How well verified is the theory of quantum electrodynamics (QED)? If you ask this to most physicists one of the first things that comes to their mind is the agreement of QED’s theoretical calculation of the anomolous magnetic moment of the electron and the extremely precise measurement of this moment. In fact, last night, while I was spending my time usefully watching the Colbert Report on Comedy Central, guest Brian Greene brought up exactly this example (well he didn’t exactly say this is what he was talking about, but it was pretty clear. The interview, by the way, was pretty funny.)
The electron magnetic moment anomoly is [tex]$a={g-2 over 2}$[/tex], measuring the deviation of the electron magnetic moment from it’s unperturbed g value of 2. Experiments done here at the University of Washington by the Dehmelt group in the late eighties gave an experimentally determined value of the anomoly of [tex]$a=1159652188.4(4.3) times 10^{-12}$[/tex] where number in parenthesis is the error. Now that’s a pretty precise measurement! On the other side of the physics departments, theorists have calculated the a value of the anomoly in quantum electrodynamics. This calculation yields an expression for the anomoly in powers of the fine structure constant. This requires calculating Feynman diagrams to eighth order in perturbation theory. The current theoretical calulculation yields an expression, to eighth order of
[tex]$a_{th}=A_2 left({alpha over pi}right)+ A_4 left({alpha over pi}right)^2+ A_6 left({alpha over pi}right)^3+ A_8left({alpha over pi}right)^4$[/tex]
where
[tex]$A_2=0.5$[/tex]
[tex]$A_4=0.328478965579 dots$[/tex]
[tex]$A_6=1.181241456 dots$[/tex]
[tex]$A_8=- 1.7366(384) $[/tex]
The first three of these terms is basically ananlytically known (i.e. can be readily obtained from functions which we can numerically calculate to any desired accuracy) and the last term, which has an error in it, is obtained by a numerical evaluation. So how well do theory and experiment agree? Well we need a value of the fine structure constant! There are many experiments which can be used to determine the fine structure constant. Among the best are experiments done using the quantum Hall effect and yield [tex]$alpha^{-1}=137.0360037(33) [2.4 times 10^{-8}]$[/tex] where the number in bracket is a fractional uncertainty. Using this value of the fine structure constant in the perturbative expansion for the theoretical expression give [tex]$a_{th}=1159652153.5 (1.2)~(28.0) times 10^{-12}$[/tex] where the number in the first parenthesis is the error from the theory calculation and the second is the error comming from the uncertainty in the value of the fine structure constant.
So, now returning to the question I started with, how well verified is QED? Well in the regime where these experiments have been preformed the results agree to an amazying precision. And when explaining this to the public, it is certainly valid to count the number of digits to which this calculation agrees with experiment. But for me, I’m more confortable saying that the above discussion shows that we’ve verified quantum electrodynamics to eighth order in perturbation theory (or to fourth order in its coupling constant.) Why do I prefer this? Well mostly because, as I understand it, modern particle theory basically says that QED must be an effective field theory for some deeper theory. Or in other words, it can’t be QED all the way down. Thus it seems more proper to ask how far down the perturbation ladder we’ve verified QED. And again, while eigth order may not sound as amazing as ten, eleven, or twelve digits of precision, it still is an amazing verfication.
And anyway, who says we should be using base ten for our measure of precision? Me, I’m in a computer science department, so it seems that base two would be much better (and you might even convince me that natural logarithms are even better.)
O brave new quantum world!
Nice post. You mention the “fine structure constant.” Feynman was intrigued by this dimension-less ratio of parameters. Is there still any mystery associated with this? I forget why Feynman was so fascinated with it.
Many have been fascinated with the fine structure constant. What is kind of funny is that there have been lots of numerology, messing around with ratios and such, trying to explain the fine structure constant’s value. What makes this funny is that the fine structure constant isn’t really a constant, in a strict sense. What I mean by this is that at higher energy the fine structure constant doesn’t have the same value as at zero (low) energy (which is where the value of the f.s.c. I talked about is measured and used.)
I’m not sure what fascinated Feynman about the fine structure constant, exactly. It could have been lots of things considering its central stage in QED (it is the coupling constant between light and matter.)
You will also find that many physicist love the number 137 because it is close to one over the fine structure constant. Just the other day I was staying in room 139 and I remember wondering whether this corresponded to a higher energy value of the fine structure constnat (turns out it doesn’t, I think, the fine structure constant grows as a function of the energy scale, so one over the fine structure constant shrinks.) If QED were an exact theory, then the fine structure constant would blow up at a finite energy!
Also note that there is a great use of the fine structure constant for different length scales: the Bohr radius times the fine structure constant gives the Compton wavelength of an electron and the Compton wavelength of an electron times the fine structure constant equals the classical radius of the electron. Prety cool
I think order is more important. If the fine structure constant was .1, you’d have the same order agreement but fewer decimals! But I’ll bet the general public is more comfortable with the number of digit agreement.