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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 ΞΊ:𝔀×𝔀→R\kappa: \mathfrak{g} \times \mathfrak{g} \to \R on its Lie algebra 𝔀\mathfrak{g}.

  • A GG-principal bundle PP over MM.

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

S YM(Ο‰):=12∫ Mβ€–F Ο‰β€– ΞΊ 2. S_{YM}(\omega) := \frac{1}{2} \int_M \| F_{\omega} \|_\kappa^2.
Remark

Above we have used the following notation:

  • F Ο‰βˆˆΞ© 2(M,Ad(P))F_{\omega} \in \Omega^2(M,\mathrm{Ad}(P)) is the curvature of Ο‰\omega.

  • Ad(P):=PΓ— Ad𝔀\mathrm{Ad}(P) := P \times_{\mathrm{Ad}} \mathfrak{g} is the adjoint bundle.

  • β€–Οˆβ€– ΞΊ 2:=ψ∧ ΞΊβ‹†ΟˆβˆˆΞ© 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, Ο†βˆˆΞ© p(M,V)\varphi \in \Omega^p(M,V), ψ∈Ω q(M,W)\psi\in\Omega^q(M,W) and f:VΓ—Wβ†’Uf: V \times W \to U is a linear map, we have Ο†βˆ§ fψ∈Ω 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 ω⋆F Ο‰=0 and D Ο‰F Ο‰=0, \mathrm{D}^{\omega}\star F_{\omega} = 0 \quad\text{ and }\quad \mathrm{D}^{\omega}F_{\omega}=0,

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

Definition

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

Remark

Let g:P→Pg: P \to P be a gauge transformation.

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

  • One can identify gg with a smooth map g˜:Pβ†’G\tilde g: P \to G, namely by g=r g˜g=r_{\tilde g}, i.e. g(p)=pβ‹…g˜(p)g(p) = p \cdot \tilde g(p) for all p∈Pp\in P.

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

    g *ψ=ρ(g˜ βˆ’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 *Ο‰)=S YM(Ο‰) S_{YM}(g^{*}\omega) = S_{YM}(\omega)

for all gauge transformations g:P→Pg:P \to P.

Proof

We have g *Ο‰=Ad g˜ βˆ’1(Ο‰)βˆ’g˜ *ΞΈΒ―g^{*}\omega = \mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta and g *Ξ©=Ad g˜ βˆ’1(Ξ©)g^{*}\Omega = \mathrm{Ad}_{\tilde g}^{-1}(\Omega). Under the isomorphism Ξ© k(P,Ad)β‰…Ξ© 2(M,Ad(P))\Omega^k(P,\mathrm{Ad}) \cong \Omega^2(M,\mathrm{Ad}(P)) this corresponds to F g *Ο‰=Ad g(F Ο‰)F_{g^{*}\omega} = \mathrm{Ad}_{g} (F_{\omega}). Since the bilinear form ΞΊ\kappa is Ad\mathrm{Ad}-invariant by assumption,

β€–F g *Ο‰β€–=Ad g(F Ο‰)∧ ΞΊAd g(F Ο‰)=F Ο‰βˆ§ ΞΊF Ο‰=β€–F Ο‰β€–. \| 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)β‰…PΓ—R\mathrm{Ad}(P) \cong P \times \R, so that Ξ© k(M,Ad(P))β‰…Ξ© k(M)\Omega^k(M,\mathrm{Ad}(P)) \cong \Omega^k(M) and Ξ© Ad k(P,𝔀)β‰…Ξ© Ad k(P)\Omega^k_{\mathrm{Ad}}(P,\mathfrak{g})\cong \Omega^k_{\mathrm{Ad}}(P). In particular, F Ο‰βˆˆΞ© 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 Ο‰=d\mathrm{D}^{\omega}=\mathrm{d} on Ξ© Ad k(P)\Omega^{k}_{\mathrm{Ad}}(P).

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

    d⋆F Ο‰=0 and dF Ο‰=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:VΓ—Vβ†’Rh: V \times V \to \R.

  • a representation ρ:GΓ—Vβ†’V\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:Vβ†’Rf: 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 Ο•:Mβ†’PΓ— ρV\phi: M \to P \times_{\rho} V. Its action functional is

S 𝒯(Ο‰,Ο•):=∫ Mβ€–D Ο‰(Ο•)β€– h 2+⋆(fβˆ˜Ο•). 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 *Ο‰,g *Ο•)=S 𝒯(Ο‰,Ο•) S_{\mathcal{T}}(g^{*}\omega,g^{*}\phi) = S_{\mathcal{T}}(\omega,\phi)

for all gauge transformations g:P→Pg:P \to P.

Proof

One calculates that g *Ο‰=Ad g˜ βˆ’1(Ο‰)βˆ’g˜ *ΞΈΒ―g^{*}\omega = \mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta, where g˜:Pβ†’G\tilde g:P \to G is the smooth map associated to gg via g(p)=pβ‹…g˜(p)g(p) = p\cdot \tilde g(p). Further, g *ψ=ρ(g˜ βˆ’1)(ψ)g^{*}\psi = \rho(\tilde g^{-1})(\psi). A computation shows

d(ρ(g˜ βˆ’1)(ψ))=ρ(g˜ βˆ’1)(dψ)+g˜ *θ¯∧ dρρ(g˜ βˆ’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:π”€βŠ—Vβ†’V\mathrm{d}\rho: \mathfrak{g} \otimes V \to V. Now we compute

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

=dρ(g˜ βˆ’1,ψ)+g *Ο‰βˆ§ dρρ(g˜ βˆ’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˜ βˆ’1,dψ)+g˜ *θ¯∧ dρρ(g˜ βˆ’1,ψ)+(Ad g˜ βˆ’1(Ο‰)βˆ’g˜ *ΞΈΒ―)∧ dρρ(g˜ βˆ’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˜ βˆ’1,dψ)+ρ(g˜ βˆ’1,Ο‰βˆ§ dρψ)\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}\psi) + \rho(\tilde g^{-1},\omega\wedge_{\mathrm{d}\rho} \psi)

=ρ(g˜ βˆ’1,d Ο‰Οˆ).\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 Ο‰\omega fixed and consider

S Ο‰(Ο•):=S 𝒯(Ο‰,Ο•) S_{\omega}(\phi) := S_{\mathcal{T}}(\omega,\phi)

as a matter field in an β€œexternal gauge field”, or consider the combined action functional

S(Ο‰,Ο•):=S YM(Ο‰)+S 𝒯(Ο‰,Ο•). 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 Ο‰=0\omega=0. A scalar field is field for MM of matter type (R,h,id,f)(\R,h,\id,f) where f(x):=βˆ’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,Ο•)=12∫ Mβ€–dΟ•β€– h 2+⋆m 2Ο• 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

(β–΅+m 2)Ο•=0, (\triangle + m^2)\phi = 0\text{,}

where β–΅:=δ∘d:Ξ© k(M)β†’Ξ© k(M)\triangle := \delta \circ \mathrm{d}: \Omega^k(M) \to \Omega^{k}(M) is the Laplace operator and Ξ΄:=⋆d⋆\delta := \star \mathrm{d} \star is the exterior coderivative.

Example

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

Let PP be a U(1)U(1)-principal bundle over MM. A field of charge n∈Zn \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)Γ—Cβ†’C\rho_n: U(1) \times \C \to \C is defined by ρ n(z,zβ€²):=z nzβ€²\rho_n(z,z') := z^n z' and f(z):=βˆ’12m 2|z| 2f(z) := -\frac{1}{2}m^2|z|^2. The action functional is

S(Ο‰,Ο•)=12∫ Mβ€–D ωϕ‖ 2+⋆m 2β€–Ο•β€– 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

(β–΅ Ο‰+m 2)Ο•=0, (\triangle^{\omega} +m^2) \phi = 0\text{,}

where β–΅ Ο‰:=Ξ΄ Ο‰βˆ˜D Ο‰\triangle^{\omega} :=\delta^{\omega}\circ \mathrm{D}^{\omega} is the covariant Laplace operator and Ξ΄ Ο‰:=⋆D ω⋆\delta^{\omega} := \star \mathrm{D}^{\omega} \star is the exterior covariant coderivative.

Matter Fields with Spin

The Klein-Gordon equations found above are – unlike the SchrΓΆdinger 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 β€œgravity” 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 vβŠ—w+wβŠ—v+2⟨v,wβŸ©β‹…1v \otimes w + w \otimes v + 2 \left \langle v,w \right \rangle\cdot 1, where βŸ¨βˆ’,βˆ’βŸ©\left \langle -,- \right \rangle is the Minkowski scalar product of signature (p,q)(p,q).

  • The map vβ†¦βˆ’vv \mapsto -v extends to an anti-automorphism Ξ±:C(p,q)β†’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+qβ†’R p+q\R^{p+q}\to \R^{p+q} that preserve the product βŸ¨βˆ’,βˆ’βŸ©\left \langle -,- \right \rangle. We define

    Spin(p,q):={v 1β‹…...β‹…v 2r∈C(p,q)∣v i∈R p+q,β€–v iβ€–=1,r∈N}. 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 Ξ›:Spin(p,q)β†’SO(p,q)\Lambda: Spin(p,q) \to SO(p,q) by Ξ›(Ο†)v=Ξ±(Ο†)vΟ† βˆ’1\Lambda(\varphi)v = \alpha(\varphi) v \varphi^{-1}. This gives a central extension

    1β†’Z 2β†’Spin(p,q)β†’SO(p,q)β†’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)βŠ— 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 trwΒ―) 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 Ξ£\Sigma:

    C(p,q)=Ξ£βŠ•...βŠ•Ξ£. \C(p,q) = \Sigma \oplus ... \oplus \Sigma \text{.}
  • The representation Ξ£\Sigma is isometric with respect to HH. If p+qp+q is odd, k=2 (p+qβˆ’1)/2k=2^{(p+q-1)/2}, and Ξ£\Sigma is irreducible. If p+qp+q is even, k=2 (p+q)/2k=2^{(p+q)/2}, and Ξ£=Ξ£ +βŠ•Ξ£ βˆ’\Sigma = \Sigma^{+} \oplus \Sigma^{-} with Ξ£ Β±\Sigma^{\pm} irreducible and dim CΞ£ Β±=2 (p+qβˆ’2)/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 Ξ»:SMβ†’FM\lambda: SM \to FM such that Ξ»(Xβ‹…Ο†)=Ξ»(X)β‹…Ξ›(Ο†)\lambda(X\cdot\varphi) = \lambda(X)\cdot\Lambda(\varphi) for all X∈SMX\in SM and all Ο†βˆˆSpin(p,q)\varphi\in Spin(p,q).

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

    Θ:=dΞ› βˆ’1(Ξ» *ΞΈ)∈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)Γ—Vβ†’V\rho: C(p,q) \times V \to V be a representation, with VβŠ‚C(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:=SMΓ— ρVV M := SM \times_{\rho} V.

  • Clifford multiplication is a map

    TMβŠ—VMβ†’VM:uβŠ—s↦uβ‹…s. 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 X∈SMX\in SM and v∈Vv\in V. We consider the orthonormal frame Ξ± X:=Ξ»(X):TMβ†’R n\alpha_X := \lambda(X): TM \to \R^n. Then, Ξ± X(u)∈C(p,q)\alpha_X(u) \in \C(p,q) and

    uβ‹…s:=(X,Ξ± X(u)β‹…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:Ξ“(M,VM)β†’Ξ“(M,VM):Οˆβ†¦βˆ‘ ie iβ‹…D 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 i∈TMe_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 VβŠ‚C(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∧ hβ‹†Οˆ+⋆f∘ψ. 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=Ξ£ Β±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=Ξ£ +βŠ•Ξ£ βˆ’V=\Sigma^{+} \oplus \Sigma^{-}, so that dim C(V)=4\dim_{\C}(V)=4. The function ff is taken to be f(v)=βˆ’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=Ξ£ +βŠ•Ξ£ βˆ’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 Ξ³:C(1,3)β†’Gl(4,C)\gamma: \C(1,3) \to \mathrm{Gl}(4,\C). Consider the R\R-linear map

    Ξ³:R 4β†’Gl(4,C):v↦(0 vβ€² vβ€³ 0), \gamma: \R^4 \to \mathrm{Gl}(4,\C): v \mapsto \begin{pmatrix} 0 & v' \\ v'' & 0 \end{pmatrix}\text{,}

    where

    vβ€²:=(v 0+v 3 v 1βˆ’iv 2 v 1+iv 2 v 0βˆ’v 3 ) and vβ€³=(v 0βˆ’v 3 βˆ’v 1+iv 2 βˆ’v 1βˆ’iv 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 Ξ³(v)β‹…Ξ³(v)=βˆ’βŸ¨v,v⟩I 4\gamma(v) \cdot \gamma(v)= - \left \langle v,v \right \rangle I_4. The Clifford algebra has a universal property that implies that Ξ³\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 Ξ³\gamma-matrices, Ξ³ k:=Ξ³(e k)\gamma_k := \gamma(e_k).

  • The restriction of the representation Ξ³\gamma to Spin(1,3)Spin(1,3) is a representation ρ:Spin(1,3)Γ—C 4β†’C 4\rho: Spin(1,3) \times \C^4 \to \C^4. It splits into a direct sum of two representations equivalent to Ξ£ +\Sigma^+ and Ξ£ βˆ’\Sigma^-. Using ρ∘α=Ξ±\rho\circ\alpha = \alpha, one checks using the above definition of the group homomorphism Ξ›:Spin(p,q)β†’SO(p,q)\Lambda: Spin(p,q) \to SO(p,q) that

    Ξ³(Ξ›(Ο†)v)=ρ(Ο†)Ξ³(v)ρ(Ο†) βˆ’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)β†’SL(2,C):v 1β‹…...β‹…v 2r↦v 1β€²v 2β€³v 3β€²β‹…...β‹…v 2rβ€³. 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)β†’Gl(4,C):A↦(A 0 0 A *βˆ’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 trΞ³ 0wΒ―. 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=MΓ—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:Mβ†’C 4\psi: M \to \C^4. The Dirac operator is now Dψ=Ξ³ iβˆ‚ iψD \psi = \gamma^{i}\partial_i \psi, where Ξ³ i:=Ξ· ikΞ³ k\gamma^i := \eta^{ik}\gamma_k. Now, the Dirac equation is

Ξ³ iβˆ‚ 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

Ξ± kβˆ‚ kψ+imψ=0 \alpha^k\partial_k\psi +im\psi=0

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

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

where β–΅=βˆ‚ kβˆ‚ k\triangle= \partial^k\partial_k. If ψ\psi is a solution to the first equation,

Ξ± jβˆ‚ jΞ± kβˆ‚ kψ=Ξ± jβˆ‚ j(βˆ’imψ)=βˆ’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 Ξ± iΞ± j=Ξ· ij\alpha^{i}\alpha^{j}=\eta^{ij}. This can only be satisfied for matrices, so that better Ξ± iΞ± j=Ξ· ijI n\alpha^{i}\alpha^{j}=\eta^{ij}I_n. Since Ξ· ijI n\eta^{ij}I_n is a symmetric matrix, this can be written as

12(Ξ± iΞ± j+Ξ± jΞ± i)=Ξ· ij. \frac{1}{2}(\alpha^{i}\alpha^j + \alpha^j\alpha^i)= \eta^{ij}\text{.}

The smallest matrices satisfying this relation are the above β€œgamma matrices”. If m=0m=0, there is a solution in dimension two, the β€œPauli matrices”.

Remark

For a general spacetime MM, and unlike in the previous remark, D 2β‰ β–΅D^2 \neq \triangle. Rather, D 2D^2 is given by the Lichnerowiz formula which has in fact been proved first by SchrΓΆdinger. So, Dirac’s motivation actually fails for β€œcurved 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 1Γ— MP 2P_1 \times_M P_2 is a principal (G 1Γ—G 2)(G_1 \times G_2)-bundle over MM, denoted P 1∘P 2P_1 \circ P_2.

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

    Ο‰ 1βˆ˜Ο‰ 2:=pr 1 *Ο‰ 1βŠ•pr 2 *Ο‰ 2∈Ω 1(P 1∘P 2,𝔀 1βŠ•π”€ 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 1∘P 2P_1 \circ P_2.

  • Suppose VV is a vector space and ρ k:G kβ†’Gl(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 1∈G 1g_1\in G_1 and g 2∈G 2g_2 \in G_2. Then, ρ 1×ρ 2\rho_1 \times \rho_2 is a representation of G 1Γ—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)β†’Gl(V)\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V) a representation with VβŠ‚C(p,q)V \subset \C(p,q), and ρ P:Gβ†’Gl(V)\rho_P: G \to \mathrm{Gl}(V) a commuting representation, the associated bundle VP:=(SM∘P)Γ— (ρ SM×ρ P)VVP := (SM \circ P) \times_{(\rho_{SM} \times \rho_P)} V still has a Clifford multiplication. For Ο‰\omega a connection on PP, one can define a Dirac operator

D Ο‰:Ξ“(M,Ξ£M∘P)β†’Ξ“(M,Ξ£M∘P):Οˆβ†¦βˆ‘ ie iβ‹…D Ξ˜βˆ˜Ο‰(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 SM∘PSM \circ P of type (V,h,ρ SM×ρ P,f)(V,h,\rho_{SM} \times \rho_P,f), where VβŠ‚C(p,q)V \subset \C(p,q) and HH is given as before. Its action functional is

S(Ο‰,ψ):=∫ MD Ο‰Οˆβˆ§ hβ‹†Οˆ+⋆f∘ψ. 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)β†’Gl(V)\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V) is some representation, and ρ P:U(1)β†’Gl(V)\rho_P: U(1) \to \mathrm{Gl}(V) is given by complex multiplication with z nz^{n}, where n∈Zn\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 Ο‰Οˆ+imψ=0. D^{\omega}\psi + im\psi = 0\text{.}

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

D Ο‰Οˆ=Ξ³ i(βˆ‚ i+A i)ψ. D^{\omega}\psi = \gamma^{i}(\partial_{i} + A_i)\psi\text{.}

This gives the β€œDirac equation” one usually finds in a textbook.

:---
and
content: ee βˆ‡\nabla ψ\psi HH
: density squared component density squared + potential density
L=L = R(e)vol(e)+R(e) vol(e) + ⟨F βˆ‡βˆ§β‹† eF βˆ‡βŸ©+\langle F_\nabla \wedge \star_e F_\nabla\rangle + (ψ,D (e,βˆ‡)ψ)vol(e)+ (\psi , D_{(e,\nabla)} \psi) vol(e) + βˆ‡HΒ―βˆ§β‹† eβˆ‡H+(Ξ»|H| 4βˆ’ΞΌ 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 BΓ€r, 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 Fields, Springer, 1999.

Last revised on April 6, 2018 at 10:16:14. See the history of this page for a list of all contributions to it.