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\begin{document}
\begin{center}

University of California at San Diego -- 
Department of Physics --
{\bf Put your name on your work.}

{\Large\bf  Physics 215B QFT Winter 2025}\\
{\Large\bf Assignment 1 \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Tuesday, January 14, 2025}



%I will add another problem later today.
\bigskip
\hrule
\begin{enumerate}



\item {\bf Brain-warmer.}  
Convince yourself that 
$$ \(\partial_g\)^n e^{ - 1 / g } |_{g=0} = 0 ~~\forall n .$$
This means that a function of the form $ e^{ - 1/g} $ does not have a useful series expansion
about $g=0$.




\item {\bf Scale invariant quantum mechanics.}

Consider the action for one quantum variable $r$ with $ r>0 $ and 
% \in \IR^+$ 
%$$ S[r] = \int dt \( \half \dot r^2 - V(r) \) ,~~~~V(r) = { \lambda \over r^2}~.$$
$$ S[r] = \int dt \( \half m \dot r^2 - V(r) \) ,~~~~V(r) = { \lambda \over r^2}~.$$


\begin{enumerate}


\item 
%\item 
Show that the (non-relativistic) mass parameter $m$ can be eliminated by a multiplicative redefinition of the field $r$ or of the time $t$.
As a result, convince yourself that the physics of interest here should only depend on 
the combination $m\lambda$.
Show that the coupling $m\lambda$ is dimensionless: $ [m\lambda]=0$.


\item Show that this action is {\it scale invariant}, i.e., show that 
the transformation 
%\cor
\be\label{eq:scale} r(t) \to s^\alpha \cdot r(st)~~~\ee
%\be\label{eq:scale} r(t) \to r_s(t_s) \equiv s^\alpha \cdot r(t)~~~, ~~~~ t \to t_s = t/s \ee
%\cob
(for some $\alpha$ which you must determine),
(with $s \in \IR^+$) is a symmetry.  Find the associated Noether charge $ \CD$.
For this last step, it will be useful to note that the infinitesimal version of 
\eqref{eq:scale} is ($ s = e^{a}, a \ll 1 $)
%\cor 
$$ \delta r(t) = a \(  \alpha + t {d \over dt} \) r(t) . $$
%$$ \delta r(t) = a \alpha r(t) . $$


\item Find the position-space Hamiltonian $\HH$ governing the dynamics of $r$.  
Show that the Schr\"odinger equation is Bessel's equation
$$ \( - 
%\cor \half \cob 
{\partial_r^2\over 2m} + { \lambda \over r^2 } \) \psi_E(r) = E \psi_E(r) .$$
%\cor 
Show that the Noether charge associated $\CD$ with scale transformations ($\equiv$ dilatations) satisfies:
 $ [ \CD, \HH] =  { -} \ii \HH$.  This equation says that the Hamiltonian has a 
 definite scaling dimension, \ie\ that its scale tranformation is
 $ \delta \HH = \ii a [ \CD , \HH ] = { +}  a \HH$. \cob
 Note that you should not need to use arcane facts about Bessel functions,
 only the asymptotic analysis of the equation, in subsequent parts of the problem.

 

\item Describe the behavior of the solutions to this equation 
as $ r \to 0$.
[Hint: in this limit you can ignore the RHS.  Make a power-law ansatz: $\psi(r) \sim r^{\Delta}$ and find $\Delta$.]


\item What happens if $  2m \cob \lambda < - {1\over 4} $ ?
It looks like there is a continuum of negative-energy solutions (boundstates).
This is another example of a {\it too-attractive} potential.





\item  A hermitian operator has orthogonal eigenvectors.
We will show next that to make $\HH$ hermitian when $ 2m \cob \lambda < - {1\over 4}$,
we must impose a constraint on the wavefunctions: 
\be\label{eq:self-adjoint} \( \psi^\star_E \partial_r \psi_E - \psi_E \partial_r \psi_E^\star \)|_{r=0}= 0 ~.\ee
There are two useful perspectives on this condition: one is that 
the LHS is the probability current passing through the point $r=0$.  
If it is nonzero, it means probability is leaking into a hole in the world at $r=0$.  

The other perspective is the following.  
Consider two eigenfunctions: $$ \HH \psi_E = E \psi_E ,~~\HH \psi_{E'} = E'\psi_{E'} . $$
Multiply the first equation by $\psi_{E'}^\star$ and integrate;
multiply the second by $\psi_{E}^\star$ and integrate;
take the difference (maybe take the complex conjugate of the second term).  Show that the result is a boundary term which must vanish when $ E = E'$.



\item Show that the condition 
\eqref{eq:self-adjoint} is empty for $ 2m \cob \lambda > -{1\over 4}$.
We must impose a further condition
on the eigenfunctions for $ 2m \cob \lambda <  - {1\over 4}$ in order to satisfy \eqref{eq:self-adjoint}.  
[It will help to impose a boundary condition at $r=\eps$ for some UV cutoff $\eps$.
Notice that this regulator explicitly breaks scale invariance.]
Show that the resulting spectrum of boundstates 
has a {\it discrete} scale invariance.

[Cultural remark: For some reason I don't know, restricting the Hilbert space in this way is called a
{\it self-adjoint extension}.]


\item{} [Extra credit] Consider instead a 
particle moving in $\IR^d$ with a central $1/r^2$ potential, $r^2 \equiv \vec x \cdot \vec x$,
$$ S[\vec x] = \int dt  \(  
 \half m  \cob
%\half
\dot{\vec x}  \cdot \dot{\vec x}  - {\lambda \over r^2} \) .$$
Show that the same analysis applies 
(\eg\ to the s-wave states)
with minor modifications.  

[A useful intermediate result is the following representation of (minus) the laplacian in $\IR^d$:
$$ \vec p^2 =- { 1\over r^{d-1} } \partial_r \(r^{d-1} \partial_r \) + {\hat L^2 \over r^2} , ~~~
\hat L^2 \equiv \half \hat L_{ij} \hat L_{ij}, ~~ L_{ij} = - \ii \( x_i \partial_j - x_j \partial_i \) , $$
where $ r^2 \equiv x^i x^i$.
By `s-wave states' I mean those annihilated by $ \hat L^2$.]


\end{enumerate}

\end{enumerate}




%\end{enumerate}
\end{document}
