From Special Relativity to Feynman Diagrams.pdf

Causality principle as explained in chap1 11 the

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causality principle, as explained in Chap.1 . 11 The requirement of commutativity of two observables separated by a space-like distance is referred to as the principle of microcausality . It is also worth noting that this result is guaranteed by the cancelation of the contributions from D ( ± ) ( x y ) in the commutator, related in turn to the presence of positive and negative energy solutions ˆ φ + ( x ), ˆ φ ( x ). The very presence of these two solutions and, in particular, of the negative energy ones, so embarrassing for the classical Klein–Gordon equation, is therefore essential for the consistency of the quantum field theory. 12 For the sake of completeness we now show that D + ( 0 , x y ) is different from zero, and give its explicit expression. D + ( 0 , x y ) = c d 3 p ( 2 π ) 3 1 2 E p e i p · ( x y ) . (11.121) Using polar coordinates for the variable p , we have d 3 p = | p | 2 sin θ d | p | d θ d ϕ, so that: c D + ( 0 , x y ) = 2 ( 2 π ) 3 ( 2 π) 0 d | p || p | 2 2 E p 1 1 d ( cos θ) e i | p || x y | cos θ = 1 ( 2 π) 2 0 d | p || p | 2 2 E p e i | p || x y | e i | p || x y | i | p || x y | = 1 ( 2 π) 2 0 d | p || p | E p | x y | sin | p | | x y | 11 We note that the requirement of causality refers to observables and in general the field operators are not necessarily observables. However quite generally observables in a physical system are constructed in terms of local functions of the field variables so that the requirement of causality can be expressed in terms of the fields themselves. 12 The requirement of commuting operators for space-like separations is also called “locality” and, correspondingly, the quantum field theory is referred to as a “local theory” . Locality assures that the results of two measures made at a space-like distance cannot have any influence on one another, there being no correlation between the two events.
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388 11 Quantization of Boson and Fermion Fields = 1 ( 2 π) 2 c 0 d | p || p | | p | 2 + m 2 c 2 sin | p | | x y | 1 | x y | = 1 ( 2 π) 2 m | x y | K 1 mc | x y | . ( 11 . 234 ) where K n ( z ) are the modified Bessel functions of the second type and we have used the general formula 0 dzz sin ( bz ) z 2 + γ 2 e β z 2 + γ 2 = b γ b 2 + β 2 K 1 γ b 2 + β 2 . (11.123) In our case we have z = | p | , γ = mc , b = | x y | , β = 0 . The asymptotic behavior of K 1 ( z ) as z → ∞ is K 1 ( z ) = π 2 z e z 1 + O 1 z e z , (11.124) and therefore for large space–time separation | x y | → ∞ D + ( 0 , x y ) c m | x y | e mc | x y | , (11.125) that is D + is sensibly different from zero only within spatial distances of the order of the Compton wave-length λ = / mc of the particle. 11.4.1 Green’s Functions and the Feynman Propagator The invariant D -functions discussed in the previous paragraph are strictly related to another invariant function which plays a major role in the theory of interacting fields: the Feynman propagator function D F ( x y ). It is defined to be the vacuum expectation value of the so called time-ordered product : D F ( x y ) = c 0 | T ˆ φ( x ) ˆ φ ( y ) | 0 , (11.126) where T ˆ φ( x ) ˆ φ ( y ) = ˆ φ( x ) ˆ φ ( y ) x 0 > y 0 , ˆ φ ( y ) ˆ φ( x ) y 0 > x 0 .
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