mu } \end{flalign} into the parameters \(\omega _{\mu \nu } = - \omega _{\nu \mu }\) and the generators \(J^{\nu \mu } = -J^{\mu \nu }\) of the infinitesimal transformations. Taking the \({}^{\rho }_{~~\sigma }\) entry on both sides of the matrix equation (5.4), one finds that the generator \(J^{\nu \mu }\) is represented by the matrix

\begin{flalign} \label {eqn:vectorrepresentation} (J^{\nu \mu })^{\rho }_{~~\sigma } = \ii \,\big (\eta ^{\nu \rho }\,\delta ^{\mu }_{\sigma }-\eta ^{\mu \rho }\,\delta ^{\nu }_{\sigma }\big )\quad . \end{flalign} These generators satisfy the following commutation relations

which characterize the behavior of infinitesimal Lorentz transformations. In mathematical terminology, the infinitesimal symmetry generators and their commutation relations form a so-called Lie algebra, which is why (5.6) is often called the Lorentz Lie algebra. The matrix representation of the generators given in (5.5) is the so-called vector representation of the Lorentz Lie algebra; this terminology makes sense because (5.5) describes the infinitesimal Lorentz transformations \((\delta _\omega A)^\rho = \frac {\ii }{2}\,\omega _{\mu \nu }\,(J^{\nu \mu })^{\rho }_{~~\sigma }\,A^\sigma =\omega ^{\rho }_{~~\sigma }\,A^\sigma \) of tangent vectors \(A^\mu \in \bbR ^d\).

The spinor representation is a different representation of the Lorentz Lie algebra (5.6) that we shall now describe. To construct this representation, one starts from a more fundamental object, the so-called Clifford algebra. This is the complex algebra generated by symbols \(\gamma ^\mu \), for \(\mu =0,1,\dots ,d-1\), that satisfy the anticommutation relations

involving on the right-hand side the inverse Minkowski metric \(\eta ^{\mu \nu }\). With some efforts that go beyond the scope of this module, one can show that, up to equivalence given by change vector space bases, there exists a unique irreducible complex representation of the Clifford algebra. This means that we can and we will, without loss of generality, think of the gamma-matrices \(\gamma ^\mu \) as complex \(N\times N\)-matrices acting on a complex vector space \(\bbC ^N\), whose dimension \(N= 2^{\lfloor \frac {d}{2}\rfloor }\) is determined by the spacetime dimension. This vector space \(\bbC ^N\) is called the space of Dirac spinors. The relevance of the Clifford algebra is that the appropriately normalized commutators of the gamma-matrices

define a representation of the Lorentz Lie algebra, i.e.

\begin{flalign} \label {eqn:spinorLiegenerator} [S^{\mu \nu },S^{\rho \sigma }] =\ii \,\big (\eta ^{\nu \rho }\,S^{\mu \sigma } -\eta ^{\mu \rho }\,S^{\nu \sigma } -\eta ^{\nu \sigma }\,S^{\mu \rho } + \eta ^{\mu \sigma }\,S^{\nu \rho } \big )\quad . \end{flalign} To prove this claim, let us first note that, using the Clifford algebra relations (5.7), we can write

\begin{flalign} \label {eqn:spinorgeneratoralternative} S^{\mu \nu } = \frac {\ii }{4}\,\big ( \gamma ^\mu \gamma ^\nu - \gamma ^{\nu }\gamma ^\mu \big ) = \frac {\ii }{4}\,\big ( 2 \,\gamma ^\mu \gamma ^\nu - \{ \gamma ^{\nu },\gamma ^\mu \}\big ) = \frac {\ii }{2}\,\gamma ^\mu \gamma ^\nu - \frac {\ii }{2} \,\eta ^{\mu \nu }\quad . \end{flalign} Using again the Clifford algebra relations, we compute

\begin{flalign} \nn [S^{\mu \nu },\gamma ^\rho ] &= \frac {\ii }{2} \, [\gamma ^\mu \,\gamma ^\nu ,\gamma ^\rho ] = \frac {\ii }{2}\, \big (\gamma ^\mu \,\gamma ^\nu \,\gamma ^\rho - \gamma ^\rho \,\gamma ^\mu \,\gamma ^\nu \big )\\ \nn &=\frac {\ii }{2}\,\big ( \gamma ^\mu \,\gamma ^\nu \,\gamma ^\rho + \gamma ^\mu \,\gamma ^\rho \,\gamma ^\nu - \{\gamma ^\rho ,\gamma ^\mu \}\,\gamma ^\nu \big )\\ \nn &= \frac {\ii }{2}\,\big ( \gamma ^\mu \,\gamma ^\nu \,\gamma ^\rho - \gamma ^\mu \,\gamma ^\nu \,\gamma ^\rho + \gamma ^\mu \,\{\gamma ^\rho ,\gamma ^\nu \} - \{\gamma ^\rho ,\gamma ^\mu \}\,\gamma ^\nu \big )\\ &= -\ii \,\big (\eta ^{\mu \rho } \,\gamma ^\nu - \eta ^{\nu \rho }\,\gamma ^\mu \big ) = - (J^{\mu \nu })^{\rho }_{~~\sigma }\,\gamma ^\sigma \quad ,\label {eqn:spinorgeneratorgamma} \end{flalign} which together with the Leibniz rule for the commutator implies (5.9) by a short calculation

\begin{flalign} \nn [S^{\mu \nu },S^{\rho \sigma }] &= \frac {\ii }{2}\,[S^{\mu \nu },\gamma ^\rho \,\gamma ^\sigma ]\\ \nn &= \frac {\ii }{2}\,\big ([S^{\mu \nu },\gamma ^\rho ] \,\gamma ^\sigma + \gamma ^\rho \,[S^{\mu \nu },\gamma ^\sigma ]\big )\\ \nn &= -\frac {\ii ^2}{2}\,\big (\eta ^{\mu \rho }\,\gamma ^\nu \,\gamma ^\sigma - \eta ^{\nu \rho }\,\gamma ^\mu \,\gamma ^\sigma + \eta ^{\mu \sigma }\,\gamma ^\rho \,\gamma ^\nu - \eta ^{\nu \sigma }\,\gamma ^\rho \,\gamma ^\mu \big )\\ &= \ii \,\big (\eta ^{\nu \rho }\,S^{\mu \sigma } -\eta ^{\mu \rho }\,S^{\nu \sigma } -\eta ^{\nu \sigma }\,S^{\mu \rho } + \eta ^{\mu \sigma }\,S^{\nu \rho } \big )\quad . \end{flalign}

The representation (5.8) of the Lorentz Lie algebra on \(\bbC ^N\) is called the Dirac spinor representation. The action of a finite Lorentz transformation on such spinors \(\psi \in \bbC ^N\) is given by exponentiation

\begin{flalign} \label {eqn:Diracrepresentation} \psi ~\longmapsto ~\psi ^\prime \,:= \, e^{\frac {\ii }{2}\omega _{\mu \nu } S^{\nu \mu }} \psi \quad . \end{flalign} A slightly unpleasant feature of the Dirac spinor representation is that it is not a unitary representation. (In fact, the Lorentz Lie algebra (5.6) does not admit any finite-dimensional unitary representations.) This means that the standard complex inner product \(\langle \psi ,\tilde {\psi }\rangle := \psi ^\dagger \,\tilde {\psi }\) on \(\bbC ^N\) is not invariant under the Lorentz transformations (5.13), hence it cannot be used later to build a Poincaré invariant action functional for the Dirac field. This issue can be traced back to the Clifford relations (5.7), which can be written equivalently for the individual gamma-matrices as

\begin{flalign} \label {eqn:CliffordalgebraComponents} (\gamma ^0)^2 \,=\, -1 ~~,\quad (\gamma ^i)^2 \,=\, 1~~,\quad \gamma ^\mu \,\gamma ^\nu \,=\,-\gamma ^\nu \,\gamma ^\mu ~~~\text {(for $\mu \neq \nu $)}\quad . \end{flalign} Because \(\gamma ^0\) squares to \(-1\), this matrix cannot be chosen to be Hermitian, but it turns out that it can be chosen to be anti-Hermitian. The other \(\gamma ^i\)’s can be chosen to be Hermitian matrices, which allows us to demand the following (anti-)Hermiticity conditions

\begin{flalign} (\gamma ^0)^\dagger = -\gamma ^0~~,\quad (\gamma ^i)^\dagger = \gamma ^i\quad . \end{flalign} Using the Clifford algebra relations in the form of (5.14), we can write these (anti)-Hermiticity conditions in closed form as

\begin{flalign} \label {eqn:gammaantihermiticity} (\gamma ^\mu )^\dagger = \gamma ^0\,\gamma ^\mu \,\gamma ^0 = -(\ii \gamma ^0)\,\gamma ^\mu \,(\ii \gamma ^0)\quad . \end{flalign} The second expression is more useful because the matrix \((\ii \gamma ^0)\) has the following pleasant properties

\begin{flalign} (\ii \gamma ^0)^2 = 1\quad ,\qquad (\ii \gamma ^0)^\dagger = (\ii \gamma ^0)\quad , \end{flalign} which we will frequently use in our calculations below. From (5.16) one immediately deduces that the generators (5.8) satisfy

\begin{flalign} (S^{\mu \nu })^\dagger = -\frac {\ii }{4}\,\big [(\gamma ^\nu )^\dagger ,(\gamma ^{\mu })^\dagger \big ] = - (\ii \gamma ^0) \,S^{\nu \mu }\,(\ii \gamma ^0)=(\ii \gamma ^0) \,S^{\mu \nu }\,(\ii \gamma ^0)\quad . \end{flalign} At the level of finite transformations, one then finds that

\begin{flalign} \label {eqn:spinortransformationdagger} \Big (e^{\frac {\ii }{2}\omega _{\mu \nu }\,S^{\nu \mu }}\Big )^\dagger = (\ii \gamma ^0)~ e^{-\frac {\ii }{2}\omega _{\mu \nu }\,S^{\nu \mu }}~(\ii \gamma ^0)\quad , \end{flalign} which shows our claim that (5.13) is not a unitary representation. This expression however motivates the definition of the Dirac adjoint

and its associated Dirac inner product

which have the following pleasant Lorentz transformation properties

\begin{flalign} \label {eqn:Diracinnerproductinvariant} \overline {\psi }^\prime \,=\, \overline {\psi }\, e^{-\frac {\ii }{2}\omega _{\mu \nu }\,S^{\nu \mu }} \quad ,\qquad \overline {\psi }^\prime \,{\tilde {\psi }}^\prime \,=\, \overline {\psi }\,\tilde {\psi }\quad . \end{flalign} In particular, in contrast to the standard complex inner product \(\psi ^\dagger \,\tilde {\psi }\), the Dirac inner product is Lorentz invariant. To conclude our discussion of the Dirac spinor representation, we note that the identity in (5.11) exponentiates to

\begin{flalign} e^{\frac {\ii }{2}\omega _{\mu \nu }\,S^{\nu \mu }} \,\gamma ^\rho e^{-\frac {\ii }{2}\omega _{\mu \nu }\,S^{\nu \mu }} = \big (e^{-\omega }\big )^\rho _{~~\sigma }\, \gamma ^\sigma \quad . \end{flalign} This identity expresses that the gamma-matrices intertwine between the spinor and vector representations of the Lorentz group. In particular, it implies that the tuple of complex numbers \(\overline {\psi }\,\gamma ^\rho \,\tilde {\psi }\), for \(\rho =0,\dots ,d-1\), transforms in the vector representation, i.e.

\begin{flalign} \label {eqn:Diracgammavector} \overline {\psi }^\prime \, \gamma ^\rho \, {\tilde {\psi }}^\prime =\overline {\psi }\, e^{-\frac {\ii }{2}\omega _{\mu \nu }\,S^{\nu \mu }} \,\gamma ^\rho \, e^{\frac {\ii }{2}\omega _{\mu \nu }\,S^{\nu \mu }}\,\tilde {\psi } = \big (e^{\omega }\big )^{\rho }_{~~\sigma }\,\overline {\psi }\,\gamma ^\sigma \,\tilde {\psi } = \Lambda ^\rho _{~~\sigma }\,\overline {\psi }\,\gamma ^\sigma \,\tilde {\psi }\quad . \end{flalign} These are now all the ingredients we need to introduce the classical Dirac field in the next section.

Explicit formulas for gamma-matrices and spinors in \(d=4\) dimensions:

To get a better understanding of the concept of Dirac spinors, we shall spell out the case of \(d=4\) dimensional Dirac spinors in a fully explicit form. The space of Dirac spinors in \(d=4\) is given by the \(N=2^{\lfloor \frac {d}{2}\rfloor }=4\)-dimensional complex vector space \(\bbC ^4\). A choice of representation for the gamma-matrices, which we recall is unique up to equivalence, is given by the following block \(4\times 4\)-matrices

\begin{flalign} \label {eqn:gammad=4} \gamma ^0 = -\ii \begin{pmatrix} 0 & 1_{2\times 2}\\ 1_{2\times 2} & 0 \end {pmatrix}\quad ,\qquad \gamma ^i = -\ii \begin{pmatrix} 0 & \sigma ^i\\ -\sigma ^i & 0 \end {pmatrix}\quad , \end{flalign} where \(1_{2\times 2}\) denotes the \(2\times 2\) identity matrix and \(\sigma ^i\) are the three Pauli matrices

\begin{flalign} \label {eqn:Paulimatrices} \sigma ^1 = \begin{pmatrix} 0&1\\ 1&0 \end {pmatrix}\quad ,\qquad \sigma ^2 = \begin{pmatrix} 0 & -\ii \\ \ii & 0 \end {pmatrix}\quad ,\qquad \sigma ^3 = \begin{pmatrix} 1&0\\ 0&-1 \end {pmatrix}\quad . \end{flalign} Recall that the Pauli matrices satisfy the following multiplication property

\begin{flalign} \sigma ^i\,\sigma ^j = \delta ^{ij} + \ii \,\epsilon ^{ijk}\,\sigma ^k\quad , \end{flalign} which implies in particular the commutation and anticommutation relations

\begin{flalign} [\sigma ^i,\sigma ^j] = 2\,\ii \,\epsilon ^{ijk}\,\sigma ^k\quad ,\qquad \{\sigma ^i,\sigma ^j\} = 2\,\delta ^{ij} \quad . \end{flalign} Using these identities, one easily checks that (5.25) satisfies the Clifford algebra relations (5.7). For the Lie algebra generators (5.9), one finds

\begin{flalign} S^{0i} = -S^{i0} = \frac {\ii }{2}\,\begin{pmatrix} \sigma ^i & 0 \\ 0 & -\sigma ^i \end {pmatrix} \end{flalign} and

\begin{flalign} S^{ij} = -\frac {1}{2} \,\begin{pmatrix} \epsilon ^{ijk}\,\sigma ^k & 0\\ 0 & \epsilon ^{ijk}\,\sigma ^k \end {pmatrix}\quad . \end{flalign} Exponentiating these generators, we observe that Dirac spinors transform under Lorentz boosts according to

\begin{flalign} \psi ~\longmapsto ~\psi ^\prime = \begin{pmatrix} e^{\frac {\omega _{0i}}{2} \,\sigma ^i} & 0 \\ 0 & e^{-\frac {\omega _{0i}}{2} \,\sigma ^i} \end {pmatrix}\,\psi \end{flalign} and under spatial rotations according to

\begin{flalign} \psi ~\longmapsto ~\psi ^\prime = \begin{pmatrix} e^{\frac {\ii }{4}\omega _{ij}\,\epsilon ^{ijk}\,\sigma ^k } & 0 \\ 0 & e^{\frac {\ii }{4}\omega _{ij}\,\epsilon ^{ijk}\,\sigma ^k } \end {pmatrix}\,\psi \quad . \end{flalign}

We conclude this section by making an observation that will become relevant later when we discuss the standard model of particle physics. From our explicit Lorentz transformation formulas (5.29) for Dirac spinors, we immediately see that the Dirac spinor representation in \(d=4\) is reducible. Indeed, writing

\begin{flalign} \label {eqn:chiraldecomposition} \psi = \begin{pmatrix} \psi _L^{}\\ \psi _R^{} \end {pmatrix} \end{flalign} in terms of \(2\)-component spinors \(\psi _L^{}\in \bbC ^2\) and \(\psi _R^{}\in \bbC ^2\), we see that Lorentz transformations don’t mix among \(\psi _L^{}\) and \(\psi _R^{}\) because (5.29) is block diagonal. The \(2\)-component spinors \(\psi _{L/R}^{}\in \bbC ^2\) are called the left/right handed Weyl (or chiral) spinors. The existence of Weyl spinors is a feature linked to the fact that the spacetime dimension \(d=4\) is even. The Weyl spinors can be defined more intrinsically in terms of the Clifford algebra by introducing the product of gamma-matrices

\begin{flalign} \gamma _5 \,:= \, -\ii \,\gamma ^0\,\gamma ^1\,\gamma ^2\,\gamma ^3\quad , \end{flalign} which in our representation (5.25) reads as

\begin{flalign} \gamma _5 \,=\,\begin{pmatrix} 1_{2\times 2} & 0 \\ 0 & -1_{2\times 2} \end {pmatrix} \quad . \end{flalign} Introducing the projectors

\begin{flalign} P_{L}^{} \,:=\, \frac {1+\gamma _5}{2} \,=\, \begin{pmatrix} 1_{2\times 2} & 0\\ 0 & 0 \end {pmatrix}\quad ,\qquad P_{R}^{} \,:=\, \frac {1-\gamma _5}{2} \,=\, \begin{pmatrix} 0 & 0\\ 0 & 1_{2\times 2} \end {pmatrix}\quad , \end{flalign} one obtains the following more intrinsic characterization of left/right handed spinors

\begin{flalign} \psi = \begin{pmatrix} \psi _L\\0 \end {pmatrix} \quad &\Longleftrightarrow \quad P_R^{}\psi = 0\quad ,\\ \psi = \begin{pmatrix} 0\\ \psi _R \end {pmatrix} \quad &\Longleftrightarrow \quad P_L^{}\psi = 0\quad . \end{flalign}