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\renewcommand{\@oddfoot}{--- ~March, 2006~ ---\hfil
\raisebox{0.3ex}{\tiny --- ~D.~A.~Arbatsky~ ``TCP II: Proof of the
uncertainty principle''~
---}\hfil --- ~\thepage~ ---}
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\title{The certainty principle II\\
Proof of the uncertainty principle}
\author{D.~A.~Arbatsky\footnote{ http://daarb.narod.ru/ , http://wave.front.ru/ }}
\date{March, 2006}
\maketitle

\begin{abstract}
A more detailed derivation of the Heisenberg uncertainty principle
from the certainty principle is given.
\end{abstract}

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\paragraph{Introductory remarks.}
After publication of the paper~\cite{Arb2005tcp} I received many
letters, including those with references. In this connection, I
think it is necessary to specify the following:
\begin{enumerate}
\item
The metric introduced by me (``quantum angle'') is known to
mathematicians from 1904 as {\it Fubini-Study metric}.
\item
The correct ``uncertainty relation'' (in fact, {\it certainty}
relation) for the quantities energy - time was first suggested by
Mandelshtam and Tamm~\cite{MandTamm1945}.
\item
Mandelshtam and Tamm studied a quantum system in the Schr\"odinger
representation and did not use group theory methods. For this
reason they could not understand that their result has more
general character.
\item
In contrast, I used group theory formulations. And implied that
the system can be studied, in particular, in the representation of
relativistic canonical quantization~\cite{Arb2005wircq}. This
allowed me to formulate the certainty principle and to suggest
more general inequalities.
\end{enumerate}

Some of my critics refused to believe that the Heisenberg
uncertainty principle is really a consequence of the certainty
principle. They claimed that ``this can not be so, because this
can never be so''.

Nevertheless, this is so. And here I give a more detailed
explanation.

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\paragraph{Derivation of the uncertainty principle.}
Suppose that for a given quantum system we succeeded to find some
observable $X$, that can be considered in some sense a
``coordinate operator''.

Suppose that $X$ is a self-adjoint operator with continuous
spectrum, $X=X^*$. Let us denote $\Omega_{(a,b)}$ its spectral
projector\footnote{Roughly speaking, spectral projector is an
operator nullifying wave function in $X$-representation outside of
the given interval.} for an arbitrary real interval $(a,b)$.

Suppose also that we have a self-adjoint operator $P$, $P=P^*$,
such that for any $a$, $b$ and $\delta x$ we have equality:
\[
 e^{+\,i\,\delta x\,P/\hbar} \,
  \Omega_{(a+\delta x,b+\delta x)} \,
    e^{-\,i\,\delta x\,P/\hbar} \,=\,
 \Omega_{(a,b)} \ ,
\]
i.~e. $P$ is a ``generator of spectral shifts'' for $X$. As we
know, such an operator is usually an operator of momentum.

Suppose now that the system is in state $\rangle$,
$\langle\,|\,\rangle=1$.

The quantity $\langle\,\Omega_{(a,b)}\,\rangle$, obviously,
defines the probability to find the system inside the interval
$(a,b)$. Let us define such $l$ and $r$, that
\[
 \textstyle
 \langle\,\Omega_{(-\infty,l)}\,\rangle
 \,=\,
 \langle\,\Omega_{(r,+\infty)}\,\rangle
 \,=\,
 \frac12-\frac12\sqrt{1-\cos^21} \,\approx\,
 0,07926\dots
\]
It is easy to see, that $l$ and $r$ exist\footnote{But, generally
speaking, they are not unique. In order to eliminate this
non-uniqueness, it is convenient to choose $l$ maximum of the
possible, and $r$~--- minimum. Then the distance $r-l$ will be
minimum.}. So, the quantity $\delta_\rangle X = r-l$ can be
naturally called ``uncertainty'' of the coordinate $X$.

\mytheorem{}
 {\it The following inequality takes place (the uncertainty principle):
\begin{equation}
 \begin{array}{|ccc|}
 \hline
 \phantom{1} & & \phantom{1}\\
  & \delta_\rangle X \Delta_\rangle P \geqslant \hbar & \\
 \phantom{1} & & \phantom{1}\\
 \hline
 \end{array}
 \label{Certainty}
\end{equation}}

In order to prove this theorem let us first estimate the scalar
product of the vector $\rangle$ and the shifted vector
$e^{-\,i\,\delta_\rangle X\,P/\hbar} \,\rangle$:
\[
 \big|\,
  \langle\,
   |\,
   e^{-\,i\,\delta_\rangle X\,P/\hbar}
  \,\rangle
 \,\big|
 \,=\,
 \big|\,
  \langle\,
   (\,\Omega_{(-\infty,r)} + \Omega_{(r,+\infty)}\,)
   \,
   e^{-\,i\,\delta_\rangle X\,P/\hbar}
  \,\rangle
 \,\big|
 \,=\,
\]
\[
 \,=\,
 \big|\,
  \langle\,
   \,\Omega_{(-\infty,r)}\, e^{-\,i\,\delta_\rangle X\,P/\hbar}
  \,\rangle
   \,+\,
  \langle\,
   \Omega_{(r,+\infty)}\, e^{-\,i\,\delta_\rangle X\,P/\hbar}
  \,\rangle
 \,\big|
 \,\leqslant\,
\]
\[
 \,\leqslant\,
 \big|\,
  \langle\,
   \,\Omega_{(-\infty,r)}\, e^{-\,i\,\delta_\rangle X\,P/\hbar}
  \,\rangle
 \,\big|
   \,+\,
 \big|\,
  \langle\,
   \Omega_{(r,+\infty)}\, e^{-\,i\,\delta_\rangle X\,P/\hbar}
  \,\rangle
 \,\big|
 \,=\,
\]
\[
 \,=\,
 \big|\,
  \langle\,
   \,(\Omega_{(-\infty,r)})^2\, e^{-\,i\,\delta_\rangle X\,P/\hbar}
  \,\rangle
 \,\big|
   \,+\,
 \big|\,
  \langle\,
   (\Omega_{(r,+\infty)})^2\, e^{-\,i\,\delta_\rangle X\,P/\hbar}
  \,\rangle
 \,\big|
 \,=\,
\]
\[
 \,=\,
 \big|\,
  \langle\,
   \,\Omega_{(-\infty,r)}\,
   e^{-\,i\,\delta_\rangle X\,P/\hbar}
   \,\Omega_{(-\infty,l)}
  \,\rangle
 \,\big|
   \,+\,
 \big|\,
  \langle\,
   \Omega_{(r,+\infty)}\,
   e^{-\,i\,\delta_\rangle X\,P/\hbar}
   \,\Omega_{(l,+\infty)}
  \,\rangle
 \,\big|
 \,\leqslant\,
\]
\[
 \,\leqslant\,
  \langle\,
   \Omega_{(-\infty,r)}
  \,\rangle^{1/2}\,
  \langle\,
   \Omega_{(-\infty,l)}
  \,\rangle^{1/2}\,
   \,+\,
  \langle\,
   \Omega_{(r,+\infty)}
  \,\rangle^{1/2}\,
  \langle\,
   \Omega_{(l,+\infty)}
  \,\rangle^{1/2}\,
 \,=\,
\]
\[
 \textstyle
 \,=\,
 \sqrt{\frac12+\frac12\sqrt{1-\cos^21}}
 \cdot
 \sqrt{\frac12-\frac12\sqrt{1-\cos^21}}
 \ + \qquad\qquad\qquad\qquad
\]
\[
 \textstyle
 \qquad\qquad\qquad\qquad + \,
 \sqrt{\frac12-\frac12\sqrt{1-\cos^21}}
 \cdot
 \sqrt{\frac12+\frac12\sqrt{1-\cos^21}}
 \,=\,
 \cos 1
 \ .
\]
Here, coming from the fifth to the sixth line, we estimated both
terms by the Cauchy-Bunyakovsky-Schwarz inequality.

So, for the quantum angle between the initial and the shifted
vectors we have estimation:
\[
 \angle\,\bigl(\
  \rangle
  \ , \
  e^{-\,i\,\delta_\rangle X\,P/\hbar}\,\rangle
 \ \bigr)
 \ \geqslant\ 1 \ .
\]
But it means that under the action of
 $e^{-\,i\,\delta_\rangle X\,P/\hbar}$
the vector $\rangle$ changes {\it
substantially}~\cite{Arb2005tcp}.

Applying the {\it certainty}\/ principle~\cite{Arb2005tcp}, we
directly get~(\ref{Certainty}). $\blacksquare$

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\paragraph{Discussion.}
Historically, it stacked up so~\cite{HilgUff2001} that now in all
textbooks the Heisenberg uncertainty principle is illustrated with
the help of the Kennard inequality,
\[
 \Delta_\rangle X\,\Delta_\rangle P \,
 \geqslant\, \frac\hbar2 \ \ ,
\]
which is not identical with~(\ref{Certainty}). Its undoubted
virtue is that its proof is easier to understand for a person, who
just starts to study quantum mechanics. Nevertheless, I believe
that~(\ref{Certainty}) is more fundamental.

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\paragraph{Acknowledgements.}
In closing I want to thank T.~A.~Bolokhov, A.~V.~Ossipov,
E.~V.~Aksyonova, A.~Yu.~To\-sche\-vi\-ko\-va, A.~Kleyn, M.~Gatti,
A.~K.~Pati, and P.~Enders for helpful discussions, references and
help in procuring literature.

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\begin{thebibliography}{99}

\bibitem{Arb2005tcp}
D.~A.~Arbatsky ``The certainty principle'' (2005), \\
 http://daarb.narod.ru/tcp-eng.html ,
 http://wave.front.ru/tcp-eng.html ,
 arXiv:quant-ph/0506165 . \\{}
 [Rus.: D.~A.~Arbatsky \rql Printsip opredelyonnosti\rqr{} (2005),
 \\
 http://daarb.narod.ru/tcp-rus.html ,
 http://wave.front.ru/tcp-rus.html ,
 arXiv:quant-ph/0506165 .]

\bibitem{MandTamm1945}
 L.~I.~Mandelshtam, I.~E.~Tamm, ``The uncertainty relation
between energy and time in nonrelativistic quantum mechanics'', J.
Phys. (USSR) {\bf 9}, 249-254 (1945). \\{}
 [Rus.: L.~I.~Mandelshtam, I.~E.~Tamm \rql Sootnoshenie
neopredelyonnosti energiya-vremya v nerelyativistskoy kvantovoy
mekhanike\rqr, Izv. Akad. Nauk SSSR (ser. fiz.) {\bf 9}, 122-128
(1945), in~\cite{Mand1947}, p. 306-315.]

\bibitem{Mand1947}
L.~I.~Mandelshtam, {\it Polnoe sobranie trudov}, v.~2, M.-L.
(1947).

\bibitem{Arb2005wircq}
D.~A.~Arbatsky ``What is ``relativistic canonical
quantization''?'' (2005), \\
 http://daarb.narod.ru/wircq-eng.html ,
 http://wave.front.ru/wircq-eng.html ,
 arXiv:math-ph/0503029 . \\{}
 [Rus.: D.~A.~Arbatsky \rql Chto takoe \rql relyativistskoe
 kanonicheskoe kvantovanie\rqr{}?\rqr{} (2005), \\
 http://daarb.narod.ru/wircq-rus.html ,
 http://wave.front.ru/wircq-rus.html ,
 arXiv:math-ph/0503029 .]

\bibitem{HilgUff2001}
J.~Hilgevoord, J.~Uffink ``The uncertainty principle'', in {\it
The Stanford encyclopedia of philosophy}, ed. E.~N.~Zalta (2001),
 http://plato.stanford.edu/archives/win2001/entries/qt-uncertainty/ .

\end{thebibliography}

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