Casimir-like Effect In Periodic Boundary Conditions

In1948Casimir predicted that, in the electromagnetic vacuum, because of zero-pointquantum fluctuations, there would be an attractive force between two closely neutral parallelconductive plates. This force is called Casimir force. Since it was predicted, the Casimir effecthas played an very important role in various fields of physics, such as Quantum Field Theory,Condensed Matter Physics, Atomic Physics, Physical Biology, Astronomy and Cosmology.In1924,Bose proposed a new theory of photon. He supposed that photon is identicalparticle which is not conservation of number. Therefore, he deduced the Plank blackbodyradiation law, and introduced a statistical method to describe the behavior of photon. In1925,Einstein generalized Bose’s theory to particle whose mass is not zero, and predicted if thetemperature is low enough, all bose particles will condense at the lowest level, this is calledBose-Einstein Condensation (BEC). It is considered as one of the greatest scientificdiscoveries in the21st century, and it has become a global focus in physical area.In the paper, I briefly review both the theories and experiments about the Casimir effectand Bose-Einstein Condensation. Then I review the calculation of the Casimir-like force ofzero-temperature dilute Bose-Einstein Condensation for periodic boundary conditions. At theend of this paper, I calculate the Casimir-like force with another method, and obtain theBogoliubov correction. It not only provides a new method to understand the BEC, but alsohas value for the measurement of Casimir-like force of BEC.