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If we write some quantum field in a form using creation and annihilation operators we are, in a way, doing a Fourier series with annihilation and creation operators being coefficients. So, if they are a coefficients shouldn't they have some structure? I mean if I do a Fourier series of some function I expect that coefficients also look like functions in k-space. So, what do these operators have besides commutation relations to define the structure of the field?

SuperCiocia
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Žarko Tomičić
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  • They not really the coefficients though. If you write a field in terms of these operators the coefficients are the first quantized wave functions. – Jan Bos Oct 03 '19 at 15:42
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    2nd quantization is after all is said and done, about promoting Fourier/ Sturm-Liouville coefficients in some complete basis, to fully-fledged operators – lurscher Oct 03 '19 at 16:39
  • But what is the difference between one scalar field and another if we write them both in terms of these operators? I dont see how would we differentiate between fields. Jan Bos, can you elaborate on your comment? – Žarko Tomičić Oct 03 '19 at 18:08
  • Perhaps this question might help. Every field φ(x) acting on the vacuum is a bit like a wavepacket, with a corresponding wavefunction. You'd normally consider just plane waves for the free field, but sticking a k-dependent profile coefficient in the k-integral will give you a different field with a different wavepacket profile/wavefunction. – Cosmas Zachos Aug 05 '20 at 20:06

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