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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?

Qmechanic
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Jason
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  • Related, and also. – Cosmas Zachos May 30 '19 at 16:45
  • Linked. – Cosmas Zachos May 30 '19 at 16:48
  • yes it does relate but those are not unitary operators so i cant make a unitary transformation out of them – Jason May 31 '19 at 08:13
  • @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? – Jason May 31 '19 at 13:14
  • 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. – Cosmas Zachos May 31 '19 at 15:02
  • The very question answered by my answer you are linking above has the correct expression of Schwartz's homework problem. Can you show your own work so one knows what it is you are conflicted about? I believe the unitarity stricture is a bluff and completely suprefluous, but you must convince the reader of its meaning first. – Cosmas Zachos May 31 '19 at 15:09
  • @CosmasZachos for the time being apart from unitarity the problem which i facced is that I can construct $\langle \phi | 0 \rangle$ but I cant find $|n \rangle$ in temrs of vacum state when we talk about fields so in result i cant calculate t $\langle \phi | n \rangle$ – Jason May 31 '19 at 16:04
  • @CosmasZachos If only i knew how to expand $|n \rangle$ in terms of $|0 \rangle$ 's in fock space correspondig to klein gordon free field that would be a great help – Jason May 31 '19 at 16:12
  • Well, you do know $\langle x |n\rangle$ in terms of Hermite polynomials, as in the Sakurai & Napolitano, QM, (2.3.21) book, so you need to multiplex by an infinity of oscillators constituting the field--that is just a transcription: the field is a linear expression of oscillators. – Cosmas Zachos May 31 '19 at 16:28
  • yes but how does the integral show up? and the fact that thers no square root of 2 multiply to $\phi$ as in HO case which there was a "square root of 2" multiplication for x , whats the origin of that? – Jason May 31 '19 at 16:34
  • Is this a homework problem? Show your work. – Cosmas Zachos May 31 '19 at 16:58
  • @CosmasZachos its not however may i email that to you? – Jason May 31 '19 at 17:03
  • Sorry, I don't have the time... It is a standard homework problem, 14.4.a in Schwartz. – Cosmas Zachos May 31 '19 at 18:31

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