P. V. Poluyan,
Numbers in Space
One of the Wolfgang Pauli's scientific texts begins with a remarkable
phrase: "Let us introduce, as usual, material coordinates Xk
for space and imaginary coordinate X4=
iCt for time and consider
Lawrence's transformations..." (W.
Pauli. Works on Quantum Theory. M. "Nauka", 1977, see article
"About Mathematical Matrix Theory Of Dirak", p. 5, "Lawrence's
Transformations of Dirak's Wave Functions", p. 233). The phrase
"as usual" can be considered here as a kind of a witty intellectual
provocation, which means that the above-mentioned procedure can be performed not
"as usual", but in "an unusual way". But how?
It is not difficult to say: we try to maintain the material coordinate
for time and consider 3 spatial coordinates imaginary. Then Minkowsky's
four-dimensional pseudo-euclidean continuum will transform into some unusual
variety, which we shall call "Quaternized time-space".
The appearance of the term "quaternion" here is evident: it is
easier to present 4 numbers, expressing coordinates (one material, three -
imaginary) as quaternion. But quaternion is algebraic numbers, and
four-dimensional space-time is continuum. If it is so, is there enough reasoning
to make them correspondent? We shall try to answer this question later and for
the present we shall consider quaternion time-space as some pure logical
construction, which can be seen as a whole and analyzed in particulars. It is
also important to mention that the term "space" in modern science is
not connected any more with distance measuring, and nothing disturbs us to make
a four-dimensional space, where a measure in [t] is put on the axis. But as time
is of physical character, which reflects the important aspect of reality, not
formal mathematical qualities of the made-up construction, but its physical
interpretation will be of greatest interest to us in this article.
The fact that the algebra of quaternions is not commutative leads us to
the idea that an abstract object, made-up this way, is directly connected with
quantum-mechanical peculiarities of the physical world. But let us consider
quaternion time-space as if we do not know anything about quantum mechanics. In
other words, we shall try to preserve the classical notions of time and space.
Thus we have a four-dimensional variety, where the material axis is pure
time, and the rest three ones are spatial coordinates transformed into imaginary
temporal axes. While building Minkowsky's four-dimensional pseudo-euclidean
continuum, all the coordinates were measured in [x]
as a result of multiplication of a temporal coordinate and coefficient C
which is velocity of light [m/s].
That is why in our quaternion time-space a 'one-measurement' is achieved in
analogical way: Multiplication of imaginary spatial coordinates and some
coefficient S, measured in [s/m].
Sometimes it is considered for the interpretation of Minkowski's
continuum that the shift of t into x
with the help of the co-efficient C
is of no importance- this is a strange illusion, because time cannot be
physically equal to the temporal extension. If we take C
for a unity, measure [t] and [x] will not disappear because of that. The same is with the
statements like "only spatial-temporal interval is truly important",
"space and time are united in their nature" and so on, which are more
philosophic statements than physical. That is why it is extremely important that
we choose a different one-measure system in our quaternion time-space: imaginary
spatial co-ordinates must be multiplied by some co-efficient S,
measured in [s/m]. And again it may seem that nothing special happens because it
is just "the reverse velocity of light". But the changing of the
co-efficient, which is not important in mathematical sense, leads to great
changes in physical sense.
The reverse velocity of light 1/c,
as real physical quantity cannot be an unknown coefficient, while the scale of
reverse velocities is irregular. In classical notion velocity is a ratio, where
the numerator is the distance segment, and the denominator is time period, time
being independent variable quantity. Then dealing with 'reverse velocity', where
the numerator and the denominator exchange their places, there appears not only
new, but also irregular measuring scale: 1[m/s] = 1[s/m], 2[m/s] = 1/2 [s/m], 3 [m/s] = 1/3
[s/m], 4 [m/s] = 1/4 [s/m], etc.
Standard mathematical analysis and pseudo-euclidian space do not
contradict each other just because space does not possesses inner metrics (as it
was underlined by Rieman), in other words a unity can be as big as possible, it
is not set as some inner measure unity of distance. In our case, velocity of
light C, which plays the part of the
co-efficient by the imaginary unity, is quite a concrete physical quantity, the
velocity of electromagnet waves. We can imagine it as some unity only
relatively. For mathematical characteristics of Minkowski's space-time it is not
essential, but in the real world this unity C is used to characterize the unique physical process, that is the
change, which is mathematically harmless, can be approved physically.
It seems that due to this reason quaternion time-space cannot be an
analogue of the four-dimensional continuum. But it easy to find the way out, if
we do not consider S to be 'reverse
velocity', but some coefficient measured in [s/m].
Let us turn from mathematics to physics. If coefficient C
in Minkowsky's pseudo-euclidean continuum is a concrete physical quantity -
velocity of light, which has in different measurement system concrete numerical
realization, in our quaternion time-space coefficient S
must be some physical constant quantity, different in its nature from velocity
of light, but having a measurement [s/m]
- a reverse one to the measurement unit off velocity. We can offer a combination
of constant h/e2 to suit
this new constant, where h is Plank's
constant, and e is the charge of an
electron. It is well known that this combination as well as C
is included in the expression of the non-measured constant of thin structure 1/a
= 137.0306... (ħ
is Plank's constant divided into 2p
I believe that is true, that quaternion time-space is a mathematical expression
of the real aspect of microphysical reality, where the constant S
= h/e2 measured in [s/m]
is as important as velocity of light for Minkowsky's four-dimensional continuum.
Of course, the author can be reproached for a kind of arbitrariness
because it is possible to construct the measure [s/m] from the constants in some other ways (e.g., by using the
gravitational constant). The only reason, by which the author is motivated here
is the desire to find the logical connexion between the quantum and relative
physics, discovering at the moment only formal deep mathematical connexion
between the global spatial-temporal picture of the world and microphysical
quantum reality, while these constants are accepted to be used to express a
size-less constant of a thin structure. The logical sense of non-measured
constant of thin structure can be seen in the fact that it shows the
correspondence between Minkowsky's continuum and quaternion time-space. I
believe Wolfgang Pauli, who insisted on theoretical grounding of physical status
of this mysterious number 137.0306... meant something of that kind.
But formal arguments are not enough here. We must show the physical
essence to discover correspondence, that is to discover the connections between
the velocity of rectilinear forward movement C and constant S,
the meaning of which is not quite clear yet. S=h/e2
is a combination of empirical constants measured in [s/m], we include it in some mathematical structure, but that has
not cleared up its meaning.
In classical physics velocity is a quantitative measure of forward
movement, which binds spatial and temporal characteristics of motion as
rectilinear forward movement. If constant S
is included in Quaternized time-space, it means that it must be also understood
as an expression of some aspect of motion, where spatial and temporal
characteristics are bound somehow. Moreover, one of the most important qualities
of Minkowsky's continuum is Lawrence's transformations, which lead to that law
of adding velocities while leaving one-measure system for the other gives us
maximum meaning for the rectilinear forward movement.
It would be logical to suppose that in quaternion time-space there is
also an analogue of Lawrence's transformations, which will let us interpret
constant S as an invariant and limit
in adding some quantities. Thus, the matter should look like a case of using 2
measures, where on the complex plane by means of pseudoe-uclidean way one
temporal and one spatial axes are being bound.
For Minkowsky's continuum an imaginary axis will be iCt - a temporal axis, for
quaternion time-space - a spatial axis iSx.
While dealing with two-dimensional case the matter does not seem difficult, as
we do not consider non-commutability (on the other hand, it is discovered that
non-commutability is directly connected with the presence of two more imaginary
While velocity of light C is
non-classical limitation of maximum velocity (velocity of signal expansion over
some distance cannot be endless), correspondently, constant S
also does not let the ratio Dx/Dt take endless meanings. But S
is a limit for "reverse velocity", and increasing of Dt/Dx means at the same time decreasing of ratio Dx/Dt. That leads us to the thought: "zero velocity" is as
unattainable as endless velocity.
Nevertheless, in case of a simplified 2-measured complex notion of
quaternion time-space, it is still not clear what measures they should be, what
is the physical meaning of "measure system" in this case? We are
expected to answer these questions.
While S is some coefficient of
proportionality between time-measurement t[s] and space-measurement x[m],
constant S as the independent
parameter expresses some aspect of motion. But while the quantity measurement
for forward rectilinear movement is the classical notion of velocity V[m/s]
and its non-classical limit C, this
new constant must be a non-classical limit of some classical movement
measurement, which is a forward movement, nevertheless. We suppose that the form
we need is rotation.
There are microphysical and mathematical premises to connect the
mentioned quantity with nothing else, but rotation.
In physics of elementary particles the existence of the so-called
isotope-transformations, which are completely the same as ordinary rotations, is
experimentally discovered. Werner Geisenberg, accounting basic symmetry groups,
places some special group next to Lawrence's group, it is the group explored by
Pauli and Gucci, which according to its structure corresponds to the group of
three-dimensional special rotations. It is isomorphous to this group and reveals
itself in appearance of the quantum number, which was discovered empirically and
which characterizes elementary particles, it is called "isospin" (W.
Geisenberg, "Quantum Theory and Material Structure" Physics and
Philosophy. Part and the Whole.
Moscow: Nauka, 1990, p. 103). Ratios,
which are the result of isotope-in-variety, are observed to calculate to within
amendments, the quantity of which is determined by the ratio e2/hC. It is
noted in the textbook that "isotope-in-variety means a special symmetry of
great interactions, which is not connected with general qualities of space and
time. Though isotope-in-variety is
discovered quite well experimentally, the qualities of symmetry connected with
it do not follow from this theory, and the nature of these qualities is not
discovered yet" ("Isotope
Spin" Physics Encyclopaedia, Moscow, 1962, Vol.2, p.143).
Mathematics-educated readers have, probably, understood that that object
which is known here as quaternion time-space is quite well-known Klifford's
algebra of the four-dimensional vector space. Its applicability in physics has
been shown several times, as well as for isospins. But the usual attitude
towards such applicability of vector algebra in non-classical physics is quite
skeptical. What has been done in that respect in France (works by G. Gusanova,
C.R. Acad and others) is regarded usually as the result of specific
interrelation of quantum physics.
Thus, the real aim of the work the author proposes to discus here is
grounding of the fundamental importance of vector algebra for studying the
Universe. The author believes that
quaternion time-space is a logically necessary element of the four dimensional
space-time (Clifford's algebra determined in the field of hyperreal numbers for
the space with time measure system), which closes spatial-temporal structural of
the universe, and the division of the size-less quantity into two measurable
constants determines that fragment of the universe where physical processes take
place in time. The author attempts
to show that vector algebra is not just a specific mathematical language to
reform the well-known physical data, but, on the contrary, it appears so
logically and naturally on the basis of the classical Decartes space pseudo-euclidian
Minkowski's continuum is built.
The applicability of the mentioned approaches is doubted by many thanks
to the standard notions of limits and infinitesimal. The author thinks that
while the logical non-contradictoriness of the non-standard analysis has already
been proved by Abraham Robinson's works, nothing disturbs us to re-realize the
standard notions of interrelations infinitely big numbers and infinitesimal.
This is what happens, when the four-dimensional space-time is closed with
quaternion time-space into one whole unity. And this really happens.