EPR, time, irreversibility and causality

Causality is the relationship between causes and effects. Following Relativity, any cause of an event must always be in the past light cone of the event itself, but causes and effect must always be related to some interactions. In this paper, causality is developed as a consequence of the analysis of the Einstein, Podolsky and Rosen paradox. Causality is interpreted as the result of the time generation due to irreversible interactions of real systems among them. Time results as a consequence of irreversibility, so any state function of a system in its space cone, when affected by an interaction with an observer, moves into a light cone or within it, with the consequence that any cause must precede its effect in a common light cone.

that is not in the past light cone of that event. Moreover, a cause cannot have an effect outside its future light cone [34,35]. Consequently, an event cannot produce any effect if it is outside of the future light cone of another event [36].
In the second law of thermodynamics, an arrow of time is defined. But, also, causality, just because the connection between a cause and an effect, requests a direction of time [37]: this property is a characteristic just only of this physical theory. The connection between causality and entropy has recently been pointed out [37], with the consequence of defining time as the metric of causality. Moreover, time has been highlighted to be discrete in nature [37].
In summary, it seems plausible to study how to link together the concepts of time, irreversibility, and causality. The aim of this paper is to suggest a possible approach to link together the concepts of time, irreversibility, and causality, by starting from some considerations on Einstein, Podolsky and Rosen (EPR) paradox.

Materials and Methods
Since 1927, Bohr developed the principle of complementarity, a fundamental theory of quantum mechanics, based on observation and measurement [38]. The principle points out that, considering two quantum systems A and B, the measurement on one of them, for example A (or B) involves a physical interaction with the experimental setup, that affects both systems. This phenomenon is uncontrollable, even if it can be predicted statistically.
Since 1935, Einstein, Podolsky and Rosen developed some criticisms on the Bohr results. The EPR paradox presents some consequences on the foundation of quantum mechanics, even if many other interesting problems were originated by this criticism. In particular the problem of the collapse of the wave function, which represents a current open problem.
In 1935 [39], Schrödinger introduced the definition of entangled states, as the quantum pure states, |Φ , from an ensamble of systems, that cannot be represented by tensorial products of eigenstates of the states themselves, which analytically results: where ⊗ is the tensorial product, and |ψ are the states in the Hilbert space H. A state is entangled if and only if it cannot be factorized.
During a measurement, a full wave function |Φ collapses into an eigenstate, |φ i , of the state bases, such that [40]: These relations express analytically the effect of the interaction between the system and the experimental set up.
Recently, the definition of time [29,41] has been introduced by considering an analysis of photon-atomic-electron interaction, in relation to the irreversibility [42][43][44] based on an engineering thermodynamic viewpoint [45][46][47][48]. Time is conjectured to be related both to the entropy production and to the entropy production rate: this result agrees to the approach of Planck and Einstein, who have pointed out that the law of evolution of a system is precisely the law of evolution of entropy [49,50]. The problem of linking macroscopic to microscopic approaches is of interest in the analysis of various problems; indeed, it has been pointed out that macroscopic and microscopic approaches are two complementary tools [51], for studying the complex problems, where both the approaches coexist [5][6][7], as, for example, the relations between quantum mechanics and classical physics, matter-radiation interaction, electrodynamics of the Wheeler-Feynman model, nanothermodynamics, etc. At atomic level, photons can be absorbed by the atomic or molecule electrons, and an electronic energy transition occurs between the energy levels of two atomic stationary states. Then, the photons can be also emitted by the excited electrons when they jump down into the energy level of the original stationary state. During this phenomenon, the electrons seem to follow a reversible energetic path, because they come back to the original stationary state of low energy level [52][53][54][55][56]. Indeed, when we consider a single atom or molecule, the energy perturbation of the center of mass is of the order of 10 −13 J, while a usual energy for the electron transition, between two atomic or molecule levels, is of the order of 10 −8 J, with an excited state lifetime of the order of 10 −8 s [55]; consequently, the approximation, of not considering the effect of the atomic nucleus, is introduced, and we need to underline that it is only an approximation [52][53][54][55][56], that cannot be considered in an analysis of the irreversibility which requires some considerations just on the role of the nucleus during the photon-atomic electron interaction [45][46][47]57]. As a consequence of the interaction between the atomic or molecule electron and the photon, a footprint occurs in the atom or molecule. The results obtained in Refs. [29,45,46] point out that the interaction between a photon and an electron in an atom affects the energy level both of the electron and of the center of mass of the atom in accordance with the theoretical and experimental results summarised in Refs. [42-44, 56, 58]. So, the macroscopic irreversibility is the result of the microscopic irreversibility due to the photon-electron interaction, which is the interaction between the environmental electromagnetic waves and the matter. Following the results obtained in the thermodynamic analysis of electromagnetic fields [59], this interaction can be expressed in term of the entropy production and of the entropy production rate. But, the ratio between the entropy production and of the entropy production rate are a time. In analogy with analytical mechanics, where position and velocity can be used as independent variables for the state space, we introduce entropy the entropy production σ and the entropy production rate Σ as the independent variables of the state space Ω = {(σ, Σ)}, used to study the behaviour of the photon-atomic electron interaction. So, we can introduce the definition of time in relation as follows: [41,60], holds to the definition of time interval as [41,60]: Now, the entropy production rate can be written in relation to the electromagnetic waves as follows [59]: where E el is the electric field, B m is the magnetic field, c is the velocity of light, a universal constant in the Universe, ε 0 is the electric permittivity in vacuum and µ 0 is magnetic permeability in vacuum, A is the area of the border of the thermodynamic control volume, and T 0 is the environmental temperature. The entropy production can be related to the analysis of irreversibility [42][43][44] of the interaction between a photon and an atomic electron [41,60]. Here, the fundamental results are summarised in order to be used in the thermodynamic analysis of EPR paradox. To do so, we consider a photon which incomes to the atomic electron. For simplicity, we consider a Hydrogen-like atom, which is an open system from a thermodynamic viewpoint. The incoming photon has an energy E γ = hν and a momentum  [45,55,62]. But, if we develop a thermodynamic analysis of irreversibility, we must take into account this effect, so that the final energy of the atom, after the photon has been absorbed, results [45,55,62]: where M is the mass of the atom, and, in an analogous way, when the photon is emitted, it follows [45,55,62]: Consequently, we expect a footprint in the atom [45,46,62] because, if we don't use the approximation of neglecting the effect on the center of mass, the interaction, between a photon and an electron in an atom, affects both the energy level of the electron and the energy level of the center of mass of the atom. Here, we stress that this effect is well known in quantum physics, even if the approximation of neglecting it is usually accepted, because of the small energy contribution of the center of mass. The quantum state function, after this interaction, solution of the Schrödinger equation, can be obtained by the usual quantum mechanical approach [61,63]. So, it was analytically shown that the macroscopic irreversibility is the consequence of the microscopic irreversibility due to the interaction photon-electron, or from a macroscopic point of view, between the electromagnetic waves and the matter. The fundamental state function, before the interaction, solution of the Schrödinger equation can be obtained by the quantum mechanics [61,63]: where φ is the wave function of the electron, r = r N −r e are the relative coordinates, with r N the coordinates of the atomic nucleus and r e the coordinates of the atomic electron, and [45,46,62]: where u CM is the versor of the nucleus momentum, E CM = P 2 /2M is the kinetic energy of the center of mass, R = (m N r N + m e r e )/(m N + m e ) is the coordinates of the center of mass before photon-atomic electron interaction, M is the total mass of the atom, m e is the mass of the electron and m N is the mass of the nucleus, P is the momentum of the center of mass, = h/2π where h is the Palnck constant, and u CM is the versor of the momentum of the nucleus. Then, the photon incomes to the atomic electron which jumps from the fundamental state into an excited energy state and then it jumps down to the fundamental state, with the emission of a new photon. The fundamental state function, after this interaction, solution of the Schrödinger equation can be obtained by the quantum mechanics [29,45,46,55,61,63]: with φ wave function of the electron and [45,46,55,62]: where E ph is the energy of the incoming photon, and u c = c/c is the versor of propagation of the electromagnetic wave, with c the velocity of light and c its value. A quantum thermodynamic approach to this photon-atomic electron interaction, allows us to prove that this atomic process leaves the footprint [29,45,46]: where H is the Hamiltonian of the photon-atomic electron interaction, i.e. from a macroscopic point of view, the interaction between electromagnetic wave and matter. Then, it is possible to evaluate the entropy production as follows [29,45,46]: Some considerations may be introduced [64]: • Time is the result of the irreversibility; • Locally, entropy can decrease, but the entropy production (due to irreversibility) must always increase, with the consequence that time can only increase.

Results
In relation to the EPR paradox, our result can be summarised stating that without any interaction, time cannot exist, as previously suggested as a consequence of the interaction between electromagnetic waves and matter. Following the original EPR gedanken-experiment [65], we consider two spin-1/2 particles, generated by the same source, A and B, which move in opposite directions. These particles can be detected only if they interact with an experimental setup. Until they interact, from the previous considerations, they are only in spatial dimensions, because time flow starts only from their interaction with an observer. So, independently by their distance, the two systems are not aware of being separate, because they aren't subjected to time dimension. Before the interaction with the observer, their state function is [66]: So, as a consequence of the interaction of one of the particle with the set up, the state function collapses into a particular value, determined by the interaction, and, as a consequence of the interaction, also time begins to flow. So, during the collapse, the particles can influence one another, because still outside of time, up to the end of the interaction, so, for example The results obtained confirm the Bohr approach, but highlight also the fundamental role of the space-time, obtained by Einstein, and the recent relation between time, causality and space-time [67]. Indeed, the proof suggested points out that the conditioning of the measurements is due to the zero value of the time dimension unless one of the system interacts with the experimental setup. Consequently, it follows how irreversibility represents a constraint in the interactions, conditioning the behaviour of correlated systems.
So, causality, as seen by an observer, is related to the time generation due to irreversible interactions of real systems among them. If the events are correlated at the start, independently from their location, they maintain the correlation, while if they aren't correlated, they could be correlated only after an interaction. In particular, Equation (3) points out that without interaction time doesn't exist, so, it would be impossible to observe an order in the events. But, after any interaction, time occurs and the order of the events can be pointed out. Consequently, also the relations between two events can be highlighted, with particular regards to their causal relations. Moreover, in the theory of special relativity causality is related to simultaneous observerdependent [68]. Consequently, following special relativity, the cause must precede its effect in accordance to all inertial observers; moreover, in general relativity, the effect must belong to the future light cone of its cause, even if the spacetime is curved [34]. These statements mean that the cause and its effect are separated by a timelike interval, and the effect belongs to the future of its cause [34]. Here, we conjecture that time interval is originated only during the interaction between a system and an observer, so a timelike interval occurs only after this interaction. So, before this interaction, all the systems are in a 'contemporary' state.

Discussion and Conclusions
In Newtonian physics [69], an effect cannot occur before its cause [70]. In Special and General Relativity this statement has been improved by stressing that an effect cannot occur from a cause that is not in the back light cone of that event [71]. These results are the consequence of finite speed of light and of the fact that the speed of light is the maximum velocity in our Universe, so that no information can be transferred at a velocity higher than the one of the light.
Moreover, the concept of causality has deeply been improved by relating it to the meaning of the simultaneous observer-dependent [72], which states that the cause must always precede its effect in accordance with all inertial observers. Consequently, the cause and its effect are separated by a time-like interval in the space-time [71,72] and a signal could be changed between these two related events at less than the speed of light.
Then, in quantum field theory, causality is closely related to the principle of locality, which is still under study because it depends on the interpretation of quantum mechanics with particular regards to quantum entanglement and Bell's Theorem [73]. Recently, in causal dynamical triangulation [67], causality has been related to the foundation of the space-time geometry [74].
All these viewpoints can be summarised by the results here obtained. Indeed, in relation to the link between the thermodynamic statistical analysis of the irreversible paths and their stochastic order [75], the energy flow, between system and environment, has been shown to select and shape the paths [17,[19][20][21][22][23][24][76][77][78][79]. As a consequence of this interaction, irreversibility occurs also at atomic dimension. But, just this result allows us to approach the problem of causality, by starting from the analysis of the EPR paradox. In this way, time, as a sequence of ordered events, is the consequence of the causality, as the result of irreversibility, because of the interaction between the atomic electrons and the electromagnetic waves in the environment, due to the non-equilibrium state of our Universe [30,47,57,60,64,80,81].