  We investigate the Hamiltonian formulation of $f(T)$ gravity and find that
there are five degrees of freedom. The six first class constraints
corresponding to the local Lorentz transformation in Teleparallel gravity
become second class constraints in $f(T)$ gravity, which leads to the
appearance of three extra degrees of freedom and the violation of the local
Lorentz invariance in $f(T)$ gravity. In general, there are $D-1$ extra degrees
of freedom for $f(T)$ gravity in $D$ dimensions, and this implies that the
extra degrees of freedom correspond to one massive vector field or one massless
vector field with one scalar field.
