## January 8, 2022

### Spinor Helicity Variables in QED

I’m teaching Quantum Field Theory this year. One of the things I’ve been trying to emphasize is the usefulness of spinor-helicity variables in dealing with massless particles. This is well-known to the “Amplitudes” crowd, but hasn’t really trickled down to the textbooks yet. Mark Srednicki’s book comes close, but doesn’t (IMHO) quite do a satisfactory job of it.

Herewith are some notes.

The first step in constructing perturbation theory is to quantize the free fields. Following Weinberg and Srednicki, I’m using the “mostly-plus” signature convention (my 2-component spinor conventions are those of Dreiner et al if you define the macro \def\signofmetric{1} in the LaTeX file). For $k^2=0$, we can define helicity spinors

(1)$(k\cdot\sigma)_{\alpha\dot\beta}= -\lambda_\alpha\lambda^\dagger_{\dot\beta},\qquad (k\cdot\overline{\sigma})^{\dot\alpha\beta} = -\lambda^{\dagger\dot\alpha}\lambda^\beta$

which allow us to straightforwardly canonically-quantize.

### Spin-1/2

For a Weyl fermion, $\mathcal{L}= i\psi^\dagger\overline{\sigma}\cdot\partial \psi$ the general solution to the equations of motion is \begin{aligned} \psi_\alpha(x)&=\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}\lambda_\alpha \left(\xi^\dagger_{\vec{k}}e^{-ik\cdot x}+\eta_{\vec{k}}e^{ik\cdot x}\right)\\ \psi^\dagger_{\dot\alpha}(x)&=\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}\lambda^\dagger_{\dot\alpha}\left(\eta^\dagger_{\vec{k}}e^{-ik\cdot x}+\xi_{\vec{k}}e^{ik\cdot x}\right) \end{aligned} The Equal-Time Anti-Commutation Relations $\{\psi_\alpha(\vec{x},0),\psi^\dagger_{\dot\beta}(\vec{x}',0)\}=\sigma^0_{\alpha\dot\beta}\delta^{(3)}(\vec{x}-\vec{x}')$ become the canonical anti-commutation relations \begin{aligned} \{\xi_{\vec{k}},\xi^\dagger_{\vec{k}'}\}&= {(2\pi)}^3 2|\vec{k}| \delta^{(3)}(\vec{k}-\vec{k}')\\ \{\eta_{\vec{k}},\eta^\dagger_{\vec{k}'}\}&= {(2\pi)}^3 2|\vec{k}| \delta^{(3)}(\vec{k}-\vec{k}')\\ \end{aligned} for creation and annihilation operators for fermions of definite helicity.

The upshot, after tracking this through the LSZ reduction formula, is that external fermion lines are contracted with the corresponding helicity spinor ($\lambda_i$ or $\lambda^\dagger_i$) depending on the helicity of the $i^{\text{th}}$ incoming/outgoing particle. When we take the absolute square of the amplitude, we use (1) to rewrite $\lambda_i\lambda^\dagger_i=-k_i\cdot\sigma$, etc.

### Spin-1

There’s a certain amount of hand-wringing associated to quantizing the free Maxwell Lagrangian, $\mathcal{L} = -\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}$ If we take the canonical variables to be $A^\mu$ and $\pi_\mu =\frac{\delta \mathcal{L}}{\delta\partial_0 A^\mu}$, then the gauge-invariance entails that the symplectic structure is degenerate ($\pi_0$ vanishes identically). The usual approach is to fix a gauge (Weinberg and Srednicki use Coulomb gauge) and then work very hard (replacing Poisson brackets with Dirac brackets, because the constraints are 2nd class, …).

On the other hand, if we

1. realize that the phase space is the space of classical solutions and
2. introduce spinor helicity variables, as before,

it’s easy to write down the general solution to the equations of motion

(2)\begin{aligned} F^{\mu\nu}(x)= \frac{1}{\sqrt{2}}\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}&\left(\lambda\sigma^{\mu\nu}\lambda\varepsilon^\dagger_{-}(\vec{k})+ \lambda^\dagger\overline{\sigma}^{\mu\nu}\lambda^\dagger \varepsilon^\dagger_{+}(\vec{k})\right)e^{-ik\cdot x}+\\ &+\left(\lambda\sigma^{\mu\nu}\lambda\varepsilon_{+}(\vec{k})+ \lambda^\dagger\overline{\sigma}^{\mu\nu}\lambda^\dagger \varepsilon_{-}(\vec{k})\right)e^{ik\cdot x} \end{aligned}

The (non-degenerate) symplectic structure on the space of classical solutions leads to the Equal-Time Commutation Relations

(3)$[F_{0i}(\vec{x},0),F_{j k}(\vec{x}',0)]=i\left(\delta_{i k}\frac{\partial}{\partial x^j}-\delta_{i j}\frac{\partial}{\partial x^k}\right)\delta^{(3)}(\vec{x}-\vec{x}')$

which, in turn, give the canonical commutation relations \begin{aligned} [\varepsilon_+(\vec{k}),\varepsilon^\dagger_+(\vec{k}')]&={(2\pi)}^3 2|\vec{k}| \delta^{(3)}(\vec{k}-\vec{k}')\\ [\varepsilon_-(\vec{k}),\varepsilon^\dagger_-(\vec{k}')]&={(2\pi)}^3 2|\vec{k}| \delta^{(3)}(\vec{k}-\vec{k}')\\ \end{aligned} of the creation and annihilation operators for photons of definite helicity.

Unfortunately, to couple to charged matter fields, we need an expression for $A^\mu$, not just $F^{\mu\nu}$, so (2) does not quite suffice for our purposes. But, again, helicity spinors come to the rescue.

Introduce a fixed fiducial null vector $\check{k}^2=0$ and the corresponding helicity spinors $(\check{k}\cdot\sigma)_{\alpha\dot\beta}= -\mu_\alpha\mu^\dagger_{\dot\beta},\qquad (\check{k}\cdot\overline{\sigma})^{\dot\alpha\beta} = -\mu^{\dagger\dot\alpha}\mu^\beta$ We then can write

(4)\begin{aligned} A^\mu(x)&= \frac{1}{\sqrt{2}}\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}\left(\frac{\mu^\dagger\overline{\sigma}^\mu\lambda}{\mu^\dagger\lambda^\dagger}\varepsilon^\dagger_{-}(\vec{k})+ \frac{\mu\sigma^\mu\lambda^\dagger}{\mu\lambda} \varepsilon^\dagger_{+}(\vec{k})\right)e^{-ik\cdot x}+\\ &\qquad+\left(\frac{\lambda^\dagger\overline{\sigma}^\mu\mu}{\lambda\mu}\varepsilon_{-}(\vec{k})+ \frac{\lambda\sigma^\mu\mu^\dagger}{\lambda^\dagger\mu^\dagger} \varepsilon_{+}(\vec{k})\right)e^{ik\cdot x}\\ &=\frac{1}{\sqrt{2}}\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}\left(\frac{\mu^\dagger\overline{\sigma}^\mu\lambda}{\mu^\dagger\lambda^\dagger}\varepsilon^\dagger_{-}(\vec{k})+ \frac{\mu\sigma^\mu\lambda^\dagger}{\mu\lambda} \varepsilon^\dagger_{+}(\vec{k})\right)e^{-ik\cdot x}\\ &\qquad-\left(\frac{\mu\sigma^\mu\lambda^\dagger}{\mu\lambda}\varepsilon_{-}(\vec{k})+ \frac{\mu^\dagger\overline{\sigma}^\mu\lambda}{\mu^\dagger\lambda^\dagger} \varepsilon_{+}(\vec{k})\right)e^{ik\cdot x}\\ \end{aligned}

which satisfies $\partial\cdot A=0$ and (exercise for the reader) \begin{aligned} \partial^\mu A^\nu-\partial^\nu A^\mu&= \frac{1}{\sqrt{2}}\int\frac{d^3\vec{k}}{{(2\pi)}^3 2|\vec{k}|}\left(\lambda\sigma^{\mu\nu}\lambda\varepsilon^\dagger_{-}(\vec{k})+ \lambda^\dagger\overline{\sigma}^{\mu\nu}\lambda^\dagger \varepsilon^\dagger_{+}(\vec{k})\right)e^{-ik\cdot x}+\\ &\qquad+\left(\lambda\sigma^{\mu\nu}\lambda\varepsilon_{+}(\vec{k})+ \lambda^\dagger\overline{\sigma}^{\mu\nu}\lambda^\dagger \varepsilon_{-}(\vec{k})\right)e^{ik\cdot x}\\ &=F^{\mu\nu}(x) \end{aligned} as before. Together, these ensure that changing the reference momentum $\check{k}$ changes $A^\mu(x)$ by a harmonic gauge transformation.

To completely justify (4), we choose R-$\xi$ gauge, and use BV-BRST quantization, but that’s the subject for another blog post.

Here, it suffices to say that the Feynman rules contract every external photon line with a $\frac{\mu\sigma^\mu\lambda^\dagger}{\mu\lambda}$ or a $\frac{\mu^\dagger\overline{\sigma}^\mu\lambda}{\mu^\dagger\lambda^\dagger}$, depending on the helicity of the incoming/outgoing photon. We’re free to make any choice of reference momentum $\check{k}$ that we want, but verifying that the final answer is independent of $\check{k}$ is a nice check on our calculations.

Notoriously, Lorentz gauge $\partial\cdot A = 0$ does not completely fix the gauge: we can still shift $A_\mu\to A_\mu+\partial_\mu f$, where $f$ is any solution to the scalar wave equation, $\square f = 0$.

Posted by distler at January 8, 2022 11:47 AM

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### Re: Spinor Helicity Variables in QED

How come Peskin and Schroeder is no longer being used?

Posted by: David on January 8, 2022 10:28 PM | Permalink | Reply to this

### Peskin and Schroeder

Just because I’m not using it doesn’t mean it’s “not being used”. I’m sure it is still, by far, the best-selling QFT textbook.

Posted by: Jacques Distler on January 8, 2022 11:17 PM | Permalink | PGP Sig | Reply to this

### Re: Peskin and Schroeder

Yeah, that is what I meant. It seems like P and S is the most popular choice, so I was just wondering why you decided to buck the tradition.

Posted by: David on January 8, 2022 11:46 PM | Permalink | Reply to this

### Re: Peskin and Schroeder

The first choice (which immediately eliminates half of the possible textbooks) is the choice of signature conventions.

I learned QFT in the “mostly-minus” signature. But, when I teach String Theory, or when I teach GR, “mostly-plus” is the default signature convention, and I wouldn’t consider any other choice. So I was slightly (but only slightly) leaning towards “mostly-plus” for this QFT course.

Of the textbooks I looked at, Srednicki’s came closest to what I wanted to do in the course. And he uses “mostly-plus”, so that basically clinched things for me.

As it turns out, I’m not actually following Srednicki all that closely, either, but by then my choice was made …

Posted by: Jacques Distler on January 9, 2022 12:52 AM | Permalink | PGP Sig | Reply to this

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