New Year’s Resolution

Happy New Year everyone! I’ll be starting my PhD at Queen Mary, University of London next week and I’ve decided to pick up my blog activity. I’ll try to post a quick article every couple of days detailing something I’ve learnt, or any interesting thoughts I’ve had. Over time I might build up to longer reviews.

If there are any specific requests for theoretical physics material you’d like to see covered, either at a layman or a research level then please do comment them below!

Why does Spontaneous Symmetry Breaking Depend on Energy Scale?

In the Standard Model we tend to argue that electroweak symmetry is broken below a certain (large) energy scale, yielding the and bosons and the photon. The usual argument for spontaneously symmetry breaking relies on a Higgs potential of the form

and the argument follows from the degeneracy of the lowest energy state.

Remark: Strictly speaking we really ought to be talking about the effective potential, which takes into account radiative corrections. The lowest energy state would then be the vacuum expectation value of the field. I’ll treat the rigorous foundations of SSB in a future post, a topic which is often overlooked in lectures and textbooks!

Critically the standard argument makes no mention of energy scale. So why should we expect it to play a role. The answer is twofold.

Firstly we must remember that we can never observe a full theory. In fact our observations are at best approximations to our theory. This slightly backwards way of looking at things aids our understanding of spontaneous symmetry breaking. At low energies we see the “Mexican hat” clearly, and observe it’s effects in experiments. But as we go to higher energies we “zoom out” on the axis. This means that the Mexican hat effect is no longer “visible”. From an experimental perspective it’s effects are dominated by other parameters such that they are unobservable. To all intents and purposes the symmetry remains unbroken.

In some theories there’s something more fundamental which averts the SSB scenario at high energies. Recall that renormalization causes the coupling constants to run. In particular our mass pseudoparameter can actually change sign at high energies. This eliminates the vacuum degeneracy in the (effective) potential. To see an example, look at Figure 13 in Ben Allanach’s SUSY notes.

I’m grateful to Zac Kenton for a fruitful discussion over a (much-needed) coffee.

Non-Perturbative QFT: A Loophole?

I’ve been musing on yesterday’s post, and in particular a potential loophole in my argument. Recall that the whole shebang hinges on the fact that is smooth but not analytic. We were interpreting as the coupling constant in some theory, say QED. But hang about, surely we could just do a rescaling to remove this behaviour? After all the theory is invariant classically under such a transformation.

But it turns out that this kind of rescaling is anomalous in (most) quantum field theories. Recall that renormalization endows quantum field theories with a function, which determines the evolution of coupling constants as energy changes. The rescaling will only remain a symmetry if is globally zero. Otherwise the rescaling only superficially eliminates the non-perturbative effect – it will reappear at different energies!

This raises a natural question: can you have instantons in finite quantum field theories? By definition these have function zero. Naively we might expect scale invariance to kill non-perturbative physics. A popular finite theory is SYM, which crops up in AdS/CFT. A quick google suggests that my naive thinking is wrong. There are plenty of papers on instantons in this theory!

There must be a still deeper level to non-perturbative understanding. Sadly most physics papers gloss over the details. Let’s keep half an eye out for an explanation!

Non-Perturbative Effects in Quantum Field Theory

In elementary QFT we only really know how to solve free theories. These require quantizing infinitely many harmonic oscillators, which is an easy problem. But the real world is described by interacting theories with Lagrangians like

in which the fields are mixed up together. The standard way of dealing with these is to write them as

and then split the path integral as

The first factor explicitly yields free theory propagators. The second factor may now be expanded as a Taylor series in some coupling constant . Often this is trivial since appears in precisely to first order. For example in QED we have

with being the electric charge. This means that the perturbation expansion is exactly the usual exponential series.

At present it looks like perturbation theory must capture all information in a QFT, provided you sum up all the terms. However, we’ve missed a crucial mathematical point. We’ll view the scattering amplitude for a given process as a complex-valued function of a real coupling ; that is to say . We construct this function in (somewhat) trivial way from smooth functions, so it should be smooth.

However, smoothness does not guarantee analyticity! Put another way, we cannot be sure that the Taylor series for will converge for . Even if it does, it’s not guaranteed to converge to the function itself. This means that there are effects that lie outside the realm of perturbation theory, even if you could sum every term.

Let’s take an example. Suppose for instance that has a dependence. If you’re balking at this “unphysical” choice, then rest assured that such Lagrangians do crop up. Now try to compute the Taylor expansion around for . A little thought shows that this is identically zero.

Such non-perturbative effects require new approaches. One fruitful method involves solving the full theory classically then then considering quantum perturbations around such solutions. For Yang-Mills theories in particular this yields the notion of an instanton.

Finally let’s go back to QED and work out whether we should see non-perturbative effects there. Although not immediately as obvious as the simple example, one can argue that the perturbation series has radius of convergence. We follow Dyson and observe that if the radius of convergence were not zero, we’d be able to “reverse” the electric field by flipping the sign of . This would render the vacuum unstable against decays into electrons and positrons, since these would repel rather than annihilate within a “Heisenberg allowed” time period. To rule out this option, we must take formally.

This argument is not mathematically rigorous, and I don’t know whether one can make it so. Please comment if you know! Thankfully there’s a more modern perspective that sheds new light. We can view the sickness in QED in the context of Wilsonian renormalization. In particular, the theory has no well-defined high energy limit, as the coupling constants flow to infinite values at finite energy. This Landau pole behaviour is perhaps evidence that non-perturbative effects rule high energy QED. Again, I don’t whether this argument can be made watertight, so please take a rain-cheque on that one!

Javascript Namespaces in Meteor

I’ve just been watching this excellent EventedMind presentation to get my head around routing for Meteor. I’ve realized that a better way to manage my code (particularly for subscriptions) is to harness the power of javascript namespaces. For example

Subscriptions = {
posts: Meteor.subscribe('posts')
};

Just keeps everything much cleaner, and prevents quite likely name conflicts. I think I’ll be using this from now on in the ongoing Fixer’s Dream development!

Meteor App Liveblog (Entry 6)

Ba-da-boom! And that’s my time up. I think I’m going to give myself a little bonus time later this evening, so watch out for a final round-up then.

Where do we stand? I’ve integrated the bootstrap popover and remove functionalities, which was pleasingly simple. Next up is the accounts feature. I’ve hit one of the hazards of working with a beta technology, namely that some packages mysteriously don’t work. A workaround is to temporarily downgrade using meteor update --release.

To summarize: there’s a bit of a learning curve with Meteor at the start, particularly if you aren’t familiar with Mongo and Handlebars. But once you’ve entered into the right spirit, it’s really good fun! I’ll see over the coming days how it stands up to more complex tasks. I still reckon MEAN is the way to go for massive scalability. But Meteor is great for smaller projects, and I daresay more enjoyable!

Meteor App Liveblog (Entry 5)

So we’re into the last few hours and sadly haven’t quite got as far as I hoped. But… we have got a very simple prototype up, allowing you to add people to the database. The next couple of hours will probably involve

• providing full info in popovers
• including a delete functionality
• implementing a login (and proper security)
• adding the “concerts” page

What have I learnt in the last few hours? Here are the highlights

• Stylus is awesome. Use it.
• Don’t try to use form tags. It’s not worth the trouble.
• You can (and should) split up your client, server and public files into so-named directories. See the parties example. Goodness knows why this isn’t the default directory structure!
• The hr tag is useful for splitting up sections

Hope some of that’s useful! If you want to check out the prototype as it develops, bookmark http://fixersdream.meteor.com/.

Meteor App Liveblog (Entry 4)

Oh dear! We really aren’t going very fast. I blame the need to get familiar with MongoDB. I’ve taken a bit of time to find a decent GUI to visualize what’s going on. Okay – I know this isn’t strictly necessary. But for a first time Meteor user like me it’s quite nice. Here’s my recommendations

• Grab a copy of GenghisApp. It seems to start natively at IP address 0.0.0.0:5678.
• Add the mongo database. You can find the correct URI by typing meteor mongo in your app directory
• Tada! You’ll now be able to navigate around (and add/change) your collections.

Time to get some logic going, and basic Bootstrap styling. Once that’s in place I’ll put the current progress onto the .meteor.com servers so you can follow along.

Meteor App Liveblog (Entry 3)

Slow progress! Mainly because I didn’t quite get the Coffeescript syntax right when I rewrote the JS file, and I forgot how useless Coffeescript errors can be. Couple of things I should’ve done

• Remember that coffeescript comments start with a #
• Add the Sublime Coffee Compile plugin for slightly better error messages than Meteor gives you
• Compare and contrast my code with this correct example.

Meteor App Liveblog (Entry 2)

So I’ve got meteor installed and the app created. But it wasn’t quite as plain sailing as the docs make it seem. Here’s some tips

• you might need to add/usr/local/bin to your $PATH if it isn’t there already
• the simplest way of shoehorning in Coffeescript seems to be adding the package with meteor add then changing the js file to a .coffee one and altering the syntax.
• It annoys me that meteor doesn’t timestamp things in bash. So use this instead
  meteor | gawk '{ print strftime("%H:%M:%S"), $0; fflush(); }'

Also Bootstrap is installed nicely. Though it seems I might need a custom package for more control over which JS files get included. I’ll probably get to play around with the unofficial Meteorite package manager later.

So. On with the coding!