mathematical physics - Can auxiliary fields be thought of as Lagrange multipliers?

In the BRST formalism of gauge theories, the Lautrup-Nakanishi field $B^a(x)$ appears as an auxiliary variable $$\mathcal{L}_\text{BRST}=-\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}+\frac{1}{2}\xi B^a B^a + B^a\partial_\mu A^{a\,\mu}+\partial_\mu\bar\eta^a(D^\mu\eta)^a,$$ and in the superfield formalism of SUSY, the field $F(x)$ also appears as an auxiliary variable: $$\mathcal{L}_\text{SUSY}=\partial_\mu \phi\partial^\mu\phi+i\bar\psi^\dagger\bar\sigma^\mu\partial_\mu\psi+F^*F+\ldots\,.$$

It is very tempting to view $B^a(x)$ and $F$ as Lagrange multipliers since their equations of motion leads to constraints. But, these variables do not enter into the Lagrangian linearly, like a conventional Lagrange multiplier. Rather, they enter into the Lagrangian quadratically.

However, in Kugo and Ojima's paper Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theores (1978), they refer to the $B^a(x)$ fields as the 'Lagrange Multiplier' fields (p.1882).

So my question is: Can these auxiliary fields be viewed as Lagrange multipliers? and in what ways do they behave differently/similar to the conventional Lagrange multipliers that enter into the function linearly?

Answer

By definition, Lagrange multipliers are only coefficients that enter the extremized quantity (action) etc. linearly – and that multiply constraints. In some exceptional cases, an auxiliary field could enter in this way. However, they typically appear in a more complicated way and bilinear terms in the auxiliary fields are a rule rather than an exception. So strictly speaking, they're not Lagrange multipliers. But they are very similar. If no derivatives of these objects appear in the action, they're also "non-dynamical" (not involving time derivatives) and the variation with respect to them implies "non-dynamical" i.e. algebraic equations of motion.

Note that in the normal treatment of extremization, we introduce Lagrange multipliers because we want to extremize the quantity given the assumption that another quantity or other quantities are kept fixed. "Kept fixed" is translated as "conservation laws" into the physics jargon. However, in physics, we rarely consider conserved quantities that are conserved because the conservation law is explicitly written down as an independent constraint. Instead, in physics we usually discover conservation laws nontrivially – the conserved quantity has to be determined by a somewhat non-trivial procedure due to Emmy Noether out of a symmetry. In almost all physical theories, conservation laws are non-trivial consequences of some other, "more elementary" equations of physics.