An Analysis On Ward Identity For Multi-Field Inflation

Given a correlation function (or n-point function), can the corresponding nature of space-time be determined ? To answer this question it is required to derive the Ward Identity (WI), analyse the symmetries and arrive at the law of conservation. Modus operandi involves Lie differentiating two-point function considering the symmetry to be non-anomalous. The WI so obtained is shown to form a Lie algebra which determines the nature of space-time. Solving the identity results in a law of conservation, which physically explains the reason for WI to form an algebra and contains in it an equation of motion for four-point function. As a special case, a relation between mass and potential involving the spatial derivatives of four- and five- point function is obtained. Finally, the conservation equation is exploited to get the probability amplitude for the two-point function which shows how correlation functions provide an opportunity to probe the fundamental laws of physics.