vacuum amplitude



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In quantum field theory (and string theory) the scattering amplitudes (string scattering amplitudes) “where nothing external scatters,” hence for no incoming and no outgoing states, are called vacuum amplitudes.


As generating functionals for all other amplitudes

As functions of source fields, the vacuum amplitudes, or rather the vacuum energy, serve as the generating functionals for all correlators/n-point functions (e.g. Scrucca, 1.6).

One loop contribution and zeta functions

The 1-loop vacuum amplitudes are regularized traces over Feynman propagators/Dirac propagators. These are the incarnations of special values of zeta functions, L-functions and eta functions in physics:

context/ θ\theta ζ\zeta (= of θ(0,)\theta(0,-)) L zL_{\mathbf{z}} (= of θ(z,)\theta(\mathbf{z},-)) η\eta
/ θ(z,τ)=Tr(exp(τ(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of τ\mathbf{\tau} of Σ\Sigma (hence of ) and / z\mathbf{z}analytically continued of ζ(s)=Tr reg(1(D 0) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued of in z\mathbf{z}: L z(s)Tr reg(1(D z) 2) s= 0 τ s1θ(z,τ)dτL_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tauanalytically continued of in z\mathbf{z} η z(s)=Tr reg(sgn(D z)|D z|) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s pvL z(1)=Tr reg(1(D z) 2)pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / fermionic pvη z(1)=Tr reg(D z(D z) 2)pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / 12L z (0)=Z H=12lndet reg(D z 2)-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)
θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) of over J(Σ τ)J(\Sigma_{\mathbf{\tau}}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})
for a (for ) and (in )
for a , (being the L zL_{\mathbf{z}} for z=0\mathbf{z} = 0 the ) L zL_{\mathbf{z}} of a z\mathbf{z}, expressible “in coordinates” (by ) as a finite-order (for 1-dimensional representations) and generally (via ) by an (for higher dimensional reps) \cdot
for \mathbb{Q} (z=0\mathbf{z} = 0)/ (z=χ\mathbf{z} = \chi a ) (being the L zL_{\mathbf{z}} for z=0\mathbf{z} = 0) of a z\mathbf{z} , expressible “in coordinates” (via ) as a (for 1-dimensional Galois representations) and generally (via ) as an

Vanishing of the vacuum amplitude and supersymmetry

In the presence of supersymmetry 1-loop vacuum amplitudes are typically supposed to vanish.

For the type II superstring, see e.g. (Palti). For the heterotic superstring see e.g. Han 89.

In view of the above relation of 1-loop vacuum amplitudes to special values of L-functions such vanishing reminds one of the Riemann hypothesis. See (ACER 11).


For particles

Discussions for particles includes

  • Claudio Scrucca, Advanced quantum field theory pdf

  • Ori Yudilevich, Calculating Massive One-Loop Amplitudes in QCD, Utrecht 2009 (pdf)

  • Robbert Rietkerk, One-loop amplitudes in perturbative quantum field theory, Utrecht 2012 (pdf)

For strings

Lecture notes for 1-loop vacuum amplitudes for strings (vacuum string scattering amplitudes) include

  • José Edelstein, Lecture 8: 1-loop closed string vacuum amplitude, 2013 (pdf)

  • Eran Palti, The IIA/B superstring one-loop vacuum amplitude (pdf)

  • Seung Kee Han, Vanishing vacuum amplitude of four-dimensional heterotic string theory compactified on N=2 superconformal field theory, Phys. Rev. D 39, 2322 – Published 15 April 1989 (web)

And with relation to the Riemann hypothesis:

  • Carlo Angelantonj, Matteo Cardella, Shmuel Elitzur, Eliezer Rabinovici, Vacuum stability, string density of states and the Riemann zeta function,JHEP 1102:024,2011 (arXiv:1012.5091)

Last revised on January 20, 2018 at 17:13:29. See the history of this page for a list of all contributions to it.