The Dirac Electron (changes) in nLab

Context

Physics

∞\infty-Chern-Weil theory

• Idea
• Yang-Mills Theory
• General Matter Fields
• Matter Fields with Spin
  • Free Spinors
  • Charged Spinors
• Related concepts
• References

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×𝔤→R\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 ω∈Ω 1(P,𝔤)\omega \in \Omega^1(P,\mathfrak{g}) on PP. The action functional is

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

Definition

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

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:P→Pg: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 ω∈Ω 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:

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

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×V→Rh: V \times V \to \R.

• a representation ρ:G×V→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:V→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 ϕ:M→P× ρV\phi: M \to P \times_{\rho} V. Its action functional is

S 𝒯(ω,ϕ):=∫ M‖D ω(ϕ)‖ h 2+⋆(f∘ϕ). 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:P→Pg: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˜:P→G\tilde g:P \to G is the smooth map associated to gg via g(p)=p⋅g˜(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:𝔤⊗V→V\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.

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∫ M‖dϕ‖ 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 ▵:=δ∘d: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 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)×C→C\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∫ M‖D ωϕ‖ 2+⋆m 2‖ϕ‖ 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 ω∘D 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.

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 v⊗w+w⊗v+2⟨v,w⟩⋅1v \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 v↦−vv \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+q→R 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 2r∈C(p,q)∣v i∈R p+q,‖v i‖=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 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

  1→Z 2→Spin(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+q−1)/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+q−2)/2\dim_{\C}\Sigma^{\pm}=2^{(p+q-2)/2}.

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 V⊂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∧ h⋆ψ+⋆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=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.

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 V⊂C(p,q)V \subset \C(p,q) and HH is given as before. Its action functional is

S(ω,ψ):=∫ MD ωψ∧ h⋆ψ+⋆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)→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 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 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 3,1→C 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.