What is "momentum density" and why it important to QFT? 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?
 A: Comments to the question (v2):


*

*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.$^1$ 

*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.

*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. 

*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.

*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.
