Download Presentazione di PowerPoint - Le siècle d`Albert Einstein

La Siècle d’Albert Einstein - Julliet 2005 - Palais de l’Unesco - PARIS
The time
shared by
Quantum Mechanics and Relativity
Giuseppe Guzzetta – Università di Napoli ‘Federico II’ (now retired)
“La geometria è infinita perché ogni quantità
continua è divisibile in infinito per l'uno e per
l'altro verso. Ma la quantità discontinua
comincia all'unità e cresce in infinito, e,
com'è detto, la continua quantità cresce in
infinito e diminuisce in infinito.”
Leonardo da Vinci (1492-1516) Codex M, 18 r (Institute de
France).
“Geometry is infinite because any continuous quantity is
divisible to the infinity on both sides. On the contrary, the
discontinuous quantity begins with the unity and grows to
the infinity, and, as above stated, the continuous quantity
grows to the infinity and decreases to the infinity.”
In his well-known lecture, Riemann
considered continuous and discrete
manifoldnesses, respectively made of
‘specialisations’ respectively called ‘points’
and ‘elements’.
“If in the case of a notion whose specialisations form a continuous
manifoldness, one passes from a certain specialisation in a definite way
to another, the specialisations passed over form a simply extended
manifoldness (einfach ausgedehnte Mannigfaltigkeit), whose true
character is that in it
a continuous progress from a point is possible only
on two sides, forwards or backwards.”
Bernhard Riemann (1854) Über die Hypothesen, welche der Geometrie zu Grunde liegen
Giving as granted that a continuous progress from
a given ‘point’ is possible both forwards and
backwards, time is currently considered as a simply
extended manifoldness.
As a consequence, the change of sign of t has the
meaning of ‘time reversal’.
Both Leonardo da Vinci and Riemann ignored the existence of
manifoldnesses in which both ‘points’ and ‘elements’ can be
identified. Propagation of plane polarized light in an optically active
medium is a suitable example of these continuous-discrete
manifoldnesses.
Pseudoscalar quantities such as the one I considered
form a simply extended manifoldness in which a
continuous progress from a point, or an element, is
possible
only on one side: either forwards or backwards.
Such a manifoldness is characterized by
continuity, periodicity and handedness
After failing in my last try to build a time
machine able to bring me back in the past, I
began suspecting that it is an impossible task.
Believing in God and (not without reservations)
in Einstein, I decided to assume as a working
hypothesis that a time interval in the Minkowski
space-time should intrinsically be a
pseudoscalar quantity.
At such a condition, time could be considered
as a simply extended manifoldness in which a
continuous progress from an instant is possible
only on one side.
As we will see later on, some evidence
supporting the validity of such a hypothesis
can be found within the formalism of
Special Relativity.
Anyhow, time-periodicity and timehandedness may be better recognized
finding out of Special Relativity an
explanation for the special connection
between space and time on which Special
Relativity is grounded.
Instead of unstructured mass points ‘being’ in the
Minkowski space-time, I will consider material
particles ‘becoming’ in a three dimensional Euclidean
space, assuming that they are characterized by a local
circulatory motion, such as the one attributed to the
electron and known as Zitterbewegung.
Therefore, a particle undergoing a change of position
should be imagined as a mass point moving at the light
speed along a cylindrical helix.
In other words, the ‘becoming’ of a material particle
should be thought of as consisting in its unceasing
change of position and/or angular position at the
universal rate of becoming c.
It follows that the speed component along the circular
orbit normal to the direction of the particle
displacement dl (that is, the rate of change of angular
position) is
so that:
For a particle with spin
angular momentum
For a particle whose position changes at constant speed v
(1) both s and ct are pseudoscalar quantities,
(2) the change of sign of ct involves the inversion of the space
coordinates,
(3) the change of sign of a pseudoscalar quantity involves the
change of sign of all the pseudoscalar quantities.
For a particle whose position changes at constant speed v
When l and ct change, the pseudoscalar
displacement s remains invariant!
So, one may seize the opportunity…....
………of considering ct as one of the four
dimensions of the Minkowski space-time and
ds as a ‘space-time interval’.
Of course, the equation
will become
Obviously, the pseudoscalar character of ds and
cdt should be recognizable in some way even in
the formalism of Special Relativity.
So, the question is:
how an intrinsically
psudoscalar quantity can
be recognized in the
formalism of Special
Relativity?
The answer is:
For the Minkowski space-time, the pseudoscalar
square root of the determinant of the metric tensor
is an imaginary number.
Therefore, pseudoscalar quantities,
such as ds and cdt, must be
represented by pure imaginary
numbers.
It follows that the equation
is algebraically, but not tensorially equivalent to
the equation
More in general, the signature of the space-time in Special
Relativity is (+ + + -) and not (- - - +).
Concluding
(1)- Time does not need any external “time
arrow” to surrogate its intrisic
one-wayness
(2)- The operation t - t does not involve
time reversal;
it only involves space inversion and
change of sign of all the pseudoscalar
quantities.
(3)- Even the space-time of
Special Relativity, while defined
independently from its ‘content’,
would have no reason to exist
without it.
(4)- The roots of the dualism
wave-particle
must be searched in the blending of
continuity and discreteness
which characterizes the pseudoscalar
time of Quantum Mechanics and
Relativity
at
g
Weyl Hermann, Space-Time-Matter, English translation by Henry L.
Brose, Dover Publications, New York, 1952(p.109)
Conception of Tensor-density
If Wdx, in which dx represents briefly the element of
integration dx1, dx2, … , dxn, is an invariant integral, then W is
a quantity dependent on the co-ordinate system in such a way
that, when transformed to another co-ordinate system, its value
become multiplied by the absolute (numerical) value of the
functional determinant. If we regard this integral as a measure
of the quantity of substance occupying the region of
integration, then W is its density. We may, therefore, call a
quantity of the kind described a scalar-density.
This is an important conception, equally as valuable as the
conception of scalars; it cannot be reduced to the latter. In
an analogous sense we may speak of tensor-densities as well
as scalar-densities.
w
H
Foundations of Physics, V ol. 28, No. 7, 1998
The Arrow of Time in the Equations of Motion
Fritz Rohrlich
Received November 5, 1997
It is argued that time’s arrow is present in all equations of
motion. But it is absent in the point particle approximations
commonly made. In particular, the L orentz-Abraham- Dirac
equation is time-reversal invariant only because it
approximates the charged particle by a point. But since
classical electrodynamics is valid only for finite size
particles, the equations of motion for particles of finite size
must be considered. Those equations are indeed found to
lack time-reversal invariance, thus ensuring an arrow of time
Similarly, more careful considerations of the equations of
motion for gravitational interactions also show an arrow of
M
Giving as granted that a continuous
(discontinuous) progress from a given
‘point’ (or ‘element’) is possible both
forwards and backwards, time is currently
considered as a simply extended continuous
(discontiuous) manifoldness.
As a consequence, the change of sign of t
has the meaning of ‘time reversal’.