# Eigenfunctionals and their application in physics

Is there any sensible meaning of the term eigenfunctionals? The object I want to describe is a solution to the following equation

$${\mathscr D}_x F[g] = f(x) F[g]$$

where $${\mathscr D}_x$$ is an operator that maps functionals to functionals (e.g. a functional derivative $$\frac{\delta}{\delta g(x)}$$), $$F$$ is an eigenfunctional of $${\mathscr D}_x$$, $$f(x)$$ is a corresponding eigenfunction, and $$g(x)$$ is some other function. Of course, this is rather a try to make sense of this term or to motivate thinking about it than a proper definition since I cannot find any. Additionally, is there an application of this concept in physics, e.g. when using functional methods in the path integral formalism of quantum field theory? A special case I encountered during studying correlation functions would be:

$$F[\phi(x)] = \exp \left( \int dx \, \phi(x) \, h(x) \right)$$

$$\frac{\delta F[\phi(x)]}{\delta \phi(y)} = h(y) F[\phi(x)]$$

However, a generalization of the idea would probably be fruitful.

• This is a Schwinger-Dyson equation. See this PSE post for the standard use in physics. – AccidentalFourierTransform Oct 23 '19 at 23:24
• In the Schrödinger picture of a quantum field theory, one has wavefunctionals of the fields which are eigenfunctionals of the Hamiltonian. Here is a nice introduction: sciencedirect.com/science/article/pii/055032138590210X – Seth Whitsitt Oct 23 '19 at 23:31
• Thank you, accidentalfouriertransform, for the reference to the great post, which will be helpful in understanding the Schwinger-Dyson equation! However, I rather intended to get some insights into the mathematical properties of the more general equation I proposed. Do you know of another application of this kind of equation? – Carlo Tasillo Oct 23 '19 at 23:34
• @SethWhitsitt: Thanks for pointing out the connection to the Schrödinger picture of QFT which I haven't heard of before. I will check if I will be able to find the mathematical foundation for its defining equation which should be equivalent to my question. – Carlo Tasillo Oct 23 '19 at 23:48
• @CarloTasillo My answers to the questions physics.stackexchange.com/q/441251 and physics.stackexchange.com/q/439434 both mention this kind of formulation (Schrodinger functional formulation of QFT). – Chiral Anomaly Oct 24 '19 at 13:06