The Three Worlds of Penrose: Strings or Rotating Vectors?

The paper examines some basic concepts of the quantum theory. It is concluded that through the concepts of rotating vectors in the complex plane and statistical formulation of quantum uncertainty the wave function ψ has its own well defined physical meaning. The approach of the present paper evi-dences once more the tight link between quantum theory and relativistic theory.


Introduction
Penrose wondered about the ability of theoretical models of describing the universe through numbers as if the mathematics was already there, whereas Wigner questioned about the unreasonable effectiveness of mathematics in physical sciences; eventually, quoting Einstein, how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? It is surprising the fact that just three brilliant minds to whom we owe crucial contributions to the advancement of science agreed about the necessity of explaining the success of their own work [1] [2].
Penrose, in particular, identified the three fundamental steps of any science, i.e. hypothesis, observation, validation; accordingly, with reference to three ideal worlds early sketched in [3]. He postulated the existence of: • an ideal and perfect world, timeless and reminiscent of that hypothesized by Plato, exists beyond our real and imperfect world whereas visible things of the physical reality are copies of non-visible perfect things of the ideal world; appropriate boundary initial conditions. The answer to the previous questions is fairly simpler considering instead the quantum world: its non-locality and non-reality appear weird if referred to the common sense of the classical physics, but actually they are rational consequences of the faintish and abstract meaning of uncertainty.
3) Eventually the relativity enters into the scheme thanks to its double link with the quantum and classical worlds. In effect the relativity, at least as early formulated by Einstein, is basically classical physics extraordinarily enriched by the concept of 4D covariance and finite light speed; yet it is connected to the mental and physical worlds via the respective limits 0 →  and c → ∞ from the quantum and relativistic sides [6].
The three worlds are thus tentatively linked as in Figure 1 on the basis of the aforesaid points. Penrose worried that the weak point of his scheme is the just link between abstract ideas and matter, despite His ideas are certainly amazing [7]: the special and general relativity is the large, the quantum theory is the small, both are deep-rooted in the abstract concept of human mind. Yet the considerations of the present paper shed some light in this respect via the agnostic yet heuristic concept of space-time uncertainty underlying the physical reality around us.
The present paper summarizes also some ideas already published elsewhere in order to make the text as self-contained as possible.

Wave and Corpuscular Quantum Mechanics
Several papers early published [8] [9] have shown the chance of describing the quantum systems via the statistical formulation of uncertainty only being not essential for the purposes of the present paper. It is worth noticing instead that the product of range sizes x p δ δ should be actually written x p δ δ ⋅ , where the scalar consists of an arbitrary number of space coordinates; this remark is of interest for the string theory [11] [12], yet this heuristic implication of (2.1) is waived in the following theoretical frame. Considering one space dimension only simplifies the exposition of the model without loss of generality as concerns the purposes of the paper; even the formal notation is simplified writing throughout simply p δ instead of Then it is required 0 0 2 , , , h x n n p p δ λ ω ε ε λ π = = − − =  (2.5) i.e. the elementary length  to be repeated n times along a circumference of ra- it has been shown in the quoted papers [8] [9]. It is easy to acknowledge that the step from (2.5) to (2.6) is not merely formal but has instead crucial conceptual significance: it rises two fundamental problems.
The first is that of understanding what actually ω means, i.e. what does in fact rotate with this frequency corresponding to the time lapse t δ . Everybody has the perception of time lapse through the daily steps of life; nevertheless nobody cares about the fact that each one of these steps could be also related to some aspect of the quantum world if regarded as reciprocal frequencies, as this mental extrapolation is less intuitive than any act of practical meaning. Yet just this step seemingly innocuous links classical and quantum physics: the former introduces the frequency as time − are length 3 /time [10].
Indeed the second (2.8) reads once more p δ and δε are related likewise in (2.1), but now n does no longer appear, being replaced by the proportionality factor m linking mv and 2 mv . The connection between quantum world and classical world could not be more immediate than the one shown here after getting rid so easily of n and  .
From a formal point of view, the classical physics requires neither n → ∞ alone nor 0 →  alone but appears when these limits, which in fact exist separately, merge into a unique finite limit From these remarks starts the discussion of the two aspects of quantum world, previously concerned by the concepts of corpuscular and wave quantum mechanics in connection with Penrose's worlds. It is interesting in this respect the chance of describing the quantum particles via an unquestionably abstract and immaterial idea: a set of two-dimensional rotating vectors in the complex plane.

Wave Quantum Mechanics and Special Relativity
Particularly interesting for the purposes of this subsection is the mathematical treatment exposed in [15] about the behavior of a great number N statistically relevant of two-dimensional vectors rotating in a , x y complex plane. Let the time evolution of these vectors be described by the function and let the orthogonal components of each vector be identified by the real and imaginary parts x ψ and y ψ of the complex function of modulus ψ = Ψ ; then with the usual notation for these orthogonal components where F is a function to be determined. Let all vectors j ψ have equal modulus and be aligned at 0 t = ; they are also allowed to rotate in this plane for example To find a reasonable form of ψ , which justifies by analogy that of the various The double sign, in principle not to be excluded as formerly introduced since (2.12), emphasizes that either chance allowed for  depends on whether 0 t δ  . In other words, regardless of any relativistic assumption, (2.1) is enough to conclude that t and t − included in In other words time and time −1 of (2.16) turn now into  and κ having physical dimensions length and length −1 ; i.e. it is possible to write are the non-relativistic operators of energy and momentum usually postulated through the position δ → ∂ . Of course both δ and ∂ have the usual meaning of change of something as a function of something else, i.e. of time coordinate t or space coordinate x; so the previous symbol → is justified owing to the arbitrariness of range sizes t δ and x δ , which can even be thought tending to 0 x δ → and 0 t δ → as particular cases.
It is essential to remark that the signs ± in (2.25) are not necessarily corresponding as initially introduced in the separate (2.21) and (2.24), being the former related to that of t δ the latter to the component of x v along x. Actually it is easy to realize that the signs must be just opposite, whence the notation.
The left hand side reads in both cases 2 1 −   by dimensional reasons, i.e. it represents just the energy range that however appears in turn at the right hand side as Eventually it is also worth remarking that in principle are also admissible Equating the real and imaginary parts of this equation one finds the fact that a unique value of t δ fulfills both equations , t const t ω ω δ ω + = (2.28) i.e. just that already found in (2.20), means that regardless of the particular o t both real and imaginary parts of (2.27) reduce to identities with the unique t ω δ previously introduced, as it is easy to verify. In effect, with 0 const < this definition of time range takes the expected form t t const ω δ ω = − : on the one hand t appears as a local time coordinate exactly as any x within x δ , on the other hand since in principle 0 const ≤ it appears that t δ → ∞ for t → ∞ and also 0 t δ → for t const ω → whatever ω might be via an appropriate value of the arbitrary const . Analogous reasoning holds for x δ , but of course neither the time range size nor the space range size can vanish because it would contradict (2.1): it would require 0 n = i.e. no allowed states at all.
Hence this approach shows why (2.21) and (2.23) link the formalism based directly on (2.1) and that based on the wave equation ψ ; indeed the fact that  and p appear related to the operators (2.25) acting on ψ is nothing else but the energy and momentum of a free particle expressed according to the operator formalism of wave mechanics via the complex function ψ .
i.e. the ratio ω ω′ defines ω′ without loss of generality via (GRQ). Let be the power series expansion of ω′ around an arbitrary c ω with constant coefficients a and b and f; so (2.29) at the first order, i.e.
which is possible with an appropriate value of ω′′ that defines the validity of the truncation of series expansion (2.30), then  Consider to this purpose the second equality (2.10); owing to (2.16) and (2.11) it reads

35) Journal of Applied Mathematics and Physics
Note that the denominator of both equations is surely positive, so it is interesting to examine the signs of the respective numerators.
As concerns * S itself it is easy to write the first (2.32) as * re im S s is = + , whose real and imaginary parts are To investigate the properties of these four equations is enough a simple assumption; write reasonably   (iii) ( ) In other words if all vectors rotate with a unique ω , then they remain aligned likewise as at being  the eigenvalue of (2.21) in agreement with (PT2). Here the ± signs of (2.12) are omitted assuming that they combine with the corresponding  od t δ . Hence Then, in particular, the last equality yields . y n n t const As the time t is here a dynamical variable likewise the space coordinates, i.e. it ranges in principle from 0 to ∞ , it follows that δε i.e., with trivial manipulations, Hence, if c is an invariant then (2.57), (2.58) define four relativistic invariants x t δ δ , (2.59), (2.60) and (2.61). Also note that 0 p → for 0 v → but exists the finite limit By implementing the condition (2.61) to calculate g v , instead of the phase velocity λν , one finds the group velocity of the whole wave packet defined by δε and p δ .

Corpuscular Quantum Mechanics and Special Relativity
In the previous subsection (2.6) have been implemented to infer the basic equa-  ;

S. Tosto Journal of Applied Mathematics and Physics
It is really remarkable that the results of this section coincide with that of the previous subsections even without involving  ; indeed  has been introduced in (2.70) for brevity only, it could have been replaced by any other constant with the same physical dimensions. This explains why the wave quantum mechanics is seemingly extraneous to the relativity owing to its evanescent character. However the considerations of the previous section show that the wave approach is essential to demonstrate just the contrary: (2.50), (2.53) and (2.52) are appropriate to obtain the results (2.55) to (2.64). In fact (2.1) are anyway the common conceptual basis of both wave and corpuscular quantum physics and relativity.

Specific Examples of Quantum Systems
So far have been introduced equations that involve the quantum/relativistic definitions of energy and momentum. Now we concern specifically how these dy- The scalar product on the right hand side defines thus and in turn defines a new component ( ) . The generalization of this result to the case of the spin-orbit coupling L S ± is concerned in [17]. Note only that in fact L is arbitrary integer; hence, when considering     i.e. k′ is numerically equal to 1 with good approximation. This shows the sought numerical relationship between e and G through the dimensional factor Note that the result (3.14) is consistent with a different choice of 4 a and 5 a in (3.9) 2 3 4 3 being v an arbitrary velocity. These positions seem apparently weird, however a trivial dimensional analysis shows that both are consistent with (3.10) likewise as the previous (3.11). In effect it is immediate to calculate These results suggest in turn how to describe another system with different interaction. Indeed the electromagnetic interaction was implied by the coefficient 4 a only, without which this result is a mere kinetic term. This means that Journal of Applied Mathematics and Physics if instead of considering x E via the terms 4 a and 5 a one considers, for example, the terms 1 a and 2 a along with 5 a one should concern another kind of system physically sensible. This is highlighted in the next example.  with the second position (3.19) to allow a non-electromagnetic interaction driven system. It is immediate to verify once more that δε has a minimum as a function of p δ , i.e. with obvious notation The relevant result is that all quantum numbers that characterize the quantum physics, including the spin, are deductible from (2.1) without additional hypotheses. Also, note that now ω is not introduced with the fundamental meaning characterizing the steps (2.3) to (2.6) that imply the Planck and De Broglie definitions as corollaries; here ω is mere notation formally summarizing the shortcut ratio K/m, yet the concept of mass is still saved in this approach. In other words K/m is an experimental definition still including the mass, well different from a conceptual renounce of the mass in fact replaced by the fundamental constant  . These examples of calculations show that the coefficients in (3.7) and (3.8) are not mere numbers, rather they have a well identifiable physi-Journal of Applied Mathematics and Physics cal meaning; hence it is reasonable to expect that other phenomena, e.g. the binding energy at nuclear or subnuclear distance, can be described through higher order coefficients.

Rotating Vectors and Special Relativity
Is it possible to highlight a direct connection between the rotating vectors introduced in section 2.2 and the special relativity? The answer should be positive because from the initial (2.9) have been inferred (   The results shortly sketched here as a straightforward consequence of the S. Tosto Journal of Applied Mathematics and Physics quantum uncertainty will be further concerned in the next section too with direct concern to the concept of space time curvature.

Discussion
The (iii) (2.7) plugs both cases (i) and (ii) into the general relativity, as in fact (2.7) is the essence itself of the covariance inherent (2.1). Also, finite c is required in (2.2) in order to obtain (2.1) as a corollary [10].
Anyone who would inquire how (2.1) change in different reference systems, easily acknowledges that actually such a question is out of place. Indeed x p n δ δ ′ ′ ′ =  and x p n δ δ =  are actually identities even in different reference systems: n and n' are not specific values, rather they symbolize sets of integers, so to any n of (2.1) corresponds another n' without changing anything, i.e. the sets are physically indistinguishable like the respective reference systems themselves.
The uncertainty introduced and implemented in its most agnostic form proposes ranges not directly related to the reality because nothing is knowable about them. Moreover Section 2 has remarked that Equation (2.1) move the physical Journal of Applied Mathematics and Physics terpreting its escaping appearance. The relativity enters into the scheme thanks to its double link with the quantum and classical worlds. It has been shown that relativistic Universe consisting of orbiting systems, black holes and so on corresponds to large numbers of states, i.e. large quantum numbers. In effect the relativity, at least as early formulate by Einstein, is basically classical physics; the concept of 4D covariance is connected to the mental and physical worlds via the respective classical limits 0 →  on the quantum side and c → ∞ on the relativistic side. The three worlds are thus linked as in Figure 6. Penrose worried that the weak point of his scheme is the link between abstract ideas and matter; yet the considerations of Section 2, shortly implemented here, shed some light in this respect according to the title of Penroses book [7]: the special and general relativity are the large, the quantum theory is the small, the deep-rooted space-time uncertainty is the underlying the human mind. In fact the impossibility of specifying anything about the uncertainty ranges means that no information is preceding the Equation (  , .
x y x y a b a b a n n n b n n n x y x y Compare this result with the standard formula of Laplace curvature, regarding x δ and y δ as curvature radii in two arbitrary axes defined in an appropriate primed and unprimed reference systems. These reference systems are actually undefined and conceptually undefinable for two reasons.
On the one hand it is possible to regard the left and right hand sides of (4.1) with a unique physical meaning referred to the aforesaid reference systems.
On the other hand, x δ and y δ introduce explicitly the arbitrary lengths implies p δ  because of (2.1) and thus the rising of a force field in x δ : but in one case the observer is at rest with respect to the reference system, whereas in the other case the observer moves solidally with the mobile boundary coordinate he is sitting on. So the observer cannot distinguish whether: (i) he is at rest whereas p δ  is due to an external force that stretches or squeezes the range size acting on either boundary coordinate or (ii) p δ  the observer is accelerating in a non-inertial reference system. Clearly this is Einsteins equivalence principle, according which no physical difference distinguishes an accelerated reference system or an external gravity force field; in the present context this principle is actually a corollary of the indistinguishability of the aforesaid situations because nothing is knowable about any uncertainty range size. For this reason the present model includes in a natural and self consistent way the extension to the relativistic world, while preserving however the elusive features of the quantum world. The fact that the concept of "distance" is actually unphysical, it would require knowing two points in the space time thus violating (2.1), has significant implications: for example the EPR paradox shouldn't even be formulated, being unphysical the concept of "superluminal" distance itself [20]. Yet, since the curvatures defined in this way are actually non-calculable specifically, it is not surprising that neither the force z F nor the local energy ε within δε are in fact definable. It is possible however to say that z F is proportional via δε to the local curvatures corresponding to all possible random values of x and y falling within x δ and y δ ; so δε , whatever its size might be, is the related range of curvature energies. Once more, likewise as in (3.7) and (3.8), the coefficients a and b characterize the specific cases. For example one recognizes the Laplace-Young capillarity equation in the particular case where a b A γ = = , being γ the surface tension of a liquid and A an arbitrary extent of curved surface, in which case z F A yields the well known capillary pressure. Also, identifying δε with min δε of (3.14) one infers that the gravity force is directly related to the curvature  of space time just defined Journal of Applied Mathematics and Physics ( ) dynamical variable δ as done throughout in this paper. Just the uncertainty properties of the ranges justify the chance of defining the derivatives as ratio of two ranges: the tensor formalism concerns by definition local quantities that must be formulated via covariant calculus required "a priori"; here, the covariancy is inherent the uncertainty ranges themselves, and thus does not need any appropriate mathematical formalism.
The fact that appears in (2.48) the ratio δ δ is not merely formal just because of the quantum uncertainty. On the one hand, being the range sizes arbitrary, it can be regarded as ∂ ∂ as a limit case with all implications of the standard wave quantum mechanics, see for example the Equation (2.25). Note however that ∂ ∂ is uniquely defined by the dependence of the function at numerator upon the dynamical variable at denominator, δ δ is mere ratio of two separate uncertainty ranges that can be handled algebraically as distinct range sizes; only when requiring both sizes tending to zero as a particular case,

Conclusions
The present model has shown that indeed even ψ and * ψ have their own Journal of Applied Mathematics and Physics physical meaning resulting from the idea of rotating vectors in the complex plane: from (2.10) and (2.11) up to (2.21) and (2.25) following the concepts of ψ and * ψ and then, thanks to (3.22) to (3.24), that of * ψ ψ as well. The Equation (2.9) is indeed a sort of "precursor" of the wave functions, while implying itself well identifiable thermodynamic properties. This conclusion is supported by the essence of Equation (2.31) to (2.38), as if they would concern indeed a lattice of physical particles rather than a collection of abstract rotating vectors.
The results of Section 3.1 and 3.2 show that wave and corpuscular quantum physics are distinct but concurrent topics to infer relativistic results too; the subtle wave/corpuscle dualism of matter, and likewise the quantization itself, do not contradict the relativistic features of the Universe.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.