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We are interested in the rate, so we also divide by the total time interval, and get4 2 34|12 1 transition probability = . 58) Both the numerator and denominator are singular. In the numerator we will get the square of the delta function, (2π )4 δ4 (k1 + k2 − k3 − k4 ) 2 = (2π )4 δ4 (0) × (2π )4 δ4 (k1 + k2 − k3 − k4 ) . e. 5 The norms of the states in the denominator are also singular. 61) by a similar argument. The transition rate is thus transition rate = (2π )4 δ(4) (k1 + k2 − k3 − k4 ) × 34|12 2 4 1 .

73) Do this by writing out the fields into creation and annihilation pieces. (b) State Wick’s theorem for the time-ordered product of an arbitrary number n of fields. (c) What can you say about the vacuum expectation value of the time-ordered product of an odd number of fields, 0| T φˆ ( x1 ) · · · φˆ ( x2n+1 ) |0 . odd number 3. Symmetry factors What are the symmetry factors for the following diagrams? 4. Feynman diagrams Consider the scalar field theory with both a φ4 and φ3 interaction term, S= 44 1 1 g λ d4 x − (∂µ φ)(∂µ φ) − m2 φ2 − φ3 − φ4 .

In the case of a field theory, we will be interested in “paths” for which the field φ(t, x ) approaches zero at spacelike and timelike infinity. In the path integral, we implement this using a simple trick. Consider for simplicity the free field theory of a real scalar. The claim is that we have the following correspondence between the vacuum-to-vacuum amplitude and a path integral expression, 0|0 ↔ D φ exp i S[φ] − h¯ h¯ d4 x φ( x )2 . 21) We will of course again have to make precise what we mean with the integral over field configurations on the right hand side, and how we can compute it.

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A guide to quantum field theory by Peeters K.


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