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By P. W Bridgman

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I" •f I n r e n o r m a l i s i n g t h e p h o t o n p r o p a g a t o r , t h e c o n d i t i o n Z = \e a s a n e c e s s a r y p h y s i c a l c o n d i t i o n , after w h i c h the corrections to the external p h o t o n lines d i s a p p e a r automatically. F o r m a l l y , h o w e v e r , the s i t u a t i o n i s s i m i l a r f o r b o t h p h o t o n a n d e l e c t r o n e x t e r n a l l i n e s : w h e n Z 9^ 1, t h e w a v e a m p l i t u d e e„ o f a r e a l p h o t o n , w i t h c o r r e c t i o n s , w o u l d b e m u l t i p l i e d b y VZ.

W e shall also need t h e second derivative o f the polarisation operator. 13) o n c e m o r e i n a s i m i l a r m a n n e r , w e h a v e ; ;? - " An dk^dk" _ . 12). §106. 1), b u t i n t h e H a m i l t o n i a n H = H Q - I - V w h i c h c o n v e r t s t o t h e Heisenberg representation o f operators w e have also t h e interaction between t h e electrons §106 Electron and t h e e x t e r n a l propagators in an external field 403 field: \ eJAJ''d^x+e^A^^^}>'d^x. 1) Since t h e e x t e r n a l field m a k e s s p a c e a n d t i m e n o l o n g e r h o m o g e n e o u s , t h e p r o p a g a t o r ^{x, x') w i l l n o w d e p e n d o n t h e t w o a r g u m e n t s x a n d x' s e p a r a t e l y a n d n o t o n l y o n t h e diiference x—x'.

_ . - ( ^ ^ » , ^ . 8) gives A comparison o f t h e t w o f o r m u l a e confirms t h e limiting expression w r i t t e n above f o r 0{p) ( t h e first t e r m i n ( 1 0 7 . 9 ) ) , a n d s h o w s t h a t * s ; -• - Z ' • ^/Z^. 15) W e shall n o w show that, w h e n t h e h m i t i n g f o r m o f the electron propagator is k n o w n , there is n o need t o estabUsh a n y f u r t h e r c o n d i t i o n s f o r t h e vertex operator. 16) w h i c h r e p r e s e n t s t h e s c a t t e r i n g o f a n e l e c t r o n i n a n e x t e r n a l field ^ ' ^ * ( A : ) , i n t h e first o r d e r w i t h r e s p e c t t o t h e field, a n d t a k i n g a c c o u n t o f a l l r a d i a t i v e c o r r e c t i o n s .

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A sophisticate's primer of relativity by P. W Bridgman


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