quantum mathematics for physics -First example : the Lorentz-Einstein tranformation

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quantum mathematics for physics -First example : the

Lorentz-Einstein tranformation

André Avramesco

To cite this version:

ccsd-00000324 (version 1) : 25 Apr 2003

Quantum Mathematics for Physics

First Example : the Lorentz-Einstein

Transformation

A. Avramesco,

Le Chˆateau, F-25290 Ornans a.avramesco@wanadoo.fr

R´esum´e

It is shown that a fundamental unit, denoted by c, naturally arises from quantum mathematical representations, together with a number of important formulae.

PACS number : 00.

Keywords : quantum (or discrete) mathematics ; graphs

Foreword

... nature is so constituted that it is possible logically to lay down such strongly determined laws that within these laws only completely determined constants oc-cur (not constants, therefore, whose numerical values could be changed without destroying the theory). (cf. reference)

perceiving and conceiving the physical world. The aim is to show that space and time are nothing but by-products of deeper aspects of reality, so that another and more general frame has to be put forward for physical thinking. In this context a serious bibliography would be longer than the exposition itself. But it is necessary to emphasize that, in a sense, the following work does nothing but take into account the most general experience : the history of those successive condensed summaries of experiments, which are called physical theories. From Riemann (the first to clearly mention the founda-tion of space upon a “discrete manifold” ) to Einstein (all along his life and work, but perhaps with special clarity in the 1954 appendix to The mean-ing of relativity), through Weyl, Eddmean-ington and Louis de Broglie to mention the best-known only, the search for another scheme (or “language”) than space and time has been ceaseless, and probably nothing but the orthodox reluctance to think in more experimentally dictated and theoretically sim-pler terms can explain that it has not been made explicit earlier. There lies the main difficulty. At first, it may appear surprising that the use of deeper concepts (events and their relations) gives as a consequence the more cus-tomary thinking (space and time). Yet for some time already, the distinction has been made clear between the reality of events and the description by coordinates.

because many readers in the present time are not conversant with the sim-plest discrete topology : graphs, for which however several good textbooks already exist -.

1

Physical and mathematical elements

The simplest grid is given on fig. 0. It can represent, in classical language, a solid straight line with regularly spread identical punctual atoms ; each atom exchanges immediately reflected signals with its two neighbours, and the absorptions and reflections at all atoms of even abscissa are simultaneous (which implies the similar property for all atoms of odd abscissa). If P is the chosen origin, the line t = 0 is the vertical through P, the “trajectory” of the corresponding atom is the horizontal of constant (reduced) abscissa x/c = 0, and the arrows represent the signals between atoms.

But the same grid can be understood in a different way. Two kinds of arrows may relate a pair of points in this acyclic directed graph, and correspondingly two integers are attached to each point (or vertex) of this graph :

– a forward arrow (F) increases both integers by one

– a backward arrow (B) increases the first integer by one, and decreases the other by one.

In short : the grid is fundamental, because it can be read in terms of events and their relations - and these are essential, coordinates are not : here lies the turning point.

In this line of thinking, two kinds of remarks should be underlined in view of the necessary change of habits and consequently for a more comfortable understanding of what follows, which amounts to an exit from the space and time language :

– a) about the physics : the notion of signal is replaced by the one of re-lation, and it is a quantum relation ; only quantum relations now have phys-ical reality, whereas intermediate continuous values of t and x/c have none : neither do the “angles” in the figure, the “lengths” of arrows, nor in general the continuous metrics.

– b) about the mathematics : as has just been mentioned, the “length” of arrows can be changed in many ways ; but in order to ease the transla-tion from quantum to continuous representatransla-tions, all figures here keep the arrows tilted by an angle +π/4 for the forward, −π/4 for the backward ones. In this way the drawings will conform with the following fact : in continuous geometrical representations, the axes of coordinates (constant t and constant x/c) are not orthogonal in the general case, but their bisecting lines (t = ± x/c) should be ;

also, although the figures in a way speak for themselves, some reminders of elementary definitions about graphs can be given as follows :

– a graph is conceived as a set of points (or vertices) and “edges” (in fact : pairs of points) ; an “edge” is a direct relation between two points ; if two points of the graph are given, there is or there is not an “edge” (a direct relation) between them - it is 1 or 0 : quantum ;

– the graphs hereafter are all directed : if there is an “edge” (a direct relation) between two points M and N, the corresponding pair is ordered ; for this reason, arrow will be used instead of “edge” ;

- the graphs are acyclic : following the arrows, it is impossible to start from a given point and come back to it - as one consequence : there cannot exist an arrow from M to N and from N to M ;

will become apparent in the figures.

2

Limit velocity c and special relativity

The fact, that the length of arrows is unimportant in the essence of the interpretation, can be now used in the following way. The frame-graph of fig. 0 has been presented as the image of a solid straight line. What happens if two similar straight lines (one “red”, the other “blue”) slide along each other ?

Consider first the phenomena in classical language. When two punctual atoms, one in each of the lines, coincide, the corresponding event is chosen as origin for both unidimensional solids - this initial coincidence is the C0,0

of the following figures -. At that time, the two atoms each send one signal to each neighbour of their own frame. Since the velocity of light is absolute, the wave fronts towards the right and towards the left will be common to both systems, but of course the durations and distances will be different. For instance, the first coincidence C1,0 on the right wave front occurs when this

front simultaneously reaches a red atom after N< red intervals, and a blue

atom after N> blue intervals. Then, in conformity with the fundamental

reciprocity of relativity, the wave front towards the left also reaches a new atom after N< intervals in the blue line, and N> in the red one.

and Cn,p+1 -).

In the red elementary mesh, in accordance with the description given above in classical terms, there are N< forward arrows from C0,0 to C1,0 (or

from Cn,p to Cn+1,p ), and N> backward arrows from C0,0 to C0,1 (or from

Cn,pto Cn,p+1). The rest of the vertices in the mesh is made clear in fig. 2 - in

the drawings : N< = 3, N>= 5 -. The squares of fig. 0 are now changed into

rectangles, since the “lengths” of F’s and B’s (in the geometrical drawing) are not equal any more.

And in the blue elementary mesh (fig. 3), in conformity with the fun-damental reciprocity, the roles of the two N’s are exchanged - so are the relative lengths of the F’s and B’s.

Many difficulties might be overcome if one bears in mind the following remark. Ascending arrows in fig. 2 and 3 are called “forward”. This in fact destroys the perfect reciprocity, but conforms with established habits : for it is customary to consider that the x-axis in “the moving frame” and in “the fixed one” are similarly directed, whereas perfect reciprocity would imply that for two “trains” passing each other (in “opposite” directions), of course the forward for one would be the backward for the other (without reference to any railroad or “fixed” frame) ; then the really reciprocal velocity would be the same for both trains, instead of changing sign from one frame to the other.

Let now the two integers, attached to the definition of F and B arrows, be evaluated with C0,0as origin (cf. fig. 1 and §1). The first integer increases

with each new event, and should be interpreted as the time t. The second integer increases following each F, but decreases following each B : it should be interpreted as the (reduced) abscissa x/c. The indexr will be attached

to the red frame-graph (tr and xr/c ), the indexb (tb and xb/c ) to the blue

2.2 Results

R1

The velocity corresponding to signals is the same in both graphs, its value is 1 in natural units, and it is at the same time absolute and limit : for the velocity must be defined as the ratio of the distance (absolute value of difference of abscissae) to the duration, and following the arrows the time always increases, whereas the distance can decrease (not for pure “signals”). This limit will be denoted by c to show that it implies two distinct types of counts.

R2

For Cn,p , counting F’s and B’s,

in the red graph

tr = nN<+ pN> , xr/c = nN<− pN> ,

whilst in the blue graph

tb = nN>+ pN<, xb/c = nN>− pN<.

These values can be interpolated for intermediate event-points : but then they lose part of their meaning, and they have no meaning “on” the “arc” of the drawing corresponding to an arrow : each arrow is, it must be stressed again, a quantum - it is a relation, an ordered pair of points, not a line in some mathematical continuous space -. This does not imply a (universal) quantum of time (or space), as is already evident from the fact that coin-cidences can correspond to quite different elementary meshes : and it will appear again in the more strictly quantum elaboration.

R3

Let ∆ denote an increase of a variable in general. For a line of red constant abscissa, ∆xr/c = 0, so that if ∆n and ∆p are the corresponding

∆n. N< − ∆p. N> = 0 or

∆p / ∆n = N< / N> .

This fundamental ratio, N< / N> , will be denoted by χ (already at

this stage, it can be seen as a Doppler ratio since it corresponds to the two different evaluations of local times between two coincidences of signals). Now if w = v/c denotes the reduced velocity (of any punctual atom, or line of constant abscissa) of the red frame-graph with respect to the blue one, it must be defined by the corresponding increases of

xb /c and tb

that is

w = (∆xb/c)/(∆tb)

or, according to the definitions of xb /c et tb and using the above values of

∆n and ∆p

w = (∆xb/c)/(∆tb) = [(∆n.N>− (∆p.N<)]/(∆n.N>+ ∆p.N<)

and after dividing numerator and denominator by (∆n.N>)

w = (1 − χ2)/(1 + χ2),

with the reciprocal result

χ = [(1 − w)/(1 + w)]1/2

which is the conventional and irrational aspect of the Doppler ratio. R’3

R4

In the formulae written at the beginning of R2, n and p can be elimi-nated, using the fundamental ratio χ :

(tr+ xr/c)/(tb+ xb/c) = (tb− xb/c)/(tr− xr/c) = χ (L-E) .

This entails the “invariance of ds2_”,

t2

r− x2r/c2= t2b− x2b/c2

but the natural results in terms of graphs are all rational, and radical signs appear only with conventional continuous habits. For instance,

(tr+ xr/c) = χ.(tb+ xb/c)

(tr− xr/c) = (1/χ).(tb− xb/c)

hence by (half) sum and subtraction the formulae allowing the change of co-ordinates : and the writing of the Lorentz-Einstein transformation is then made completely explicit - it can be rewritten in the conventional way using the formula above giving χ as a function of w = v/c.

R5

In view of space-time rewriting, the ratio χ is essential. But as can be seen, the C coincidences (common vertices, image of common events : not simple superpositions in the drawings) need not be as dense and regular as above, and the graphs need not be infinite. From this, a first generalization follows : let

χ1/2= f1/f2

f1 number of forward arrows between two C’s in a first graph, f2 the

this last to a third one (as just remarked, all the coincidences need not be common to the three graphs). Then

f1 = χ1/2.χ2/3.f3

or

χ1/3= χ1/2.χ2/3 ( χ )

and since the ratios of backward and forward arrows are inverse

(1/χ)1/3= 1/(χ1/3) = 1/(χ1/2.χ2/3) = (1/χ1/2).(1/χ2/3).

In this way the group character of Lorentz-Einstein transformations is made trivial - and equation ( χ ) also gives the formula for the composition of (reduced) velocities.

In order to see more completely the relation to the physical context, one step further can be taken concerning the χ’s. All frame-graphs above are, in space translation, one-dimensional, with their lines of constant abscissae (images of identical punctual atoms) at equal spacing. In the corresponding figures, instead of the red frame-graph, let a green one be considered with N< = 6 (instead of 3), and N> = 10 (instead of 5), the (real, physical)

common vertices-coincidences with the blue system being the same. This of course means that the spacing between the green lines of constant time and abscissa will be half the red equivalent : it is not difficult to generalize further.

3

Hints at further generalizations

representations are almost straightforward, and will be given in the second part of this text : but two more remarks might be welcome.

First, it is of course simpler to begin with only one dimension in space, hence only F and B arrows. But this restriction is not necessary, and if it is put aside, the three-dimensional character of the natural space of represen-tation appears as a result of graph properties - instead of being a thrown-in axiom.

Second, consider again fig. 0. There is no materialized horizontal line corresponding to x/c = 0, hence the punctual atom does not seem to ex-ist between P and the next crossing of signals on the right [vertex (2,0)] : which of course is not acceptable. The apparent difficulty is lifted by the comparison between fig. 4 and fig. 5. In fig. 4, the elementary (tilted) square of fig. 0 is simply reproduced : nothing happens, no event, between the left and right vertices. In fig. 5 on the contrary, a (very) great number of ele-mentary events (“confined signals”) indicate that an atom corresponds to a complicated internal subprogram, where many things do happen between two exchanges with other atoms.

All this is elaborated in the following part : the quantum formulae of Planck, Einstein and de Broglie then appear as simply and naturally as did those of Lorentz-Einstein above. And this is not the end of the extensions of graph representations.

Reference

Figure captions

Fig. 0 : Solid straight line of identical punctual atoms in its natural rest frame.

Fig. 1 : Staking of coincidences. Fig. 2 : Elementary red mesh. Fig. 3 : Elementary blue mesh.

Fig. 4 : Element between intermediate vertices in the geometry of the rest frame (cf. fig. 0) : the line of constant abscissa (image of a punctual atom) has no internal “structure”.

constant tb

constant xb

C

C

C

C

(again N

=3, N

=5 - but exchanged roles)