”Derrick’s theorem” Consider the functional
\[ E [ \phi ] = \int d ^ { d } x \left( \frac { 1 } { 2 } K _ { a b } ( \phi ) \sum _ { i = 1 } ^ { d } \partial _ { i } \phi ^ { a } \partial _ { i } \phi ^ { b } + V ( \phi ) \right) ,\tag{20} \]
where \(\phi = \phi ( x _ { 1 } , . . . , x _ { d } ) \in C ^ { 2 } ( \mathbb { R } ^ { d } ) , K _ { a b } ( \phi )\) is a positive-definite matrix for any \(\phi ,\) i.e.,
\[ K _ { a b } ( \phi ) \partial _ { i } \phi ^ { a } \partial _ { j } \phi ^ { b } \geqslant 0 ,\tag{21} \]
where the equality implies \(\phi = 0\) , and \(V ( \phi ) \geqslant 0 , V ( \phi ) = 0 \Rightarrow \phi = 0\) Suppose \(\phi _ { 0 } ( x )\) is a non-zero extremum of \(E [ \phi ]\) . Consider the configurations of the form \(\phi _ { \lambda } ( x ) = \phi _ { 0 } ( \lambda x )\) , obtaining from \(\phi _ { 0 } ( x )\) by stretching the coordinates by a factor of λ.
- Show that \(E ( \lambda ) \equiv E [ \phi _ { \lambda } ]\) must satisfy
\[ \left. \frac { d { E } } { d { \lambda } } \right| _ { \lambda = 1 } = 0 .\tag{22} \]
- Using the notations
\[ \Gamma = \int d ^ { d } x \frac { 1 } { 2 } K _ { a b } ( \phi _ { 0 } ) \sum _ { i = 1 } ^ { d } \partial _ { i } \phi _ { 0 } ^ { a } \partial _ { i } \phi _ { 0 } ^ { b } , ~ \Pi = \int d ^ { d } x V ( \phi _ { 0 } ) ,\tag{23} \]
show that the above relation implies
\[ ( 2 - d ) \Gamma - d \Pi = 0 .\tag{24} \]
- Give a conclusion about the existence of \(\phi _ { 0 }\) , if
\[ ( \mathrm { a } ) d > 2 \qquad \quad ( \mathrm { b } ) d = 2 \qquad \quad ( \mathrm { c } ) d = 1 \]