Can a primary constraint contain spatial derivative of the field?

Question

I am currently studying the Hamiltonian formulation of GR and I have problems understanding this definition of primary constraint.

In the textbooks, primary constraint occurs when a momentum conjugate by definition is not invertible for the corresponding velocity, but primary constraint reads $$\phi(p,q)=0,$$ which does not contain spatial derivative of $q$, namely $\partial_iq$.

My question is then if the conjugate momentum by definition gives you a formula as $$\phi(p,q,\partial_iq)=0$$ can we call it a primary constraint?

Answer 1

Yes, in field theory the primary constraints can contain spatial$^1$ derivatives of the canonical fields.

One example is when Legendre transforming the Nambu-Goto (NG) string action, see e.g. this Phys.SE post.

One way to heuristically justify this is to view field theory as point mechanics where the spatial coordinates serve as continuous indices of the canonical variables. The spatial derivatives can be understood as spatial differences if we discretize spatial (but not temporal) directions.

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$^1$ OP seems already aware that the presence of temporal derivatives of the canonical fields in a primary constraint would render the (singular) Legendre transformation incomplete.