TP fractures of L2-3 are somewhat common. They are typically a result of blunt impact to the back of the lumbar spine but these fractures can also result from avulsion (pulling off) from contraction of the psoas muscle. TP fractures typically do not heal with bone to bone union but with fibrous union. Normally, after 6-8 weeks, these fractures are not painful.
There is most likely another source of pain from your lumbar spine if you have chronic pain in your lower back.
These fractures are not related to carpel tunnel syndrome. If however you fell onto your hands in the course of the impact, this could cause carpel tunnel syndrome.
Dr. Corenman