The timeThe time

shared byshared byQuantum Mechanics and RelativityQuantum Mechanics and Relativity

Giuseppe Guzzetta – Università di Napoli ‘Federico II’ (now retired)

La Siècle d’Albert Einstein - Julliet 2005 - Palais de l’Unesco - PARIS

““Geometry is infinite because any continuous quantity is Geometry is infinite because any continuous quantity is divisible to the infinity on both sides. On the contrary, the divisible to the infinity on both sides. On the contrary, the discontinuous quantity begins with the unity and grows to discontinuous quantity begins with the unity and grows to the infinity, and, as above stated, the continuous quantity the infinity, and, as above stated, the continuous quantity

grows to the infinity and decreases to the infinity.grows to the infinity and decreases to the infinity.””

““La geometria è infinita perché ogni quantità La geometria è infinita perché ogni quantità continua è divisibile in infinito per l'uno e per continua è divisibile in infinito per l'uno e per

l'altro verso. Ma la quantità discontinua l'altro verso. Ma la quantità discontinua comincia all'unità e cresce in infinito, e, comincia all'unità e cresce in infinito, e,

com'è detto, la continua quantità cresce in com'è detto, la continua quantità cresce in infinito e diminuisce in infinitoinfinito e diminuisce in infinito.”

Leonardo da Vinci (1492-1516) Leonardo da Vinci (1492-1516) Codex M, 18 rCodex M, 18 r (Institute de (Institute de France).France).

In his well-known lecture, Riemann In his well-known lecture, Riemann considered considered continuouscontinuous and and discretediscrete

manifoldnesses, respectively made of manifoldnesses, respectively made of ‘specialisations’ respectively called ‘‘specialisations’ respectively called ‘pointspoints’ ’

and ‘and ‘elementselements’’..

“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 oron two sides, forwards or backwardsbackwards..””

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.

PPseudoscalar quantities seudoscalar quantities such as the one I considered such as the one I considered form a sform a simply extended manifoldness in which a imply extended manifoldness in which a

continuous progress fromcontinuous progress from a point, or an element, isa point, or an element, is possiblepossible

only on only on one sideone side:: either either forwards orforwards or backwards.backwards.

Such a manifoldness is characterized bySuch a manifoldness is characterized by

continuitycontinuity, , periodicityperiodicity andand handednesshandedness

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 apseudoscalar 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 time-handedness 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 the ‘becoming’ of a material particle should be thought of as consisting in its unceasing should be thought of as consisting in its unceasing change change of of positionposition and/or and/or angular positionangular position at theat the

universal rate of becoming cuniversal 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.

is algebraically, but not tensorially equivalent to the equation

It follows that 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-densityIf 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 time. The existence of arrows of time in quantum dynamics is also emphasized.

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’.