ITS CRITIQUE1
By IRVING STEIN
1 INTRODUCTION
In classical physics, either Newtonian or Einsteinian, space and time are realities independent of each other; i.e., there is no correlation or functional relation between the values of one and the other. Objects exist "in" space and time and are defined as space functions of time "associated" with a mass; that is, it is in the existence of objects that space and time find a correspondence. Space, time, mass, and object are taken as fundamental realities in terms of which other concepts such as velocity, etc. are defined.
The functions themselves are analytic, thus resulting in a deterministic physics. The cardinality of the sets of both space and time points or values is nondenumerable; however, the condition of analyticity allows the definition of an object to be specified by a denumerable infinity of conditions within an interval, D t > 0; this can be done either by the specification of the derivatives at a given time or by specification of the function values.
However, from such an ontology it is not possible to derive the Theory of Special Relativity; in order to establish the Theory of Relativity it is required to introduce phenomena such as the constancy of the velocity of light in all galilean frames or the experimental result that no object apparently has a speed greater than that of light; there is nothing in the classical concepts of space, time, or object from which a maximum speed, the essential basis of relativity, can be inferred. That is, whether or not there exists radiation, or for that matter, gravitational waves, the Lorentz Transformation still holds. A space-time or energy-momentum Lorentz Transformation is of sufficient generality so that in pre-quantum physics it cannot depend on or be a consequence of any particular or contingent phenomena but must arise out of the very nature of an object itself in space and time. That is, it is not that the Theory of Relativity has its source in any particular phenomena, but rather that all phenomena that are defined in terms of space-time or energy-momentum must satisfy the Lorentz Transformation. In this sense, the source of the theory is yet to be found. This point is emphasized by Einstein himself in a letter to Sommerfeld3: "A physical theory can be satisfactory only if its structures are composed of elementary foundations. The theory of relativity is just as little ultimately satisfactory as, for example, classical thermodynamics was before Boltzmann had interpreted the entropy as probability." Einstein, as did Poincare, called such a theory of elementary foundations a "constructive" theory. . In this sense there still remains the problem of re-understanding the nature of these concepts and transforming them so that it is they themselves that give rise to the theory of relativity; this is the meaning of what Einstein meant by a constructive theory; and this is what we do in this work.
The amazing thing about such a theory as developed here is that not only can we then derive the theory of relativity, but find it as the necessary bridge to quantum mechanics.
Such classical concepts as space, time, and classical object as described above cannot give rise to either relativity or quantum mechanics. Relativity theory arises not out of classical physics but as a restraint, arising out of electromagnetism, on the classical concepts of space and time. However the existence of electromagnetism, as stated above, is irrelevant to the classical concepts of space and time. The postulate for the Theory of Special Relativity that all objects must have a speed less than that of electromagnetic waves such as light tells us nothing about the nature of space, time, or classical object from which such a limitation must arise
In this work I investigate what such a revision must be so that from it can arise the space-time and momentum-energy Lorentz transformations. It will be shown that this revision is based upon the concept of what I call precursor objects. From an analysis based on the nature of these objects we will be able to derive the Lorentz Transformation, both the space-time and, after showing the necessity of the concept of mass, the momentum-energy forms.
The contradiction between classical and quantum theory is even more glaring. There appears to be no way that quantum mechanics can arise from the classical concepts of space, time, and object. In quantum mechanics classical concepts , such as frames of reference, position, and velocity have no meaning except insofar as wave functions are referred to classical kinematic variables. Thus, although the wave functions of quantum mechanics can be defined in terms of position and often refer to position and galilean frames of reference, the objects that quantum mechanics deals with do not normally have the properties of position or velocity, and therefore can not be referred to frames of reference, either spatial or velocidal; in fact, the very concept of object needs to be revised, not only for quantum mechanics, but also for relativity.
Thus, despite the overwhelming success of the theory for the derivation of relativity, it is, at best, only a basis from which to derive quantum mechanics. Therefore, we subject it to a critical analysis, removing all assumptions, assumptions that also exist in classical, or pre-quantum physics, that cannot be justified. The resulting bare-boned physics, based on what I call nonspace, turns out to be the ontology, the basis, of quantum mechanics. That is, the precursor objects turn out to be the precursor of the even more fundamental concept of nonspace, which is nothing but all possible Feynman integral "paths," and thus the spin of objects. These "paths" derive from the precursor objects. From this ontology and the nature of measurement we indicate how, first the Schroedinger, and then the Dirac, equations were derived in reference [1].
II RELATIVITY
In order to develop the theory of relativity arising directly out of a transformed concept of object we must first clearly understand the fundamental nature of the Theory of Relativity. It is not so much a new "relativity principle," since it also holds in Newtonian physics , it is rather based on the fact that all objects have a maximum speed; such a fact derives either from experiment or from the logical consequences of electromagnetic theory as Einstein showed. Based on this restriction it is then found that the principle of relativity holds for all classical situations, even for quantum mechanical wave functions although not for the quantum mechanical objects themselves, as stated in the introduction. It is this apparently very strange restriction that is put onto the concept of object from which the Theory of Relativity arises. The question for us here is: is it possible to modify the concept of object so that the very concept of object itself will give rise to the Theory of Relativity? The answer is yes. We do this by introducing the concept of a precursor object. But the precursor object not only allows us to derive the Theory of Relativity, but it also is fundamental to the derivation of quantum mechanics.
1. We define a precursor object as a binary object, an object having two values, +c or -c.
2. A value of the precursor object is defined at each
value of time. Thus, at each value of time there is defined a value + c
or - c value of the precursor object.
3. These values are distributed randomly
over the set of time points so that the mean over any time interval is
zero and the standard deviation over any interval is c.
4. I now define on any time interval, D
t > 0, a sequence of precursor values containing a fraction, p, of + c
precursor values and a fraction, q, of - c precursor values, so that p
+ q =1. The average , v, of the sum of these values, is then defined
as v(t0
,
D
t) = p(t0
,
D
t) – q(t0
,
D
t), where
t0
is a limit point.
5. Furthermore, we define the function, p(t0 , D t), so that it has a limit at t = t0 ; that is,
6. There are, of course, an infinity of sequences, {f(ti)} , that satisfy the condition
8. Just as in the usual formulation of classical physics where the sum of velocities is defined as the sum of two distance intervals at the same time, so here also the addition of any terms of the two sequences, {f(ti)} and {g(tj)} must also be at the same time; that is, i = j. Therefore, in order to add two sequences of distributions , p1(t0 , Dt) and p2(t0 , Dt), we must chose the sequences so that they have like terms defined at the same times, i = j.
9. Since the sum of the two sequences is also to be a binary sequence it is required to define the addition of like terms such that + c Å +c = +c and - c Å - c = -c. And since at any time instant the precursor value is either + c or - c, there cannot be a + c from one sequence and a - c from the other sequence at any given time. Thus, in the sum, there is, once again, only + c and - c terms. That is, in order to be able to add two sequences it is necessary to define a random binary object on the time instances as we have done.
10. We can now express p(t) and q(t), in terms of the average,v(t), where p(t) + q(t) = 1 for all t of the sequence ;
11. Now, as stated above, we define an element of the
third sequence to be +c if there is a +c at the same time in both the first
and second sequences. Thus the probability of there being a +c from each
of the first two (independent) sequences is p1
p2 and the probability
of there being a -c is q1
q2. Since an element
in the third sequence is +c if there is a +c in each of the first two sequences
and –c if there is a –c in each of the first two sequences, and no other
contributions to the sum sequence, we conclude that the fractions of +c’s
and –c’s in the third sequence are:
12. This, of course holds not only within the interval, D t , but also at the limit point, t0 . Thus, at some point of time, t0, there exists not only a precursor value, +c or –c, but also the limits of the average of these values, one each for any sequence, and also of the sum. The limit of the velocity of the sum at the limit point, t0, can now be evaluated from the probabilities p3 and q3 . In expressing the above equations in terms of the averages we get,
13. This equation is reminiscent of the "sum of addition formula" of special relativity. And it is true that the precursor object in being able to produce this formula has penetrated to the heart of the physics of special relativity. However, a Lorentz transformation is first a space-time transformation and we have no concept of space yet. Therefore we require a way to develop the concept of space. We do this by identifying the concept of the average of the precursor values, v(t0), with the concept of the classical velocity, v(t0) ; (we now use the same notation) However, classical physics requires the velocity to be continuous (in fact, analytic). Thus, our concept of the average of the precursor values, v(t0), must also be continuous (analytic). This means that there must be an interval, D t, in which the
average--- or that which is now the same--- the velocity, is defined and continuous. Thus, its integral exists and we can write,
Since the integrand is the velocity, the integral then is a space interval; we see that position is relative..
14. The limit of the average of a sequence of precursor objects, in now having the restriction on it as an analytic space function of time, thus defines not only velocity but a space interval and thus can define a classical relativistic object except for one further required development -- the v's developed above are not relative to other v's; they are absolute In order to define classical physics and derive the Lorentz Transformation, the concepts of galilean frames of reference and relative velocity must be introduced. We define the concept of the relative velocity, v(t), by defining the value of one average function, vi(t), with respect to the value of another average function, vj(t). When defined, such a function will be designated vi / j (t) .
15. The first step in such a process is to define the
value of v relative to that of v0 = 0 having the same value;
that is v1/0
=
v1
- v0
=
v1.
Then if v1
= v1/0
,
v2
= v1/0
,
and v3
=
v2/0
,
from the above formula for the absolute velocities we have,
In words, this says that the sum of two such functions, one, v1 , relative to v0 = 0 and the other, v2 , relative to the first, v1 , gives a value v2/0 , by the above formula, of the second, v2 , relative to v0= 0..
16. If we now solve for v2/1 , we then get the relative value between any two functions in terms of their values relative to v0 = 0 , That is,
17. Therefore, we now can write,
.
.
Thus, whether or not velocity is absolute or relative, position is relative.
Then, if v(t) is constant, we
can write, after integrating , x-x0
=
v(t-t0)
19. Since, the concept of space has no been defined, we use it in its differential form to rewrite the above formula for relative velocity as,
where v 1/0 is taken as a constant. We then write,,
, and therefore,
where g may be a function
not only of v1/0
but
possibly also of x and t. We can show, however, that it is not a function
of x or t in the following manner.
20. Recall from the meaning and definition of "relative" that vi/j = - vj/i. Therefore, we can write ,
Solving for v2/1 , we get.
from which we get,
Substituting these expressions for dx and dt into the equations ,
If v1/0 º v, then integration gives us,we get
.
If we take x’0 , t’0= (0, 0) then we simplify to
Thus, based on the concept of a binary variable, the precursor object, I have been able to derive the space-time Lorentz transformation.
20. I now recapitulate what has been done so far: On each element of the set of real numbers (time), {t}, is defined a random binary variable, ± c, which I call a precursor object. On each element of the set of real numbers, {t}, is also defined v(t0)=lim v(t0,D t) where D t® 0. That is, v(t0) is the limit of the average of a denumerable set of
± c points in D t as D t® 0. It is seen that v(t0)<c except where, with a probability of zero, v(t0) =c. Thus, for any precursor variable sequence, that is, for any random binary "walk," we define a limit to the average of its values. The limit of the magnitude of such an average will always be less than c
22. A function, p(t), is defined as
this is the fraction of v(t) = + c at points, t Î D t, as D t® 0. If the function q is defined as q = 1- p, so that p + q =1, then q is the fraction, similarly, of -c values at points, t, in D t as D t approaches zero; thus both v(t) and a precursor value are defined at every value of the set of reals, {t}.
23. However, it is still necessary to define the addition of two velocities. Suppose that we have two such functions defined over the same domain of real numbers, v1(t) and v2(t). How can their addition be defined? I have shown that apparently the only reasonable way results in the formula,
24. The reason why v3£ v1+v2 clearly comes out in the definition of the addition of the c’s. Although the precursor object, even if it could be defined, would be highly discontinuous, v(t) can be continuous and even analytic. We assume that it is. In defining a function v(t), or equivalently p(t), on the precursor object, we take a step towards defining the classical relativistic object. If we now define v(t) or p (t) as analytic, we then have taken the next step in defining the classical relativistic object.
25. The last step in defining the classical relativistic object is to show how to redefine its velocity function as a relative average function, vi/j. The above addition formula is then shown to be able to be written in terms of the relative average function as
and v is one of the above velocities, taken as a constant.
27. It is in this manner that I put sufficient restrictions on the precursor object, a binary random walk object, to transform the time sequence of a precursor object into a classical relativistic object in classical Einsteinian space-time.
28. A classical relativistic object is defined
as a mass "associated" with a space function of time, where space is derived
eventually from an analytic function of time, v(t), which is defined in
terms of random binary variable, ± c.
It is this basis from which the space-time transformation, x’= g
(x-vt), 
which
we now call the Lorentz Transformation, is derived. No reference
need be made to any concept other than that of the concept of the precursor
object and the usual continuity restrictions on it. The concept of a classical
Newtonian object is also defined as a mass "associated" with
an analytic function of time, which, however, is not based on a precursor
function. Therefore, in this case there is no limiting velocity and the
space-time transformation is galilean.
29. Only when we get to quantum mechanics does the concept of mass arise from the very nature of the object itself. However, even before we get to quantum mechanics we can show that the Lorentz Transformation, when applied to more than a single object, requires the concept of mass. This requirement depends not directly on the precursor variable but on the Lorentz Transformation.
30. Suppose that we write the Lorentz Transformation of space-time points as a rotation in a Minkowski (pseudo-Euclidean) space. Then, the space-time four-vector, u, can be written as,
and the normalized difference between any two space -time points as,
where m is the unit four-vector.
,
of two objects, m
and
m
where
v
=
-v
. Then if the mass
is not introduced, we have,
This, of course, cannot be since the left-hand side of the equation is a constant and the right-hand side is a variable depending upon the velocity, v. It is at this point that another variable must be introduced. Therefore, instead of equation (9) we introduce the concept of mass and write,
Then, once again, if v1= -v2 and m1= m2, we have m0= g m. Thus, if the system, as defined by m0, is constant, then m, the mass of each object must be a variable. It is in this manner that the concept of mass is first introduced.
31. In summary: the concept of the highly discontinuous
two-valued object, ± c, which does not
occur in ordinary classical physics is the basis of classical relativistic
physics. What has been done in this work so far is to find the "underpinning"
to the classical concepts of space and (relative) velocity. Fundamental
to and "underneath" the continuous nature of space and velocity is a highly
discontinuous discreteness. This discreteness arises from the random
binary precursor object.. Because of this, the construction of the concept
of object as done here results in the Lorentz transformation without any
reference to any particular phenomena other than the very "nature" of itself.
However, as will see in the next section, the precursor object is just
the classical incursion to an even more fundamental reality, that of nonspace.
III QUANTUM MECHANICS
32. Nevertheless, there are serious problems in
the theory presented above insofar as it still is a deterministic theory,
even with the "underpinning" precursor variable. A deterministic theory,
by its very nature, is dependent upon making a statement about the present
based on claims on the future
even
though it appears to be the other way around. Thus, in the theory presented
above, all of the v(t) averages approach a limit,v(t0),
at times, t, not only before, but also after t0.
How can something in the future determine something preceding it? Likewise,
if v(t0) is to be
continuous or analytic at t = t0
, then there are restrictions on v(t0),
not only from a former but also from a later time. Furthermore, if v(t)
is analytic in some region, then the value of the function at any point
in the region determines the function not only in the future but also in
the past. Such a physics has little justification outside of the fact that
apparently it has worked very well, at least until the discovery and development
of quantum mechanics. Another objection to the above theory is that for
a given relativistic object, even for one whose entire trajectory is defined
over all of time, or even only at an instant of time, there are an infinity
of precursor object sequences from which the same v(t0)
could arise; the theory at best, then, is incomplete—there is no way in
this theory that a unique velocity (sequence) function can be determined
or is there any rationale why it should be one function instead of another.
33. A third objection is that although the theory requires objects to have mass, it requires the existence of two or more objects to show this; that is, the existence of mass is not a consequence of the nature of the object itself.
34. If therefore one gives up all arbitrary preferences for one result instead of another, if one then allows all possibilities, that is, if one attempts to be as consistent as possible in constructing a theory that does not make the present depend on the future and is as complete as possible, then what kind of theory would result? Would there even be the possibility of a theory? And how would it be done?
35. In order to produce a theory with no preferences, that is, a theory where the present is not dependent on the future or the past and is furthermore complete, there could be no sequence, no function, v(t0), at the basis of the theory; any function, even one with p = q = 50% has a preference; this preference being simply that 50% of the ± c values are preferred +c and 50% are preferred -c. No preference means that at each value of time, the object is both +c and -c. That is, at every value of time, the object assumes all possible values. No average limit or, for that matter, no limit of any kind would be approached. Thus, since there would be no average or instantaneous velocity, there could be no space. An object in such a state is called a nonspace.
Eventually, this must be the ontology, or "non-ontology" upon which physics is based. But since the only kind of measuring instruments available are classical objects, objects that make position coincidences or distance measurements, any definition or expression of the nature of the nonspace must be in terms of classical objects.
36. This now is the clue to the relationship between the precursor variable and the concept of object as nonspace. The precursor object is only the precursor to an even more basic concept; that is, that at every value of time, at every instant of time, there is a not a precursor object of a value +c or -c. but an object, which I call a nonspace, defined by the fact that at every instant of imaginary time its values are both +c and -c. On the other hand the precursor object value, +c or -c, is one of the nonspace or object values when a successful measurement is made on it. If no measurement is made then each nonspace element of +c and -c mitoses into a pair of +c and -c elements continuously and randomly in imaginary time so that the number of nonspace values at any instant of imaginary time increases linearly. Only if a successful measurement is made do space and time reappear and the process begins anew. Thus, the precursor object values only appear as a result of measurement. This is how the precursor object leads us to the fundamental reality of the nonspace.1 It should be noted that the "width" of the wave function solutions of the Schroedinger equation with no potential, even though a non-relativistic equation, increases linearly with time, thus indicating, nevertheless, a relativistic source.4
37. A successful measurement is defined as the position coincidence of two objects; an unsuccessful measurement is defined as a failed position coincidence of two objects where it was possible to have a successful measurement. If there is a successful measurement, the object that was a nonspace is no longer the same nonspace since it now does have a position in coincidence with that of the classical object.; that is, a classical object is a highly specialized nonspace. This means that now there is a distance defined, a distance between the original position and the final position. This distance divided by the elapsed time may, if we wish, be called the velocity. But no velocity or position of the nonspace object existed during the time between the two position measurements. However, an unsuccessful measurement does not mean that nothing has occurred to the nonspace; on the contrary, the nonspace is transformed into another nonspace, one, however, that is not a space point. An example of this is an electron or other fundamental particle passing through two or more slits; the electron nonspace has been transformed to another nonspace other than a space point. Normally when I use the term nonspace, I will not mean a classical or space object.
38. The value of mass is defined by the average time of reversal of the nonspace for zero distance "moved" during any time. If this time of reversal is close to zero, the nonspace is essentially a classical object. Thus, between reversals of the nonspace, space and time are identical in magnitude, and the mass is defined by this space or time magnitude. That is, to this degree and in this manner, space, time, and mass are identical; they differ only in the way described .
39. In this way, by successful measurements, nonspaces become classical objects; if there is interaction during measurement so that the nonspace becomes part of the measuring agent; e.g. an electron becoming part of a molecular structure, it remains a classical object.
40. On the basis of the concept of nonspace defined here, both the Schroedinger and Dirac equations have been derived. The Schroedinger equation is derived on the basis that two successive position measurements made on an object define a single "step" in space; from this it immediately follows, as shown in [1], that the nonspace width or "dispersion" in terms of space increases linearly with time and not as the square root of time as in classical physics. From this, we then show that nonspace time is imaginary; the Heisenberg Uncertainty Principle then follows as does the Schroedinger equation [2]. Using these results and a detailed counting of the continuous steps of the nonspace. one can then derive the Dirac Equation. As shown in [1], this work converges with Feynman’s path integral concept and uses the significant calculations made by Jacobson and Schulman
41. However, it should be pointed out that all these equations do is, on the basis that the future can tell us absolutely nothing about the present (at least for noninteracting objects), translate the initial conditions into the results of future measurements. That is, the initial state of a nonspace object can be generally specified by an array of classical objects such as a grating. Then, if there is equal preference for all consequent states, which means that the future cannot determine the present, the probability density, and only the probability density, of the results of (position) measurements at any future time is determined. That is, the calculation of this probability density of position, which is expressed by the Schroedinger and Dirac equations, is not a statement about physics but rather the logical consequences of the physics stated above as a function of time. That is, from the lack of any preference and the consequent establishment of imaginary time and nonspace as the basic ontology of physics, one can derive the "kernel", the Green’s type function that carries the initial condition of an object into the results of later measurements on the object. It is only the restriction on the average time of reversal of the nonspace, which defines the mass, which enters the derivation as physics.
42. The ± c values of the precursor variable are the eigenvalues of the relativistic velocity operator in quantum mechanics. Furthermore, in 3-dimensions the precursor objects are the basis of the concept of spin.5
43. The physics that results from the answer to all of
these questions promulgated at the beginning of this section is quantum
mechanics.
1V CONCLUSION
The source of both relativity and quantum mechanics has
been discovered. This source is what I call nonspace, indicated
by the concept of the precursor object, and is simply an object of "all
possibilities" defined in terms of positions of classical objects and is
inferred from the concept of the precursor object.. Since the reversal
time of the nonspace, which defines the mass of an object, can approach
zero, classical objects and space can exist and thus nonspace can be defined
in the only way possible, that is, in terms of classical objects.
The proofs of the statements on quantum mechanics and detailed material on nonspace and measurement are presented in the book, The Concept of Object as the Foundation of Physics, 1996, Peter Lang. An excellent paper containing references to related work is V. A. Karmanov, Physics Letters A 174 (1993) 371-376 .
1 This work is an extension of the work presented in my book, The Concept of Object as the Foundation of Physics, Peter Lang, 1996
2 This work is restricted to a single dimension, non-interacting object.
3 Quoted in Jack Stachel, Einstein’ Miraculous Year, Princeton University Press, 1998, p. 19
4 See any first year text on quantum mechanics, such as Schiff's, Bohm's, et. al; also the derivation in reference [1].
5 T. Jacobson, J. Phys. A: Math. Gen. 17 (1984)
2433-2451
