vacuum energy




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Theta functions



Given a Feynman propagator 1H\frac{1}{H}, then the corresponding vacuum energy ZZ is the logarithm of the functional determinant det regdet_{reg} of HH

Zlogdet regH. Z \coloneqq -log\,det_{reg} H \,.


In terms of special values of the zeta function

The vacuum energy is equivalently the special value of the zeta function ζ H\zeta_H of HH given by the derivative at 0:

Z=12ζ H (0). Z = -\frac{1}{2}\zeta_H^\prime(0) \,.

See at zeta function of an elliptic differential operator – Functional determinant for more.

In terms of the path integral and relation to generating function

Traditionally the vacuum energy is expressed in terms of a hypothetical path integral. (As opposed to the above zeta-function formalization this is not rigorous, but it serves to give the idea of why this is the vacuum energy and the the zeta-function expression may be taken to be the rigorous definition of the path integral heuristics.)

By analogy with finite-dimensional Gaussian integrals (see at Feynman diagram – For finitely many degrees of freedom) one expects that the Wick rotated vacuum amplitude version of the path integral (no field insertions, no boundary conditions) is

ϕFieldsexp(S H(ϕ))Dϕ=(det regH) 1/2. \underset{\phi \in \mathbf{Fields}}{\int} \exp(- S_H(\phi)) D\phi = (det_reg H)^{-1/2} \,.


logϕFieldsexp(S H(ϕ))Dϕ=12logdet regH log \underset{\phi \in \mathbf{Fields}}{\int} \exp(- S_H(\phi)) D\phi = -\frac{1}{2}log\, det_{reg} H

is the generating functional for n-point functions. (…)

e.g. (Scrucca, section 1.6, Edelstein 13, page 2)

As holomorphic potential for Determinant line bundle


h12det regH h \coloneqq \frac{1}{2} det_{reg}H

as a hermitian structure on a holomorphic line bundle, hence, locally, as the absolute value-squared of the unit section ϕ i\phi_i of a holomorphic line bundle with respect to a local trivializing section (see at Chern connection).

h| U i=ϕ i 2. h|_{U_i} = {\Vert \phi_i \Vert}^2 \,.

Then this line bundle is the determinant line bundle of HH. (Quillen 85), review includes (Freed 87, p. 18, Qiu 12, section 2.8.1).

The Chern connection is

A=log(h)=12det regH=Z A = \partial log(h) = \partial \frac{1}{2} det_{reg}H = \partial Z

and the curvature differential 2-form is

F=i¯Z. F = i \bar \partial\partial Z \,.

eh? Something wrong with the factors of 1/21/2 here…


The solution to the strong CP problem via axions

In the solution to the strong CP problem via axions it is analysis of the dependency of the vacuum energy of Yang-Mills theory on the theta-angle θ\theta which is argued to show that the axion expectation value θ=a\theta = \langle a \rangle vanishes (Vafa-Witten 84).

See at axion – As a solution to the strong CP-problem.


For instance

  • Claudio Scrucca, section 1.6 in Advanced quantum field theory pdf

  • José Edelstein, page 2 of Lecture 8: 1-loop closed string vacuum amplitude, 2013 (pdf)

  • Daniel Quillen, Determinants of Cauchy-Riemann Operators over a Riemann Surface, Functional Anal.

    Appl. 19 (1985) 31.

  • Daniel Freed, On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (revised pdf, dg-ga/9505002)

  • Jia Qiu, section 2.8 of Lecture notes on topological field theory arXiv:1201.5550

The application to the axion solution to the strong CP problem is due to

Last revised on November 9, 2018 at 11:01:01. See the history of this page for a list of all contributions to it.