Another one on Lorentz Invariance

In summary, the author argues that if <m|A^0(t,0)|n>=B by Lorentz invariance, then <m|A^mu(t,0)|n>=(B/p^0)*p^mu. They ask for clarification on the circumstances under which this holds and how it is derived. They also question why p^mu is the chosen momentum 4-vector and whether this relation applies to all operators forming a 4-vector. The author later clarifies that A^mu is a conserved current and asks for an explanation of the argument that <0|A^mu(t,0)|p> = (B/E) * p^mu.
  • #1
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I recently read an author making the following argument in QFT:
if <m|A^0(t,0)|n>=B then <m|A^mu(t,0)|n>=(B/p^0)*p^mu by Lorentz invariance. Can anybody tell me under which circumstances this holds and how it comes about? I understand that <m|A^mu(t,0)|n> had to transform as a 4-vector but why should it be the momentum 4-vector?
 
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  • #2
Who's [itex] A^{\mu} [/itex] ? What does it stand for ?
 
  • #3
That is part of my question: Does the relation hold for arbitrary operators which form a 4-vector? I think not, but maybe I am wrong. The context in which I saw the argument, A^mu was a conserved current but otherwise arbitrary. Does that help?
 
  • #4
Ok I think I understand the argument partly now and must admit that the question did not really make sense the way I phrased it earlier. What I should have asked is: Suppose we have <0|A^0(t,0)|p> = B where <0| is the vacuum |p> is an eigenstate of momentum p and A^0(t,0) is the zero component of an operator forming a 4-vector evaluated at x_i=0. Given this does it then follow that <0|A^mu(t,0)|p> = (B/E) * p^mu ? I think it should and I think the argument goes something like this: the LHS only really depends on p^mu and therefore the only 4-vector that can possibly appear on the RHS is the momentum 4-vector p^mu. However this argument is not totally clear to me as A^mu itself appears on the LHS and so why should the RHS not be something proportional to that? Anyway if you can explain this argument to me pleae let me know.
 

1. What is Lorentz Invariance?

Lorentz Invariance is a fundamental principle in physics that states that the laws of nature remain the same for all observers who are moving at a constant velocity. This means that the laws of physics do not change under transformations between different inertial frames of reference.

2. Why is Lorentz Invariance important?

Lorentz Invariance is important because it is a fundamental principle that underlies many key theories in physics, such as special relativity and quantum field theory. It allows for the consistency and predictability of physical phenomena across different frames of reference.

3. How is Lorentz Invariance related to Einstein's theory of relativity?

Einstein's theory of special relativity is based on the principle of Lorentz Invariance. In his theory, he proposed that the laws of physics should be the same for all observers in inertial frames of reference, and this led to the development of the famous equation E=mc².

4. Can Lorentz Invariance be violated?

There have been theories and experiments that suggest that Lorentz Invariance may be violated at very small scales, such as in quantum gravity or high-energy particle interactions. However, these violations have not been conclusively proven and Lorentz Invariance remains a key principle in modern physics.

5. How is Lorentz Invariance tested?

Lorentz Invariance can be tested through experiments that measure the properties of particles, such as their mass, energy, and momentum, in different frames of reference. If the results are consistent, then Lorentz Invariance is upheld. There are also ongoing efforts to test Lorentz Invariance at extremely small scales using advanced technologies and experiments, such as with gravitational wave detectors.

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