## LATEX

### The collapse equation

The unitary and deterministic time evolution of an isolated physical system ${\cal S}$ obeys the Schrodinger equation until a single collapse event (caused by a measurement) occurs at some instant $t=T$. We will assume that the collapse process is not instantaneous but instead it lasts for a finite duration $\tau$ of the order of the Planck time. During the collapse process between $T$ and $T+\tau$ we will further assume that the time evolution of the physical system ${\cal S}$ is controlled by a modified Schrodinger equation with a Hamiltonian $H(t)$ given by
\begin{eqnarray}
i\hbar\frac{\partial }{\partial t}|\psi(t)\rangle=H(t)|\psi(t)\rangle+|\phi(t)\rangle~,~T\le t\le T+\tau.\label{mod}
\end{eqnarray}
The collapse Hamiltonian ${\cal H}(t)$, as opposed to the standard Hamiltonian $H(t)$, is not required to be a hermitian operator yielding therefore a non-unitary time evolution  and it is also generally a non-local and stochastic operator. The instantaneous character of the collapse is relaxed by the assumption that $\tau\neq 0$.

Furthermore, we will assume that during the collapse between $T$ and $T+\tau$ the dominant force is actually provided by the collapse Hamiltonian ${\cal H}(t)$ and not by $H(t)$.

The state vector $|\phi(t)\rangle$ encodes the collapse dynamics and it is postulated for simplicity to be given in terms of a  collapse Hamiltonian ${\cal H}(t)$ by the simple rule
\begin{eqnarray}
|\phi(t)\rangle={\cal H}(t) |\psi(t)\rangle.\label{colla2}
\end{eqnarray}
The state vector $|\psi^{\prime}(t)\rangle$ before $t=T$ and after $t=T+\tau$ is a solution of the ordinary Schrodinger equation, viz
\begin{eqnarray}
i\hbar\frac{\partial }{\partial t}|\psi^{\prime}(t)\rangle=H(t)|\psi^{\prime}(t)\rangle.
\end{eqnarray}
The collapse is characterized by a preferred-basis with complete and orthonormal states denoted by $|\phi_n\rangle$. At the end of the collapse at the instant $T+\tau$ the state of the system is given precisely by \begin{eqnarray}
|\psi(T+\tau)\rangle=\langle\phi_k|\psi(T+\tau)\rangle |\phi_k\rangle.\label{colla}
\end{eqnarray}
In other words, the state of the system ${\cal S}$ collapses to the preferred-basis state $|\phi_k\rangle$. We expand the state $|\phi(t)\rangle$ in the preferred-basis as
\begin{eqnarray}
|\phi(t)\rangle&=&\sum_n\langle\phi_n|\phi(t)\rangle|\phi_n\rangle\nonumber\\
&=&\sum_{n,m} \langle\phi_n|{\cal H}(t)|\phi_m\rangle\langle\phi_m|\psi(t)\rangle|\phi_n\rangle.
\end{eqnarray}
Hence
\begin{eqnarray}
\langle\phi_n|\phi(t)\rangle
&=&\sum_{m} \langle\phi_n|{\cal H}(t)|\phi_m\rangle\langle\phi_m|\psi(t)\rangle.
\end{eqnarray}
First orientation we will consider first the case of a diagonal collapse Hamiltonian, i.e. $\langle\phi_n|{\cal H}(t)|\phi_m\rangle=i\hbar \lambda_n(t) \delta_{n,m}$. In other words, the preferred-basis $|\phi_n\rangle$ is the eigenbasis of the collapse Hamiltonian ${\cal H}(t)$ with complex eigenvalues given precisley by $\lambda_n(t)$. We get then
\begin{eqnarray}
\langle\phi_n|\phi(t)\rangle
&=&i\hbar \lambda_n(t)\langle\phi_n|\psi(t)\rangle.
\end{eqnarray}
In general, this equation should be viewed as a parametrization of the collapse state $|\phi(t)\rangle$ in terms of the state vector $|\psi(t)\rangle$, the preferred-basis states $|\phi_n\rangle$ and the functions $\lambda_n(t)$. In other words, the collapse Hamiltonian ${\cal H}(t)$ does not need to be diagonal and $\lambda_n(t)$ are not its complex eigenvalues but only a characterization of the collapse state $|\phi(t)\rangle$ which does not need to be of the form (\ref{colla2}) altogether.

The modified Schrodinger equation (\ref{mod}) can then be rewritten in the form
\begin{eqnarray}
i\hbar\frac{\partial }{\partial t}|\psi(t)\rangle=H(t)|\psi(t)\rangle+i\hbar\sum_n \lambda_n(t)\langle\phi_n|\psi(t)\rangle|\phi_n\rangle.
\end{eqnarray}
At the beginning  of the collapse at $t=T$ we must clearly have the unmodified Schrodinger dynamics, viz $|\phi(T)\rangle=0$, and since  generally $\langle\phi_n|\psi(T)\rangle\neq 0$, we must then have at $t=T$ the condition
\begin{eqnarray}
\lambda_n(T)=0.
\end{eqnarray}
At the end of the collapse at $t=T+\tau$ we must also have the unmodified Schrodinger dynamics and thus
\begin{eqnarray}
\lambda_n(T+\tau)\langle\phi_n|\psi(T+\tau)\rangle=0.\label{res1}
\end{eqnarray}
For $n=k$ we have $\lambda_k(T+\tau)=0$ but for $n\neq k$ we can either have $\lambda_n(T+\tau)=0$  or $\langle\phi_n|\psi(T+\tau)\rangle=0$.

The switching on and off of the collapse Hamiltonian at $t=T$ and $t=T+\tau$ is similar to what happens in second order phase transitions where the modified Schrodinger equation appears as the renormalization group equation which controls this dynamics.

We can rewrite the modified Schrodinger equation in the form

\begin{eqnarray}
i\hbar\frac{\partial }{\partial t}\langle\phi_n|\psi(t)\rangle=\langle\phi_n|H(t)|\psi(t)\rangle+i\hbar\lambda_n(t)\langle\phi_n|\psi(t)\rangle.\label{schr}
\end{eqnarray}
We will focus on the time evolution of the components $n\neq k$ where $k$ labels the collapsed state as indicated by equation (\ref{colla}). The solution is immediately given by
\begin{eqnarray}
\langle\phi_n|\psi(t)\rangle&=&\exp\big(\int_T^tdt_1\lambda_n(t_1)\big)\bigg[\langle\phi_n|\psi(T)\rangle\nonumber\\
&+&\int_T^tdt_1\exp\big(-\int_T^{t_1}dt_2\lambda_n(t_2)\big)\langle\phi_n|\frac{H(t_1)}{i\hbar}|\psi(t_1)\rangle\bigg].
\end{eqnarray}
By imposing (\ref{colla}) we obtain the condition
\begin{eqnarray}
\langle\phi_n|\psi(T+\tau)\rangle&=&\exp\big(\int_T^{T+\tau}dt_1\lambda_n(t_1)\big)\bigg[\langle\phi_n|\psi(T)\rangle\nonumber\\
&+&\int_T^{T+\tau}dt_1\exp\big(-\int_T^{t_1}dt_2\lambda_n(t_2)\big)\langle\phi_n|\frac{H(t_1)}{i\hbar}|\psi(t_1)\rangle\bigg]=0.\label{cond}
\end{eqnarray}
As we have already discuss the collapse Hamiltonian ${\cal H}(t)$ is the dominant force during the collapse process between $T$ and $T+\tau$. Thus, towards the end of the collapse we expect that the collapse state $|\phi(t)\rangle$ dominates the modified Schrodinger equation, viz
\begin{eqnarray}
i\hbar\frac{\partial }{\partial t}|\psi(t)\rangle\simeq |\phi(t)\rangle~,~t\longrightarrow T+\tau.\label{schr1}
\end{eqnarray}
This means in particular that during the collapse between the instants $T$ and $T+\tau$ we can treat the collapse Hamiltonian ${\cal H}(t)$ as the unperturbed Hamiltonian while the Hamiltonian $H(t)$ should be considered  a small perturbation around it. If we simply set $H(t)=0$ in equation (\ref{cond}) and then use the fact that we have generally  $\langle\phi_n|\psi(T)\rangle\ne 0$ we see immediately that we must have the condition
\begin{eqnarray}
\exp\big(\int_T^{T+\tau}dt_1\lambda_n(t_1)\big)=0\Rightarrow \int_T^{t}dt_1\lambda_n(t_1)\longrightarrow -\infty~,~t\longrightarrow T+\tau.
\end{eqnarray}
We get then the behavior \cite{Mei}

\begin{eqnarray}
\lambda_n(t)=\frac{\alpha_n}{(T+\tau-t)^{\beta_n}}~,~t\longrightarrow T+\tau.\label{res2}
\end{eqnarray}
The coefficients $\alpha_n$ and $\beta_n$ are such that (where ${\rm Real}$ denotes the real part)
\begin{eqnarray}
{\rm Real}(\alpha_n)\lt 0~,~\beta_n\gt 1.\label{res3}
\end{eqnarray}
Equations (\ref{res2}) holds only for $n\neq k$ since $\lambda_k(T+\tau)=0$ and $\langle\phi_k|\psi(T+\tau)\rangle\neq 0$ as we have already seen from equations (\ref{colla}) and (\ref{res1}). From the two results (\ref{res1}) and (\ref{res2}) we conclude that the components with $n\neq k$ of the wave function $|\psi(t)\rangle$ at the end of the collapse must vanish as desired, viz
\begin{eqnarray}
\langle\phi_n|\psi(T+\tau)\rangle=0.\label{res4}
\end{eqnarray}
We check now explicitly that the behavior of $\langle\phi_n|{H(t)}|\psi(t)\rangle$ is sub-dominant at the end of the collapse. We start by assuming the converse proposition, i.e. $\langle\phi_n|{H(t)}|\psi(t)\rangle$ is much larger than $i\hbar\lambda_n(t)\langle\phi_n|\psi(t)\rangle$. Thus by assuming near $t=T+\tau$ the behavior
\begin{eqnarray}
\langle\phi_n|{H(t)}|\psi(t)\rangle=\gamma_n(T+\tau-t)^{\delta_n}
\end{eqnarray}
and neglecting $i\hbar\lambda_n(t)\langle\phi_n|\psi(t)\rangle$ we obtain the behavior
\begin{eqnarray}
\langle\phi_n|\psi(t)\rangle=-\frac{1}{i\hbar}\frac{\gamma_n}{\delta_n+1}(T+\tau-t)^{\delta_n+1}.
\end{eqnarray}
By substituting this solution back into the modified Schrodinger equation (\ref{schr}) and using the behavior (\ref{res2}) we obtain a contradiction for all values of $\beta_n$ unless $\beta_n=1$ which is forbidden by equation (\ref{res3}).

For $\beta_n\gt 1$ we obtain then a contradiction which means that the initial proposition is not correct. In other words, $\langle\phi_n|{H(t)}|\psi(t)\rangle$ is indeed much smaller than $i\hbar\lambda_n(t)\langle\phi_n|\psi(t)\rangle$ at the end of the collapse and as a consequence the modified Schrodinger equation (\ref{schr}) in the limit $t\longrightarrow T+\tau$ reduces to
\begin{eqnarray}
i\hbar\frac{\partial }{\partial t}\langle\phi_n|\psi(t)\rangle\simeq i\hbar\lambda_n(t)\langle\phi_n|\psi(t)\rangle~,~t\longrightarrow T+\tau.
\end{eqnarray}
This is precisely equation (\ref{schr1}).

For the value $\beta_n=1$ we have found that $\langle\phi_n|{H(t)}|\psi(t)\rangle$ is much larger than $i\hbar\lambda_n(t)\langle\phi_n|\psi(t)\rangle$. To see that the value $\beta_n=1$ can not be allowed we substitute the behavior (\ref{res2}) with $\beta_n=1$ into the modified Schrodinger equation (\ref{schr}), expand about $t=T+\tau$, use the result (\ref{res4}) and then integrate  both sides of the modified Schrodinger equation to obtain
\begin{eqnarray}
\langle\phi_n|\psi(t)\rangle=-\frac{1}{i\hbar}\frac{1}{1+\alpha_n}\langle\phi_n|{H(T+\tau)}|\psi(T+\tau)\rangle(T+\tau-t).
\end{eqnarray}
This vanishes in the limit $t\longrightarrow T+\tau$ but not sufficiently fast. Indeed, we also compute
\begin{eqnarray}
\lambda_n(t)\langle\phi_n|\psi(t)\rangle=-\frac{1}{i\hbar}\frac{\alpha_n}{1+\alpha_n}\langle\phi_n|{H(T+\tau)}|\psi(T+\tau)\rangle.
\end{eqnarray}
By using the condition (\ref{res1}) we see that $\langle\phi_n|{H(T+\tau)}|\psi(T+\tau)\rangle=0$. This indicates that $\langle\phi_n|{H(t)}|\psi(t)\rangle$ is actually of the same order as $i\hbar\lambda_n(t)\langle\phi_n|\psi(t)\rangle$ which contradicts the initial proposition. Hence the value $\beta_n=1$ is disallowed.

We have obtained then for $n\neq k$ the crucial result

\begin{eqnarray}
\langle\phi_n|{H(T+\tau)}|\psi(T+\tau)\rangle=0.
\end{eqnarray}
By using (\ref{colla}) we obtain the eigenvalue equation
\begin{eqnarray}
{H(T+\tau)}|\phi_k\rangle=v_k|\phi_k\rangle.
\end{eqnarray}
In other words, the preferred-basis states $|\phi_k\rangle$ are nothing else but the eigenvectors of the standard Hamiltonian $H(t)$ at the end of the collapse.

%\cite{Mei}
\bibitem{Mei}
S.~Mei,
On the origin of preferred-basis and evolution pattern of wave function,''
arXiv:1311.4405 [quant-ph].