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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.
(see: the main article about Yang-Mills theory)
Under a spacetime we understand a smooth, oriented, pseudo-Riemannian manifold.
A Yang-Mills theory over a spacetime $M$ is:
A Lie group $G$, called the gauge group, together with an $\mathrm{Ad}$-invariant scalar product $\kappa: \mathfrak{g} \times \mathfrak{g} \to \R$ on its Lie algebra $\mathfrak{g}$.
A $G$-principal bundle $P$ over $M$.
A gauge field is a connection $\omega \in \Omega^1(P,\mathfrak{g})$ on $P$. The action functional is
Above we have used the following notation:
$F_{\omega} \in \Omega^2(M,\mathrm{Ad}(P))$ is the curvature of $\omega$.
$\mathrm{Ad}(P) := P \times_{\mathrm{Ad}} \mathfrak{g}$ is the adjoint bundle.
$\| \psi \|_{\kappa}^2 := \psi \wedge_{\kappa} \star \psi \in \Omega^n(M)$. In general, if $U,V,W$ are vector spaces, $\varphi \in \Omega^p(M,V)$, $\psi\in\Omega^q(M,W)$ and $f: V \times W \to U$ is a linear map, we have $\varphi \wedge_{f} \psi \in \Omega^{p+q}(M,U)$.
$\star$ is the Hodge-star operator? determined by the metric on $M$.
The Euler-Lagrange equations determined by the above action together with the Bianchi identity are called Yang-Mills equations:
where $\mathrm{D}^\omega$ denotes the covariant derivative.
A gauge transformation is a smooth bundle morphism $g: P \to P$.
Let $g: P \to P$ be a gauge transformation.
If $\omega$ is a connection on $P$, then $g^{*}\omega$ is another connection on $P$.
One can identify $g$ with a smooth map $\tilde g: P \to G$, namely by $g=r_{\tilde g}$, i.e. $g(p) = p \cdot \tilde g(p)$ for all $p\in P$.
The pullback along a gauge transformation restricts to an automorphism of $\Omega^k_{\rho}(P,V)$. In terms of the associated map $\tilde g$, we have
The Yang-Mills action functional $S_{YM}$ is gauge-invariant, i.e.
for all gauge transformations $g:P \to P$.
We have $g^{*}\omega = \mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta$ and $g^{*}\Omega = \mathrm{Ad}_{\tilde g}^{-1}(\Omega)$. Under the isomorphism $\Omega^k(P,\mathrm{Ad}) \cong \Omega^2(M,\mathrm{Ad}(P))$ this corresponds to $F_{g^{*}\omega} = \mathrm{Ad}_{g} (F_{\omega})$. Since the bilinear form $\kappa$ is $\mathrm{Ad}$-invariant by assumption,
Let $M$ be a spacetime. A classical electromagnetic field theory over $M$ is a Yang-Mills Theory over $M$ with gauge group $G=U(1)$. In more detail:
for an electromagnetic field theory given by a $U(1)$-bundle $P$ over $M$, we have $\mathrm{Ad}(P) \cong P \times \R$, so that $\Omega^k(M,\mathrm{Ad}(P)) \cong \Omega^k(M)$ and $\Omega^k_{\mathrm{Ad}}(P,\mathfrak{g})\cong \Omega^k_{\mathrm{Ad}}(P)$. In particular, $F_{\omega} \in \Omega^2(M)$.
Since $U(1)$ is abelian, $\mathrm{d}(\mathrm{Ad}) = 0$ and so $\mathrm{D}^{\omega}=\mathrm{d}$ on $\Omega^{k}_{\mathrm{Ad}}(P)$.
Thus, the Yang-Mills equations reduce to Maxwell’s equations for an electromagnetic field on $M$:
Let $G$ be a gauge group. A matter type for $G$ is a tuple $(V,h,\rho,f)$ consisting of:
a finite-dimensional real vector space $V$ called the internal state space.
a scalar product $h: V \times V \to \R$.
a representation $\rho: G \times V \to V$ that is isometric with respect to $h$ i.e. $h(\rho(g)(v),\rho(g)(w)) = h(v,w)$.
a smooth function $f: V \to \R$ that is $\rho$-invariant, i.e. $f(\rho(g)(v))=f(v)$.
Let $P$ be a principal $G$-bundle over $M$, and let $\mathcal{T} =(V,h,\rho,f)$ be a matter type for $G$. A field for $P$ of type $\mathcal{T}$ is a smooth section $\phi: M \to P \times_{\rho} V$. Its action functional is
The action functional $S_{\mathcal{T}}$ is gauge invariant, i.e.
for all gauge transformations $g:P \to P$.
One calculates that $g^{*}\omega = \mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta$, where $\tilde g:P \to G$ is the smooth map associated to $g$ via $g(p) = p\cdot \tilde g(p)$. Further, $g^{*}\psi = \rho(\tilde g^{-1})(\psi)$. A computation shows
where $\mathrm{d}\rho: \mathfrak{g} \otimes V \to V$. Now we compute
$\quad\quad\mathrm{d}^{g^{*}\omega}(g^{*}\psi)$
$\quad\quad\quad\quad= \mathrm{d}\rho(\tilde g^{-1},\psi) + g^{*}\omega \wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi)$
$\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}\psi) + \tilde g^{*}\bar\theta\wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi) + (\mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta) \wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi)$
$\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}\psi) + \rho(\tilde g^{-1},\omega\wedge_{\mathrm{d}\rho} \psi)$
$\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}^{\omega}\psi)\text{.}$
Since $h$ is invariant, the invariance of the first term follows. The invariance of the second term is clear.
One can either keep a connection $\omega$ fixed and consider
as a matter field in an “external gauge field”, or consider the combined action functional
(Scalar particle in an external, trivial gauge field)
We consider $G=\left \lbrace e \right \rbrace$, $P=M$, so that necessarily $\omega=0$. A scalar field is field for $M$ of matter type $(\R,h,\id,f)$ where $f(x) := -\frac{1}{2}m^2x^2$ and $h(x,y) = x y$. The action functional is
The Euler-Lagrange equation is the Klein-Gordon equation
where $\triangle := \delta \circ \mathrm{d}: \Omega^k(M) \to \Omega^{k}(M)$ is the Laplace operator and $\delta := \star \mathrm{d} \star$ is the exterior coderivative.
(Charged particle in an electromagnetic field, e.g. a $\pi^{-}$-meson)
Let $P$ be a $U(1)$-principal bundle over $M$. A field of charge $n \in \Z$ is a field for $P$ of matter type $(\C,h,\rho_n,f)$, where $\rho_n: U(1) \times \C \to \C$ is defined by $\rho_n(z,z') := z^n z'$ and $f(z) := -\frac{1}{2}m^2|z|^2$. The action functional is
The Euler-Lagrange equation is covariant Klein-Gordon equation
where $\triangle^{\omega} :=\delta^{\omega}\circ \mathrm{D}^{\omega}$ is the covariant Laplace operator and $\delta^{\omega} := \star \mathrm{D}^{\omega} \star$ is the exterior covariant coderivative.
The Klein-Gordon equations 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.
We recall some facts about Clifford algebras and the spin group.
We denote by $C(p,q)$ the Clifford algebra on $\R^{p,q}$, i.e. the quotient of the tensor algebra of $\R^{p+q}$ by the ideal generated by $v \otimes w + w \otimes v + 2 \left \langle v,w \right \rangle\cdot 1$, where $\left \langle -,- \right \rangle$ is the Minkowski scalar product of signature $(p,q)$.
The map $v \mapsto -v$ extends to an anti-automorphism $\alpha: C(p,q) \to C(p,q)$, whose eigenspace decomposition yields the usual $\Z_2$-grading on $C(p,q)$.
We have $\dim C(p,q) = 2^{p+q}$.
The Clifford algebra inherits a bilinear form $H(v,w) := (v^{tr}w)_0$, where $()^{tr}$ is the anti-automorphism of the tensor algebra that reverts the order of tensor products, and $()_0$ denotes the degree 0 part.
We denote by $SO(p,q)$ the group of linear maps $\R^{p+q}\to \R^{p+q}$ that preserve the product $\left \langle -,- \right \rangle$. We define
Then, we define a group homomorphism $\Lambda: Spin(p,q) \to SO(p,q)$ by $\Lambda(\varphi)v = \alpha(\varphi) v \varphi^{-1}$. This gives a central extension
We denote by $\C(p,q) := C(p,q) \otimes_{\R} \C$ the complexification of the Clifford algebra. The bilinear form $H$ on $C(p,q)$ extends to a sesquilinear form $H$ on $\C(p,q)$ defined by $H(v,w)=(v^{tr}\bar w)_0$.
Multiplication in $\C(p,q)$ restricts to an action of $\spin{p,q}$ on $\C(p,q)$. One can decompose $\C(p,q)$ into $k$ copies of a subrepresentation $\Sigma$:
The representation $\Sigma$ is isometric with respect to $H$. If $p+q$ is odd, $k=2^{(p+q-1)/2}$, and $\Sigma$ is irreducible. If $p+q$ is even, $k=2^{(p+q)/2}$, and $\Sigma = \Sigma^{+} \oplus \Sigma^{-}$ with $\Sigma^{\pm}$ irreducible and $\dim_{\C}\Sigma^{\pm}=2^{(p+q-2)/2}$.
We also need some facts about spin structures.
Let $M$ be a spacetime with pseudo-Riemannian metric of signature $(p,q)$. We denote by $FM$ be the principal $SO(p,q)$-bundle over $M$ of orthonormal frames, the frame bundle.
A spin structure on $M$ is a principal $Spin(p,q)$-bundle $SM$ over $M$ together with a bundle morphism $\lambda: SM \to FM$ such that $\lambda(X\cdot\varphi) = \lambda(X)\cdot\Lambda(\varphi)$ for all $X\in SM$ and all $\varphi\in Spin(p,q)$.
Let $\theta \in \Omega^1(FM,\mathfrak{so}(p,q))$ be the Levi-Cevita connection on $FM$. Then,
is a connection on $SM$.
Finally, we recall the definition of the Dirac operator.
Let $\rho: C(p,q) \times V \to V$ be a representation, with $V \subset \C(p,q)$. Note that in the above realization of the group $Spin(p,q)$, the representation $\rho$ restricts to a representation of $Spin(p,q)$. The spinor bundle is the vector bundle $V M := SM \times_{\rho} V$.
Clifford multiplication is a map
It is defined as follows. We write $s=(X,v)$ with $X\in SM$ and $v\in V$. We consider the orthonormal frame $\alpha_X := \lambda(X): TM \to \R^n$. Then, $\alpha_X(u) \in \C(p,q)$ and
One can show using above-listed properties of the Clifford algebra that this definition does not depend on the choice of the representative $(X,v)$.
The Dirac operator is
where $e_i\in TM$ runs over a local orthonormal basis.
Let $M$ be a spacetime with spin structure $SM$, and considered as a $Spin(p,q)$ as a Yang-Mills theory over $M$. A free spinor is a field for $SM$ of type $(V,h,\rho,f)$, where $V \subset \C(p,q)$, the scalar product $h$ is
and $\rho$ is the restriction of the multiplication in $\C(p,q)$ to $Spin(p,q)$. The action functional is
The Euler-Lagrange equation determined by the action functional $S(\psi)$ is the Dirac equation
(Weyl spinors)
We assume spacetime to have even dimension. Weyl spinors have $V=\Sigma^{\pm}$, with the sign corresponding to left/right-handed spinors. Thus, $\dim_{\C}(V)=2$. Further $f=0$ (they are massless). In the standard model, neutrinos are left-handed Weyl spinors.
(Dirac spinors)
We assume spacetime to have signature $(1,3)$. Dirac spinors have $V=\Sigma^{+} \oplus \Sigma^{-}$, so that $\dim_{\C}(V)=4$. The function $f$ is taken to be $f(v)=-mh(v,v)$. In the standard model, electrons are Dirac spinors.
In the physical literature, the picture is slightly different: The representation space $V$ of a spinor is not a subspace of the Clifford algebra, but rather $\C^n$. One can think about this as a further association of $\C^n$ to the Clifford bundle $VM$ using a representation of $\C(p,q)$ on $\C^n$. Below we describe this in the case of the electron, i.e. $(p,q)=(1,3)$ and $V= \Sigma^{+} \oplus \Sigma^{-}$. Another difference is here that instead of $Spin(1,3)$, physicists often use the (non-canonically) isomorphic group $SL(2,\C)$.
One starts with the following representation $\gamma: \C(1,3) \to \mathrm{Gl}(4,\C)$. Consider the $\R$-linear map
where
It satisfies $\gamma(v) \cdot \gamma(v)= - \left \langle v,v \right \rangle I_4$. The Clifford algebra has a universal property that implies that $\gamma$ extends uniquely to a representation of $\C(1,3)$. The images of the standard basis vectors $e_0,...,e_3$ are often called $\gamma$-matrices, $\gamma_k := \gamma(e_k)$.
The restriction of the representation $\gamma$ to $Spin(1,3)$ is a representation $\rho: Spin(1,3) \times \C^4 \to \C^4$. It splits into a direct sum of two representations equivalent to $\Sigma^+$ and $\Sigma^-$. Using $\rho\circ\alpha = \alpha$, one checks using the above definition of the group homomorphism $\Lambda: Spin(p,q) \to SO(p,q)$ that
The above mentioned identification between $Spin(1,3)$ and $SL(2,\C)$ is
Under this isomorphism, $\rho$ becomes
Under this identification, the splitting of $\rho$ into a direct sum yields the defining representation, often called $D^{(1/2,0)}$ and its conjugate, often called $D^{(0,1/2)}$.
Finally, the bilinear form $H$ becomes
If $M=\R^{1,3}$ one can take the trivial spin structure $SM=M \times SL(2,\C)$. It has a canonical global section, so that a spinor $\psi$ can be identified with a map $\psi: M \to \C^4$. The Dirac operator is now $D \psi = \gamma^{i}\partial_i \psi$, where $\gamma^i := \eta^{ik}\gamma_k$. Now, the Dirac equation is
Let’s go back to Dirac’s orginial motivation. Dirac was looking for a first order differential equation
for functions $\psi: \R^{4} \to \C^n$, whose solutions are automatically solutions of the Klein-Gordon equation
where $\triangle= \partial^k\partial_k$. If $\psi$ is a solution to the first equation,
This is the Klein-Gordon equation, if $\alpha^{i}\alpha^{j}=\eta^{ij}$. This can only be satisfied for matrices, so that better $\alpha^{i}\alpha^{j}=\eta^{ij}I_n$. Since $\eta^{ij}I_n$ is a symmetric matrix, this can be written as
The smallest matrices satisfying this relation are the above “gamma matrices”. If $m=0$, there is a solution in dimension two, the “Pauli matrices”.
For a general spacetime $M$, and unlike in the previous remark, $D^2 \neq \triangle$. Rather, $D^2$ is given by the Lichnerowiz formula which has in fact been proved first by Schrödinger. So, Dirac’s motivation actually fails for “curved spacetimes”.
(Bundle Splicing)
Consider principal $G_k$-bundles $P_k$ over $M$, for $k=1,2$. The fibre product $P_1 \times_M P_2$ is a principal $(G_1 \times G_2)$-bundle over $M$, denoted $P_1 \circ P_2$.
If $\omega_1$ and $\omega_2$ are connections on $P_1$ and $P_2$, respectively, then
is a connection on $P_1 \circ P_2$.
Suppose $V$ is a vector space and $\rho_k: G_k \to \mathrm{Gl}(V)$ are representations, such that $\rho_1(g_1) \circ \rho_2(g_2) = \rho_2(g_2) \circ \rho_1(g_1)$ for all $g_1\in G_1$ and $g_2 \in G_2$. Then, $\rho_1 \times \rho_2$ is a representation of $G_1 \times G_2$ on $V$.
Let $M$ be a spacetime with spin structure $SM$ and let $P$ be a Yang-Mills theory over $M$ with gauge group $G$. For $\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V)$ a representation with $V \subset \C(p,q)$, and $\rho_P: G \to \mathrm{Gl}(V)$ a commuting representation, the associated bundle $VP := (SM \circ P) \times_{(\rho_{SM} \times \rho_P)} V$ still has a Clifford multiplication. For $\omega$ a connection on $P$, one can define a Dirac operator
Let $M$ be a spacetime with spin structure, let $P$ be a Yang-Mills theory with gauge group $G$ over $M$, and let $\rho_P$ be a representation of $G$ on $V$ commuting with $\rho_{SM}$. A charged spinor is a field for $SM \circ P$ of type $(V,h,\rho_{SM} \times \rho_P,f)$, where $V \subset \C(p,q)$ and $H$ is given as before. Its action functional is
(Spinor in an electromagnetic field)
Here, $\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V)$ is some representation, and $\rho_P: U(1) \to \mathrm{Gl}(V)$ is given by complex multiplication with $z^{n}$, where $n\in \Z$ is the charge of the spinor. Obviously $\rho_{SM}$ and $\rho_P$ commute. The Euler-Lagrange equation is
If $M=\R^{1,3}$ one can take $SM=M \times SL(2,\C)$. The canonical global section identifies $\psi$ with a smooth function $\psi: \R^{3,1} \to \C^4$ and the connection $\omega$ with a 1-form with components $A_{i}$. Then,
This gives the “Dirac equation” one usually finds in a textbook.
: | - | - | - | |
---|---|---|---|---|
and | ||||
content: | $e$ | $\nabla$ | $\psi$ | $H$ |
: | density | squared | component density | squared + potential density |
$L =$ | $R(e) vol(e) +$ | $\langle F_\nabla \wedge \star_e F_\nabla\rangle +$ | $(\psi , D_{(e,\nabla)} \psi) vol(e) +$ | $\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$ |
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 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.