(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Quantum Field Theory: Field Operators and Lorentz invariance

Hi there,

I am currently working my way through a book an QFT (Aitchison/Hey) and am a bit stuck on an important step in the derivation of the Feynman Propagator. My problem is obviously that I am not a hard core expert of relativitiy :)

Actually, I haveTWOquestions on the same matter.

The central quantity is the Feynman Propagator

[tex] <0|T([\hat{\phi}(x_1) \hat{\phi}(x_2)|0> [/tex]

where the [tex] \hat{\phi} [/tex] are scalar field operators and T is the time-ordering operator and the x are 4-vectors.

The point of interest is now this quantity's Lorentz Invariance.

The book says: "If the two points [tex]x_1[/tex] and [tex]x_2[/tex] are separated by atime-likeinterval ([tex] (x_1 - x_2)^2 > 0)[/tex] then the time ordering is Lorentz invariant;this is because no proper (doesn't change the sense of time) Lorentz transformation can alter the time-ordering of time-like separated events."

It goes on:

"The fact that time-ordering is invariant for time-like separated events is what guarantees that we cannot influence our past, only our future"

First question:The first (italic) part sounds suspiciously self-evident, but how can that be quickly shown mathematically? And for the second part: I would say: the fact that it is Lorentz invariant means that one can not think of a coordinate frame where the events change their order of time. Is that right?

Now the book goes on and treats the case ofspace-like ([tex] (x_1 - x_2)^2 < 0)[/tex]separated events. The book says it can be shown that it can be shown that the two field operatorsalways commutein this case. I tried to show that following a hint:

Commutator of 2 scalar Field Operators of the same kind:

.. with 3-vectors and time component treated seperatly.

[tex] D(x_1, x_2) = [\hat{\phi}(x_1, t_1), \hat{\phi}(x_2, t_2)] [/tex]

I could show that this can be written as

... with x and k being again 4-vectors.

[tex] D(x_1, x_2) = \int \frac{d^3 k}{(2 \pi)^3 2E} [ e^{-ik\cdot(x_1-x_2)} - e^{ik\cdot(x_1-x_2)}][/tex]

The right side is obviouslyLorentz invariant. The book now hints that this fact is enough to show that in this case [tex] D(x_1, x_2)[/tex] actually alwaysvanishes.

Second question:How is Lorentz invariance enough to show that?

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# Quantum Field Theory: Field Operators and Lorentz invariance

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