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\begin{document}

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\section*{vacuum energy}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{vacua}{}\paragraph*{{Vacua}}\label{vacua}

[[!include vacua -- contents]]

\hypertarget{theta_functions}{}\paragraph*{{Theta functions}}\label{theta_functions}

[[!include theta functions - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{InTermsOfSpecialValuesOfZetaFunctions}{In terms of special values of the zeta function}\dotfill \pageref*{InTermsOfSpecialValuesOfZetaFunctions} \linebreak
\noindent\hyperlink{in_terms_of_the_path_integral_and_relation_to_generating_function}{In terms of the path integral and relation to generating function}\dotfill \pageref*{in_terms_of_the_path_integral_and_relation_to_generating_function} \linebreak
\noindent\hyperlink{as_holomorphic_potential_for_determinant_line_bundle}{As holomorphic potential for Determinant line bundle}\dotfill \pageref*{as_holomorphic_potential_for_determinant_line_bundle} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{the_solution_to_the_strong_cp_problem_via_axions}{The solution to the strong CP problem via axions}\dotfill \pageref*{the_solution_to_the_strong_cp_problem_via_axions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Given a [[Feynman propagator]] $\frac{1}{H}$, then the corresponding \emph{vacuum energy} $Z$ is the [[logarithm]] of the [[functional determinant]] $det_{reg}$ of $H$

\begin{displaymath}
Z \coloneqq -log\,det_{reg} H
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{InTermsOfSpecialValuesOfZetaFunctions}{}\subsubsection*{{In terms of special values of the zeta function}}\label{InTermsOfSpecialValuesOfZetaFunctions}

The vacuum energy is equivalently the [[special values of L-functions|special value]] of the [[zeta function of an elliptic differential operator|zeta function]] $\zeta_H$ of $H$ given by the [[derivative]] at 0:

\begin{displaymath}
Z = -\frac{1}{2}\zeta_H^\prime(0)
  \,.
\end{displaymath}
See at \emph{\href{zeta+function+of+an+elliptic+differential+operator#FunctionalDeterminant}{zeta function of an elliptic differential operator -- Functional determinant}} for more.

\hypertarget{in_terms_of_the_path_integral_and_relation_to_generating_function}{}\subsubsection*{{In terms of the path integral and relation to generating function}}\label{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 \hyperlink{InTermsOfSpecialValuesOfZetaFunctions}{above} zeta-function formalization this is not rigorous, but it serves to give the idea of \emph{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 \emph{\href{Feynman+diagram#ForFinitelyManyDegreesOfFreedom}{Feynman diagram -- For finitely many degrees of freedom}}) one expects that the [[Wick rotation|Wick rotated]] [[vacuum amplitude]] version of the [[path integral]] (no [[field (physics)|field]] insertions, no [[boundary field theory|boundary conditions]]) is

\begin{displaymath}
\underset{\phi \in \mathbf{Fields}}{\int} 
   \exp(- S_H(\phi))
  D\phi
  = 
  (det_reg H)^{-1/2}
  \,.
\end{displaymath}
Therefore

\begin{displaymath}
log
    \underset{\phi \in \mathbf{Fields}}{\int} 
   \exp(- S_H(\phi))
  D\phi
  =
  -\frac{1}{2}log\, det_{reg} H
\end{displaymath}
is the [[generating functional]] for [[n-point functions]]. (\ldots{})

e.g. (\hyperlink{Scrucca}{Scrucca, section 1.6}, \hyperlink{Edelstein13}{Edelstein 13, page 2})

\hypertarget{as_holomorphic_potential_for_determinant_line_bundle}{}\subsubsection*{{As holomorphic potential for Determinant line bundle}}\label{as_holomorphic_potential_for_determinant_line_bundle}

Regard

\begin{displaymath}
h \coloneqq \frac{1}{2} det_{reg}H
\end{displaymath}
as a hermitian structure on a [[holomorphic line bundle]], hence, locally, as the [[absolute value]]-squared of the unit [[section]] $\phi_i$ of a [[holomorphic line bundle]] with respect to a local trivializing section (see at [[Chern connection]]).

\begin{displaymath}
h|_{U_i} = {\Vert \phi_i \Vert}^2
  \,.
\end{displaymath}
Then this line bundle is the [[determinant line bundle]] of $H$. (\hyperlink{Quillen85}{Quillen 85}), review includes (\hyperlink{Freed87}{Freed 87, p. 18}, \hyperlink{Qiu12}{Qiu 12, section 2.8.1}).

The [[Chern connection]] is

\begin{displaymath}
A = \partial log(h) = \partial \frac{1}{2} det_{reg}H = \partial Z
\end{displaymath}
and the [[curvature]] [[differential 2-form]] is

\begin{displaymath}
F = i \bar \partial\partial Z
  \,.
\end{displaymath}
\begin{quote}%
eh? Something wrong with the factors of $1/2$ here\ldots{}


\end{quote}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{the_solution_to_the_strong_cp_problem_via_axions}{}\subsubsection*{{The solution to the strong CP problem via axions}}\label{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]] $\theta$ which is argued to show that the [[axion]] expectation value $\theta = \langle a \rangle$ vanishes (\hyperlink{VafaWitten84}{Vafa-Witten 84}).

See at \emph{\href{axion#AsASolutionToTheStrongCPProblem}{axion -- As a solution to the strong CP-problem}}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{[[vacuum]]}

\begin{itemize}%
\item [[vacuum state]], [[Hadamard vacuum state]]


\item [[interacting vacuum]]


\item [[vacuum expectation value]], [[vacuum amplitude]], [[vacuum fluctuation]]


\item [[vacuum energy]]


\item [[vacuum diagram]]


\item [[vacuum diagram]]


\item [[thermal vacuum]], [[KMS state]]


\item [[vacuum stability]]


\item [[false vacuum]], [[tachyon]], [[Coleman-De Luccia instanton]]


\item [[theta vacuum]]


\item [[perturbative string theory vacuum]]


\item [[landscape of string theory vacua]]



\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For instance

\begin{itemize}%
\item Claudio Scrucca, section 1.6 in \emph{Advanced quantum field theory} \href{http://itp.epfl.ch/webdav/site/itp/users/181759/public/aqft.pdf}{pdf}


\item [[José Edelstein]], page 2 of \emph{Lecture 8: 1-loop closed string vacuum amplitude}, 2013 (\href{http://www-fp.usc.es/~edels/Strings/Lect8Str.pdf}{pdf})


\item [[Daniel Quillen]], \emph{Determinants of Cauchy-Riemann Operators over a Riemann Surface}, Functional Anal. Appl. 19 (1985) 31.


\item [[Daniel Freed]], \emph{On determinant line bundles}, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987, (revised \href{http://www.math.utexas.edu/~dafr/Index/determinants.pdf}{pdf}, \href{http://arxiv.org/abs/dg-ga/9505002}{dg-ga/9505002})


\item Jia Qiu, section 2.8 of \emph{Lecture notes on topological field theory} \href{http://arxiv.org/abs/1201.5550}{arXiv:1201.5550}



\end{itemize}
The application to the [[axion]] solution to the [[strong CP problem]] is due to

\begin{itemize}%
\item [[Cumrun Vafa]], [[Edward Witten]], \emph{Parity Conservation in Quantum Chromodynamics} Phys. Rev. Lett. 53, 535 (1984) (\href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.53.535}{publisher})

\end{itemize}


\end{document}