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I've seem in several books and lecture notes the quantization of the free KG field, and perhaps because I'm a kind of person that feels umconfortable with "hand waving" constructions, I still feel the need for a somwhat a little more rigorous approach which I found nowhere.

To summarize, the way this is done everywhere is:

  1. First take the Fourier transform of the KG equation $\Box\phi = 0$ and realize that in Fourier space we get the time evolution of uncountably many harmonic oscillators, that is, the equation $\partial_t \hat{\phi}(t,\mathbf{p})+\omega_p^2 \hat{\phi}(t,\mathbf{p})=0$ with $\mathbf{p}$ fixed.
  2. From this one directly states that the quantum field is $$\phi(x,t)=\int \dfrac{d^3 p}{(2\pi)^3}(a_p e^{-i p_\mu x^\mu}+a_p^\dagger e^{ip_\mu x^\mu})$$

    importantly here no one has ever defined the operators $a_p$. At this point one claims that this is just a consequence of the field being equivalent to uncountably many decoupled oscillators and claims one is just using the results from QM of the harmonic oscillator, but not much of a precise construction is made.

  3. After that derives that the commutation relations for $\phi,\pi$ are equivalent to the commutation relations $[a_p,a_q^\dagger]=i\delta(p-q)$ and $[a_p,a_q]=[a_p^\dagger,a_q^\dagger]=0$ working formaly. Remember that neither $\phi(x)$, nor $a_p$, nor the space where they act was ever defined.

  4. One just claims that it is obvious that the space where these operators are acting is a Fock space (though nothing is said about the fact that the usual Fock space is built over countably many Hilbert spaces, while here we have uncountably mane oscillators in the Classical Picture as explained in (1)). Another issue that is never tackled is that the Fock space requires tacking the symmetric or antisymmetric tensor product, and it is not made clear which of them and how this was derived.

  5. To make tings worse, with having never defined $a_p$ one just claims that there is one state $|0\rangle$ such that $a_p |0\rangle = 0$ and that for each $p$ fixed the results for the ladder operators from the Harmonic Oscilator can be carried over and adapted.

I mean, I understand that rigor is complicated in QFT. I've read a little bit about that. But this is another story: here things are coming out of thin air!

As an example, I don't have a problem with Dirac's formalism in QM, even though making it rigorous is a lot complicated, but I'm fine with it because the assumptions are made clear in most QM books and the derivations are almost always done without anything coming out of thin air. The derivation of the spectrum and eigenstates of the SHO for example is carried out in detail and in a prceise fashion in many books.

Now thiz quantization procedure has a lot of gaps that are not explained. The relation between the field, the harmonic oscilators and the Fock space is used all the time, but never made precise. One just claims things without much explanation.

What is really going on here? How can all this construction and these relations be made precise? What can be done to at least make the assumptions clear and the derivation also clear? How can we construct all this in a more comprehensive manner?

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  • $\begingroup$ Try this book , it's the QFT analogue of Dirac's masterpiece "The Principle of Quantum Mechanics". $\endgroup$ – Count Iblis Mar 29 '17 at 3:25
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    $\begingroup$ Do you remember how the simple harmonic oscillator is quantized? $\endgroup$ – Prahar Mar 29 '17 at 3:30
  • $\begingroup$ Yes @Prahar, however I can't see one precise connection. My point is: when quantizing the SHO in QM, we assume that we have one state space $\mathcal{E}$ with operators $X,P$ satisfying $[X,P]=i\hbar$ together with a representation $X|x\rangle=x|x\rangle$ st. we can represent kets $|\psi\rangle$ by functions $\psi(x)=\langle x |\psi\rangle$. These structures are assumed given (this can be seem as one implicit usage of the Stone-von Neumann theorem I believe). Then in terms of these we define $a$ and $a^\dagger$ to factorize $H$. $\endgroup$ – user1620696 Mar 29 '17 at 4:07
  • $\begingroup$ We then prove lots of properties of these operators that allows to find the eigenvalues and eigenstates of $H$. But we can only do this because we have defined $a$ and $a^\dagger$ in a space assumed to be given in terms of operators we already have. In other words there is a clear starting point from where everything is derived and that is $\mathcal{E}$, the operators $X,P$ and the representation $|x\rangle$. I can't see this clear structure in quantizing the free scalar field. $\endgroup$ – user1620696 Mar 29 '17 at 4:14
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    $\begingroup$ If you don't like handwaving, read Weinberg's books;) $\endgroup$ – Peter Kravchuk Mar 29 '17 at 7:36
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You are asking for a precise, i.e. mathematically rigorous, construction of free scalar QFT and yet you seem to have only read references by physicists rather than mathematical physicists who have sorted this out a long time ago. The reference by 't Hooft suggested in the comments will not help you much in this regard. You can find a precise treatment of canonical quatization of the free scalar Boson field in for instance:

  1. Volume 2 of "Methods of Modern Mathematical Physics" by Reed and Simon, see in particular Section X.7 (1975 edition).
  2. The book "Quantum Physics, A functional Integral Point of View" by Glimm and Jaffe, see in particular Chapter 6 (1987 edition).
  3. "Quantum Mechanics and Quantum Field Theory, a Mathematical primer" by Dimock, see in particular Section 5.4 and Chapter 8 (2011).
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