Marco Pettini
email: pettini@cpt.univ-mrs.fr

Quantum entanglement: proposal for a new kind of experiment

This work aims at exploring whether the nonlocal correlations due to quantum entanglement could exist without nonlocal causation. This is done with the aid of a toy model which is based on a simple but decidedly unorthodox physical hypothesis: the nonlocally correlated outcomes of the wavefunction collapse of two entangled particles is driven by an exchange of information between them. Of course exchange of information between spacelike separated entities is forbidden in 3+1 space-time, thus we can wonder whether or not a way out could be found by invoking the existence of some hidden field propagating in an extra time dimension to make the mentioned exchange of information occurring at finite velocity through this extra dimension. Only the description of the wavefunction collapse would be affected by the existence of this extra-time dimension. The model proposed here is defined by borrowing Bohm-Bub's proposal to describe the wavefunction collapse by means of nonlinear (non-unitary) dynamical equations and then elaborating this approach for an entangled system. The model so obtained is just speculative without any claim of being robust against any criticism, nevertheless, it satisfies the purpose of giving the possibility to the hypotheses formulated above to be verified experimentally; in fact, it makes the proposal of an experiment potentially interesting which would otherwise be immediately dismissed as manifestly trivial. The proposed experiment would consist of checking the possible violation of Bell's inequality between two identical but independent systems under appropriate conditions. Beyond its theoretical interest, entanglement is a key topic in quantum computing and quantum technologies, so any attempt to gain a deeper understanding of it could be useful.

I. INTRODUCTION

Quantum nonlocality, related with non factorisability of the wavefunction of a composite system, leads to a seemingly paradoxical situation concerning the mutual action between particles or quanta, when a local and causal viewpoint is assumed. The supposed incompleteness of quantum mechanics, motivated by the existence of the "spooky action at a distance" represented by the nonlocality paradox of the Einstein, Podolski, and Rosen (EPR) gedankenexperiment, was refuted by Bohr's argument maintaining that a composite system must always be regarded as an indivisible totality which in principle cannot be subdivided into independently existing units.

Einstein's claim of incompleteness of quantum mechanics became susceptible of an experimental confirmation or refutation after John Bell's proposal of a quantitative criterion [1] to test the completeness of the theory versus the need of the so-called hidden variables to recover a locality condition fulfilling Einstein's causality. Under a suitable locality assumption, Bell's theorem states that local hidden-variables theories are constrained by some given inequality, and some predictions of quantum mechanics can violate this inequality. A pioneering experimental implementation of a test based on Bell's criterion was put forward by Clauser, Horne, Shimony and Holt [START_REF] Clauser | Proposed Experiment to Test Local Hidden-Variable Theories[END_REF] and led to a confirmation of quantum mechanics through the violation of Bell's inequality. However, having held fixed the orientation of the polarizers during the experimental run of this and other experiments made the results not conclusive because in the case of static experiments both the locality condition and the validity of Bell's inequality can be questioned. Bell thus insisted on the opportunity of performing experiments according to the Bohm-Aharonov proposal [START_REF] Bohm | Discussion of experimental proof for the paradox of Einstein-Podolsky-Rosen[END_REF] of changing the settings of the instruments while the correlated particles were in flight. In so doing, the locality condition is entailed by Einstein's causality forbidding faster-than-light interactions.

Then a strict and undeniable violation of Bell's inequality was given by Alain Aspect's experiments [START_REF] Aspect | Proposed experiment to test the separable hidden-variables theories[END_REF][START_REF] Aspect | Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment -A new violation of Bell inequalities[END_REF] where each single channel polarizer was replaced by a fast switching device redirecting the incident light beam to a differently oriented polarizer. The detection events on opposite sides of the experimental apparatus were separated by a space-like interval, thus free of locality loopholes. Further confirmations of the violation of Bell's inequality were then obtained at increasingly large spatial separations [START_REF] Weihs | Violation of Bell's Inequality under Strict Einstein Locality Conditions[END_REF][START_REF] Tittel | Violation of Bell Inequalities by Photons More Than 10 km Apart[END_REF][START_REF] Scheidl | Violation of local realism with freedom of choice[END_REF][START_REF] Hensen | Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km[END_REF]. Therefore, quantum nonlocality is undeniably a property of our physical world, but Bohr's viewpoint can sound tautologic and unsatisfactory compared to our description of physical phenomena based on events located in space and evolving in time, so that the existence of correlations between events separated by an arbitrarily large space-like interval is a conundrum that goes against our intuitive perception of the physical world. Of course, not only quantum mechanics compels us to give up our intuitive perception of the physical world, nevertheless we can wonder whether -to some extent -we can recover a "causal" description of quantum nonlocality by paying a different price with respect to Bohr's claim of indivisibility of quantum systems.

The price to pay in the framework proposed in the present work is to consider the existence of an extra time dimension. In what follows there is no pretence of proposing something that could qualify as a new theory, rather, by means of a toy model we want to move some first steps in exploring whether nonlocality can be given a "more familiar" interpretation by considering that while performing the measurement of an entangled state this appears as actually composed of independently existing units that exchange information via a subquantum field X (x, t, τ ) that propagates at finite velocity through an extra dimension of temporal kind τ .

I.1. A quick digression on extra time-dimensions

The existence of extra dimensions, beyond the 3+1 with which we perceive the physical world, has entered theoretical physics with the formulation of the five-dimensional Kaluza-Klein theory (KKT), a classical unified field theory of gravitation and electromagnetism.

With a few exotic exceptions [START_REF] Rubakov | Do we live inside a domain wall?[END_REF][START_REF] Visser | An Exotic Class of Kaluza-Klein Models[END_REF], the fifth dimension -of space kind -in the KKT is compactifed under the so-called cylinder condition. The KKT is considered a precursor of string theory, where resorting to extra space dimensions is deemed natural and necessary [START_REF] Rizzo | Pedagogical Introduction to Extra Dimensions[END_REF]; also in this context the extra dimensions are curled up and microscopic. Now, what appears nonlocal in a 3+1 space-time could appear as such after projection from a higher dimensional space-time. This possibility has been suggested by considering extra dimensions of space kind surmising that "...while the usual fields only "live" in 3 + 1 dimensions, the collapse involves also other dimensions, eventually being induced by "some field" propagating also in these extra-dimensions" [START_REF] Genovese | Can quantum non-locality be connected to extra-dimensions?[END_REF].

We might wonder why not considering the extra dimension of temporal kind. Actually, extra dimensions of time have been avoided because of several reasons. Among the oth- ers, in [START_REF] Tegmark | On the dimensionality of spacetime[END_REF] it was claimed that with more than one time dimension, the partial differential equations for fields would be of ultrahyperbolic kind lacking the hyperbolicity property that enables observers to make predictions. However, it has been later proved that the initial value problem for ultrahyperbolic equations, with data posed on an initial hypersurface of mixed space and timelike signature, is well-posed [START_REF] Craig | On determinism and well-posedness in multiple time dimensions[END_REF]. Based on this work, the author of Ref. [START_REF] Weinstein | Multiple time dimensions[END_REF] proved that against to conventional beliefs, a well-posed initial value problem exists entailing deterministic, stable evolution for theories in multiple time dimensions. Moreover, the author puts forward the following intriguing idea: "quantum mechanics predicts nonlocal entanglement between the properties of a given field at various locations in space. On the other hand, any formulation of fundamental physics with multiple times is somehow non-trivial. In fact, as clearly summarized in Ref. [START_REF] Bars | Extra Dimensions in Space and Time[END_REF], naive attempts to add extra time dimensions to existing frameworks have led to two major kinds of discouraging and seemingly unavoidable difficulties: the appearance of ghosts and violations of causality. Ghosts are quantum states of systems occurring with negative probability. Causality violations are exemplified by the so called "grandfather paradox": two-dimensions of time by allowing time travels would make possible to kill one's ancestors before having being born, thus entailing an absurdity. This notwithstanding, an extensive work (an excerpt of which can be found in Refs. [START_REF] Bars | Theories with two times[END_REF][START_REF] Bars | String and particle with two times[END_REF][START_REF] Bars | Two-time physics in field theory[END_REF][START_REF] Bars | Survey of Two-Time Physics[END_REF][START_REF] Bars | The standard model of particles and forces in the framework of 2T-physics[END_REF][START_REF] Bars | Interacting two-time physics field theory with a BRST gauge invariant action[END_REF]) by Itzhak Bars and co-workers on a new gauge (symplectic) symmetry, called Sp(2, R), overcomes the mentioned problems and uniquely leads to the formalism of 2T-physics. In this two time theory the additional time dimension is compactified and infinitesimally small. It is worth mentioning that a new phase of matter [START_REF] Dumitrescu | Dynamical topological phase realized in a trapped-ion quantum simulator[END_REF] has recently been observed which seems to occupy two temporal dimensions. This new phase of matter was obtained in a quantum computer after emitting pulsed light on its qubits in a sequence following that of Fibonacci. This experiment builds on earlier work that proposed the creation of something called a quasi-crystal in time [START_REF] Else | Long-lived interacting phases of matter protected by multiple time-translation symmetries in quasiperiodically driven systems[END_REF]. Whether or not this result suggests the real existence of an additional time dimension is being discussed with a cautious attitude [START_REF] Sumner | Strange New Phase of Matter Created in Quantum Computer Acts Like It Has Two Time Dimensions[END_REF]. In what follows we need to invoke the existence of a non-compact, arbitrarily large, extra time dimension. In this case, in order to avoid time loops, and thus to exorcise the specter of the "grandfather paradox", we will assume that only the field describing the subquantum level depends on two time dimensions since only the wavefunction collapse would be affected by the extra temporal dimension. It is worth mentioning that in Ref. [START_REF] Visser | An Exotic Class of Kaluza-Klein Models[END_REF] in place of considering a compact extra space dimension, in an exotic version of Kaluza-Klein models an alternative has been explored in which the extra dimension is neither compact nor even finite, and particles are gravitationally trapped near a four-dimensional submanifold of the higher dimensional spacetime. In general, the warping of an extra dimension ζ added to {x µ } is expressed as ds 2 = e -f (ζ) g µν dx µ dx ν + dζ 2 [START_REF] Randall | Large Mass Hierarchy from a Small Extra Dimension[END_REF][START_REF] Randall | An Alternative to Compactification[END_REF], where f (ζ) is a suitable function.

Similarly but with more details, in Ref. [START_REF] Dvali | Constraints on Extra Time Dimensions[END_REF] the authors investigate the phenomenology of extra time dimensions in presence of constraints that localize the standard model particles in the extra times, allowing them to move freely in our 3+1 space-time. This entails the breaking of the translation invariance in the extra time dimensions generating a Goldstone boson that propagates in all the space-time dimensions and is viewed as a tachyonic mode from our 3+1 space-time perspective. The presence and the meaning of tachyonic modes in Kaluza-Klein models with extra time dimension(s) has been discussed from different viewpoints, see for instance Refs. [START_REF] Ynduráin | Disappearance of matter due to causality and probability violations in theories with extra timelike dimensions[END_REF][START_REF] Erdem | Reconsidering extra time-like dimensions[END_REF].

The formulation of exotic KKTs can inspire further developments of the present proposal, as is sketchily shown in the Appendix, though for the moment we cannot go into much details.

On the other hand, in Section II.3 we suggest that the fundamental idea proposed here could be tested against an experiment, if such an experiment proved feasible and produced the hypothesized result, then it would be worthwhile and necessary to go beyond the toy model discussed below.

II. NONLINEAR EQUATIONS FOR WAVEFUNCTION COLLAPSE

As is well known, two postulates of quantum mechanics are somewhat conflicting because on the one side the time evolution of the state vector of a given system is described by a linear, unitary operator and, on the other side, the result of the measure of an observable projects the system into the subspace relative to the eigenvalue/eigenstate of the measured observable according to the result obtained. Thus, measuring an observable entails the socalled collapse of the state vector (or wavefunction collapse), a discontinuous breach of the unitary evolution of a quantum system, actually a nonlinear time evolution. This topic has given rise to several discussions and interpretations (see for instance Ref. [START_REF] Bassi | Dynamical reduction models[END_REF]), however, for our purpose we resort to the proposal described in the following section.

II.1. The Bohm-Bub theory in a nutshell

Our starting point is the Bohm-Bub non-unitary evolution model of the quantum state of a system during the measurement interaction with a macroscopic system. Bohm and Bub assume a non-unitary modification of the Schrödinger equation that reads [START_REF] Bohm | A Proposed Solution of the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory[END_REF]

∂Ψ(x, t) ∂t = B(x, t) - i ĤΨ(x, t) (1)
where B(x, t) has to take into account what happens during a measurement. In particular, during the interaction with a measuring apparatus the term B(x, t) is assumed to be much larger than the standard unitary one yielding nonlinear dynamical equations that describe the wavefunction collapse. This term, with Ψ(x, t) = i ψ i (t)φ i (x), after Ref. [START_REF] Bohm | A Proposed Solution of the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory[END_REF] and also after a more refined derivation [START_REF] Tutsch | Mathematics of the Measurement Problem in Quantum Mechanics[END_REF], is assumed to be

B(x, t) = γ i ψ i (t)φ i (x) j |ψ j (t)| 2 (R i -R j ) (2)
where

R i = |ψ i (t)| 2 /|ξ i (t)| 2
and ξ i (t) are the components of the state vector of a hidden variable. Therefore, the Schrödinger equation becomes

dψ i (t) dt = γψ i (t) j |ψ j (t)| 2 (R i -R j ) - i j H ij ψ j (t) (3)
and in the continuum case

∂ψ(x, t) ∂t = γψ(x, t) dy|ψ(y, t)| 2 [R(x, t) -R(y, t)] - i Ĥψ(x, t) . (4)
Applied to the special case of a dichotomic observable

|ψ = ψ + (t)|S + + ψ -(t)|S - (5)
after the introduction of a hidden state [START_REF] Bohm | A Proposed Solution of the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory[END_REF]

ξ| = ξ + (t) S + | + ξ -(t) S -|
whose components are randomly distributed hidden variables, the wavefunction collapse is described by the model equations

dψ + (t) dt = γ (R + -R -) ψ + (t)J -- i [ H ++ ψ + (t) + H +-ψ -(t)] (6)
dψ

-(t) dt = γ (R --R + ) ψ -(t)J + - i [ H --ψ -(t) + H -+ ψ + (t)] (7)
where

J ± = |ψ ± (t)| 2 , R ± = |ψ ± (t)| 2 /|ξ ± (t)| 2
, and the randomly distributed hidden variables are assumed constant under the assumption of an impulsive measurement. During the measurement time the interaction with the apparatus is assumed to be so large that the effects of the usual Schrödinger part -i.e. of the undisturbed system -can be neglected.

Hence, by multiplying the first equation by ψ + and the second by ψ -one is left with the following nonlinear equations

dJ + dt = 2γ (R + -R -) J + J - (8)
dJ - dt = 2γ (R --R + ) J -J + (9)
whence d(J + +J -)/dt = 0 so that |ψ remains normalized during measurement, and rewriting these equations as

d log J + dt = 2γ (R + -R -) J - ( 10
)
d log J - dt = 2γ (R --R + ) J + ( 11
)
where γ is always positive, if initially R + > R -and J -= 0 then J + increases and J - decreases until J + = 1 and J -= 0 since J + + J -= 1; as a consequence the final state after the measurement of |S is |S + . Conversely, if initially R -> R + and J + = 0 then the final state after the measurement of |S is |S -. The evolution of |ψ during the measurementand thus the final outcome of the measure -depends on the values that the hidden random variables ξ + (t) and ξ -(t) had immediately before the measurement, the outcome of which is therefore unpredictable.

II.2. Extending Bohm-Bub's model to entangled particles

Let us now consider two particles, each one described by a dichotomic observable, that is

|Φ (1) = ψ (1)

+ (t)|φ

(1) + + ψ

(1)

-(t)|φ (1) - ( 12
)
|Φ (2) = ψ (2)
+ (t)|φ (2) + + ψ (2)
-(t)|φ (2) - , (13)
an entangled state of these particles is described by the Bell state

|Ψ = 1 √ 2 (|φ (1) + |φ (2) - -|φ (1) -|φ (2)
+ ) ; [START_REF] Genovese | Can quantum non-locality be connected to extra-dimensions?[END_REF] as is well known, this means that a pair of entangled entities (particles, photons) must be considered a single non-separable physical object and it is impossible to assign local physical reality to each entity; this is a direct consequence of the formalism which implies that no physical theory explaining correlations between distant events by means of locality conditions can reproduce the quantum probabilities of the outcomes of experiments. This is certainly true in the four dimensional space-time, but we can hypothesize that what appears nonlocal in a 3+1 space-time can appear as such after projection from a higher dimensional space-time, as discussed in Section I.1.

Before proceeding to extend the Bohm-Bub's model to entangled particles, a premise is necessary concerning the interaction of a microscopic system [as the one described in Eq.( 14)]

with the measuring apparatus (a macroscopic system). The latter can be thought of a manybody system described by a wave function factorized into a product of localized states of its constituent particles. When a microsystem combines with such a macroscopic system, a process of spontaneous localization occurs in the microsystem as it has been proposed in Ref. [START_REF] Baracca | On Some New Notions Concerning Locality and Nonlocality in the Quantum Theory[END_REF]. A few years later a more elaborated theory was proposed by Ghirardi, Rimini and Weber (GRW) to describe the spontaneous decoherence process of the quantum state describing a system with an arbitrary number of degrees of freedom N [START_REF] Ghirardi | Unified Dynamics for Microscopic and Macroscopic Systems[END_REF], the decoherence time scale being proportional to 1/(N λ GRW ), with λ GRW ∼ 10 -16 s -1 , thus getting very short for a macroscopic system for which N is large.

Let us now assume a time dependent γ in B(x, t) of the form γ(t) = N (t)λ GRW , where

N (t) is the number of particles with N (t) = N 0 + N M A Θ(t -t 0 )
where N 0 is the number of particles of the quantum system, N M A is the number of particles of the measuring apparatus, Θ(t -t 0 ) is a Heaviside step function, and t 0 is the time at which the measurement on the quantum system is performed.

As N M A is a macroscopic number, along the same line of thought proposed in Refs. [START_REF] Baracca | On Some New Notions Concerning Locality and Nonlocality in the Quantum Theory[END_REF] and [START_REF] Ghirardi | Unified Dynamics for Microscopic and Macroscopic Systems[END_REF] we can assume that at t 0 a complete factorization of the composite state vector |Ψ |Ψ M A takes place, where |Ψ is the microscopic system state vector and |Ψ M A is the macroscopic state vector of the measuring device. In other words, when the microscopic system comes into contact with the measuring apparatus it becomes part of an overall system with a macroscopic number of degrees of freedom subject to the above mentioned decoherence mechanism. Therefore, since at the measuring apparatuses the particles are separated by a space-like interval, thus are distinguishable and non-interacting, the state vector ( 14) factors into the product of the single particle states given above

|Ψ = |Φ (1) |Φ (2) (15)
and -at the same instant of time t 0 -in the Schrödinger equation Ĥ = Ĥ1 ⊗ Ĥ2 splits as

Ĥ = Ĥ1 ⊗ Î2 + Î1 ⊗ Ĥ2 (16)
then the non-unitary evolution of the whole system (microscopic quantum system plus mea-suring apparatus) will read

dψ (1) + (t) dt = γ(t) R (1)
+ -R

(1)

-+ R (2) --R (2) + ψ (1) + (t)J (1)
--i H 1++ ψ

(1)

+ (t) + H 1+-ψ (1)
-(t) dψ (1)
-(t) dt = γ(t) R (1)
--R

(1)

+ + R (2)
+ -R (2) - ψ (1)
-(t)J (1)
+ - i H 1-+ ψ (1)
+ (t) + H 1--ψ (1)
-(t) dψ (2)
+ (t) dt = γ(t) R (2)
+ -R

-+ R

(1)

--R (1) + ψ (2)
+ (t)J (2)
-- i H 2++ ψ (2)
+ (t) + H 2+-ψ (2)
-(t) dψ (2)
-(t) dt = γ(t) R (2)
--R

+ + R

(1)

+ -R (1) - ψ (2)
-(t)J (2)
+ - i H 2-+ ψ (2)
+ (t) + H 2--ψ (2)
-(t) (17)
where

J (j) ± = |ψ (j) ± (t)| 2 , R (j)
± = |ψ (j) ± (t)| 2 |ξ (j) ± (t)| 2 , j = 1, 2 (18)
according to the initial sign of the terms in the first parenthesis of the r.h.s. of each one of the equations above, it is immediately evident that if the first particle collapses to the state |+ the second one collapses to |-and viceversa. In analogy with the original Bohm-Bub's model, the ξ

± (t) are assumed to be hidden random variables, almost everywhere non vanishing, and constant during measurement. The non-locality is here expressed by the fact that even if the two particles are separated by a space-like interval, the values of the quantities R (j) have an instantaneous mutual influence on the respective wavefunction collapses. Hereafter our main hypothesis enters the game. We assume that the wavefunctions collapses of the two particles -that now have their own individuality -are correlated via an exchange of information mediated by a hidden sub-quantum field X (x, t, τ ) which propagates through an extra temporal dimension described by the variable τ . We are replacing the hidden variables ξ (j) ± (t) with a completely different physical entity, that is, a real physical field X (x, t, τ ), almost nowhere vanishing (that is with the possible exception of a zero measure subset of space), a condition that could be ensured for example by the existence of a stochastic background component of the field. We tentatively assume that the dynamical evolution of the scalar field X (x, t, τ ) is described by a wave equation

∇ 2 X (x, t, τ ) - 1 w 2 ∂ 2 X (x, t, τ ) ∂τ 2 + F [ψ (j) ± (t), X (x, t, τ )] = 0 ( 19
)
accounting for information propagation at speed w by means of X (x, t, τ ) through the dimension τ , and where the inhomogeneous source term F [ψ (j) ± (t), X (x, t, τ )] should be chosen so that equation ( 19) fulfils the following requirements: i) it describes the propagation of the field X (x, t, τ ) in the form of a non-dispersive impulse carrying the information of the outcome of a measure performed at some spatial location; ii) this information is isotropically propagated in space; iii) the information impulse carried by the field X (x, t, τ ) is not attenuated with distance, and this assumption is required to comply with standard quantum non-locality which is independent of the distance between entangled entities. So we update equation [START_REF] Bars | Theories with two times[END_REF] as

∇ 2 X (r, t, τ ) - 1 w 2 ∂ 2 X (r, t, τ ) ∂τ 2 + sin[X (r, t, τ )] + F [x, ψ (j)
± (t), τ ; t 0 , τ 0 ] = 0 ( 20
)
that is a wave equation where a tentative nonlinearity is introduced to account for the propagation of a non dispersive impulse, actually a sine-Gordon soliton, in a radial direction r (under spherical symmetry), stemming from a given point x, and carrying the information coded by the source term F [x, ψ

± (t), τ ; t 0 , τ 0 ], where t 0 and τ 0 stand for the initial times. In analogy with the standard assumption [START_REF] Belinfante | A survey of hidden-variables theories[END_REF] for the evolution of ξ(t), that is, ξ(t) = 0 during an impulsive measurement, no term containing the derivative of X (x, t, τ ) with respect to our usual time is considered in the equation above. At the present state of affairs, we are tackling a toy model because we lack physical input to make more definite formulations, in particular to assign the analytical form of the function F [x, ψ (j) ± (t), τ ; t 0 , τ 0 ], nevertheless, for the moment being it is sufficient to formally enter this term in Eq. [START_REF] Bars | String and particle with two times[END_REF] to represent the source of the information carried by X (x, t, τ ). In fact, already at the present stage an experiment seems conceivable (see the next section) to test whether or not the surmised scenario corresponds to the physical reality, in case of a positive experimental outcome one could make more motivated assumptions. The collapse equations [START_REF] Weinstein | Multiple time dimensions[END_REF] are then rewritten as

dψ (1) + (t) dt P 1 = γ(t) R (1) + -R (1) -+ R (2) --R (2) + ψ (1) + (t)J (1) -- i H 1++ ψ (1)
+ (t) + H 1+-ψ (1)
-(t) (21) dψ (1) -(t) dt P 1 = γ(t) R (1) --R (1) + + R (2) + -R (2) - ψ (1) -(t)J (1) + - i H 1-+ ψ (1)
+ (t) + H 1--ψ (1)
-(t) (22) dψ
+ (t) dt P 2 = γ(t) R (2) + -R (2) -+ R (1) --R (1) + ψ (2) + (t)J (2) -- i H 2++ ψ (2)
+ (t) + H 2+-ψ (2)
-(t) (23) dψ
-(t) dt P 2 = γ(t) R (2) --R (2) + + R (1) + -R (1) - ψ (2) -(t)J (2) + - i H 2-+ ψ (2)
+ (t) + H 2--ψ (2)
-(t) (24)
where, with an abuse of notation, it is evidenced that the first two equations describe the measurement process at the space-time coordinates P 1 = (x 1 , t 0 , τ 0 + ∆τ ), and the other two equations describe the measurement process at the space-time coordinates P 2 = (x 2 , t 0 , τ 0 + ∆τ ), that is, one of the polarizers is located at x 1 and the other polarizer is located at x 2 ; both measurements are simultaneous, performed at t 0 and at τ 0 + ∆τ where ∆τ = |x 2 -x 1 |/|w| with w the velocity vector of information transfer between the particles through the extra time dimension, that is, the extra time at which a mutual exchange of information takes place. In order to describe this exchange of information between the two subsystems -operated via the sub-quantum field X (x, t, τ ) propagating through the extra time dimension τ -and in order to keep the essential of the Bohm-Bub model the variables

R (1)
± and R

(2)

± in the equations above are assumed to be of the form

R (1) ± = |ψ (1) ± (t)| 2 |X (x 2 -w∆τ, t 0 , τ 0 + ∆τ )| 2 , R (2)
± = |ψ (2) ± (t)| 2 |X (x 1 -w∆τ, t 0 , τ 0 + ∆τ )| 2 (25) and R (1) ± = |ψ (1) ± (t)| 2 |X (x 1 , t 0 , τ 0 + ∆τ )| 2 , R (2)
± = |ψ (2) ± (t)| 2 |X (x 2 , t 0 , τ 0 + ∆τ )| 2 (26)
with the assumption γ(t) = [N 0 + N M A Θ(t -t 0 )]λ GRW , where N 0 is the number of particles of the quantum system.

It is important to remark that the modification of the Schrödinger equation ( 1) is without any consequence on the standard unitary evolution of a quantum system -including the Bell state ( 14) -until γ(t) = const = N 0 λ GRW takes a large value because of the interaction with the measuring apparatus. In other words, neither the field X (x, t, τ ) nor the extra temporal dimension have any consequence on the standard unitary evolution until measurement.

In this latter case, under the hypothesis of the wavefunction factorization induced by the GRW mechanism, the evolution of the quantum system during the measurement process is described by equations ( 21)- [START_REF] Bars | Interacting two-time physics field theory with a BRST gauge invariant action[END_REF] and as in the Bohm-Bub theory it is assumed that the unperturbed quantum evolution term is negligible in comparison with the interaction term between the quantum system and the measuring apparatus. Hence, the time evolution of the wavefunctions of the system is described only by the nonlinear parts of the equations above, that is, after multiplications by the complex conjugate of the wavefunction of each equation, by the following system of equations

dJ (1) + dt = 2γ(t) R (1) + -R (1) -+ R (2) --R (2) + J (1) + J (1) - dJ (1) - dt = 2γ(t) R (1)
--R

(1)

+ + R (2) + -R (2) - J (1)
-J

(1) + ( 27)

dJ (2) + dt = 2γ(t) R (2) + -R (2)
-+ R

(1)

--R (1) + J (2) + J (2) - dJ (2) - dt = 2γ(t) R (2)
--R

+ + R

(1)

+ -R (1) - J (2)
-J

(2) + which evidently preserve the normalization of wavefunctions, that is, d(J

(1)

+ + J (1)
-)/dt = 0 and d(J

(2) + + J (2)
-)/dt = 0. Equations ( 27) are rewritten as

d log J (1) + dt = 2γ(t) R (1)
+ -R

(1)

-+ R (2)
--R

(2) +

J

(1) -

d log J (1) - dt = 2γ(t) R (1)
--R

(1)

+ + R (2) + -R (2) - J (1) + d log J (2) + dt = 2γ(t) R (2)
+ -R (2)
-+ R

(1)

--R (1) + J (2) - (28)
d log J (2) - dt = 2γ(t) R (2)
--R

+ + R

(1)

+ -R (1) - J (2) + if J (1)
-= 0 and R

(1)

+ -R (1) -+ R (2) --R (2)
+ > 0 then J

(1) + increases while J

(1)

-decreases until J

(1) + = 1 and J

(1) -= 0 so that the measure of |Φ (1) gives |φ

+ -R (1) -+ R (2) --R (2) + > 0, if J (2) + = 0 then J (2)
-increases until J

(2) -= 1 and J

(2) + = 0 so that the measure of |Φ (2) gives |φ

(2)
-. And this is the necessarily expected result.

The final outcome of the measure is determined by the evolution of |Φ (1) |Φ (2) which depends on the values that the field X (x, t 0 , τ 0 + ∆τ ) had immediately before the measurement at x 1 and x 2 , and thus it depends on the random component of the field and on the unknown details about the shaping of the information-carrying impulses. All this makes the outcome of the measurement unpredictable.

II.3. Proposal for a nonconventional experiment

Quantum mechanics predicts random results on each side of an experimental apparatus like the one sketched in Figure 2 with 1/2 probability of measuring +1 or -1, and it also predicts strong correlations between these random results. Bell's inequality gives an upper bound to the correlations predicted by local realism, whereas quantum predictions violate this inequality. A Bell test consists of measuring the correlations and comparing the results with Bell's inequality. To perform a loophole-free Bell test, the polarizer settings must be changed randomly while the photons are in flight between the source and the polarizers [START_REF] Aspect | Proposed experiment to test the separable hidden-variables theories[END_REF][START_REF] Aspect | Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment -A new violation of Bell inequalities[END_REF][START_REF] Aspect | Experimental test of Bell's inequalities using timevarying analyzers[END_REF]. The source term F [x, ψ (j) ± (t), τ ; t 0 , τ 0 ] in equation [START_REF] Bars | String and particle with two times[END_REF], even if of unspecified analytic form, amounts to assuming that the field X (x, t, τ ) encodes the information about the wavefunction collapse due to the measurement performed on a given system, and equation [START_REF] Bars | String and particle with two times[END_REF] tells that this information is spread in every spatial direction. Therefore, a collapse information could also reach another identical system driving its collapse and mimicking entanglement correlation. This is the hypothesis to be experimentally proved or disproved that also suggests how to design an appropriate experiment. In fact, let us now consider two identical systems, depicted in Figure 3, where Alice and Bob perform measurements at the ends of the first system, and Eve and Tom at the ends of the second system. Of course, the photons (or particles with spin) reaching Alice and Bob or Eve and Tom are in be such that the distance between Alice and Eve is much shorter than that between Alice and Bob and Eve and Tom, respectively. The two setups should be suitably electromagnetically shielded in order to avoid loopholes. A coincidences detector should be used to check whether or not Bell's inequality is violated when checking the correlations between the measurements performed by Alice and Eve. In case of a positive result this would lend credit to the idea surmised in the present work. (The experiment could as well be performed using particles with spin).

state superposition and constrained to fulfil a conservation law (momentum, energy, angular momentum) which is not the case of two particles reaching, say, Alice and Eve because apriori the entities (say photons, as in Figure 3) reaching Alice and Eve are independent, that is, not entangled. Now, as soon as the photon detected by Alice has "chosen" its polarization state how does the photon being detected by Bob "know" that it has no choice left for its polarization state? The orthodox reply is that they are part of the same indivisible system even if they are very far apart, but if we assume that the information about the outcome of the measurement performed by Alice is broadcasted in every spatial direction (through the extra temporal dimension), a particle belonging to an identical system, like the one reaching Eve, could be driven to collapse into the same complementary state as in Bob's of directions of polarization analyzers α, α , β, β predicts a violation of this inequality, the violation being maximal for the set of angles S QM (0, 45 o , 22.5 o , 67.5 o ) = 2 √ 2. Therefore, let us imagine to measure this quantity through the coincidences detected between Alice and

Eve -under all the standard requirements of detectors efficiency, electromagnetic shielding to avoid loopholes and so on -in case one would observe a violation of the above Bell-CHSH inequality this would support the hypothesis of the existence of an information exchangebetween the subsystems -mediated by some physical entity that we have called sub-quantum field X (x, t, τ ) propagating through an extra time dimension. This hypothesis is thus falsifiable and even if somewhat daring it cannot be a-priori discarded.

A clarification needs to be made here. Equations ( 27) and ( 29) describe the evolution of wavefunctions collapse in ordinary time after t 0 , the instant of particles interaction with the measuring devices, and "after" a mutual exchange of information in the extra time. In the case of the proposed new experiment, the surmised phenomenon of information exchange through the extra time depends entirely on the spatial arrangement of the four measuring devices. The exchange of information between Eve and Alice (assumed at the shortest distance among the four measuring devices) could conceivably destroy the entanglement with their respective partners allowing a rearrangement of the correlations among the results of the four measuring devices. A more refined analysis would be needed to describe what kind of correlations -among all these devices -could be expected with generic spatial arrangements.

Such analysis is beyond the scope of our toy model and would be motivated by a positive experimental outcome of the proposed experiment outlined above.

III. CONCLUDING REMARKS

Entanglement, that is the ability of two particles to "correlate" their behaviors even at very large distances and in the absence of any physical connection, is intrinsic to the formalism of quantum mechanics and although it is an experimentally proven reality of our world, it defies our perception of physical phenomena and the way of representing them. On the other hand, since its inception, quantum mechanics has always allowed to perform very accurate calculations regarding microscopic phenomena, providing predictions that have always been verified without exception. So why should we worry about entanglement since the theory works so well? But attempting to delve behind a formalism describing as an indivisible system two entangled objects, even if sitting at opposite borders of our galaxy, could have a twofold interest, on the one side a conceptual relevance and, on the other side, perhaps interesting implications in the field of quantum technologies and quantum computation where entanglement plays a crucial role. Therefore, any attempt at gaining a deeper understanding of quantum entanglement through different hypotheses seems worthwhile, even in case a given hypothesis is disproved by experiments, since it would thus introduce a "no go". A possible "softening" of the conundrum represented by quantum entanglement could be found by considering what is observed in our 3+1 dimensional space-time as a projection from a higher dimensional space-time. This has been suggested until recently [START_REF] Genovese | Can quantum non-locality be connected to extra-dimensions?[END_REF] Moreover, it is worth mentioning that the puzzling and counterintuitive properties of entanglement between spacelike separated quantum objects turn to unbelievable when entanglement is found between photons that never coexisted in time [START_REF] Megidish | Entanglement Swapping between Photons that have Never Coexisted[END_REF]. Out of two temporally separated photon pairs it is found that one photon belonging to the first pair can be entangled with a photon from the second pair and the first photon is detected before the creation of the second one. This experimental outcome followed a previous theoretical work where quantum interferences and violation of Bell's inequality has been studied by considering photons emitted from independent single photon sources that do not overlap in time [START_REF] Wiegner | Quantum interference and entanglement of photons that do not overlap in time[END_REF]. At least in principle, we can speculate that this phenomenon could be given a more "intuitive" explanation under the hypothesis put forward in the present paper by considering that the information carrying field X (x, t, τ ) a-priori should propagate also in our familiar time coordinate t. This is not explicitly formalized in equation ( 19) because the measuring process was assumed to take place during a very short interval of ordinary time. In fact, equation ( 19) could be expressed as a ultrahyperbolic wave equation thus propagating to the ordinary future the information of the outcome of the measure on the first photon. This information could thus drive the outcome of the measure on the second photon.

Finally, in case the proposed experiment would lend credit to the hypothesis formulated in the present work, information would be given an ontological status and in so doing this would be somehow echoing "[...] the idea that every item of the physical world has at bottom -at a very deep bottom, in most instances -an immaterial source and explanation; [...] in short, that all things physical are information-theoretic in origin..." as advocated by J.A.

Wheeler [START_REF] Wheeler | Information, Physics, Quantum: the search for links[END_REF].

IV. APPENDIX

It is not out of place to sketchily show how we can borrow from Ref. [START_REF] Visser | An Exotic Class of Kaluza-Klein Models[END_REF] -almost verbatim -the suggestion for a five dimensional space-time metric of (+ + + --) signature -with a non-compact and infinite extra time dimension -where the particles are trapped on the standard four-dimensional space-time. Thus, by simply modifying the signature of the extra dimension of the "exotic" Kaluza-Klein metric of Ref. [START_REF] Visser | An Exotic Class of Kaluza-Klein Models[END_REF], consider the infinitesimal arclength in 3+2 space-time as ds 2 = g µν dx µ dx ν = dx 2 + dy 2 + dz 2 -e 2ϕ(τ ) c 2 dt 2 -w 2 dτ 2 then the Klein-Gordon equation written on this space-time background (using the Laplace-Beltrami operator) reads

g -1/2 ∂ µ (g 1/2 g µν ∂ ν Ψ) -m 2 5 Ψ = 0 ( 33
)
where g is the determinant of the metric g µν and m 5 is a suitably defined five dimensional particle rest mass [START_REF] Visser | An Exotic Class of Kaluza-Klein Models[END_REF]. One finds

∇ 2 -e -2ϕ(τ ) 1 c 2 ∂ 2 ∂t 2 - 1 w 2 ∂ 2 ∂τ 2 Ψ - 1 w 2
∂ϕ ∂τ ∂Ψ ∂τ -m 2 5 Ψ = 0 [START_REF] Bassi | Dynamical reduction models[END_REF] which has the form a ultrahyperbolic equation [START_REF] Weinstein | Multiple time dimensions[END_REF] for which a solution can be found in the form Ψ = e -i(ωt-k•x) e -ϕ(τ )/2 Φ(τ ) [START_REF] Bohm | A Proposed Solution of the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory[END_REF] where Φ(τ ) satisfies the equation

1 w 2 ∂ 2 ∂τ 2 + ( 1 2 φ + 1 4 φ2 -ω 2 e -2ϕ ) + m 2 5 + k 2 Φ = 0 ( 36
)
then with a convenient choice of ϕ(τ ), the factor e -ϕ(τ )/2 -which is called warp factordetermines the degree of warping along the extra time dimension. These are just a few hints on how to imagine an infinite temporal extra-dimension where the sub-quantum field X (x, t, τ ) is fully extended whereas particles are trapped around τ = 0. The field X (x, t, τ ) is assumed to be a real physical entity, therefore in principle it could enter the 3 + 2 spacetime metric g µν similarly to the electromagnetic vector potential in Kaluza-Klein theories.

This could be done so as to find a field equation for X (x, t, τ ) as that in Eq. [START_REF] Bars | String and particle with two times[END_REF]. However, how to choose such a metric, how to choose a suitable and physically meaningful function ϕ(τ ), and how to make a connection with the 2T physics developed by Itzhak Bars and co-workers remain far beyond the aim of the present work.

Figure 1 .

 1 Figure 1. Schematic representation of a snapshot at measuring time. Alice and Bob are at a spacelike distance (light cones are represented). A pulse-shaped signal of sub-quantum field X (x, t, τ ) propagates through the extra time dimension τ carrying the information of the outcome of a measurement done by Alice.

  [...] The sort of constraint explored in this essay, one arising from the presence of extra time dimensions, exhibits one sort of nonlocality, but there are other sorts as well, [...] what I have called "nonlocality without nonlocality", meaning nonlocal correlations without nonlocal causation."

Figure 2 .

 2 Figure 2. An apparatus for performing a Bell test via quantum correlation measurement. A source S emits a pair of entangled photons γ 1 and γ 2 in a singlet state |Ψ and their linear polarizations are measured by polarizers p 1 and p 2 . Each polarizer has two output channels, labeled +1 and -1. In this schematic diagram a representation of the switching devices, performing a change of the settings of the polarizers while the photons are in flight between the source and the polarizers, is omitted.

Figure 3 .

 3 Figure 3. Two identical and standard apparatuses -used to check Bell's inequality violation in coincidences measurement of entangled particles -should be operated at the same time. The two sources emit entangled photons γ 1 and γ 2 whose polarizations are analysed by the polarizers p 1 and p 2 . Each polarizer has two possible outputs indicated as +1 or -1. The spatial arrangement should

  by invoking the existence of extra dimensions of space kind, in fact, two points very far apart in the 3+1 space-time can be very close one another in a higher dimensional embedding space. In the present work we have suggested a different scenario by invoking the existence of an extra dimension of time kind in order to describe nonlocality without nonlocality, that is, nonlocal An experimental test that would have no reason to be performed, being manifestly trivial in the absence of any thinkable reason to doubt of its triviality. Of course, the model put forward in the present work can be criticized from many different viewpoints but, being not conflicting with the robust theoretical framework of quantum mechanics, it is intended to provide the mentioned thinkable reason to stimulate the interest of some experimentalists, and, possibly, some constructive theoretical contribution. In fact, the proposed experiment is clearly defined and not affected by the unspecified details of the phenomenological toy model.

correlations without nonlocal causation, in the words of the author of Ref.

[START_REF] Weinstein | Multiple time dimensions[END_REF]

. Aiming at depicting a possible scenario of this kind, we have borrowed the longstanding Bohm-Bub's phenomenological proposal for a non-unitary dynamical description of the wavefunction collapse

[START_REF] Bohm | A Proposed Solution of the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory[END_REF][START_REF] Tutsch | Mathematics of the Measurement Problem in Quantum Mechanics[END_REF]

, and, after suitable elaboration, we have outlined a toy model whose function is to motivate an experiment to test a physical hypothesis: the existence of a sub-quantum field carrying information.

ACKNOWLEDGMENTS

The author wishes to thank Roger Penrose for an interesting and useful discussion held at the Arts Centre De Brakke Grond, Amsterdam, in 2014, a discussion that has been the remote origin of the present work. Useful comments and suggestions emerged during several discussions with Giulio Pettini, Gabriele Vezzosi, Matteo Gori, Roberto Franzosi, Guglielmo Iacomelli, and Jack Tuszynski. The author thanks Stefano Ruffo for having brought to his attention the paper in Ref. [37].

[1] J.S. Bell, On the Einstein-Podolsky-Rosen Paradox, Physics 1, 195 (1964).

case. Therefore, let us imagine an experimental setup where the measurement apparatuses of Alice and Eve are spatially close to one another and both of them much farther from the apparatuses of their companions Bob and Tom, so that the extra-time interval ∆ 1 τ needed by the hypothetical field X (x, t, τ ) to exchange information between Alice and Bob on one side, and Eve and Tom on the other side, is much longer than the interval ∆ 2 τ needed to exchange information between Alice and Eve. Consequently, the central hypothesis of the present work boils down to rewriting the collapse equations [START_REF] Sumner | Strange New Phase of Matter Created in Quantum Computer Acts Like It Has Two Time Dimensions[END_REF] as

dJ

where enters the Bell-CHSH (Clauser, Horne, Shimony, Holt) [START_REF] Clauser | Proposed Experiment to Test Local Hidden-Variable Theories[END_REF][START_REF] Aspect | Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment -A new violation of Bell inequalities[END_REF] inequality

which is satisfied by realistic local theories, and a-priori should be also fulfilled by non entangled particles as those reaching Alice and Eve. Quantum mechanics for various combinations


Jérôme Blaizot
email: jerome.blaizot@insa-lyon.fr

Thibaut Chaise
email: thibaut.chaise@insa-lyon.fr

Daniel Nelias
email: daniel.nelias@insa-lyon.fr

Michel Perez
email: michel.perez@insa-lyon.fr

CHARACTERIZATION AND MODELLING OF TENSILE FLOW BEHAVIOR OF NI BASE ALLOY 690 AT VARIOUS TEMPERATURES AND STRAIN RATES

ABSTRACT

Pressurized Water Reactor components are welded by Gas Tungsten Arc Welding (GTAW). To achieve good corrosion resistance and mechanical properties, Ni base alloy 690 is used to manufacture these components. The understanding of physical phenomena involved during welding and the prediction of induced residual stresses are crucial to guarantee high quality of these components. Welding induces drastic changes in the microstructure of the molten zone and heat-affected zone of metallic alloys especially for multi-pass welding. These changes may deteriorate the mechanical properties of the assembly. In order to reproduce the complex thermo-mechanical loading occurring within the heat affect zone, experiments on a thermomechanical simulator Gleeble 3500 have been carried out. In order to characterize the base alloy, isothermal tensile tests have been performed at various strain rates and temperatures (from 25 to 1100°C). A constitutive law has been proposed to predict the mechanical properties under different strain rates and temperatures. Tensile tests have also been performed after several thermal cycles to understand the effect of welding on mechanical properties of Ni alloy 690. In parallel, grain size evolution and carbide precipitation have been characterized and correlated to measured mechanical properties.

NOMENCLATURE

INTRODUCTION

Pressurized Water Reactor components are welded by Gas Tungsten Arc Welding (GTAW). To achieve good corrosion resistance and mechanical properties, Ni base alloy 690 is used to manufacture these components. The understanding of physical phenomena involved during welding and the prediction of induced residual stresses are crucial to guarantee high quality of these components. Welding induces drastic changes in the microstructure of the molten zone and heat-affected zone of metallic alloys especially for multi-pass welding. These changes may deteriorate the mechanical properties of the assembly. In order to reproduce complex thermo-mechanical loading occurring within the heat affect zone, experiments on a thermomechanical simulator Gleeble 3500 have been conducted. In the present work, the Kocks-Mecking phenomenological approach is used to predict the stress-strain curves as a function of temperature and strain rate [START_REF] Mecking | The kinetics of flow and strain hardening[END_REF][START_REF] Estrin | A unified phenomenological description of wok hardening and creep based on oneparameter models[END_REF][START_REF] Kocks | Physics and phenomenology of strain hardening: the FCC case[END_REF].

MATERIALS AND METHODS

The material chosen for this study is a nickel based superalloy namely alloy 690 used for Pressurized Water Reactor components of nuclear power plants. This alloy contains mainly nickel, chromium, iron and minor amounts of titanium, carbon and silicon. Like austenitic stainless alloys, this alloy contains carbides at grain boundaries in the austenite matrix. Micrographic optic methods have been used to determine the initial grain size in the alloy and after tensile tests between 25 and 1000°C. The grain size and distribution have been determined by image analysis after an electrochemical etching at 2.5V during 20s in an electrolyte containing H3PO4, HNO3 and H2O. Characterization with a Scanning Electron Microscope has been performed to qualitatively quantify carbides at grain boundaries and to observe their morphologies. The composition of grain boundaries has been identified as M23C6 chromium carbides by using Transmission Electron Microscope. To reproduce complex thermo-mechanical loading occurring within the heat affected zone, tensile tests on the thermo-mechanical simulator Gleeble 3500 have been done from 25 to 1100 °C at various strain rates ranging from 10 -4 to 5.10 -3 s -1 . A constant deformation rate in the sample is imposed by using a longitudinal extensometer which operates up to 1100°C. The tensile tests were done on the alloy Ni 690 as received after the calibration of load cell and the hot extensometer. The measurement of Young's modulus was also confirmed by other method at 25°C.

EXPERIMENTAL RESULTS

Figures 1 and2 show the initial microstructure and the morphology of grain boundary carbides in the initial material. Titanium nitrides are also present in the alloy and can be seen on figure 1. Grain boundary carbides have been identified as M23C6 carbides containing mainly chromium by using TEM. As pointed out by Li et al. [START_REF] Li | The growth mechanism of grain boundary carbide in alloy 690[END_REF], M23C6 carbides are partially coherent with the matrix.

Young's modulus has been determined between 25 and 1000°C and the values measured are similar to the technical data from Special Metals for alloy 690 [2] (fig. 3). Tensile yield strength has been also determined from 25 to 1100°C at a constant strain rate of 10 -4 and 10 -3 s -1 (fig. 4). Strain rate has an influence on yield stress above 900°C. From 25 to 600°C, strain hardening is predominant and a balance between strain hardening and dynamic recovery is observed above 750°C as illustrates on figure 5. A significant change of mechanical behavior occurs between 600 and 700°C. Strain hardening and stationary stress are greatly influenced by strain rate at 750°C (fig. 6).

Fig. 5: stress-strain curves at a strain rate of 10 -4 s -1 and various temperatures.

Fig. 6: stress-strain curves at a temperature of 750°C and various strain rates.

MODELLING OF MECHANICAL BEHAVIOR

Predicting the mechanical behavior as a function of strain rate and temperature requires the determination of yield stress, stationary stress and the evolution of hardening. To predict the yield stress from 25 to 1000°C with a unique law, an empiric relationship similar to Johnson-Cook can be used. Kocks-Mecking's model is particularly interesting to predict the isotropic strain hardening [START_REF] Mecking | The kinetics of flow and strain hardening[END_REF][START_REF] Estrin | A unified phenomenological description of wok hardening and creep based on oneparameter models[END_REF][START_REF] Kocks | Physics and phenomenology of strain hardening: the FCC case[END_REF]. The Kocks-Mecking's model gives the evolution of the dislocation density as follows:

ℎ (1)
Parameters h and r are respectively the strain-hardening and dynamic recovery parameters.

Starting from this equation, the evolution of strain-hardening rate and flow stress can be determined by using Taylor's equation [START_REF] Taylor | The mechanism of plastic deformation of crystals. Part I. Theoretical[END_REF]:

(2) By using equations ( 1) and ( 2), the evolution of strain-hardening rate Θ can be determined.

Θ Θ 1
where Θ and

By integrating equation 1 and using equation 2, the macroscopic Voce's law is obtained.

! exp %

where %

As shown on figures 5 and 6, the parameter and the saturation stress depend on temperature and strain rate [START_REF] Sellars | On the mechanism of hot deformation[END_REF]. The saturation stress is given by the following equation:

& ' exp ( )* (5)
In equation 5, & is a constant, ( the activation energy and + the strain rate sensitivity coefficient. The value of the reference strain rate is 10 -4 s -1 . These coefficients have been calculated in two ranges of temperature: 25-600 and 600-1100°C. For a temperature below 600°C, the hypothetic saturation stress is not a function of the strain rate. The activation energy is low and the hypothetic saturation stress decreases slowly as shown on figure 7.

The coefficient & and + of equation 5 have been also determined between 600 and 1100°C for a strain rate ranging from 10 -4 to 5.10 -3 s -1 . An activation energy ( of 68 kJ/mol has been found, this value is greatly lower than the activation energy of selfdiffusion of nickel (219 kJ/mol) and vacancy migration (106 kJ/mol) [START_REF] De Reca | Self-diffusion of Ni in Ni-Fe alloys[END_REF][START_REF] Chambron | Etude des lacunes en très faible concentration dans l'alliage fer-nickel à 70% de nickel par la méthode de l'anisotropie magnétique[END_REF]. The influence of temperature and strain rate on in a large range of temperature is not clearly known. The parameter has been calculated between 25 and 750°C at various strain rates. Above 750°C, this parameter has not been calculated because the stress reaches the steady value at the beginning of the plastic deformation. It dramatically increases from 600 to 700°C to reach a steady value for a strain rate of about 10 -4 s -1 .

Finally, by predicting the yield stress, saturation stress and recovery parameter (fig. 78), the evolution of tensile stress can be calculated as function of temperature and strain rate (fig. 9-10). The challenge is to predict the change of mechanical behavior between 600 and 750°C by considering also the effect of strain rate on recovery parameter. The interest of this study is to keep the relative simplicity of Kocks-Mecking model which contains only one parameter that depends on temperature and strain rate.

CONCLUSIONS

Tensile flow behavior has been studied between 25 and 1100°C at various strain rates including the effect of strain rate on strain hardening and dynamic recovery. By predicting the yield stress, saturation stress and recovery parameter depending on temperature and strain rate, the tensile stress has been calculated and the results are in a good agreement with the experimental data. Now, the prospects are to consider the effect of microstructure such as grain size on mechanical behavior and a better prediction of yield stress depending on temperature, strain rate and grain size. It also necessary to compare the results with independent data.

Fig. 1 :

 1 Fig. 1: optical image of the initial microstructure of alloy 690.

Fig. 2 :

 2 Fig. 2: SEM image of carbides at grain boundaries.

Fig. 3 :Fig. 4 :

 34 Fig. 3: evolution of Young's modulus versus temperature from 25 to 1000°C.

Fig. 7 :

 7 Fig. 7: experimental values and prediction of saturation stress versus temperature at various strain rates

Fig. 9 :

 9 Fig.9: prediction of the tensile stress at a constant strain rate of 10 -4 s -1 and various temperatures and comparison with experimental data.

Fig. 10 :

 10 Fig. 10: prediction of the tensile stress at a temperature of 750°C and various strain rates and comparison with experimental data.

ACKNOWLEDGMENTS

The authors want to thank Philippe Gilles, Vincent Robin, Guillaume Tirand and Miguel Yescas (AREVA) and the AREVA-SAFRAN chair for supporting this research.


Danial Pour

Arab Icube

Matthias Spisser

Caroline Essert
email: essert@unistra.fr

Introduction of a Skip-Row pattern in complete coverage path planning for agricultural fields

Keywords: Coverage Path Planning, Precision Agriculture, Autonomous Agriculture, Vehicle Routing Problem, Wheeled robots, Path Planning, Route Planning

Over the past two decades, an evolutionary effort has been established in the agricultural sector to develop efficient autonomous systems that can carry out common infield operations including harvesting, mowing, and spraying. Increasing production while decreasing costs and environmental damages is one of the main objectives for these autonomous systems. Due to the nature of these tasks, complete coverage path planning techniques are crucial to determining the best path that covers the entire field while accounting for terrain characteristics, operational needs, and robot properties.

In this study, we propose a novel complete coverage path planning approach to define the ideal path for a wheeled robot across an agricultural field. To identify all feasible solutions satisfying a set of predefined constraints, a method based on tree exploration is first proposed that examines skip-row patterns. Second, the most optimal solution is selected by a selection method. Maximizing the covered area while minimizing overlaps, non-working path length, number of turns containing reverse moves, and overall travel time are the objectives of the selection method.

We showed on 6 real-world fields geometries that the row skip approach offered benefits in terms of reduction of the required headland size, and often helped decreasing the number of necessary reverse moves and the overlaps, while increasing the covered area.

I. INTRODUCTION

Automated systems and robots are increasingly used in every sector of industry since they boost productivity while cutting expenses. In comparison to humans, they are able to do a wider range of specialized and tedious activities with a remarkable precision and accuracy. Agriculture is no exception to this.

More food and agricultural products are required due to a growing population [START_REF] Marques | Increasing impacts of land use on biodiversity and carbon sequestration driven by population and economic growth[END_REF], yet agriculture is also known as a remarkable source of the air and environmental pollution [START_REF] Meftaul | Pesticides in the urban environment: A potential threat that knocks at the door[END_REF]. Autonomous systems capable of optimizing the cost and efficiency of in-field operations is indispensable and demands an urgent attention.

One of the main challenges to perform an autonomous operation such as harvesting, tillage, seeding, and pulverization is to find a path that optimally covers the entire field. This challenge is commonly known as Complete Coverage Path Planning (CCPP). To generate an optimal and feasible path, it is essential to consider the geometry and characteristics of the field, the robot, and the machinery i.e. implement (Imp for short) connected to the robot.

Due to the complexity of this problem, in the majority of cases, the CCPP is referred to in the literature as two distinct and sequential tasks: Coverage Path Planning (CPP) and Agricultural Routing Problem (AVRP). CPP consists in generating a set of parallel trajectories while AVRP consists in finding an optimal sequence of these trajectories for a single or multiple robots.

Generating parallel trajectories based on a reference direction or the longest edge of the field is a prevalent CPP method chosen by [START_REF] Cariou | Mobile robot trajectory planning under kinematic and dynamic constraints for partial and full field coverage[END_REF]- [START_REF] Zhou | Route planning for capacitated agricultural machines based on ant colony algorithms[END_REF]. Furthermore, improving on this method, generating parallel trajectories to a curved reference line was also proposed [START_REF] Plessen | Freeform path fitting for the minimisation of the number of transitions between headland path and interior lanes within agricultural fields[END_REF], [START_REF] Zhou | Metric map generation for autonomous field operations[END_REF]. Another improvement was to generate straight trajectories parallel to the boundary that minimized the number of trajectories [START_REF] Edwards | Route planning evaluation of a prototype optimised infield route planner for neutral material flow agricultural operations[END_REF]- [START_REF] Cao | A tractor formation coverage path planning method based on rotating calipers and probabilistic roadmaps algorithm[END_REF].

After determining the parallel lines, the generated paths often simply links them sequentially by half-turns. To determine an optimal sequence, some authors applied a genetic algorithm [START_REF] Hameed | Intelligent coverage path planning for agricultural robots and autonomous machines on threedimensional terrain[END_REF]. Zhou et al. [START_REF] Zhou | Route planning for capacitated agricultural machines based on ant colony algorithms[END_REF] applied the ant colony optimization. Jeon et al. [START_REF] Jeon | Design and validation testing of a complete paddy field-coverage path planner for a fully autonomous tillage tractor[END_REF] examined sequential and gathering patterns. In the gathering pattern, the distance between two consecutive trajectories is around half of the width of the field.

All of these methods, however, perform CCP and AVRP separately as two distinct problems. In a previous work, we proposed a one-step CCPP algorithm [START_REF] Arab | Complete coverage path planning for wheeled agricultural robots[END_REF] to cover convex and concave fields entirely, including the headlands, for field operations in which the implement is in contact with the ground when it is engaged. We demonstrated that a one-step CCPP has the potential to find some interesting solutions that are not possible to find with a two-step CCPP. This approach provides optimal solutions with a sequential pattern of adjacent tracks and headland tracks.

However, in some cases, it may be interesting to consider skipping every other track: with some vehicles that have a large turning radius, this method may reduce the size required for headlands or avoid reversing. A human-operated vehicle will prefer to avoid reversing and will find it difficult to skip rows. But for an automated system or robot, this is a viable option that has the potential to reduce the headlands area, which are generally under-performing parts of the land due to the soil compaction.

Based on our previous works [START_REF] Arab | Complete coverage path planning for wheeled agricultural robots[END_REF], in this paper we propose an extension of our one-step CCPP approach. In Section II, we first recall the main principles of our method. Then, we explain how the algorithm takes into account row skipping in the sequence of trajectories. In Section V, we detail our experiment and results, before discussing and concluding them.

II. TREE-BASED INTELLIGENT SEARCH

Our approach, explained in more detail in [START_REF] Arab | Complete coverage path planning for wheeled agricultural robots[END_REF], is performed in three primary steps: 1) preprocessing 2) exploration 3) Selection of optimal solutions. The preprocessing step determines headlands, trajectories within headlands, as well as turning spaces needed for traveling from one headland to another. Considering the accessible edges of the field also known as access segments, entry points are also determined.

The exploration algorithm finds every potential solution, while respecting some predefined constraints, and stores them in a solution space. A solution is a path i.e. a sequence of trajectories that begins at an entrance, covers the field entirely, and ends on one of the access segments. Finally, in the last step, the cost of each solution is computed and the one that has the lowest cost is selected.

In the extended algorithm, the first (preprocessing) step remains identical to our previous method. The exploration step is modified to generate solutions with a skip-row pattern instead of a sequential pattern. In a skip-row pattern two consecutive working trajectories in the main part of the field are not adjacent. As an exception, they may be adjacent when they are located close to a border of the field. This is useful to cover previously skipped tracks. In order to take into account and minimize the number of turns that include reverse moves as a soft constraint, the last step (selection of optimal solutions) is also slightly modified.

Headlands, that are used for performing half-turns, are computed adjacent to the borders of the field. Performing halfturns causes unworked areas in headlands as well as gaps before and after each half-turn due to the distance needed for lowering/raising the implement properly. Therefore, p trajectories (inner trajectories) are considered inside a headland for covering unworked areas caused by half-turns. One trajectory outside and adjacent to each headland is also considered to cover the unworked areas caused by gaps. These trajectories are known as gap covering trajectories.

During the preprocessing step, turning spaces are also computed around each corner of the field borders to ensure feasible turns from one headland to another. For further detail on the preprocessing step, we refer the readers to our previous approach [START_REF] Arab | Complete coverage path planning for wheeled agricultural robots[END_REF].

III. EXPLORATION ALGORITHM

The inputs of the exploration algorithm are the result of the preprocessing, the entry points, a set of hard constraints, γ on , γ of f , ℓ t , ℓ o and a coverage threshold, where γ on and γ of f represent, respectively, the minimum turning radius of the robot while its implement is on (lowered into the soil and activated) and off (raised from the ground). ℓ t represents the distance needed for lowering/raising the implement properly. ℓ o represents the offset between the robot and its implement.

Starting from an entrance, the exploration algorithm progressively constructs and explores a tree where each sequence of connected nodes represent a possible sequence of trajectories. A sequence of trajectories that satisfies the constraints, has a coverage rate greater or equal to the coverage threshold ∆ cov and ends on an access segment, is stored as a solution in the solution space.

Let us recall the hard constraints, defined as follows:

• the robot and its implement must remain inside the field • crossing a previously worked area with a new trajectory, during which the implement is off, is forbidden • overlaps in the main part of the field, i.e. outside the headlands and the gap covering trajectories is forbidden • the total overlap area caused by all trajectories of a sequence must not exceeds a global overlap threshold ∆ global that is a percentage of the field area • to avoid unwanted local loops, the overlap of a trajectory of a sequence with its ancestors in the same sequence must not exceed a percentage of the trajectory's worked area (∆ local ). to let the robot find an access segment and exit the field, once the coverage rate exceeds ∆ cov , this constraint is neglected • the robot must travel at least ∆ min dist while its implement is on to authorize another trajectory for which the implement is not on. The main reason is to avoid lowering the implement to the ground and raising it for short distances After initializing a tree with an entry point as its root, the construction and the exploration of the tree is performed for each unvisited leaf node as a depth first exploration.

As illustrated in Fig. 1, for each leaf node N p a ray r p is constructed based on its location and the direction of the robot at this location. The intersection of ray r p with an inner border of a headland and/or with a turning space defines the next nodes that will be generated and might be added to the tree after being validated by the hard constraints.

In case ray r p intersects with a turning space, trajectories for traveling from one headland to another are generated as detailed in our previous approach [START_REF] Arab | Complete coverage path planning for wheeled agricultural robots[END_REF].

In case ray r p intersects with an inner border of a headland, first a sequence of three trajectories are generated to reach the corresponding inner border. As illustrated in Fig. 1, this sequence contains a gap trajectory of length ℓ t from N p to N c1 for lowering the implement, a working trajectory from N c1 to N c2 and an inverse gap trajectory of length ℓ t + ℓ o from N c2 to N c3 for raising the implement. Afterwards, four -Fig. 1: Trajectory and tree representation of back-and-forth moves. The purple point represents an exit node. For readability purposes, only inner border of two headlands are represented side rays r ( r1), r ( r2), r ( l1), and r ( l2) are defined. They are parallel to r p , with two on each side of r p , and are located at a respective distance of w and 2w from r p . The closest rays r ( r1) and r l1 correspond to the adjacent tracks, while rays r ( r2) and r l2 correspond to a track skipping move. If the intersections of the most distant rays r ( r2) and r l2 with an inner border both exist, two corresponding nodes N c6 and N c7 are generated. Otherwise, nodes N c4 and N c5 are also generated at the intersection with r ( r1) and r l1 . Finally a turn from N c3 to each of these new nodes is generated and they are added to the tree after being validated.

When the intersection of either N c6 or N c7 with an inner border exist, it means that taking an adjacent track is needed to fill all previous skips.

The node generation scheme is repeated for all unvisited leaves of the tree. If an access segment is nearby and the coverage rate is already satisfied, a possible sequence of trajectories to exiting the field is also generated. The sequence of nodes N p , N c1 , N c2 , N c3 and N c9 in Fig. 1 illustrate an example of this case.

IV. SELECTING THE MOST OPTIMAL SOLUTION

Assuming that e entrances are proposed in the preprocessing step, the exploration algorithm is performed for each of the entrances. The results of all explorations are stored in a single solution space. Afterwards, coverage rate S cov , overlap rate S ovl , non-working distance S nwd , operation time S otm and number of turns having reverse moves S rvs are computed for each solution and normalized by [START_REF] Marques | Increasing impacts of land use on biodiversity and carbon sequestration driven by population and economic growth[END_REF].

S =

S -S min S max -S min [START_REF] Marques | Increasing impacts of land use on biodiversity and carbon sequestration driven by population and economic growth[END_REF] where S min and S max represent the minimum and maximum value of the corresponding metric over all solutions of the solution space. For a solution i.e. a sequence of trajectories, the coverage and overlap rates are respectively computed as the sum of worked and overlap area caused by each trajectory. The operation time for each solution is computed by [START_REF] Meftaul | Pesticides in the urban environment: A potential threat that knocks at the door[END_REF].

S otm = L on V on + L of f V of f + L gap V gap (2)
where L on , L on and L gap are respectively the length of all trajectories during which the implement is on, of and in transition (from on to off or vice versa). Accordingly V on , V of f and V gap are the average speed of the robot when its implement is in on, off and in transition. Non-working distance for the corresponding solution is then computed as L of f + L gap .

Considering C = (1 -S cov S ovl S nwd S otm S rvs ) and W = (W cov W ovl W nwd W otm W rvs ) where W cov , W ovl , W otm , W nwd , and W rvs are weights given as input for the corresponding soft constraint, the final cost of each solution is computed by [START_REF] Cariou | Mobile robot trajectory planning under kinematic and dynamic constraints for partial and full field coverage[END_REF].

f = CW ⊺ W cov + W ovl + W otm + W nwd (3)
Finally the solution that has the lowest cost is represented as the most optimal solution found by our approach.

V. RESULTS

To compare the presented approach (Skip-Row pattern CCPP, SR-CCPP for short) against our previous CCPP approach (simple CCPP), six real-world fields were selected. The area of these fields varies from 4.34 to 8.23 hectares (see Fig. 2). Both approaches were implemented and run on an Intel Xeon(R) W-2135 CPU @ 3.70GHz × 12 with 32GB RAM.

To compare the final cost of the most optimal result found by each approach, the solution spaces acquired by their exploration algorithms were combined and the selection method was applied on the combined solution space.

To compare SR-CCPP and simple CCPP, we considered two different value for γ of f ; 2m and 3m, while other parameters remain the same. The number of trajectories within a headland was set to two (p = 2). Other parameters are given in 3 summarizes numerical results of these comparisons and Fig. 4 illustrate the most optimal results for Field #6.

VI. DISCUSSION

Let us recall that for performing turns and half-turns, our approach first tries to employ only forward moves by applying the method proposed by [START_REF] Dubins | On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents[END_REF]. If it is not possible, turns that include reverse moves are also considered. This second type of turns are generated by the method proposed by [START_REF] Reeds | Optimal paths for a car that goes both forwards and backwards[END_REF].

Table 3a summarizes the results obtained by the presented approach (SR-CCPP) and our previous approach (simple CCPP) while γ of f = 2m. According to these results, in general SR-CCPP performed better than simple CCPP for five fields out of six. For Fields #1, #2 and #3, SR-CCPP achieved a better coverage rate. For Fields #1, #2, #3, #5 and #6, SR-CCPP did considerably less overlaps. However, Increasing γ of f to three meters, SR-CCPP performed better for all cases. As summarized in Table 3b, skipping feature of SR-CCPP made it able to use less turns including reverse moves for all fields. It also decreased the overlap rate by almost 1% or even more, for the last four fields. However, skipping tracks also caused more non-working traveled distance.

As can be noticed in Fig. 4, when skipping rows, turns use less space in the headlands. A tight turn will tend to have a bulbous shape that occupies more space, whereas a wider turn may be flatter. This highlights the ability of the row skip pattern to decrease the size required for headlands. Given that a headland is generally a less productive area of the field, this can potentially increase the productivity of the field.

In general, depending on the features of the robot, the field shape and accessibility, the headlands width and the goals of optimization, one method might outperform another. For instance, considering the minimization of non-working traveled distance as the primary goal of the optimization by modifying the weight of constraints, simple CCPP may perform better. As a result, allowing the decision-making algorithm to try each approach, or to combine them into a more sophisticated CCPP, could improve the effectiveness of the final method for finding a good solution for a variety of machinery, operations and field shapes.

VII. CONCLUSION

In this paper, a one-step CCPP method that can provide a coverage path with a skip-row pattern is proposed. The presented approach maximizes the covered area while minimizing overlaps, non-working path length, number of turns containing reverse moves and overall travel time. It covers the headlands automatically while consider the geometry of both the robot and its implement. We compared the proposed approach with our previous approach and we showed that considering both sequential and skip-row pattern increases the possibility of finding the optimum solution. It also increases the ability of a CCPP approach to find a proper solution for a variety of parameters and field shapes.

Fig. 2 :

 2 Fig. 2: Green segments are accessible edges of the field. Entrances are represented by red arrows

Fig. 3 :

 3 Fig. 3: Numerical results. The exploration time is referred to as time for short. Rvs represents the number of turns that include reverse moves and L nw represents the non-working traveled distance

Fig. 4 :

 4 Fig. 4: Most optimal results obtained for Field #6 while γ of f = 2m. The black arrows indicate where the robot enters and exits

Table Parameter Value

 Parameter

			Parameter	Value
	w	3m	∆ global	5%
	γon	15m	∆ local	95%
	Von	3.5m/s	∆ min dist	4m
	Vgap	2.5m/s	Wcov	0.65
	V of f	1.5m/s	W ovl	0.15
	ℓt	2m	W nwd	0.05
	ℓo	2m	Wotm	0.05
	∆cov	96%	Wrvs	0.10

TABLE I :

 I Inputs and parameters

	Field Approach time (s) Coverage Overlap Rvs L nw (m)	f
	#1	SR-CCPP 160.59	98.93%	1.23%	2	1184, 17 0.110
		CCPP	9.73	98.80%	2.27%	4	3005.92	0.281
	#2	SR-CCPP 131.47 CCPP 10.01	98.80% 98.21%	2.05% 3.58%	0 2	1485.36 0.118 1433.84 0.336
	#3	SR-CCPP 282.28 CCPP 18.65	98.96% 98.86%	1.39% 2.15%	0 0	1565.47 0.118 1303.98 0.136
	#4	SR-CCPP CCPP	27.58 7.63	97.52% 98.23%	2.46% 2.11%	0 1	1990.87 1957.83 0.168 0.363
	#5	SR-CCPP 222.69 CCPP 10.99	99.00% 99.05%	0.31% 2.38%	0 2	1249.25 0.074 1402.57 0.180
	#6	SR-CCPP 335.91 CCPP 31.23	98.71% 98.81%	1.20% 3.15%	0 0	1314.63 0.105 1194.58 0.135
				(a) γ of f = 2m
	Field Approach time (s) Coverage Overlap Rvs L nw (m)	f
	#1	SR-CCPP	58.08	98.61%	1.68%	10	2916.47 0.193
		CCPP	14.14	98.42%	0.23%	57	882.89	0.199
	#2	SR-CCPP CCPP	35.92 8.70	98.65% 98.75%	2.01% 3.31%	48 71	1415.98 0.201 1080.41 0.224
	#3	SR-CCPP CCPP	38.48 9.01	98.73% 98.66%	3.39% 4.35%	34 74	1206.01 0.193 1117.74 0.312
	#4	SR-CCPP 132.44 CCPP 23.96	99.08% 97.48%	2.28% 4.24%	10 103 1485.31 2013.52 0.137 0.585
	#5	SR-CCPP CCPP	98.97 10.54	98.99% 99.20%	0.31% 2.09%	4 80	1243.58 0.088 1093.59 0.188
	#6	SR-CCPP CCPP	51.86 13.73	98.56% 98.76%	1.20% 3.12%	48 48	1222.81 0.140 969.11 0.220

(b) γ of f = 3m

This work was funded by research grant CIFRE 2019/1084 from Technology & Strategy Group (T&S) and the French National Association for Research and Technology (ANRT).