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Quantum mathematics for physics:second example :
elementary quantum formulae
André Avramesco
To cite this version:
ccsd-00000327 (version 1) : 25 Apr 2003
Quantum Mathematics for Physics
Second Example : the Elementary
Quantum Formulae
A. Avramesco,
Le Chˆateau, F-25290 Ornans [email protected]
Abstract
It is shown that a fundamental unit, denoted by h, naturally arises from the quantum mathematical representations already used for the Lorentz-Einstein transformations in the First Example (cf.reference). PACS number : 00.
Keywords : quantum (or discrete) mathematics ; graphs
Foreword
The first part of this publication (cf.reference) has established the bridge between continuous description and quantum frame for physics (the same figures and captions are used here). The general idea is to preserve the images of events and of their relations, but to put aside the coordinates and their axes – in conformity with the initial, and essential, intuitions of Einstein –. A summary here would be inefficient : but perhaps a kind of dictionary can bring closer to the new conception and axiomatics.
1
Elementary lexicon and comments
The image of physical phenomena is now given by acyclic directed graphs. The vertices of the graphs correspond to physical events and to some situ-ationsin x and t of continuous descriptions.
The directed edges, or arrows, correspond to the ordered relations between events and to the signals of continuous descriptions : but each arrow is a quantum, and there is no intermediate value on the arc in the drawing of graphs. For the sake of simplicity, the theory is at first restricted to one dimension : there are only two kinds of arrows, F’s (forward ones) and B’s (backward).
transformations appear with a remarkable straightforwardness if the notion of coincidence is used : a coincidence simply is an event common to (at least) two physical systems, and its image naturally is a vertex common to (at least) two graphs. This is very different from continuous descriptions, where any situation (x, t) is regarded as physically meaningful and hence mixed up with events except in rare criticisms such as those by Einstein himself, and perhaps still more clearly by Eddington.
A number of consequences of the definitions and axioms dictated by experiment have been deduced. Perhaps the first to be recalled is the fun-damental property of reciprocity, which is easily formulated in the case of coincidences separated by only one kind of arrows. In this case, let two graphs be given (see fig. 1, 2, 3) ; if for them there exist two consecutive coincidences Cn,p and Cn+1,p, then there are also two (possible)
consecu-tive coincidences Cn,p and Cn,p+1 , such as :
given the shortest path from Cn,p to Cn+1,p in the two graphs, if in one
of these graphs (red) this shortest path includes N< F’s, and in the other
(blue) N> F’s, then the shortest path from Cn,p to Cn,p+1 includes N> B’s
in the red graph, and N< B’s in the blue one.
Using this fundamental reciprocity (the relative Doppler effect is the same when observed from any of the two graphs or physical systems), it is immediate to complete the elementary mesh with the four coincidences
Cn,p, Cn+1,p, Cn,p+1, Cn+1,p+1. The translation from quantum to continuous
description ensues, with
χ = [(1 − w)/(1 + w)]1/2= N
</N> (Doppler ratio)
and
w = v/c = (1 − χ2)/(1 + χ2) (reduced velocity).
But even these first formulae show that the translation is not faithful. Space-time parameters depend upon the ratio of N< and N> , not upon their
values. The velocity describes in the same way very different coincidences, very different sequences of events common to various physical systems. As a matter of fact, the discrepancy of space-time translation goes much farther than that. Only two coincidences C0,0 and Cn,p are enough to allow the
1) it corresponds to an infinity of superfluous (continuous) values of coordinates without physical significance
2) on the other hand it cannot detect the dissimilarities between very dif-ferent physical developments (very difdif-ferent coincidences), and so unavoid-ably “hides” parameters with no equivalent in space-time language : for the coincidences giving the same χ, v or w = v/c etc. might not be as trivially spread as has been put forward in the periodic graphs of the first examples previously analysed, and in the corresponding figures.
One more preliminary is necessary in view of quantum generalizations. As has been just underlined, the coincidences can be spread in many dif-ferent ways. Consider again fig. 4 and 5. In fig. 4, there are no internal events between the left and right vertices, which correspond to the emissions-absorptions of signals by the same punctual atom. In fig. 5 on the contrary, there is a (very) great number of such internal events : a (very) great num-ber of things do happen in the subprogram starting from (O1) from A to B
(here of course B indicates the vertex in the figure, not a backward arrow) , and these intermediate quanta materialize the existence of the atom between the exchanges with its neighbours. It is clear then that the grid of events is (very much) finer and (indeed) not so regular as the one initially considered. Once again : the ratio χ allowing the comparison of times and spaces will be the same for two analogous frame-graphs (images of two analogous solid straight lines) – but the atoms are no longer punctual.
Intuitively, it is partly possible to translate this in conventional language : an atom is now conceived as a subprogram of confined signals. If the frame of reference is changed, the frequencies are greater in one direction (ratio 1/χ), and lesser in the other direction (ratio χ). Then the formulae for en-ergy etc. follow in an elementary way. But it is necessary, and easy, to build them up in the new conception itself.
For this, in conformity with the rest of this exposition, the physical sys-tems and corresponding frame-graphs are kept two-dimensional, hence one-dimensional in space, and in the drawings the F and B arrows are respec-tively tilted by the angles + and π/4. Since the number of internal vertices-events (like T in fig. 5) is (much) greater than the one corresponding to signals towards neighbouring atoms (external relations, as from A to (S), or as the one coming from (S) to B in the same drawing), the arrows between internal events must be represented as much smaller than the others. The corresponding length – which has a very simple classical interpretation – actually is a matter of relative frequency of arrows of the two types, the changes in t and x/c are no longer as simple as on fig. 0, and the difference is essentially a matter of topology of the graph, as follows.
be on the other side. On the contrary if, in a drawing, vertices of both kinds are mixed on the same part of “half square” in which AB is a “diagonal”, then the lines corresponding to arrows give crossings without significance, crossings in the drawings which do not correspond to vertices-events – a well-known discussion in graph affairs.
With this last preliminary, the path to quantum generalizations is clear again : it will now follow more closely the way to the rebuilding of formulae – but not to the importance of the ideas themselves.
2
Notion of quantum of action h
In the one-dimensional case examined in this text, a physical system can be defined as a (relatively dense) subprogram of a frame-graph with as many F as B arrows: this is enough to define a kind of line “parallel” to an axis x/c = constant (in other words : this is enough to define a rest frame-graph). A very simplified but good example is the subprogram on the right of (O1)
in fig. 5 : and it is easily generalized to any periodic subprogram obeying the fundamental condition of equality of numbers of F’s and B’s. Let h be the value of a certain quantity attached to one arrow, and let a “big” unit of time be chosen along a line x/c = constant (hence something like a “horizontal” row of very many squares of the kind of fig. 4 and 5). Let ν0 be the total number of arrows corresponding to this unit for the
considered periodic physical system. Define E0 as
E0 = ν0h
so that if t0 is the time, measured in the chosen “big” unit, the total number
of arrows for the system during t0 is
S = ν0h.t0= E0.t0.
This number is independent of the frame-graph in which it is evaluated – for important reasons, this S must have the sign opposite to the conventional least action –.
Recall now the formulae for the change of frame-graph, or Lorentz-Einstein transformation, written in the first part of this text as
(tr+ xr/c) = χ.(tb+ xb/c)
(tr− xr/c) = (1/χ).(tb− xb/c)
and replace the index r by 0, while the index b simply is suppressed. This
gives
t0 = 1/2.(1/χ + χ)t − 1/2.(1/χ − χ)x/c
x0/c = 1/2.(1/χ + χ)x/c − 1/2.(1/χ − χ)t.
E = 1/2.(1/χ + χ)E0, pc = 1/2.(1/χ − χ)E0
this can be rewritten as
E0t0= Et − px
E0x0/c = Ex/c − pct.
It is already evident that E0 = ν0h or E = νh is the Planck-Einstein formula,
with the natural variation when the frame is changed. Then, since E0 is
the total rest energy of the considered system, the corresponding Compton length is
λ0 = hc/E0= c/ν0
and
– p0 = 0 : the total momentum of a system is 0, since by definition
there are as many F’s as B’s (again the fundamental condition : as many “confined signals” in the two directions), but this transforms as
– p = 1/(2c).(1/χ − χ).hc/λ0 = h/λ , now the Einstein-de Broglie
for-mula and the tie with the Compton length.
The interpretation with χ as the Doppler ratio is straightforward but the reading of these formulae in terms of relative frequencies of arrows in the graphs (hence in the drawings : relative “lengths”) is still more instructive. The reader will perhaps be pleased to compare all these expressions to the irrational Lorentz contractions etc. – which is immediate according to the lexicon given in section 1 here, especially the equation for χ. If this last coefficient is ' 1, that is
N<' N>, v/c ' 0, E ' E0 ,
and since by identity E2
− E02 = (pc)
2 or
(pc)2
= (E − E0).(E + E0) ' (E − E0).2E0
it is possible to rewrite some formulae in order to answer frequently asked questions. From the last approximation, it follows that
(E − E0) ' p2/[2(E0/c2)].
In other words, the increase of density of quanta when changing the frame (the kinetic energy), draws attention, in the case of “slow movements”, to the coefficient E0/c2. The fundamental character of this coefficient could
3
Hints at further developments
It is amusing to ponder over the case of sequences Σ of arrows of the same kind (“pure signals”). For these, there is of course no rest frame, since a frame is defined as a subprogram with as many F’s as B’s. But the relative frequency, the relative number of arrows during any part of the subprogram from which a Σ is emitted, does exist. This amounts to saying that there is a well defined frequency (of emission), but no rest energy for Σ .
With the same simplicity, the fact that a subprogram is easily seen as periodic (more precisely : corresponds to a doubly periodic grid of real or virtual coincidences), is mathematically synonymous to the existence of functions of grids such as exp[i.(2π/h).(at − bx)] and for one system there are as many grids as other systems, so such functions have to be combined to describe the possible evolution.
But the point of view here has been that – at least in the frame-graph of one system –, it is possible to compare phenomena at a given time. In fact, if the notion of unfolding of a graph, of a real program, is better defined, the vertices common to different subprograms cannot be supposed to exist so to speak “in advance” according to as elementary rules as the ones used in all the figures here – in other words : in the physical, real directed graph theory, the notions corresponding to space and time do not play at all as symmetric a part as has been considered above. A notion corresponding to the “present” should be redefined by relative priorities of subprograms, and this is a point of view rather different from the building of “simultaneity”, which is the root of the illusions of space.
These essential points explain why the two examples of Lorentz-Einstein transformations and elementary quantum formulae cannot be directly ex-tended to more general mechanics. For instance, graph theory explains sim-ply why the natural embedding space is R3 : if different frame-graphs are
embedded in the same space, the “state” or “present time” of each should be first put into correspondence with all others (which crudely results in a non-directed graph), and this correspondence sends back to the already mentioned problems of unfolding of graphs and “present”, and so leads to other works.
But the whole of experience includes its history, and this shows that new ideas, however simple and directly fertile, are not always immediately wel-come. Yet perhaps some readers at least would be willing to pay attention to the following provisional conclusion : even if the use of graph theory were to be restricted to the above results, these would be sufficient counterexamples against the assertions of impossibility of another “language” than space and time to think physical reality and theory.
Reference
A. Avramesco (Quantum Mathematics for Physics, First Example : the Lorentz-Einstein transformation) http://ccsd.cnrs.fr/HAL (ccsd 00 000 324)
