  Polymer quantization was discovered during the construction of Loop Quantum
Cosmology. For the simplest quantum theory of one degree of freedom, the
implications for dynamics were studied for the harmonic oscillator as well as
some other potentials. For more degrees of freedom, the possibility of
continuous, kinematic symmetries arises. While these are realised on the
Hilbert space of polymer quantum mechanics, their infinitesimal versions are
not supported. For an invariant Hamiltonian, these symmetry realizations imply
infinite degeneracy suggesting that the symmetry should be spontaneously or
explicitly broken. The estimation of symmetry violations in some cases have
been analysed before. Here we explore the alternative of shifting the arena to
the distributional states. We discuss both the polymer quantum mechanics case
as well as polymer quantized scalar field.
