Foundations o f Physics, Vol. 6, No. 1, 1976
Giidel Axiom Mappings in Special Relativity and Quantum-Electromagnetic Theory William M. Honig 1 Received January 4, 1974 Exponential mappings into an imaginary space or number field .for the axioms o f a theory, which are in the fbrm o f propositional constants and variables, make possible: (a) an understanding o f the meaning and differences between the Lorentz transformation constants, such that their p v d u c t is still equal to one, but the axioms at each end o f the transformations are logically inverse and separately consistent," (b) an interpretation o f the psi function phase factor which is part o f the axiom E = hf," (c) the unification of" the quantummechanical psi function and the electromagnetic wave fimction. Thus, those statements whose mechanisms are unknown (the axioms o f the theory) are to be assigned the axiom propositional number symbol 0 and are to be associated with the complex probability e w, which is a uniform ]'actor o f the energy equations expressing the physical state. Such probabilistic axiom functions can be associated with both the special theory o f relativity and the quantum-electromagnetic theory.
1. I N T R O D U C T I O N T h e classical laws o f physics are u s u a l l y w r i t t e n as e q u a t i o n s d e s c r i b i n g physical reality, w i t h t h e i r a x i o m s g i v e n in a n i n f o r m a l m a n n e r as a s e p a r a t e set o f s t a t e m e n t s a n d / o r w i t h t a c i t o r i m p l i c i t s t a t e m e n t s t h a t are c o n s i d e r e d to be o b v i o u s . T h e s e a x i o m s f o r the classical e q u a t i o n s o f physics are to be k e p t in m i n d w h e n i n t e r p r e t i n g the m e a n i n g o f the " p h y s i c a l " e q u a t i o n s . T h i s h a d n o t b e e n a p r o b l e m u p to the a d v e n t o f the special t h e o r y o f r e l a t i v i t y ( S T R ) o r q u a n t u m m e c h a n i c s ( Q M ) , T h i s is b e c a u s e classically
z Electrical Engineering Department, Western Australian Institute of Technology, South Bentley, Western Australia. 37 © 1976 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming,recording, or otherwise, without written permission of the publisher.
38
Honig
the axioms appeared to be obvious and did not change in any way during the application of the equations describing physical reality. In what follows, the definition is made that the "physical" meaning is that which can be deduced from the axioms of a theory, and "nonphysical" meaning is that which is given by the axioms themselves. In this way, statements with "physical" meaning can always be deduced from axioms. The axioms--those statements having "nonphysical" meaning--connote that there is no statement from which these axioms can be deduced. This means that the mechanism for an axiom is unknown. The fact that this is unknown is an important and commonly agreed upon piece of information. Such an ad hoc procedure is causal and quite clear but not particularly necessary with respect to classical physics. When considering STR and quantumelectromagnetic theory (QM-EMT), however, the usefulness for this procedure will become evident. This is because there is something in the content of STR and QM-EMT which requires such a procedure for their deeper understanding. It is shown here that STR and Q M - E M T each contains a double set of axioms, where each set is self-consistent but the double set is not. The member sets of the double set, however, are inverse with respect to each other. In a formal sense they are the negation of each other, but under the imaginary exponential mappings of the axiom symbols to be described, the axioms become axiom functions which are shown to be inverse with respect to each other. Both STR and QM-EMT handle this difficulty by the method of symmetric and conjugate transformations, with respect to "physical" and "nonphysical" variables (axioms), respectively. This can be explained by showing how this occurs first in STR and then in QM-EMT. Some remarks on QM and an outline of the general method will first be given.
2. REMARKS ON QM AND OUTLINE OF THE GENERAL METHOD Much has been written on the meaning, if any, of the psi function of QM. (1) The original definition was given by Born (2) that physical meaning can only be associated with ~b*~b or ~b*A~b, where ~b and ~b* are the psi function and its conjugate and A is an operator representing a "physical" variable with respect to a particle (electron). The use of
(1) (for the one-dimensional case) defines the probability (real and in the range 0 <~ p ~< 1), in the yon Mises and Reichenbach sense, for the full range of u, the variable of the representation. For the particle, existence in the range
GOdel Axiom Mappings in Special Relativity and Quantum-ElectromagneticTheory
39
of u has the probability one, and nonexistence has the probability zero, so that
dp = ¢ * ¢ du
(2)
(A} = f ¢ * A ¢ du
(3)
and
where (A} is the expectation value of A, and where the representation of with orthogonal functions makes it possible to treat many features of the behavior of particles and their interactions with electromagnetic fields. One may conclude, however, in view of the many efforts to find a meaning for the psi function, that "physical" meaning can only be associated with ¢ * ¢ or ¢ * A ¢ as in the equations above. The use of probability in the real number range, zero to one, as in Eqs. (1) and (2) means that exact magnitudes cannot be deduced for both values of canonically conjugate variables, becaues of our ignorance of the exact conditions that QM treats. The attempts to find meaning for the psi function have been confronted with the difficulty that if ignorance is to be associated with ¢, then ¢ should have different values for different investigators, depending on the amount of knowledge which each investigator would have about the same situation. Since a science should not be subjective, efforts to explain ¢ have not been successful. The meaning for the psi function and the necessity for complex variables for its expression is still a pressing and reasonable question. This question can be resolved in an objective way by dividing our ignorance into two parts: first, that which is associated with the magnitude of "physical" variables which are well defined and where the well-known concepts of probability apply (where the probability has the range 0 ~< p ~ 1); and, second, the objective ignorance that is associated with the unknown mechanisms of the axioms of the theory. That is, while the reason for the axiom of a theory is always unknown and although it is always tentative, it is, however, always agreed upon and commonly true that the reason or mechanism for an axiom is unknown. Ignorance of the cause of an axiom, just as' ignorance of the magnitude of a "physical" variable, can also define a probability, but it must be different in character from p (where 0 ~ p ~ 1) given above. This difference can be given in a distinctive way by noting that an axiom statement can be symbolized and the symbols can be mapped into an imaginary number field. If an axiom statement is symbolized as 0, which is a propositional or sentential constant in the Russell-Frege-Tarski sense/3m then under an exponential mapping of the above-mentioned imaginary number field 151 iO exp~ ei 0
(4)
40
Honig
An equation for the energy of a situation can be written in the form D = 0
(5)
where all terms are on one side of the equation. One may combine Eq. (5) with the expression on the right side of (4), so that the resulting equation will express both the physical equation (5) and also carry along an axiom on which it is based; thus D e i° = 0
(6)
where D = 0 and e ~° =/= 0. The axioms, in view of Eqs. (5) and (6), are to be associated with probability in the following way. The axioms 0 which have been mapped to iO and to ee° may be recovered by means of the inverse mappings: ei° i->-~n iO ~/-!E%0
(7)
and the probability p', associated with an axiom, is defined as p ' ==- e i°
(8)
0 = (1/i)[lnp']
(9)
so that
The character of 0, i.e., how it is constructed, will be illustrated in the next section, but, if one adopts the propositional calculus of Russell-Frege-Tarski, then the use of propositional variables can also be included. (a,~) Therefore, if a statement 0 is made which is an axiom or even part of an axiom and which applies to a variable u, then the complete statement is
ou
(lo)
where 0 is the propositional constant and u is the propositional variable. In this case, the statement Ou maps as follows: Ou ~
iOu ~
e i°~
(11)
where the last term on the right is also an axiom probability p ' as in Eq. (8). The recovery of the axiom 0 from the p' function (10) is given by 1
i 6u [p'] = 0
(12)
Equations such as (4) and (7) will be called GSdel axiom mappings. GSdel has shown that the statements deducible from axioms and the axioms themselves are both susceptible to a unified logical treatment and analysis.
G6del Axiom Mappings in Special Relativity and Quantum-Elee~'omagnetic Theory
41
His famous theorem, (6,7) which considers whether it is possible to deduce the consistency of a set of axioms, clearly illustrates this idea. He symbolized both the axioms and the theorems (the deductions from the axioms) of a mathematical system and then he mapped a / / o f the symbols into the positive integer number field. By means of the properties of the number field (the mathematical structure of the numbers) and the properties of the symbols which are mapped into the number field, both mathematical and metamathematical, he was able to make statements on this problem. Although the conclusion of his examination is not of direct interest here, it is summarized because it is illustrated in the statements, Eqs. (21) and (22), of the next section: The consistency of a set of axioms, that is, whether one can or cannot derive opposing statements (theorems) from the same set of axioms, is not decidable within the number field into which the axioms and their theorems have been mapped. In this paper, his ideas are applied in the way that is illustrated by Eq. (6). The mappings are not the same as G6del's, but they nevertheless exist in Eq. (6), where the axioms are in the exponential axiom function with the imaginary argument and the "physical" variables are deductions from axioms and represented by the factor D of Eq. (6). I f an axiom is characterized by 0, then, as shown in the following section, the negation of an axiom can be derived and characterized as --0. The transformation or mapping from an unprimed to a primed onedimensional variable x can be given under the axiom 0 as x'
=
A x e i°
(13)
where A is a factor due to axioms that do not change in all the mappings: x--+
x'--+
x
(14)
The unchanged axiom A, therefore, may be removed from all expressions such as Eq. (13) and carried along in a separate informal manner as in classical physics. Equation (13) carl now be rewritten as
(15)
x' = xd °
for the transformation x -+ x'. For the transformation x' --+ x the expresison is x = x'e -i° (16) This is based on the fact that 0 is to apply to the x -~ x' transformation, and - - 0 is to apply to the x' -~ x transformation. For the mapping x-+
x~-~
x
(17)
42
Honig
one gets X ~
X r e -iO - - ~ x e i O c -iO
-:
X
(18)
For the mapping x' -+ x --7 x'
(19)
one gets X ' "-~ X e iO ---+ X r e - i ° e iO ~
X'
(20)
Thus the two transformations above yield the correct forms for x and x', respectively.
3. APPLICATION TO STR Using the terminology of Pauli, the axioms of STR are(8): (I) the postulate of relativity (to be described), and (11) the postulate of the constancy of the velocity of light. The methods of the previous section will be illustrated with respect to I. Remarks on its application with respect to II will conclude this section. One may now examine the well-known Lorentz transformations from the unprimed to the primed system and vice versa (see Fig. 1). The dimensions are x, y, z, t and x', y', z', t', respectively. There has always been present for each set of transformations a uniform factor of the equations. These 5i l:u a:l:io~ 1
gra.~e A
.j,
Frame ]~
Si~u~tio~ ?-
[ 3
Fr~e
A
Fig. 1.
"J'
Frame B
Lorentz frames in STR.
G6del Axiom Mappings in Special Relativity and Quantum-ElectromagneticTheory
43
factors were called k(v) and k(--v) by Pauli and I and 1-1 by Poincar6 for the transformations x~-+ xi' and x i ' - + x~, respectively, where the subscript identifies the particular dimension involved. Moller and Lorentz denoted the same factors by k and l, respectively. These factors, the Lorentz transformation constants, will be given a different designation here: g(v) and g(--v), in order, we hope, to reduce the confusion. It will now be shown how the axiom I can be symbolically characterized and how it can confer a meaning on the Lorentz transformation constants. Figure 1 shows the two frames (primed and unprimed) which can exist in two different situations. Figure 1 illustrates the meaning of axiom I. In words, axiom I can be given for Fig. 1 in the following manner: Situation l:
A is stationary.
Situation 2:
A is moving.
B is moving. (21) B is stationary.
The description of each situation can be rendered into a symbotogy if we use the logical apparatus of Russell-Frege-Tarski to define the following glossary (refer to Fig. 1). Let one set consist of: A and B, so that ~-~A = B, and ~ - ~ B = A, where ~ is " n o t " or negation, so that i f A = ~ = ~--~B,then B = - - ~ = ~-~A, where minus is used for convenience later. Let another set consist of [is] and [is not], so that [is] = ~ [ i s not], and ~ [ i s not] =- [is], so that if [is] = /~ = ~ [ i s not], then [is not] = --/3 =
~[is]. Let another set consist of: [stationary] and [moving], so that ~ [ s t a t i o n a r y ] = [moving], and ~ [ m o v i n g ] = [stationary], so that if [stationary] = ~, = ~ [ m o v i n g ] , then [moving] = --Y -- ~ [ s t a t i o n a r y ] . Using the above symbols, situation 1 m a y be described as ~f17 and [ - - c q / ? [ - - y ] , both of which are ~/37,, since [ - - ~ ] / 3 [ - - 7 ] - - ~ / 3 7 ; and situation 2 m a y be described as c~fl[--7] and [-- c~]/3y, b o t h of which are -- a/37, for similar reasons as above. N o w if aft), ~ 0 for situation 1, then --7/37 = - - 0 for situation 2. One m a y satisfy oneself by m a p p i n g situation 1 directly into situation 2 under the negation operation: A is stationary.
B is moving.
B is not moving.
A is not stationary.
B is stationary.
A is moving.
44
Honig
The mapping from the first to the second line is a negation operation. The mapping from the second to the third line is an identity mapping, which is made by using the glossary in order to bring the final result on the third line to that which corresponds to situation 2 as given by the second line of (21), which proves that situation 1 maps identically to situation 2 under the negation operation. The Lorentz transformations from the stationary (unprimed) system (x, y, z, t) to the moving (primed) system (x', y', z', t') are (for situation 1) 1)2
g(v)(x -- vt)/(1
XI z
--
cO
y, _ g(~) y z' - - g ( v ) z
7)/('->,
t'=
The substitution (primed to unprimed) for situation 2 is x =g(--v)(x'--
vt')
1 --~i-!
=g(--v)
g(v)x=
x
y = g ( - - v ) y ' = g ( - - v ) g(v) y = y z =g(--v)
z' = g ( - - v )
t = g ( - - v ) \{r - -
g(v)z=
vx' ~//
c--ff-,llll.,
z
v ~ ~/~
-- -~]
= g(--v)g(v)t=t
At this point it is clear that if one makes g(v) = e i°
and
g(--v)
= e -i°
(22)
then one has carried out the method of the previous section. Also g(v) g ( - - v )
= ei°e -i° = e ° = 1
(23)
and although the product of g(v) and g ( - - v ) is equal to unity, as is well known, each of the factors is not. This way of characterizing the g factors above incorporates the axioms relating to the result of each of the two transformations which have been called situations 1 and 2. Each of the g factors is no longer equal to unity, as called for by others, and the Lorentz transformation equations can now be characterized in the probabilitsic way given by Eq. (8): x = ei°[ ],
x ' = e-i°[ ]
y = d°[ ],
y'=
z = d°[ ],
z' = e-i°[ ]
t = el°[ ],
t'=
e-iO[ ]
e-i°[ l
Gfidel Axiom Mappings in Special Relativity and Quantum-Electromagnetic Theory
45
The meaning of 0 -- 0 : 0 is that the axioms 0 and - - 0 no longer require consideration after the transformations x -+ x' --+ x
and
x' --+ x --+ x'
have been made. The general significance of this is that the first axiom of STR (I), viz. there is no absolute motion, is truly an axiom, in that there is no known mechanism or statement from which axiom I can be derived. In addition, STR incorporates, as stated in the introduction, a double set of axiomatic statements that are mutually inconsistent or negative with respect to each other. That is, the equations above now contain the factors e ~° and e -~°, corresponding to the different axioms relevant to the transformations xi --+ x~' and x~' -+ x~, respectively. This also demonstrates that the negation of an axiom symbol generates the inverse axiom. Of course, this depends on having a glossary such as was used here, and this glossary can be considered as a set of superaxioms. The fact that axioms and their negatives have been involved in the Lorentz transformation occasions the remark that the first axiom of STR (I) is not merely a statement about physical reality, but is a special application of logic, since an opposing set of axioms has been used for situations 1 and 2 here described. The application of these ideas to the second axiom of STR, involving the constancy of the speed of light, requires a detailed examination of how axiom II can be related to a non-Euclidean, four-dimensional metric for a space in which electromagnetic waves are stationary. ~9~These considerations will be deferred to another paper, because the purpose here is to show that the same type of probabilistic axiom functions (i.e., e ~°, etc.) can exist both in STR, as has been shown, and in Q M - E M T . In the latter case, the QM probabilistic function (the psi function), and the E M T probabilistic function (the E M T wave function), will be shown to be similar to the functions e ~° and e -~° which have been discussed.
4. A P P L I C A T I O N TO Q M - E M T A precise description is necessary of the conditions under which Gbdel axiom mappings are to be made for Q M and EMT. Tile axiom to be mapped is the Planck energy relation E = hf This relation will treat the physical case of an electron and an electromagnetic wave. The mathematical models for the axiom mappings will treat analogous entities called "particle" and "wave," respectively. The correspondence between electron and EM wave on the one hand, and that between "particle" and "wave" on the other hand, is not identical. The exact relation between these sets of concepts must first be specified.
46
Honig
First, in the physical case the motion of the electron is to be rectilinear (including rectilinear acceleration and deceleration). The EM wave considered is that which is generated by the above motion. The EM wave, therefore, is EM dipole field radiation. The concept of "particle" is that of a localized entity having a mass m with energy equivalent m c L The general meaning for "particle," however, can be described in a number of ways. It can represent merely the energy of a stationary electron (mass m0). It can represent the sum of the kinetic energy and the stationary mass energy, where such a mass for the electron is m. It can also represent the mass equivalent energy of the kinetic energy alone of the electron, where such a mass is m'. In addition to this, if the electron has two successive values of kinetic energy E l , E2 (El > E~), where each of these values of kinetic energy is represented by the "particle" masses m' and m", respectively, then the difference of these kinetic energies E1 -- E2 can also be represented by a "particle" with the mass m" = m' -- m". The way in which the concept "wave" can be related to EM waves is more difficult. A discussion of EM waves is given here that, together with Ref. 10, describes how EM waves can be considered as assemblies of discontinuous entities. " W a v e " will then be identified as a finite number of such entities, m~ This requires that a physical model for free space be used consisting of two oppositely charged superposable fluids. In free space, with no EM waves present, the separate fluids, each with opposing polarity and equal charge density, cause a net neutrality for the space. EM waves can be considered as propagating disturbances in the relative densities of the two fluids. In particular, for the EM waves here considered, which have been restricted to EM dipole radiation fields, the EM waves take a special physical form. They can be shown to correspond to assemblies of separate toroidal vortices in the charged two-fluid space model. They can be pictured as the wellknown Hertzian illustrations for such fields. (1°~ Dipole EM fields can be considered as a contiguous assembly of such toroidal vortices. They initially girdle the electron but then move out in a radial direction at the speed c; during this motion, the cross section of the toroidal vortex, which is initially circular, becomes deformed into the kidney-shaped field distributions that are shown in the Hertzian pictures. (1°~ The toroidal vortex, which is well known as a discontinuous phenomenon in single-fluid hydrodynamics, (lz~ is here considered to be a tor0idal flow of charge imbalance of the two-fluid model. Each such separate vortex is a half-cycle phenomenon. A finite number of these vortices arranged in a contiguous fashion as in the Hertzian pictures (~°) can be used to represent a finite EM wave. The relation E = hf (24)
G6del Axiom Mappings in Special Relativity and Quantum-Electromagnetic Theory
47
as used by Planck and Einstein, defines quantal chunks of energy E, at any frequency f, and the word photon has been commonly used for the discrete energy h f a t a particular EM frequency. The procedure quantizes EM energy so that any larger amount of energy EL at a f r e q u e n c y f i s
(25)
EL = nhf = nE
where n is the number of photons of energy E that add up to the energy EL • This procedure leads to the difficult question of how continuous EM waves can be described. The model for EM waves used here is discontinuous, and the smallest indivisible entity out of which waves can be constructed is the half-cycle toroidal vortex. Extended waves in space are assemblies of these vortices. Each such vortex will have a minimum energy equal to the magnitude of h/2 and the relation (24) still holds (see Ref. 10 for discussion of this point). The relation (25) can be rewritten (26)
EL = n h f = nn'h/2
where n' = 2 f = the number of such vortices per second corresponding to the frequency f, or we can write (27)
E1 = nTh/2
as the quantal energy relation for the energy of a finite EM wave train, where nr = nn' is the total number of vortices. In this model EM waves travel at the speed c and the radial thickness of the half-cycle vortex is equal to the half-wavelength of the EM radiation. The vortex possesses qualities of both continuity and discontinuity; it is continuous inside and discontinuous at its edges. Figure 2 shows the magnitude of E and H, the electric and magnetic fields that would be seen with respect to a single vortex in free space; it is half of a sine wave. In both cases, the horizontal axis is either proportional to t (time) at a fixed value of r or proportional to r (radial distance from the origin of the dipole radiation fields) at a fixed value of t. There are a number of laws and theorems that describe how vortices behave with respect to other vortices in a single-fluid
£ H or
Fig. 2. E and H for electromagnetic photon fields.
48
Honig
model and where action at a distance is communicated by means of the fluid, a31 The rotational flow inside the vortex generates irrotational flow outside the vortex, so that mechanical interaction can be communicated at a distance because of these flows. These last remarks indicate that the use of these vortices should provide a satisfying model for EM waves, because it should be possible to show, using the yon K a r m a n vortex stability theorem, that stable assemblies of contiguous and expanding toroidal vortices will exist, and that such a model reconciles many of the paradoxical facts which are known about EM waves and their interactions. Notice that the relation (27) is independent of the EM frequency. This means that the frequency or wavelength is a parameter that can be determined by the conditions of the acceleration or deceleration of the electron that generates the EM radiation. It is also to be noted that, as explained elsewhere, (1°~ the minimum mass equivalent energy for each half-cycle vortex is approximately 10.48 g, which means, when compared to the electron mass of 10.27 g, that the minimum EM vortex energy is less by a factor of 10 -21. This minimum energy for the vortex can also be called the photon mass, because the Proca and EM wave equations apply to it and because the efforts of others to measure such a mass for the photon have given results that lie in the same range that obtains for these vortices. (1°~ It is well known that EM dipole fields are generated by the rectilinear acceleration and deceleration of electrons. If one tries to use this behavior as a model for the conversion of the kinetic energy of an electron into an EM wave, one fails, because successive alternate accelerations and decelerations are required to generate a multicycle EM wave. However, a Gedanken experiment capable of showing such a conversion is the following: Suppose that an electron in rectilinear motion makes a so-called "elastic" collision with an impenetrable barrier. If it had a velocity v before the collision, then its velocity is --v after the collision. I f this successive deceleration (to zero) and acceleration (to --v) generates two half-cycle vortices, then, first, this means that truly elastic electron collisions are never possible, because once the EM vortices are generated by the electron at the barrier they would spread out from the point of impact; and, second, there would always be a difference between the incident and the reflected velocities that would define an energy decrement equal to the energy of the created EM vortices. There is, of course, no such behavior which has been noted with respect to electrons. The minimum energy for these vortices, however, corresponds to 10-4ag or 10 -15 eV. This means that a total of 1015 such "elastic" collisions would be necessary before an amount of energy equal to 1 eV could be noticed. Such a mechanism can provide a deterministic interpretation for the double slit experiment. Although this discussion has been carried out in a deterministic fashion, the generation of the E M vortices is a sub-quantum mechanical
G6del Axiom Mappings in Special Relativity and Quantum-Electromagnetic Theory
49
phenomenon. Furthermore, it is nonlinear in that once the vortices are formed, their behavior is independent of the electron that generated the vortex. The spreading of the vortex field distribution through larger and larger volumes of space with time shows that this entity is nonlocal with respect to the electron generating the vortex. The measurement of these low energies requires that experiments be done which are capable of resolving energy increments of 10 -15 eV in times of the order of one half-cycle of the EM frequency. The idea that EM waves can be considered as assemblies of discontinuous entities (toroidal vortices) will not be pursued further here (see Refi 10), but it should be remarked that such entities may be prime candidates for the "hidden variables." This is because according to the remarks above these entities can satisfy the well-known yon N e u m a n n and Bell arguments; an objective prediction that EM entities of 10 -z5 eV exist and are nonlocal will satisfy these arguments. " W a v e " is now taken as referring to a precise number of such toroidal vortices. This will now make clear the construction of the two axiomatic situations analogous (actually isomorphic) to the two situations described in the previous section with respect to STR: Situation 1:
"Particle" has E l .
" W a v e " has E2.
Situation 2:
"Particle" has E2.
" W a v e " has E1.
(28) The expressions (28) imply a glossary that is isomorphic to that given in the previous section on STR, but with the following possible mapping of the above terms with respect to the terms which are initially shown in (21): "Particle" --~ A "Wave" ~ B has --~ is
(29)
E1 --+ stationary E2 ~ moving With the above mapping the previous glossary can now be used with respect to the expressions (28). The following additional remarks help to describe (28). For both situations: "particle" rest energy is E 0 ; its rest mass is m 0 . For situation I: "particle" kinetic energy is E1 ; its mass equivalent is m'. For situation 2: "particle" kinetic energy is E2 ; its mass equivalent is m". F o r both situations E1 ~ E2 • The difference E 1 -- E2 has the mass equivalent m". It is also AE, where AE = nrh/2, that is, it corresponds to n~ toroidal vortices. In going from situation l to situation 2 some of the kinetic energy (AE) of the "particle" has been converted to an increased energy (AE) for the "wave." In going from situation 2 to situation 1, the reverse happens. 825/6/I-4
50
Honig
The descriptions (28) are unnecessarily cumbersome, assuming as they do the initial and final values of the masses (energies) of the "particle" and the initial and final values of the energies (mass equivalents) of the "wave." A simpler description is now given containing only the essential features: Situation 1:
"Particle" has A E .
Situation 2:
"Wave" has A E .
but where "particle" = m" = E1 -- E~ = A E , and where "wave" = a number of toroidal vortices nr with energy A E , where AE = nrh/2
(In subsequent equations E will be used for A E . ) Since situation 1 above refers to the "particle" and situation 2 refers to the "wave," then if these situations are called x and x', respectively, the mappings x ~ x' ~ x
(30)
are: "Particle" converted to "wave." "Wave" converted to "particle." For the mappings x' --~ x --~ x' (31) they are: "Wave" converted to "particle." "Particle" converted to "wave." The mappings (30) begin and end with the "particle," whereas the mappings (31) begin and end with the "wave." If the QM ~b function is to apply to the manner in which electrons interact with EM waves, then the mapping x - , x' proceeds under x exv(i~/~)> x' (32) where e x p ( i E t / h ) is also the axiom function defined in (11). On the other hand, if the EM wave function ~b' is to apply to the manner in which EM waves interact with electrons, then the mapping x' --~ x is under the ~b' function x'
exv(-i~)~, x
(33)
and exp(--ioJt) is also an axiom function, as per (11). The signs of the exponents in the above two axiom functions are opposite for the same reasons as in (18), (20), and (22). [As will be explained subsequently, the mapping (30) is for QM and the mapping (31) is for EMT.] The mapping (30), which brings x to back to x, should be characterized by the resulting axiom e i° ( = 1); or, since the successive mappings are e iE~/~ and e -i°~, their product, as in the Lorentz transformations of the last section and (23), gives e i(etlr~-°m = e i° = 1
(34)
G6del Axiom Mappings in Special Relativity and Quantum-Electromagnetic Theory
51
and for the mappings (31) ei(-~+Et/~ ~
e i° =
1
(35)
from which, by equating exponents, one gets for both cases E = h~o
(36)
Using the symbolism of the previous section, let the equality sign in (34) be symbolized e ~ to stand for [--] (37) so that (34) becomes e i(eA-~)~+~+°
(38)
but in this symbol notation the -~ is merely the statement that the symbols on both sides of the ~- sign stand next to each other. Therefore, (38) becomes e i~E/n-~)t~°
(39)
The axiom E = hco can also be written as E / h -- ~o = 0
(40)
Since (39) and (40) are the same statements, the result is that -~ [ = ]
(41)
e" ---* {=]
(42)
e ~ = e~
(43)
so that one obtains
which is most peculiar. This is not investigated any further here; we merely note that ~ and e ~ both stand for [=]. (See Appendix for additional remarks.) (m The mapping x --> x ' is the total conversion of "particle" with mass m " , having the mass equivalent energy E, into the set of nr vortices having the same total energy. It says, therefore, that all of the "particle" has been converted into "wave." The mapping x ' - - + x is the total conversion of "wave" with its nr vortices into the "particle" with the mass m " . In a physical sense the mapping x --> x' describes the fact that a moving electron has a portion of its kinetic energy E (equal to A E of the earlier discussion) converted to a finite EM wave (the nr vortices). The mapping x'--> x describes the adsorption by the electron of a portion E ( ~ A E ) of the energy of an already existing EM wave and the fact that this energy is equal to the energy of the nr vortices. This physical description refers, however, to the more complete set, the situations represented by (28). Thus the axiomatic functions e i e t A and e -io~ should
52
Honig
refer only to portions of the complete axiom E = h~o. The mechanism for the conversion of the kinetic energy of an electron to EM wave energy and vice versa is, of course, unknown, but the foregoing should make clear how the axiom and the axiomatic functions ¢ and ¢' apply to the logical models which can be constructed and which illustrate the meaning of the axioms. Note again that in this case we find an unknown mechanism for the axiom.
5. Q M - E M T A X I O M F U N C T I O N S W I T H C O M P L E X A R G U M E N T S Expressing the argument of (38) by the single symbol B, where B = Ot, we get, on using (38), B = (Et/h -- co) t~O = (E/h -- co) tc&t = Ot
(44)
since 0t = 0. If E/h = 01 and --co = 02 , then 0102~0 --~ 0
(45)
and 01 and 02 are only part of the axiom 0, which need not be pure numbers as is 0. Since E/h and co have the dimensions of inverse time, only 01t and 02t are pure numbers. If (45) is rewritten as 01tO2to~O --~ 0
(46)
then it can be seen that the complete axiom 0 is Et/h -- o~t = 0
or
E = tiw
and contains the two numbers 01t and Od, each of which can be considered as part of the total axiom that can lie on either side of the a symbol; i.e., for E/h = co, expression (46) can be rearranged 01t~O~' t --+ 0
(47)
where 02' = --02, 01t :- Et/h, and 02t = --cot. Since Oat, 02t, and O~'t are in the form (10), they should be considered in themselves as axioms or portions of axioms to which the variable t applies. In order to find a consistent definition for the above numbers, the following interpretation is made. A mapping should exist Y _exv(~et/rO> y,
exv(-~zt/r~) ~. y
(48)
where this mapping (the entire expression) is isomorphic to the mappings shown in (30); using (32) and (33), we find X
exp(iEt/fi)~. X'
exp(iEt/fi)__>.X
(49)
G6de! Axiom Mappings in Special Relativity and Quantum-Electromagnetic Theory
53
which can be seen if the relation E = hco or E/h = co is used as an identity and one substitutes for E/h = co in the second axiom the function e -~zt/~ in (48). Therefore, y--~ y ' is x--~ x', the conversion of "particle" to "wave"; and y ' -+ y is x' -+ x, the conversion of "wave" to "particle"; as has been previously described. The meaning of the Q M and EM wave functions are: e~E~/~ is conversion of "particle" to "wave," which is also e~°'~; but e-~E~/r~ is conversion of "wave" to "particle," which is also e-i~'~; and where e~Et/n and e-~et/r~ are the QM axiom functions and e i°)~ and e - / ~ are the EM axiom functions. QM and E M T do not consider such conversions, but is each a separate field of study. Up to now the conversions for QM may be termed: "particle" to unspecified object; unspecified object to "particle," for the mappings
X exp(iEt/n)> X' exp(-izt/f~)*-Y
(49a)
F o r E M T it is: "wave" to unspecified object; unspecified object to "wave," for the mappings x ' exv(-io~t), x exp(i~.~) x ' (49b) The reason for this tautological discussion is that the occurrence of the exponential factors has an important significance in QM and EMT. They occur when energy is being calculated. In fact, existence of an entity, whether it be vortex or particle, is defined only by its energy. The Schr6dinger equation --ih O~//~t = e ~
(50)
holds for E as a function of a one-dimensional variable u, i.e., for E(u). The term on the left is the operation that must be performed on the axiom function ~ in order to extract the "physical" variable E, the kinetic energy of the particle. This operation extracts from ~b the parameter E which is part of the axiom E - hco. The extraction is from the partial axiom form O~t. Let us rewrite the term on the right-hand side of (50): E(u)~b .... E(u) e ~et/~ ~ e~eie/r~ =- e ~+iB
(51)
where c~ :A [=], E ( u ) = e% Et/h = ft. Now the axiom function e ~ is a constant in these considerations; the axiom does not change. The previous definitions of probability are repeated: 1. "Physical" probability: probability in the range zero to unity to be applied to the magnitude of a well-defined ("physical") variable. 2. "Nonphysical" probability: axiomatic probability = d o, where the axiom is the symbol 0, and that which is unknown is the mechanism for 0.
54
Honig
If, as per statements 1 and 2 above, one defines "physical" probability = pp "nonphysical" probability = pnp = e~¢ then both probability functions can be represented by a single probability function p~:
E(u) P~ -~ PpPnv ~ Pp ei~ =
-
E- r
• e i~"
=
e~+i~ ---
Er
(52)
where E r = j" E(u) du. The meaning of P r = ( Pt du d
(53)
is that it contains the probability associated with both statements 1 and 2, but since the axiom function does not change, E(u) P r ~ ei~ f - - E r " du - - et~e °
(54)
Therefore, as per (1) and (3), which are in terms of the Born definitions f P r du = l e i~
(55)
and f A e ~ du = ( A ) e iB
6. C O N C L U D I N G
(56)
REMARKS
(a) The combined mathematical treatment of both the axioms and the "physical" variables (which are deduced from the axioms and their glossaries) should be a fruitful approach for the mathematical unification of QM and EMT. This mathematical unification should be accompanied by more inclusive physical models. The GSdel axiom mapping technique did indeed serve as a guide for the construction of the physical model which was actually used here for EM waves: the two-fluid hydrodynamic vortices. This model, however, is not as yet complete, although evidently it can serve for the construction of a two-fluid hydrodynamic model for the electron. The major problem will be to prove the mathematical invariance of the two-fluid hydrodynamic models and to interpret this result physically. In this connection, the ideas of Segal, Inonu, and Wigner on group contractions should provide the formalism for a physical two-fluid hydrodynamic interpretation of invariance.
Giidel Axiom Mappings in Special Relativity and Quantum-Electromagnetic Theory
55
(b) The meaning for e ~+IBof the previous section is such that operations that extract e ~ from this function is a statement equivalent to the conservation of energy [see (52) and (53)]. (c) The operation on the QM axiom function eiEt/r~ which extracts the axiom as in (12) is ln p ~t
~ (ln 4 ) =
~t
~ (i~t)
~
----
iE
= ~- ~:ico
The term (D/et)(ln p) vaguely resembles the Boltzmann definition of entropy, i.e., the time rate of change of the logarithm of a probability. This implies that the above equation is an abstract form for a generalized version of the second law; not as it is applied in thermodynamics, but as a more general concept which states that the result of applying the above operation to that in (12) is to extract from, or insert into, the mathematical formalism of a theory the significant axiom of the theory. (d) Although the discussion has been carried out in QM for an axiom function e~e~/n, it is easy to see that a mapping can be performed from the energy-time to the position-momentum representations, so that the axiom function becomes ei~/h and the total axiom is p = hk, wherep is momentum and k is wave number. This is also implied in Fig. 2, which shows that the horizontal axis can be either time or distance. It can also apply to the angle-angular momentum representation. (e) The QM wave-particle dualism is duplicated in STR with the dualism: stationary-moving; as has been shown by means of the glossary mappings.
APPENDIX
Equation (43) consists of three symbols: e ~, ~, and [=], all of which stand for the same thing; or 27 different combinations of these symbols can stand for (43). Also, if one asks what value of ~ will satisfy (43), the difficulties which arise might be solved if one makes the following condition a requirement of a solution: The value of ~ must not lie in the number range of E (and therefore off, since E = hf). This is in order that a single numerical value will not correspond to two unrelated symbols. A unique numerical designation for ~ is necessary so as not to cause confusion. Since the range of E (and o f f ) is 0 ~< E, f ~ 0% it looks as though there is no place to fit ~. We make the conjecture here that at least in the formal sense it may be possible to use Cantor numbers for this purpose.
56
Honig
it is well k n o w n (14,15) that the first transfinite ordinal co satisfies relations with the f o r m e-~ = co (A.1) Therefore, the substitution o f ~ for c~ and for [ : ] in Eqs. (38), (39), (44), etc. will, according to the rules with which Eq. (A.1) is defined, also make 0 a transfinite ordinal: [E = hf] +-+ o2_ +-+ 0
(A.2)
The forms e i° can also be given a set-theoretic meaning which derives f r o m the power set axiom. ~1~) F o r discrete sets, the set of all subsets (the power set PS) of a discrete set T is PS :
2t
(A.3)
where t is the cardinality of T and 2 is the cardinality (and ordinality) of the finite set (1, 0) and is a way of expressing the discreteness of 7". Here 1 is taken as the presence of a member of T and 0 as its absence. I f one tries to express continuous sets in a similar manner, the set (1, A) can be defined, where, for example, 1 is a point on a continuous interval and A is a literal infinitesimal along that interval: A = lim(1/n) for increasing integral n, such as has been discussed by Robinson. ~6~ The n-fold Cartesian p r o d u c t set o f (I, 0) can be shown to have the cardinality 2, and for the set (1, A) the same procedures yield e. Thus relation (A.3) can be replaced for a continuous set by PS -- e ~ (A.4) where t is the cardinality of that set. The forms e i°, therefore, can be understood in the light o f the above as the set o f all subsets o f a continuous set 0. In physics this would imply that d ° is a f o r m which stands for all of the cases to which 0, a continuous set, appliesY r~ The use o f i is, therefore, to be taken as designating an axiom.
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
T. Bastin, Beyond Quantum Theory, Cambridge Univ. Press (1971). M. Born, Z. Phys. 38, 803 (1926); Atomic Physics, Blackie & Son (1944), pp. 140-141. R. Wilder, Foundations of Mathematics, Wiley (1952), Chapter 9. A. Tarski, Introduction to Logic, Oxford (1965), Chapters 1, 2, and 6. W. M. Honig, Bull. A P S 18, 144 (1973); Bull. A P S 12, 123 (1967). E. Nagel and J. Newman, Godel's Proof, New York Univ. Press (1960). M. Davis, The Undecidable, Raven Press (1965), pp. 4--87. W. Pauli, Theory of Relativity, Pergamon (1965), pp. 3-11. W. M. Honig, Bull. A P S 19, 108 (1974); Int. J. Theoret. Phys., in press; Found. Phys., submitted.
Giidel Axiom Mappings in Special Relativity and Quantum-Electromagnetic Theory 10. 11. 12. 13. 14. 15. 16. 17.
57
W. M. Honig, Found. Phys. 4, 367 (1974). W. M. Honig, Bull. APS 18, 44 (1973). H. Lamb, Hydrodynamics, Dover (1945), Chapter 7. L. Milne-Thomson, Theoretical Hydrodynamics, Macmillan (1961), Chapters 12 and 13. A. A. Fraenkel, Abstract Set Theory, North-Holland (1966), pp. 158-159. W. Sierpinski, Cardinal and Ordinal Numbers, Hafner (1958), pp. 315-316. A. Robinson, Non-Standard Analysis, North-Holland (1966). W. M. Honig, Int. Y. Theoret. Phys., submitted.
