- #1

Haorong Wu

- 413

- 89

Next, we will utilize the 3+1 decomposition on the action to separate the time variable ##t##, yielding $$S=\int dt \int dx^3 L(t, x, y, z, f).$$

My problem is that could we identify ##\int dx^3 L(t, x, y, z, f)## as a new Lagrangian ##L'(t, x, y, z, f)## such that only ##t## is identified as the variable and ##x##, ##y##, ##z##, ##f## are functions on ##t##. If this is valid, then we can write the canonical momentum conjugate to ##f## as ##\pi_f=\frac {\partial L'}{\partial (\partial_t f)}##.

cf. A master equation for gravitational decoherence: probing the textures of spacetime section 2.1 to 2.2.

Many thanks.