Mass of field for small fluctuations

[h0dgey84bc]

Hi,

If I have a Lagrangian that looks like $$L=-\frac{1}{2} \partial^a{\phi}\partial_a{\phi}-\frac{1}{2} \mu \phi^2 -\frac{1}{24}\lambda \phi^4$$.
Where $$\lambda>0$$

Then how do I figure out the mass for small fluctuations about $$\phi=0$$ ?

I don't think I really understand what it means for some parameter to represent the mass. I mean without the phi^4 term, it would just lead to KG equation and obviously there $$\mu$$ represents the mass when you find the dispersion relation.
Should I just therefore find the equations of motion, then plug in some superposition type solution to find out the dispersion relation, and thus find a term that I would normally call the mass in a relation of the form $$E^2=P^2+m^2$$? or is there something else to this?

I'm not really sure how to incorporate the phi=0 expansion into this, I was originally thinking just Taylor expand but then I would have expanded about $$\phi(0,\vec(0))$$, as oppose to the trivial $$\phi=0$$ solution.

Thanks for any help

 

[torquil]

I will only comment in the context of classical field theory, because I think it is not necessary to mention detailed aspects of QFT.

I think you already have the correct idea. The mass of the field is determined by its behaviour for small values of the field \phi. This means that you can simply neglect the quadric term.

If you want to actually do a Taylor expansion, you can construct the action integral and do a Taylor expansion with functional derivatives around \phi=0, but the end result will be the same.

Lots of more complicated answers can be given, involving quantization, representation theory of the Lorentz-group, mass renormalization with bare masses and dressed masses, singularities of propagators, and so on. But I believe what I mentioned above is a pretty reasonable definition in the context of classical field theory. If you are interested in more detailed aspects, consult Weinbergs first QFT book.

One thing to watch out for with this kind of approximation. After you have assumed that \phi is small, you have to make sure that the solutions in the approximative theory don't grow to become large... That would be inconsistent. But in this case, your approximation of small \phi leads to a linear equation of motion, so all is well!

Torquil

 

[Entropia]

!

Hello,

Thank you for your question. The mass term in a Lagrangian represents the energy required to excite a field, and is related to the strength of the interaction between the field and its corresponding particle. In this case, the mass term \mu \phi^2 is responsible for the self-interaction of the field \phi. The larger the value of \mu, the stronger the self-interaction and thus the larger the mass.

To find the mass for small fluctuations about \phi=0, you can first expand the Lagrangian using a Taylor series. This will give you an expression for the potential energy V(\phi) as a function of \phi. The mass term can then be found by taking the second derivative of V(\phi) at \phi=0.

Alternatively, you can find the equations of motion for the field \phi and then use a superposition of solutions to find the dispersion relation. This will give you a term that can be interpreted as the mass in the relation E^2=P^2+m^2. The advantage of this method is that it allows you to take into account the effects of all terms in the Lagrangian, including the \phi^4 term.

I hope this helps clarify the concept of mass in this context. Please let me know if you have any further questions. Keep up the good work in your research!