nLab vacuum amplitude

Idea

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.

Properties

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/function field analogy	theta function $\theta$	zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)	L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)	eta function $\eta$	special values of L-functions
physics/2d CFT	partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$	analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$	analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$	analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$	regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)		zeta function of an elliptic differential operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$	zeta function of a Riemann surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)	Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)		class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$	Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)	Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)	Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

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).

Related concepts

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

• Riemann hypothesis and physics

References

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)