Vacuum expectation value

As vacuum expectation value in is quantum field theory the expected value of an operator in the state of lowest energy of a system, its basic or vacuum state , respectively. If the quantum vacuum is and an operator, then the vacuum expectation value of the operator,, is defined as: ${\displaystyle |\Omega \rangle }$ ${\displaystyle O}$ ${\displaystyle \langle O\rangle }$

${\displaystyle \langle O\rangle =\langle \Omega |O|\Omega \rangle }$

The vacuum expectation values ​​of field operators or of products of several of them are of particular importance. As a rule, the vacuum expectation value of an individual field operator is zero (equivalent to the statement that no field exists in the quantum vacuum), but in the case of spontaneous symmetry breaking , the vacuum expectation value assumes a value other than zero. The most important case of such a spontaneous symmetry breaking is the Higgs mechanism ; the vacuum expectation value of the Higgs field is ${\displaystyle \phi }$

${\displaystyle \langle \phi \rangle ={\frac {v}{\sqrt {2}}}={\frac {246\,{\text{GeV}}}{\sqrt {2}}}}$.

The square root of two in the denominator is convention. Sometimes it is also referred to as the vacuum expectation value. ${\displaystyle v}$

The vacuum expectation value of a time-ordered product of field operators can be reduced to a sum of vacuum expectation values ​​of a product of two field operators using Wick's theorem .

literature

• Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).
• Expected vacuum values . In: Lexicon of Physics . Spectrum Academic Publishing House, Heidelberg 1998 ( Spektrum.de ).