What is "momentum density" and why it important to QFT?

Question

I am reading Quantum Field Theory for the Gifted Amateur. On page 98, they provide a summary of a basic canonical quantization procedure:

• Step I: Write down a classical Lagrangian density in terms of the field. This is the creative part because there are lots of possible Lagrangians. After this step, everything else is automatic.
• Step II: Calculate the momentum density and work out the Hamiltonian density in terms of fields.
• Step III: Now treat the fields and momentum density as operators. Impose commutation relations on them to make them quantum mechanical.
• Step IV: Expand the field in terms of creation/annihilation operators. This will allow us to use occupation numbers and stay sane.
• Step V: That's it. Congratulations, you are now the proud owner of a working quantum field theory, provided you remember the normal ordering interpretation.

I don't understand what momentum density is or why it comes up at this point in the quantization process. If by momentum, they mean like the operator $\hat{p}$, what about the position operator $\hat{x}$? Why isn't there a position density operator needed too? Everything else in the procedure makes sense to me except Step II. I assume Hamiltonian density is the Hamiltonian counterpart to Lagrangian density.

Can someone explain what momentum density is and why it's needed at this step in the procedure?

Answer 1

Comments to the question (v2):

1. A field $\phi^{\alpha}:[t_i,t_f]\times \mathbb{R}^3\to \mathbb{R}$ is the field-theoretic version of a (generalized) position variable $q^i:[t_i,t_f]\to \mathbb{R}$ in point mechanics. Note that the physical position space $\mathbb{R}^3$ typically plays very different roles in field theory and in point mechanics.$^1$

2. Momentum density $\pi_{\alpha}$ is the natural field-theoretic generalization of momentum variable $p_i$ from point mechanics. The (Lagrangian) momentum densities are $$\tag{1}\pi_{\alpha}~:=~\frac{\partial{\cal L}}{\partial\dot{\phi}^{\alpha}}$$ in analogy with $$ \tag{2}p_i~:=~\frac{\partial L}{\partial\dot{q}^i}$$ in point mechanics.

3. Note that there is another notion of momentum $P^i=T^{0i}$ coming from the stress-energy-momentum tensor $T^{\mu\nu}$, cf. e.g. this Phys.SE post.

4. One should perform a Legendre transformation $$\tag{3}\dot{\phi}^{\alpha} \quad\longleftrightarrow\quad \pi_{\alpha}$$ to get to the Hamiltonian formulation. Note in particular that (Hamiltonian) momentum densities $\pi_{\alpha}$ are independent variables.

5. The Hamiltonian formulation is needed$^2$ in order to impose the canonical commutation relations (CCRs) necessary for quantization.

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$^1$ The notion of spacetime, position and field can more generally be defined with the help of differential geometry and the notion of a manifold.

$^2$ Here we only discuss the traditional approach. For manifestly covariant Hamiltonian formulation, see also e.g. this and this Phys.SE posts.