Functional Integral Representation for Relativistic Schrodinger Operator Coupled to a Scalar Bose Field with $P(\varphi )$ Interaction

In this paper the system of a semi-relativistic particle interacting with a scalar Bose field is investigated. The ultraviolet cutoff condition is imposed on the Bose field. In the main theorem, the functional integral representation of the semi group generated by the total Hamiltonian with $P(\varphi )$ interaction is obtained.