How does Perturbation theory account for interactions in QED?

[rodsika]

Many of you stated how ad hoc is QFT as the field is supposed to be non-interacting yet how could they get an incredibly accurate value of calculated magnetic moment of the electron of value 1.0011596522 compared to measured 1.00115965219 with accuracy to better than one part in 10^10, or about three parts in 100 billion!

How does Perturbation theory really work (I don't know how exactly the power series work but let's use QED as example to illustrate the concept)? I know the Fine Structure Constant or coupling constant is 1/137 which is acquired from actual measurement and can't be calculated.

Now without Perturbation, the fields are not supposed to interact, so what value do you get? Would it be zero?

After adding the first series. You get the initial value of 1.00 (how do you get this from 1/137?)
After adding the fourth series. You get the value of 1.0011596522

How did they do the series expanding to get the fourth series amount?

Are the above steps correct? Then it is really ad hoc since you just do power series expanding and don't really solve for the interacting fields. By the way.. in the magnetic moment of the electron. What is interacting there since there is no external field?

I'd like to understand how ad hoc is perturbation theory as it is used in almost all aspects of physics. Thanks a lot!

 

[DarMM]

It's really nothing more, at the formal level, than expanding a given quantity in powers of the coupling constant.

After adding the first series. You get the initial value of 1.00 (how do you get this from 1/137?)

Well you get it from performing the relevant first order calculation, explaining how this is done is the content of most introductory books of quantum field theory such as Peskin and Schroder.

 

[rodsika]

It's really nothing more, at the formal level, than expanding a given quantity in powers of the coupling constant.


Do you know an introductory lesson for doing power series without any knowledge of mathematics?


Well you get it from performing the relevant first order calculation, explaining how this is done is the content of most introductory books of quantum field theory such as Peskin and Schroder.

I only finished high school algebra. Would anyone be so kind as to share how the coupling constant was able to arrive at the right magnetic moment of the electron in the fourth series? Not detailed explanation but high school friendly. Thanks in advance!

 

[Physics Monkey]

The unperturbed value of g/2 is 1 not zero (g is the g-factor that enters the dipole moment). The value of 1 is given by non-interacting physics i.e. by the dirac equation. The first correction is [tex] \alpha/2 \pi [/itex] (see http://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment).

 

[rodsika]

The unperturbed value of g/2 is 1 not zero (g is the g-factor that enters the dipole moment). The value of 1 is given by non-interacting physics i.e. by the dirac equation. The first correction is [tex] \alpha/2 \pi [/itex] (see http://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment).

Ok. In the power series, do they make use of the measured value of 1.00115965219? If not, why is the power series so accurate as to be close to it at 1.0011596522?

 

[The_Duck]

It's presumably accurate because the theory is a good description of reality. The fact that calculations like this come out so close to the measured results is what gives us confidence in QED as a good theory.

 

[rodsika]

It's presumably accurate because the theory is a good description of reality. The fact that calculations like this come out so close to the measured results is what gives us confidence in QED as a good theory.

But these were not calculated from first principle. It uses the value of the dirac equation as 1 before the power series starts. The power series is just approximate solution and this doesn't prove QED is original.

 

[The_Duck]

The Dirac equation can be derived from first principles, so using the magnetic moment implied by the Dirac equation as a first approximation is still working from first principles. It's true that the value we calculate from QED using perturbation theory is only approximate, but the approximation is extremely accurate--to ten decimal places or something--which is how we are able to compare it to the similarly accurate experimental measurements. If we had the time, we could do a more intensive calculation and improve the perturbation theory prediction, adding more decimal places, but this will only be necessary when we have more precise experiments to compare to.

 

[rodsika]

The Dirac equation can be derived from first principles, so using the magnetic moment implied by the Dirac equation as a first approximation is still working from first principles. It's true that the value we calculate from QED using perturbation theory is only approximate, but the approximation is extremely accurate--to ten decimal places or something--which is how we are able to compare it to the similarly accurate experimental measurements. If we had the time, we could do a more intensive calculation and improve the perturbation theory prediction, adding more decimal places, but this will only be necessary when we have more precise experiments to compare to.

What is the difference between Perturbation series used in the determination of the magnetic moment of the electron and another case where for example two electrons are interacting. In the former case, the Dirac Equation produces 1.0 and after the fourth series, it comes up with 1.0011596522. How about in two electrons that are interacting. Quantum field theory being non-interacting says the electrons would just pass through each other. So what is the equivalent of the 1.0 in Dirac Equation in this case (or the initial value)? Would you solve for the dirac equations of the two electrons separately producing let's say 5.0 and then doing perturbation series on the interactions producing say 7.0?