# Help understanding a term in the derivation of #\Pi(\vec(x),t)# for KG eq.

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• TroyElliott
In summary, when deriving the term ##\int_{-\infty}^{\infty}d^{3}y \nabla_{y}(\delta(\vec{x}-\vec{y}))\nabla\phi(y)## for the Klein-Gordon equation, we use integration by parts to obtain the term ##-\int_{-\infty}^{\infty}d^{3}y \delta(\vec{x}-\vec{y})\nabla^{2}\phi(y)##. This relies on the assumption that ##\nabla\Phi## vanishes at infinity for the integral to make sense. The boundary term should be zero because this is the definition of the derivative for
TroyElliott
TL;DR Summary
I need help understanding a term that appears in the derivation of the time dependent conjugate momentum of the Klein-Gordon field
When deriving ##\Pi(\vec{x},t)## for the Klein-Gordon equation (i.e. plugging ##\Pi(\vec{x},t)## into the Heisenberg equation of motion for the scalar field Hamiltonian), we come across a term that is the following
##\int_{-\infty}^{\infty}d^{3}y \nabla_{y}(\delta(\vec{x}-\vec{y}))\nabla\phi(y)##

We are then told "using integration by parts we get the following"

##\int_{-\infty}^{\infty}d^{3}y \nabla_{y}(\delta(\vec{x}-\vec{y}))\nabla\phi(y) = -\int_{-\infty}^{\infty}d^{3}y \delta(\vec{x}-\vec{y})\nabla^{2}\phi(y).##

For this to be true, the the boundary term ##\delta(\vec{x}-\vec{y})\nabla\phi(y)## evaluated at ##\infty## and at ##-\infty##, must be zero. Are we assuming that the gradient of the the field goes to zero at infinity? And what about the delta function term when it becomes ##\delta(\vec{x}-\infty)##? This blows up at the the boundary at infinity. Any thoughts on why this boundary term should be zero is much appreciated!

Thanks!

I guess the simplest answer is that this is just the definition of the derivative for a ##\delta## distribution. This definition is of course designed to be compatible with partial integration if instead of the ##\delta## there would be an actual function. It does not really make sense to plug in numbers in the argument of ##\delta## like you would have to do for the boundary terms. We still have to assume that ##\nabla\Phi## vanishes at infinity for the integral to make sense in the first place.

## 1. What does #\Pi(\vec(x),t)# represent in the derivation of the KG equation?

#\Pi(\vec(x),t)# represents the energy-momentum tensor, which is a mathematical object used to describe the distribution of energy and momentum in a physical system. In the derivation of the KG equation, it is used to represent the energy and momentum of a scalar field.

## 2. Why is the KG equation important in physics?

The KG equation is important in physics because it describes the behavior of scalar fields, which are fundamental in many areas of physics such as quantum mechanics and general relativity. It also has applications in fields such as cosmology and particle physics.

## 3. What is the significance of the term #\Pi(\vec(x),t)# in the KG equation?

The term #\Pi(\vec(x),t)# represents the kinetic energy of the scalar field, which is an important factor in understanding its dynamics. It also plays a crucial role in determining the evolution of the field over time.

## 4. How is the energy-momentum tensor related to the KG equation?

The energy-momentum tensor is used in the derivation of the KG equation to describe the energy and momentum of the scalar field. It is also used in the equation itself to represent the kinetic energy of the field.

## 5. Can the KG equation be applied to other types of fields besides scalar fields?

Yes, the KG equation can be applied to other types of fields such as vector fields and spinor fields. However, the specific form of the equation may vary depending on the type of field being studied.

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