12
$\begingroup$

When we canonically quantize the scalar field in QFT, we use a Lorentz invariant integration measure given by $$\widetilde{dk} \equiv \frac{d^3k}{(2\pi)^3 2\omega(\textbf{k})}.$$

How can I show that it is Lorentz invariant?

$\endgroup$

closed as off-topic by akhmeteli, tpg2114, Abhimanyu Pallavi Sudhir, Waffle's Crazy Peanut, Qmechanic Nov 3 '13 at 18:52

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – akhmeteli, tpg2114, Abhimanyu Pallavi Sudhir, Waffle's Crazy Peanut, Qmechanic
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 2
    $\begingroup$ Hi Ome. If you haven't already done so, please take a minute to read the definition of when to use the homework tag, and the Phys.SE policy for homework-like problems. $\endgroup$ – Qmechanic Nov 3 '13 at 13:37
  • 6
    $\begingroup$ I'd say that this question very much falls under the category homework. $\endgroup$ – Danu Nov 3 '13 at 14:30
  • 1
    $\begingroup$ Related: physics.stackexchange.com/q/53534 $\endgroup$ – joshphysics Nov 3 '13 at 18:23
  • 1
    $\begingroup$ This is certainly homework. Don't feel demeaned by your question being tagged as such, it really means that your question is just about problem-solving. You can improve your question, by the way, by showing your effort, and asking a conceptual question. Your question may then be reopened. If you do that, just ping me here, using "@DIMension10", and I'd vote to reopen. I'm not the downvoter, by the way. $\endgroup$ – Abhimanyu Pallavi Sudhir Nov 5 '13 at 14:48
  • 1
    $\begingroup$ +1 for this question and flagging the moderator, for the following reason: "This question belongs to the important conceptual (as well as technical indeed) questions and obstacles that usually hinders the understanding of QFT. It certainly is useful to a broad public!" $\endgroup$ – Noix07 Oct 12 '18 at 8:21
17
$\begingroup$

To show that this measure is Lorentz invariant you first need to explicitly write your integral as an integral over mass shell in 4D k-space. This could be done by inserting Dirac delta function $\delta[k^\mu k_\mu-m^2]$ and integrating over the whole 4D space.

Then you could apply the following transformations: \begin{align} \theta(k_0)\cdot\delta[k^\mu k_\mu-m^2] &= \theta(k_0)\cdot\delta[k_0^2-|\mathbf{k}|^2-m^2]\\ &=\theta(k_0)\cdot\delta\left[(k_0-\sqrt{|\mathbf{k}|^2+m^2})(k_0+\sqrt{|\mathbf{k}|^2+m^2})\right]\\ &=\frac{\delta\left[k_0-\sqrt{|\mathbf{k}|^2+m^2}\right]}{2\,k_0}, \end{align} where Heaviside function $\theta(k_0)$ is used to select only future part of the mass shell.

$\endgroup$
  • $\begingroup$ You nean $\theta(k_0) \delta(k^2 - m^2)$ since $\omega_{\vec{k}} = \sqrt{ |\vec{k}|^2 + m^2 } $. $\endgroup$ – nervxxx Nov 3 '13 at 15:59
  • $\begingroup$ @nervxxx: Thanks! I mistakenly thought we were talking about massless (or EM) field. Edited accordingly. $\endgroup$ – user23660 Nov 3 '13 at 16:23

Not the answer you're looking for? Browse other questions tagged or ask your own question.