A functional theory, in the context of physics, refers to a theoretical framework that is based on functionals. In physics, functionals are mathematical objects that assign a number to a function or a set of functions. They can be thought of as generalizations of functions, where instead of mapping a single input value to an output value, they map entire functions to numbers.
Functional theories are often employed in the study of fields and systems that exhibit continuous variations, such as classical mechanics, quantum mechanics, and quantum field theory. These theories use functionals to describe the behavior of physical quantities, such as the action, energy, or probability, in terms of functions that depend on variables such as space, time, or field configurations.
In classical mechanics, for example, the principle of least action is formulated in terms of a functional called the action functional. It assigns a value to each possible path taken by a particle or a system, and the principle states that the actual path followed by the system is the one that minimizes the action. By considering functionals, one can obtain the equations of motion that govern the dynamics of a system.
In quantum mechanics, functionals play a fundamental role in the path integral formulation, where the probability amplitude for a particle to travel from one state to another is expressed as a sum over all possible paths. The path integral involves a functional integration over the space of all paths, with a weight given by the action functional.
Functional theories also find applications in quantum field theory, where functionals are used to describe the behavior of fields and the associated particle interactions. The functional approach allows for a unified treatment of particles and fields, where the dynamics of the fields are described by functionals that depend on the field configurations.
Overall, functional theories provide a powerful mathematical framework for describing physical systems with continuous variations and have found wide-ranging applications in various areas of physics. They allow for a systematic and elegant treatment of complex physical phenomena by considering functionals that capture the essential properties of the underlying systems.