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By Lomonaco S.J.

The new program of the foundations of quantum mechanics to cryptography has ended in a amazing new size in mystery communique. due to those new advancements, it truly is now attainable to build cryptographic verbal exchange platforms which discover unau- unauthorized eavesdropping should still it ensue, and which offer a warrantly of no eavesdropping may still it no longer take place.

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15. 16. Prove that 0| T (ψ(x)Γ ψ(y)) |0 is equal to zero for Γ = {γ5 , γ5 γμ }, while for Γ = γμ γν one gets the result −4imgμν ΔF (y − x). 17. The Dirac spinor in terms of two Weyl spinors ϕ and χ is of the form ψ= ϕ −iσ2 χ∗ . (a) Show that the Majorana spinor equals ψM = χ −iσ2 χ∗ . (b) Prove the identities: ψ¯M φM = φ¯M ψM , ψ¯M γ μ φM = −φ¯M γ μ ψM , ψ¯M γ5 φM = φ¯M γ5 ψM , ψ¯M γ μ γ5 φM = φ¯M γ μ γ5 ψM , ψ¯M σμν φM = −φ¯M σμν ψM . (c) Express the Majorana field operator, ψM = √12 (ψ + ψc ) using creation and annihilation operators of a Dirac field.

Assume that the scalar field is massless. Prove that the obtained expression satisfies the Klein–Gordon equation. Chapter 7. 15. Calculate 0| φ(x1 )φ(x2 )φ(x3 )φ(x4 ) |0 for a free scalar field. 16. Find 0| φ(x)φ(y) |0 in two dimensions, for a massless scalar field. 17. Prove the relation ( x + m2 ) 0| T (φ(x)φ(y)) |0 = −iδ (4) (x − y) . 18. The Lagrangian density of a spinless Schr¨odinger field ψ, is given by L = iψ † 1 ∂ψ − ∇ψ † · ∇ψ − V (r)ψ † ψ . ∂t 2m (a) Find the equations of motion. (b) Express the free fields ψ and ψ † in terms of creation and annihilation operators and find commutation relations between them.

2 n The γ = 0, 5772 is the Euler–Mascheroni constant. 1. The most frequently used parameterizations are: ψ(n + 1) = 1 + 1 1 = AB 1 =2 ABC 1 dx 0 1−x dx 0 dz 0 1 , [xA + (1 − x)B]2 1 . H) • Cutkosky rule for computing discontinuity of any Feynman diagram contains the following steps: 1. Cut through the diagram in all possible ways such that the cut propagators can be put on–shell. 2. For each cut, make the replacement p2 1 → (−2iπ)δ (4) (p2 − m2 )θ(p0 ) . − m2 3. Sum the contributions of all possible cuts.

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