Quantum Correlations: Entropy, Wave/Corpuscle Dualism, Bell Inequality

The paper shows that the second law of thermodynamics and Pauli principle are implications of the Bell inequality.

3) is separately reasonable; as such, their worth is based on the chance of being contextually deductible along with other physical laws simply via the "ab initio" heuristic model [1]. It could seem weird, however, the further step of merging them in (1.4) and next splitting again this latter as in (1.5).
On the one hand, the mere numerical correspondence All considerations exposed in the following sections aim to clarify these points.
Purpose of the paper is to show that the thermodynamic functions can be inferred directly from quantum first principles that concurrently imply relativistic outcomes as well.
The model highlights a further series of evidences additional to that carried outed in [1] to show the actual reasonableness of (1.5): this is the new contribution of this paper with respect to the previous one. The following considerations aim also to highlight the physical essence of these implications.
With this premise, after a short remind of elementary statistical concepts in Section 2, the positions (1.5) are first concerned in the subsections 3 and 4 in a mere formal way through simple algebraic steps to infer preliminary corollaries, whose physical meaning is next exposed in the Sections 5 and 6. In particular the sections 3 to 5 are not a superfluous list of results already known; looking for further concepts already known while obtaining contextually also unprecedented results. The text is organized in order to be as self-contained as possible.

Mathematical Tools
The quantum uncertainty equations read * * * * x p t x p n t n n n n n ε δ δ δεδ inferred themselves as corollaries in [1]. The stars label arbitrary rational numbers expressing the range sizes as * n times the respective Planck units, n is an arbitrary integer.
where gj v is the j-th group velocity of the wave packet related to the respective state of the system concurring to define S and S δ . Also, analogous considerations hold for where const is a quantity not dependent on the summation index j. Here appears the average normalized probability Π Θ plus a constant. So S Θ is the dimensionless Boltzmann entropy, usually expressed with positive sign as appears here the invariant interval one infers that if S has statistical meaning, then the same must hold also for g v c ; in effect it will be also shown in the next Equation (3.3). This conclusion and the meaning of the factor 0 v are further concerned soon below.
Although reasonable, these preliminary remarks and that of section 1 are examined in the following three introductory sections. To carry out the next considerations are enough simply the average definitions of thermodynamic functions; in particular g v is enough for the present purposes whatever the statistical distribution law of group velocities around this mean value might be.

Preliminary Corollaries of (1.5)
Relevant implications of both (1.5) are immediately recognizable writing the proportionality factor γ has been purposely introduced for sake of generality. Thus, whatever γ might actually be, it must be also true that as it is immediate to verify.
the last step has implemented explicitly the general definition of range  , , whence the possible correlation chances that agree with (2.9) and imply with the help of (1.1) and first (1.5) If so, then S should be a relativistic invariant. In effect, this conclusion agrees with that early inferred by Planck [3], who concluded that "the entropy of the body does not depend on the choice of the reference frame"; i.e. the disorder degree of a body appears the same to two inertial observers in reciprocal motion.
Consider now the first equality (3.8) suggests that Θ defined in R implies regarding The connection of these considerations with the quantum theory is immediate with the help of (3.7) and (2.1), simply noting that (3.7) defines an invariant Thus, writing identically On the one hand δε ε ε = −  ; thus, being correspondingly the first equation splits as follows n ω  at the right hand side compatible with any arbitrary integer n.
Hence to any particular * * g n ω  corresponds the value * g ε , i.e.
Being the boundaries of δε arbitrary means that actually n can take any integer value. A reasoning exactly similar implements the further chance of inferring from (GGG) where g κ is actually the so called wave vector. Thus, as before, * * * As a preliminary conclusion, therefore, the hints prospected in the present section by both (1.5) appear sensible enough and stimulating to deserve further investigation, in particular as concerns the direct link between thermodynamics and relativity. As the quantum definition (1.1) of entropy is already evident through its statistical meaning related to the probability of allowed states, actually this link is reasonably expected to highlight the connection between quantum probabilistic and relativistic theory. These non trivial conclusions justify the reasoning hitherto carried out via its possible implications.
Note eventually that this result indeed agrees with that inferable in an analogous way from (3.5), which implies it "a fortiori". As this inequality holds even multiplying both sides by an arbitrary coefficient 0 where θ represents an arbitrary energy factor. (3.13) is significant because it implies (3.14) that defines S as the ratio of two energies and evidences its fundamental property 0, .
In this respect, note that (3.14) is not "ad hoc" position: implementing once more (1.5), it is easy to acknowledge that S defined by (3.14) is actually the same function introduced in (1.1). Indeed, proceeding in analogy with (3.1), one finds that just (1.5) allow writing where S is in fact an energy ratio; regarding of course Θ still as that introduced in (3.7), this suggests the correlations So, whatever * ε and ** ε might be, the proportionality constant θ justifies in principle the link between S ε and pv S of (3.14) to that hitherto concerned via (1.1) only; are crucial in this respect the positions (1.5).
As a closing remark, note that in general owing to (3.14) it is possible to expand in series (3.15) as ( ) recalling now (2.4) and (2.5), average both sides and implement 2 The importance of this result and (3.15), better explained in the next section, appears even here regardless of any information or hypothesis about g ε and g g p v ; it is simply due to 0 g v c ≤ ≤ allowing (1.3) and (1.4). Precisely for this reason it holds for an isolated system not subjected to any external action that would necessarily require an appropriate external energy ext ε exchange to be introduced in the previous equations; this action could perturb and modify the spontaneous evolution of the system leading, as so far described, to (3.15).
In the present model, owing in particular to both positions (1.5), quantum, relativistic and thermodynamic results merge in a natural and elementary way.
The definition (3.8) of entropy as ratio of two square lengths is seemingly weird; actually however it is not so, the next section shows that this definition is closely related to two significant relativistic corollaries.

Black Hole Surface Entropy and Red Shift
It is known that the Hawking surface entropy of a black hole is in fact the ratio of square lengths [4], like (3.8). So it is sensible to start just from this equation and write thus, being by definition g s δ the difference between two arbitrary space In the present context, regard once more  is an arbitrary length in the reference system R were is defined the rest mass r m ; in the last step r γ =   is the Lorentz contraction of r  in another inertial reference system R′ moving with respect to R at arbitrary rate v corresponding to that defined by γ . So to show that effect of an average gravitational potential difference δϕ , while instead the deterministic approach of general relativity yields the frequency shift of one photon traveling between the local points 1 s and 2 s in the presence of a central gravity field; this information is replaced here by a statistical reasoning leading to the same conclusion, now however probabilistic and non deterministic thus fully compatible with the ideas of quantum theory.

Thermodynamic Corollaries
All considerations of the previous section did not need introducing explicitly the microstates appearing at the right hand sides of (1.2); this is done now. As a preliminary consideration note that owing to (1. through which is easily proven the aforesaid statistical meaning of g ε and g p on the one hand and g v on the other hand, see (3.3) and (3.4); owing to (2.5), it is possible to regard now also (3.2) as in order to obtain from (5.2)  The last equation at constant P is the well known thermodynamic definition of entropy, whereas (5.14) is the first law taking  as heat affecting internal energy change U δ and work δ  done on/from the system. To clarify the link between  and L put L X = +  , being X a positive or negative quantity to be specified. So in (5.10) 0 U L X S θ Further comments on these results are superfluous; the fact of having introduced the factor θ in (3.14) and (5.5) is thus not merely formal.
Two significant implications of these results allow to highlight more clearly the thermodynamic meaning of (3.15).
1) The second line of Equation ( Here appears again, but now explicitly, that 0 S δ ≥ implies no work done on or performed by the system, which is one condition for an isolated system. to an arbitrary initial disordered state.

Quantum Correspondences: Bell Inequality
After having introduced a self consistent landscape of relativistic properties of thermodynamic functions linking macro and quantum micro states of physical systems, is justified the further extension of these results to better emphasize the quantum basis unifying all concepts of the present theoretical model.
Consider (1.4) recalling that all terms have probabilistic meaning and range as in (1.3); write then (1.4) as 1 1 log .
This equation can be identically rewritten as The notation is justified because * δΘ reads i.e. With these premises, it this possible to highlight more Expressively (7.5) introducing Bell's language: the aim is to show without "ad hoc" hypotheses that the notation emphasizes that S depends on an appropriate parameter X, which reasonably is a representative feature of the specific system; e.g. X depends on whether the system is crystal lattice or gas or liquid or amorphous material. The notation ( ) S is quite obvious. Take N as a fixed parameter, which for simplicity of notation will be thus omitted in the following, whereas instead d j and o j affect of course the resulting S ∆ concerned here. Note merging both possible chances notZ and Z. So, summing and subtracting ( ) , X Z Π , the difference of probabilities at the right hand side is summarized by the unique not term at the left hand side. Thus, whatever X might be, 2) An analogous reasoning holds to handle the term In summary, Collecting (7.6), (7.7) and (7.8), by comparison with (7.5) it is possible to write three possible correspondences linking the three ranges (7.5 A crucial aspect of this result is that 0 S δ ≥ is calculated as difference between S of a system with N states and another d S of a system with d j N ≤ allowed states; this shows that in fact the entropy tends to increase along with the number of states, in turn corresponding to an increased total disorder of the whole system. Obviously the positions (7.2) concern an isolated system, because they do not involve explicitly any external energy that could modify them: in effect according to (7.5) all addends are self-defined uniquely through the allowed states of the system, regardless of any possible external energy,  or  , that in principle can modify the spontaneous evolution of the system explicitly inherent the right hand side of (5.8) and (5.9).
It is known that the Bell inequality is the key to prove the non locality and reality of the quantum world; hence, recalling also the previous results, the correlations (1.5) are to be considered at this point definitively proven.
The correlations (TP0) deserve a further comment.
Summing these inequalities one finds for the aforesaid reasons, (7.13) reads Thus are possible two chances only: the former trivially means that the probability of occupancy of a given quantum state is identically null, empty state, the latter that admits the occupancy of a particle whose X and Y are uniquely selected, concern explicitly the existence of a property of particles we call "spin", nevertheless it states the principle according which in Nature must exist two ways to fill the allowed states of a physical system. Regarding Z as a two valued property compliant with (7.12) would seem seemingly a weird "ad hoc" hypothesis, but knowing the existence of the spin this position becomes a fundamental statement to acknowledge the two different ways to fill the allowed states of physical systems in agreement with (6.5). Also this point has been better explained in [1].

Further Implications of (1.5)
If it is true that the entropy is the "arrow of time", then once more (1.5) should be adequate to infer an interesting implication: to describe the evolution of the Universe. This is the purpose of this last section.
Rewrite the previous (3.7) as follows the first equality reminds the starting point (3.7) in R, the second equality is (3.7) with time and space ranges t δ ′ and g s δ ′ defined in another inertial reference system R′ , the third equality rewrites the second one emphasizing explicitly time dilation and space contraction of the initial respective ranges in R with respect to that in R′ . Implement now the new position

Discussion
The present paper must be regarded as an extension of the previous [1]. Most of the considerations hitherto carried out show that (1.4), formally legitimate, has actual physical worth and that its splitting into separate correlations (1.5) implies sensible outcomes. The probabilistic meaning of both (1.3) explains why (1.4) is more than mere algebraic unification of two equations seemingly different and highlights the correlations (1.5). Although (1.1) have been contextually inferred in the quoted paper through different arguments, it has been remarked for shortness in section 1 that the first (1.3) is simply the widely accepted velocity dependency of mass even though its inherent physical meaning concerns the actual probability that one particle exhibit dual corpuscular or wave like properties. The question about what has to do the wave/corpuscle dual behavior of a single particle with the probabilistic concept of order/disorder of a system of particles, has adequate statistical answer suggested by their form of alternative probabilities concurring to the respective certainties: the chance of quantum energy fluctuation implies actually both of them, as it appears in (1.1) through r m m > and in (3.11). The position (2.5) has made acceptable the idea of linking the average properties of a whole system to that of its j-th microstates, being these states representative of the macroscopic behavior even concerning one particle only as a limit case. In effect (1.4) regards dual behavior and entropy increase of matter as two different aspects of a unique probabilistic cause: the evolution of an undefined state merging both probabilities (1.3) brings to a new state revealed by and accessible to the experiment. So one particle appears as corpuscle or wave through its r m m or g v c depending on what the experiment is aimed to measure; similarly where matter appears ordered or disordered with probabilities S Θ or o S Θ via a microstructural observation. In fact, if the quantum world is non-local and non-real until an experiment is carried out, there is no reason to reject the idea that even order and disorder are undefined states that take physical meaning under the experiment. This consideration reminds for example Schrödinger's cat, with the dead cat described by 0 g v v ≡ ≡ to which corresponds dynamical mass m equal to its rest mass r m . This delicate conceptual point deserves special attention and explains the strategy followed in the previous and present paper: to reproduce well known results or infer crucial implications confirming the validity of (1.5) and related equations. Although several results have been already inferred in the quoted paper, it appeared appropriate to propose here further results to obtain more significant confirms.
The novelty of this study is the way to identify the common root underlying quantum physics, thermodynamics and relativity implementing the premises introduced in section 1; formulas and concepts are in turn part of a broader is not surprising, therefore, the plot of Figure 1 involving directly the entropy. Just in this respect the present model appears significant: a subtle connection is evidenced between quantum interpretation of De Broglie waves, statistical meaning of entropy and concept of invariance of the special relativity, which in turn implies Lorentz transformations, velocity dependence of mass and definition of thermodynamic functions. Also two results of general relativity, for brevity the Hawking entropy and the red shift only, have been included in section 4 as straightforward and natural corollaries of these premises. This connection compels accepting even the statistical meaning of invariant interval and Lorentz transformations evident since the early Equations (2.8) and (2.9). Consider in particular the entropy, which is known to be a fundamental principle of thermodynamics; it appears here closely related to other fundamental principles of Nature like the aforesaid dual behavior of matter and the Pauli principle via Bell's inequality. Thermodynamics was born as conceptual extrapolation of the technology of heat engines; with the contribution of the quantum theory, e.g. see Boltzmann's H theorem and Maxwell equations, it became a fundamental Science itself. The present model introduces the basic principles of thermodynamics without implementing neither concepts of heat cycles nor intuitive assumptions like the energy conservation, in fact expressed through the first law. Actually it appears reductive to introduce fundamental concepts like S and 0 S δ ≥ merely examining the Carnot cycle; as it is known that the Bell inequality is the key to prove the non locality and reality of the quantum world, (7.9) and (7.14) of the section 7 appear the most appropriate way to understand what really the entropy is and thus the correlations (1.5), which at this point are to be considered definitively proven.

Conclusions
The present model unifies within a common conceptual frame relativity, thermodynamics, quantum theory and, via the Bell inequality, dual wave/corpuscle behavior of matter and Pauli principle; electron diffraction, "Aufbau" of electrons in atoms and molecules and red shift appear rooted in a unique principle, the quantum uncertainty. Usually, the Bell inequality is synonym of EPR paradox; here however it appears as a law of thermodynamics.
In general, it is reductive and misleading to think quantum theory or relativity or thermodynamics as separate disciplines of physics; trying to merge all of them does not imply necessarily more difficult approach, but an in-depth investigation on their shared root. In effect, the laws of Nature reveal unexpected links and common implications that solve problems apparently different through simple arguments and, mostly important, elementary mathematical formalism.

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