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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?

Qmechanic
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rainman
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    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. – Qmechanic Nov 03 '13 at 13:37
  • @Qmechanic: I was unaware of the policy. I have removed the tag as it is not a hw. – rainman Nov 03 '13 at 14:28
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    Related: https://physics.stackexchange.com/q/53534/ – joshphysics Nov 03 '13 at 18:23
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    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. – Abhimanyu Pallavi Sudhir Nov 05 '13 at 14:48
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    +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!" – Noix07 Oct 12 '18 at 08:21
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    Let me now clarify why this is of great conceptual importance and subtlety: it seems at first that one integrates on $\mathbb{R}^3$ the usual vector spaces. This does make sense but first, it is not a stable subspace of $\mathbb{R}^4$ under boosts and secondly the measure would not be invariant. (I suggest that the people who closed the question explicitly write down the image measure). The crucial point is that one integrates on a Lorentz invariant submanifold (the set of 4-vectors $k^{\mu}$ such that $k^{\mu} k_{\mu}=m^2$). So now we now see that we are dealing with integration of... – Noix07 Oct 12 '18 at 08:35
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    a function on a 3 dimension submanifold of $\mathbb{R}^4$. Second difficulty is that the Lebesgue measure on the latter space induces a measure on the submanifold. From here on, I do not know the mathematical details, but there are several visions: the answer below uses the Dirac distribution and the formula $\delta(f(x))= \sum_{x_i \in f^{-1}(0)} \frac{\delta(x-x_i)}{f'(x_i)}$ which in itself is already worth an explanation. The second vision is to stay in the framework of measure, for example ???-Stietjes measures. – Noix07 Oct 12 '18 at 08:47
  • For those who like to have a general and slightly more abstract vision, the very natural question is how to find an invariant measure on the orbits (here the mass-shell hyperboloid) under a group (here Lorentz), orbits which are submanifold of some manifold that is equipped with a measure. – Noix07 Oct 12 '18 at 08:58
  • (Absolute value missing in my formula with delta in previous² comment) – Noix07 Oct 12 '18 at 09:05
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    link. Possible ref. on the mathematical aspect "A course on integration, Nicolas Lerner" (2014), §5.5 p.238. – Noix07 Oct 12 '18 at 12:56
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    Homework 1.4.7 of Robin Ticciati's book seems to suggest that this can be done directly by evaluating the determinant of the Jacobian and factorizing $\Lambda$ appropriately. Does anybody know how to do this? – Ivan Burbano Feb 13 '19 at 01:12

1 Answers1

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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.

user23660
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