Michel Paty

Are quantum systems physical objects with physical properties ?

Despite its power as the conceptual basis for a huge range of physical phenomena in atomic and subatomic physics, quantum mechanics still suffers from a lack of clarity regarding the physical meaning of its fundamental theoretical concepts such as those of quantum state and of quantum theoretical quantities or variables, dealt with by the known mathematical-theoretical rules. These concepts have generally been considered as not giving a direct description of physical systems, for they do not correspond to what is usually understood by "physical states" or "physical properties", notably characterized by definite numerical values such as those obtained from measurement. The situation has been tentatively expressed in terms of various "interpretations", conceived not only with regard to the physical meanings of mathematical quantities, but also in connexion with philosophical statements about "physical reality". The question of whether quantum theoretical quantities describe or not definite physical systems existing in nature has been therefore generally considered as escaping the possibilities of physics, because of the definitions that are commonly taken for "physical state" and for "physical quantity". We sketch the main conceptions on this problem and propose a possible way out of the puzzle, in terms of an extension of the meaning given to the concepts of physical state and physical quantity of a system, which would allow, without any theoretical change in quantum mechanics, to speak consistently of real quantum systems as having definite physical properties.

Contents 1. Introduction. The physical interpretation of quantum mechanics and the statue with feet of clay 2. States, variables and physical meaning. The quantum algorithm 3. The difference between quantum mechanics and the classical theories : mathematical formalism and physical interpretation 4. Measurement process and the classical to quantum barrier 5. The superposition principle and quantum properties 6. Probabilities and the single event description problem 7. Coherent superposition states observed to propagate must be physical 8. Extension of the meaning for quantities : surpassing the merely numerically valued

INTRODUCTION. PHYSICAL INTERPRETATION OF QUANTUM MECHANICS AND THE STATUE WITH FEET OF CLAY

There is no need to recall the power of quantum theory -the theory of quantum physics -in affording knowledge of atomic and subatomic matter, from the constitution of the bodies of our environment as molecular associations of atoms up to the deep structure of matter in general, that of atomic nuclei, of elementary particles actually or «virtually» contained in the latter, and that as well of cosmic objects and of the Universe considered in its «elementary stage», providing the germs of its future states, as manifested in the primordial phases of cosmology. Clearly, this powerful theory yet remains incomplete or unachieved with regard to such an unsually ambitious project that notwithstanding is generally considered legitimate and within the reach of human thought. Further endeavours in this direction will anyhow be based on present quantum theory as a well established body of knowledge, and we shall stay, in the following, within the scope of its present characteristics taken for granted.

There is however a paradox in the situation of quantum theory, that is related with the so-called «foundational problems of quantum mechanics». The paradox is that this theory, so powerful in describing matter, is insecure about the exact meaning of its statements : its description of «elements of the material world» or «physical states» (atoms, elementary particles, interaction fields, etc.) makes use of «state functions» and «magnitudes», more specifically «state functions of quantum systems» and «quantum magnitudes», that have very precise mathematical expressions and are submitted to definite rules of utilization as dynamical variables, but whose physical meaning is far from being direct and remains today a disputed question among physicists.

It is true that quantum theory is by now much wider than quantum mechanics in the restricted sense, for it not only includes further theoretical models for atomic and nuclear properties, but has been extended, from a more fundamental point of view, into quantum field theory, from quantum electrodynamics (QED) 1 to electroweak and chromodynamic gauge field theories 2 . In the quantum-field theoretical sense the concept of «state function» is not as simple as in the genuine quantum mechanics, where it is a solution of Schrödinger or Dirac equations expressing the laws of the physical system under consideration. Instead of being merely, as in the latter, a function in the usual sense, taking numerical values with the variables, a state function in quantum field theory is an operator, like other quantum magnitudes in general. The problem of its physical meaning or content in relation to a physical system or phenomenon is therefore essentially the same as that of quantum mechanics since its beginnings.

1 On quantum electrodynamics, see for instance, Schwinger [1958] (technical fundamental papers), [START_REF] Feynman | QED. The Strange Theory of Light and Matter[END_REF] (popular exposition) ; for its history, see [START_REF] Schweber | QED and the Men who Made it[END_REF]. See also Leite Lopes & Escoubès [1995]. 2 See, for example, Leite [START_REF] Lopes | Gauge Field Theories. An Introduction[END_REF], [START_REF] Bimbot | Vingt cinq années d'évolution de la physique nucléaire et des particules[END_REF].

One may say, in this respect, that notwithstanding the sophisticated further theoretical elaborations of quantum physics, quantum mechanics continues to be the basic frame of thought for the exploration of all kinds of quantum phenomena. The success obtained has, in effect, confirmed its heuristic power and considerably enlarged its basis.

The paradox we alluded to can now be formulated in the following terms : quantum physics can describe and predict an incredibly large amount and variety of properties that are ultimately referred to «objects of the world», and at the same time it is not legitimated in any claim for its theoretical conceptual entities (state functions and magnitudes of physical systems) to fully represent or describe something that could be consistently called «physical objects». For physical objects must have definite physical properties that can be ascribed to them unambiguously, and this is not the case for quantum systems whose properties are, in general, only contextual, depending on the type of observation one decides to perform about them, and also, as it is commonly said, on they being observed or not.

It has been a general opinion up to now that quantum physics as it stands has not provided a satisfactory way out of such a paradox, that would make of it a giant statue with feet of clay («un colosse aux pieds d'argile»). For some people, this was an indication for the need of another theoretical description, alternative to standard quantum theory, in which physical states would obtain a direct theoretical expression and the relevant dynamical variables, or theoretical magnitudes, would directly correspond to physical properties. For others, the demand for quantum states and magnitudes to describe definite physical systems actually existing (or «real» systems) escapes the possibilities of physics, this being an evidence for new conceptions about the nature of human knowledge. Both positions leave us fundamentally unsatisfied regarding the aim and nature of physics, and we are pushed to inquire more thoroughly the meaning of such basic concepts of quantum theory as state function and quantum magnitude with regard to physical phenomena and systems.

We shall see that among the various possible readings of these notions according to the various interpretations there is one which might well deserve to be favoured from the point of view of intelligibility, supported by familiarization with and practice of the «quantum (theoretical) tool» as well as by recent experimental developments on the production and analysis of simple specific quantum phenomena. Besides hypothetical alternative reformulations and «orthodox» interpretations of quantum theory, there seems to be room for a direct and consistent physical interpretation in objective terms (those of a theory that describes physical objects having physical properties), provided one extends the physical meaning of concepts such as state function and magnitude from mere numerically valued quantities to more complex mathematical forms.

STATES, MAGNITUDES AND PHYSICAL MEANING. THE QUANTUM ALGORITHM

What kind of a theory quantum mechanics is, and what it is aimed at, in descriptive terms, this story is well-known and we shall only remind of its main lines insofar as they let us see how it happened that states and magnitudes have become problematic when considered for the quantum domain, although they seem deeply rooted in «what is naturally given to be treated by physics» (if we commit ourselves to use this «puritan» periphrasis, in order to avoid speaking crudely of «the physical reality»). The quantum domain in physics has revealed itself step by step since the very beginning of XXth century on an essentially «phenomenological» basis, before being expressed theoretically by quantum mechanics.

Phenomena having their source in the radiative and atomic properties of matter could not be accounted for by the then available classical physical theories (electromagnetism, thermodynamics and statistical mechanics) and needed the elaboration of new conceptualized properties that somewhat distorted the usual ones. Such were the discontinuity in energy of radiation and of atoms (expressed by means of the fundamental Planck's constant 3 ), as manifested in the black-body radiation law, in the photoelectric effect, in specific heats and in the atomic-level structure ; in the wave-corpuscle dual property of light radiation and of matter elements 4 , showing up particularly in interference phenomena (experiments with «double slit» diaphragms, crystal diffraction of light, of X-rays, later on of electrons and many other quantum particles, including atoms). Indistinguishability of identical quantum particles (or «systems») and their «quantum statistical» behaviour («Bose-Einstein» for integer-spin particles and «Fermi-Dirac» for half-integer ones) are also remarkable non-classical features of the description of phenomena : the first appeared to be the deep reason of Planck's radiation law and the second explained Pauli exclusion principle, responsible of the inner electronic structure of atoms and the periodical classification of chemical elements. Must be added to these the probabilistic character of intensity distributions in radioactive decreases and in atomic transition processes, which turned out to be much more fundamental than one had imagined first, and as an originary property.

Though formulated in a somewhat theoretical fashion, these statements on proper characteristics of quantum phenomena and systems were the most direct expressions possible for factual situations met with in this field. As Einstein -one of the most perspicace pionneers in establishing quantum physicsused to say, these stated «factual» or «empirically given» properties were compelling in that they gave evidence for a radical break with the previous, classical, theories of matter and radiation, and were at the same time both imperative and unavoidable conditions that any future theory of the phenomena would have to integrate. This integration would have been performed, it was generally hoped, when one could refer to some fundamental physical principles structuring a theory that would be intelligible in terms of description of physical systems and of the properties attached to them, as it had always been the case in 3 E.g. in the Planck-Einstein relation between energy (E) and frequency () for atomic energy exchanges and for radiation : ; with Planck constant ergs.sec.

4

To the preceding relation another one is added between wave length and momentum : .

physics.

Quantum mechanics is the physical theory that has been elaborated from those «conceptualized factual» features as a «closed», self-consistent theoretical structure. It revealed itself as predictive, leading to new statements about phenomena that were verified. Such were, among others, the Heisenberg relations between the widths of spectral distributions of «conjugate» quantities 5 , the explanation of the tunnel effect for alpha particles in a potential well due to the probability interpretation of the «wave» or «state function», the calculation of the levels of the hydrogen atom, etc.

Taken in its elaborated formulation 6 , quantum mechanics is based, as all other physical-mathematical theories, on a lagrangian-hamiltonian formalism yielding an equation for the dynamical variables that is linear and whose solutions determine the possible states of a physical system (for instance, the electronic levels of the hydrogen atom). But, except for this (essential) feature, quantum mechanics differs from all physical theories previously known with respect to the correspondence that relates the theoretical magnitudes to the physical properties of the system. The state of a system is represented by a «state function» or «state vector» 7 , defined in a (mathematical) Hilbert space 8 , solution of the dynamical state equation 9 . The state function obeys the «superposition principle» : any linear superposition of the eigenfunctions is also a possible solution of the equation. As to the magnitudes or dynamical variables attached to the system, they are represented mathematically by linear (hermitian) operators acting on these state functions or vectors. The value of a given magnitude (A) for a state considered (for example, momentum, or spin, etc.) is the eigenvalue (a, a number) of its operator (A) corresponding to the eigenvector () representing the state (i.e. A = a ).

Rules for the physical interpretation of those mathematically defined magnitudes are also part of the quantum algorithm. They were not formulated as correspondences of mathematical quantities with physical ones as in the previous theories of physics, but with observed physical ones, notably through measurement processes. The square of the modulus of the state function gives the probability of the system to be (or rather «to be found») in the corresponding state, which means therefore that the state function itself has the meaning of an «amplitude of probability», an unusual and «purely abstract» entity.

5

For example, between position (x) and impulsion or momentum (p) : (with ).

6

After the first elaborations of «wave mechanics» by Erwin Schrödinger (Schrödinger [1926]) and of «matrix mechanics» or quantum mechanics in the strict sense by Max Born, Werner Heisenberg, Pascual Jordan and Paul A. M. Dirac, which were proved to be equivalent as for their results (for historical descriptions, see [START_REF] Jammer | The Conceptual Development of Quantum Mechanics[END_REF]Jammer [ , 1974]], Mehra & Rechenberg [1982[START_REF]On the problem of hidden variables in quantum mechanics[END_REF]), quantum mechanics was formulated in a more rigorous mathematical way notably by Paul Dirac, John von Neumann, David Hilbert, Hermann Weyl. See Dirac [1930], von [START_REF] Neumann | Mathematische Grundlagen der Quantenmechanik[END_REF], Weyl [1932] and [START_REF] Pauli | General Principles of Quantum Mechanics[END_REF].

7

It is often called «wave function». «State function» (or «state vector») is a more adequate expression for the representation of a system that is not reduced to a mere wave.

8

The Hilbert space is built up by functions  whose squared modulus ( ) is integrable. 9 I.e. Schrödinger's equation in non relativistic quantum mechanics and Dirac's equation in the relativistic case.

The measurement of a quantum magnitude related with a physical system through an appropriate experimental arrangement yields statistically the various eigenvalues corresponding to the system's state function, each with the probability for this state. Such a formulation avoids (and even rejects) the association of a definite physical quantity to a given system, except for quantities obeying a superselection condition (corresponding to one unique value determination) : a quantum system can only have a probability for a given state, and a given quantity (though numerically valued) is not a property of a system but only the possible result of a measurement with the appropriate probability. Quantum magnitudes are therefore not thought as existing by themselves, but as conditional properties, depending of the observational arrangement : for that reason they have most usually been referred to as «observables».

This way of formulating a physical theory was new in physics when it was proposed as a solution of the difficulty to interprete directly the «wave» function and the other quantum theoretical quantities (or variables) in terms of physical entities, a difficulty that had appeared in Shrödinger's «wave equation» and Heisenberg's «transition matrices». The dispersion in time of a wave along its path forbade to consider the  state function of wave mechanics, describing a physical system having also particle properties, to be the amplitude of a wave (or even the amplitude for a wave packet) 10 . The same consideration applies to a system whose state function has the form of a linear superposition of «eigen» or basis states : only one of these basic states or components of the superposition (statistically distributed) would correspond to the result of an experiment performed according to an appropriate preparation for a set of compatible magnitudes, and be «physical» at the same time. The  function was therefore to be undertsood as a «probability amplitude», an abstract concept, not considered as truly physical. As for the operator-magnitudes of matrix or quantum mechanics, they had been explicitely introduced to get amplitude transitions between atomic energy levels without worrying about unobervable (in principle) trajectories 11 .

Other features of the quantum algorithm complete those already mentioned and we add a few of them. Only a (complete) set of «compatible» magnitudes (whose operators commute among themselves) can give the complete represention of the states of the system. Incompatible (or conjugate) sets of quantities (represented by non commuting operators) lead to the Heisenberg type inequalities between the corresponding classical quantities (the so-called «uncertainty» or «indeterminacy» relations), and give rise to independent («complementary») descriptions of the system, through independent projection bases for the state function. This property of quantum theory is sometimes referred to as «complementarity», but this word is more adequate when used to qualify the classical corresponding representations, as we shall see, and the word «complementarity» characterizes more adequately the philosophy of knowledge elaborated by Niels Bohr.

To sum up the status of the standard physical interpretation, wave mechanics proposed a wave equation without real waves, whereas quantum 10 See Schrödinger [1926] and also various contributions in Bitbol & Darrigol [1993].

11 [START_REF] Born | La mécanique des quanta[END_REF]. See also [START_REF] Heisenberg | Der Teil und der Ganze[END_REF], chapter 6. mechanics permitted to calculate atomic transitions without a spatial representation of these transitions (trajectories), and therefore the object of the quantum theoretical description was, so to speak, evasive (unless, but we shall come to it later on, one could imagine a kind of physical object different from the classical ones of wave or particle). In the usual conception of quantum mechanics, the wave function or the state function used as the description of a physical quantum state cannot be regarded as a direct representation of the state, and its relationship with the physical state is only formally analogous to that of a wave amplitude with its wave : a similar equation holds for both, but they possess different physical meanings, namely a «direct» meaning for the second (classical) one, and a quite indirect meaning for the first (quantum) one. This interpretation problem stands today as seventy years ago. Nevertheless the generalized use of quantum concepts through quantum theory has led since that time to some physical and conceptual clarifications that might help not to solve the problem given in these (restrictive) terms, but to formulate it in a different way and thus perhaps avoid and resolve the difficulty.

THE DIFFERENCE BETWEEN QUANTUM MECHANICS AND THE CLASSICAL THEORIES : MATHEMATICAL FORMALISM AND PHYSICAL INTERPRETATION

The difference between quantum mechanics and the previous physical theories has been frequently referred to stressing the particularly abstract and highly mathematical concepts of the first one -as if the concept of the second were more «concrete» or direct. Niels Bohr used to say that the mathematics used in classical physics reduced to real numbers whereas quantum physics makes use in an essential way of the pure imaginary number i = , that prominently figures in Schrödinger's equations and, above all, in the «uncertainty» Heisenberg's relations between conjugate quantities, as if i so to speak was the mathematical counterpart of «complementarity»12 . Actually, it does not seem that the high degree of mathematical abstraction by itself makes the difference : when new concepts entered physics, it used to be in a mathematical form whose «direct» character looked often problematic in first inspection, needing interpretation and starting philosophical debates (consider for instance continuous quantities as «simple» as space and time coordinates when they took the form of differentials, from the end of XVIIth century up to mid-XVIIIth).

The difference between the classical and the quantum domain of physics consists not so much in the nature of the mathematics used than in the physical meaning of their use : mathematical concepts and relations in classical physics referred «directly» (whatever be their degree of abstraction) to entities having a physical existence and physical properties, whereas in the standard presentation of quantum mechanics they are referring only indirectly to physics. One might propose that the physico-mathematical formalism had a useful (and even powerful) drift effect on quantum theory, but a high price had to be paid for it : it seemed necessary to give up the hope to get from quantum theory a direct intelligibility of the quantum physical phenomena and of the physical systems responsible of these. Intelligibility, it was said, was provided by the mathematical scheme plus rules of physical interpretation, and would not deal with physical existing quantities but with observational data and ultimately with observed quantities and measurement processes.

At this stage, one may wonder whether one cannot dissolve this difference. There is, indeed, a general feeling among physicists that one should have a unique way to think in physics. The passage from the classical to the quantum, and vice-versa, considered in physics as well as in its different modes of representation, have remained from the beginning the most basic concern for the interpretation problems in quantum mechanics. Underlying this difference -that is effective at the physical conceptual level -, is a questioning about the aim of physics and the nature of its object. Such questions have more than a flavour of epistemology and philosophy, and whether one accept it or not, the «new deal» for physical knowledge was actually left to philosophical choices. In the debate about the interpretation of quantum theory, positions on determinism, subjectivism, observationalism, empiricism, realism have continuously interfered with the discussions on conceptual and theoretical problems.

Those who accept this situation, according to Bohr's line and with some differences from Heisenberg and Born to Dirac, admit that the traditional conception of physical knowledge (originated in classical physics) based on realism, objectivity and determinism had to be abandoned and to be replaced by a new one. In the new conception one must admit that the properties of a physical system cannot be considered independently of the conditions of its observation, which in turn necessarily brings us ultimately to macroscopic observation and measurement devices. For these are adapted to our senses, and relative to quantities of classical physics, that are supposed to be more natural than any other one.

From this point of view, the traditional question concerning the nature of the system described by a state function is turned round, and importance is assigned, theoretically and conceptually, not to an indescribable hypothetical physical system as considered for itself but to the very description and its operational character. The state function would definitely not represent a system but rather the knowledge that one is able to get about it through interaction with the system or, in other words, the catalogue of its observed properties. (But note at the same time how much difficult it is to avoid using the word system, that is : object). This operational approach of the meaning of quantum mechanics and of its state function puts measurement, instead of physical existing (or «real») systems, into the center of the theoretical representation.

Those who, on the contrary, do not accept this «interpretation» situation, replace the «subjective» conception by an objective one according to which the quantum theoretical representation refers to physical objets manifesting themselves in the phenomena and to which one should attribute properties. Their objection against quantum mechanics is both epistemological (as directed against the standard interpretation) and theoretical (quantum mechanics, in their view, does not fully describe its legitimate objects ans should be completed in that sense). To strenghten their position, they would have to find a more satisfactory quantum theory : this is a more deterministic for those (who invoke following Louis de Broglie, David Bohm, Jean-Pierre Vigier and others, supplementary «hidden» dynamical variables 13 ), or a more realistic one for others (who claim, with Einstein, the need for a representation of real physical states having properties of their own, deeper than the mechanistic concepts on which quantum mechanics was built in his opinion 14 ). But none has so far been able to overcome the limitations of quantum theory as it stands.

Quantum physics seems therefore unable by itself to solve the problem of the physical meaning of its own theoretical quantities. One would have to invent and accept a radically new view about knowledge in general to give full legitimity to the powerful quantum mechanical description. Such was the dogma stated by the «orthodox» Copenhagen interpretation, and one may think that it was a philosophical response to a physical problem that stopped too early the search for solutions (such was Einstein's belief). We shall not enter this philosophical debate here, and try to restrict ourselves to the consideration of the physical meaning of quantum theoretical quantities, calling on as few external considerations as possible.

Generally speaking, the various positions in the quantum debate can be displayed as possible responses to the question of the price which must be paid if one wants to maintain the intelligibility of quantum phenomena. It will depend on what one is ready to give up and what not. Taking quantum mechanics as it stands as a satisfactory theory 15 , an alternative to the Copenhagen philosophical response would be to ask : at which cost is it possible to maintain the requirement of a direct theoretical description of physical systems (a claim usually called «realism») ? It would obviously involve a change in our understanding of the physical concepts relative to a «direct theoretical description». We shall include such a possibility among the existing positions of the interpretation problem, although it has seldom been considered explicitly. It is, altogether, often implicitly admitted by today physicists even when they pay lip service to the complementarity interpretation. To pass to the explicit formulation would require to give a new precise conceptual definition of a physical magnitude and of a quantum state. These are generally conceived, by subjective as well as by objective approaches, as identical to quantities given by measurement.

MEASUREMENT PROCESS AND THE CLASSICAL TO QUANTUM BARRIER 13

Louis de Broglie proposed in 1926 to consider, with his «double solution theory» a non linear state equation. His simplified «pilote wave» theory has been independently rediscovered and perfectioned in 1952 by David Bohm in a way that is not contradictory with the results of standard quantum theory, as it includes non locality. See [START_REF] Bohm | Wholeness and the implicate order[END_REF], Bell [1987].

14 [START_REF] Einstein | Quantenmechanik und Wirklichleit[END_REF][START_REF] Einstein | Quantenmechanik und Wirklichleit[END_REF][START_REF] Einstein | Quantenmechanik und Wirklichleit[END_REF]. See Paty [1995Paty [ , 1999]].

15

For nuances, see what we wrote in beginning.

The problem of measurement in quantum mechanics has generally been considered as determining all the other aspects of the «interpretation». This problem is at the same time that of the relationship between the quantum phenomena and the macroscopic and classical ones, and in particular it has to do with the transfer of informations or of structuration from one domain to the other. This relationship and this transfer go in both ways, from the classical to the quantum, if we consider that observation with macroscopic instruments yields informations in classical terms that are integrated into the knowledge of the quantum systems, and from the quantum to the classical, if we prefer to consider in physics structuration aspects prior to information ones and admit that classical systems and properties are derived in principle from the properties of their quantum constituents. One can indeed consider the «measurement problem» from either points of view.

First, from the classical to the quantum knowledge, a proper conception or theory of quantum measurement would constitute a basis for a sound quantum theory. The strict Copenhagen conception proposes measurement by classical systems as reference and thus does not need a theory of measurement. Measurement is the scheme from which knowledge is acquired, and quantum properties need to be defined and determined through classical concepts, i.e. those of responses from macroscopic apparatus (taking into account the specific quantum-mechanical features).

As it was formulated in the set of quantum theoretical definitions and rules, the measurement process of quantum systems by classical aparatuses revealed a difficulty with causality. The state equation is causal with respect to the state function, for it describes the evolution in time of the latter when we consider it «in itself», independently of any act of observation. But, according to the quantum rules, when the system is submitted to an observation or measurement processs, its state function reduces from the coherent quantum superposition into one of its components, and this appears to be the case in actual experiments. This reduction seems to be at random, for when the same experiment is performed a number of times under the same conditions on the same system, one recognizes that reduction has occured to all the possible states, with a statistical law corresponding to the probabilities of the states.

In the orthodox interpretation, as we said, this was not a problem, but a mere question of definition : indeed, each observation or measurement defines anew the state of the system. There is no need for a theory of measurement, because measurement participates in the definition of physical quantum systems, once one admits that there is definitely -as derived from quantum phenomena -, no clear separation between the object and the subject. For Niels Bohr there was a barrier between the quantum and the classical domain, that was due to measurement, made necessary to know from perception. It resulted from there, in his view, that the knowledge of quantum phenomena cannot grasp these directly as they are, but has always to refer to classical descriptions.

A variant of the «subjective» view admits considering measurement processes as an interaction between quantum system (whatever it be) and the classical apparatus (see, for example, von Neumann's theory of measurement, and the theory of Daneri, Loinger and Prosperi 16 ).

«Reduction» was a problem to those who were not content with the «subjective» conception of the state function and wished a description of the physical system itself : such an instantaneous projection or «collapse» to one of the possible states was difficult to admit because, precisely, it would be acausal. It has been the purpose of the attempts at an «objective quantum theory of measurement» to give an account in purely causal terms of the transition. Objective conceptions require also in general a description of the interaction between the quantum system and quantum parts of the macrosccopic measurement device, yielding finally the observed «one basis state reductions» associated with the appropriate probabilities as frequencies of events 17 . Some objective conceptions consider «no-reduction processes», where the observed reduction of the state function to one of its components on the prepared basis is not due to a physical interaction, but this component is just choosen. The reason or the modality of the choice differs with the various proposed theories : it may be a mere perspective effect, like in Hugh III Everett's «relative state» theory of measurement, or in David Bohm's causal and non-local theory with hidden parameters 18 .

Note that today physicists working on the developments of dynamical quantum theories consider the second point of view, i.e. of the passage from quantum representation to classical one, to be more fundamental than the first, in agreement with the most current theoretical problem for physics, namely the unifying programme for the elementary theory of matter. We shall stop here with noticing the following : even without disposing of a quantum theory of measurement, the practical interpretation rule of the quantum algorithm that connects observed classical quantities with quantum ones is sufficient to yield, at the present stage, a representation of physical quantum systems, whatever be their «reality», on which one can rely and formulate such a programme.

If this is the case, one would admit implicitly that a «quantum theory of measurement with the help of classical instruments» is not necessarily a requisit for physics to progress further. One would assume, at least provisionally, that the present quantum theory would not have to be changed if the representation of quantum states was a direct one and the quantum entities would describe objects existing in a quantum world of an indirect access. This is not to deny the persistence of the problem of «quantum measurement», but to relativize it and to put it more into the epistemological interpretation side than into the theoretical physical description. Viewed from both sides, the passage from one domain or representation to the other is a reduction, but of a different kind. Reduction of (classically described) macroscopic matter to its (quantum) elementary constituents for the second, reduction of the specific quantum features to classical (complementary) description for the first.

The conceptions or theories of quantum measurement, the subjective 16

Von [START_REF] Neumann | Mathematische Grundlagen der Quantenmechanik[END_REF], [START_REF] Daneri | Quantum theory of measurement and ergodiicity conditions[END_REF]. 17

The litterature on this subject is enormous. A rather representative sample is given in Wheeler & Zurek [1983].

18

The original papers (published, respectively, in 1957 and 1952), have been reprinted in Wheeler & Zurek [1983] and [START_REF] Bohm | Wholeness and the implicate order[END_REF]. See also Bell [1987].

as the objective ones, have in common to admit as physical state only those «basis states» that are given as results of measuring quantities. These quantities have been chosen at a «preparation» stage, defined by the type of experimental arrangement, and these data obtained are, by definition, numerical values. Take, for example, a particle of spin J = , for which an apparatus is designed to measure the z-component of the spin. (This choice, allowed by the compatibility of both J and quantities, i. e. the commutation of the corresponding operators, precludes any simultaneous determination of the other components, since and or do not commute and are therefore incompatible quantities).

A measurement of the component of the spin with the help of a Stern-Gerlach magnetic apparatus will yield either , corresponding to a spin up state, or , corresponding to a spin down state. If the beam of particles is polarized with spin up (and conversely with spin down), the measurement of it will yield as only result (conv. ). If the beam is not polarized, measurement of will give with the 50 % probability the value and with 50 % probability the value . The state function before detection is a superposition , but the measurement process reduces (or projects) it only on one of the states of the superposition : only the latter states are generally considered physical. The physical state before measurement is supposed to be unknown, determined only at the moment of measurement. For some (like Heisenberg), this determination occurs through the interaction of the superposed state with the measurement device, and there is no other way of speaking of a physical state. A system can be spoken of and described only when it is measured and not before.

On may ask how the connection between measurement and knowledge actually works. We have first to specify the physical meaning of the  state function. Within the theory, it represents the system under study. The state function is given a determined precise form and content from experiments measuring the relevant quantities (A) : from the numerical values (corresponding to the solutions of the eigenvalue equation (A = a ), associated with given frequencies (= ) (in statistcal experiments), one gets the weighted (vectorial) sum of the corresponding states ( ) 19 , that describes the state in the theory (object state of the theory). Note that this state is an invariant in the following sense : it remains the same for another choice of basis states (defined by another set of commuting quantities) ; it is an invariant in the vectorial sense (in Hilbert space).

From the point of view of quantum theory dealing with quantum systems, this invariant or basis independent character is important in that it liberates the state from its contextual determinations. Some tension or duality of concern between the measurement approach and the theoretical one can be diagnosed here, for one might ponder about the meaning of the reconstituted invariant state function in theory construction if it does not correspond to a physical state. Present quantum physics (of fundamental interactions) seems (fortunately) to have forgotten that, according to the standard interpretation, its favorite conceptual and theoretical tool has no meaning as a physical entity, but only its contexual projections have. One is therefore tempted to think that the standard interpretation of quantum mechanics has been restricted by limiting itself 19 With an adequate normalisation factor.

to the supposedly unavoidable transition from the classical to the quantum domains, when the fundamental issue is on the quantum level itself where theoretical physics is actually operating with a Hilbert vector entity, associating it with a theoretical physical content.

Clearly, the reconstitution of the state function from measurement is an intellectual procedure that is no more situated at the level of perception that had been claimed as refererential to measurement processes. For theoretical knowledge does not stop at the perceptive level but requires understanding, in order to formulate relations of concepts. In other words, our knowledge is not (and has never been, even in classical physics) restricted to our observations and measurements, but requires mental, symbolic, theoretical constructions. Now we turn, for some time, away from the measurement problem, and inquire further the physical meaning of the state function.

THE SUPERPOSITION PRINCIPLE AND QUANTUM PROPERTIES

When physicists speak today of a (quantum) «elementary particle» (for example, a proton, or a quark), they implicitely mean that it is described by «quantum numbers» that are «eigenvalues» of the operators representing the adequate physical quantities. But this representation is at the same time fundamentally based on the superposition principle, which is generally given in the epistemological debate on quantum physics as a purely mathematical and formal property. It is therefore of a primary importance to understand what is the relation of this «formal» principle to physical phenomena and systems : whether it is purely formal and indirect or direct (albeit mathematical), that it to say entailing physical properties which can only be related to it. Considering a physical system and its description by the state function, one charges, so to speak, the principle of superposition for the responsibility to make impossible any «direct physical interpretation» of the state function in «classical terms» ; but one gives to it as well, from the point of view of the mathematical formalization, the credit of being responsible for all the specific characteristics of quantum systems. It is the most economical mathematical expression of these characteristics, a trait which, in theoretical physics, is generally attributed to a physical principle. The quantum debate nowadays cannot avoid any more to consider this problem, as many recent physical results have reinforced the physical predictive power of the superposition principle. We shall mention some of them, although rather briefly, as illustrations of the possibly more and more physical character of the state function 20 .

Let us begin by noting, as all quantum concepts are parts of an imbricated whole, that the difficulty to consider a (quantum) physical «particle» state as a linear superposition of its various possible (eigen or basis) states was 20 A more detailed analysis is proposed in a forthcoming publication (Paty [in press, c]). mainly due to the connection of this property with non-locality. It is now admitted as a fact that non-locality is inherent to quantum systems (see further). Other direct consequences of the superposition principle have been shown to be fully physical, notwithstanding their initially apparently purely formal character inside quantum theory. I mean by physical the correspondence with existing reported phenomena at the quantum level, and which can eventually manifest themselves also at the macroscopic level.

Indistinguishability of identical quantum systems is such a property. The state function of a system constituted by identical quantum particles is constructed from those of the constituents ( and ) through the superposition principle so as to get the quantum statistical behaviour : it must be symmetrical with respect to permutation for bosons ( = ) and antisymmetrical ( = -) for fermions. These «formal conditions», imposed by strong albeit limited original data (as recalled above : Plancks radiation law, electronic occupations of atomic levels or Pauli's exclusion principle), entail enlarged powerful physical predictions, as consequences of the mathematical form of the state functions.

For a system of identical bosons, there is a possibility to accumulate an arbitrary number of them in the same state. Physical phenomena theoretically predicted as consequences of this simple or «economic» theoretical feature have all been observed, and so are optical pumping and laser effect, supraconductivity (trough Cooper's electron pairs), superfluidity and, even more striking, Bose-Einstein condensation. There, tenths of thousands atoms of a given metal are accumulated in the «zero point energy» state, very close to absolute zero temperature, as has been recently observed (up to a macroscopic scale, the condensate instantaneously occupying all the space in its reach and climbing on the vessel's walls) 21 . For a system of identical fermions, the fact that a given state can be occupied by just one system at most has consequences not only at the atomic level (hence governing nothing less than all the organization of matter) but also at a highly macroscopic one as well, such as the structure of degenerate stars : white dwarfs and neutrons stars maintain their equilibrium between the gravitational tendency to collapse and the degeneracy pressure of electrons in the first case and of neutrons in the second (resulting from the impossibility for them to fall in the same state).

It is actually tempting to see in such a theoretical feature as indistinguishability of identical quantum particles, which enables us to predict and understand such rich and uncommon physical phenomena that are otherwise unconceivable, the mark of a theoretical description of «objects» by the state function which is more rigid than usual. This might look paradoxical for a description that has been, on the contrary, considered for a long time as lacking knowledge (beyond visualizability : remind the claims of subjectivism, of indeterminism, and of uncompleteness as well). In the quantum domain, the state function of a physical system covers all the properties that can be attributed to that system.

Inverting the usual way of thinking, we might say that, in this sense, quantum mechanics provides more knowledge about the described physical systems than classical physical theories used to do : in the latter distinguishability of identical particles or systems might be seen as nothing more than the expression of our ignorance about them (what allows to distinguish them is not given by the theory). Quantum theory provides in this respect a complete equivalence between the theoretical tool and the physical object, as there is contained no more in the object of description than in its theoretical description (on the contrary, classical physics takes the objects it describes as empirically given, be they a falling stone or the Moon). This may seem a paradox for the common sense, but the understanding of phenomena requires it…(so does it seem to me).

Non separability, or to be more precise, non-local separability, is another characteristics of quantum systems which is directly connected with the theoretical properties of the state function and which has been shown to correspond to a factual property. Quantum (sub-)systems that are initially correlated in one state and are thenafter dissociated maintain the same correlation in future independently of their space separation. This trait of quantum theory had been pointed out by Einstein, in his attempts to fix the description of quantum systems by their state function independently of measurement («EPR argument») 22 . Einstein felt the necessity to invoke a principle of separability guaranteeing the dynamical independence of spatially distant subsystems, knowing nevertheless that such a principle did not exist in quantum theory. He showed that, requiring this principle, quantum mechanics could not describe individual subsystems but only statistical ensembles 23 . Schrödinger spoke, for his part, of an «entanglement» of quantum systems. Bohr objected that one cannot consider the systems independently of their measurement conditions and therefore the measurement of one of the EPR subsystems is also a measurement of the other. The correlation of the systems was in principle indissociable from that of the measurement apparatus : there was no point in speaking of the systems by themselves 24 .

An important step was accomplished by Bell's theorem on locality that demonstrated an incompatibility between quantum mechanics and «local hidden variables», extended afterward in a more general fashion to any kind of «locality», corresponding to Einstein's «principle of separability» 25 . This incompatibility was tested later on by correlation experiments on distant systems that concluded in favour of quantum mechanics 26 . Hence non-local separability of quantum systems was establishedas a physical fact. One must first observe, however, that it was present in the theoretical formalism, as an impossibility to factorize the state functions of the subsystems, and to extract one of them from the overall system formed initially (unless one changes the initial conditions, and the state function 22 [START_REF] Einstein | Can quantummechanical description of physical reality be considered complete ?[END_REF].

23

This conclusion comes out particularly clear in [START_REF] Einstein | Quantenmechanik und Wirklichleit[END_REF][START_REF] Einstein | Quantenmechanik und Wirklichleit[END_REF][START_REF] Einstein | Quantenmechanik und Wirklichleit[END_REF]. See Paty [1995 and in press, a].

24 [START_REF] Bohr | Can quantum-mechanical description of physical reality be considered complete ?[END_REF].

25 [START_REF] Bell | On the Einstein-Podolsky-Rosen paradox[END_REF]. This historical paper is reproduced in Bell [1987]. Further results were obtained in particular by [START_REF] Greenberger | Going beyond Bell's theorem[END_REF], [START_REF] Mermin | Quantum Mystery Revisited[END_REF]. For more details, see, for example, d 'Espagnat [1984'Espagnat [ , 1994]], Paty [1986 and in press, c], . of the overall system). Hence its theoretical meaning was dependent on the meaning one attributed to the state function, so that it has been understood in various manners.

The evidence for distant quantum correlations has been often considered as a victory of Bohr's interpretation of quantum mechanics (an operationalist view) over Einstein's «realistic» demand. It was, for sure, a victory of quantum mechanics over a principle of separability external to it. But quantum mechanics is a physical theory and should not be identified with Bohr's interpretation or philosophy. And there is implied, in the correlations between systems, more than there was in Bohr's response, which would not have been able, by itself, to specify the quantum correlations through which non-local separability revealed itself as a property of quantum systems 27 .

From the objectivist and realist point of view, a variety of conclusions have been given about the meaning of the quantum correlation (apart from pure refusal or doubt about the validity of Bell's theorem, or of the results of the correlation experiments). For some people who take localization in space as a requisite for physics, this property of quantum systems is a dynamical one, which reintroduces some kind of instantaneous action at a distance of the newtonian type, one being left with a problem of compatibility with special relativity. In such a case the theoretical description by the state function as it stands would be insufficient.

But there is, from an objectivist or realist point of view, another, simpler and direct, way to live with this property and consider at the same time that the state function represents theoretically a physical system. Admitting nonlocal separability as a basic fact of quantum physics (in the same sense as the relativity principle, for instance, is a basic fact for classical field theories), quantum theory appears, precisely, perfectly adequate to give account of it : there is no need to require any additional property (such as a principle of separability or locality) taken from outside of the theory. Quantum systems being defined by quantum theory, their attributable properties are covered by their state function ; they are naturally extended in space and entangled with the other subsystems conative to them (and special relativity stays outside of this, as quantum systems are not material points in space but extended systems) 28 . Again, the state function is sufficient by itself to represent and describe the fundamental property under consideration.

PROBABILITIES AND THE SINGLE EVENT DESCRIPTION PROBLEM

The state function is an amplitude of probability ; with respect to observed or measured quantities, its modulus squared gives the frequencies of the possible (or eigen-) values for the corresponding eigenstate. In the context of the standard interpretation referring to observation, probability and statistics were usually identified 29 . Actually one can recognize a difference between them, from the point of view of their functions : probability (derived from a «probability amplitude», which is a physical concept, not a mathematical one) has its meaning inside physical theory, while statistics refer to experiments and measurements.

This conceptual difference might be useful to clarify the status of the description of individual systems in quantum physics, a question that has been considered unimportant in the standard interpretation insofar as counting quantum particles was considered requiring a measurement and therefore perturbating the state (and finally one could never ascertain whether one deals with only one particle). For example, interference patterns produced by having quantum particles interfere in a two-slit interferometer are always obtained statistically, although the process is attributed, from the behaviour of the state function, to a self-interference of quantum particles 30 . (Remember that it is this simple quantum phenomena that had led Born to his probabilistic interpretation). The statistical meaning of the state function permitted, in the Bohrian perspective of complementarity, to reconcile the wave and particle aspects of quantum phenomena. If these dual aspects are incompatible for individual (single) systems, they can be assumed for ensemble of systems. One may ask, however, what would happen if there was no need for a complementary description, which would happen if individual phenomena could be considered independently, i.e. without reduction onto classical features. This is indeed possible, since about two decades, thanks to the production of extremely rarefied beams of quantum particles (photons, electrons, neutrons, atoms…) with a high time resolution 31 . One knows that particles have crossed the interferometer one by one, without counting them on their path. Instead of doing one experiment with many (say N) quantum particles, one is now able to perform N independent experiments with a single quantum particle (identical in all experiments). One obtains, at the end, the same interference pattern, with the difference that one is sure that every individual system has equally contributed to form the final pattern. If the impact on the screen appears for each one at random, the accumulated impacts reproduce the statistical distribution law.

Each phenomenon related to an individual system is a quantum phenomenon, collected on the screen through a classical measurement process (the quantum particle impact on a grain of the sentitive matter of the screen). One is left with the usual problem of the quantum measurement process : identical quantum systems provide, after detection, different results, but endowed with probabilities corresponding to the amplitude of probability of their state of superposition.

The observed diffraction phenomenon of single quantum particles, yields also probabilities having a meaning for individual events. On the whole, 29 See, for example, the conceptions shared on this point by both Einstein and Born, the inventor of the «probabilistic» (or equivalently «statistical») interpretation of the state function [START_REF] Einstein | Briefwechsel 1916-1955, Nymphenburger Verlagshandlung[END_REF]). 30 [START_REF] Dirac | The principles of quantum mechanics[END_REF].

this phenomenon provides another evidence for a direct physical meaning of the probability amplitude, that is of the state function.

COHERENT SUPERPOSITION STATES OBSERVED TO PROPAGATE MUST BE PHYSICAL

When physicists consider a physical system as represented by a state function, they do not consider the latter as a pure mathematical entity or a catalogue of informations. They treat it as if it meant really a physical system propagating. They think physically and only afterwards do they eventually interprete in the standard «authorized» way. In particular, they treat as physical a state whose state function is a coherent superposition of basis states related to a given «preparation» (or choice of compatible magnitudes represented by commuting operators). As a whole, this state vector is invariant under a change of the basis corresponding to a preparation with another (incompatible) set of magnitudes. Being a mathematical invariant, this state vector (or state function) is theoretically meaningful, much more than its projected components.

It has been claimed in the standard interpretation, however, that a coherent superposition has no direct physical meaning as it does not correspond to direct possible observation. By definition and construction, direct observation can only detect one of the basis states, from a given numerical value for each relevant magnitude. This doctrine would, therefore, deny the name of physical state for a state of superposition, even if there would be experimental evidence for it, and experimental evidence can also be obtained from indirect observation. Now, such an evidence has been recently reported as «decoherence experiments». A twostates Rydberg atom (superposition quantum state) coupled with a two-values electromagnetic field with very few photons («mesoscopic» state) realizing a kind of «Schrödinger's cat experiment» 32 has been detected as a superposition state in its propagation inside the experimental arrangement, during a small finite time, before its decoupling (or decoherence, within the macroscopical environment of the measurement apparatus 33 ). This experiment can be considered as an evidence for the physical propagation of a coherent superposition state. It give other indications on the «measuremeent process» and «reduction of the state function»decoherence apparently arises from a thermodynamical process, which we shall not discuss 34 .

We content ourselves here with the new fact of a visualization of a state of coherent superposition propagating through space, and note that superposition states observed to propagate must be physical.

8.

32 [START_REF] Schrödinger | Die Gegenwärtige Situation in der Quantenmechanik[END_REF]. 33 [START_REF] Haroche | Experiments with single atoms in a cavity : entanglement, Schrödinger's cats and decoherence[END_REF].

34

See [START_REF] Zurek | Decoherence and the Transition from Quantum to Classical[END_REF], Omnès [1994].

EXTENSION OF THE MEANING FOR QUANTITIES : BEYOND THE MERELY NUMERICALLY VALUED

From the preceding argumentation, the description of a quantum system by the state function of quantum theory looks in many respects as highly powerful for predicting as well as for explaining physical phenomena in the quantum domain. Everything in our understanding would suggest considering this mode of description as a physically direct one : the state function as it stands represents a physical system, and this has dynamical properties given by the quantum magnitudes as expressed by the formalism. The only restriction to that «simple» view is that these theoretical concepts have not the meaning of simple numerically valued quantities as one is used to in «pre-quantum» physics. But we know that the quantum domain of phenomena exceeds the classical one and that the understanding of the former must be different in some way from the latter. In contrast to the proposed view, the standard interpretation of quantum mechanics considered as untouchable the classical understanding of physical concepts that they referred to a primacy of measurement procedures, and preferred to give up the idea of «real physical systems» having properties.

Understanding, as we practice it in quantum physics, cannot be reduced to perception, and physical description not to observation. We may choose another way that would provide the quantum concepts a direct physical meaning, related to the main function of physical quantities which is to express physical relations among them, and consider their connection to observation and measurement as secondary and indirect. For this, we would have simply to extend the usual concepts of physical state and physical quantity of a system, which have been up to now restricted to definite numerical values, so as to include the mathematical forms met with in quantum physics, i.e. state vector superpositions, matrix operators. This means to recognize that the understanding of the quantum domain is given through the quantum theory and that the physical thought has been driven by the formalism of quantum theory in a more powerful way than it was initially thought. It can be stated also that the power of quantum physics comes from its mathematical expression and not from its «interpretation» in terms of knowledge of the natural world in general.

The intuition of some of the founders of quantum theory had led them to think in such a way, up to the point where they considered the connection of the quantum to classical, that they gave way to the then current observationalist conception. Consider Dirac's notion of q-numbers as an extension of usual cnumbers 35 , or his conception of theoretical research in physics, as he formulated it when he was discovering the concept of antiparticle to which the mathematical form of his equation of electron motion was leading : The "most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical 35 See Darrigol [1992].

entities…" 36 .

The conclusion or the moral of our story about the physical direct meaning of the quantum fundamental concepts of state function and quantity is that the clue is contained in the mathematical formalism. But this was not obvious from the start, for it is not enough to dispose of mathematical relationships convenient for the solution of a physical problem to be insured that the elements of these relations can be identified with physical entities. However, one may consider that today quantum phenomena as displayed above justify a simpler and coherent representation in such terms.

Thus a solution of the «unsolved problem» of the physical meaning of quantum concepts and magnitudes seems to emerge, if its consistence is fully confirmed. This solution would be more an example for a historical situation where a problem considered as having no solution in the existing narrow frame of thought can actually get one by simply widening the meaning of its concepts. Such a situation has been quite frequent in the history of mathematics (think only of the various extensions of the concept of number, which were the only way to get rid of the mysteries of the irrational and later on of the imaginary quantities), and also occurred in theoretical and mathematical physics as well.

SeeBohr [1958].

Griffin, Snoke & Stringari [1995],[START_REF] Cornell | The Bose-Einstein condensate[END_REF].

[START_REF] Freedman | Experimental test of localhidden variable theories[END_REF],[START_REF] Aspect | Trois tests expérimentaux des inégalités de Bell par mesure de polarisation de photons[END_REF].

 Paty [1995 and in press, a].

Paty [1986].

See, in particular,[START_REF] Pflegor | Interference of independent photon beam[END_REF],[START_REF] Grangier | Etude expérimentale de propriétés non-classiques de la lumière ; interférences à un seul photon[END_REF].

[START_REF] Dirac | The principles of quantum mechanics[END_REF]. My emphasis (M.P.)


Salwa Baabou
email: baabousalwa@gmail.com

François Bremond
email: francois.bremond@inria.fr

Awatef Ben Fradj
email: benfradj_awatef@yahoo.fr

Mohamed Amine Farah
email: med.farah@yahoo.fr

Abdennaceur Kachouri
email: abdennaceur.kachouri@enis.rnu.tn

Hand-Crafted System for Person Re-Identification:A Comprehensive Review

Keywords:

In video surveillance, Person Re-Identification(Re-ID) consists in recognizing an individual who has already been observed (hence the term Re-Identification) over a network of cameras. Usually, the person Re-Id system is divided into two stages: i)constructing a person's appearance signature by extracting feature representations which should be robust against pose variations, illumination changes and occlusions and ii)Establishing the correspondence/matching between feature representations of probe and gallery by learning similarity metrics or ranking functions. A gallery is a dataset composed of images of people with known IDs whereas a probe is collected of detected persons with unknown IDs from different cameras. Specifically, the process of person Re-Identification aims essentially at matching individuals across non-overlapping cameras at different instants and locations. However, the matching is challenging due to disparities of human bodies and visual ambiguities across different cameras. This paper provides an overview of hand-crafted system for person Re-identification, including features extraction and metric learning as well as their advantages and drawbacks. The performance of some state-ofthe-art person Re-ID methods on the commonly used benchmark datasets is compared and analyzed. It also provides a starting point for researchers who want to conduct novel investigations on this challenging topic.

I. INTRODUCTION

Person Re-Identification aims to match individuals appearing across non-overlapping camera networks. This task has attracted more interest over the recent few years. The goal is to return a list of probabilistic matched images ranked by degree of similarity.

In the literature, to tackle the Person Re-Identification, two major directions are taken into consideration which aim to: Extract robust and invariant feature representations for both probe and gallery images and then learn specialized distance metrics to person matching using this representation. This Kind of approaches is called Appearance-based.

Appearance-based methods can be divided into two categories. For the first category, color and texture based features are widely used [START_REF] Farenzena | Person re-identification by symmetry-driven accumulation of local features[END_REF][START_REF] Liu | Person re-identification: What features are important?[END_REF]. However, these features representations are sensitive to pose and illumination change, which may result in larger intra-person variation (difference between features of the same person) than inter-person variation (difference between features of different persons). Besides, in order to improve the recognition accuracy, low level image features and attribute or shape information have been applied in conjunction with color or texture features. For the second group of methods [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF][START_REF] Dikmen | Pedestrian recognition with a learned metric[END_REF], to guarantee more reliable matching, feature transformations or distance metrics are learned. Support Vector Machine (SVM) with ranking [START_REF] Prosser | Person reidentification by support vector ranking[END_REF] and transfer learning [START_REF] Dikmen | Pedestrian recognition with a learned metric[END_REF] have also been proposed to obtain better matching correspondence. The distance metric learning based methods have been proved to be effective in matching person images [START_REF] Bedagkar-Gala | A survey of approaches and trends in person re-identification[END_REF]. However, the drawback of these methods is that the learned model tends to overfit the training data. This paper is organized as follows. An overview of a person Re-Identification system is described in section 2: Re-ID diagram, the main challenges, principle methods and a description of the hand-crafted system for person Re-ID. The experimental results and performances of the state-of-the-art Re-ID methods which were evaluated on iLIDS, ViPeR, CAVIAR and 3DPeS datasets are shown in section 3. Section 4 concludes the paper.

II. PERSON RE-IDENTIFICATION: AN OVERVIEW

A. System Design

Fig. 1 shows the diagram of person Re-Identification system. It starts with automatic person detection. In recent years, most of the existing person re-identification works have ignored this step and assume perfect pedestrian detection. However, perfect detection is impossible in real scenarios and misalignment can seriously affect the person Re-ID performance. Therefore this factor should be carefully studied in future works. In order to build a strong visual signature of people appearances, people have to be accurately detected and tracked, so the step of person tracking should be also taken into consideration. However, person detection and multiple person tracking are difficult problems with their own hurdles. Significant amount of work has gone into the problem of person detection over the years as well as Multiple Object Tracking (MOT) within a single camera's FOV which has also been widely researched, but sustained tracking under varying observation environments remains an open problem. For feature extraction and descriptor generation, the most commonly used features are color, shape, position, texture, and soft-biometry. The adopted feature is determined by different factors. On one side, the signature should be unique and discriminative enough which can lead to the selection of biometry or soft-biometry features. On the other side, camera resolution, computational load and other implementation issues can prevent or limit their usage and more generic features are required.

B. Main challenges of Person Re-ID

The main problem in Re-ID resides in the variation in a person's appearance across different cameras.

A typical Re-ID system may have an image (single shot) or a video (multi-shot) as input for feature extraction and signature generation. Thus, the first step in Re-ID is to learn a person's visual signature or model and then compare the two models to get either a match or a non-match. Extracting a reliable signature depends on the availability of good observations. Besides, faulty trajectory estimation and incorrect detections introduce errors in signature generation and extraction that affect the Re-ID quality. The most obvious and simplest signature of a person is characterized by features like color, texture and shape. However, these features are hardly unique, not descriptive enough and prone to variations. Color/texture varies due to cross view illumination variations, pose variations, view angle or scale changes in multi-camera settings. To solve this problem, the equalization between cameras is needed. Different camera geometries also make shape descriptors less discriminative. A subject may be fully or partially occluded by other subjects or carrying items that lead to errors in matching between tracklets. Furthermore, some works in person Re-ID used body-parts methods (such as SDALF, MPMC…) to solve the issue of signature alignment but this problem is still difficult and not efficient as these methods require real detections and many annotations. We can also cite the low image quality as another problem in person Re-ID where the captured image of a person may suffer from low resolution, noise or blur due to limited imaging quality of surveillance cameras. All these issues may affect the performance of person Re-ID which is still not robust enough to guarantee high accuracy in practice.

C. Principle Methods of Person Re-ID

In general, the existing methods predominating the person Re-ID area can be classified into two major categories: Singleshot and Multi-shot person Re-Identification. The first category only analyzes a single image for each subject assuming no tracking information is available. The second one assumes multiple images are available for each person through tracking. These existing methods are divided into two groups: unsupervised and supervised approaches. Unsupervised methods mainly focus on feature design and feature extraction and do not require manually labeling training samples. However, Supervised methods generally require the assistance of manually labeled training samples which lead to better performance. Most existing works [START_REF] Farenzena | Person re-identification by symmetry-driven accumulation of local features[END_REF][START_REF] Dikmen | Pedestrian recognition with a learned metric[END_REF][START_REF] Prosser | Person reidentification by support vector ranking[END_REF][START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF][START_REF] Munaro | One-shot person re-identification with a consumer depth camera[END_REF] choose training and test samples from the same camera views. Only very recently, people started to study the cases when training and testing samples are from different camera views [START_REF] Liu | Person re-identification: What features are important?[END_REF].

Unsupervised methods include: Symmetry-Driven Accumulation of Local Features (SDALF) [START_REF] Farenzena | Person re-identification by symmetry-driven accumulation of local features[END_REF], Biologically inspired features and Covariance descriptors (BiCov) [START_REF] Munaro | One-shot person re-identification with a consumer depth camera[END_REF], Local Descriptors encoded by Fisher Vectors combined with other features (eLDFV) [START_REF] Munaro | One-shot person re-identification with a consumer depth camera[END_REF],etc. Among Supervised methods, we can cite: Ensemble of Localized Features learned with AdaBoost (ELF) [START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF], distance metric learning for Large Margin Nearest Neighbor classification (LMNN) [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF], Information Theoretic Metric Learning (ITML) [19], Pairwise Constrained Component Analysis (PCCA) [START_REF] Mignon | PCCA: A new approach for distance learning from sparse pairwise constraints[END_REF], Large Margin Nearest Neighbor with Rejection (LMNN-R) [START_REF] Dikmen | Pedestrian recognition with a learned metric[END_REF], supervised Local Descriptors encoded by Fisher Vectors (sLDFV) [START_REF] Ma | Local descriptors encoded by Fisher vectors for person re-identification[END_REF],etc.

III. HAND-CRAFTED SYSTEM FOR PERSON RE-ID

A. Features Extraction and descriptor Generation

State-of-the-art descriptors have been reviewed from two different viewpoints, namely the kind of body model and the kind of features used to represent a person.

Each body part (or the whole image of the individual, if no body part subdivision model is used) is described using one or more different global, local or patch-based features.

That's why, appearance-based methods rely mainly on designing discriminative features such as viewpoint invariance features, low-dimensional discriminative features [START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF], combination of local and global features, accumulation of multiple features [START_REF] Farenzena | Person re-identification by symmetry-driven accumulation of local features[END_REF], bio-inspired features, Fisher vector encoded features and attribute-based features [START_REF] Munaro | One-shot person re-identification with a consumer depth camera[END_REF]. To select the most descriptive features for person Re-ID, the work in [START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF] used Adaboost to learn effective representations from a set of local features. In [START_REF] Farenzena | Person re-identification by symmetry-driven accumulation of local features[END_REF] the authors showed a strategy based on the localization of perceptual relevant human parts, driven by asymmetry/ symmetry principles to suggest a method called Symmetry Driven Accumulation of Local Features (SDALF), which is robust to background clutter. A model in a covariance metric space is proposed in [START_REF] Bak | Learning to match appearances by correlations in a covariance metric space[END_REF] in order to extract features from different regions of each person which should be matched specifically. The work in [START_REF] Martinel | Re-identify people in wide area camera network[END_REF] presented a discriminative signature from multiple local features and designed a distance measure by exploiting different body parts. A Local Maximal Occurrence (LOMO) feature is proposed [START_REF] Liao | Person re-identification by local maximal occurrence representation and metric learning[END_REF], in which the horizontal occurrence of local features is used to achieve a constant representation against view changes. Attribute-based features are more robust to image translations comparing with low-level color and texture features. Recently, the authors in [START_REF] Li | A richly annotated dataset for pedestrian attribute recognition[END_REF] gathered a richly annotated dataset for pedestrian attribute.

B. Distance Metric learning

Many distance metric learning and matching process have been proposed for person Re-Identification. These methods aim at learning the best metric between appearance features of the same pedestrian across camera pairs. Support Vector Machine and boosting [START_REF] Gray | Viewpoint invariant pedestrian recognition with an ensemble of localized features[END_REF] have been widely used, which cast the problem into two-class or multiclass classification problem. However, in [START_REF] Prosser | Person reidentification by support vector ranking[END_REF] the person Re-ID scheme is considered as a ranking problem which trained a primal RankSVM ranker and tried to find a linear function to weigh the absolute difference between samples. In [START_REF] Weinberger | Distance metric learning for large margin nearest neighbor classification[END_REF] the Large-Margin Nearest Neighbor metric (LMNN) is proposed which put a limited area (perimeter) for the matched pairs of subjects whilst punishes the unmatched ones, LMNN is time consuming and suffer from the problem of overfitting. In [START_REF] Dikmen | Pedestrian recognition with a learned metric[END_REF] in order to improve on previous results, a reject option for unfamiliar matches, as an LMNN variant, named LMNN-R was introduced. To achieve encouraging re-identification performance, several distance metric learning methods have also been proposed, such as Relative Distance Comparison (RDC), Pairwise Constrained Component Analysis (PCCA) [START_REF] Mignon | PCCA: A new approach for distance learning from sparse pairwise constraints[END_REF] and Local Fisher Discriminant Analysis (LFDA) [START_REF] Pedagadi | Local fisher discriminant analysis for pedestrian re-identification[END_REF]. However, these methods are prone to overfitting problems especially in large scale and high dimensional learning scenarios. Information Theoretic Metric Learning (ITML) [19] and Logistic Discriminant Metric Learning (LDML) [START_REF] Bedagkar-Gala | A survey of approaches and trends in person re-identification[END_REF] have as well been applied. A strategy called Keep It Simple and Straightforward Metric (KISSME) has been proposed to learn a distance metric from equivalence constraints. This method is based on the class of Mahalanobis distance functions which generalizes Euclidean distance. KISSME does not rely on complex optimization and computationally expensive iterations. However, because of the intra-class and inter-class variation, the Mahalanobis metric is more suitable for person Re-ID problem. An effective method called Geometric Preserving Local Fisher Discriminant Analysis (GeoPLFDA) was proposed in [19].The method combines LFDA with geometric preserving method which uses a nearest neighbor graph to approximate local manifold. LFDA [START_REF] Pedagadi | Local fisher discriminant analysis for pedestrian re-identification[END_REF] provides descriptive information by separating differently labeled samples and gathering similarly labeled ones together. Taking the benefit of the complementarily between them, the proposed method achieves considerable advance over state-of-the-art approaches.

Most previous distance metrics are learned by supervised approaches and they are not practical in real-world applications in which the data comes in without any manully labeling efforts. Furthermore, a drawback of the distance metric learning methods is the overfitting problem caused by the small sample size (SSS) problem in person Re-ID where the number of samples per subject is fewer than the dimension of the feature.

There are also some other works in the literature using neural network models to address the person Re-ID problem [18]. A Set Label Model was presented in [START_REF] Mignon | PCCA: A new approach for distance learning from sparse pairwise constraints[END_REF] which applies Neighborhood Component Analysis (NCA) and Deep Belief Network (DBN) on the features of the query and gallery images to improve person Re-Identification performance.

IV. PUBLIC DATASETS AND EVALUATION METRICS

A. Benchmark Datasets Several frequently used datasets have been adopted for evaluation of Re-ID approaches, namely iLIDS,VIPeR, ETHZ , CAVIAR and 3DPeS. Additional datasets should be mentioned in this context like V-47 ,GRID ,Chokepoint,Terrascope, Person RE-ID(PRID), SAIVT, CUHK02 and Sarc3D. The details of the most commonly used datasets are given in TABLE I.However, the estimation of long period Re-ID requires data to be collected over several days using the same or different set of cameras. None of the available datasets offers such instances of people collected on different days. A recent RGB-D person Re-ID dataset captures depth information that can be used for the evaluation of depth based features for Re-ID.As RGB-D datasets we can cite the BIWI RGBD-ID Dataset and the IAS-Lab RGBD-ID Dataset.

For More details and descriptions of these datasets, we refer to survey [START_REF] Bedagkar-Gala | A survey of approaches and trends in person re-identification[END_REF].

B. Evaluation Metrics

The metric known as the Cumulative Matching Characteristic (CMC) curve is the most widely used evaluation metric for performance of person Re-ID.

Since person Re-ID is considered as a ranking problem, this metric is adopted where each element in the gallery is ranked based on its comparison to the probe. The probability that the correct match is ranked the same as or less than a particular value is plotted against the size of the gallery set .The Synthetic Re-ID Rate (SRR) curve is derived from the CMC curve in order to evaluate the performance of the simultaneously matching multiple probe images of the gallery. It gives the probability that any of the given fixed number of matches is correct. The normalized Area Under the CMC curve (nAUC) and Rank 1 recognition rate is also an important performance metric. The nAUC is the probability that the Re-ID system will produce a true match over a false (incorrect) match. However, these metrics are inadequate for evaluating the ability of the system to determine if a probe ID exists in the gallery or not (novelty detection).

C. Performance on the iLIDS dataset

The iLIDS dataset contains 476 images (128x64 pixels) of 119 pedestrians taken from 2 non-overlapping cameras and captured at a busy airport arrival hall. We randomly select the images of 30 and 50 people to form the test set and the rest of images are used for training. Fig. 2 proves the CMC results of LMNN, ITML,KISSME, PCCA, LFDA and GeoPLFDA when the rank r is set to 1,5,10 and 20.

D. Performance on the ViPeR dataset

The ViPeR dataset contains 632 people taken outdoor with 2 images (128x48 pixels) for each pedestrian. The images of 316 and 532 people are selected for the testing set and the rest for the training set. Fig. 3 shows the top ranked matching rate(%)on ViPeR dataset when the rank r is set to 1,5,10 and 20.

E. Performance on the CAVIAR dataset

The dataset contains 1220 images with 10 to 20 images for each person. The minimum and maximum sizes of the images are 17x39 and 72x144, respectively. We selected 18 and 54 people for the test and the remaining images were used for training. Fig. 4 indicates the CMC results when the ranks are 1,5,10 and 20.

F. Performance on the 3DPeS dataset

The 3DPeS dataset contains 1011 images of 192 persons (each person has 2 to 26 images) captured from 8 outdoor different cameras viewpoints. We selected 48 and 143 people for the test and the rest of images were used for training. Fig. 5 shows the CMC results when the ranks are 1,5,10 and 20.

G. Discussion

From the above experiments and among the comparing methods, LMNN, ITML, KISSME and PCCA, we notice that these methods necessitate an iterative optimization scheme which is very expensive on both memory and computation. GeoPLFDA take over from the merits of LFDA and can efficiently find the optimal solution without any iteration, that's why we observe that GeoPLFDA provide the best rank 1 matching rate and high performance compared with other methods .On ViPeR dataset, the results showed 27% and 12,8% when p is set to 316 and 532 respectively. For the 3DPeS dataset, GeoPLFDA attained also the best rank 1 matching rate (50.3% and 32.6% when p is set to 18 and 54, respectively) which proves that this method can measure the distance between probe and gallery images effectively. We can see then that LFDA with geometric preserving projection is more efficient than performing it alone.

Finally, on all the four datasets, when the value of p increases, less data are available for training thus learning becomes more difficult. Therefore the matching becomes harder due to large test set. As a result, the matching rate decreases with the increase of p.

V. CONCLUSION

This paper provided a survey of current approaches for constructing appearance descriptors by extracting feature representations from person detections which are discriminative and robust against illumination change, occlusion and pose variations.

State-of-the-art appearance-based methods as well as matching methods using distance metric learning or ranking classifiers for person Re-Identification were reviewed. We tried to provide a comprehensive review and we also highlight some important issues of person Re-ID that may attract the attention of researches in the future. In summary, as the process of person Re-Identification is significantly a challenging field due to high intra-class variation and inter-class similarity with vast opportunities for improvements and research, we can say that the integration of discriminative and robust feature learning, detector/tracking optimization, and the emergence of large-scale datasets and deep learning systems will lead to a successful person reidentification system. We hope that this paper will be useful as reference for anyone willing to work on this interesting topic.

Fig. 1 .

 1 Fig. 1. Re-ID system Diagram.

Fig. 2 .

 2 Fig. 2. CMC RESULTS ON ILIDS DATASET. P IS THE SIZE OF THE GALLERY SET.(P=30 AND P=50)

Fig. 3 .

 3 Fig. 3. CMC RESULTS ON VIPER DATASET.(P=316 AND P=532)

Fig. 4 .

 4 Fig. 4. CMC RESULTS ON CAVIAR DATASET.(P=18 AND P=54)

Fig. 5 .

 5 Fig. 5. CMC RESULTS ON 3DPES DATASET.(P=48 AND P=143)

TABLE I .

 I SIZE OF SOME COMMONLY USED DATASETS.