Student Friendly Quantum Field Theory

[sandy stone]

The thread https://www.physicsforums.com/threads/qft-operators-time-space-asymmetry.906369/ contains the first recommendation I have seen in these forums for Klauber's book, and instead of hijacking that thread I thought I might ask a question here. I find the book more readable than many for someone at my level, but the author admits he has some minority views regarding normal ordering and vacuum energy. Do any of the mentors find his opinions far enough off the mainstream to not recommend the book?

 

[vanhees71]

Well, if he doesn't normal order even the free Hamiltonian of any QFT, he'll get an infinite vacuum energy. On the other hand normal ordering the free Hamiltonian leads to a finite vacuum energy (set to 0 as it should be) and to a finite energy of any well-defined Fock state, which makes much more sense than the theory without normal ordering.

On the other hand, normal ordering is not necessary, if you discuss renormalization anyway, and you have to to get finite results, and then you can as well renormalize the effective quantum action as the scattering amplitudes. I don't think that this makes Klauber's book a nogo. However, I'm not familiar enough with it to say anything qualified in its favor or disfavor.

My favorite as an introductory textbook is

M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University press

 

[sandy stone]

As I understand it, Klauber's position is that normal ordering ignores the fact that the commutators of the reversed terms are non-zero. Of course, as you point out, normal ordering does give sensible answers. It seems a puzzle.

 

[Demystifier]

If the difference between normal ordered and default ordered Hamiltonian is just a constant, then both orderings lead to the same physical results. That's the case in all examples studied in Klauber's book. But in some cases (not studied in the book) the difference may depend on some dynamical quantities, and in such cases normal ordering usually gives wrong results.

 

[vanhees71]

But in some cases (not studied in the book) the difference may depend on some dynamical quantities, and in such cases normal ordering usually gives wrong results.

Do you have an example?

 

[Demystifier]

Do you have an example?

Sure, Casimir effect:
https://arxiv.org/abs/1702.03291
When the difference between the two orderings is dynamical (as in Casimir effect), then the corresponding energy is real but cannot be properly interpreted as zero-point energy.

Note also that for two-point functions
$$\langle 0|\phi(x)\phi(y)| 0\rangle \neq \langle 0|:\phi(x)\phi(y):| 0\rangle$$
The left-hand side is an important physical quantity in QFT, while the right-hand side is zero. The vacuum-expectation value of the Hamiltonian can often be expressed in terms of such a two-point function.