In other words, it's reasonnable that every component u of such a wacuum statisfy
the wave equation:
| 1 | |
¶ 2 u |
= |
¶ 2 u |
+ |
¶ 2 u |
+ |
¶ 2 u |
|
 |
|
|
|
|
| c 2 |
¶ t 2 |
¶ x 2 |
¶ y 2 |
¶ z 2 |
| |
| | = | DU |
[D is the Laplacian operator] |
Any covariant derivative of a combination of such components is a linear combination of components and
their derivatives so that if you apply it twice, the above equation is retrieved.
This imposes a definite internal symmetry among components. Conversely, only those
operators which respect the wave equation are called covariant.
A single component forms what's called a scalar field (spin 0).
A 4-component field is either a vector field (spin-1 like the electromagetic field)
or a spinor field (spin ½ as used in the dirac equation for the electron).
The leeway in building such things is provided by 4-dimensial dirrerential operator that
vanish (in 3 dimensions, the divergence of a rotational and the rotational of a gradient both vanish).
Empty Space is not Empty (4:45)
by Derek Muller (Veritasium, 2013-04-30).
Zero-Point Energy Demystified (10:08)
by Matt O'Dowd (2017-11-08).
