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## Main Question or Discussion Point

The fundamental question about the nature of inertial mass is not solved up to now.

Success of Newton's conception of physical force influencing on a separated body may be explained by the fact that the \emph{geometric counterpart to the force - acceleration in some inertial frame} was found with the simplest relation to the mass of a body. The class of the inertial systems contains (by a convention) the one unique inertial system - the system of remote stars. On the abstract mathematical level arose ``space" - the linear space with appropriate vector operations on forces, momenta, velocities, etc. General relativity and new astronomical observations concerning accelerated expansion of Universe show that all these constructions are only a good approximation, at best.

The line of Galileo-Newton-Mach-Einstein argumentations made accent on some absolute global reference frame associated with the system of remote stars. This point of view looks as absolutely necessary for the classical formulation of the inertial principle itself. This fundamental principle has been formulated, say, ``externally", i.e. as if one looks on some massive body perfectly isolated from Universe. In such approach only ``mechanical" state of relative motion of the body has been taken into account.

Nevertheless, Newton clearly saw some weakness of such approach. His famous example of rotating bucket with water shows that there is an absolute motion since the water takes on a concave shape in any reference frame. Here we are very close to different - ``internal" formulation of the inertia principle and, probably, to understanding of the quantum nature of inertial mass. Namely, the ``absolute motion" should be turned towards not outward, to distant stars, but inward -- to the body deformation.

In fact one should take into account that external force not only change the inertial character of its motion: motion with the constant velocity transforms to accelerated motion; moreover -- the body deforms.

Two aspects of a force action - acceleration relative inertial reference frame and

deformation of the body are very important already on the classical level as it has been shown by Newton's bucket rotation. But second aspect is especially important for quantum ``particles" since the

acceleration requires a good localization in space-time but just such a localization is very problematic in quantum theory.

Nevertheless, almost all discussions in foundations of quantum theory presume that

space-time structure is close with an acceptable accuracy to Minkowski

geometry and may be used without changes in quantum theory up to Planck's

scale or up to topologically different space-time geometry of string theories.

Under such approach, one loses the fact that space-time relationships and

geometry for quantum objects should be reformulated totally \emph{at any space-time distance} since from the quantum point of view such fundamental dynamical variables as ``time-of-arrival" \cite{A98} and position operator in Foldy-Wouthuysen (F.-W.)

\cite{FW} and Newton-Wigner \cite{NW} representations are state-dependent \cite{Le1,Le2,Le3,Le4,Le5,Le6,Le7}. Therefore space-time itself should be built in the frameworks of new

``quantum geometry".

In such a situation one should make accent on the second aspect of the force action -- body deformation.

It means that microscopically the state of body is changed. In fact it is already a \emph{different body}, with different temperature, etc., \cite{Le6}. It means that

under inertial motion one has opposite situation -- the internal state of the body does not change, i.e. body is self-identical during inertial space-time motion. In fact this is the basis of all classical physics. Generally, space-time localization being treated as ability of coordinate description of an object in classical relativity closely connected with operational

identification of ``events" \cite{Einstein1}. It is tacitly assumed that

all classical objects (frequently represented by material points) are

self-identical and they can not disappear because of the energy-momentum

conservation law. The inertia law of Galileo-Newton ascertains this self-conservation ``externally". This means that objectively

\emph{physical state of body (temporary in somewhat indefinite sense) does

not depend on the choice of the inertial reference frame}. One may accept this statement as an ``internal" formulation of the inertial law that should be of course formulated mathematically. I put here some plausible reasonings leading to such a formulation.

Up to now the localization problem of quantum systems in the space-time is connected

in fact with the fundamental classical notion of potential energy and force.

Einstein and Schr\"odinger already discussed the inconsistency of usage such

purely classical notions together with the quantum law of motion and the concepts

of ``particle" and ``acceleration" as well (see one of the letter of Einstein

to Schr\"odinger \cite{Ein_Schr} and the article \cite{Ein_de_Broglie}). But

these messages are mostly did not get attention by the physical community.

Newton's force is the physical reason for the \emph{absolute} change of the character of motion realized in space-time \emph{acceleration} that serves as geometric

counterpart to force (curvature of the world line in Newtonian space-time

is now non-zero). However there is no adequate geometric notion in quantum

theory since, for example, the notion of trajectory in space-time of quantum system was systematically banned. In some sense the energy of interaction

expressed by $H_{int}$ is analoge of a classical force. Generally,

this interaction leads to the absolute change (deformation) of the quantum

state \cite{Le6} (remember: quantum state is the state of motion

\cite{Dirac}). Motion takes the place in state space modeled

frequently by some Hilbert space. But there is no geometric

counterpart to $H_{int}$ in such functional space. In order to establish the geometric counterpart to $H_{int}$ it is useful

initially to clarify the important question: what is the quantum content

of classical force, if any?

Let me use a small droplet of mercury as simple example of macroscopic system \cite{Le6}. The free droplet of mercury is in the state

of the motion ``whether it be of rest, or of moving uniformly forward in a

straight line". This statement literally means that physical states of the droplet (its internal degrees of

freedom) in rest and in uniform motion in a straight line are physically

non-distinguishable. The force applied to the droplet breaks the inertial character of its motion and deforms its surface

tension, changes its temperature, etc. In fact an external force perturbs

Goldstone's modes supporting the droplet as a macro-system \cite{TFD} and micro-potentials acting on any internal quantum particle, say, electrons inside of the droplet. It means that quantum states and their deformations may serve as a

``detector" of the ``external force" action on the droplet.

This deformation being discussed from the quantum point of view gives the alternative way for the connection of a forse action with a \emph{new geometric counterpart of interaction - coset structure of the quantum state space}.

It means that instead of absolute external reference frame of remote stars

one may use ``internal", in fact -- a quantum reference frame \cite{Le1,Le2,Le3,Le4,Le5,Le6,Le7}.

Therefore it is

reasonable to use quantum state deformations instead of classical acceleration,

since generally acceleration depends on mass, charge, etc., that is

impossible establish a pure space-time invariant (geometric) counterpart

of a classical force independent on material body. Only a classical gravitation

force may be geometrized assuming the gravitation field may be replaced locally by

an accelerated reference frame since in general relativity the gravitation

and reference frame are locally non-distinguishable.

There is, however, a more serious reason why space-time acceleration is not a

so good counterpart of the force.

The physical state of the droplet freely falling in the gravitation field

of a star is non-distinguishable from the physical state of the droplet in

an remote from stars area. It means that from the point of view of the ``physical state" of the droplet, the class of the inertial systems may be

supplemented by a reference frame freely falling in a gravitation field.

Therefore macroscopic space-time acceleration cannot serve as a discriminator of physical state of body \cite{Le5}. Thus, instead of choosing, say, the system of distant stars as an ``outer" absolute reference frame

\cite{Einstein2} the deformation of quantum state of some particle of the droplet may be used. It means that the deformation of quantum motion in quantum state space serves as an ``internal detector" for ``accelerated" space-time motion.

I have assumed that same approach may be applied to single quantum electron whose model is a dynamical process in the state space of its spin/charge degrees of freedom \cite{Le2}. Then the \emph{deformations of quantum motion generated by the coset action in the quantum state space will be used as an internal counterpart of a self-interacting electron in dynamical space-time}. It means nothing but in the developing theory \emph{a distance between quantum states in the state space should replace a distance between ``bodies" in space-time as the primary geometric notion}.

A simple formulation of the quantum inertia law as the affine parallel transport of energy-momentum in the projective Hilbert state space has been proposed \cite{Le1}.

I hope it paves the way to clarification

the old problem of inertial mass and such ``fictitious" forces as, say,

centrifugal force. Shortly speaking the inertia and inertial forces are

originated not in space-time but it the space of quantum states since they

are generated by deformation of quantum states as a reaction on external interaction.

\vskip .5cm

\begin{thebibliography}{}

\bibitem{A98}

Aharonov Y. et al, Mearurement of Time-Arrival in Quantum Mechanics,

Phys. Rev., {\bf A57},4130 (1998).

\bibitem{FW}

Foldy L.L., Wouthuysen S.A., Phys. Rev., {\bf 78}, 29 (1950).

\bibitem{NW}

Newton T.D. and Wigner E.P., Localized States for Elementary Systems,

Rev. Mod. Phys., {\bf 21}, No.3, 400-406 (1949).

\bibitem{Le1}

Leifer P., The quantum content of the inertia law and field dynamics, arXiv:1009.5232v1.

\bibitem{Le2}

Leifer P., Self-interacting quantum electron, arXiv:0904.3695v3.

\bibitem{Le3}

Leifer P., Superrelativity as an Element of a Final Theory, Found. Phys. {\bf 27},

(2) 261 (1997).

\bibitem{Le4}

Leifer P., Objective quantum theory based on the $CP(N-1)$ affine gauge field,

Annales de la Fondation Louis de Broglie, {\bf 32}, (1) 25-50 (2007).

\bibitem{Le5}

Leifer P., State-dependent dynamical variables in quantum theory, JETP Letters,

{\bf 80}, (5) 367-370 (2004).

\bibitem{Le6}

Leifer P., Inertia as the ``threshold of elasticity" of quantum states,

Found.Phys.Lett., {\bf 11}, (3) 233 (1998).

\bibitem{Le7}

Leifer P., An affine gauge theory of elementary particles,

Found.Phys.Lett., {\bf 18}, (2) 195-204 (2005).

\bibitem{Einstein1}

Einstein A., Ann. Phys. Zur Electrodynamik der bewegter K\"orper, {\bf 17},

891-921 (1905).

\bibitem{Ein_Schr}

Einstein A., Letter to Schr\"odinger from 22. XII. 1950.

\bibitem{Ein_de_Broglie}

Einstein A., ``Einleitende Bemerkungen \"uber Grundbegriffe" in ``Louis de Broglie,

physicien et penseur", Paris, pp. 4-14, (1953).

\bibitem{Dirac}

Dirac P.A.M., The principls of quantum mechanics, Fourth Edition, Oxford, At the

Clarebdon Press, 1958.

\bibitem{TFD}

H. Umezawa, H. Matsumoto, M. Tachiki, Thermo Field Dynamics and Condensed States,

North-Holland Publishing Company, Amsterdam, New York, Oxford, 504 (1982).

\bibitem{Einstein2}

Einstein A., Ann. Phys. Die Grunlage der allgemeinen Relativit\"atstheorie,

{\bf 49}, 769-822 (1916).

Success of Newton's conception of physical force influencing on a separated body may be explained by the fact that the \emph{geometric counterpart to the force - acceleration in some inertial frame} was found with the simplest relation to the mass of a body. The class of the inertial systems contains (by a convention) the one unique inertial system - the system of remote stars. On the abstract mathematical level arose ``space" - the linear space with appropriate vector operations on forces, momenta, velocities, etc. General relativity and new astronomical observations concerning accelerated expansion of Universe show that all these constructions are only a good approximation, at best.

The line of Galileo-Newton-Mach-Einstein argumentations made accent on some absolute global reference frame associated with the system of remote stars. This point of view looks as absolutely necessary for the classical formulation of the inertial principle itself. This fundamental principle has been formulated, say, ``externally", i.e. as if one looks on some massive body perfectly isolated from Universe. In such approach only ``mechanical" state of relative motion of the body has been taken into account.

Nevertheless, Newton clearly saw some weakness of such approach. His famous example of rotating bucket with water shows that there is an absolute motion since the water takes on a concave shape in any reference frame. Here we are very close to different - ``internal" formulation of the inertia principle and, probably, to understanding of the quantum nature of inertial mass. Namely, the ``absolute motion" should be turned towards not outward, to distant stars, but inward -- to the body deformation.

In fact one should take into account that external force not only change the inertial character of its motion: motion with the constant velocity transforms to accelerated motion; moreover -- the body deforms.

Two aspects of a force action - acceleration relative inertial reference frame and

deformation of the body are very important already on the classical level as it has been shown by Newton's bucket rotation. But second aspect is especially important for quantum ``particles" since the

acceleration requires a good localization in space-time but just such a localization is very problematic in quantum theory.

Nevertheless, almost all discussions in foundations of quantum theory presume that

space-time structure is close with an acceptable accuracy to Minkowski

geometry and may be used without changes in quantum theory up to Planck's

scale or up to topologically different space-time geometry of string theories.

Under such approach, one loses the fact that space-time relationships and

geometry for quantum objects should be reformulated totally \emph{at any space-time distance} since from the quantum point of view such fundamental dynamical variables as ``time-of-arrival" \cite{A98} and position operator in Foldy-Wouthuysen (F.-W.)

\cite{FW} and Newton-Wigner \cite{NW} representations are state-dependent \cite{Le1,Le2,Le3,Le4,Le5,Le6,Le7}. Therefore space-time itself should be built in the frameworks of new

``quantum geometry".

In such a situation one should make accent on the second aspect of the force action -- body deformation.

It means that microscopically the state of body is changed. In fact it is already a \emph{different body}, with different temperature, etc., \cite{Le6}. It means that

under inertial motion one has opposite situation -- the internal state of the body does not change, i.e. body is self-identical during inertial space-time motion. In fact this is the basis of all classical physics. Generally, space-time localization being treated as ability of coordinate description of an object in classical relativity closely connected with operational

identification of ``events" \cite{Einstein1}. It is tacitly assumed that

all classical objects (frequently represented by material points) are

self-identical and they can not disappear because of the energy-momentum

conservation law. The inertia law of Galileo-Newton ascertains this self-conservation ``externally". This means that objectively

\emph{physical state of body (temporary in somewhat indefinite sense) does

not depend on the choice of the inertial reference frame}. One may accept this statement as an ``internal" formulation of the inertial law that should be of course formulated mathematically. I put here some plausible reasonings leading to such a formulation.

Up to now the localization problem of quantum systems in the space-time is connected

in fact with the fundamental classical notion of potential energy and force.

Einstein and Schr\"odinger already discussed the inconsistency of usage such

purely classical notions together with the quantum law of motion and the concepts

of ``particle" and ``acceleration" as well (see one of the letter of Einstein

to Schr\"odinger \cite{Ein_Schr} and the article \cite{Ein_de_Broglie}). But

these messages are mostly did not get attention by the physical community.

Newton's force is the physical reason for the \emph{absolute} change of the character of motion realized in space-time \emph{acceleration} that serves as geometric

counterpart to force (curvature of the world line in Newtonian space-time

is now non-zero). However there is no adequate geometric notion in quantum

theory since, for example, the notion of trajectory in space-time of quantum system was systematically banned. In some sense the energy of interaction

expressed by $H_{int}$ is analoge of a classical force. Generally,

this interaction leads to the absolute change (deformation) of the quantum

state \cite{Le6} (remember: quantum state is the state of motion

\cite{Dirac}). Motion takes the place in state space modeled

frequently by some Hilbert space. But there is no geometric

counterpart to $H_{int}$ in such functional space. In order to establish the geometric counterpart to $H_{int}$ it is useful

initially to clarify the important question: what is the quantum content

of classical force, if any?

Let me use a small droplet of mercury as simple example of macroscopic system \cite{Le6}. The free droplet of mercury is in the state

of the motion ``whether it be of rest, or of moving uniformly forward in a

straight line". This statement literally means that physical states of the droplet (its internal degrees of

freedom) in rest and in uniform motion in a straight line are physically

non-distinguishable. The force applied to the droplet breaks the inertial character of its motion and deforms its surface

tension, changes its temperature, etc. In fact an external force perturbs

Goldstone's modes supporting the droplet as a macro-system \cite{TFD} and micro-potentials acting on any internal quantum particle, say, electrons inside of the droplet. It means that quantum states and their deformations may serve as a

``detector" of the ``external force" action on the droplet.

This deformation being discussed from the quantum point of view gives the alternative way for the connection of a forse action with a \emph{new geometric counterpart of interaction - coset structure of the quantum state space}.

It means that instead of absolute external reference frame of remote stars

one may use ``internal", in fact -- a quantum reference frame \cite{Le1,Le2,Le3,Le4,Le5,Le6,Le7}.

Therefore it is

reasonable to use quantum state deformations instead of classical acceleration,

since generally acceleration depends on mass, charge, etc., that is

impossible establish a pure space-time invariant (geometric) counterpart

of a classical force independent on material body. Only a classical gravitation

force may be geometrized assuming the gravitation field may be replaced locally by

an accelerated reference frame since in general relativity the gravitation

and reference frame are locally non-distinguishable.

There is, however, a more serious reason why space-time acceleration is not a

so good counterpart of the force.

The physical state of the droplet freely falling in the gravitation field

of a star is non-distinguishable from the physical state of the droplet in

an remote from stars area. It means that from the point of view of the ``physical state" of the droplet, the class of the inertial systems may be

supplemented by a reference frame freely falling in a gravitation field.

Therefore macroscopic space-time acceleration cannot serve as a discriminator of physical state of body \cite{Le5}. Thus, instead of choosing, say, the system of distant stars as an ``outer" absolute reference frame

\cite{Einstein2} the deformation of quantum state of some particle of the droplet may be used. It means that the deformation of quantum motion in quantum state space serves as an ``internal detector" for ``accelerated" space-time motion.

I have assumed that same approach may be applied to single quantum electron whose model is a dynamical process in the state space of its spin/charge degrees of freedom \cite{Le2}. Then the \emph{deformations of quantum motion generated by the coset action in the quantum state space will be used as an internal counterpart of a self-interacting electron in dynamical space-time}. It means nothing but in the developing theory \emph{a distance between quantum states in the state space should replace a distance between ``bodies" in space-time as the primary geometric notion}.

A simple formulation of the quantum inertia law as the affine parallel transport of energy-momentum in the projective Hilbert state space has been proposed \cite{Le1}.

I hope it paves the way to clarification

the old problem of inertial mass and such ``fictitious" forces as, say,

centrifugal force. Shortly speaking the inertia and inertial forces are

originated not in space-time but it the space of quantum states since they

are generated by deformation of quantum states as a reaction on external interaction.

\vskip .5cm

\begin{thebibliography}{}

\bibitem{A98}

Aharonov Y. et al, Mearurement of Time-Arrival in Quantum Mechanics,

Phys. Rev., {\bf A57},4130 (1998).

\bibitem{FW}

Foldy L.L., Wouthuysen S.A., Phys. Rev., {\bf 78}, 29 (1950).

\bibitem{NW}

Newton T.D. and Wigner E.P., Localized States for Elementary Systems,

Rev. Mod. Phys., {\bf 21}, No.3, 400-406 (1949).

\bibitem{Le1}

Leifer P., The quantum content of the inertia law and field dynamics, arXiv:1009.5232v1.

\bibitem{Le2}

Leifer P., Self-interacting quantum electron, arXiv:0904.3695v3.

\bibitem{Le3}

Leifer P., Superrelativity as an Element of a Final Theory, Found. Phys. {\bf 27},

(2) 261 (1997).

\bibitem{Le4}

Leifer P., Objective quantum theory based on the $CP(N-1)$ affine gauge field,

Annales de la Fondation Louis de Broglie, {\bf 32}, (1) 25-50 (2007).

\bibitem{Le5}

Leifer P., State-dependent dynamical variables in quantum theory, JETP Letters,

{\bf 80}, (5) 367-370 (2004).

\bibitem{Le6}

Leifer P., Inertia as the ``threshold of elasticity" of quantum states,

Found.Phys.Lett., {\bf 11}, (3) 233 (1998).

\bibitem{Le7}

Leifer P., An affine gauge theory of elementary particles,

Found.Phys.Lett., {\bf 18}, (2) 195-204 (2005).

\bibitem{Einstein1}

Einstein A., Ann. Phys. Zur Electrodynamik der bewegter K\"orper, {\bf 17},

891-921 (1905).

\bibitem{Ein_Schr}

Einstein A., Letter to Schr\"odinger from 22. XII. 1950.

\bibitem{Ein_de_Broglie}

Einstein A., ``Einleitende Bemerkungen \"uber Grundbegriffe" in ``Louis de Broglie,

physicien et penseur", Paris, pp. 4-14, (1953).

\bibitem{Dirac}

Dirac P.A.M., The principls of quantum mechanics, Fourth Edition, Oxford, At the

Clarebdon Press, 1958.

\bibitem{TFD}

H. Umezawa, H. Matsumoto, M. Tachiki, Thermo Field Dynamics and Condensed States,

North-Holland Publishing Company, Amsterdam, New York, Oxford, 504 (1982).

\bibitem{Einstein2}

Einstein A., Ann. Phys. Die Grunlage der allgemeinen Relativit\"atstheorie,

{\bf 49}, 769-822 (1916).