Degree of freedom count in linear gauge invariant PDE systems

Abstract: We consider not necessarily Lagrangian partial linear differential equations (PDE) with constant coefficients. Einstein proposed a definition of the"strength" of such a field theory that defines its degree of freedom (DoF). Einsteinian strength is based on the asymptotic number of free Taylor series coefficients of bounded degree in the general solution of the PDE system. The direct count of DoF in this way is a complex and technically demanding process. Proceeding from Einsteinian strength of equations and making use of commutative algebra tools, we deduce another DoF count recipe which is formulated in terms of orders of the field equations, their gauge symmetries and gauge identities. This DoF count recipe covers the case of reducible gauge field theories, and it is easy to use. We begin with interpreting the matrix of the system as a linear map between polynomial modules. First, proceeding from Einsteinian definition, we derive an explicit formula for DoF as the multiplicity of a certain extension module. Second, we prove another explicit formula for DoF in terms of orders of equations and gauge generators. A notable consequence of this formula is that two Hermitian conjugate systems have identical DoF. Every classical field theory defines the BRST complex which has the natural grading known as the ghost number. We equip this complex with another grading, which we call differential order. This grading is 1 for every space time-derivative, while all thefields, ghosts, and antifields are assigned with this degree in a certain way, depending on their ghost number and the orders of equations and gauge generators. We compute the Euler characteristic of the BRST complex with respect to this new grading. This provides homological interpretation of DoF for linear gauge system as minus the residue at infinity of the logarithmic derivative of the Euler characteristics for its BRST complex.