What is the dynamic equation for a complex field in four dimensional space-time?

[Sojourner01]

Homework Statement

Consider a complex field f (x, t) defined over the full four dimensional space-time. The Lagrangian is:
$$\frac{i}{2}(f*\frac{df}{dt}-f\frac{df*}{dt}) - \frac{1}{2}[(\frac{df*}{dx})(\frac{df}{dx})+(\frac{df*}{dy})(\frac{df}{dy})+(\frac{df*}{dz})(\frac{df}{dz})] - V(x,y,z)f*f$$

Determine the dynamic equation for this complex field. There are two ways to deal with
complex fields: One is to treat the real and imaginary parts as two independent fields; the
other, and much more useful, approach is to treat field f (x, t) and its complex conjugate field
f (x, t) as the two independent fields. Comment on this dynamical system.

Homework Equations

The euler-lagrange equation, presumably

The Attempt at a Solution

It's very time-consuming to type out the obvious result of plugging into the euler-lagrange equation. Basically - should I be getting something resembling the Klein-Gordon equation? I'm getting two more or less identical dynamical equations; one for f and one for f* - should this be combined for f*f?

 

[Sojourner01]

Nobody?

Ok, well, my main problem is that I can't see why the lagrangian is selected the way it is; the closest functional form I can derive is:

$$L=\nabla(SS*) + \frac{d(SS*)}{dt} - V(x,y,z)SS*$$

This is obviously missing the correct coefficients. The question asks what the dynamical system represents - since I don't understand why this lagrangian was chosen and why the invariant is field x conjugate, I'm stumped.

 

[Sojourner01]

Is this because nobody knows the answer, I haven't phrased the question properly, or I haven't done the work for it to be worth your time?

OK, I'll give you a little more working. I'm getting:

$$\frac{1}{2} \nabla^{2}f* - \frac{i}{2}\frac{df*}{dt} - Vf* = 0$$

Does this look right? If so, what does it represent? I'm stumped.

 

[George Jones]

Is this because nobody knows the answer, I haven't phrased the question properly, or I haven't done the work for it to be worth your time?

OK, I'll give you a little more working. I'm getting:

$$\frac{1}{2} \nabla^{2}f* - \frac{i}{2}\frac{df*}{dt} - Vf* = 0$$

Does this look right? If so, what does it represent? I'm stumped.

I haven't done the calculation, but it looks like you have the non-relativistic Schrödinger equation.

Are you sure about all the factors of 2?

 

[Sojourner01]

Positive - the approach we're taking in class - rightly or wrongly, it doesn't sound too rigorous - is to insert coefficients by hand afterwards.

It had occurred to me that it did look like the time-dependent Schrödinger equation - but surely in order to derive that you'd have to use a spinor field, not a complex scalar, since that's what it operates on? I was under the impression that scalar fields in conjugate pairs were supposed to yield the klein-gordon equation, and that's the bulk of an example given later in our course. Actually, doing the derivation is ok, but the crux of the question is explaining the result, and that's what's baffling me.

 

[George Jones]

Positive - the approach we're taking in class - rightly or wrongly, it doesn't sound too rigorous - is to insert coefficients by hand afterwards.

Then, assume $V$ is real, take the complex conjugate conjugate of your equation, and set $1 = \hbar = 2m$ and $U = 2V$.

It had occurred to me that it did look like the time-dependent Schrödinger equation - but surely in order to derive that you'd have to use a spinor field, not a complex scalar, since that's what it operates on?

Not for a hypothetical spinless particle. In any case, in non-relativistic QM, spin is added as an afterthough by tensoring with a 2-dimensional spin space.

I was under the impression that scalar fields in conjugate pairs were supposed to yield the klein-gordon equation, and that's the bulk of an example given later in our course.

Yes, the right complex Lagrangian spits out the charged Klein-Gordon equation. I've been following your thread from the start, but:

1) your Lagrangian didn't look close enough to the standard Lagrangian to give the Klein-Gordon equation
2) I didn't recognize your Lagrangian;
3) I was too Lazy to work out the equation of motion.

Actually, doing the derivation is ok, but the crux of the question is explaining the result, and that's what's baffling me.

I recognized the Schrödinger equation in post #3, and then I went to one of the only relevant references that I have at home - Field Quantization by Greiner and Reinhardt. They use a Lagrangian similar (not exactly the same) to yours to derive the Schrödinger equation.

 

[Sojourner01]

Many thanks for your help, I think I'm starting to get the hang of this now.

Are these standard sorts of questions to be asked? I get the impression that PF contributors are more familiar with lagrangians in the context of classical mechanics, whereas the faculty here seem to be of the opinion that it's useful only as a route to QFT and don't teach any as a classical mechanics class whatsoever. It isn't even offered on the BSc course and those graduates see no lagrangian or hamiltonian mechanics at all.

 

[George Jones]

Are these standard sorts of questions to be asked?

Do you mean here at PF, or in courses, or both?

I can't remember if I ever had to derives the non-relativistic Schrödinger equation from a Lagrangian, but I do remember seeing Lagrangian for the electromagnetic field, which leads to Maxwell's equations. I think that the idea behind the Schrödinger equation example is to show that a very familiar equation can be derived from an appropriate Lagrangian.

I get the impression that PF contributors are more familiar with lagrangians in the context of classical mechanics,

Here in the homework section. I think many people who contribute to High Energy, Nuclear, Particle Physics, to Beyond the Standard Model, to Special & General Relativity, and maybe to other forums here are familiar with Lagrangians for lots of different situations.

whereas the faculty here seem to be of the opinion that it's useful only as a route to QFT and don't teach any as a classical mechanics class whatsoever. It isn't even offered on the BSc course and those graduates see no lagrangian or hamiltonian mechanics at all.

To me, this seems really bizarre.

Advanced courses classical mechanics give opportunities to introduce powerful, abstract theoretical techniques (Lagrangians, Hamiltonians, canonical transformations, Poisson brackects, Noether's theorem, Hamilton-Jacobi theory, etc.) in a somewhat familiar setting. All of these things were in a couple of classical mechanics courses that I had to take as an undergrad.

Lagrangian stuff certainly is very, very useful in QFT and particles physics, but Lagrangian/Hamiltonian/variational stuff has uses all over the spectrum. For example, Brandon Carter was the first person to finds solutions to geodesic orbits around rotating black holes, and he did this by separating the Hamilton-Jacobi differential equation.

Finally, a dissadent view from Roger Penrose (Road to Reality, page 491):

"However, I must confess my unease with this as a fundamental approach. I have difficulties in formulating my unease, but it has something to do with the generality of the Lagrangian approach, so that little guidance may be provided towards finding the correct theories. Also the choice of Lagrangian is often not unique, and sometimes rather contrived - even to the extent of undisguised complication.There tends to be a remoteness from actual 'hands-on understanding, particularly in the case of Lagrangians for fields. ... Langrangians for fields are undoubtedly useful as mathematical devices, and they enable us to write down large numbers of suggestion for physical theories. But I remain uneasy about relying upon them too strongly in our searches for improved fundamental physical theories."

 

[Sojourner01]

To me, this seems really bizarre.

I thought so too. Chatting to various staffers at this year's christmas party, the general opinion is that there is a great deal more to cover these days than there was when they were taking their first degrees. As a result, mechanics, statistics and so on get pushed aside and more time is allotted to QM, lasers, condensed matter, chaos theory, computational techniques and other such oddities. Perhaps one of the biggest drains is the focus on computers - I've taken three computational classes in a year before, because they're convinced that it's desperately needed, and for some reason it's incredibly time-consuming to teach programming to students who know nothing about it.