Deduction of formula for Lagrangian density for a classical relativistic field

[StenEdeback]

Homework Statement

Deduction of formula for Lagrangian density for a classical relativistic field

Relevant Equations

See the attached file

Hi,

I am reading Robert D Klauber's book "Student Friendly Quantum Field Theory" volume 1 "Basic...". On page 48, bottom line, there is a formula for the classical Lagrangian density for a free (no forces), real, scalar, relativistic field, see the attached file.
I like to understand formulas which are new to me, so I have searched internet for the deduction of that formula, without success.
I would be grateful for info where I can find such a deduction. It feels necessary to see it before I go on reading Klauber's book.
Thank you in advance, as we say here in Sweden!Sten Edebäck

 

[vanhees71]

Just derive the equation of motion for the field from the Euler-Lagrange equations. What do you get?

 

[StenEdeback]

Thank you!
OK, I should be able to do that, but I seem to have got stuck, just looking at the Euler-Lagrange equations and not getting any further. It could be psychological - having pondered this for some time my energy has gone low.

 

[StenEdeback]

Well, I would be grateful to get a hint where to find this deduction, since my mind seems to be blocked. :)

 

[vanhees71]

The Euler-Lagrange equations read
$$\partial_{\mu} \frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)}=\frac{\partial \mathcal{L}}{\partial \phi}.$$

 

[StenEdeback]

Thank you!
Yes, I know the Euler-Lagrange equation for fields, and I can even derive it from variation of the action.
But trying to get from there to the formula in the attachment of this thread has got me stuck for some reason. Such things happen sometimes, and more frequently when you study on your own for fun like I do. Then Physics Forums can be a lifeline for me, hopefully this time too.

 

[PeroK]

Well, I would be grateful to get a hint where to find this deduction, since my mind seems to be blocked. :)

The terms in the Lagrangian broadly relate to the kinetic, (inherent) potential and rest-mass energy of the particle. If you generate the Euler-Lagrange equations, you should get the equations of motion you are looking for.

In many cases, the Lagrangian itself is to some extent derived by educated guesswork.

 

[StenEdeback]

Thank you!
I have the Euler-Lagrange equations, and I can derive them too, but I cannot derive the Lagrangian specified in the attachment of this thread.

 

[PeroK]

but I cannot derive the Lagrangian specified in the attachment of this thread.

What's your starting point? Derive from what?

 

[StenEdeback]

Well, I don't really know my starting point. I want to find out why the Lagrangian density is as specified in the attachment of this thread. And I cannot see how it satisfies the Euler-Lagrange equations.

 

[PeroK]

Well, I don't really know my starting point. I want to find out why the Lagrangian density is as specified in the attachment of this thread. And I cannot see how it satisfies the Euler-Lagrange equations.

A Lagrangian doesn't satisfy the E-L equations, it generates them. And, as I said above, writing down a Lagrangian is often educated guesswork. Especially for QFT.

 

[vanhees71]

I don't understand the problem. Is it about taking the derivatives? So here's the calculation. The Lagrangian is
$$\mathcal{L}=\frac{1}{2} (\partial_{\mu} \phi) (\partial^{\mu} \phi) -\frac{m^2}{2} \phi^2.$$
That's a Lorentz scalar you can build from a scalar field and its four-gradient, and restricted to having a free field, i.e., the Lagrangian being bilinear that's basically all you can write down.

Now we evaluate the Euler-Lagrange equations:
$$\frac{\partial \mathcal{L}}{\partial \phi}=-m^2 \phi, \quad \frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)}=\partial^{\mu} \phi$$
So the equation of motion reads
$$\Box \Phi=-m^2 \phi,$$
i.e., the Klein-Gordon equation, which is compatible with the dispersion relation expected from the Einstein-de Broglie relations $E=\hbar \omega$, $\vec{p}=\hbar \vec{k}$, i.e., the plane-wave solutions have to fulfill
$$E^2=m^2 + \vec{p}^2,$$
where I've used natural units, $\hbar=c=1$.

That justifies the Ansatz for the Lagrangian of a free scalar field.

 

[StenEdeback]

I don't understand the problem. Is it about taking the derivatives? So here's the calculation. The Lagrangian is $$\mathcal{L}=\frac{1}{2} (\partial_{\mu} \phi) (\partial^{\mu} \phi) -\frac{m^2}{2} \phi^2.$$ That's a Lorentz scalar you can build from a scalar field and its four-gradient, and restricted to having a free field, i.e., the Lagrangian being bilinear that's basically all you can write down. Now we evaluate the Euler-Lagrange equations: $$\frac{\partial \mathcal{L}}{\partial \phi}=-m^2 \phi, \quad \frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)}=\partial^{\mu} \phi$$ So the equation of motion reads $$\Box \Phi=-m^2 \phi,$$ i.e., the Klein-Gordon equation, which is compatible with the dispersion relation expected from the Einstein-de Broglie relations $E=\hbar \omega$, $\vec{p}=\hbar \vec{k}$, i.e., the plane-wave solutions have to fulfill $$E^2=m^2 + \vec{p}^2,$$ where I've used natural units, $\hbar=c=1$. That justifies the Ansatz for the Lagrangian of a free scalar field.

I don't understand the problem. Is it about taking the derivatives? So here's the calculation. The Lagrangian is $$\mathcal{L}=\frac{1}{2} (\partial_{\mu} \phi) (\partial^{\mu} \phi) -\frac{m^2}{2} \phi^2.$$ That's a Lorentz scalar you can build from a scalar field and its four-gradient, and restricted to having a free field, i.e., the Lagrangian being bilinear that's basically all you can write down. Now we evaluate the Euler-Lagrange equations: $$\frac{\partial \mathcal{L}}{\partial \phi}=-m^2 \phi, \quad \frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)}=\partial^{\mu} \phi$$ So the equation of motion reads $$\Box \Phi=-m^2 \phi,$$ i.e., the Klein-Gordon equation, which is compatible with the dispersion relation expected from the Einstein-de Broglie relations $E=\hbar \omega$, $\vec{p}=\hbar \vec{k}$, i.e., the plane-wave solutions have to fulfill $$E^2=m^2 + \vec{p}^2,$$ where I've used natural units, $\hbar=c=1$. That justifies the Ansatz for the Lagrangian of a free scalar field.

I don't understand the problem. Is it about taking the derivatives? So here's the calculation. The Lagrangian is
$$\mathcal{L}=\frac{1}{2} (\partial_{\mu} \phi) (\partial^{\mu} \phi) -\frac{m^2}{2} \phi^2.$$
That's a Lorentz scalar you can build from a scalar field and its four-gradient, and restricted to having a free field, i.e., the Lagrangian being bilinear that's basically all you can write down.

Now we evaluate the Euler-Lagrange equations:
$$\frac{\partial \mathcal{L}}{\partial \phi}=-m^2 \phi, \quad \frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)}=\partial^{\mu} \phi$$
So the equation of motion reads
$$\Box \Phi=-m^2 \phi,$$
i.e., the Klein-Gordon equation, which is compatible with the dispersion relation expected from the Einstein-de Broglie relations $E=\hbar \omega$, $\vec{p}=\hbar \vec{k}$, i.e., the plane-wave solutions have to fulfill
$$E^2=m^2 + \vec{p}^2,$$
where I've used natural units, $\hbar=c=1$.

That justifies the Ansatz for the Lagrangian of a free scalar field.

Thank you very much!
Now it is crystal clear to me. I have actually seen and understood this before, and it is not complicated. I was confused by the way the formula was written in the attachment of this thread. As you say, there is no problem. I have experienced "chess blindness", and apparently there is such a thing as "physics blindness". About a couple of times a year I get stuck like I did this time, and most often it is just some misunderstanding on my side. Then it is very valuable to get support from Physics Forums. Studying on my own I depend on it.
Again, thank you so much!

Sten Edebäck

 

[Orodruin]

Well, I would be grateful to get a hint where to find this deduction, since my mind seems to be blocked. :)

What do you mean by ”deduce”? How do you ”deduce” F=ma in Newtonian mechanics? In the end, it is a definition that gives you the equations of motion and the question is rather if it describes a physical system or not.

 

[StenEdeback]

Well, "deduction" may not be a good word for it. My question is really: "Why do you specify the Lagrangian with the formula in the attached file? And how do you prove that it is useful, that is describes a physical system?"

 

[PeroK]

"Why do you specify the Lagrangian with the formula in the attached file?

Because it works.

And how do you prove that it is useful, that is describes a physical system?"

Generate the E-L equations, then do an experiment to test the validity of these equations.

 

[vanhees71]

Indeed, you use the Lagrangians that give the right relativistic field equations of motion. The latter are "deduced" from the demand of Poincare invariance, i.e., consistency of the theory is the relativistic spacetime model. It leads to the characterization of fields in terms of mass and spin. Together with microcausality this establishes the type of equations you get for the quantum fields for particles with a given mass and Spin. The elementary particles, as described by the Standard Model, are all spin-1/2 fermions, described by the Dirac equation (leptons and quarks), spin-1 bosons (all of them being gauge fields of the fundamental electroweak (W, Z and photons) and strong (gluons) interactions), (at least) one spin-0 boson (the Higgs boson).

 

[StenEdeback]

Thank you! I have understood now that you make an "educated guess" to find the Lagrangian and then check if it works. My problem is basically that I need knowledge of classical relativistic fields, so I have now started to read the book "Special relativity and classical field theory" by Leonard Susskind and Art Friedman, and that has already spread some light over my foggy thoughts. So, to study Quantum Field Theory I first need to study some Classical Field Theory. It is the same situation as when I started to study String Theory which I was curious about. After a while I found that I needed knowledge about Quantum Field Theory. So I started reading at the top and then needed to go down stepwise. It is still very interesting and fun. And Physics Forums have knowledgeable and helpful people, which is very valuable to me. Thank you so much again!

 

[PeroK]

There is a series of lectures on QFT by Tobias Osborne online here:

 

[vanhees71]

Thank you! I have understood now that you make an "educated guess" to find the Lagrangian and then check if it works. My problem is basically that I need knowledge of classical relativistic fields, so I have now started to read the book "Special relativity and classical field theory" by Leonard Susskind and Art Friedman, and that has already spread some light over my foggy thoughts. So, to study Quantum Field Theory I first need to study some Classical Field Theory. It is the same situation as when I started to study String Theory which I was curious about. After a while I found that I needed knowledge about Quantum Field Theory. So I started reading at the top and then needed to go down stepwise. It is still very interesting and fun. And Physics Forums have knowledgeable and helpful people, which is very valuable to me. Thank you so much again!

Another very valuable book as a bridge from classical to quantum field theory is

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles, Springer, Wien (2001).

 

[StenEdeback]

Thank you very much! I will look at the book you recommend after I have read the book by Leonard Susskind and Art Friedman which is probably more basic.

 

[apostolosdt]

Looking into your opening post, I get the impression you’re asking not about the equation of motion of the Lagrangian shown in the attached image, but how one derives that Lagrangian. Of all the responses to your post, at least one is addressing your question—that getting a Lagrangian is usually educated guesswork. And, of course, that is correct.

So, I'll give a try, too. You probably will remain with lots of questions unanswered, especially when coming to details, but hopefully, you will be less confused.

The best source I have found for your inquiry is Zee’s QFT in a nutshell, 2nd edition. On pp. 18, he attempts to take the reader from a silly but instructive example of what a field is to a more serious mathematical description. He presents a “modern view” which he calls the “Landau-Ginzburg view.”

I am most certain Zee is adapting a theory by Landau and Ginzburg for superconductivity to his pedagogical task. Anyway, Zee states that, under Landau-Ginzburg's view, one starts with the kind of symmetry he (she) wants to impose on the physical problem. So, for elementary particles, it is Lorentz invariance; for solid state, it is translational symmetry in the crystal lattice, etc. Then, says Zee, one must decide on a field that, under the selected symmetry, will transform in some desired way. In the case of your actual example, you choose a real scalar field $\phi$. The next step is to write down the action. That’s a step we usually take for granted. An action is a space-time integral on a Lagrangian; when we apply certain mathematical operations called variational methods, we derive the Euler-Lagrange equations, from them the equation of motion, etc.

Therefore, however one puts it, it is the Lagrangian one wishes to construct. Practicing physicists say constructing a Lagrangian is an art as much as science! Still, there are also rules to observe with a Lagrangian. For instance, we expect it to be a function of our field(s), but what about the field’s derivatives? Will the Lagrangian contain first-order, second-order of any order? Here again, Zee’s text comes to the rescue. “No more than two time derivatves,” he claims, “for we don’t know how to quantize such actions.”

I’ve written all that to provoke your appetite. Please, read the original Zee!

There is much more to the topic, of course. I cannot cover even a tiny fraction of them, for I don’t know that much myself. But there are numerous books that will answer many of your questions. In fact, it is good practice at that level of yours to get used to thumbing through advanced texts and read only the absolute essentials. For example, at that point, I might suggest that you just read pp. 79-82 from Peskin & Schroeder’s QFT book. You will find in their discussion why the ways the various fields in a Lagrangian that interact with each other cannot be infinite in number—-that would have been a disaster in our attempts to understand physical phenomena. But, please, don't fret about the heavy mathematics in the text. Just read the concepts between the equations!

I hope that will help a little. Enjoy your studies!

 

[StenEdeback]

Thank you for your long and interesting answer! I will look at both Zee's book and Peskin and Schroeder's book after reading the more basic book about classical fields by Susskind and Friedman. I have finally understood that finding the Lagrangian is a trial and test business. Yes, I enjoy my studies. I like the beautiful mathematics of theoretical physics, and the connection to our daily reality. Studying on my own and trying advanced topics without reading more basic things first leads of course to the need for stepping back to basics now and then, but that is fine to me. I can often get by without assistance, but sometimes I get stuck like this time, and then Physics Forums is a safe haven to me.

 

[vanhees71]

Finding the Lagrangian is NOT just "trial and test", but it's based on the symmetries of special-relativistic spacetime, i.e., one uses group theory to find the representations of this symmetry group and writes down the most general Lagrangian with some other constraints, i.e., that it is a local function of the fields and its first derivatives. Doing this leads you to socalled "effective field theories", which are valid below some energy-momentum scale, $\Lambda$, and you expand the S-matrix in powers of $p/\Lambda$, where $p$ is the typical energy-momentum scale of the scattering processes you want to describe. Such theories have an infinite number of "low-energy constants", i.e., you need more and more the more terms in the expansion you take into consideration.

Then there is a special subclass of QFTs, the so-called Dyson-renormalizable theories, where you only need a finite number of parameters (masses, coupling constants) at all orders of perturbation theory (which is organized as an expansion of powers of the coupling constants or, more elegantly, in powers of $\hbar$). This constrains the possible Lagrangians further to contain only parameters with energy-momentum dimensions $\geq 0$.

Further constraints come from other empirically found symmetries involving charge-conservation laws or gauge invariance.

 

[StenEdeback]

Thank you vanhees71 for your very interesting comment! I hope to get to these things eventually, though obviously it will take some time. I have studied some group theory but not used it much so far in connection with physics. I have read David J Griffiths' three excellent books about quantum mechanics, elementary particles, and electrodynamics, and also some more general relativity which I started studying for fun in my youth at the Royal Institute of Technology here in Stockholm, Sweden. Now as a pensioner my interest in physics has gone flourishing again, but my energy has limits and studying takes time. But it is fun!

 

[Orodruin]

which I started studying for fun in my youth at the Royal Institute of Technology here in Stockholm, Sweden

Out of interest, who gave the course at that time?

 

[StenEdeback]

Hi Orodruin! I think the name of the professor who gave the course in relativity was Ramsgaard. This was in the sixties (I am old, 78). I did most of the studies on my own, with tensor calculus and general relativity, and then I worked as assistant teacher of Mechanics and computer programming, and I studied extra also other kinds of mechanics like strength of materials. I considered going on with an academic career but decided to work with computers and began with IBM.

 

[malawi_glenn]

I also want to do physics just for fun in my 70's you are an inspiration @StenEdeback !

 

[StenEdeback]

Thank you malawi_glenn for your kind words! I wish you luck in your future studies!

Sten edebäck

 

[Orodruin]

I think the name of the professor who gave the course in relativity was Ramsgaard.

Oh, that is a tragic story right there.

I suspected it may have been Ramgard based on your age. Parts of his lecture notes (particularly those in vector analysis) live on to this day in one form or another. I never met him personally as it was significantly before my time. I did at some point extend and rewrite his lecture notes on tensor analysis though.