Return to Question

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.

In the physics literature, in particular in QFT, one sometimes says that if the unknown function $u$ satisfies boundary conditions then $u=0$.

Question. What exactly boundary conditions guarantee vanishing of $u$?

I am mainly interested in such conditions of physical interest, say used in QFT.

UPDATE. Below is a concrete example of a situation of interest. In comments I briefly describe yet another situation which is typical for QFT and more interesting, but mathematically unrigorous. As I was told, the second situation is not appropriate for this site.

Consider now the classical (i.e. not quantized) free electromagnetic field $A_\nu$. It satisfies the equation $$\Box A_\nu-\partial_\nu (\partial\cdot A)=0.$$ It is gauge invariant, i.e. $A_\nu+\partial_\nu f$ satisfies the same equation for any function $f$. Under a change under such a gauge transformation we may assume that $\partial\cdot A=0$ (Lorentz gauge). Then the equations become $$\Box A_\nu=0,\,\,\, \partial\cdot A=0.$$ It is still invariant under adding to $A_\nu$ the expression $\partial_\nu f$ where $\Box f=0$. What are the natural boundary conditions on all the electromagnetic fields which would guarantee that $f=0$?

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.

In the physics literature, in particular in QFT, one sometimes says that if the unknown function $u$ satisfies boundary conditions then $u=0$.

Question. Under what decay assumptions at infinity are the solutions of the above equation necessarily trivial?

I am mainly interested in such conditions of physical interest, say used in QFT.

UPDATE. Below is a concrete example of a situation of interest. In comments I briefly describe yet another situation which is typical for QFT and more interesting, but mathematically unrigorous. As I was told, the second situation is not appropriate for this site.

Consider now the classical (i.e. not quantized) free electromagnetic field $A_\nu$. It satisfies the equation $$\Box A_\nu-\partial_\nu (\partial\cdot A)=0.$$ It is gauge invariant, i.e. $A_\nu+\partial_\nu f$ satisfies the same equation for any function $f$. Under a change under such a gauge transformation we may assume that $\partial\cdot A=0$ (Lorentz gauge). Then the equations become $$\Box A_\nu=0,\,\,\, \partial\cdot A=0.$$ It is still invariant under adding to $A_\nu$ the expression $\partial_\nu f$ where $\Box f=0$. What are the natural boundary conditions on all the electromagnetic fields which would guarantee that $f=0$?

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.

In the physics literature, in particular in QFT, one sometimes says that if the unknown function $u$ satisfies boundary conditions then $u=0$.

Question. What exactly boundary conditions guarantee vanishing of $u$?

I am mainly interested in such conditions of physical interest, say used in QFT.

UPDATE. A concrete example of a situation of interest is classical (i.e. not quantized) free electromagnetic field $A_\nu$. It satisfies the equation $$\Box A_\nu-\partial_\nu (\partial\cdot A)=0.$$ It is gauge invariant, i.e. $A_\nu+\partial_\nu f$ satisfies the same equation for any function $f$. Under a change under such a gauge transformation we may assume that $\partial\cdot A=0$ (Lorentz gauge). Then the equations become $$\Box A_\nu=0,\,\,\, \partial\cdot A=0.$$ It is still invariant under adding to $A_\nu$ the expression $\partial_\nu f$ where $\Box f=0$. What are the natural boundary conditions on all the electromagnetic fields which would guarantee that $f=0$?

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.

In the physics literature, in particular in QFT, one sometimes says that if the unknown function $u$ satisfies boundary conditions then $u=0$.

Question. What exactly boundary conditions guarantee vanishing of $u$?

I am mainly interested in such conditions of physical interest, say used in QFT.

UPDATE. Below is a concrete example of a situation of interest. In comments I briefly describe yet another situation which is typical for QFT and more interesting, but mathematically unrigorous. As I was told, the second situation is not appropriate for this site.

Consider now the classical (i.e. not quantized) free electromagnetic field $A_\nu$. It satisfies the equation $$\Box A_\nu-\partial_\nu (\partial\cdot A)=0.$$ It is gauge invariant, i.e. $A_\nu+\partial_\nu f$ satisfies the same equation for any function $f$. Under a change under such a gauge transformation we may assume that $\partial\cdot A=0$ (Lorentz gauge). Then the equations become $$\Box A_\nu=0,\,\,\, \partial\cdot A=0.$$ It is still invariant under adding to $A_\nu$ the expression $\partial_\nu f$ where $\Box f=0$. What are the natural boundary conditions on all the electromagnetic fields which would guarantee that $f=0$?

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.

In the physics literature, in particular in QFT, one sometimes says that if the unknown function $u$ satisfies boundary conditions then $u=0$.

Question. What exactly boundary conditions guarantee vanishing of $u$?

I am mainly interested in such conditions of physical interest, say used in QFT.

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.

In the physics literature, in particular in QFT, one sometimes says that if the unknown function $u$ satisfies boundary conditions then $u=0$.

Question. What exactly boundary conditions guarantee vanishing of $u$?

I am mainly interested in such conditions of physical interest, say used in QFT.

UPDATE. A concrete example of a situation of interest is classical (i.e. not quantized) free electromagnetic field $A_\nu$. It satisfies the equation $$\Box A_\nu-\partial_\nu (\partial\cdot A)=0.$$ It is gauge invariant, i.e. $A_\nu+\partial_\nu f$ satisfies the same equation for any function $f$. Under a change under such a gauge transformation we may assume that $\partial\cdot A=0$ (Lorentz gauge). Then the equations become $$\Box A_\nu=0,\,\,\, \partial\cdot A=0.$$ It is still invariant under adding to $A_\nu$ the expression $\partial_\nu f$ where $\Box f=0$. What are the natural boundary conditions on all the electromagnetic fields which would guarantee that $f=0$?