In this thesis we first study the quantization of a massive open bosonic string in the presence of a background B-field and then survey associated Casimir effect. Initially, using In the next step, we use these oscillation frequencies to find zero-point energy of bosonic string; obviously it would be infinite , in fact, as a consequence of definition of energy in quantum field theory, as we know. To find the convergent part of zero-point energy (that is also called the Casimir energy ), we use a powerful mathematical technique namely the Abel-plana formula. We see the Casimir energy of our massive bosonic string is a function of background B-field as well as mass and length of the string; in fact, B-field plays a role only in the constant part of Casimir energy that can be neglected. Casimir force is simply achieved deriving Casimir energy to the length of string; we will see the force does not depend on B-field, as is expected. Then we plot the casimir force to the length, for several value of mass and show the force has a logical behavior at large value of mass or length of the string. Finally, employing a useful expansion namely the heat kernel expansion we find divergent parts of the zero-point energy and will have a glance at the connection between these divergences and the theme of renormalization and provide some suggestions for more studies. Keywords : Dirac constraints, Reduced phase space, Symplectic approach, Casimir force, Heat kernel expansion.
