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\makeatletter
\renewcommand{\@oddfoot}{--- ~August, 2006~ ---\hfil
\raisebox{0.3ex}{\tiny --- ~D.~A.~Arbatsky~ ``The certainty
principle (review)''~ ---}\hfil --- ~\thepage~ ---}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% TABLE

\newcommand{\titu}{%
\begin{tabular}{c}
 {\bf uncertainty principle}\\
 \footnotesize (Heisenberg, 1927)
\end{tabular}}

\newcommand{\titc}{%
\begin{tabular}{c}
 {\bf certainty principle}\\
 \footnotesize (Arbatsky, 2005)
\end{tabular}}

\newcommand{\xpuk}{%
\begin{tabular}{c}
 $ \Delta_\rangle X\,\Delta_\rangle P \,\geqslant\, \frac\hbar2 $\\
 \footnotesize (Kennard, 1927)
\end{tabular}}

\newcommand{\xpua}{%
\begin{tabular}{c}
 $ \delta_\rangle X\,\Delta_\rangle P \,\geqslant \hbar $\\
 \footnotesize (Arbatsky, 2006)
\end{tabular}}

\newcommand{\xpco}{%
\begin{tabular}{c}
 $ |\delta x|\, \Delta_\rangle P \geqslant \hbar $\\
 \footnotesize (Arbatsky, 2005)
\end{tabular}}

\newcommand{\xpca}{%
\begin{tabular}{c}
 $\Delta_\rangle (\,
   \delta x_1\,P_1\, +\,
   \delta x_2\,P_2\, +\,
   \delta x_3\,P_3\,
   ) \geqslant \hbar $\\
 \footnotesize (Arbatsky, 2005)
\end{tabular}}

\newcommand{\pjuj}{%
\begin{tabular}{c}
 $\displaystyle
 \Delta'_\rangle\Phi \, \Delta_\rangle J \,\geqslant\,
 \frac\hbar2
 \,
 \left[\,
  1\, -\, \frac3{\pi^2}\, (\Delta'_\rangle\Phi)^2
 \,\right]$\\
 \footnotesize (Judge, 1964)
\end{tabular}}

\newcommand{\pjua}{%
\begin{tabular}{c}
 $\delta_\rangle\Phi \ \geqslant\
 \min\ (\ \hbar\,/\,\Delta_\rangle J\ ;\ \pi\ )$\\
 \footnotesize (Arbatsky, 2006)
\end{tabular}}

\newcommand{\pjco}{%
\begin{tabular}{c}
 $ |\delta\varphi|\, \Delta_\rangle J \geqslant \hbar $\\
 \footnotesize (Arbatsky, 2005)
\end{tabular}}

\newcommand{\pjca}{%
\begin{tabular}{c}
 $\Delta_\rangle (\,
   \delta\varphi_1\,J_1\, +\,
   \delta\varphi_2\,J_2\, +\,
   \delta\varphi_3\,J_3\,
   ) \geqslant \hbar$\\
 \footnotesize (Arbatsky, 2005)
\end{tabular}}

\newcommand{\tecmt}{%
\begin{tabular}{c}
 $|\delta t|\, \Delta_\rangle H \geqslant \hbar$\\
 \footnotesize (Mandelshtam, Tamm, 1945)
\end{tabular}}

\newcommand{\alca}{%
\begin{tabular}{c}
 $\Delta_\rangle\,
   (\,-\,\delta x_\mu\, P_\mu\, +\,
      \frac12\,
      \delta\omega_{\mu\nu}\, J_{\mu\nu} \,) \geqslant \hbar$\\
\footnotesize (Arbatsky, 2005)
\end{tabular}}

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\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{The certainty principle (review)}
\author{D.~A.~Arbatsky\footnote{ http://daarb.narod.ru/ , http://wave.front.ru/ }}
\date{August, 2006}
\maketitle

\begin{abstract}
The certainty principle (2005) allowed to generalize and unify
both the Heisenberg uncertainty principle (1927) and
the Mandelshtam-Tamm relation (1945). It turned out to be applicable
to any quantum systems, including relativistic quantum fields.
One can even suppose, that it will finally turn out to be more
fundamental than quantum mechanics and quantum field theory.

In the frame of orthodox quantum theory the certainty principle can
be considered as mathematical theorem. Exactly this approach
is assumed as a basis in this review. And with it the Heisenberg
uncertainty principle and the Mandelshtam-Tamm relation are derived
as consequences of the certainty principle. Another interesting
consequence appears to be a simple uncertainty relation for angle
and angular momentum, which, nevertheless, was not known before.

To wide extent, the certainty principle is not only a mathematical
theorem, but also a gnoseological principle. This principle
substantially complements the philosophy of quantum mechanics.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Historical comments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Uncertainty principle.}
A reader interested in the history of the {\it uncertainty}
principle should read, for example, the review \cite{HilgUff2001}.
Here I give only some notes, that are important for the following
recital.

The uncertainty principle was suggested by Heisenberg in 1927
\cite{Heis1927}. Heisenberg formulated it as follows:
\begin{itemize}
\item
The more precisely the position is determined, the less precisely
the momentum is known in this instant, and vice versa.
\end{itemize}
For uncertainties of coordinate and momentum the following
relation was suggested:
\begin{equation}
 \Delta_\rangle X\,\Delta_\rangle P \,\sim\, \hbar \ .
 \label{HeisRel1}
\end{equation}

Considering concrete examples, Heisenberg gave just qualitative
formulation. But for the notion of ``uncertainty'' he did not give
exact definition.

Soon Kennard \cite{Kennard1927} gave exact mathematical
formulation for the case of coordinate and momentum. Assuming
``commutation relation'' $[X,P]=i\hbar$, he has shown that for an
arbitrary quantum state $\rangle$ the following relation takes
place:
\begin{equation}
 \Delta_\rangle X\,\Delta_\rangle P \,\geqslant\,
 \frac\hbar2 \ .
 \label{KennardRel}
\end{equation}
And here
\[
 \Delta_\rangle X =
   \left\langle\bigl(\,
    X-\langle X\rangle
   \,\bigr)^2\right\rangle^{1/2}
 \ , \qquad
 \Delta_\rangle P =
   \left\langle\bigl(\,
    P-\langle P\rangle
   \,\bigr)^2\right\rangle^{1/2}
 \ .
\]
It means that, in accordance with probabilistic interpretation of
quantum mechanics, it was suggested to understand the
uncertainties of coordinate and momentum as standard deviations of
these observables.

Because of simplicity of the proof, the relation
(\ref{KennardRel}) became conventional mathematical expression of
the uncertainty principle and appeared in all textbooks on quantum
mechanics. All criticism (including that about correspondence to
practical experiment) \cite{HilgUff2001} was mainly ignored. But
soon we will see that the certainty principle allows to get
alternate inequalities, describing the uncertainty principle,
which turn out to be more sapid from practical point of view.

Later the mathematical proof was generalized in many ways. And it
was extended to other pairs of non-commuting observables. So, in
order to avoid ambiguity in the term {\it uncertainty principle},
let us give here some other, more up-to-date and general,
formulation of this principle\footnote{Such a formulation not only
allows to better understand, what exactly is ``uncertain'', but
also allows to clearly see the contrast with the {\it certainty}
principle.}:
\begin{itemize}
\item
If one tries to describe the dynamical state of a {\it quantum}
particle by methods of {\it classical} mechanics, then precision
of such description is limited in principle. The {\it classical}
state of the particle turns out to be {\it badly} defined. This
uncertainty can be mathematically expressed by different
inequalities, describing spreads of values of observables that
have semiclassical limit.
\end{itemize}

Of course, Heisenberg could not formulate the uncertainty
principle in this form. It would contradict the logic of
historical moment. In 1927 physics came from classical mechanics
to quantum, and the uncertainty principle was considered as a way
from the old theory to new.

But today, when quantum mechanics is already well-established, the
uncertainty principle is just a method of qualitative estimation
of precision of semiclassical approximation. And there are no
reasons to think that this principle is more fundamental than
mathematical formalism of quantum mechanics.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Mandelshtam-Tamm relation.}
So far as for coordinate and momentum there is an uncertainty
relation like (\ref{HeisRel1}), general arguments, connected with
the theory of relativity,\footnote{But soon we will see that these
arguments are {\it incorrect}, because existence of the
uncertainty principle is connected with specific peculiarities of
{\it non-relativistic} approximation.} point out that, seemingly,
an uncertainty relation for time and energy like the following
should exist:
\begin{equation}
 \Delta_\rangle T\,\Delta_\rangle H \,\sim\, \hbar \ .
 \label{TH1}
\end{equation}
Heisenberg himself suggested such a relation. But even in that
time it was clear \cite{Pauli1933} that explanation of such a
relation must be more difficult, because there is no such a
quantum-mechanical observable as ``time'', and it is unclear what
should be understood as ``uncertainty'' of time.

Nevertheless, physicists wanted to believe that the uncertainty
principle was a {\it fundamental} physical principle, and attempts
to ground relation (\ref{TH1}) did not stop up to now. These
attempts mainly tried to show that the notion ``uncertainty of
time'' was meaningful. And it was usually made by analysis of
measurement process, i.~e. to analysis of interaction of quantum
particle with apparatus like Heisenberg microscope. (a detailed
review with critical analysis see in \cite{Voron1981})

I cannot say that those attempts were unsuccessful. Nevertheless,
I think that such an approach does not correspond to general
methodology of quantum mechanics. The matter is that analytical
formalism of quantum mechanics is oriented to studying quantum
systems as independent physical objects. This analytical formalism
does not study details of interaction of apparatuses with the
quantum system.

In 1945 Mandelshtam and Tamm \cite{MandTamm1945} gave rigorous
mathematical formulation  of relation\footnote{Here it is written
a little differently, so that connection with further discussion
would be clearer.}
\begin{equation}
 |\delta t|\, \Delta_\rangle H \geqslant \hbar \ .
 \label{MandTammRel}
\end{equation}
Mandelshtam and Tamm named this relation ``the uncertainty
relation between energy and time in nonrelativistic quantum
mechanics'', because, obviously, they thought that they found
mathematical expression of the uncertainty principle for energy
and time. Correspondingly, further attempts to ground {\it
uncertainty} relation for energy and time were generally attempts
to show that the quantity $\delta t$ in Mandelshtam-Tamm relation
can be understood in some sense as ``uncertainty'' of time.

But, I believe that there are no reasons convincing enough to call
Mandelshtam-Tamm relation as {\it uncertainty} relation. As it was
explained above, the uncertainty principle is not more fundamental
than mathematical formalism of quantum mechanics itself. And what
is more, analysis of relativistic quantum systems shows that this
principle is {\it less} fundamental. Therefore there are no
reasons to think that fundamental uncertainty relation like
(\ref{TH1}) must exist.

Soon we will see that Mandelshtam-Tamm relation in the case of
closed systems is a consequence of other physical principle, the
{\it certainty} principle, which is more fundamental than the
Heisenberg uncertainty principle. And the quantity $\delta t$ in
the inequality (\ref{MandTammRel}), if we make transformation from
Schr\"odinger to Heisenberg representation (or to
RCQ-representation), is a logarithmic coordinate on Poincare
group. For some purposes this coordinate can be interpreted as
``uncertainty of time'', but, generally speaking, there is no need
in such interpretation.

As regards non-closed systems, with arbitrary dependence of
Hamiltonian on time, Mandelshtam-Tamm relation should be
considered as relation for the speed of quantum evolution. And it
is quite independent relation, not mathematically equivalent to
the certainty relation for time and energy.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Certainty principle.}

When we raise a question about application of ideas connected with
the theory of relativity in quantum mechanics, it is worth to
think about what we know about relativistic invariance of quantum
theory at all.

The group of invariance of Minkowski space\footnote{Of
four-dimensional space-time of the special theory of relativity.}
is Poincare group. With invariance with respect to this group of
laws of nature, in general, and of quantum mechanics, in
particular, is connected the notion of relativistic invariance.

All known physically important quantum systems, to which action of
Poincare group can be applied, are quantum fields (or their
subsystems). And before 2002 mathematical construction of quantum
fields was made by such recipes\footnote{Recipes, because formal
mathematical process did not exist.}, which made action of
Poincare group on space of states of quantum field hidden. In
fact, physicists were satisfied by statement that Poincare group
acts on space of states not-explicitly, by some quite complicated
formulas.

When relativistic canonical quantization (RCQ) appeared
\cite{Arb2002} (for introduction review \cite{Arb2005wircq} is
recommended) it became clear how action of Poincare group is
transferred from Minkowski space to the space of states of quantum
field.\footnote{Specific feature of RCQ method is construction of
quantum fields in the frame of four-dimensional geometry of
Minkowski space, without artificial separation of space and time.}
It became also obvious the following:
\begin{itemize}
\item
The notion of coordinate as quantum-mechanical observable for
relativistic systems is not natural at all, not even
non-fundamental. (Attempts in the past to introduce coordinates
for some systems, for example \cite{NewtWig1949}, seem to be quite
artificial.)
\item
For this reason the uncertainty principle {\it can not} be
fundamental, in the sense of its generalization for relativistic
quantum theory.
\end{itemize}

Using methodology of RCQ, I have formulated \cite{Arb2005tcp} the
{\it certainty} principle:
\begin{itemize}
\item
If one describes the dynamical state of a quantum particle
(system) by methods of {\it quantum} mechanics, then the {\it
quantum} state of the particle (system) turns out to be {\it well}
defined. This certainty of the quantum dynamical state means that
``small'' space-time transformations can not substantially change
the quantum state. And for the case of Poincare group, for
transformations, that can substantially change the quantum state,
we have estimation\footnote{Here and later we imply relativistic
summation by pairs of the same Greek indices:
\[
 a_\mu\, b_\mu = a_0\, b_0 - a_1\, b_1 - a_2\, b_2 - a_3\, b_3 \ .
\].}:
\begin{equation}
 \textstyle
 \Delta_\rangle\,
   (\,-\,\delta x_\mu\, P_\mu\, +\,
      \frac12\,
      \delta\omega_{\mu\nu}\, J_{\mu\nu} \,) \geqslant \hbar \ ,
 \label{CertPoin}
\end{equation}
Here $P_\mu$ is a vector operator of energy-momentum, $J_{\mu\nu}$
is a tensor operator of the four-dimensional angular momentum,
$\delta x_\mu$ and $\delta\omega_{\mu\nu}$ are the standard
logarithmic coordinates of the Poincare group.
\end{itemize}

So far as the certainty principle is universal, it works well also
in usual quantum mechanics, including {\it non-relativistic}. So,
it seems to be surprising that it was not formulated earlier.
Nevertheless, there are two explanations for this.

First, this principle is formulated naturally with basic notions
of RCQ. In particular, it is most natural in formulation of the
certainty principle to understand the notion of {\it dynamical
state} within RCQ interpretation. This difficulty, seemingly, was
the main.

Second, in order to come to the certainty principle from the usual
quantum mechanics, it is necessary to use the theory of quantum
angle (Fubini-Study metric). But existence of this metric, though
not difficult to prove, is not obvious. This metric was discovered
only in 1905 by Fubini \cite{Fubini1904} and Study
\cite{Study1905}, and most physicists do not know about it.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Mathematical formalism}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Quantum angle (Fubini-Study metric).}
Consider the unit sphere $\|x\|=1$ in the usual three-dimensional
Euclidean space $\mathbb R^3$. The distance measured on the
surface of the sphere between any two points is equal to the angle
between rays starting from the center of coordinates and coming
through these points.

So, the connection of angles with rays on the sphere makes obvious
the important property of angle: it satisfies the ``triangle
inequality'':
\begin{equation}
 \angle(a,c) \leqslant \angle(a,b) + \angle(b,c) \ .
 \label{TrianIneq}
\end{equation}

It is known, that angle in this example is expressed through the
scalar product $\langle\cdot|\cdot\rangle$ in the space
 $\mathbb R^3$ by formula:
\[
 \angle(a,b)\, = \,\arccos\, \langle a|b\rangle \ .
\]
Here in the right part of the formula vectors, whose ends are in
the points of intersections of the rays $a$ and $b$ with the unit
sphere, are denoted by the same letters as rays. Possible values
of angle in this example are in the range from $0$ to $\pi$.

Let us now consider angles not between rays, but between lines
coming through the center of coordinates. In this case possible
values of angle are in the more narrow range from $0$ to $\pi/2$.
The formula for the angle in this case is slightly changed:
\begin{equation}
 \angle(a,b)\, = \,\arccos\,\bigl| \langle a|b\rangle \bigr| \ .
 \label{QAngle}
\end{equation}

Will the angle satisfy the triangle inequality (\ref{TrianIneq})
in this case? The answer to this question does not seem to be
immediately obvious, because for every line we have two points on
the sphere. But elementary logical reasoning allows to reduce this
example to the previous. The answer turns out to be positive.

Consider now the case of the three-dimensional complex Hilbert
space $\mathbb C^3$. Instead of real lines in this case we
obviously have complex lines, i.~e. one-dimensional complex
subspaces. Every such a subspace intersects with the unit sphere
even not in two, but in infinite number of points. These points
are different from each other by complex factor, whose absolute
value is equal to one.

The angle in this case is defined by the same formula
(\ref{QAngle}), but the scalar product here corresponds to the
space $\mathbb C^3$. Possible values of angle, obviously, also are
in the range from $0$ to $\pi/2$.

Will angle in this case satisfy the triangle inequality
(\ref{TrianIneq})? The answer in this case turns out to be
positive also. This fact was proved by Fubini \cite{Fubini1904}
and Study \cite{Study1905} (quite simple proof is also given in
\cite{Arb2005tcp}), so the angle between complex lines in Hilbert
space is called {\it Fubini-Study metric}.

Of course, the fact that triangle inequality takes place cannot
depend on the dimensionality of the Hilbert space, because when we
check the triangle inequality we always can consider the
corresponding three-dimensional subspace.

So far as in quantum mechanics the space of states of a quantum
system $\mathcal H$ is complex Hilbert space, Fubini-Study metric
can be naturally considered as a measure of difference of quantum
states. In this context we will call this metric {\it quantum
angle}, because it allows to form natural collocations like
``quantum angular speed''.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Quantum angular speed.}
Let now the vector $r$ depend on the real parameter $t$:
$t\in\mathbb R$, $r(t)\in\mathcal H$, $\|\,r(t)\,\|=1$.

Let us define {\it the quantum velocity} $v(t)$ by the formula:
\[
 v(t) = \dot r(t) =
    \lim_{\delta t\to0}
    \frac{r(t+\delta t)-r(t)}{\delta t} \ .
\]

Let us define also {\it the quantum angular speed} $\omega(t)$ by
the formula:
\[
 \omega(t) = \lim_{\delta t\to0}
     \frac{\angle\,\big(\,r(t+\delta t)\,,\,r(t)\,\big)}{|\,\delta t\,|} \ .
\]

In order to express $\omega(t)$ through $v(t)$ let us decompose
$v(t)$ into the two orthogonal components:
\[
 v_\parallel(t) = r(t)\,\langle r(t)|v(t)\rangle \ , \qquad
 v_\perp(t) = v(t) - v_\parallel(t) \ .
\]

This decomposition formally looks the same as the corresponding
decomposition in the real case, which is well known in kinematics
of point. But we should not rely on this analogy too much. The
matter is that in the real case existence of the parallel
component of velocity is necessarily connected with growth of the
norm of the vector. But in the complex case under consideration
vector is supposed to have always unit norm. Existence of parallel
component of velocity turns out connected with the possibility of
multiplication of the vector by complex phase factor, that is
in absolute value equal to one. Nevertheless, the following theorem is true

\mytheorem{}
 {\it The quantum angular speed is equal to the norm of the orthogonal
component of the quantum velocity:
\begin{equation}
 \omega(t)=\|\,v_\perp(t)\,\| \ .
 \label{QAngSpeedQVel}
\end{equation}}

Before proving this theorem let us note that in the real case
equality (\ref{QAngSpeedQVel}) is {\it definition} of angular
speed $\omega$. And angle in the real case is {\it defined} as
integral of angular speed along extreme arc. After that
trigonometrical function $\arccos$ is {\it defined} for {\it
finite} angles.

Straightforward transfer of such logic to the case of complex
Hilbert space is problematic, because there are always many
extreme arcs, and none of them is so simple that appearance of
$\arccos$ function would be geometrically obvious.

So, here we need to study directly infinitesimal  angles, i.~e.
work around singular point of $\arccos$ function. In order to
escape from this, let us use Parseval equality to change $\arccos$
to $\arcsin$:
\[
 \angle\,\big(\,r(t+\delta t)\,,\,r(t)\,\big) \,=\,
 \arccos\,\big|\,\langle\,r(t+\delta t)\,|\,r(t)\,\rangle\,\big| \,=\,
\]
\[
 \,=\,
 \arcsin\,
   \big\|\,
   r(t+\delta t) -
   r(t) \,\langle\, r(t) \,|\, r(t+\delta t) \,\rangle
   \,\big\| \,=\,
\]
\[
 \,=\,
   \arcsin\,
    \big\|\,
    r(t+\delta t) \,-\, r(t) \,-\,
    r(t) \,
     \langle\,
        r(t) \,|\, r(t+\delta t)-r(t)
     \,\rangle
    \,\big\|
 \,=\,
\]
\[
 \,=\,
   \arcsin\,
    \big\|\,
    v(t)\,\delta t \,+\, o(\delta t) \,-\,
    r(t) \,
     \langle\,
        r(t) \,|\, v(t)\,\delta t + o(\delta t)
     \,\rangle
    \,\big\|
 \,=\,
\]
\[
 \,=\,
   \arcsin\,
    \Big\|\,
     \big(v(t)\,-\,
     r(t) \,
     \langle\,
        r(t) \,|\, v(t)
     \,\rangle\big)
     \,\delta t \,+\, o(\delta t)
    \,\Big\|
 \,=\,
\]
\[
 \,=\,
   \arcsin\,
    \big\|\,
    v_\perp(t)\,\delta t \,+\, o(\delta t)
    \,\big\|
 \,=\,
   \arcsin\,
    \big(\,
    \|v_\perp(t)\|\,|\delta t| \,+\, o(\delta t)
    \,\big)
 \,=\,
\]
\[
 \,=\,
 \|v_\perp(t)\|\,|\delta t| \,+\, o(\delta t) \ .
\]
Dividing by $|\delta t|$, we get (\ref{QAngSpeedQVel}).
$\blacksquare$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Continuous unitary groups.}
Suppose now that we have in the space of states of quantum system
a self-adjoint operator $A=A^*$. The set of unitary operators of
type $U(\delta s)=e^{-\,i\,\delta s\,A/\hbar}$ (where $\delta s$
runs the set of all real numbers, and $\hbar$ is fixed positive
real number (in quantum mechanics this is Planck's constant))
forms strongly continuous one-parameter unitary group:
 $U(\delta s_1)\,U(\delta s_2)\,=\, U(\delta s_1 + \delta s_2)$.
The operator $A$ here is called {\it generator} of this group.

And suppose that the state vector is changed by the action of this
group:
\[
 r(\delta s) \,=\, |\,\delta s\,\rangle \,=\,
 U(\delta s)\,\rangle \,=\,
 e^{-\,i\,\delta s\,A/\hbar} \,\rangle \ .
\]
Here $|\,\delta s\,\rangle\in\mathcal H$ is another notation of
the vector with parameter equal to $\delta s$;
 $\rangle\in\mathcal H$ is a fixed ket-vector.

Suppose now that the function $r(\delta s)$ is differentiable.
Then the quantum velocity is expressed by the formula:
\[
 \textstyle
 v(\delta s) \,=\,
 \frac1{i\hbar}\, A \, e^{-\,i\,\delta s\,A/\hbar} \,\rangle \,=\,
 \frac1{i\hbar}\, A \, |\,\delta s\,\rangle \ .
\]

The mean of the operator $A$ does not depend on time:
\[
 \overline A \,=\,
 \langle\,\delta s\,|\, A \, |\,\delta s\,\rangle \,=\,
 \langle\,e^{+\,i\,\delta s\,A/\hbar} \, A \,
    e^{-\,i\,\delta s\,A/\hbar} \,\rangle \,=\,
 \langle\, A \, e^{+\,i\,\delta s\,A/\hbar} \,
    e^{-\,i\,\delta s\,A/\hbar} \,\rangle \,=\,
 \langle\, A \, \rangle \ .
\]

Therefore the components of the quantum velocity can be written
just as:
\[
 \textstyle
 v_\parallel(\delta s) \,=\,
 |\,\delta s\,\rangle \, \langle\,\delta s\,|\,
 \frac1{i\hbar}\, A \, |\,\delta s\,\rangle \,=\,
 \frac1{i\hbar}\,\overline A \, |\,\delta s\,\rangle
\]
\[
 \textstyle
 v_\perp(\delta s) \,=\,
 \frac1{i\hbar}\,\left(A-\overline A\right) \,
 |\,\delta s\,\rangle \ .
\]

The quantum angular speed turns out to be independent of the
parameter also:
\[
 \textstyle
 \omega(\delta s)
 \,=\,
 \|\,v_\perp(\delta s)\,\|
 \,=\,
 \frac1\hbar\,
 \langle\,\delta s\,|\,
    \left(A-\overline A\right)^2
 \, |\,\delta s\,\rangle^{1/2}
 \,=\,
 \frac1\hbar\,
 \langle\,
    e^{+\,i\,\delta s\,A/\hbar} \,
    \left(A-\overline A\right)^2 \,
    e^{-\,i\,\delta s\,A/\hbar}
 \,\rangle^{1/2}
 \,=\,
\]
\[
 \textstyle
 \,=\,
 \frac1\hbar\,
 \langle\,
    \left(A-\overline A\right)^2 \,
    e^{+\,i\,\delta s\,A/\hbar} \,
    e^{-\,i\,\delta s\,A/\hbar}
 \,\rangle^{1/2}
 \,=\,
 \frac1\hbar\,
 \langle\,
    \left(A-\overline A\right)^2
 \,\rangle^{1/2}
 \,=\,
 \frac1\hbar\, \Delta_\rangle A  \ .
\]
Here $\Delta_\rangle A$ is a short notation for the standard
deviation of $A$ in the state $\rangle$.

So, we have proved the important

\mytheorem{}
 {\it Standard deviation of the generator of one-parameter unitary
group is equal to the quantum angular speed, multiplied by
Planck's constant $\hbar$. And both quantities remain conserved
under the action of the group.}

Determine now, how different can become the initial $\rangle$ and
final $|\,\delta s\,\rangle$ vectors. The triangle inequality, in
usual way, is generalized to arbitrary curves \cite{Arb2005tcp}.
Therefore, the angle between initial and final vectors cannot be
greater than increment of parameter, multiplied by quantum angular
speed:
\begin{equation}
 \textstyle
 \angle\,\big(\,|\,\delta s\,\rangle\,,\,\rangle\,\big) \,\leqslant\,
 |\delta s| \, \frac1\hbar\, \Delta_\rangle A  \ .
 \label{AngEst}
\end{equation}

For physical applications it is convenient to introduce some
qualitative border, when two state vectors can be considered
different {\it substantially}. So, we will say that two vectors
are different {\it substantially}, if the angle between them is
greater or equal to $1$.

Let us reformulate inequality (\ref{AngEst}) as the following
theorem.

\mytheorem{} (certainty principle)
 {\it So that under the action of strongly continuous one-parameter
unitary group $U(\delta s)=e^{-\,i\,\delta s\,A/\hbar}$ the
initial state vector $\rangle$ changes substantially, it is
necessary to satisfy the inequality:
\begin{equation}
 |\delta s|\, \Delta_\rangle A \geqslant \hbar \ .
 \label{Certainty1}
\end{equation}}

This theorem is easily generalized to the case when the group is
many-parameter. Namely, if $\delta s$ and $A$ have matrix indices,
implying summation, then the inequality (\ref{Certainty1}) can be
re-written for this case a little differently:
\[
 \Delta_\rangle (\,\delta s_j\, A_j\,) \geqslant \hbar \ .
\]
And here $\delta s_j$ are the so called logarithmic coordinates of
the group.

It is worth also to mention the following. When proving the
theorem we supposed that quantum velocity is well-defined, i.~e.
the corresponding derivative of the vector with respect to the
parameter exists and belongs to the Hilbert space. But, if the
generator is an unbounded operator, for some state vectors it can
be not so. Nevertheless, the statement of the theorem turns out to
be true even in this case, because in this case the value
$\Delta_\rangle A$ turns out to be infinite. Therefore, when
parameter is not equal to zero, in the left part of the inequality
we have infinity (which is, of course, greater than Planck's
constant).

So, the statement of the theorem turns out to be true for {\it
any} state vector $\rangle$ and {\it any} (self-adjoint) generator
$A$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Subgroups of the Poincare group}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The theorem presented at the end of the previous section is the
most general formulation of the certainty principle. But in
different situations the unitary group under consideration can be
representation of some concrete physical symmetry group. The most
important example is Poincare group, for which the general
certainty relation (\ref{CertPoin}) was already presented above.
This group has some physically important subgroups, which are so
practically important, that we will discuss them here in more
detail.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Space translations.}
It is known that generators of three parameter group of space
translations are components of the vector operator of momentum
$P_1$, $P_2$ and $P_3$. Consequently, the certainty relation is
written as
\begin{equation}
 \Delta_\rangle (\,
   \delta x_1\,P_1\, +\,
   \delta x_2\,P_2\, +\,
   \delta x_3\,P_3\,
 ) \geqslant \hbar  \ .
 \label{CertTrans}
\end{equation}
Here numbers $\delta x_1$, $\delta x_2$ and $\delta x_3$ form
components of the vector that defines the translation.

If we suppose now that $\delta x_2=\delta x_3=0$, and also omit
the index, then the inequality (\ref{CertTrans}) can be written
as:
\[
 |\delta x|\, \Delta_\rangle P \geqslant \hbar \ .
\]

Suppose now, that for the system under consideration we were able
to find  some observable $X$, that can be considered in some sense
a ``coordinate operator''. This supposition is accepted as
undoubted in non-relativistic quantum mechanics. In contrast, in
relativistic quantum mechanics there is no natural notion of
coordinate.\footnote{This is true even for Dirac's electron.
Superposition of positive- and negative-frequency solutions of
Dirac's equation does not have any physical meaning, and therefore
there is no observable described by ``operator of multiplication
by variable $x$''.} It is possible, for example, to form from
generators of the Poincare group some vector operator, to which in
classical mechanics coordinates of the center of mass correspond.
But it should be understood that it will have some unusual
properties. For example, its coordinates will not commute with
each other.

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 $X$, being a coordinate, behaves in usual way
under action of translations, namely, 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 $X$.

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
\[
 \textstyle
 \langle\,\Omega_{(-\infty,l)}\,\rangle
 \,=\,
 \langle\,\Omega_{(r,+\infty)}\,\rangle
 \,=\,
 \frac{1-\sin1}2
 \,\approx\, 0,07926\dots
\]
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{}(uncertainty principle)
 {\it The following inequality takes place:
\begin{equation}
 \delta_\rangle X \Delta_\rangle P \geqslant \hbar \ .
 \label{Uncertainty}
\end{equation}}

In order to prove this theorem let us use (two times) the triangle
inequality for the quantum angle:
\[
 \angle\,\bigl(\
  \rangle
  \ , \
  e^{-\,i\,\delta_\rangle X\,P/\hbar}\,\rangle
 \ \bigr)
 \ \geqslant\
 \angle\,\bigl(\
  \Omega_{(-\infty,r)}\,\rangle
  \ , \
  \Omega_{(r,+\infty)}\,e^{-\,i\,\delta_\rangle X\,P/\hbar}\,\rangle
 \ \bigr)
 \,-\,
 \qquad\qquad\qquad\qquad\qquad\qquad
\]
\[
 \qquad\qquad\qquad\qquad\qquad\qquad
 \,-\,
 \angle\,\bigl(\
  \rangle
  \ , \
  \Omega_{(-\infty,r)}\,\rangle
 \ \bigr)
 \,-\,
 \angle\,\bigl(\
  e^{-\,i\,\delta_\rangle X\,P/\hbar}\,\rangle
  \ , \
  \Omega_{(r,+\infty)}\,e^{-\,i\,\delta_\rangle X\,P/\hbar}\,\rangle
 \ \bigr)
 \,=\,
\]
\[
 \textstyle
 \,=\,
 \frac\pi2
 \,-\,
 \arcsin\sqrt{\frac{1-\sin1}2}
 \,-\,
 \arcsin\sqrt{\frac{1-\sin1}2}
 \,=\,
 \frac\pi2
 \,-\,
 \left(\frac\pi4-\frac12\right)
 \,-\,
 \left(\frac\pi4-\frac12\right)
 \,=\,
 1 \ .
\]
But this means that under action of
 $e^{-\,i\,\delta_\rangle X\,P/\hbar}$
the vector $\rangle$ changes substantially.

Applying  the {\it certainty} principle, we directly get
(\ref{Uncertainty}). $\blacksquare$

So, the certainty principle allows to get the alternate inequality
(\ref{Uncertainty}) describing the uncertainty principle. This
inequality is not equivalent to the Kennard inequality
({\ref{KennardRel}).

From formal mathematical point of view, none of the inequalities
({\ref{KennardRel}) and (\ref{Uncertainty}) is stronger than the
other.

But it is not difficult to see that the fraction of the quantities
$\Delta_\rangle X$ and $\delta_\rangle X$ always satisfies the
inequality:
\[
 \frac{2\,\Delta_\rangle X}{\delta_\rangle X} \,\geqslant\,
 \sqrt{1-\sin1} \,\approx\, 0,398\dots
\]
Therefore, if we weaken the Kennard inequality ({\ref{KennardRel})
by the additional factor $0,398$ (which is usually not very
important for qualitative estimations), then it becomes weaker
than (\ref{Uncertainty}).

On the other hand, the fraction
 $\Delta_\rangle X/\delta_\rangle X$ can be arbitrarily great,
or even infinitely great, if the wave packet is badly localized.
In such a situation the Kennard inequality ({\ref{KennardRel}), in
contrast to the inequality (\ref{Uncertainty}), does not give any
estimation for the uncertainty of momentum at all.

So, the inequality (\ref{Uncertainty}) is more informative than
({\ref{KennardRel}) for qualitative estimations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Space rotations.}
The generators of the three-parameter group of space rotations are
components of the vector operator of angular momentum $J_1$, $J_2$
and $J_3$. Consequently, the certainty relation is written as
\[
 \Delta_\rangle (\,
   \delta\varphi_1\,J_1\, +\,
   \delta\varphi_2\,J_2\, +\,
   \delta\varphi_3\,J_3\,
 ) \geqslant \hbar  \ .
\]
Logarithmic coordinates have here simple geometric interpretation:
rotation, described by coordinates $\delta\varphi_1$,
$\delta\varphi_2$ and $\delta\varphi_3$, can be performed as
rotation around the vector with these coordinates by angle equal
to the length of this vector.

If we suppose now that $\delta\varphi_2=\delta\varphi_3=0$ and
omit the index, then we have:
\[
 |\delta\varphi|\, \Delta_\rangle J \geqslant \hbar \ .
\]

Let us try to get an uncertainty relation by the same scheme as in
the case of the group of space translations.

First of all, it is necessary to note, that {\it uncertainty}
relation in the form ``product of uncertainty of angle and
uncertainty of angular momentum is greater or equal to Planck's
constant'' can not be suggested, because in a state, when
uncertainty of angular momentum is equal to zero (an eigenstate),
``uncertainty of angle'' should not be greater than $2\pi$.

Furthermore, here we meet the difficulty that it is impossible to
introduce a ``good'' self-adjoint operator of angle. The number of
operators turns out to be infinite (if at least one is defined),
but as soon as we fix the choice of one of them by some condition
(for example, that its spectrum is inside the interval from $0$ to
$2\pi$), it turns out to have quite complicated behaviour under
rotations, and analysis of formulas becomes difficult.

So, we will not choose here one from the infinite set of operators
of angle, but will imply that we talk about the whole infinite
set. For this infinite set we will use symbolic notation $\Phi$.

Such an approach does not meet difficulties, because when deriving
uncertainty relation we in fact operate with spectral measure (the
set of spectral projectors). And this spectral measure for all
operators of angle is the same in essence. Just argument of this
measure is not an interval of the real line, as in the case of
coordinate operator, but a segment of circle. And real coordinate
on the circle should be understand up to addition of $2\pi$. So,
the spectral measure satisfies relation:
\[
 \Omega_{(a+2\pi,b+2\pi)} = \Omega_{(a,b)} \ .
\]

Suppose also that the operator $J$ of angular momentum is a
``generator of spectral rotations'' for $\Phi$:
\[
 e^{+\,i\,\delta\varphi\,J/\hbar} \,
  \Omega_{(a+\delta\varphi,b+\delta\varphi)} \,
    e^{-\,i\,\delta\varphi\,J/\hbar} \,=\,
 \Omega_{(a,b)} \ .
\]

The quantity $\langle\,\Omega_{(a,b)}\,\rangle$, obviously,
defines the probability to find the system inside the angle
segment $(a,b)$. Suppose now that there is such a small segment
$(l,r)$, where mainly the probability to find the system is
concentrated. Namely, suppose that the probability to find the
system in the alternate segment is:
\[
 \textstyle
 \langle\,\Omega_{(r,l+2\pi)}\,\rangle
 \,=\,
 \frac{1-\sin1}2
 \,\approx\, 0,07926\dots
\]
If $r$ and $l$ were chosen so that the quantity
 $\delta_\rangle \Phi = r-l$ is minimum, then
$\delta_\rangle \Phi$ can be naturally called ``uncertainty'' of
the angle $\Phi$.

\mytheorem{} (uncertainty principle for angle and angular
momentum)
 {\it The following inequality takes place:
\begin{equation}
 \delta_\rangle\Phi \ \geqslant\
 \min\ (\ \hbar\,/\,\Delta_\rangle J\ ;\ \pi\ ) \ .
 \label{UncertRot}
\end{equation}}

Consider the case, when $\delta_\rangle\Phi\,\leqslant\,\pi$. In
this case the rotated angle segment $(l,r)$ will not overlap with
the non-rotated one, and we can apply the same estimation
technique for quantum angle as in the case of translations.

So, let us use (two times) the triangle inequality for the quantum
angle:
\[
 \angle\,\bigl(\
  \rangle
  \ , \
  e^{-\,i\,\delta_\rangle\Phi\,J/\hbar}\,\rangle
 \ \bigr)
 \ \geqslant\
 \angle\,\bigl(\
  \Omega_{(l,r)}\,\rangle
  \ , \
  \Omega_{(r,r+\delta_\rangle\Phi)}\,e^{-\,i\,\delta_\rangle\Phi\,J/\hbar}\,\rangle
 \ \bigr)
 \,-\,
 \qquad\qquad\qquad\qquad\qquad\qquad
\]
\[
 \qquad\qquad\qquad\qquad\qquad\qquad
 \,-\,
 \angle\,\bigl(\
  \rangle
  \ , \
  \Omega_{(l,r)}\,\rangle
 \ \bigr)
 \,-\,
 \angle\,\bigl(\
  e^{-\,i\,\delta_\rangle\Phi\,J/\hbar}\,\rangle
  \ , \
  \Omega_{(r,r+\delta_\rangle\Phi)}\,e^{-\,i\,\delta_\rangle\Phi\,J/\hbar}\,\rangle
 \ \bigr)
 \,=\,
\]
\[
 \textstyle
 \,=\,
 \frac\pi2
 \,-\,
 \arcsin\sqrt{\frac{1-\sin1}2}
 \,-\,
 \arcsin\sqrt{\frac{1-\sin1}2}
 \,=\,
 \frac\pi2
 \,-\,
 \left(\frac\pi4-\frac12\right)
 \,-\,
 \left(\frac\pi4-\frac12\right)
 \,=\,
 1 \ .
\]
But this means that under action of
 $e^{-\,i\,\delta_\rangle\Phi\,J/\hbar}$
the vector $\rangle$ is changed substantially.

Applying the {\it certainty} principle, we directly get
(\ref{UncertRot}). $\blacksquare$

We see that the certainty principle allows to easily get also the
simple {\it uncertainty} relation for angle and angular momentum.

It should be noted that attempt \cite{Judge1964} to get an
uncertainty relation for angle and angular momentum by analogy
with the Kennard inequality ({\ref{KennardRel}) leads to quite
complicated formulas. Namely, if we fix the self-adjoint operator
of angle $\Phi$ by the condition, that its spectrum is the
interval $[-\pi,\pi]$, and define the ``uncertainty'' of angle by
the formula:
\[
 \Delta'_\rangle\Phi =
  \min_{\delta\varphi}
  \sqrt{\langle\,
   e^{+\,i\,\delta\varphi\,J/\hbar} \,
   \Phi^2 \,
   e^{-\,i\,\delta\varphi\,J/\hbar}\,\rangle} \ ,
\]
then we have the uncertainty relation:
\begin{equation}
 \Delta'_\rangle\Phi \, \Delta_\rangle J \,\geqslant\,
 \frac\hbar2
 \,
 \left[\,
  1\, -\, \frac3{\pi^2}\, (\Delta'_\rangle\Phi)^2
 \,\right]
 \ .
 \label{UncJudge}
\end{equation}
For investigation of semiclassical limit the inequality
(\ref{UncertRot}) turns out to be more convenient than
(\ref{UncJudge}), because allows to use weaker conditions for
localization of wave packet.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph{Time shifts.}
Poincare group includes as a subgroup one-parameter group of
shifts in time. It is natural to expect that the certainty
principle has corresponding mathematical expression in this case.
And this is really so.

But our discussion here is complicated by the circumstance that
all educational literature on both classical and quantum mechanics
is oriented to resolving of problems of dynamics only. And the
notion of dynamical state is necessarily attributed to some moment
of time. Such approach hides relativistic invariance. So, the
notion of time shifts becomes senseless.

Completely self-consistent, from this point of view, are invariant
Hamiltonian formalism (in the case of the classical mechanics) and
relativistic canonical quantization (RCQ) (in the case of
relativistic quantum field theory). In these approaches the theory
includes not only time shifts, but also Lorentz boosts. But, so
far as most physicists are not well-acquainted with such
approaches now, here we will discuss some intermediate approach
that is in essence based on Heisenberg representation. (Lorentz
boosts  we will not consider in this review at all.)

Usually, dynamical state $S_t$ in some moment of time $t$ is
understood as some set of values, that can be measured in the
moment $t$ by some set of apparatuses, and that define the whole
subsequent evolution of the system (i.~e. analogous sets of values
in subsequent moments of time). In classical mechanics dynamical
state is usually described either by set of coordinates and
velocities (Lagrangian formalism) or by set of coordinates and
momenta (Hamiltonian formalism). In other words, dynamical state
in a given moment of time is defined by a point on some manifold,
which is called phase space. In quantum mechanics dynamical state
is described by vector in Hilbert space.

When we have the question about description of the group of space
motions on the space of dynamical states in some moment of time,
we usually think in the following way. Suppose a set of physical
apparatuses $K$, which measures the full set of dynamical
variables of the system under consideration, under action of some
motion $G$ transferred to $K'$, $G:K\to K'$. And suppose for some
dynamical state of the system $S$ we have dynamical state $S'$,
such that the set of apparatuses $K'$, which measures parameters
of the state $S'$, gives the same results as $K$, which measures
parameters of the state $S$. Then we say that under action of the
transformation $G$ state $S$ transfers to $S'$. Symbolically we
can write it as:
\[
 G:K\to K' \ ,\ K(S)=K'(S')
 \qquad\Longrightarrow\qquad
 G:S\to S' \ .
\]
Exactly in this way the action of space translations and rotations
is introduced, described above.

It seems, it would be natural to apply this definition to the
group of time shifts:
\[
 G:K_{t_1}\to K_{t_2} \ ,\ K_{t_1}(S_{t_1})=K_{t_2}(S_{t_2})
 \qquad\Longrightarrow\qquad
 G:S_{t_1}\to S_{t_2} \ .
\]
But this definition turns out to be not very useful, because it
turns out to be purely formal and with no connection with
dynamics.

Let us improve a little the definition of dynamical state given
above. Let us suppose that if some dynamical state $S_{t_1}$ is
transferred by evolution to $S_{t_2}$, then $S_{t_1}$ and
$S_{t_2}$ describe the same dynamical state. In other words,
dynamical state is just some abstract set of values (not
necessarily connected with measurements in a concrete moment of
time) defining evolution of the system. When we talk about quantum
systems, we can think that dynamical state is just described by
Heisenberg vector of state.

If the system is closed, its Hamiltonian does not depend on time,
and it commutes with space motions. Therefore the theory of space
motions, described in the previous sections, can be applied in
this case. And what is more, the group of time shifts is joined
here, which is described by the family of operators:
\[
 U(\delta t) = e^{-\,i\,\delta t\,(-H)/\hbar} \ .
\]
It should be noticed that generator of this group is Hamiltonian
with sign ``minus''. An operator, describing action of the group
under consideration, is reciprocal of the operator, describing
evolution in Schr\"odinger equation. We can say that here the
group acts actively, but in the Schr\"odinger representation it
acts passively.

The certainty relation for the group of time shifts takes the
form:
\begin{equation}
 |\delta t|\, \Delta_\rangle H \geqslant \hbar \ .
 \label{CertTH}
\end{equation}
If we look at this relation from the point of view of the
Schr\"odinger representation (but doing so we lose connection with
Poincare group), then this relation turns out to be exactly the
Mandelshtam-Tamm relation \cite{MandTamm1945} for the speed of
quantum evolution.

So, the Mandelshtam-Tamm relation for the case of closed systems
is a consequence of the certainty principle.

Let us consider now non-closed systems. The Mandelshtam-Tamm
relation for the speed of quantum evolution (derived in the
Schr\"odinger representation) is naturally generalized for this
case. Let us denote Hamiltonian for this case as
$H^{\mathrm{full}}$. So far as the quantity
 $\Delta_{|t\rangle} H^{\mathrm{full}}$, generally speaking,
depends on time, it should be just averaged:
\begin{equation}
 |\tau|\, \overline{\Delta_{|t\rangle} H^{\mathrm{full}}}
 \geqslant \hbar \ .
 \label{MTFull}
\end{equation}
Here $\tau$ is the time interval of evolution (it does not have
meaning of logarithmic coordinate on any group).

The certainty principle in this situation, formally, turns out to
be not applicable. But, in practical situations, when we consider
non-closed quantum systems, their Hamiltonian can be usually
represented as sum of two terms, the Hamiltonian of ``free''
system and Hamiltonian of interaction ``with external classical
field'':
\[
 H^{\mathrm{full}} = H + H^{\mathrm{int}} \ .
\]
And usually the Hamiltonian of interaction $H^{\mathrm{int}}$ can
be ``turned off''.

In this case it is natural to connect the notion of dynamical
state of the system with its evolution when Hamiltonian of
interaction is turned off (though, the state can be considered in
a concrete moment of time, without turning off the Hamiltonian of
interaction). Correspondingly, the certainty relation
(\ref{CertTH}) remains the same. Its meaning becomes somewhat more
abstract, with no direct connection with the Mandelshtam-Tamm
relation (\ref{MTFull}).

In the end, let us note that Mandelshtam and Tamm derived their
relation by direct estimation of the scalar product of the initial
and final vectors of state. They did not use the theory of
Fubini-Study metric. Such a derivation turns out to be
substantially shorter. But geometrical nature of the result turns
out to be quite hidden. This additionally complicates
interpretation (in addition to the more serious disadvantage:
interpretation based on the Schr\"odinger equation).

Derivation of the Mandelshtam-Tamm relation (in the frame of the
Schr\"odinger equation), based on Fubini-Study metric, was given
later by Anandan and Aharonov \cite{AnanAhar1990}. They did not
give the reference to Mandelshtam and Tamm. So, in the western
literature this relation is often erroneously called the
Anandan-Aharonov relation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In closing I want to thank A.~Kleyn, S.~Lorenz, T.~A.~Bolokhov,
A.~V.~Ossipov, E.~V.~Aksyonova, A.~Yu.~Tos\-che\-vi\-ko\-va,
M.~Gatti, and A.~K.~Pati for helpful discussions, references and
help in procuring literature.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{thebibliography}

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