The Euler Lagrange equations for this Lagrangian density L _ 0 are, with \ xi ( x, y, t ) representing either \ varphi or \ zeta:
22.
This will certainly be true if the Lagrangian density \ mathcal { L } is left invariant, but it will also be true if the Lagrangian changes by a divergence,
23.
The three constants of the theory, G, \ mu and \ omega, are promoted to scalar fields by introducing associated kinetic and potential terms in the Lagrangian density:
24.
The Lagrangian density is the wedge product of that of ordinary Chern Simons theory with the holomorphic ( 3, 0 )-form, which exists in the Calabi-Yau case.
25.
Let us refer to such a regularization as " the minimal realistic regularization ", and start searching for the corresponding, modified free-field parts of the QED Lagrangian density.
26.
The evolution operator is obtained in the interaction picture where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:
27.
This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning " n & # 770; " = constant at infinity.
28.
Since linear combinations of these quantities are also Lorentz invariant, this leads naturally to the Lagrangian density for the Dirac field by the requirement that the Euler Lagrange equation of the system recover the Dirac equation.
29.
Characteristic of field theories, the dynamics of the field strength are summarized by a suitable Lagrangian density and substitution into the Euler Lagrange equation ( for fields ) obtains the equation of motion for the field.
30.
Let \ mathcal { L } _ \ mathrm { M } represent the Lagrangian density of matter and \ mathcal { L } _ \ mathrm { G } represent the Lagrangian density of the gravitational field.
