  Semiclassical trace formulas are examined using phase space path integrals.
Our main concern in this paper is the Maslov index of the periodic orbit, which
seems not fully understood in previous works. We show that the calculation of
the Maslov index is reduced to a classification of connections on a vector
bundle over $S^{1}$ with structure group $Sp(2n,R)$. We derive a formula for
the index of the n-repetition, and show that a Bohr-Sommerfeld type
quantization condition including quadratic fluctuation around the orbit is
derived using this formula.
