## July 30, 2024

### The Zinn-Justin Equation

A note from my QFT class. Finally, I understand what Batalin-Vilkovisky anti-fields are for.

The Ward-Takahashi Identities are central to understanding the renormalization of QED. They are an (infinite tower of) constraints satisfied by the vertex functions in the 1PI generating functional $\Gamma(A_\mu,\psi,\tilde\psi,b,c,\chi)$. They are simply derived by demanding that the BRST variations

(1)$\begin{split} \delta_{\text{BRST}} b&= -\frac{1}{\xi}(\partial\cdot A-\xi^{1/2}\chi)\\ \delta_{\text{BRST}} A_\mu&= \partial_\mu c\\ \delta_{\text{BRST}} \chi &= \xi^{-1/2} \partial^\mu\partial_\mu c\\ \delta_{\text{BRST}} \psi &= i e c\psi\\ \delta_{\text{BRST}} \tilde{\psi} &= -i e c\tilde{\psi}\\ \delta_{\text{BRST}} c &= 0 \end{split}$

annihilate $\Gamma$: $\delta_{\text{BRST}}\Gamma=0$ (Here, by a slight abuse of notation, I’m using the same symbol to denote the sources in the 1PI generating functional and the corresponding renormalized fields in the renormalized action $\mathcal{L}= -\frac{Z_A}{4}F_{\mu\nu}F^{\mu\nu} + Z_\psi \left(i\psi^\dagger \overline{\sigma}\cdot(\partial-i e A)\psi+ i\tilde{\psi}^\dagger \overline{\sigma}\cdot(\partial+i e A)\tilde{\psi} -Z_m m(\psi\tilde{\psi}+\psi^\dagger\tilde{\psi}^\dagger) \right) +\mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}$ where $\begin{split} \mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}&= \delta_{\text{BRST}}\frac{1}{2}\left(b(\partial\cdot A+\xi^{1/2}\chi)\right)\\ &=-\frac{1}{2\xi} (\partial\cdot A)^2+ \frac{1}{2}\chi^2 - b\partial^\mu\partial_\mu c \end{split}$ They both transform under BRST by (1).)

Posted by distler at 1:56 PM | Permalink | Followups (1)

## September 2, 2023

### dCS

For various reasons, some people seem to think that the following modification to Einstein Gravity
(1)$S= \int \tfrac{1}{2} d\phi\wedge *d\phi + \tfrac{\kappa^2}{2} *\mathcal{R} + {\color{red} \tfrac{3 \phi}{192\pi^2 f}Tr(R\wedge R)}$
is interesting to consider. In some toy world, it might be1. But in the real world, there are nearly massless neutrinos. In the Standard Model, $U(1)_{B-L}$ has a gravitational ABJ anomaly (where, in the real world, the number of generations $N_f=3$)
(2)$d * J_{B-L} = \frac{N_f}{192\pi^2} Tr(R\wedge R)$
which, by a $U(1)_{B-L}$ rotation, would allow us to entirely remove2 the coupling marked in red in (1). In the real world, the neutrinos are not massless; there’s the Weinberg term
(3)$\frac{1}{M}\left(y^{i j} (H L_i)(H L_j) + \text{h.c.}\right)$
which explicitly breaks $U(1)_{B-L}$. When the Higgs gets a VEV, this term gives a mass $m^{i j} = \frac{\langle H\rangle^2 y^{i j}}{M}$ to the neutrinos, So, rather than completely decoupling, $\phi$ reappears as a (dynamical) contribution to the phase of the neutrino mass matrix
(4)$m^{i j} \to m^{i j}e^{2i\phi/f}$
Of course there is a CP-violating phase in the neutrino mass matrix. But its effects are so tiny that its (presumably nonzero) value is still unknown. Since (4) is rigourously equivalent to (1), the effects of the term in red in (1) are similarly unobservably small. Assertions that it could have dramatic consequences — whether for LIGO or large-scale structure — are … bizarre.

#### Update:

The claim that (1) has some observable effect is even more bizarre if you are seeking to find one (say) during inflation. Before the electroweak phase transition, $\langle H \rangle=0$ and the effect of a $\phi$-dependent phase in the Weinberg term (3) is even more suppressed.

1 An analogy with Yang Mills might be helpful. In pure Yang-Mills, the $\theta$-parameter is physical; observable quantities depend on it. But, if you introduce a massless quark, it becomes unphysical and all dependence on it drops out. For massive quarks, only the sum of $\theta$ and phase of the determinant of the quark mass matrix is physical.
2 The easiest way to see this is to introduce a background gauge field, $\mathcal{A}$, for $U(1)_{B-L}$ and modify (1) to
(5)$S= \int \tfrac{1}{2} (d\phi-f\mathcal{A})\wedge *(d\phi-f\mathcal{A}) + \tfrac{\kappa^2}{2} *\mathcal{R} + {\color{red} \tfrac{3 \phi}{24\pi^2 f}\left[\tfrac{1}{8}Tr(R\wedge R)+d\mathcal{A}\wedge d\mathcal{A}\right]}$
Turning off the Weinberg term, the theory is invariant under $U(1)_{B-L}$ gauge transformations $\begin{split} \mathcal{A}&\to \mathcal{A}+d\chi\\ \phi&\to \phi+ f \chi\\ Q_i&\to e^{i\chi/3}Q_i\\ \overline{u}_i&\to e^{-i\chi/3}\overline{u}_i\\ \overline{d}_i&\to e^{-i\chi/3}\overline{d}_i\\ L_i&\to e^{-i\chi}L_i\\ \overline{e}_i&\to e^{i\chi}\overline{e}_i\\ \end{split}$ where the anomalous variation of the fermions cancels the variation of the term in red. Note that the first term in (5) is a gauge-invariant mass term for $\mathcal{A}$ (or would be if we promoted $\mathcal{A}$ to a dynamical gauge field). Choosing $\chi = -\phi/f$ eliminates the term in red. Turning back on the Weinberg term (which explicitly breaks $U(1)_{B-L}$) puts the coupling to $\phi$ into the neutrino mass matrix (where it belongs).
Posted by distler at 1:34 PM | Permalink | Followups (3)

## February 12, 2023

### MathML in Chrome

Thanks to the hard work of Frédéric Wang and the folks at Igalia, the Blink engine in Chrome 109 now supports MathML Core.

It took a little bit of work to get it working correctly in Instiki and on this blog.

• The columnalign attribute is not supported, so a shim is needed to get the individual <mtd> to align correctly.
• This commit enabled the display of SVG embedded in equations and got rid of the vertical scroll bars in equations.
• Since Chrome does not support hyperlinks (either href or xlink:href attributes) on MathML elements, this slightly hacky workaround enabled hyperlinks in equations, as created by \href{url}{expression}.

There are a number of remaining issues.

• Math accents don’t stretch, when they’re supposed to. Here are a few examples of things that (currently) render incorrectly in Chrome (some of them, admittedly, are incorrect in Safari too):

$\mathbf{V}_{1} \times \mathbf{V}_{2} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix}$

$\left\vert\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}\right\vert\le \left\vert\frac{z-a}{1-\overline{a}z}\right\vert$

$\widetilde{PGL}(N)$

$\begin{matrix} P_1(Y) &\to& P_1(X) \\ \downarrow &\Downarrow\mathrlap{\sim}& \downarrow \\ T' &\to& T \end{matrix}$

$p_3 (x) = \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}}$

• This equation $\boxed{(i\slash{D}+m)\psi = 0}$ doesn’t display remotely correctly, because Chrome doesn’t implement the <menclose> element. Fixed now.

But, hey, this is amazing for a first release.

#### Update:

I added support for \boxed{} and \slash{}, both of which use <menclose>, which is not supported by Chrome. So now the above equation should render correctly in Chrome. Thanks to Monica Kang, for help with the CSS.

## October 17, 2022

### Fine Structure

I’m teaching the undergraduate Quantum II course (“Atoms and Molecules”) this semester. We’ve come to the point where it’s time to discuss the fine structure of hydrogen. I had previously found this somewhat unsatisfactory. If one wanted to do a proper treatment, one would start with a relativistic theory and take the non-relativistic limit. But we’re not going to introduce the Dirac equation (much less QED). And, in any case, introducing the Dirac equation would get you the leading corrections but fail miserably to get various non-leading corrections (the Lamb shift, the anomalous magnetic moment, …).

Instead, various hand-waving arguments are invoked (“The electron has an intrinsic magnetic moment and since it’s moving in the electrostatic field of the proton, it sees a magnetic field …”) which give you the wrong answer for the spin-orbit coupling (off by a factor of two), which you then have to further correct (“Thomas precession”) and then there’s the Darwin term, with an even more hand-wavy explanation …

So I set about trying to find a better way. I want use as minimal as possible input from the relativistic theory and get the leading relativistic correction(s).

Posted by distler at 9:00 PM | Permalink | Followups (2)

## July 15, 2022

### HL ≠ HS

There’s a nice new paper by Kang et al, who point out something about class-S theories that should be well-known, but isn’t.

In the (untwisted) theories of class-S, the Hall-Littlewood index, at genus-0, coincides with the Hilbert Series of the Higgs branch. The Hilbert series counts the $\hat{B}_R$ operators that parametrize the Higgs branch (each contributes $\tau^{2R}$ to the index). The Hall-Littlewood index also includes contributions from $D_{R(0,j)}$ operators (which contribute $(-1)^{2j+1}\tau^{2(1+R+j)}$ to the index). But, for the untwisted theories of class-S, there is a folk-theorem that there are no $D_{R(0,j)}$ operators at genus-0, and so the Hilbert series and Hall-Littlewood index agree.

For genus $g\gt0$, the gauge symmetry1 cannot be completely Higgsed on the Higgs branch of the theory. For the theory of type $J=\text{ADE}$, there’s a $U(1)^{\text{rank}(J)g}$ unbroken at a generic point on the Higgs branch2. Correspondingly, the SCFT contains $D_{R(0,0)}$ multiplets which, when you move out onto the Higgs branch and flow to the IR, flow to the $D_{0(0,0)}$ multiplets3 of the free theory.

What Kang et al point out is that the same is true at genus-0, when you include enough $\mathbb{Z}_2$-twisted punctures. They do this by explicitly calculating the Hall-Littlewood index in a series of examples.

But it’s nice to have a class of examples where that hard work is unnecessary.

## April 11, 2022

### Monterey and Samba

I reluctantly upgraded my laptop from Mojave to Monterey (macOS 12.3.1). Things have not gone smoothly. My biggest annoyance, currently, is with Time Machine.

Posted by distler at 1:48 AM | Permalink | Followups (1)

## 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.

Posted by distler at 11:47 AM | Permalink | Followups (4)

## March 21, 2021

### Cosmic Strings in the Standard Model

Over at the n-Category Café, John Baez is making a big deal of the fact that the global form of the Standard Model gauge group is $G = (SU(3)\times SU(2)\times U(1))/N$ where $N$ is the $\mathbb{Z}_6$ subgroup of the center of $G'=SU(3)\times SU(2)\times U(1)$ generated by the element $\left(e^{2\pi i/3}\mathbb{1},-\mathbb{1},e^{2\pi i/6}\right)$.

The global form of the gauge group has various interesting topological effects. For instance, the fact that the center of the gauge group is $Z(G)= U(1)$, rather than $Z(G')=U(1)\times \mathbb{Z}_6$, determines the global 1-form symmetry of the theory. It also determines the presence or absence of various topological defects (in particular, cosmic strings). I pointed this out, but a proper explanation deserved a post of its own.

None of this is new. I’m pretty sure I spent a sunny afternoon in the summer of 1982 on the terrace of Café Pamplona doing this calculation. (As any incoming graduate student should do, I spent many a sunny afternoon at a café doing this and similar calculations.)

Posted by distler at 12:27 AM | Permalink | Followups (9)

## January 6, 2020

### Entanglement for Laymen

I’ve been asked, innumerable times, to explain quantum entanglement to some lay audience. Most of the elementary explanations that I have seen (heck, maybe all of them) fail to draw any meaningful distinction between “entanglement” and mere “(classical) correlation.”

This drives me up the wall, so each time I am asked, I strive to come up with an elementary explanation of the difference. Rather than keep reinventing the wheel, let me herewith record my latest attempt.

Posted by distler at 2:30 PM | Permalink | Followups (17)

## February 28, 2019

### Instiki 0.30.0 and tex2svg 1.0

Instiki is my wiki-cum-collaboration platform. It has a built-in WYSIWYG vector-graphics drawing program, which is great for making figures. Unfortunately:

• An extra step is required, in order to convert the resulting SVG into PDF for inclusion in the LaTeX paper. And what you end up with is a directory full of little PDF files (one for each figure), which need to be managed.
• Many of my colleagues would rather use Tikz, which has become the de-facto standard for including figures in LaTeX.

Obviously, I needed to include Tikz support in Instiki. But, up until now, I didn’t really see a good way to do that, given that I wanted something that is

1. Portable
2. Secure
Posted by distler at 2:21 PM | Permalink | Followups (5)

## February 15, 2019

### Brotli

I finally got around to enabling Brotli compression on Golem. Reading the manual, I came across the BrotliAlterETag directive:

Description: How the outgoing ETag header should be modified during compression
Syntax: BrotliAlterETag AddSuffix|NoChange|Remove

with the description:

Append the compression method onto the end of the ETag, causing compressed and uncompressed representations to have unique ETags. In another dynamic compression module, mod_deflate, this has been the default since 2.4.0. This setting prevents serving “HTTP Not Modified (304)” responses to conditional requests for compressed content.
NoChange
Don’t change the ETag on a compressed response. In another dynamic compression module, mod_deflate, this has been the default prior to 2.4.0. This setting does not satisfy the HTTP/1.1 property that all representations of the same resource have unique ETags.
Remove
Remove the ETag header from compressed responses. This prevents some conditional requests from being possible, but avoids the shortcomings of the preceding options.

Sure enough, it turns out that ETags+compression have been completely broken in Apache 2.4.x. Two methods for saving bandwidth, and delivering pages faster, cancel each other out and chew up more bandwidth than if one or the other were disabled.

Posted by distler at 9:47 AM | Permalink | Followups (1)

## December 27, 2018

### Python urllib2 and TLS

I was thinking about dropping support for TLSv1.0 in this webserver. All the major browser vendors have announced that they are dropping it from their browsers. And you’d think that since TLSv1.2 has been around for a decade, even very old clients ought to be able to negotiate a TLSv1.2 connection.

But, when I checked, you can imagine my surprise that this webserver receives a ton of TLSv1 connections… including from the application that powers Planet Musings. Yikes!

The latter is built around the Universal Feed Parser which uses the standard Python urrlib2 to negotiate the connection. And therein lay the problem …

## February 24, 2017

### Responsibility

Many years ago, when I was an assistant professor at Princeton, there was a cocktail party at Curt Callan’s house to mark the beginning of the semester. There, I found myself in the kitchen, chatting with Sacha Polyakov. I asked him what he was going to be teaching that semester, and he replied that he was very nervous because — for the first time in his life — he would be teaching an undergraduate course. After my initial surprise that he had gotten this far in life without ever having taught an undergraduate course, I asked which course it was. He said it was the advanced undergraduate Mechanics course (chaos, etc.) and we agreed that would be a fun subject to teach. We chatted some more, and then he said that, on reflection, he probably shouldn’t be quite so worried. After all, it wasn’t as if he was going to teach Quantum Field Theory, “That’s a subject I’d feel responsible for.”

This remark stuck with me, but it never seemed quite so poignant until this semester, when I find myself teaching the undergraduate particle physics course.

Posted by distler at 6:13 PM | Permalink | Followups (29)

## December 4, 2016

### MathML Update

For a while now, Frédéric Wang has been urging me to enable native MathML rendering for Safari. He and his colleagues have made many improvements to Webkit’s MathML support. But there were at least two show-stopper bugs that prevented me from flipping the switch.

Posted by distler at 2:56 PM | Permalink | Followups (8)

## June 14, 2016

### Coriolis

I really like the science fiction TV series The Expanse. In addition to a good plot and a convincing vision of human society two centuries hence, it depicts, as Phil Plait observes, a lot of good science in a matter-of-fact, almost off-hand fashion. But one scene (really, just a few dialogue-free seconds in a longer scene) has been bothering me. In it, Miller, the hard-boiled detective living on Ceres, pours himself a drink. And we see — as the whiskey slowly pours from the bottle into the glass — that the artificial gravity at the lower levels (where the poor people live) is significantly weaker than near the surface (where the rich live) and that there’s a significant Coriolis effect. Unfortunately, the effect depicted is 3 orders-of-magnitude too big.

Posted by distler at 4:40 PM | Permalink | Followups (2)