QFT: Invariant Measures & Rotational Invariance Explained

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In summary, Coleman talks about the importance of invariance in a calculation in order to prove unitarity, and uses the example of rotational invariance. He then goes on to explain that when performing a change of variable in an integral, you must include the Jacobian determinant of the transformation.
  • #1
whynothis
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I am reading through sidney colemans lectures on QFT and I am stuck on what seem to be a silly question: He talks about the fact that the measure used in a calculation should be invariant in order to prove unitarity and later on that operators transform properly. He uses the example of rotational invariance.

[tex]U^{-1}(R)\int d^{3}k|k><k| U(R)[/tex]
[tex]\int d^{3}k |R^{-1}k><R^{-1}k|[/tex]

Change of variable: Rk' = k
Now this is the part I don't get (I must be confused)
[tex]d^{3}k' = d^{3}k[/tex]
It seems to me like there should be a factor of R or something. However, the strange thing is that R is a matrix (isn't it?) so I don't really get it. Can someone explain what is going on here?
 
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  • #2
When in doubt, use brute force. Grind it out. That is, here, compute the appropriate Jacobean, and thus prove the equality. (Done in countless texts.)
Regards,
Reilly Atkinson
 
  • #3
whynothis said:
[Coleman] talks about the fact that the measure used in a calculation should be invariant [...]

[tex]U^{-1}(R)\int d^{3}k|k><k| U(R)[/tex]
[tex]\int d^{3}k |R^{-1}k><R^{-1}k|[/tex]

Change of variable: Rk' = k
Now this is the part I don't get (I must be confused)
[tex]d^{3}k' = d^{3}k[/tex]

In general, when you perform a change of variable in an integral, you
must include the Jacobian determinant of the transformation. In this case
the Jacobian is [itex]| \partial k'_i/\partial k_j |[/itex].
(Consult Wiki for more detail.)

In the current case, [itex]d^3k[/itex] is an infinitesimal volume element
in 3-momentum space, and are preserved by rotation transformations,
so the Jacobian turns out to be 1. But that's not necessarily so in more
general transformations.
 
  • #4
whynothis said:
I am reading through sidney colemans lectures on QFT and I am stuck on what seem to be a silly question: He talks about the fact that the measure used in a calculation should be invariant in order to prove unitarity and later on that operators transform properly. He uses the example of rotational invariance.

[tex]U^{-1}(R)\int d^{3}k|k><k| U(R)[/tex]
[tex]\int d^{3}k |R^{-1}k><R^{-1}k|[/tex]

Change of variable: Rk' = k
Now this is the part I don't get (I must be confused)
[tex]d^{3}k' = d^{3}k[/tex]
It seems to me like there should be a factor of R or something. However, the strange thing is that R is a matrix (isn't it?) so I don't really get it. Can someone explain what is going on here?

From the point of view of group theory, what he used is the so-called rearrange lemma. It's most easily to be understood in finite dimensional.
For example, you have a cyclic group [tex] G = \{e,a,a^2\}[/tex] with [tex]a^3 = e[/tex]. Let [tex]g\in G[/tex], for example, say [tex] g = a[/tex], then [tex]g\{e,a,a^2\} = \{a,a^2,e\}[/tex], meaning, [tex]g[/tex] operates on the all group element would reproduce all group elements. Hence, if we consider a summation over the group elements, say [tex]\sum_{g\in G}f(g'g)[/tex] where [tex]f[/tex] is a function of the group element. By the rearrangement lemma, we may safely rewrite the summation as [tex] \sum_g f(g) [/tex]

In the continuous group case, the rearrangement lemma is somewhat more involved. Since we have infinitely many ways to parametrize a Lie group, so we have to be more careful. We define that, a parametrisation [tex] g(\xi) [/tex], together with a weight function [tex] \rho_g(\xi) [/tex] such that the following equation holds
[tex] \int dg f(g) = \int dg f(s^{-1}g) [/tex]
where [tex]s\in G[/tex] and [tex] dg = \rho_g(\xi)d\xi [/tex] is called to provide an invariant measure. And one can prove that the weight function for SO(3) group is just 1. (It happens to be the Jacobian factor).
 
  • #5
ismaili said:
We define that, a parametrisation [tex] g(\xi) [/tex], together with a weight function [tex] \rho_g(\xi) [/tex] such that the following equation holds
[tex] \int dg f(g) = \int dg f(s^{-1}g) [/tex]
where [tex]s\in G[/tex] and [tex] dg = \rho_g(\xi)d\xi [/tex] is called to provide an invariant measure.
Is this the Haar measure?
 
  • #6
looks like it, if you nomalize it to 1 when you integrate 1 over the whole manifold (group).
 
  • #7
Thanks everyone for the help. I was forgetting about the fact that the determinant of the rotation matrix was 1... oops. Thanks for the further insight ismaili that is very interesting and I will have to look further into your comment.
 

What is QFT?

QFT stands for quantum field theory. It is a theoretical framework that combines quantum mechanics and special relativity to describe the behavior of particles in a field.

What are invariant measures in QFT?

Invariant measures are mathematical tools used in QFT to describe the symmetries of a system. They are used to determine the probabilities of different outcomes in a system that is invariant under certain transformations.

What is rotational invariance in QFT?

Rotational invariance is a type of symmetry in QFT where the laws of physics remain the same under rotations of space. This means that the results of experiments should not depend on the orientation of the coordinate system used to describe the system.

How does QFT explain rotational invariance?

QFT explains rotational invariance through the use of mathematical tools such as Noether's theorem and gauge symmetries. These tools allow for the description of rotational invariance in terms of conserved quantities and transformations that leave the system unchanged.

What are some practical applications of QFT?

QFT has many practical applications, including particle physics, condensed matter physics, and cosmology. It is used to study the behavior of subatomic particles, understand the properties of materials, and investigate the origins and evolution of the universe.

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