We can construct the unitary transformation for change of basis from $x$ to number operator $n$ in harmonic oscillator by using $a|0\rangle=0$ and then multiply $\langle x|$ to the both side and calculating $\langle x | 0 \rangle$ and then find $\langle x | p \rangle$ and finally use the completeness relation to find the transformation.

Now I would like to somehow find the transformation for a free, real Klein-Gordon Lagrangian with $[\phi(x)]$ being the field operator. How can we construct a unitary transformation from occupation number $|n \rangle$ basis to $|\phi \rangle$ basis?

  • $\begingroup$ Related, and also. $\endgroup$ Commented May 30, 2019 at 16:45
  • $\begingroup$ Linked. $\endgroup$ Commented May 30, 2019 at 16:48
  • $\begingroup$ yes it does relate but those are not unitary operators so i cant make a unitary transformation out of them $\endgroup$
    – Jason
    Commented May 31, 2019 at 8:13
  • $\begingroup$ @CosmasZachos I studied all of those you've mentioned, however specifically talking i couldn't follow the generalization you've mentioned at the end of this to find the QFT version of it. could you please be more specific about how to achieve such formula? $\endgroup$
    – Jason
    Commented May 31, 2019 at 13:14
  • $\begingroup$ First, you must appreciate that the standard transition from the Fock vacuum to $|x\rangle$ is not and cannot be made unitary. Do you understand why? As indicated here, you are solving problem 14.4.a in Schwartz's QFT standard book. $\endgroup$ Commented May 31, 2019 at 15:02


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