# First quantization version of quantum field theory

In quantum mechanics, we have the word second quantization for identical particles. However, when dealing with localized states, first quantization version of quantum mechanics is also very convenient. I wonder if there is a first quantization version of QFT which is handy in dealing with, for example, two particles that are far away from each other.

Well, I think most answers missed my point, but I do learned a lot from your answers. Now I put it in another way. Is there a way to describe two identical but localized particles without symmetrize the states like what we do in basic QM

• The word second quantization is an ugly misnomer that needs to die. All it is is performing canonical quantization of a classical field theory as opposed to a classical non-field theory, so there cannot be a "first quantization" of a QFT, since "first quantization" yields QM, not QFT. – ACuriousMind Jul 5 '14 at 3:01
• In peskin's introduction to QFT, page 103, he decides to use a notation as $|k_A k_B>$. If there is no way to distinguish A and B, I cannot imagine what state is the notation denotes in QFT. – Chong Wang Jul 5 '14 at 3:19
• @ACuriousMind : you really have a particular dislike for the name...it is just a name, and if used properly I think it is quite convenient ("the second quantization of $\sqrt{k^2+m^2}$" is, in my opinion, more understandable than "the canonical quantization of the Hamiltonian for a free Klein-Gordon field"). The fact here is that the terms first and second quantization are not used in the most proper way. – yuggib Jul 5 '14 at 9:20
• See this E. Witten's lecture about first vs second quantization in QFT and String: youtube.com/watch?v=H0jLD0PphTY – Nogueira Jul 18 '17 at 19:34

I will try an answer, even if I'm not sure to understand fully what the question means. Using the second quantization formalism in QM is actually sometimes useful, but it is in some sense just a trick. Suppose you have $N$ identical $1$-dimensional bosons interacting by means of a two-body potential. The Hilbert space of the theory is $L^2_s(\mathbb{R}^N)$, the space of square integrable symmetric functions. The Hamiltonian is usually something like this: $$H_N=-\sum_{j=1}^N \Delta_j +\frac{1}{N}\sum_{i<j} V(x_i-x_j)$$ where $V$ is symmetric and real. You can see this operator as the restriction to the $N$-particle sector of an operator in a symmetric Fock space (second quantization formalism). The Fock space is $\Gamma_s(L^2)=\bigoplus_{n=0}^\infty L^2_s(\mathbb{R}^N)$ (with the convention $L^2_s(\mathbb{R}^0)=\mathbb{C}$) and the operator is, using the usual annihilation and creation operators, $$H=\int_{\mathbb{R}}(\nabla a)^*(x)\nabla a(x)dx +\frac{1}{2}\int_{\mathbb{R}^2}V(x-y)a^*(x)a^*(y)a(x)a(y)dxdy\; .$$ It is easy to see that $H\rvert_{L^2_s(\mathbb{R}^N)}=H_N$.

In many body theory you have a state with fixed number of particles, i.e. $\psi_N\in L^2_s(\mathbb{R}^N)$, and the evolution $U_N(t)$ generated by $H_N$ (equivalently $H$) maps $L^2_s(\mathbb{R}^N)$ on itself (preserves the number of particles). So even if you have these two ways of writing the Hamiltonian of your system, they are completely equivalent. In some sense, you have a direct sum of "theories" that do not interact, each one corresponding to a different number $n$ of particles: not only the Fock space is a direct sum, but also $H=\bigoplus_{n=0}^\infty H_n$; once the number $n=N$ is fixed and it is finite, your system will never leave the subspace $L^2_s(\mathbb{R}^N)$.

In real relativistic QFTs, the situation is radically different. The Hamiltonian does not preserve the number of particles, and has terms with an uneven number of creation and annihilation operators. So it is impossible to give a closed description of the dynamics in the subspace $L^2_s(\mathbb{R}^N)$: even if you start with a state in this space, the evolution will create and destroy particles, and you will end up with a state that spreads in general on the whole Fock space. It is therefore of no use to search for a "first quantization" (i.e. in the subspace $L^2_s(\mathbb{R}^N)$) description of QFTs.

I wonder if there is a first quantization version of QFT which is handy in dealing with, for example, two particles that are far away from each other

In the first-quantization formalism, in a many-body problem with indentical particles, you have to symmetrize (for bosons) or anti-symmetrize (for fermions) the many-body wave function.

However, it is a heavy process and we are more interesting, for instance, in knowing how many particles having some momentum, are present in a state, so the QFT Fock state notation $|n_{k_1}, n_{k_2}...n_{k_n}\rangle$ represents a state with has $n_{\vec k_1}$ particles with momentum $k_1$,$n_{\vec k_2}$ particles with momentum $k_2$, etc...

The notation $|k_A k_B\rangle$ is a shortcut for $|1_{k_A}1_{k_B}\rangle$, that is a state where you have one particle with momentum $k_A$ and one particle with momentum $k_B$.

Note that one is working with momenta, so there is no notion of localization, because a particle with a precise momentum extends across the whole space-time, this is just the consequence of the Heisenberg inequalities. So it does not make sense, for particles having a precise momentum, to speak about "two particles that are far away from each other"

Practically, to compare a theorical result to experiments, you begin with amplitudes for a process with initial and final particles having a precise momentum, then you calculate cross sections from these amplitudes, so you get probabilities that you may compare to those you find from experiments.

The issue may arise due to the unfortunate choice of terminology, 'second quantization.' When one applies canonical quantization to a field theory, e.g. with Lagrangian,

$$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}m^2\phi^2$$

we treat the original theory as a classical field theory. Hence, 'second quantization' is really applying a single quantization procedure, once. The term 'first quantization' describes quantizing a theory which is not a field, such as a classical harmonic oscillator. Hence, there is no 'first quantization' of a field theory.