By E. Elizalde, S. D. Odintsov, A. Romeo, Andrei A. Bytsenko, S. Zerbini
Discussing Zeta regularization concepts, this booklet contains chapters on: the Reimann Zeta functionality; the Casimir impact in flat space-time; and homes of the chemical capability in higher-dimensional manifolds.
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Additional info for Zeta Regularization Techiques with Applications
Example text
For all loops with momentum k, we must integrate over the momentum: d4 k/(2π)4 . This corresponds to the addition of quantum mechanical amplitudes. ◦ For fermion loops we have to take the trace and multiply it by the factor −1. Chapter 10. Processes in the lowest order of perturbation theory 57 ◦ If two diagrams differ for an odd number of fermionic interchanges, then they must differ by a relative minus sign. 1. For the process A(E1 , p1 ) + B(E2 , p2 ) → C(E1 , p1 ) + D(E2 , p2 ) prove that the differential cross section in the center of mass frame is given by dσ |p1 | 1 mA mB mC mD |M|2 , = 2 2 dΩ cm 4π (E1 + E2 ) |p1 | where iM is the Feynman amplitude.
Show that the Lagrangian density of a real scalar field can be taken as L = − 12 φ( + m2 )φ. Chapter 5. 6. Show that the Lagrangian density of a free spinor field can be taken in ¯∂ ψ − (∂μ ψ)γ ¯ ¯ μ ψ) − mψψ. 7. The Lagrangian density for a massive vector field Aμ is given by 1 1 L = − Fμν F μν + m2 Aμ Aμ . 4 2 Prove that the equation ∂μ Aμ = 0 is a consequence of the equations of motion. 8. Prove that the Lagrangian density of a massless vector field is invariant under the gauge transformation: Aμ → Aμ + ∂μ Λ(x), where Λ = Λ(x) is an arbitrary function.
3. The action of a free scalar field in two–dimensional spacetime is ∞ S= L dt −∞ dx 0 m2 2 1 ∂μ φ∂ μ φ − φ 2 2 . The spatial coordinate x varies in the region 0 < x < L. Find the equation of motion and discuss the importance of the boundary term. 4. Prove that the equations of motion remain unchanged if the divergence of an arbitrary field function is added to the Lagrangian density. 5. Show that the Lagrangian density of a real scalar field can be taken as L = − 12 φ( + m2 )φ. Chapter 5. 6. Show that the Lagrangian density of a free spinor field can be taken in ¯∂ ψ − (∂μ ψ)γ ¯ ¯ μ ψ) − mψψ.