Let \ mathcal { L } _ \ mathrm { M } represent the Lagrangian density of matter and \ mathcal { L } _ \ mathrm { G } represent the Lagrangian density of the gravitational field.
32.
To use the Dyson series to calculate anything, one needs more than a gauge-invariant Lagrangian density; one also needs the quantization and gauge fixing prescriptions that enter into the Feynman rules of the theory.
33.
For example, in the case of Newtonian gravity, the Lagrangian density integrated over spacetime gives you an equation which, if solved, would yield \ Phi ( \ mathbf { x }, t ).
34.
Dirac s two preceding remarks suggest that we should start searching for a realistic regularization in the case of quantum electrodynamics ( QED ) in the four-dimensional Minkowski spacetime, starting with the original QED Lagrangian density.
35.
Instead of " fixing " the gauge to a particular " constraint surface " in configuration space, one can break the gauge freedom with an additional, non-gauge-invariant term added to the Lagrangian density.
36.
The fundamental Lagrangian density of one's theory is presumed to lie in the subspace " Pl " 0 of polynomials which are real-valued and invariant under any unbroken non-gauge symmetry groups.
37.
Noether's theorem begins with the assumption that a specific transformation of the coordinates and field variables does not change the action, which is defined as the integral of the Lagrangian density over the given region of spacetime.
38.
In Lagrangian field theory, the Lagrangian as a function of generalized coordinates is replaced by a Lagrangian density, a function of the fields in the system and their derivatives, and possibly the space and time coordinates themselves.
39.
Using a Taylor series expansion for the second integral around the mean free-surface elevation z = 0, and only retaining quadratic terms in \ Phi and \ zeta, the Lagrangian density L _ 0 for linear wave motion becomes
40.
To obtain the field equations the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of the 4-potential " A ", and its this potential which enter the Euler-Lagrange equations.
