Partition function and vacuum energy (basic stuff)

[diegzumillo]

Hi all
This is a fairly basic QFT question but it's bothering me. And Peskin/Schroeder fails at explaining basic stuff, so here I am.

After calculating Z for a particular theory I know this can be used to calculate all kinds of correlation functions. Itself, however, is the probability amplitude of vacuum-vacuum, correct? So when a problem asks me to calculate some vacuum diagrams I simply get a few terms of Z.

My difficulty is that I'm supposed to compare with the ground state energy that I calculated using a different method (for reference, it's the perturbed oscillator, or anharmonic oscillator). And when I realized I had exactly the same expansion for both ground energy and Z my whole understanding of it fell apart.

Considering the representation of Z using the time evolution operator (exponential of Hamiltonian etc) being sandwiched between vacuum states, I was expecting to be able to relate Z and E through an exponential/logarithm. Which goes well with its relation to statistical mechanics. But I'm not seeing that happen.

 

[diegzumillo]

I can try rephrasing it.

How do I extract the ground energy from the generating function?

 

[Roy_1981]

I can try rephrasing it.

How do I extract the ground energy from the generating function?

I don't think it's that simplistic. There is a difference between the the field theory generating functional and the parittion function. In particular, in the exponent of the generating functional one has the Lagrangian, L = p \dot{q}- H, while the partition function has simply - H. There is a difference of the so called symplectic term, namely p \dot{q} and so partition function and the generating function, Z, they are legendre transforms of each other. The right quantity you need is called the "generating functional for proper vertices or 1PI (one PI) diagrams". It is usually denoted by Γ. The first/leading term in the expansion of Γgives you the vacuum energy. Not sure if Peskin has it but you can try may be Zee.

 

[Roy_1981]

Actually I found a reference for you which would make things as clear as water. Check out pages 11-12 of this field theory set of notes. It spells out how to extract vacuum energy from the generating functional in a few lines:

http://itp.epfl.ch/webdav/site/itp/users/181759/public/aqft.pdf

Please avoid Peskin/Schroeder till you are ready for your Field Theory 2, i.e. Wilson RG. Peskin-Schoeder is an abject failure of a book for beginning learners of even for advanced learners/instructors who desire a more cogent and clear representation of field theory. There are just too many good introductory books available -for the basic stuff older books are pretty good, like Mandl and Shaw or Lowell Brown or Schweber even Huang for that matter. Polchinski lamented that currently popular field theory textbooks and by which I understood Peskin, does not even teach the student about Gupta-Bleuler quantization of the EM field.

 

[diegzumillo]

Thanks! Things are starting to clear up a little. I found some more discussions online.

Please avoid Peskin/Schroeder till you are ready for your Field Theory 2, i.e. Wilson RG. Peskin-Schoeder is an abject failure of a book for beginning learners of even for advanced learners/instructors who desire a more cogent and clear representation of field theory.

Oh god yes. Unfortunately there is little I can do to avoid Peskin. It's the main book on this course and even the homeworks are made based on it (derive equation such, using method from section such etc). Being stuck with it is definitely making my learning less efficient than it could be.

 

[diegzumillo]

For future googlers reaching this thread, what finally "clicked" for me, if I can call it that, was equation 4.56 from Peskin and Schroeder. The whole chapter is frustrating and confusing so my understanding of it is still superficial. Keep in mind that it considers E0 = 0 for convenience, because we can only measure variation of vacuum energy or something like that, which is not the case on my problem, where the ground energy without perturbation is the regular oscillator's ground energy. So It would be $E_\Omega = E_0 -\frac{i}{T}$(connected vacuum diagrams), that is, without the single loop diagram. At least that's what I understand by this terminology. Also, the equation I mentioned is written slightly differently there, in a way that I don't really understand, but what I wrote here I know where it came from. He managed to pull a volume out of his... sleeve... It makes some sense if I think of the relation with statistical mechanics, but in this context, vacuum energy, what the hell is volume? the entire universe? does vacuum energy changes with expanding universe? Ah... sorry for venting my frustration here. I'm just bitter because I can't make sense of this stuff yet.

 

[Roy_1981]

For future googlers reaching this thread, what finally "clicked" for me, if I can call it that, was equation 4.56 from Peskin and Schroeder. The whole chapter is frustrating and confusing so my understanding of it is still superficial. Keep in mind that it considers E0 = 0 for convenience, because we can only measure variation of vacuum energy or something like that, which is not the case on my problem, where the ground energy without perturbation is the regular oscillator's ground energy. So It would be $E_\Omega = E_0 -\frac{i}{T}$(connected vacuum diagrams), that is, without the single loop diagram. At least that's what I understand by this terminology. Also, the equation I mentioned is written slightly differently there, in a way that I don't really understand, but what I wrote here I know where it came from. He managed to pull a volume out of his... sleeve... It makes some sense if I think of the relation with statistical mechanics, but in this context, vacuum energy, what the hell is volume? the entire universe? does vacuum energy changes with expanding universe? Ah... sorry for venting my frustration here. I'm just bitter because I can't make sense of this stuff yet.

Yes the energy is proportional to the volume since the energy density is constant. In particular for infinite volume the ground state energy is infinite or better to say they diverge with volume of the field theory. So do all the higher/excited levels, they have a term which is finite wand another term which diverges as volume. However we can only measure transitions between levels, i,e. the finite difference. This is why it makes sense to take the ground state energy to be zero. If you couple the system to gravity then ofc this vacuum or ground state energy is a measurable aka since it exerts gravitational pull. Going back to non-gravitational field theory, for a field theory in finite volume, such as field theory on a line-segment or a circle, the vacuum energy is finite and is called the Casimir energy.