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The Dirac Electron

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Contents

Idea

The purpose of this page is to explain - using appropriate mathematical terminology - Dirac’s theory of coupling particles with spin to a Yang-Mills gauge field.

We proceed in three steps: first we recall relevant facts about the gauge field itself, then we discuss charged particles in gauge fields, and finally we add spin.

Yang-Mills Theory

(see: the main article about Yang-Mills theory)

Under a spacetime we understand a smooth, oriented, pseudo-Riemannian manifold.

Definition

A Yang-Mills theory over a spacetime MM is:

  • A Lie group GG, called the gauge group, together with an Ad\mathrm{Ad}-invariant scalar product κ:𝔤×𝔤R\kappa: \mathfrak{g} \times \mathfrak{g} \to \R on its Lie algebra 𝔤\mathfrak{g}.

  • A GG-principal bundle PP over MM.

A gauge field is a connection ωΩ 1(P,𝔤)\omega \in \Omega^1(P,\mathfrak{g}) on PP. The action functional is

S YM(ω):=12 MF ω κ 2. S_{YM}(\omega) := \frac{1}{2} \int_M \| F_{\omega} \|_\kappa^2.
Remark

Above we have used the following notation:

  • F ωΩ 2(M,Ad(P))F_{\omega} \in \Omega^2(M,\mathrm{Ad}(P)) is the curvature of ω\omega.

  • Ad(P):=P× Ad𝔤\mathrm{Ad}(P) := P \times_{\mathrm{Ad}} \mathfrak{g} is the adjoint bundle.

  • ψ κ 2:=ψ κψΩ n(M)\| \psi \|_{\kappa}^2 := \psi \wedge_{\kappa} \star \psi \in \Omega^n(M). In general, if U,V,WU,V,W are vector spaces, φΩ p(M,V)\varphi \in \Omega^p(M,V), ψΩ q(M,W)\psi\in\Omega^q(M,W) and f:V×WUf: V \times W \to U is a linear map, we have φ fψΩ p+q(M,U)\varphi \wedge_{f} \psi \in \Omega^{p+q}(M,U).

  • \star is the Hodge-star operator? determined by the metric on MM.

Remark

The Euler-Lagrange equations determined by the above action together with the Bianchi identity are called Yang-Mills equations:

D ωF ω=0 and D ωF ω=0, \mathrm{D}^{\omega}\star F_{\omega} = 0 \quad\text{ and }\quad \mathrm{D}^{\omega}F_{\omega}=0,

where D ω\mathrm{D}^\omega denotes the covariant derivative.

Definition

A gauge transformation is a smooth bundle morphism g:PPg: P \to P.

Remark

Let g:PPg: P \to P be a gauge transformation.

  • If ω\omega is a connection on PP, then g *ωg^{*}\omega is another connection on PP.

  • One can identify gg with a smooth map g˜:PG\tilde g: P \to G, namely by g=r g˜g=r_{\tilde g}, i.e. g(p)=pg˜(p)g(p) = p \cdot \tilde g(p) for all pPp\in P.

  • The pullback along a gauge transformation restricts to an automorphism of Ω ρ k(P,V)\Omega^k_{\rho}(P,V). In terms of the associated map g˜\tilde g, we have

    g *ψ=ρ(g˜ 1,ψ). g^{*}\psi = \rho(\tilde g^{-1},\psi)\text{.}
Theorem

The Yang-Mills action functional S YMS_{YM} is gauge-invariant, i.e.

S YM(g *ω)=S YM(ω) S_{YM}(g^{*}\omega) = S_{YM}(\omega)

for all gauge transformations g:PPg:P \to P.

Proof

We have g *ω=Ad g˜ 1(ω)g˜ *θ¯g^{*}\omega = \mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta and g *Ω=Ad g˜ 1(Ω)g^{*}\Omega = \mathrm{Ad}_{\tilde g}^{-1}(\Omega). Under the isomorphism Ω k(P,Ad)Ω 2(M,Ad(P))\Omega^k(P,\mathrm{Ad}) \cong \Omega^2(M,\mathrm{Ad}(P)) this corresponds to F g *ω=Ad g(F ω)F_{g^{*}\omega} = \mathrm{Ad}_{g} (F_{\omega}). Since the bilinear form κ\kappa is Ad\mathrm{Ad}-invariant by assumption,

F g *ω=Ad g(F ω) κAd g(F ω)=F ω κF ω=F ω. \| F_{g^{*}\omega} \| = \mathrm{Ad}_{g} (F_{\omega}) \wedge_{\kappa} \mathrm{Ad}_{g} (F_{\omega}) = F_{\omega} \wedge_{\kappa} F_{\omega} = \| F_{\omega}\|.
Example

Let MM be a spacetime. A classical electromagnetic field theory over MM is a Yang-Mills Theory over MM with gauge group G=U(1)G=U(1). In more detail:

  • for an electromagnetic field theory given by a U(1)U(1)-bundle PP over MM, we have Ad(P)P×R\mathrm{Ad}(P) \cong P \times \R, so that Ω k(M,Ad(P))Ω k(M)\Omega^k(M,\mathrm{Ad}(P)) \cong \Omega^k(M) and Ω Ad k(P,𝔤)Ω Ad k(P)\Omega^k_{\mathrm{Ad}}(P,\mathfrak{g})\cong \Omega^k_{\mathrm{Ad}}(P). In particular, F ωΩ 2(M)F_{\omega} \in \Omega^2(M).

  • Since U(1)U(1) is abelian, d(Ad)=0\mathrm{d}(\mathrm{Ad}) = 0 and so D ω=d\mathrm{D}^{\omega}=\mathrm{d} on Ω Ad k(P)\Omega^{k}_{\mathrm{Ad}}(P).

  • Thus, the Yang-Mills equations reduce to Maxwell’s equations for an electromagnetic field on MM:

    dF ω=0 and dF ω=0. \mathrm{d}\star F_{\omega} = 0 \quad\text{ and }\quad \mathrm{d}F_{\omega}=0 \text{.}

General Matter Fields

Definition

Let GG be a gauge group. A matter type for GG is a tuple (V,h,ρ,f)(V,h,\rho,f) consisting of:

  • a finite-dimensional real vector space VV called the internal state space.

  • a scalar product h:V×VRh: V \times V \to \R.

  • a representation ρ:G×VV\rho: G \times V \to V that is isometric with respect to hh i.e. h(ρ(g)(v),ρ(g)(w))=h(v,w)h(\rho(g)(v),\rho(g)(w)) = h(v,w).

  • a smooth function f:VRf: V \to \R that is ρ\rho-invariant, i.e. f(ρ(g)(v))=f(v)f(\rho(g)(v))=f(v).

Definition

Let PP be a principal GG-bundle over MM, and let 𝒯=(V,h,ρ,f)\mathcal{T} =(V,h,\rho,f) be a matter type for GG. A field for PP of type 𝒯\mathcal{T} is a smooth section ϕ:MP× ρV\phi: M \to P \times_{\rho} V. Its action functional is

S 𝒯(ω,ϕ):= MD ω(ϕ) h 2+(fϕ). S_{\mathcal{T}}(\omega,\phi) := \int_M \;\|\, \mathrm{D}^{\omega}(\phi)\, \|^2_{h} + \star (f \circ \phi) \text{.}
Theorem

The action functional S 𝒯S_{\mathcal{T}} is gauge invariant, i.e.

S 𝒯(g *ω,g *ϕ)=S 𝒯(ω,ϕ) S_{\mathcal{T}}(g^{*}\omega,g^{*}\phi) = S_{\mathcal{T}}(\omega,\phi)

for all gauge transformations g:PPg:P \to P.

Proof

One calculates that g *ω=Ad g˜ 1(ω)g˜ *θ¯g^{*}\omega = \mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta, where g˜:PG\tilde g:P \to G is the smooth map associated to gg via g(p)=pg˜(p)g(p) = p\cdot \tilde g(p). Further, g *ψ=ρ(g˜ 1)(ψ)g^{*}\psi = \rho(\tilde g^{-1})(\psi). A computation shows

d(ρ(g˜ 1)(ψ))=ρ(g˜ 1)(dψ)+g˜ *θ¯ dρρ(g˜ 1)(ψ), \mathrm{d}(\rho(\tilde g^{-1})(\psi)) = \rho(\tilde g^{-1})(\mathrm{d}\psi) + \tilde g^{*}\bar\theta\wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1})(\psi)\text{,}

where dρ:𝔤VV\mathrm{d}\rho: \mathfrak{g} \otimes V \to V. Now we compute

d g *ω(g *ψ)\quad\quad\mathrm{d}^{g^{*}\omega}(g^{*}\psi)

=dρ(g˜ 1,ψ)+g *ω dρρ(g˜ 1,ψ)\quad\quad\quad\quad= \mathrm{d}\rho(\tilde g^{-1},\psi) + g^{*}\omega \wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi)

=ρ(g˜ 1,dψ)+g˜ *θ¯ dρρ(g˜ 1,ψ)+(Ad g˜ 1(ω)g˜ *θ¯) dρρ(g˜ 1,ψ)\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}\psi) + \tilde g^{*}\bar\theta\wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi) + (\mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta) \wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi)

=ρ(g˜ 1,dψ)+ρ(g˜ 1,ω dρψ)\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}\psi) + \rho(\tilde g^{-1},\omega\wedge_{\mathrm{d}\rho} \psi)

=ρ(g˜ 1,d ωψ).\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}^{\omega}\psi)\text{.}

Since hh is invariant, the invariance of the first term follows. The invariance of the second term is clear.

Remark

One can either keep a connection ω\omega fixed and consider

S ω(ϕ):=S 𝒯(ω,ϕ) S_{\omega}(\phi) := S_{\mathcal{T}}(\omega,\phi)

as a matter field in an “external gauge field”, or consider the combined action functional

S(ω,ϕ):=S YM(ω)+S 𝒯(ω,ϕ). S(\omega,\phi) := S_{Y\!M}(\omega) + S_{\mathcal{T}}(\omega,\phi)\text{.}
Example

(Scalar particle in an external, trivial gauge field)

We consider G={e}G=\left \lbrace e \right \rbrace, P=MP=M, so that necessarily ω=0\omega=0. A scalar field is field for MM of matter type (R,h,id,f)(\R,h,\id,f) where f(x):=12m 2x 2f(x) := -\frac{1}{2}m^2x^2 and h(x,y)=xyh(x,y) = x y. The action functional is

S(0,ϕ)=12 Mdϕ h 2+m 2ϕ 2. S(0,\phi) =\frac{1}{2} \int_M \| \mathrm{d}\phi \|_h^2 +\star m^{2}\phi^2\text{.}

The Euler-Lagrange equation is the Klein-Gordon equation?

(+m 2)ϕ=0, (\triangle + m^2)\phi = 0\text{,}

where :=δd:Ω k(M)Ω k(M)\triangle := \delta \circ \mathrm{d}: \Omega^k(M) \to \Omega^{k}(M) is the Laplace operator and δ:=d\delta := \star \mathrm{d} \star is the exterior coderivative.

Example

(Charged particle in an electromagnetic field, e.g. a π \pi^{-}-meson)

Let PP be a U(1)U(1)-principal bundle over MM. A field of charge nZn \in \Z is a field for PP of matter type (C,h,ρ n,f)(\C,h,\rho_n,f), where ρ n:U(1)×CC\rho_n: U(1) \times \C \to \C is defined by ρ n(z,z):=z nz\rho_n(z,z') := z^n z' and f(z):=12m 2|z| 2f(z) := -\frac{1}{2}m^2|z|^2. The action functional is

S(ω,ϕ)=12 MD ωϕ 2+m 2ϕ 2. S(\omega,\phi) =\frac{1}{2} \int_M \| \mathrm{D}^{\omega}\phi \|^2 +\star m^{2} \| \phi \| ^2\text{.}

The Euler-Lagrange equation is covariant Klein-Gordon equation

( ω+m 2)ϕ=0, (\triangle^{\omega} +m^2) \phi = 0\text{,}

where ω:=δ ωD ω\triangle^{\omega} :=\delta^{\omega}\circ \mathrm{D}^{\omega} is the covariant Laplace operator and δ ω:=D ω\delta^{\omega} := \star \mathrm{D}^{\omega} \star is the exterior covariant coderivative.

Matter Fields with Spin

The Klein-Gordon equation?s found above are – unlike the Schrödinger equation – of second order on time. Dirac’s motivation was to find a first order equation which upon iteration yields the Klein-Gordon equation. We first discuss free spinors (where free means that they are not coupled to an electromagnetic field, but still feel the “gravity” of the spacetime manifold), and then add the coupling.

Free Spinors

Example

We recall some facts about Clifford algebras and the spin group.

  • We denote by C(p,q)C(p,q) the Clifford algebra on R p,q\R^{p,q}, i.e. the quotient of the tensor algebra of R p+q\R^{p+q} by the ideal generated by vw+wv+2v,w1v \otimes w + w \otimes v + 2 \left \langle v,w \right \rangle\cdot 1, where ,\left \langle -,- \right \rangle is the Minkowski scalar product of signature (p,q)(p,q).

  • The map vvv \mapsto -v extends to an anti-automorphism α:C(p,q)C(p,q)\alpha: C(p,q) \to C(p,q), whose eigenspace decomposition yields the usual Z 2\Z_2-grading on C(p,q)C(p,q).

  • We have dimC(p,q)=2 p+q\dim C(p,q) = 2^{p+q}.

  • The Clifford algebra inherits a bilinear form H(v,w):=(v trw) 0H(v,w) := (v^{tr}w)_0, where () tr()^{tr} is the anti-automorphism of the tensor algebra that reverts the order of tensor products, and () 0()_0 denotes the degree 0 part.

  • We denote by SO(p,q)SO(p,q) the group of linear maps R p+qR p+q\R^{p+q}\to \R^{p+q} that preserve the product ,\left \langle -,- \right \rangle. We define

    Spin(p,q):={v 1...v 2rC(p,q)v iR p+q,v i=1,rN}. Spin(p,q) := \left \lbrace v_{1}\cdot ... \cdot v_{2r}\in C(p,q)\mid v_i \in \R^{p+q}, \| v_i \|=1,r \in \N \right \rbrace \text{.}

    Then, we define a group homomorphism Λ:Spin(p,q)SO(p,q)\Lambda: Spin(p,q) \to SO(p,q) by Λ(φ)v=α(φ)vφ 1\Lambda(\varphi)v = \alpha(\varphi) v \varphi^{-1}. This gives a central extension

    1Z 2Spin(p,q)SO(p,q)1. 1 \to \Z_2 \to Spin(p,q) \to SO(p,q) \to 1\text{.}
  • We denote by C(p,q):=C(p,q) RC\C(p,q) := C(p,q) \otimes_{\R} \C the complexification of the Clifford algebra. The bilinear form HH on C(p,q)C(p,q) extends to a sesquilinear form HH on C(p,q)\C(p,q) defined by H(v,w)=(v trw¯) 0H(v,w)=(v^{tr}\bar w)_0.

  • Multiplication in C(p,q)\C(p,q) restricts to an action of spinp,q\spin{p,q} on C(p,q)\C(p,q). One can decompose C(p,q)\C(p,q) into kk copies of a subrepresentation Σ\Sigma:

    C(p,q)=Σ...Σ. \C(p,q) = \Sigma \oplus ... \oplus \Sigma \text{.}
  • The representation Σ\Sigma is isometric with respect to HH. If p+qp+q is odd, k=2 (p+q1)/2k=2^{(p+q-1)/2}, and Σ\Sigma is irreducible. If p+qp+q is even, k=2 (p+q)/2k=2^{(p+q)/2}, and Σ=Σ +Σ \Sigma = \Sigma^{+} \oplus \Sigma^{-} with Σ ±\Sigma^{\pm} irreducible and dim CΣ ±=2 (p+q2)/2\dim_{\C}\Sigma^{\pm}=2^{(p+q-2)/2}.

Remark

We also need some facts about spin structures.

  • Let MM be a spacetime with pseudo-Riemannian metric of signature (p,q)(p,q). We denote by FMFM be the principal SO(p,q)SO(p,q)-bundle over MM of orthonormal frames, the frame bundle.

  • A spin structure on MM is a principal Spin(p,q)Spin(p,q)-bundle SMSM over MM together with a bundle morphism λ:SMFM\lambda: SM \to FM such that λ(Xφ)=λ(X)Λ(φ)\lambda(X\cdot\varphi) = \lambda(X)\cdot\Lambda(\varphi) for all XSMX\in SM and all φSpin(p,q)\varphi\in Spin(p,q).

  • Let θΩ 1(FM,𝔰𝔬(p,q))\theta \in \Omega^1(FM,\mathfrak{so}(p,q)) be the Levi-Cevita connection on FMFM. Then,

    Θ:=dΛ 1(λ *θ)Ω 1(SM,𝔰𝔭𝔦𝔫(p,q)) \Theta := \mathrm{d}\Lambda^{-1}(\lambda^{*}\theta) \in \Omega^1(SM,\mathfrak{spin}(p,q))

    is a connection on SMSM.

Remark

Finally, we recall the definition of the Dirac operator.

  • Let ρ:C(p,q)×VV\rho: C(p,q) \times V \to V be a representation, with VC(p,q)V \subset \C(p,q). Note that in the above realization of the group Spin(p,q)Spin(p,q), the representation ρ\rho restricts to a representation of Spin(p,q)Spin(p,q). The spinor bundle is the vector bundle VM:=SM× ρVV M := SM \times_{\rho} V.

  • Clifford multiplication is a map

    TMVMVM:usus. TM \otimes V M \to V M: u \otimes s \mapsto u \cdot s\text{.}

    It is defined as follows. We write s=(X,v)s=(X,v) with XSMX\in SM and vVv\in V. We consider the orthonormal frame α X:=λ(X):TMR n\alpha_X := \lambda(X): TM \to \R^n. Then, α X(u)C(p,q)\alpha_X(u) \in \C(p,q) and

    us:=(X,α X(u)v). u \cdot s := (X,\alpha_X(u)\cdot v)\text{.}

    One can show using above-listed properties of the Clifford algebra that this definition does not depend on the choice of the representative (X,v)(X,v).

  • The Dirac operator is

    D:Γ(M,VM)Γ(M,VM):ψ ie iD Θψ(e i) D: \Gamma(M,V M) \to \Gamma(M,V M) : \psi \mapsto \sum_{i} e_i \cdot \mathrm{D}^{\Theta}\psi (e_i)

    where e iTMe_i\in TM runs over a local orthonormal basis.

Definition

Let MM be a spacetime with spin structure SMSM, and considered as a Spin(p,q)Spin(p,q) as a Yang-Mills theory over MM. A free spinor is a field for SMSM of type (V,h,ρ,f)(V,h,\rho,f), where VC(p,q)V \subset \C(p,q), the scalar product hh is

h(v,w):=12(H(v,w)+H(w,v)), h(v,w):=\frac{1}{2}(H(v,w) + H(w,v))\text{,}

and ρ\rho is the restriction of the multiplication in C(p,q)\C(p,q) to Spin(p,q)Spin(p,q). The action functional is

S(ψ):= MDψ hψ+fψ. S(\psi) := \int_M D \psi \wedge_{h} \star \psi + \star f \circ \psi\text{.}
Definition

The Euler-Lagrange equation determined by the action functional S(ψ)S(\psi) is the Dirac equation

Dψ+imψ=0. D\psi + \mathrm{i}m\psi = 0\text{.}
Definition

(Weyl spinors)

We assume spacetime to have even dimension. Weyl spinors have V=Σ ±V=\Sigma^{\pm}, with the sign corresponding to left/right-handed spinors. Thus, dim C(V)=2\dim_{\C}(V)=2. Further f=0f=0 (they are massless). In the standard model, neutrinos are left-handed Weyl spinors.

Definition

(Dirac spinors)

We assume spacetime to have signature (1,3)(1,3). Dirac spinors have V=Σ +Σ V=\Sigma^{+} \oplus \Sigma^{-}, so that dim C(V)=4\dim_{\C}(V)=4. The function ff is taken to be f(v)=mh(v,v)f(v)=-mh(v,v). In the standard model, electrons are Dirac spinors.

Remark

In the physical literature, the picture is slightly different: The representation space VV of a spinor is not a subspace of the Clifford algebra, but rather C n\C^n. One can think about this as a further association of C n\C^n to the Clifford bundle VMVM using a representation of C(p,q)\C(p,q) on C n\C^n. Below we describe this in the case of the electron, i.e. (p,q)=(1,3)(p,q)=(1,3) and V=Σ +Σ V= \Sigma^{+} \oplus \Sigma^{-}. Another difference is here that instead of Spin(1,3)Spin(1,3), physicists often use the (non-canonically) isomorphic group SL(2,C)SL(2,\C).

  • One starts with the following representation γ:C(1,3)Gl(4,C)\gamma: \C(1,3) \to \mathrm{Gl}(4,\C). Consider the R\R-linear map

    γ:R 4Gl(4,C):v(0 v v 0), \gamma: \R^4 \to \mathrm{Gl}(4,\C): v \mapsto \begin{pmatrix} 0 & v' \\ v'' & 0 \end{pmatrix}\text{,}

    where

    v:=(v 0+v 3 v 1iv 2 v 1+iv 2 v 0v 3 ) and v=(v 0v 3 v 1+iv 2 v 1iv 2 v 0+v 3 ). v' := \begin{pmatrix} v_0 + v_3 & v_1 - iv_2 \\ v_1 + iv_2 & v_0 - v_3 \\ \end{pmatrix} \quad\text{ and }\quad v'' = \begin{pmatrix} v_0-v_3 & -v_1 + iv_2 \\ -v_1-iv_2 & v_0+v_3 \\ \end{pmatrix}\text{.}

    It satisfies γ(v)γ(v)=v,vI 4\gamma(v) \cdot \gamma(v)= - \left \langle v,v \right \rangle I_4. The Clifford algebra has a universal property that implies that γ\gamma extends uniquely to a representation of C(1,3)\C(1,3). The images of the standard basis vectors e 0,...,e 3e_0,...,e_3 are often called γ\gamma-matrices, γ k:=γ(e k)\gamma_k := \gamma(e_k).

  • The restriction of the representation γ\gamma to Spin(1,3)Spin(1,3) is a representation ρ:Spin(1,3)×C 4C 4\rho: Spin(1,3) \times \C^4 \to \C^4. It splits into a direct sum of two representations equivalent to Σ +\Sigma^+ and Σ \Sigma^-. Using ρα=α\rho\circ\alpha = \alpha, one checks using the above definition of the group homomorphism Λ:Spin(p,q)SO(p,q)\Lambda: Spin(p,q) \to SO(p,q) that

    γ(Λ(φ)v)=ρ(φ)γ(v)ρ(φ) 1. \gamma(\Lambda(\varphi)v) = \rho(\varphi)\gamma(v)\rho(\varphi)^{-1}\text{.}
  • The above mentioned identification between Spin(1,3)Spin(1,3) and SL(2,C)SL(2,\C) is

    Spin(1,3)SL(2,C):v 1...v 2rv 1v 2v 3...v 2r. Spin(1,3) \to SL(2,\C) : v_1 \cdot...\cdot v_{2r} \mapsto v_1'v_2''v_3'\cdot...\cdot v_{2r}''\text{.}

    Under this isomorphism, ρ\rho becomes

    ρ:SL(2,C)Gl(4,C):A(A 0 0 A *1). \rho: SL(2,\C) \to \mathrm{Gl}(4,\C) : A \mapsto \begin{pmatrix}A & 0 \\ 0 & A^{*-1} \end{pmatrix}\text{.}

    Under this identification, the splitting of ρ\rho into a direct sum yields the defining representation, often called D (1/2,0)D^{(1/2,0)} and its conjugate, often called D (0,1/2)D^{(0,1/2)}.

  • Finally, the bilinear form HH becomes

    H(v,w):=v trγ 0w¯. H(v,w) := v^{\mathrm{tr}}\gamma_0\bar w.
Remark

If M=R 1,3M=\R^{1,3} one can take the trivial spin structure SM=M×SL(2,C)SM=M \times SL(2,\C). It has a canonical global section, so that a spinor ψ\psi can be identified with a map ψ:MC 4\psi: M \to \C^4. The Dirac operator is now Dψ=γ i iψD \psi = \gamma^{i}\partial_i \psi, where γ i:=η ikγ k\gamma^i := \eta^{ik}\gamma_k. Now, the Dirac equation is

γ i iψ+imψ=0. \gamma^i\partial_i \psi + \mathrm{i} m\psi = 0\text{.}

Let’s go back to Dirac’s orginial motivation. Dirac was looking for a first order differential equation

α k kψ+imψ=0 \alpha^k\partial_k\psi +im\psi=0

for functions ψ:R 4C n\psi: \R^{4} \to \C^n, whose solutions are automatically solutions of the Klein-Gordon equation

(+m 2)ψ=0, (\triangle + m^2)\psi = 0\text{,}

where = k k\triangle= \partial^k\partial_k. If ψ\psi is a solution to the first equation,

α j jα k kψ=α j j(imψ)=m 2ψ. \alpha^{j}\partial_j\alpha^k\partial_k \psi = \alpha^{j}\partial_j(-im\psi) = - m^2\psi\text{.}

This is the Klein-Gordon equation, if α iα j=η ij\alpha^{i}\alpha^{j}=\eta^{ij}. This can only be satisfied for matrices, so that better α iα j=η ijI n\alpha^{i}\alpha^{j}=\eta^{ij}I_n. Since η ijI n\eta^{ij}I_n is a symmetric matrix, this can be written as

12(α iα j+α jα i)=η ij. \frac{1}{2}(\alpha^{i}\alpha^j + \alpha^j\alpha^i)= \eta^{ij}\text{.}

The smallest matrices satisfying this relation are the above “gamma matrices”. If m=0m=0, there is a solution in dimension two, the “Pauli matrices”.

Remark

For a general spacetime MM, and unlike in the previous remark, D 2D^2 \neq \triangle. Rather, D 2D^2 is given by the Lichnerowiz formula which has in fact been proved first by Schrödinger. So, Dirac’s motivation actually fails for “curved spacetimes”.

Charged Spinors

Remark

(Bundle Splicing)

  • Consider principal G kG_k-bundles P kP_k over MM, for k=1,2k=1,2. The fibre product P 1× MP 2P_1 \times_M P_2 is a principal (G 1×G 2)(G_1 \times G_2)-bundle over MM, denoted P 1P 2P_1 \circ P_2.

  • If ω 1\omega_1 and ω 2\omega_2 are connections on P 1P_1 and P 2P_2, respectively, then

    ω 1ω 2:=pr 1 *ω 1pr 2 *ω 2Ω 1(P 1P 2,𝔤 1𝔤 2) \omega_1 \circ \omega_2 := \mathrm{pr}_1^{*}\omega_1 \oplus \mathrm{pr}_2^{*}\omega_2 \in \Omega^1(P_1 \circ P_2,\mathfrak{g}_1 \oplus \mathfrak{g}_2)

    is a connection on P 1P 2P_1 \circ P_2.

  • Suppose VV is a vector space and ρ k:G kGl(V)\rho_k: G_k \to \mathrm{Gl}(V) are representations, such that ρ 1(g 1)ρ 2(g 2)=ρ 2(g 2)ρ 1(g 1)\rho_1(g_1) \circ \rho_2(g_2) = \rho_2(g_2) \circ \rho_1(g_1) for all g 1G 1g_1\in G_1 and g 2G 2g_2 \in G_2. Then, ρ 1×ρ 2\rho_1 \times \rho_2 is a representation of G 1×G 2G_1 \times G_2 on VV.

Remark

Let MM be a spacetime with spin structure SMSM and let PP be a Yang-Mills theory over MM with gauge group GG. For ρ SM:Spin(p,q)Gl(V)\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V) a representation with VC(p,q)V \subset \C(p,q), and ρ P:GGl(V)\rho_P: G \to \mathrm{Gl}(V) a commuting representation, the associated bundle VP:=(SMP)× (ρ SM×ρ P)VVP := (SM \circ P) \times_{(\rho_{SM} \times \rho_P)} V still has a Clifford multiplication. For ω\omega a connection on PP, one can define a Dirac operator

D ω:Γ(M,ΣMP)Γ(M,ΣMP):ψ ie iD Θω(ψ)(e i). D^{\omega}: \Gamma(M,\Sigma M \circ P) \to \Gamma(M,\Sigma M \circ P):\psi \mapsto \sum_{i} e_i \cdot \mathrm{D}^{\Theta \circ \omega}(\psi)(e_i)\text{.}
Definition

Let MM be a spacetime with spin structure, let PP be a Yang-Mills theory with gauge group GG over MM, and let ρ P\rho_P be a representation of GG on VV commuting with ρ SM\rho_{SM}. A charged spinor is a field for SMPSM \circ P of type (V,h,ρ SM×ρ P,f)(V,h,\rho_{SM} \times \rho_P,f), where VC(p,q)V \subset \C(p,q) and HH is given as before. Its action functional is

S(ω,ψ):= MD ωψ hψ+fψ. S(\omega, \psi) := \int_M D^{\omega} \psi \wedge_{h} \star \psi + \star f \circ \psi\text{.}
Definition

(Spinor in an electromagnetic field)

Here, ρ SM:Spin(p,q)Gl(V)\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V) is some representation, and ρ P:U(1)Gl(V)\rho_P: U(1) \to \mathrm{Gl}(V) is given by complex multiplication with z nz^{n}, where nZn\in \Z is the charge of the spinor. Obviously ρ SM\rho_{SM} and ρ P\rho_P commute. The Euler-Lagrange equation is

D ωψ+imψ=0. D^{\omega}\psi + im\psi = 0\text{.}

If M=R 1,3M=\R^{1,3} one can take SM=M×SL(2,C)SM=M \times SL(2,\C). The canonical global section identifies ψ\psi with a smooth function ψ:R 3,1C 4\psi: \R^{3,1} \to \C^4 and the connection ω\omega with a 1-form with components A iA_{i}. Then,

D ωψ=γ i( i+A i)ψ. D^{\omega}\psi = \gamma^{i}(\partial_{i} + A_i)\psi\text{.}

This gives the “Dirac equation?” one usually finds in a textbook.

standard model of particle physics and cosmology

theory:Einstein-Yang-Mills-Dirac-Higgs
gravityelectroweak and strong nuclear forcefermionic matterscalar field
field content:vielbein field eeprincipal connection \nablaspinor ψ\psiscalar field HH
Lagrangian:scalar curvature densityfield strength squaredDirac operator component densityfield strength squared + potential density
L=L = R(e)vol(e)+R(e) vol(e) + F eF +\langle F_\nabla \wedge \star_e F_\nabla\rangle + (ψ,D (e,)ψ)vol(e)+ (\psi , D_{(e,\nabla)} \psi) vol(e) + H¯ eH+(λ|H| 4μ 2|H| 2)vol(e) \nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)

References

Useful literature on this topic is:

  • Christian Bär, Introduction to Spin Geometry, Oberwolfach Reports 53 (2006), p. 3135-3136.

  • D. Bleecker, Gauge Theory and Variational Principles, Addison-Weasley, 1981.

  • H. Blaine Lawson Jr. , Marie-Louise Michelson, Spin geometry, Princeton Univ. Press, 1989.

  • G. L. Naber, Topology, Geometry and Gauge Dields, Springer, 1999.

Revised on January 14, 2013 18:25:20 by Urs Schreiber (203.116.137.162)