0
$\begingroup$

In quantum mechanics, we know that:

  1. The global phase of a wavefunction does not have any physical effect, but the relative difference in phase ($\Delta \phi$) among different components of the same wavefunction has physically observable effects, such as the destructive and constructive interference in the double-slit experiment.

  2. Local $U(1)$ gauge symmetry allows us to chose any value of the phase at any point we want in the wavefunction.

Then how can a $U(1)$-invariant wavefunction exhibit any observable effect of $\Delta \phi$? If any two points can have any value for their phase, then their $\Delta \phi$ will be any number, thus the double-slit experiment will have to simultaneously show a destructive interference, a fully constructive interference and anything in between. But this is not what happens.

What is the correct way to have statement 1 fit with statement 2, so that a $U(1)$-invariant wavefunction can still exhibit the observable effects of $\Delta \phi$?

Improve this question
$\endgroup$
9
  • 1
    $\begingroup$ The gauge transformation you are referring to applies to the quantum field, not the wavefunction. The field and the wavefunction are entirely different objects. $\endgroup$ Commented Mar 14 at 13:59
  • $\begingroup$ @J.Delaney PBS spacetime did an entire episode on the U(1)-invariant wavefunction: youtube.com/watch?v=V5kgruUjVBs $\endgroup$ Commented Mar 14 at 14:03
  • 2
    $\begingroup$ Popular science videos such as this one can indeed be misleading since they need to oversimplify the concepts. You shouldn't expect rigorous accuracy, especially from a 10-min. video. $\endgroup$ Commented Mar 14 at 14:45
  • $\begingroup$ 1. Gives you self interference and wavy detection patterns. 2. Reminds you that the wave function has a labile local phase at any point comporting with the electron charge. What is your strange point, and how do you absurdly connect the two? $\endgroup$ Commented Mar 14 at 15:25
  • $\begingroup$ @Cosmas Zachos the self interference is caused by ΔΦ $\endgroup$ Commented Mar 14 at 15:28

1 Answer 1

Reset to default
0
$\begingroup$

It is problematic to rewrite/read your question into something coherent and answerable. I'll try below to stress a few points which might help you get it focussed.

Your 1. covers nonrelativistic wave functions, which are not fully gauge covariant for a given time point unlike the relativistic wave functions/fields of your 2. So you are really comparing chopped apples and oranges. As you point out in 1., NR wave functions propagate and interfere in the complex plane, and develop $\Delta \phi$s internally, yielding calculable variations on the post-double-slit screen.

By sharp contrast, (2.), for relativistic wave functions/fields, U(1) covariance permits gauge-fixing for all such wavefunctions at a spacetime point, so at a given fixed gauge, different wave functions may still interfere into different complex composite wavefunctions. It is not true that phases are irrelevant and out of the calculation for any spacetime point, as in your final comment assertion,

If (in a U(1)invariant wavefunction) we are free to set any two points to have any value for their phase, then their Δ𝜙 can be any number

so we might as well be dealing with real quantities!?

There are unstated backgrounds in your question which are hardly possible to parse out... A decent mainstream text beyond verbose videos might help.

By the way, an issue of language. Relativistic wave functions are U(1) gauge covariant, if they are representing a charged particle. (Invariant signifies their formally not depending on the phase).

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.