The statement that is not correct among the given options is:

"Transverse wave can be polarized while longitudinal wave cannot be polarized."

Transverse waves are the waves in which the vibration of the particles is perpendicular to the direction of wave propagation. These waves can be polarized, which means that the vibrations of the particles can be restricted to a particular plane. For example, light waves are transverse waves and can be polarized.

On the other hand, longitudinal waves are the waves in which the vibration of the particles is parallel to the direction of wave propagation. These waves can also be polarized, and the vibrations can be restricted to a particular plane. For example, sound waves are longitudinal waves and can be polarized.

Therefore, the correct statement is "Longitudinal wave can be polarized while transverse wave cannot be polarized," and the given statement is incorrect.

The other statements are correct:

In progressive waves, the amplitude may be constant, and neighboring points are out of phase with each other. This is because the wave is traveling through the medium, and the particles at different points have different phases.

In air or other gases, a progressive antinode occurs at a displacement node, and a progressive node occurs at a displacement antinode. This is because the air particles move in the opposite direction to the wave motion at the displacement node, and they move in the same direction at the displacement antinode.

In conclusion, the correct statement is that "Longitudinal wave can be polarized while transverse wave cannot be polarized," and the other statements are correct.