Quantum Theory

Peeters K. 's A guide to quantum field theory PDF

By Peeters K.

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In an experiment, we will have one beam of particles of momentum k1 colliding with a beam of particles of momentum k2 . We will be interested in computing the transition rate to an outgoing state, with one bunch of particles of momentum k3 and one of momentum k4 . 5. 5 Cross sections and decay rates and k4 ). We computed it in the previous section, and will from now on denote it by 34|12 , † † 34|12 = 0| T aˆ out (k3 ) aˆ out (k4 ) aˆ in (k1 ) aˆ in ( k 2 ) |0 4 = 34|12 × (2π )4 δ4 (k1 + k2 − k3 − k4 ) ∏ √ i =1 1 .

1) all paths X (τ ) We have parameterised the path by some parameter τ which will have to be specified later. Of course, the number of paths is typically infinite, so the sum is generically an integral over a (very large) space of paths, and making sense of the ‘sum’ is often a challenging task. The so-called path integral formalism of quantum mechanics states that we can find an expression for the phase purely in terms of the classical paths, and then compute the amplitude as a weighted sum over classical objects, instead of dealing with operators and state vectors in Hilbert space (as we have done until now).

Now there are two more things to do. Firstly, in a real experiment we can never measure particles with exactly fixed momenta. What we should do is collect all particles in some small range of three-momenta. In a square box of size V = L3 , three-momenta are quantised as k = (2π n)/L. 63) ∑ → (2π )3 d3 k . n We should thus multiply our rate by a factor of this type for each outgoing particle. Secondly, we need to normalise our probability by dividing by the incoming flux. 4 All manipulations here are rather singular if we consider an infinitely extended space-time, so we will temporarily put things in a space-time box of finite three-volume V and time extent T, and take the limit V → ∞, T → ∞ at the end.

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A guide to quantum field theory by Peeters K.


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