Question

A3.3 Show directly that if ф (x) satisfies the Klein-Gordon equation then ф(A-1x) also satisfies this equation for any Lorentz transformation A.

Answer 1

ANSWER :-

Given that ,

is the klein Gordan equation.

Φ(z)

Here we need to show that $\Phi (A^-^1x)$ also satisfies for any lorentz transformation A.

Let us suppose that $\Phi$ , are the elements in lorentz equation.

From, klein gordan equation,

we have,  $\Phi A = x$

$\Phi = x \left (\frac{1}{A} \right )$

$\Phi = x A^-^1$

and also we already know that
Φ(z)

(xA ) TA-

$\Phi (xA^-^1)$ satisfies the equation directly in lorentz transformation A.

Hence proved.

Thank you