Student Friendly Quantum Field Theory

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Discussion Overview

The discussion revolves around the readability and content of Klauber's book on Quantum Field Theory (QFT), particularly focusing on his views regarding normal ordering and vacuum energy. Participants explore the implications of these views in the context of QFT theory and its applications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that Klauber's book is more accessible for readers at their level but questions whether his minority views on normal ordering and vacuum energy are too far from mainstream to warrant a recommendation.
  • Another participant argues that failing to normal order the free Hamiltonian results in infinite vacuum energy, while normal ordering leads to finite vacuum energy and more sensible results.
  • It is mentioned that normal ordering may not be necessary if renormalization is applied, suggesting that Klauber's book may still be valuable despite its unconventional views.
  • A participant highlights that Klauber's position suggests normal ordering overlooks non-zero commutators of reversed terms, creating a puzzle regarding its necessity.
  • Concerns are raised about scenarios where the difference between normal ordered and default ordered Hamiltonians may depend on dynamical quantities, potentially leading to incorrect results in those cases.
  • One participant provides the Casimir effect as an example where the difference between orderings is dynamical, indicating that the energy cannot be interpreted as zero-point energy.
  • It is noted that the vacuum-expectation value of the Hamiltonian can often be expressed in terms of two-point functions, which differ significantly between normal and default ordering.

Areas of Agreement / Disagreement

Participants express differing views on the implications of Klauber's approach to normal ordering and its effects on physical results. There is no consensus on whether his views are too far from mainstream or if they hold merit in specific contexts.

Contextual Notes

Some limitations are noted regarding the assumptions made about normal ordering and its implications in various scenarios, particularly in relation to dynamical quantities and their effects on physical interpretations.

sandy stone
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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?
 
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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
 
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.
 
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.
 
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Demystifier said:
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?
 
vanhees71 said:
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.
 
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