# Book recommendations for second quantization

I am trying to familiarize myself with the ideas of second quantization. However, the literature that I can find online seems only to outline the tools of this formalism of quantum mechanics.

There is very little description about the origin of this formalism and how it connects to other problems in physics. For example, I know that I have solved the harmonic oscillator problem in QM using ladder operators, but I do not understand why the mathematics of the quantum harmonic oscillator should be the same as the mathematics I use to describe many body systems.

Are there fundamental symmetries? Are all objects in the universe oscillators in some sense? I was hoping that someone could recommend a good text for an introduction to this second quantization formalism. The other problem is that I am an undergraduate student without a knowledge of QFT, thus I seek a text which does not rely on a detailed knowledge of this area.

Moreover, since second quantization (apparently) has a broader relevance than QFT, particularly in condensed matter physics, cold atom theory, and quantum simulation, it would be good to get an introduction that is focused on where undergraduate courses drop off and relevant to the various different fields that use the formalism.

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Piotr Chankowski's lecture notes on second quantization (23 pp) seem to deliver an accessible yet rigorous presentation:

• it starts from the description of $$N$$ distinguishable particles on $$\mathcal{H}^{\otimes N}$$, goes on to the description of $$N$$ bosonic/fermionic undistinguishable particles on the symmetric/alternating subspace $$\mathcal{H}^{\otimes N, \text{sym/alt}}$$, then justifies why an arbitrary number of particles should be described on the $$\bigoplus_{N \geq 0}$$ of those;

• the occupation number basis and the ladder operators are constructed and illustrated on simple examples (compensating for the not always completely clear explanations of the required combinatorial factors); I would give it extra credit for not fixing once and for all a 1-particle ONB, so that change of basis/representation and the creation/annihilation of arbitrary 1-particle states are naturally covered; formulas are written in a unified way for bosons/fermions when possible;

• the lifting of operators, in particular the Hamiltonian, from the 1-particle theory to the multi-particles one is explained, as well as the construction of possible interaction terms;

• finally, it is shown how the formalism can arise from the quantization of coupled harmonic oscillators (in the spirit of phonons).

Although it is part of a larger introduction to QFT, this chapter appears to be effectively standalone (save a couple side notes pointing to subsequent chapters for additional details).

My only gripe with it is in the first few sentences, where the author goes out of his way to insist that second quantization would have nothing to do with quantizing a second time. This is a widely held attitude: although the second-quantization formalism can in fact be obtained by "quantizing" the wave-equation of the first-quantized theory (see this answer), it seems to have been "forgotten", and texts which acknowledge it tend to be much more advanced and/or older (hence a much harder read, as they use old-fashioned notations and concepts).

You may find some of the quantum optics texts dealing with the Second Quantisation of the Electromagnetic Field a gentler introduction, for example:

These will deal of course with the nonrelativistic electromagnetic field. The justification for all this (at least for the nonrelativistic EM field) is not as lofty as any of your suggestions: one simply recognizes that solutions to Maxwell's equations are superpositions of harmonic oscillators (e.g. superpositions of plane waves with time-harmonic dependence), so one replaces each mode of the EM field with a quantum harmonic oscillator. Quantum harmonic oscillators are easy systems to do many body problems for; if we have a many body Hamiltonian comprising non-interacting QHOs:

$$\hat{H} = \hbar \sum_j \omega_j \left(\hat{a}_j^\dagger \hat{a}_j + \frac{1}{2}\right)\quad\quad\quad(1)$$

We can make the QHOs interact in intuitively clear and simple ways: we simply add a term of the form $\hbar\,\kappa_{\ell,m} \left(\hat{a}_\ell^\dagger \hat{a}_m + \hat{a}_m^\dagger \hat{a}_\ell\right)$ to model an interaction between the $\ell^{th}$ and $m^{th}$ oscillator. The term $\hat{a}_\ell^\dagger \hat{a}_m$ pulls one photon out of oscillator $m$ and put it into oscillator $\ell$, and the interaction terms always hang out in Hermitian conjugate pairs so as to keep the Hamiltonian Hermitian (thus the time evolution operator $\exp(i\,\hbar^{-1}\,\hat{H}\,t)$ unitary) - this is exactly the same as for classical harmonic oscillators where the time evolution operator must also be unitary to conserve energy. If we have a general coupled QHO Hamiltonian:

$$\hat{H} = \hbar \sum_j \omega_j \left(\hat{a}_j^\dagger \hat{a}_j + \frac{1}{2}\right) + \hbar \sum_{j > k} \omega_{j, k}\, \kappa_{j,k}\, \left(\hat{a}_j^\dagger \hat{a}_k + \hat{a}_k^\dagger \hat{a}_j\right)\quad\quad\quad(2)$$

then we can do a simple orthogonal transformation and diagonalise it to the form in Eq. (1), just as we can diagonalised a coupled system of classical harmonic oscillators and find the normal modes. Once we have done the analogous thing in the quantum case, we have an equivalent set of noninteracting oscillators.

Is the whole universe made of oscillators? As far as I understand QFT (and, beyond quantum optics, my understanding is patchy), well the universe is thought of in QFT as being made of fields very like the second quantized EM field that interact. We now don't think of them as nonrelativistic QHOs but more abstractly, where only ladder operators and their particles remain.

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I would suggest looking at "Condensed Matter Field Theory" by Altland and Simons, or perhaps even the first chapter of "Quantum Many-particle Systems" by Negele and Orland.

• Maybe I can comment on Altland&Simons: while the book is otherwise exceptionally good I find the start of the second quantisation chapter not very accessible to say the least. Having gone back to look at it for a second time I find some bits of it even circular. The applications part of the chapter is great again though. – Wolpertinger May 9 '17 at 20:19
• I personally found the discussion of second quantization in both of these books incredibly and surprisingly obscure. – DanielSank May 9 '17 at 21:36