- #1

Haorong Wu

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Hi, there. I am reading

## \begin{align} i \frac \partial {\partial t} \phi \left ( \bf {x}, t\right ) &=\left [ \phi \left ( \bf x, t \right ), \int d^3 x^{'} \{ \frac 1 2 \pi ^2 \left ( \bf { x^{'}}, t \right ) +\frac 1 2 \left ( \nabla\phi \left ( \bf { x^{'}}, t \right ) \right ) ^2 + \frac 1 2 m^2 \phi ^2 \left ( \bf { x^{'}}, t \right ) \} \right ] \nonumber \\ &=\int d^3 x^{'} \left ( i \delta ^{(3)} \left ( \bf x - \bf x^{'} \right ) \pi \left ( \bf x^{'} , t \right ) \right ) \nonumber \\ &= i \pi \left ( \bf x , t\right ) \nonumber\end{align} ##;

and

## \begin{align} i \frac \partial {\partial t} \pi \left ( \bf {x}, t\right ) &=\left [ \pi \left ( \bf x, t \right ), \int d^3 x^{'} \{ \frac 1 2 \pi ^2 \left ( \bf { x^{'}}, t \right ) + \frac 1 2 \phi \left ( \bf x^{'} ,t \right ) \left ( - \nabla^2 + m^2 \right ) \phi \left ( \bf x^{'} ,t \right ) \} \right ] \nonumber \\ &=\int d^3 x^{'} \left (- i \delta ^{(3)} \left ( \bf x - \bf x^{'} \right ) \left ( - \nabla^2 + m^2 \right ) \phi \left ( \bf x^{'} ,t \right ) \right ) \nonumber \\ &= -i \left ( - \nabla^2 + m^2 \right ) \phi \left ( \bf x ,t \right ) \nonumber\end{align} ##.

The first equation is a piece of cake. But I have some problems about the second equation.

First, the Hamiltonian is given by ##H=\int d^3 x^{'} \{ \frac 1 2 \pi ^2 \left ( \bf { x^{'}}, t \right ) +\frac 1 2 \left ( \nabla\phi \left ( \bf { x^{'}}, t \right ) \right ) ^2 + \frac 1 2 m^2 \phi ^2 \left ( \bf { x^{'}}, t \right ) \} ##. Then from ##\frac 1 2 \left ( \nabla\phi \left ( \bf { x^{'}}, t \right ) \right ) ^2 ## how to derive ##\frac 1 2 \phi \left ( \bf { x^{'}}, t \right ) \left ( - \nabla ^2 \right ) \phi \left ( \bf { x^{'}}, t \right ) ##?

Second, I do not know how to derive the second line of the second equation. why does the factor ##\frac 1 2## disappear?

Also I find that the mathematics in the book is quite difficult when it comes to covariant and contravariant tensors , even though I have learned the concepts of them from special relativity. Is there any useful material I could read?

Thanks!

*An Introduction to Quantum Field Theory*by Peskin and Schroeder. I am confused about some equations in section 2.4 The Klein-Gordon Field in Space-Time. It computes the Heisenberg equations of ##\phi \left ( x \right )## and ##\pi \left ( x \right)## as (in page 25)## \begin{align} i \frac \partial {\partial t} \phi \left ( \bf {x}, t\right ) &=\left [ \phi \left ( \bf x, t \right ), \int d^3 x^{'} \{ \frac 1 2 \pi ^2 \left ( \bf { x^{'}}, t \right ) +\frac 1 2 \left ( \nabla\phi \left ( \bf { x^{'}}, t \right ) \right ) ^2 + \frac 1 2 m^2 \phi ^2 \left ( \bf { x^{'}}, t \right ) \} \right ] \nonumber \\ &=\int d^3 x^{'} \left ( i \delta ^{(3)} \left ( \bf x - \bf x^{'} \right ) \pi \left ( \bf x^{'} , t \right ) \right ) \nonumber \\ &= i \pi \left ( \bf x , t\right ) \nonumber\end{align} ##;

and

## \begin{align} i \frac \partial {\partial t} \pi \left ( \bf {x}, t\right ) &=\left [ \pi \left ( \bf x, t \right ), \int d^3 x^{'} \{ \frac 1 2 \pi ^2 \left ( \bf { x^{'}}, t \right ) + \frac 1 2 \phi \left ( \bf x^{'} ,t \right ) \left ( - \nabla^2 + m^2 \right ) \phi \left ( \bf x^{'} ,t \right ) \} \right ] \nonumber \\ &=\int d^3 x^{'} \left (- i \delta ^{(3)} \left ( \bf x - \bf x^{'} \right ) \left ( - \nabla^2 + m^2 \right ) \phi \left ( \bf x^{'} ,t \right ) \right ) \nonumber \\ &= -i \left ( - \nabla^2 + m^2 \right ) \phi \left ( \bf x ,t \right ) \nonumber\end{align} ##.

The first equation is a piece of cake. But I have some problems about the second equation.

First, the Hamiltonian is given by ##H=\int d^3 x^{'} \{ \frac 1 2 \pi ^2 \left ( \bf { x^{'}}, t \right ) +\frac 1 2 \left ( \nabla\phi \left ( \bf { x^{'}}, t \right ) \right ) ^2 + \frac 1 2 m^2 \phi ^2 \left ( \bf { x^{'}}, t \right ) \} ##. Then from ##\frac 1 2 \left ( \nabla\phi \left ( \bf { x^{'}}, t \right ) \right ) ^2 ## how to derive ##\frac 1 2 \phi \left ( \bf { x^{'}}, t \right ) \left ( - \nabla ^2 \right ) \phi \left ( \bf { x^{'}}, t \right ) ##?

Second, I do not know how to derive the second line of the second equation. why does the factor ##\frac 1 2## disappear?

Also I find that the mathematics in the book is quite difficult when it comes to covariant and contravariant tensors , even though I have learned the concepts of them from special relativity. Is there any useful material I could read?

Thanks!