Aude Caussarieu

Andrée Tiberghien

When and Why Are the Values of Physical Quantities Expressed with Uncertainties? A Case Study of a Physics Undergraduate Laboratory Course

Keywords: first-year university physics teaching, laboratory course, measurement uncertainties, physics, teaching design

The understanding of measurement is related to the understanding of the nature of science-one of the main goals of current international science teaching at all levels of education. This case study explores how a first-year university physics course deals with measurement uncertainties in the light of an epistemological analysis of measurement. The data consist of the course documents, interviews with senior instructors, and laboratory instructors' responses to an online questionnaire. During laboratory work, uncertainties are expressed in the large majority of the measurements made by the students but only in less than half of their calculation results. The instructors' expectations are that students systematically estimate uncertainties so that they become aware that measurements and calculations are never exact. However, since uncertainties are not specified for the values given in the laboratory guides, uncertainties are often missing from the results of students' calculations. The potential side effects of students' measurement understanding are discussed and suggestions for improvements are proposed.

Introduction

Students' understanding of the nature of science (NOS) is a major goal of contemporary science education [START_REF] Bartos | Teachers' knowledge structures for nature of science and scientific inquiry: Conceptions and classroom practice[END_REF][START_REF] Urhahne | Conceptions of the nature of science-Are they general or context specific[END_REF]. However, NOS content to be taught has been debated amongst science education researchers because there is lack of consensus amongst philosophers [START_REF] Abd-El-Khalick | Teaching with and about nature of science, and science teacher knowledge domains[END_REF][START_REF] Duschl | Two views about explicitly teaching nature of science[END_REF][START_REF] Irzik | A family resemblance approach to the nature of science for science education[END_REF]. [START_REF] Salter | What students say versus what they do regarding scientific inquiry[END_REF] found six recurring NOS themes to be taught at school in published literature: empiricism, process, tentativeness, subjectivity, context, and creativity. Measurement uncertainties are central in the validation process of knowledge production and the tentativeness of the knowledge claims in science; therefore, students need to understand these uncertainties to construct adequate views of NOS.

This case study documents how uncertainty of measurements is incorporated into a first-year introductory physics course at a large French university and what related outcomes are expected by the course and laboratory instructors. Course documents, interviews of senior instructors, and questionnaire surveys of laboratory instructors were used to construct, illustrate, and justify assertions about the central research focus. A background about NOS, measurements, and uncertainty serves as a foundation for the interpretative framework for this case study.

Nature of Science and Scientific Measurements

Numerous science education studies have explored students' conceptions about NOS [START_REF] Deng | Students' views of the nature of science: A critical review of research[END_REF][START_REF] Lederman | Nature of science: Past, present, and future[END_REF] and data processing [START_REF] Buffler | The development of first year physics students' ideas about measurement in terms of point and set paradigms[END_REF]Day & Bon, 2011;[START_REF] Deardorff | Introductory physics students' treatment of measurement uncertainty[END_REF][START_REF] Séré | Learning the statistical analysis of measurement errors[END_REF]. These studies have shown that most students at all teaching levels hold inadequate views about NOS. Students often think that scientific knowledge is absolute and that a scientist's primary goal is to uncover natural laws and discover truths. The naive, positivist science epistemology often embedded in physics teaching may even prevent students from developing modern NOS views [START_REF] Sin | Epistemology, sociology, and learning and teaching in physics[END_REF]. The students' understanding of data processing aspects of NOS has been investigated mainly in the context of replicated measurements [START_REF] Buffler | The development of first year physics students' ideas about measurement in terms of point and set paradigms[END_REF]Day & Bon, 2011;[START_REF] Séré | Learning the statistical analysis of measurement errors[END_REF]. One result from these studies is that students tend to choose the mode of a series of measurements rather than the mean of the set of measurements. Another finding is that students believe a true measurement exists and that proper use of high-quality measurement devices with no human error will provide exact measurements of this true value [START_REF] Deardorff | Introductory physics students' treatment of measurement uncertainty[END_REF]. The choice of the repeated value is often explained as a consequence of the belief in the true value: because one can measure the true value then, within a series of measurements, one should find more often the true value; other values are obtained because of some avoidable error. [START_REF] Buffler | The relationship between students' views of the nature of science and their views of the nature of scientific measurement[END_REF] found a correlation between a naïve view on NOS and a naïve view on measurement.

Recent studies on measurement teaching have dealt with the presentation and evaluation of innovative teaching sequences [START_REF] Allie | Teaching measurement in the introductory physics laboratory[END_REF][START_REF] Lippmann-Kung | Teaching the concepts of measurement: An example of a concept-based laboratory course[END_REF]. These courses were designed to teach measurement explicitly while taking into account the students' difficulties involving measurement; such instructional approaches have been shown to improve students' measurement understanding. However, to our knowledge, there has been no study about the effective teaching practices of university physics instructors regarding measurement based on the underlined epistemic choices situated within the academic physics discipline.

Epistemological Analysis of Physics Measurement

Measurement is at the very heart of all empirical sciences since knowledge production requires comparing experiment results to theoretical claims: "The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific 'truth'." (Feynmann, 1965, p. 1-1). However, physical measurements are always limited by several sources of uncertainty. Two such sources are: (a) material objects and phenomena vary and, thus, replicated measurements of the same physical quantity may differ; and (b) measurement instruments are limited by a finite resolution since they all rely on a physical comparison with a reference quantity. A third source of uncertainty is related to the difficulty in discerning the limits of an object or a phenomenon. For example, in an optical experiment in which a lens projects a real image of an object onto a screen, it is very difficult to tell the precise position of the image. The range of positions in which the image looks equally in focus-because the light is polychromatic and the lens is not a perfect thin lens-illustrates an important source of uncertainty.

Measurements of quantities in physics are made to allow comparisons. Hence, it is recommended to estimate the uncertainty associated with a measurement and to indicate it by reporting the result of a measurement with two numbers, as in . One number refers to the value of the measurand (i.e., the quantity being measured) whereas the other refers to the uncertainty in the value of the measurand. The value of the measurement should be written with a reasonable number of significant figures. However, this notation is not the only one that can be found in scientific literature or in science textbooks. The notations used in physics publications can be grouped into three categories:  Interval notations: When the information is given with two numerical values that specify the measurement and its uncertainty, as in .  Point notations: When the information is given with only one numerical value of the measurement with no explicit reference to the uncertainty, as in m.s -2 .  Approximate notations: When the information is given with one numerical value accompanied by a symbol ( or ) or an adjective (e.g., about or approximately), as in s, which acknowledges the existence of uncertainty within the last significant figure.

The interval notation is obviously the more precise notation. When physicists use the point or approximate notation, they have in mind the order of magnitude of the uncertainty to choose the number of significant figures, which reflect the precision of the methods and techniques used.

Interpretation and estimation of uncertainties can be made within two different frameworks: the historical classical approach (sometimes called the error approach) and the contemporary uncertainty approach. [START_REF] Bevington | Data reduction and error analysis for the physical sciences[END_REF] and [START_REF] Taylor | An introduction to error analysis: The study of uncertainties in physical measurements[END_REF] provided extensive descriptions of the classical approach to measurement in which uncertainty is defined as the difference between the measurement and the true value of a given physical quantity. The International Organization for Standardization published the Guide to the Expression of Uncertainty in Measurement (GUM; Group 1 of the Joint Committee for Guides in Metrology, 2008) that describes the uncertainty approach in which uncertainty is defined as a "parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand" (GUM, 2008, p. 14). When an experiment is replicated and the result of the measurement varies, both approaches estimate uncertainties in the same way. When the uncertainty cannot be estimated through statistical methods, the two approaches have different ways of estimating uncertainties but their final numerical values are somewhat close. On the contrary, the rules for combining different sources of uncertainties and for estimating uncertainty in the result of a calculation involving values of other physical quantities are very different. The classical approach estimates the upper limit of the absolute value of the total error with a cumulative process whereas the uncertainty approach takes into account the possibility that one uncertainty source offsets another. Hence, the two approaches refer to different propagation laws. These two approaches can be seen as having different epistemological families. The classical approach may be associated with a positivist view of science in which laws are to be unveiled and true values exist; the uncertainty approach seems closer to a modeling approach where "scientists use [probabilistic] models to represent aspects of the world for specific purposes" and interpret their results (Giere, 2004, p. 742).

This epistemological analysis reveals that the existence of unavoidable uncertainty in measurement implies that uncertainties are estimated when numerical values are to be used in a comparison. When precise uncertainty estimations have been carried out, the numerical value of a physical quantity is written in interval notation.

Research Questions

The purpose of this study is to investigate the teaching about measurement offered in a first-year physics laboratory course that has been redesigned and implemented in a large university. The following research questions (RQs) involving specific elements and explicit/implicit rules guided the investigation: RQ1: What are the elements of measurement teaching in this course? RQ2: When does the laboratory guide ask the students to estimate measurement uncertainties?

RQ3a: What are the instructors' goals regarding measurement and uncertainty teaching? RQ3b: How do instructors justify the actual handling of uncertainties during laboratory work?

Methods

A multi-method case study was used to address the research questions. Case studies require a well-defined context to establish boundaries for rich data collection, interpretation of the data, and limited generalization of results.

Context of the Study

The RQs were investigated in the framework of an introductory physics course for students majoring in biology at an open university in a large French city. The recently redesigned and implemented laboratory course included lectures, tutorial sessions, and laboratory sessions. This multi-section physics course is offered during the fall and spring semesters with a total enrolment of about 500 students. The university and course are not selective and do not require entrance examinations for program entry.

The teaching team consists of supervising professors and about 30 instructors of whom one-third is renewed each year. A senior professor supervises the whole course and two other professors are in charge of the electricity laboratory and the optics laboratory sessions. The lectures are given by three senior instructors. Every instructor has responsibility for a tutorial group (i.e., optic or electricity-radioactivity) and six laboratory sessions.

A major revision of the laboratory part of the course (i.e., experiments to be performed and laboratory guides) occurred in 2010 after the Physics Department received new equipment. An educational design team composed of a small group (n ~ 3-4) of the instructors conducted the revision with three major aims guiding the redesign of the course. First, they wanted to harmonize the content of the laboratory work with what was addressed during lectures and tutorials. Second, they wanted to address the needs of the biology instructors concerning the physics concepts. Third, they wanted to specifically address the teaching of measurement and measurement uncertainties. Some adjustments to the original version of the course materials were implemented over the next two years to reach the current version, which is studied here.

The course consists of 12 lectures (1h30 each), 12 tutorials (1h30 each), and 6 laboratory sessions (2h each). On average, students are expected to attend 6.5 hours of teaching sessions (1 lecture, 2 tutorials, and 1 laboratory session) each week. The lectures and tutorials deal with the basics of geometrical optics, electricity, and radioactivity; the bi-weekly laboratory sessions deal only with geometrical optics (3 sessions) and electricity (3 sessions). The subjects addressed during the laboratories and tutorials are aligned with the lectures and the needs expressed by the biology instructors.

Data Collection

The corpus of data for this study consists of the written documents for the course given to the students, responses to three individual interviews with instructors, and the answers of other instructors to a short online questionnaire.

Written documents. Analyses of the course documents provided to students in the course digital workspace were helpful in exploring the first two RQs. The elements of measurement knowledge taught in the lecture slides, the tutorial booklet, the pre-laboratory assignments, and the laboratory guides were analyzed to address RQ1 (What are the elements of measurement teaching in this course?). RQ2 (When does the laboratory guide ask the students to estimate measurement uncertainties?) was explored using the tasks students were expected to perform during laboratory work described in the laboratory guides.

Lecture slides. There are 12 lectures in this course; four deal with electricity, two with radioactivity, and six with optics. Lectures are supported by 12 slide sets that are the same across the different lecture halls, which are available online for student use. There is one supplementary set of 16 slides dealing with measurement uncertainties that is no longer addressed during the lectures because of a reduction in the teaching hours allocated to this course. Students are encouraged to download these slides to read and study at home before the first laboratory session.

Tutorials booklet. During the tutorial sessions, students are divided into groups of 20 to 35 students. Students work on exercises developed by the design team in booklets distributed to each student at the beginning of the course; these booklets are also available online. There is one booklet dealing with electricity and radioactivity and another dealing with geometrical optics. Each booklet starts with one exercise on error estimations; however, after this initial exercise, errors are not mentioned again. There is one instructor per tutorial group whose role is to help the students and to give the answers to the exercises.

Laboratory guides. A 34-page booklet distributed to each student at the beginning of the course contains an introduction to the laboratory course and the six guides corresponding to each session. Each guide provides instructions and commentaries on the specific investigation. During the session, students follow the instructions and fill in the blanks by writing down their measurements and answering short questions; Figure 1 provides a sample question. Pre-laboratory assignments. The restructuring of the course required that the laboratory sessions were shortened from three to two hours. Therefore, pre-laboratory assignments were designed to contain most of the theoretical developments present in the previous laboratory guides. They often contain background concerning uncertainty estimations and about ten questions to help students understand the experimental tasks to be conducted. These pre-laboratory assignments are not printed; students can find them on the digital workspace. Students are encouraged to prepare for the laboratory sessions with these assignments but they are neither collected nor graded by the instructors; therefore, very few students actually read or use them.

Professor and instructor feedback. Interviews and a questionnaire were used to collect information from the instructors about RQ3a (What are the instructors' goals regarding measurement and uncertainty teaching?) and RQ3b (How do instructors justify the actual handling of uncertainties during laboratory work?).

Interviews. Semi-structured interviews were developed and conducted after the analysis of the teaching documents was performed with the help of three senior professors. The instructors interviewed were the current leaders of this physics teaching. They have been part of this course for more than 5 years; two were current lecturers and part of the design team. The first part of the interview asked the professors about the general instructions given to the students in the laboratory guides: 1. Why do students have to perform uncertainty estimations during laboratory work? 2. What are the specific constraints that you faced during the rewriting of the laboratory guides? 3. Why does it say in the introduction of the laboratory guides that students should give every numerical value with its associated uncertainty? 4. Is this rule the actual one used during laboratory work?

The second part of the interview asked the professors to comment on specific instructions in the laboratory guides. The interviews (~1h) were audio-recorded and transcribed.

Online questionnaire. An online questionnaire was composed of two questions on the laboratory instructors' administrative status (e.g., PhD, post doc, or permanent instructor) and their seniority in the teaching unit as background on the respondents. The next two questions focused on the laboratory work's goals to supplement the interview responses dealing with RQ3a and RQ3b. The first question about the general goals for laboratory work is taken from [START_REF] Welzel | Teachers' objectives for labwork: Research tool and cross country results (Working Paper 6[END_REF] and asks the instructors to rank-order five possible goals for laboratory work. The second question on measurement learning outcomes was an open-ended question: "Can you explain what, in your opinion, are the goals of these laboratory sessions as regards teaching measurement and uncertainties?" Some instructors did not answer the open-ended question when completing the questionnaire.

The questionnaire was sent to all instructors associated with the course during the previous academic year; only 16 instructors completed and returned the questionnaire. This low response rate (~50%) is due to the facts that it was sent after the spring semester and that about 30% of the instructors would not teach this course the following year.

Data Analysis

This introductory course is attended by a very large population (500 students), and there are many instructors (senior professors, young professors, PhD students) with a high turnover (30 instructors, a third replaced each year). The final examination is the same for all students; therefore, there is a strong concern that each student has received the same teaching. The strategy adopted by the design team consisted of writing down as much information as possible to avoid possible variations amongst the instructors' presentations.

A thematic analysis [START_REF] Braun | Using thematic analysis in psychology[END_REF] of the lecture slides, tutorial booklets, pre-laboratory assignments, and laboratory guides was used to determine the measurement framework used by the design team: Classical or uncertainty approach and the eventual choices made by the design team (RQ1). The themes were based on our epistemological analysis of measurement and uncertainty; in particular, the concepts associated with the definition of uncertainty and the rules associated with its estimation.

We looked for the context elements that are associated with the estimation of measurement uncertainties during laboratory work (RQ2). The epistemological analysis led us to distinguish among the three categories of notation for numerical values described above (i.e., interval, point, and approximate). We used these notation categories to code all the numerical values presented in the tasks of the laboratory guides. The examples from the electricity and optical laboratory guides (Fig. 1) illustrate that students may have to report a direct measurement (e.g., a reading on a scale as in Fig. 1a or on a measuring instrument as in Fig. 1b) or to calculate the value of a physical quantity from other values (e.g., Fig. 1b). They may also use numerical values given in the laboratory guide, such as the values of , , and shown in Figure 1b. Hence, we defined three context categories corresponding to the origin of the numerical value: given in the laboratory guide, measure performed by the student, or calculation performed by the student. The association of the notation of a numerical value and its origin provided an overview of the notational values and categorical origin throughout the laboratory guide.

We investigated the teaching goals of the instructors through the analysis of the interviews and the online questionnaire responses (RQ3a). We also wanted to understand the instructors' justifications for the actual practice of uncertainty estimations during laboratory work (RQ3b), which relied on the analysis of the interview responses. The responses from the interviews and the open-ended question were analyzed using an inductive qualitative research method [START_REF] Strauss | Basics of qualitative research: Techniques and procedures for developing grounded theory[END_REF] to reveal constructs emerging from the professors' and instructors' responses. Their responses were coded into themes or categories based on patterns observed through repeated words or phrases. The research data were gathered, organized, and analyzed in a computer spreadsheet program (Excel).

Results

Samples of the teaching documents and responses from the interviews and questionnaires are presented as evidence (italics) and illustrations for the assertions (boldface) related to each research question. The authors elaborate the results in normal font.

RQ1: WHAT ARE THE ELEMENTS OF MEASUREMENT TEACHING IN THIS COURSE?

Analysis of the written documents (lecture slides, laboratory tasks, and pre-laboratory assignments) revealed that the classical approach of measurement was privileged over the modern uncertainty approach, and several elements related to variability were not addressed in these documents. The lecture slides and pre-laboratory guides provided the most information about the elements of measurement uncertainty, but no elements on variability were mentioned in any of the documents analyzed.

The word error-a central characteristic of the classical approach-is used in the slides on measurement for the interpretation of the interval notation given as the result: the 'true' value of the measurement lies in the error interval given [in the result]. The concomitant use of the term uncertainty is defined in the measurement lecture slides as: the difference between the measured value and the exact value of a quantity . This definition relies on the concept of true value, which is characteristic of the classical approach of measurement. The propagation law and the error combinations made in the different documents are also based on the classical approach in which the final uncertainty is the sum of the absolute contributions of each uncertainty source, as in the example found in the lecture's slides on measurement: if then . Table 1 contains the main knowledge elements (concepts or rules) on measurement uncertainties in the classical approach and the results of the thematic analysis indicating which knowledge elements were present in the documents given to the students. Examination of Table 1 reveals that the knowledge elements on measurement and uncertainties were spread across the different documents and that 5 of the 22 elements were absent. All of the absent elements are related to the variability of physical objects or phenomena, which was not discussed in the content of the measurement teaching. Also, there is no mention of curve fitting in which multiple measurements are combined to estimate the value of a parameter. Only two sources of uncertainty are presented in this course: the measurement instrument and the difficulty involved in defining the limits of the measurand.

Most of the information on measurement is present in documents on the digital workspace that students are only encouraged to consult (e.g., lecture slides on measurement and pre-laboratory guides). The lecture slides on measurement, which might have been expected to act as a reference document for the students, are incomplete since they address only 11 of the 17 (= 22 -5) knowledge elements on measurement present in this course. Furthermore, the rules for assessing specific uncertainties, such as the uncertainty associated with the reading of a measurement instrument, are not given in the lecture slides. The rule for concluding the compatibility of two measurements was only present in the pre-laboratory guides: Measurements and are compatible with each other if the calculated relative error is less than the sum of the relative uncertainties of each measurement .

The epistemological analysis pointed out that the justification of the uncertainty estimation-necessary for making comparisons-is central to understanding the role of uncertainties in physics. We did not find any mention of the fact that error estimations are necessary to compare two numerical values in the documents provided to students. On the contrary, the introduction of the laboratory guide presents uncertainties estimations as a systematic rule justified by a practice of experimentalists: good experimentalists always give their results with an uncertainty.

RQ2: WHEN DOES THE LABORATORY GUIDE ASK THE STUDENTS TO ESTIMATE THE UNCERTAINTIES?

Students are explicitly instructed that they have to state uncertainties in all their results (measurements and calculations) whereas in fact the laboratory guide asks them to do so on most of their measurements but in less than half of their calculations. The analysis of the written documents reveals that incompatibility between expectations and outcomes is mainly due to the systematic absence of uncertainty of the numerical values given in the laboratory guide.

The introduction of the laboratory guide (a similar statement is present in the slides on measurement) states:

Important: Good experimentalists always give their results with an uncertainty. Do the same: in each laboratory session, get used to giving your measurements and calculations with an uncertainty. This last point is important for the final examination.

This statement allowed us to identify an explicit rule in this course about the expression of uncertainties by the students during laboratory work: Students should always estimate uncertainties for their results during laboratory work.

An inspection of the laboratory guide shows that in some cases students are not asked explicitly to estimate the uncertainty associated with their results. This is clearly visible in the extract of the laboratory guide provided in Figure 1b, presented earlier. We determined how widespread this oversight was by counting the different numerical values to be estimated by the students in the laboratory guides as a function of their origin (measurement or calculation) and their notation (interval, point, or approximate). The result of the count revealed 137 measurement and calculation requests distributed across interval and point notations (Fig. 2). First, Figure 2 shows that students are never expected to use approximate notation, but they are expected to use both interval and point notation. They are asked to use interval notation for 63% of the occurrences (86 of 137 cases), which is not fully consistent with the explicit rule provided in the laboratory guide's introduction. Second, it shows that students should nearly always estimate the uncertainty in the direct measurements they perform (90%,

Measure

Calculation 61 of 69 requests). Among the 8 anomalies (measures in the interval notation), four are introductory measurements (e.g., first measurement with an oscilloscope), two are measurements involved in the calculation of the power and magnification of a microscope for which students do not have the formula to propagate the uncertainty, and the last two are found in the final experiment of a very long optics practical.

Figure 2 also shows that students determine the uncertainty associated with the result of their calculations of physical quantity values less than half the time (~40%, 29 of 72 cases). Figure 1b, presented earlier, illustrates that students are not asked to estimate the uncertainty for the result of their calculation. The quantity to be evaluated through a calculation is the capacitance (C), which is a function of the charging time measured by the students and of the resistance ( whose value is given by the laboratory guide:

. The uncertainty in the value of the resistance is not given in the laboratory guide; therefore, it is not possible to estimate the uncertainty in the result of the calculation without making an assumption about the uncertainty in the resistance.

Our second claim about RQ#2 is that the main cause of absence of uncertainty on the calculations is missing information. Thus, most of the data on results notations are explained by the following modified implicit rule: Students are expected to assess uncertainties on their results whenever it is possible. We identified all the numerical values involved in calculations to be performed by the students. Some values were the result of measurements or calculations already performed and others were numerical values of physical quantities provided (given) by the laboratory guides (Table 2). We identified 48 numerical values given in the laboratory guide: 41 are written down in point notation and 7 in approximate notation. Under the label Calc1, we grouped the results of students' calculations for which there is no missing information necessary for estimating the error for the calculated result. Under the label Calc2, we grouped the cases for which some information necessary for estimating the error of the result of the calculation is missing. This information might be the uncertainty on a value necessary for the calculation or the formula to propagate the uncertainty. Finally, we counted the number of results of calculation in each category and represented them as a function of their notation (interval or point). Calc2: Result of calculation with some information missing 0 31

Table 2 shows that our claim is supported by 56 (25 + 31) out of 68 calculations requested. There are 12 results of calculation in the point notation for which students had all the information necessary to estimate the uncertainty. Three of these results are preliminary results for a future complicated calculation for which students do not know the propagation formula. The other 9 results without uncertainties where students had all necessary information are in the last quantitative experiment of the session. Insights into these cases might be found later in the consideration of RQ3b.

Hence, we can formulate the following rule: Students are expected to estimate the uncertainty of the result of their measurement whenever it is possible, but it is not possible whenever a necessary uncertainty is missing. This reconstructed rule provides an explanation for 117 of the 137 students' results.

RQ3A: WHAT ARE THE INSTRUCTORS' GOALS REGARDING MEASUREMENT AND UNCERTAINTY TEACHING?

Teaching introductory physics and related laboratories can emphasize a variety of concepts, processes, practices, and NOS ideas. The instructors' responses to the online questionnaire and interviews indicate: Teaching measurement is an important goal and instructional priority for them. These instructors want students to become aware that uncertainties are associated with every measurement and calculations using measurements. However, this priority for measurement uncertainty is embedded in the university science culture where knowledge and conceptual understanding is primary.

The online questionnaire asked the instructors to rank five different goals one may assign to laboratory work [START_REF] Welzel | Teachers' objectives for labwork: Research tool and cross country results (Working Paper 6[END_REF]: A-for the student to link theory to practice; B-for the student to learn experimental skills, including measurement; C-for the student to get to know the methods of scientific thinking; D-for the student to foster motivation, personal development, social competency; and E-for the teacher to evaluate the knowledge of the students. Sixteen instructors responded to this question. Two respondents did not follow the instructions and assigned the same rank to several goals; their responses were omitted from further consideration. Figure 3 summarizes the instructors' responses to the online questionnaire. The data in Figure 3 illustrate clearly that these instructors' priority goal of laboratory work is Goal A-to link theory to practice. The second most important goals assigned nearly equal priority are B and C, which involve knowledge on measurement and uncertainties. Goals D and E are the lowest priorities for laboratory work for these instructors; Goal E was assigned a slightly higher priority than Goal D. The emphasis (i.e., ranked first by 10 of the 14) on connecting the knowledge taught during lectures to practice in other course components may explain why there is no time in the course devoted to explicit teaching of measurement.

We applied content analysis to the responses to the open-ended question (i.e., Can you explain what, in your opinion, are the goals of these laboratory sessions as regards teaching measurements and uncertainties?) to establish which aspects of measurement teaching were actually important to these instructors. The responses were first split into segments concerning a single idea (i.e., irreducibility of the segment to keep the idea). We identified a total of 48 segments in the 14 responses with an average of segments in each instructor's response. We focused further consideration only on the 36 segments related to goals concerning uncertainty teaching. First, we reformulated the segments into a formal sentence in which the goal is expressed from the students' point of view. Whenever the original segment was linked to a descriptive verb, then we kept it to build a formal sentence of the form: "One goal is that students [+ verb + subordinate clause]." (N.B. When there was no verb in the response, only the subordinate clause was retained.) Second, we grouped together the respondents' formal sentences that were similar. It appeared that most of the groups obtained thereby corresponded to the five elements of knowledge (i.e., definition of uncertainty, source of uncertainty, assessment of uncertainty, presentation of measurement result, and comparison of results using uncertainties) as themes that aligned with the earlier content analysis of the measurement teaching.

Two groups of segments (10 + 7) did not fit with these five themes. Hence, we constructed two new themes to accommodate these results: A-uncertainty is systematically associated to measurement and B-reliability of a result depends on its uncertainty. Theme A (10 segments) is straightforward and corresponds to the knowledge of a fact that is often part of the introduction of courses or textbooks on measurement [START_REF] Taylor | An introduction to error analysis: The study of uncertainties in physical measurements[END_REF]. What is interesting is that three instructors used verbs (see/note/realize) showing that these laboratory sessions were also aimed at convincing students of the systematic existence of uncertainties. In Theme B (7 segments), instructors' responses involved critical thinking about measurement results because of limited precision of measurements. In their responses, limits are related to the precision of a result, which is defined in the classical approach as relative uncertainty. One instructor said, one goal is that students know that uncertainties are important in physics; this statement could not be classified into the existing themes but appears to capture the overall, non-exact tenor of the nature of physics and science. The number of occurrences of each segment is presented in Table 3, indicating that the new themes (A and B) correspond to important teaching goals for the instructors. Surprisingly, only three segments correspond to the assessment of uncertainties theme whereas students are expected to assess 90 uncertainties during the laboratory course (Fig. 2). Further insights into the reasons why teaching elements about uncertainties is important to the instructors can be found in the interview responses. The three instructors interviewed were asked to justify from a pedagogical point of view why students are asked to estimate uncertainties during laboratory work (RQ1). Interpretation of their responses revealed the instructors' expectation that students should be of their result and that they should invoke the existence of uncertainties to explain why their actual measurement is different from the theoretical expected value.

Overall, these results show that in the instructors' responses the emphasis is on conceptual understanding about uncertainties rather than technical skill regarding uncertainties' estimations. This finding appears to contradict the outcomes from the document analysis that shows students are requested to estimate uncertainties regardless of their utility from the physics point of view.

RQ3B: HOW DO INSTRUCTORS JUSTIFY THE ACTUAL HANDLING OF UNCERTAINTIES DURING LABORATORY WORK?

The instructors justify that students should estimate the uncertainties of their results (i.e., measurements and calculations) as a norm of what must be done during physics laboratory work. The three instructors interviewed agreed on the explicit rule that students should always estimate uncertainty in laboratory sessions, which is given in the introduction of the laboratory guide. The statement Good experimentalists always give their results with an uncertainty appears to be the justification of the rule asking students to estimate uncertainties systematically. When asked about the justification of this systematic rule, two instructors suggested that in physics it is meaningless to give a result without its associated uncertainty. The normative aspect of the practice is also strengthened by the instructors, when one stated:

Because you're doing physics, it must be meaningful, so, first you must give the result with a unit, it's not mathematics. [Then] because you are in a laboratory session, your measurement must have an associated uncertainty…. it is clearly indicated in the laboratory guide that each measurement without uncertainty will be penalized.

Another instructor stated that she justifies uncertainty estimations by her own research activity: we do experiments in China and in France and we will try to compare our results and in order to compare our results it's necessary to know how precise they are.

These instructors were probed during the interview to determine why in many cases students are not asked to estimate measurement uncertainties in which numerical values were to be written in point notation. We grouped their responses depending on the origin of the numerical value that was at stake.

Cases where the value is given. When asked about the absence of uncertainty in this case, each instructor was surprised: I had not considered that problem and No no no no no, that's not a choice; in fact, we haven't done what we should do. They immediately expressed the desire to change the laboratory guides so that the values are given with their uncertainty: What must be done is that we give in the instructions the values with their uncertainties, we must remain coherent and indeed, here we are not. Their surprise indicates clearly that there is no conscious rule that prevailed for the use of point notation for given values. Nevertheless, one instructor explained that he does not ask the students to take into account the error of the given quantity because his goal is that students understand errors associated with the reading of a digital or analogic instrument: Well, there are already all the reading uncertainties with the number of divisions; it's okay if they manage to understand this notion. I'm happy.

Cases where the value is the result of a calculation with some information missing. Most of the results of calculation expressed in point notation are explained by the instructors as the result of missing information: Since we haven't given with an uncertainty, they cannot [estimate the uncertainty of the result] and We should give them the formula, it would be easiest, that is something we can do. The instructors confirmed that the implicit rule we posit in the results to RQ2 explained the absence of uncertainty on most of the calculation results.

Cases where the value is the result of a measurement or a calculation. When the instructors realized that in some cases students are asked to express the result of their measurements or calculations without uncertainty although it is possible to estimate them, the instructors saw an incoherence that should be fixed: I didn't remember that we didn't ask for that. They subsequently justify it with three different reasons. First, some cases are explained by the instructors as if they were omissions that should be fixed: It's rather an omission. Second, there are time constraints because the calculation is performed at the end of the session: For me, I'd say that the problem is because we lack time. and We know that it is the end and that they won't finish, we don't ask them [to estimate errors]. Third, the absence of measurement errors is put down to the nature and pedagogical purpose of the task involving the measurement or the calculation as an illustration or verification of the lecture's contents. In these cases, instructors just want the students to check if their result is of the same order of magnitude (power of ten) as the theoretical value rather than if their result is precisely consistent with the theoretical value. The laboratory activity is truly applicative, it is just for illustrating the lecture but maybe there is an incoherence at this level or because the instructors want the students to be aware of the order of magnitude: It's the order of magnitude that matters, … if I find 10 microfarads, there is a problem; I mean the order of magnitude is bad. Hence, the physics context of the numerical value is finally evoked to justify those problematic cases.

It appears at this introductory-level service course that conceptual understanding about uncertainty exists in primary measurements and awareness of this uncertainty is inherent in any calculations involving these data is the essential goal rather than the actual calculation of the uncertainty using complex formulae. Nevertheless, the instructors, for the sake of coherence, are willing that uncertainties are estimated in all cases even if it implies the use of a "turnkey" formula by the students.

Conclusion and Discussion

This case study of the redesign and implementation of an introductory physics course focused on the laboratory component and specifically the teaching of measurement and measurement uncertainty, which is essential to an informed view of the nature of science (physics). The research questions systematically considered the elements of measurement uncertainty, the intentions of the redesign team in developing course materials, and the actual use of these materials by course instructors.

Results of this study show that during laboratory sessions students actually express uncertainties on nearly all their measurements and about half of their calculations. This is mainly due to the absence of uncertainty on the data given in the laboratory guide, which creates an asymmetry between data obtained by the students and values given in the instructions. The risk is that it might lead students to think that the results of measurements made by scientists or manufacturers are of a distinct nature from those made by a student with the laboratory apparatus. Furthermore, there was a misalignment of the intentions and the resources regarding the determination of uncertainty in measurements and associated calculations since instructors wanted systematic estimation of uncertainties. These findings appeared to some members of the redesign team as an incoherence that should be fixed by providing students with all information necessary to estimate each uncertainty. We agree with the instructors that all the values given in the instructions of the laboratory guides should be provided with their associated uncertainties. Hence, their absence would not limit error estimation when they are necessary and would not reinforce the myth of the true value and physics as being an exact science. It is also coherent with the fact that if one communicates the numerical value of a physical quantity without knowing the future use-like a manufacturer does-one should still indicate the associated error.

In physics research, interval notation is nearly only used when the error is necessary for the comparison of two similar values; however, during daily routine in the laboratory, researchers do not systematically estimate the uncertainties. In the course studied, errors are always supposed to be estimated, even when they are unnecessary from the physics point of view. For example, in the first optics laboratory session, students are asked to measure the diameter of a spot of light on a screen after passing through a converging lens. This diameter (about one centimeter) is then compared to the diameter of the lens (about ten centimeters). The use of a ruler was useless in comparing these diameters and concluding that the lens action is to make the light rays converge (an observation with the naked eye was sufficient), and the uncertainties estimations are all the more useless. This may lead students to see uncertainty evaluation as a routine, meaningless, useless practice. A natural perspective of this study is the analysis of the teaching situations in which error estimations are necessary. A comprehensive study of the way physicists actually deal with uncertainties in their laboratories may help build a more realistic frame for uncertainty estimation in experimental physics courses and make students aware of uncertainty in physics.

The strength of systematic uncertainties estimation is that students will have estimated a high number of uncertainties during these sessions and, therefore, should have acquired solid technical skills. Nevertheless, we showed that instructors' main goals regarding uncertainties are that students know what uncertainties are and that they get convinced that there are always uncertainties. These goals are more related to conceptual understanding than technical skills and one may wonder if the actual teaching strategy is optimal to achieve conceptual understanding. Results of this case study also revealed that the design team and instructors have chosen the classical approach to measurement, rather than the contemporary uncertainty approach, leaving aside all concepts related to the variability of physical objects or phenomena. Variations across examples of objects and occurrences of events are not addressed in the course documents. These elements are crucial to develop a full understanding of measurement uncertainties and more generally of the NOS. Hence, the overall findings of this study indicate the need for new course designs compatible with the instructors' actual goals and constraints that enable an understanding of physics uncertainty.

A comprehensive overview of physics uncertainty includes the unavoidable nature of uncertainties and the relevant use of uncertainty for situations in which they are necessary to form a conclusion from the physics point of view. Designing such a course would imply a collaborative work between instructors and researchers to construct a conceptual framework of uncertainty adapted to introductory physics students with a hierarchy of the associated objectives. The design cycle should start based on these preliminary considerations, the implementation of the designed course would allow the collection of data in that authentic context that would then lead to refine the framework and revise the course. This design-based research will help instructors to become aware of the necessity of refining and aligning their goals and the intended goals, which will lead to better achievement within the new teaching activities. At a minimum, we recommend explicit teaching of the origin of errors (i.e., measurement and variability of physical phenomena) and explicit teaching of the role of uncertainties in physics research. Contrary to the actual rule in laboratory work, we suggest that instructors limit error computations performed by students to situations in which they are necessary to form a conclusion from the physics point of view.

Figure 1 .

 1 Figure 1. Extracts from the laboratory guides: (a) an optical laboratory guide; (b) an electricity laboratory guide.

Figure 2 .

 2 Figure 2. Number of results to be estimated by students depending on their notation (interval, point, approximate) and their origin (measure, calculation). Total possible number of requests = 137.

Figure 3 .

 3 Figure 3. Goal-ranking responses to the online questionnaire by instructors (n = 14).

Table 1 .

 1 Results of the Analysis of the Content of the Measurement Teaching Provided in Comparison to the Reference Content of the Classical Approach to MeasurementNote. X = presence of this knowledge element in the corresponding written document. Missing elements are highlighted in boldface.

				Lecture	Tutorial	Pre-lab.	Lab. guide
	Elements of knowledge		slides	booklet	guide	commentary
	Concept
	1	Uncertainties (or measurement error)	Definition (with reference to the true value)	X
	2		Random error
	3		Systematic error
	4		Relative uncertainty/fractional uncertainty/precision	X		X
	5	Source of uncertainties	Variability of physical objects and phenomena
	6		Limits of objects/phenomena	X		X
	7		Limited resolution of measurement instruments			X
	8		Non accuracy of measurement instruments	X		X
	9		Instrument reading	X		X
	10	Interpretation of a measurement result		X
	Rule
	11	Assessment of uncertainties associated	Variability in the replication of a measurement
	12	with	Limits of objects/phenomena			X
	13		Manufacturer indications on the measurement			X	X
			instrument
	14		Instrument reading			X	X
	15	Propagation of measurement	Combination of different errors for the same	X		X	X
		uncertainties	measurement
	16		Simple combination of errors on different measured	X	X	X	X
			values
	17		Law of propagation of uncertainties	X
	18	Assessment of uncertainty on a	Linear regression
	19	parameter of curve fitting	Least square fitting

Table 2 .

 2 The Associated Notations of Calculation Results and Numerical Values Involved

	Notation Origin

Table 3 .

 3 Themes Identified in the Instructors' Responses Text in italics indicates elements defined in the epistemic analysis.

	Theme	Goals in instructors' responses	Segments (n)
	A	Uncertainty is systematically associated to measurement	10
	B	Reliability of a result depends on its uncertainty	7
	C	Source of uncertainty	4
	D	Comparisons	3
	F	Assessment of uncertainties	3


M Berthier
email: michel.berthier@univ-lr.fr

Geometry of color perception. Part 2: perceived colors from real quantum states and Hering's rebit

Keywords: Color Perception, Quantum states, Jordan Algebras, Quantum Rebit

Inspired by the work of Resnikoff, which is described if full details in the first part of this two-part paper, we give a quantum description of the space P of perceived colors. We show that P is the effect space of a rebit, a real quantum qubit, whose state space S is isometric to the hyperbolic Klein disk K. This chromatic state space of perceived colors can be represented as a Bloch disk of real dimension 2 that coincides with the Hering disk given by the color opponency mechanism. Attributes of perceived colors, hue and saturation, are defined in terms of Von Neumann entropy.

Introduction

"The structure of our scientific cognition of the world is decisively determined by the fact that this world does not exist in itself, but is merely encountered by us as an object in the correlative variance of subject and object" [START_REF] Weyl | Mind and nature[END_REF].

The mathematical description of human color perception mechanisms is a longstanding problem addressed by many of the most influential figures of the mathematical physics [START_REF] Weyl | Mind and nature[END_REF], [START_REF] Grassmann | Zur theorie der farbenmischung[END_REF], [START_REF] Maxwell | On color vision[END_REF], [START_REF] Schrödinger | Grundlinien einer theorie der farbenmetrik in tagessehen[END_REF]... The reader will find an overview of the main historical contributions at the beginning of [START_REF] Resnikoff | Differential geometry and color perception[END_REF] where H. L. Resnikoff points out that the space, that we denote P, of perceived colors is one of the very first examples of abstract manifold mentioned by B. Riemann in his habilitation [START_REF] Riemann | Über die hypothesen, welche der geometric zu grunde liegen[END_REF], "a pregnant remark". As suggested by H. Weyl [START_REF] Weyl | Mind and nature[END_REF], it is actually very tempting to characterize the individual color perception as a specific correlative interaction between an abstract space of perceived colors and an embedding space of physical colors. This raises the question of defining intrinsically, in the sense of Riemannian geometry, the space of perceived colors from basic largely accepted axioms. These axioms, which date back to the works of H. G. Grassmann and H. Von Helmhotz [START_REF] Grassmann | Zur theorie der farbenmischung[END_REF], [START_REF] Helmhotz | Treatise on Physiological Optics, English Trans. of First German Edition[END_REF], state that P is a regular convex cone of dimension 3. It is worth noting that convexity reflects the property that one must be able to perform mixtures of perceived colors or, in other words, of color states [START_REF] Bengtsson | Geometry of Quantum States, an Introduction to Quantum Entanglement[END_REF]. What makes the work of H. L. Resnikoff [START_REF] Resnikoff | Differential geometry and color perception[END_REF] particularly enticing is the remarkable conclusions that he derives by adding the sole axiom that P is homogeneous under the action of the linear group of background illumination changes [START_REF] Provenzi | Geometry of color perception. part 1: A homogeneous color space[END_REF]. We will discuss at the end of this work, in section 6, the relevance of this statement. To the best of our knowledge, this axiom, which involves an external context, has never been verified by psychophysical experiments.

It endows P with the rich structure of a symmetric cone [START_REF] Faraut | Analysis on Symmetric Cones[END_REF]. With this additional axiom, and the hypothesis that the distance on P is given by a Riemannian metric invariant under background illumination changes, H. L. Resnikoff shows that P can only be isomorphic to one of the two following Riemannian spaces: the product P 1 = R + ×R + ×R + equipped with a flat metric, namely the Helmhotz-Stiles metric [START_REF] Wyszecki | Color Science, Concepts and Methods, Quantitative Data and Formulae[END_REF], and P 2 = R + ×SL(2, R)/SO(2, R) equipped with the Rao-Siegel metric of constant negative curvature [START_REF] Calvo | A distance between multivariate normal distributions based in an embedding into the siegel group[END_REF], [START_REF] Siegel | Symplectic geometry[END_REF]. Let us recall that the quotient SL(2, R)/SO(2, R) is isomorphic to the Poincaré hyperbolic disk D. The first space is the usual metric space of the colorimetry, while the second one seems to be relevant to explain psychophysical phenomena such as the ones described by H. Yilmaz in [START_REF] Yilmaz | Color vision and a new approach to general perception[END_REF] and [START_REF] Yilmaz | On color perception[END_REF]. In the sequel, we focus on the latter.

The starting point of this work originates from the second part of [START_REF] Resnikoff | Differential geometry and color perception[END_REF] dedicated to Jordan algebras. Contrary to H. L. Resnikoff we suppose at first that the perceived color space P can be described from the state space of a quantum system characterized by a Jordan algebra A of real dimension 3 [START_REF] Koecher | Jordan algebras and differential geometry[END_REF], [START_REF] Koecher | The Minnesota Notes on Jordan Algebras and Their Applications[END_REF], [START_REF] Mccrimmon | Jordan algebras and their applications[END_REF], [START_REF] Mccrimmon | A Taste of Jordan Algebras[END_REF]. This is our only axiom. Jordan algebras are non associative commutative algebras that have been classified by P. Jordan, J. Von Neumann and E. Wigner [START_REF] Jordan | On an algebraic generalization of the quantum mechanical formalism[END_REF] under the assumptions that they are of finite dimension and formally real. They are considered as a fitting alternative to the usual associative noncommutative algebraic framework for the geometrization of quantum mechanics [START_REF] Carinena | Geometrization of quantum mechanics[END_REF], [START_REF] Gunson | On the algebraic structure of quantum mechanics[END_REF], [START_REF] Jordan | Ueber verallgemeinerungsmöglichkeiten des formalismus der quantenmechanik[END_REF]. Not so surprisingly in view of what precedes, A is necessarly isomorphic to one of the two following Jordan algebras: the algebra R ⊕ R ⊕ R or the algebra H(2, R) of symmetric real 2 by 2 matrices. It appears that the two geometric models of H. L. Resnikoff can be recovered from this fact by simply taking the positive cone of A. The Jordan algebra H(2, R) carries a very special structure being isomorphic to the spin factor R ⊕ R 2 . It can be seen as the non associative algebra linearly spanned by the unit 1 and a spin system of the Clifford algebra of R 2 [START_REF] Mccrimmon | A Taste of Jordan Algebras[END_REF], [START_REF] Chevalley | The algebraic theory of spinors and clifford algebras[END_REF]. The main topic of this work is to exploit these structures to highlight the quantum nature of the space P of perceived colors. Actually, the quantum description that we propose gives its full meaning to the relevant remark of [START_REF] Ashtekar | Geometry in color perception[END_REF], p. 5.

Although the geometry of the second model P 2 of H. L. Resnikoff is much richer than the geometry of the first model P 1 , very few works are devoted to the possible implications of hyperbolicity in color perception. One of the main objectives of this contribution is to show that the model P 2 is perfectly adapted to explain the coherence between the trichromatic and color opponency theories. We show that the space P is the effect space of a so-called rebit, a real quantum qubit, whose state space S is isometric to the hyperbolic Klein disk K. Actually, K is isometric to the Poincaré disk D, but its geodesics are visually very different, being the straight chords of the unit disk. Klein geometry appears naturally when considering the spin factor R ⊕ R 2 and the 3-dimensional Minkowski future lightcone L + whose closure is the state cone of the rebit. We show that the chromatic state space S can be represented as a Bloch disk of real dimension 2 that coincides with the Hering disk given by the color opponency mechanism. This Bloch disk is an analog, in our real context, of the Bloch ball that describes the state space of a two-level quantum system of a spin-1 2 particule. This suggests that the processing performed by the color opponent ganglion cells is similar to spin inversions.

Following this quantum interpretation, we give precise definitions of the two chromatic attributes of a perceived color, hue and saturation, in terms of Von Neumann entropy.

As explained by P. A. M. Dirac in [START_REF] Dirac | The Principles of Quantum Mechanics[END_REF], p. 18, physical phenomena justify the need for considering complex Hilbert spaces in quantum mechanics. Alternatively, the structures we deal with in the sequel are real and we may consider the space P as a non trivial concrete example of an effect space of a real quantum system. The reader will find more information on real-vector-space quantum theory and its consistency regarding optimal information transfer in [START_REF] Wootters | Optimal information transfer and real-vector-space quantum theory[END_REF].

Finally, since the spin factors and the corresponding Clifford algebras share the same representations (and the same squares), one may envisage to adapt to the present context the tools developed in [START_REF] Berthier | Spin geometry and image processing[END_REF] for the harmonic analysis of color images.

Mathematical preliminaries

We introduce in this section the mathematical notions needed in the sequel. They mainly concern the properties of Jordan algebras. The reader will find more detailed information on this subject in [START_REF] Koecher | Jordan algebras and differential geometry[END_REF], [START_REF] Mccrimmon | Jordan algebras and their applications[END_REF], [START_REF] Mccrimmon | A Taste of Jordan Algebras[END_REF], [START_REF] Baez | Division algebras and quantum theory[END_REF] or in the seminal work of P. Jordan [START_REF] Jordan | Über eine klasse nichtassociativer hyperkomplexer algebren[END_REF]. A Jordan algebra A is a real vector space equipped with a commutative bilinear product A × A -→ A, (a, b) -→ a • b, satisfying the following Jordan identity:

(a 2 • b) • a = a 2 • (b • a) . (1)
This Jordan identity ensures that the power of any element a of A is well-defined (A is power associative in the sense that the subalgebra generated by any of its elements is associative). Since a sum of squared observables never vanishes, one logically requires that if a 1 , a 2 ,..., a n are elements of A such that:

a 2 1 + a 2 2 + • • • + a 2 n = 0 , (2)
then

a 1 = a 2 = • • • = a n = 0.
a • b = 1 2 (ab + ba) . (3)
The spin factors form "the most mysterious of the four infinite series of [simple] Jordan algebras" [START_REF] Baez | The octonions[END_REF]. They were introduced for the first time under this name by D. M. Topping [START_REF] Topping | Jordan Algebras of Self-Adjoint Operators[END_REF] and are defined as follows. The spin factor J(V ) of a given n-dimensional real inner product space V is the Jordan algebra freely generated by V with the relations:

v 2 = v 2 , (4)
for all v in V . Due to commutativity, this implies:

v • w = v, w , (5)
for all v and w in V . Consequently, J(V ) is isomorphic to the direct sum R ⊕ V with the product:

(α + v) • (β + w) = (αβ + v, w + αw + βv) , (6)
where α and β are reals. The following result is well known [START_REF] Baez | The octonions[END_REF].

Proposition 1 Let K be the division algebra R, C, H or O. The spin factor J(K ⊕ R) is isomorphic to the Jordan algebra H(2, K).

Proof. An explicit isomorphism is given by the map:

φ : H(2, K) -→ J(K ⊕ R) (7)
α + β x x * α -β -→ (α + x + β) , (8)
with x in K and x * the conjugate of x.

Now, we focus on the algebra H(2, R) and the spin factor J(R ⊕ R), both being isomorphic to R ⊕ R 2 . The latter is equipped with the Minkowski metric [START_REF] Minkowski | The principle of relativity, english translation of Raum und Zeit[END_REF]:

(α + v) • (β + w) = αβ -v, w , (9)
where α and β are reals and v and w are vectors of R 2 . It turns out that Proposition 1 has a fascinating reformulation: the algebra of observables of a 2-dimensional real quantum system is isomorphic to the 3-dimensional Minkowski spacetime. Let us recall that the lightcone L of R ⊕ R 2 is the set of elements a = (α + v) that satisfy:

a • a = 0 , (10)
and that a light ray is a 1-dimensional subspace of R ⊕ R 2 spanned by an element of L. Every such light ray is spanned by a unique element of the form (1 + v)/2 with v a unit vector of R 2 . Actually, the space of light rays coincides with the projective space P 1 (R). In other words, we have the following result. Proposition 2 There is a one to one correspondance between the light rays of the spin factor R ⊕ R 2 and the rank one projections of the Jordan algebra H(2, R).

Proof. The correspondance is given by:

(1 + v)/2 -→ 1 2 1 + v 1 v 2 v 2 1 -v 1 , (11)
where

v = v 1 e 1 + v 2 e 2 is a unit vector of R 2 .
We will see in the next section that this result has a meaningful interpretation: there is a one to one correspondance between the light rays of the spin factor R⊕R 2 and the pure state density matrices of the algebra H(2, R).

The positive cone C of the Jordan algebra A is the set of its positive elements, namely:

C = {a ∈ A, a > 0} . ( 12
)
It can be shown that C is the interior of the positive domain of A defined as the set of squares of A. The convex cone C is symmetric: it is regular, homogeneous and self-dual [START_REF] Faraut | Analysis on Symmetric Cones[END_REF]. The positive cone H + (2, R) of the algebra H(2, R) is the set of positive-definite symmetric matrices.

Quantum preliminaries

The positive cone C is the set of positive observables. A state of A is a linear functional:

• : A -→ R , (13)
that is nonnegative: a ≥ 0, ∀a ≥ 0, and normalized: 1 = 1. Since A is formally real, the pairing:

a, b = Trace(L(a)(b)) = Trace(a • b) , (14)
is a real-valued inner product and one can identify any state with a unique element ρ of A by setting:

a = Trace(ρ • a) , (15)
where ρ ≥ 0 and Trace(ρ) = 1. Such a ρ, for A = H(2, R), is a so-called state density matrix [START_REF] Bengtsson | Geometry of Quantum States, an Introduction to Quantum Entanglement[END_REF] and describes a mixed state or mixture. Formula (15) gives the expectation value of the observable a in the state with density matrix ρ.

Regarding Proposition 1, the positive state density matrices of the algebra H(2, R) are in one to one correspondance with the elements of the future lightcone:

L + = {a = (α + v), α > 0, a • a > 0} . ( 16
)
that are of the form a = (1 + v)/2. One way to qualify states is to introduce the Von Neumann entropy [START_REF] Bengtsson | Geometry of Quantum States, an Introduction to Quantum Entanglement[END_REF], [START_REF] Fano | Description of states in quantum mechanics by density matrix and operator techniques[END_REF]. It is given by:

S(ρ) = -Trace(ρ log ρ) . (17)
It appears that S(ρ) = 0 if and only if ρ satisfies ρ • ρ = ρ. The zero entropy state density matrices characterize pure states that afford a maximum of information.

Among the other state density matrices, one is of particular interest. It is given by ρ 0 = Id 2 /2 or ρ 0 = (1 + 0)/2 (Id 2 is the identity matrix) and is characterized by:

ρ 0 = argmax ρ S(ρ) . (18)
The mixed state with density matrix ρ 0 is called the state of maximum entropy, S(ρ 0 ) being equal to log 2. It provides the minimum of information. Using [START_REF] Yilmaz | On color perception[END_REF], we have:

a 0 = Trace(a) 2 . ( 19
)
Now, given an observable a, a acts on the state • 0 by the formula:

a : • 0 -→ a • • = • 0,a , (20)
Since for any state ρ, the element 2ρ is an observable, we get:

a 0,2ρ = 2ρ • a 0 = Trace(ρ • a) = a , (21)
for all observable a. This means that any state with density matrix ρ can be obtained from the state of maximal entropy with density matrix ρ 0 using the action of the observable 2ρ. The pure state density matrices of the algebra A = H(2, R) are of the form:

1 2 1 + v 1 v 2 v 2 1 -v 1 , (22)
where v = v 1 e 1 + v 2 e 2 is a unit vector of R 2 . They are in one to one correspondance with the light rays of the 3-dimensional Minkowski spacetime, see Proposition 2.

A classical representation of quantum states is the Bloch body [START_REF] Appleby | Symmetric informationally complete measurements of arbitrary rank[END_REF]. An element ρ of H(2, R) is a state density matrix if and only if it can be written as:

ρ(v 1 , v 2 ) = 1 2 (Id 2 + v • σ) = 1 2 (Id 2 + v 1 σ 1 + v 2 σ 2 ) , (23)
where σ = (σ 1 , σ 2 ) with:

σ 1 = 1 0 0 -1 σ 2 = 0 1 1 0 , (24)
and v = v 1 e 1 + v 2 e 2 is a vector of R 2 with v ≤ 1. The matrices σ 1 and σ 2 are Pauli-like matrices. In the usual framework of quantum mechanics, that is when the observable algebra is the algebra H(2, C) of 2 by 2 hermitian matrices with complex entries, the Bloch body is the unit Bloch ball in R 3 . It represents the states of the two-level quantum system of a spin-1 2 particule, also called a qubit. In the present context, the Bloch body is the unit disk of R 2 associated to a rebit. We give now more details on this system using the classical Dirac notations [START_REF] Dirac | The Principles of Quantum Mechanics[END_REF], bra, ket, etc... Let us denote |u 1 , |d 1 , |u 2 and |d 2 the four state vectors defined by:

|u 1 = 1 0 , |d 1 = 0 1 , |u 2 = 1 √ 2 1 1 , |d 2 = 1 √ 2 -1 1 . (25)
We have:

σ 1 = |u 1 u 1 | -|d 1 d 1 | , σ 2 = |u 2 u 2 | -|d 2 d 2 | . ( 26
)
The state vectors |u 1 and |d 1 , resp. |u 2 and |d 2 , are eigenstates of σ 1 , resp. σ 2 , with eigenvalues 1 and -1.

Using polar coordinates v 1 = r cos θ, v 2 = r sin θ, we can write ρ(v 1 , v 2 ) as:

ρ(r, θ) = 1 2 1 + r cos θ r sin θ r sin θ 1 -r cos θ (27) = 1 2 {(1 + r cos θ)|u 1 u 1 | + (1 -r cos θ)|d 1 d 1 | + (r sin θ)|u 2 u 2 | (28) -(r sin θ)|d 2 d 2 |} . (29)
This gives, for instance:

ρ(1, 0) = |u 1 u 1 | = 1 0 0 0 , (30)
ρ(1, π) = |d 1 d 1 | = 0 0 0 1 , (31)
ρ(1, π/2) = |u 2 u 2 | = 1 2 1 1 1 1 , (32)
ρ(1, 3π/2) = |d 2 d 2 | = 1 2 1 -1 -1 1 . ( 33
)
More generally:

ρ(1, θ) = |(1, θ) (1, θ)| , (34)
with:

|(1, θ) = cos(θ/2)|u 1 + sin(θ/2)|d 1 . (35)
This means that we can identify the pure state density matrices ρ(1, θ) with the state vectors |(1, θ) and also with the points of the unit disk boundary of coordinate θ. The state of maximal entropy, given by the state density matrix:

ρ 0 = 1 2 1 0 0 1 , (36)
is the mixture:

ρ 0 = 1 4 |u 1 u 1 | + 1 4 |d 1 d 1 | + 1 4 |u 2 u 2 | + 1 4 |d 2 d 2 | (37) = 1 4 ρ(1, 0) + 1 4 ρ(1, π) + 1 4 ρ(1, π/2) + 1 4 ρ(1, 3π/2) , (38)
with equal probabilities. Using [START_REF] Wootters | Optimal information transfer and real-vector-space quantum theory[END_REF], we can write every state density matrix as a mixture:

ρ(r, θ) = ρ 0 + r cos θ 2 (ρ(1, 0) -ρ(1, π)) + r sin θ 2 (ρ(1, π/2) -ρ(1, 3π/2)) . ( 39
)
Such a mixture is given by the point of the unit disk of polar coordinates (r, θ). In the usual framework, that is when A is H(2, C), the three Pauli matrices are associated to the three directions of rotations in R 3 . In our case, there are only two Pauli-like matrices. The interpretation in terms of rotations ceases to be relevant since there is no space with a rotation group of dimension 2. This makes rebits somewhat strange. We show in Section 5 that the opponency color mechanism of E. Hering is given by such a rebit when dealing with chromatic opponencies.

The pure and mixed states play a crucial role in the measurements: "... that is, after the interaction with the apparatus, the system-plus-apparatus behaves like a mixture... It is in this sense, and in this sense alone, that a measurement is said to change a pure state into a mixture" [START_REF] Jammer | The Philosophy of Quantum Mechanics[END_REF] (see also the cited reference [START_REF] Landau | Das dämpfungsproblem in der wellenmechanik[END_REF]). It seems actually that the problem of measurements in quantum mechanics was one of the main motivations of P. Jordan for the introduction of his new kind of algebras: "Observations not only disturb what has to be measured, they produce it... We ourselves produce the results of measurements" [START_REF] Jammer | The Philosophy of Quantum Mechanics[END_REF] p. 161, [START_REF] Jordan | Quantenphysikalische bemerkungen über biologie und psychologie[END_REF], [START_REF] Mermin | Is the moon there when nobody looks? reality and the quantum theory[END_REF]. [START_REF] Schrödinger | Grundlinien einer theorie der farbenmetrik in tagessehen[END_REF] The Riemannian geometry of C and L + Now, we give further information on the underlying geometry of the Jordan algebra A from both the points of view discussed above, that is A as the algebra H(2, R) and A as the spin factor R ⊕ R 2 . We consider the level set:

C 1 = {X ∈ H + (2, R), Det(X) = 1} , (40)
Every X in C 1 can be written as:

X = α + v 1 v 2 v 2 α -v 1 , (41)
with v = v 1 e 1 + v 2 e 2 a vector of R 2 satisfying α 2 -v 2 = 1 and α > 0. Using the one to one correspondance:

X = α + v 1 v 2 v 2 α -v 1 -→ (α + v) , (42)
the level set C 1 is sent to the level set:

L 1 = {a = (α + v) ∈ L + , a • a = 1} , (43)
of the future lightcone L + . It is well known that the projection:

π 1 : L 1 -→ {α = 0} , (44)
defined by:

π 1 (α + v) = (0 + w) , (45)
with w = w 1 e 1 + w 2 e 2 and:

w i = v i 1 + α , (46)
for i = 1, 2, is an isometry between the level set L 1 and the Poincaré disk D [START_REF] Cannon | Hyperbolic geometry. Flavors of Geometry[END_REF].

Simple computations show that the matrix X can be written:

X =     1 + 2w 1 + (w 2 1 + w 2 2 ) 1 -(w 2 1 + w 2 2 ) 2w 2 1 -(w 2 1 + w 2 2 ) 2w 2 1 -(w 2 1 + w 2 2 ) 1 -2w 1 + (w 2 1 + w 2 2 ) 1 -(w 2 1 + w 2 2 )     , (47)
in the w-parametrization. Proposition 3 Let X be an element of C 1 written under the form (47), we have:

Trace (X -1 dX) 2 2 = 4 (dw 1 ) 2 + (dw 2 ) 2 (1 -(w 2 1 + w 2 2 ) 2 = ds 2 D . (48)
Proof. Cayley-Hamilton theorem implies the following equality, where A denotes a 2 by 2 matrix:

(Trace(A)) 2 = Trace(A 2 ) + 2Det(A) . (49)
We apply this equality to the matrix A = X -1 dX. The matrix X can be written as:

X = 1 + |z| 2 1 -|z| 2 I 2 + X 1 1 -|z| 2 , (50)
where:

X 1 = 2w 1 2w 2 2w 2 -2w 1 , (51)
and z = w 1 + iw 2 . We have:

X -1 = 1 + |z| 2 1 -|z| 2 I 2 - X 1 1 -|z| 2 , (52)
and:

dX = d 1 + |z| 2 1 -|z| 2 I 2 + dX 1 1 -|z| 2 + d(|z| 2 ) (1 -|z 2 |) 2 X 1 . (53)
Consequently:

X -1 dX = 2d(|z| 2 ) (1 -|z| 2 ) 2 I 2 - d(|z| 2 ) (1 -|z| 2 ) 2 X 1 + (1 + |z| 2 )dX 1 (1 -|z| 2 ) 2 - X 1 dX 1 (1 -|z| 2 ) 2 . ( 54
)
Since Trace(X 1 ) = Trace(dX 1 ) = 0 and Trace(X 1 dX 1 ) = 4d(|z| 2 ), then:

Trace(X -1 dX) 2 = 0 , (55)
and:

Trace (X -1 dX) 2 = -2Det(X -1 dX) = -2Det(dX) . (56)
We have also:

dX = d(|z| 2 ) (1 -|z| 2 ) 2 1 + |z| 2 + 2w 1 2w 2 2w 2 1 + |z| 2 -2w 1 (57) + 1 (1 -|z| 2 ) d(|z| 2 ) + 2dw 1 2dw 2 2dw 2 d(|z| 2 ) -2dw 1 . ( 58
)
Simple computations lead to:

Det(dX) = -4 (dw 1 ) 2 + (dw 2 ) 2 (1 -|z| 2 ) 2 , (59)
and end the proof.

This proposition means that C 1 equipped with the normalized Rao-Siegel metric, i. e. Trace (X -1 dX) 2 /2, is isometric to the Poincaré disk D of constant negative curvature equal to -1. Actually C is foliated by the level sets of the determinant with leaves that are isometric to D. This description, which is analog to the one considered by H. L. Resnikoff in [START_REF] Resnikoff | Differential geometry and color perception[END_REF], does not take into account the specific role of the state density matrices of the algebra H(2, R).

Another classical result of hyperbolic geometry asserts that the projection:

1 : L 1 -→ {α = 1} , (60)
defined by:

1 (α + v) = (1 + x) , (61)
with x = x 1 e 1 + x 2 e 2 and:

x i = v i α , (62)
for i = 1, 2, is an isometry between the level set L 1 and the Klein disk K, the Riemannian metric of which is given by:

ds 2 K = (dx 1 ) 2 + (dx 2 ) 2 1 -(x 2 1 + x 2 2 ) + (x 1 dx 1 + x 2 dx 2 ) 2 (1 -(x 2 1 + x 2 2 )) 2 , (63)
see [START_REF] Cannon | Hyperbolic geometry. Flavors of Geometry[END_REF]. An isometry between K and D is defined by:

x i = 2w i 1 + (w 2 1 + w 2 2 ) , (64)
w i = x i 1 + 1 -(x 2 1 + x 2 2 ) , (65)
for i = 1, 2. In other words, we have the following commutative diagram of isometries:

L 1 / / 1 L 1 / / π1 C 1 K / / D / / D (66)
Let us recall that the state density matrices of the quantum system we consider can be identified with the elements:

a = (1 + v)/2 , (67)
of the spin factor R ⊕ R 2 with v 2 ≤ 1. Let us denote:

1/2 : L 1/2 = {a = (α + v)/2, α > 0, a • a = 1/4} -→ {α = 1/2} , (68)
the projection given by:

1/2 ((α + v)/2) = (1 + v/α)/2 . ( 69
)
We have:

1/2 ((α + v)/2) = 1 (α + v)/2 . ( 70
)
This means that the map:

ϕ : K -→ K 1/2 , (71)
defined by:

ϕ(x 1 , x 2 ) = (x 1 , x 2 )/2 , ( 72
)
is an isometry between K and:

K 1/2 = {x/2 ∈ R 2 , x 2 < 1} , (73)
the Riemannian metric on the latter being given by:

ds 2 K 1/2 = (ϕ -1 ) * ds 2 K = (dx 1 ) 2 + (dx 2 ) 2 1/4 -(x 2 1 + x 2 2 ) + (x 1 dx 1 + x 2 dx 2 ) 2 (1/4 -(x 2 1 + x 2 2 )) 2 . ( 74
)
One can verify that L + is foliated by the level sets α = constant with leaves that are isometric to the Klein disk K. This description is more appropriate than the above one to characterize perceived colors from real quantum states since the state space S is naturally embedded in L + , see (78) below.

Let us recall some basic facts about the geometry of the Klein disk. Contrary to the Poincaré disk, the geodesics of K are straight lines and more precisely, the chords of the unit disk. An important feature of the Klein metric is that it coincides with the Hilbert metric defined as follows. Let p and q be two interior points of the disk and let r and s be the two points of the boundary of the disk such that the segment [r, s] contains the segment [p, q]. The Hilbert distance between p and q is defined by:

d H (p, q) = 1 2 log[r, p, q, s] , (75)
where:

[r, p, q, s] = q -r p -r

× p -s q -s , ( 76
)
is the cross-ratio of the four points r, p, q and s [START_REF] Beardon | The Klein, Hilbert and Poincaré metrics of a domain[END_REF] (in (76), • is the Euclidean norm).

Perceived colors and chromatic states

We describe the space P of perceived colors under the only hypothesis that P can be described from the state space of a quantum system characterized by the Jordan algebra H(2, R). As explained before, we exploit the fact that H(2, R) is isomorphic to the spin factor R ⊕ R 2 . Let us recall that this description does not involve any reference to physical colors or to an observer. The state space S is the unit disk embedded in the space of state density matrices by:

s = (v 1 , v 2 ) -→ ρ(v 1 , v 2 ) = 1 2 1 + v 1 v 2 v 2 1 -v 1 . ( 77
)
and in the Klein disk K 1/2 of the closure L + of the future lightcone L + by:

s = (v 1 , v 2 ) -→ 1 2 (1 + v) = 1/2 + (v 1 /2, v 2 /2) . ( 78
)
In order to describe perceived colors, i.e. measured colors, it is necessary to characterize all the possible measurements that can be performed on the states. We adopt here the viewpoint of the generalized probability theory [START_REF] Janotta | Generalized probability theories: what determines the structure of quantum theory?[END_REF].

We denote C(S) the state cone defined by:

C(S) =      α    v 1 v 2 1    , α ≥ 0, s = (v 1 , v 2 ) ∈ S      . ( 79
)
This cone is self-dual, that is:

C(S) = C * (S) = {a ∈ A, ∀b ∈ C(S), a, b ≥ 0} , ( 80
)
where •, • denotes the inner product of A given by [START_REF] Yilmaz | Color vision and a new approach to general perception[END_REF]. By definition, an effect is an element e of C * (S) such that e(s) ≤ 1 for all s in S. Such an effect e can be seen as an affine function e : S -→ [0, 1] with 0 ≤ e(s) ≤ 1 for all s. It is the most general way of assigning a probability to all states. Effects correspond to positive operator valued measures. In the present settings, every effect is given by a vector e = (a 1 , a 2 , a 3 ) such that:

0 ≤ e •    v 1 v 2 1    ≤ 1 , (81)
for all s = (v 1 , v 2 ) in S. The measurement effect associated to e is the operator:

E = a 3 Id 2 + a 1 σ 1 + a 2 σ 2 , (82)
that must satisfy 0 ≤ E ≤ Id 2 . This last condition implies that 0 ≤ a 3 ≤ 1, with

a 2 1 + a 2 2 ≤ a 2 3 and a 2 1 + a 2 2 ≤ (1 -a 3 ) 2 .
We denote E(S) the effect space of S, that is the set of all effects on S. Note that the so-called unit effect, e 1 = (0, 0, 1), satisfies e 1 (s) = 1 for all s in S.

A perceived color c = (a 1 , a 2 , a 3 ) is by definition an effect on S, that is an element of the effect space E(S). Since C(S) = C * (S), a perceived color c is an element of the state cone of S, this one being the closure L + of the future lightcone L + . The element c/(2a 3 ) = (a 1 /2a 3 , a 2 /2a 3 , 1/2), a 3 = 0, belongs to the Klein disk K 1/2 of L + . This suggests to define the colorimetric attributes of c as follows.

-The real a 3 , with 0 ≤ a 3 ≤ 1, is the magnitude of the perceived color c.

-The element s c = (a 1 /a 3 , a 2 /a 3 ) ∈ S is the chromatic state of c.

-A perceived color with a unit chromatic state is a pure perceived color.

-The saturation of a perceived color c is given by the Von Neumann entropy of its chromatic state.

-A perceived color whose chromatic state is the state of maximal entropy is achromatic.

Given a state (v 1 , v 2 ) ∈ S, the perceived colors which have this state as chromatic state form the intersection:

c s = E(S) ∩         a 3 v 1 a 3 v 2 a 3    , 0 ≤ a 3 ≤ 1      . ( 83
)
The maximum value of a perceived color c = (a 1 , a 2 , a 3 ) is:

   a 1 a 2 a 3    •    a 1 /a 3 a 2 /a 3 1    = a 2 1 + a 2 2 a 3 + a 3 = a 3 (1 + r 2 ) , (84)
with r 2 = (a 2 1 + a 2 2 )/a 2 3 . We must have:

0 ≤ r 2 ≤ 1 -a 3 a 3 ≤ 1 . (85)
If 0 < a 3 < 1/2, the measure of a perceived color c = (a 1 , a 2 , a 3 ) on his chromatic state (a 1 /a 3 , a 2 /a 3 ) gives the probability a 3 (1+r 2 ), this one being well defined for all 0 < r ≤ 1. In particular, primary colors are measured with the maximum probability 2a 3 . In this case, the magnitude is not high enough to allow measurements with probability 1. In this case, the perceived color is under-estimated.

If a 3 = 1/2, the measure of a perceived color c = (a 1 , a 2 , 1/2) on his chromatic state (2a 1 , 2a 2 ) gives the probability (1 + r 2 )/2. This probability is well-defined for all 0 < r ≤ 1. It is maximal, equal to 1, if and only if c is a primary color. In this case, the perceived color is ideally-estimated.

If 1/2 < a 3 < 1, the measure of a perceived color c = (a 1 , a 2 , a 3 ) on his chromatic state (a 1 /a 3 , a 2 /a 3 ) gives the probability a 3 (1 + r 2 ). This probability is well defined if and only if equation (85) is satisfied. In particular, primary colors can not be measured on their chromatic states. For instance, if a 3 = 2/3, then r should be less than or equal to √ 2/2 and perceived colors with chromatic states of norm equal to √ 2/2 are measured with probability 1. In this case, the perceived color is over-estimated.

An achromatic perceived color c = (0, 0, a 3 ) measured on a chromatic state gives the probability a 3 and this independently of the considered chromatic state. Such a perceived color does not take into account chromaticity. The unit perceived color c = e 1 is the saturated achromatic perceived color.

We explain now how to describe the Hering color opponency mechanism [START_REF] Hering | Zur lehre vom lichtsinne[END_REF], [START_REF] Krantz | Color measurement and color theory: II. opponent-colors theory[END_REF], from the rebit system introduced at the end of Section 3. Let us first quote D. H. Krantz [START_REF] Krantz | Color measurement and color theory: II. opponent-colors theory[END_REF]: "E. Hering noted that colors can be classified as reddish or greenish or neither, but that redness and greenness are not simultaneously attributes of a color. If we add increasing amounts of a green light to a reddish light, the redness of the mixture decreases, disappears, and gives way to greenness. At the point where redness is gone and greenness is not yet present, the color may be yellowish, bluish, or achromatic. We speak of a partial chromatic equilibrium, with respect to red/green... Similarly, yellow and blue are identified as opponent hues..." We rename |g = |u 1 , |r = |d 1 , |b = |u 2 and |y = |d 2 the four state vectors characterizing the rebit. The opponency mechanism is given by the two matrices σ 1 and σ 2 . More precisely, the state vector:

|(1, θ) = cos(θ/2)|g + sin(θ/2)|r , (86) satisfies:
(

1, θ)|σ 1 |(1, θ) = cos θ, (1, θ)|σ 2 |(1, θ) = sin θ . (87)
This means that if cos θ > 0, then the pure chromatic state s(θ) of the Bloch disk with coordinate θ is greenish, and if cos θ < 0, then s(θ) is reddish. For θ = π/2, or θ = 3π/2, s(θ) is achromatic in the opposition green/red. In the same way, if sin θ is positive, then s(θ) is bluish, and if sin θ is negative, then s(θ) is yellowish. For θ = 0, or θ = π, s(θ) is achromatic in the blue/yellow opposition. The phenomenon "that redness and greenness are not simultaneously attributes of a color", for instance, is a trivial consequence of the fact that (1, θ)|σ 1 |(1, θ) can not be simultaneously positive and negative. It appears in consequence that the quantum model that we propose allows to recover axiomatically that a chromatic pure state, that is a hue, is given by a pair of splittings similar to the two spin up and down inversions of a rebit. Following L. E. J. Brouwer, "Newton's theory of color analyzed light rays in their medium, but Goethe and Schopenhauer, more sensitive to the truth, considered color to be the polar splitting by the human eye" [START_REF] Brouwer | Life, art, and mysticism[END_REF] (see also [START_REF] Goethe | Theory of Colours[END_REF] and [START_REF] Schopenhauer | On Vision and Colors[END_REF]).

Group actions and homogeneity

Let us first recall that the special Lorentz group SO + (1, 2) is the identity component of the group O(1, 2), this latter being the matrix Lie group that preserves the quadratic form:

(α + v) M = α 2 -v 2 , (88)
where (α+v) belongs to the spin factor R⊕R 2 . The fact that SO + (1, 2) acts linearly on L + means that it acts projectively on the set of lines of L + and consequently on the points of the Klein disk K 1/2 , [START_REF] Ghys | Groups acting on the circle[END_REF]. Moreover, this projective action gives the isometries of K 1/2 . The subgroup of SO + (1, 2) that fixes (1 + 0) may be identified with the group of rotations SO(2) and in fact every element g of SO + (1, 2) can be decomposed in a unique way as [START_REF] Johns | Analytical Mechanics for Relativity and Quantum Mechanics[END_REF]:

g = b ζ r ξ , (89)
where b ζ is a boost map and r ξ is a proper rotation. More precisely, if we consider the coordinates (α, v 1 , v 2 ) in L + , the matrix associated to b ζ is given by:

M (b ζ ) =    cosh(ζ 0 ) ζ x sinh(ζ 0 ) ζ y sinh(ζ 0 ) ζ x sinh(ζ 0 ) 1 + ζ 2 x (cosh(ζ 0 ) -1) ζ x ζ y (cosh(ζ 0 ) -1) ζ y sinh(ζ 0 ) ζ x ζ y (cosh(ζ 0 ) -1) 1 + ζ 2 y (cosh(ζ 0 ) -1)    , (90)
where (ζ x , ζ y ) is a unit vector of R 2 and ζ 0 is the rapidity of the boost. It should be noted that the set of boosts is not a subgroup of the special Lorentz group. The matrix associated to r ξ is given by:

M (r ξ ) =    1 0 0 0 cos ξ -sin ξ 0 sin ξ cos ξ    , (91)
In order to illustrate the projective action of the boost b ζ on the Klein disk K 1/2 , let us consider a simple example (quite more complicated computations give similar results in the general case). We choose ζ x = 1, ζ y = 0, and denote ζ = tanh(ζ 0 ). The image (α, w 1 , w 2 ) of a vector (1/2, cos θ/2, sin θ/2) is given by:

     2α = cosh(ζ 0 ) + sinh(ζ 0 ) cos θ 2w 1 = sinh(ζ 0 ) + cosh(ζ 0 ) cos θ 2w 2 = sin θ . (92)
This means that the image of the boundary point (cos θ/2, sin θ/2) is the boundary point (v 1 , v 2 ) with:

         2v 1 = ζ + cos θ 1 + ζ cos θ 2v 2 = (1 -ζ 2 ) 1/2 sin θ 1 + ζ cos θ . (93)
One may notice that the map sending the point (cos θ/2, sin θ/2) to the point (v 1 , v 2 ) is an element of the group P SL(2, R).

Now, let us explain how these computations are related to the experiments 2 and 3 described by H. Yilmaz in [START_REF] Yilmaz | On color perception[END_REF]. The image of the point R = (1/2, 0) is the point R = R = (1/2, 0) (we use the notations of [START_REF] Yilmaz | On color perception[END_REF]). So, the point remains unchanged. The image of the point Y = (0, 1/2) is the point Y = (ζ/2, ( 1 -ζ 2 ) 1/2 /2) and the point Y has moved on the boundary, the angle φ of [START_REF] Yilmaz | On color perception[END_REF] (p. 12) being given by: sin φ = ζ. This means that the experiments 2 and 3 of H. Yilmaz can be interpreted as a relativistic aberration effect on colors [START_REF] Landau | The Classical Theory of Fields[END_REF].

The fact that boost maps act on pure states with P SL(2, R) transformations is not surprising in view on the following result [1] . Proposition 4 Every pure state generates a one parameter subgroup of boosts. Proof. As seen before, the state density matrix of a pure state is given by:

ρ(v 1 , v 2 ) = 1 2 (Id 2 + v • σ) , (94)
where v = (v 1 , v 2 ) is a unit vector and σ = (σ 1 , σ 2 ). The matrices σ 1 and σ 2 are symmetric traceless matrices that are usually chosen to be the two first generators [1] It is tempting to draw parallels between this results and the more general point of view of [START_REF] Connes | Von neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories[END_REF].

of the Lie algebra sl(2, R) of the group SL(2, R). Note that they do not generate a sub Lie algebra of sl(2, R). The matrix:

A(ρ, ζ 0 ) = exp ζ 0 v • σ 2 , (95)
is a symmetric element of P SL(2, R), ζ 0 being a real parameter. Let us recall that the P SL(2, R) action on H(2, R) is defined by:

X -→ AXA t . (96)
We have clearly: Det(AXA t ) = Det(X). Since σ 1 and σ 2 are elements of H(2, R), we can consider the matrices given by:

σ i -→ A(ρ, ζ 0 )σ i A(ρ, ζ 0 ) , (97)
for i = 0, 1, 2, with σ 0 = Id 2 . It can be shown that the 3×3 matrix with coefficients:

M (ρ, ζ 0 ) ij = 1 2 Trace (σ i A(ρ, ζ 0 )σ j A(ρ, ζ 0 )) , (98) is
a boost b ζ with ζ = tanh(ζ 0 )(v 1 , v 2 )
. Let us verify it on a simple example where v 1 = 1 and v 2 = 0. In this case:

A(ρ, ζ 0 ) = exp ζ 0 v 1 σ 1 2 = exp ζ 0 σ 1 2 = e ζ0/2 0 0 e -ζ0/2 . ( 99
)
We only need to compute the coefficient B(σ, ζ 0 ) i,j for i ≤ j. We have:

                   M (σ, ζ 0 ) 00 = 1 2 Trace A 2 (ρ, ζ 0 ) = cosh(ζ 0 ) M (σ, ζ 0 ) 01 = 1 2 Trace (A(ρ, ζ 0 )σ 1 A(ρ, ζ 0 )) = sinh(ζ 0 ) M (σ, ζ 0 ) 02 = 1 2 Trace (A(ρ, ζ 0 )σ 2 A(ρ, ζ 0 )) = 0 M (σ, ζ 0 ) 11 = 1 2 Trace (σ 1 A(ρ, ζ 0 )σ 1 A(ρ, ζ 0 )) = cosh(ζ 0 ) M (σ, ζ 0 ) 12 = 1 2 Trace (σ 1 A(ρ, ζ 0 )σ 2 A(ρ, ζ 0 )) = 0 M (σ, ζ 0 ) 22 = 1 2 Trace (σ 2 A(ρ, ζ 0 )σ 2 A(ρ, ζ 0 )) = 1 . (100)
This means that M (ρ,

ζ 0 ) = M (b ζ ) with ζ = tanh(ζ 0 )(1, 0), see equation (90).
One can easily verify that the image of the vector (1/2, 0, 0) of L + by the boost b ζ = tanh(ζ 0 )(1, 0) is the vector (cosh(ζ 0 )/2, sinh(ζ 0 )/2, 0). In consequence, the state of maximal entropy ρ 0 = (0, 0) is sent to the state (tanh(ζ 0 )/2, 0). This extends to general boosts.

We can summarize these computations in the following way. As before, we consider the state space S as the Klein disk K 1/2 of the closure L + of the future lightcone L + by using the map:

s = (v 1 , v 2 ) -→ 1 2 (1 + v) = 1/2 + (v 1 /2, v 2 /2) . (101)
Every pure state ρ generates a one parameter subgroup of boosts, the parameter ζ 0 being the rapidity. Actually, every boost can be obtained in this way. Boost maps act on the Klein disk K 1/2 by isometries. If we consider S as embedded in the space of state density matrices by:

s = (v 1 , v 2 ) -→ ρ(v 1 , v 2 ) = 1 2 1 + v 1 v 2 v 2 1 -v 1 , (102)
the one parameter subgroup of boosts is obtained by considering the action of P SL(2, R) on H(2, R). It is important to notice that we use only the matrices σ 0 , σ 1 and σ 2 , i.e. only information from S. Since every state can be obtained from the state of maximal entropy, boosts, or equivalently pure states, act transitively on S. However, one has to pay attention to the fact that boost maps do not form a subgroup of the special Lorentz group which is reflected by the fact that σ 1 and σ 2 do not form a sub Lie algebra of the Lie algebra sl(2, R). This point of view is quite different from the approach adopted in [START_REF] Resnikoff | Differential geometry and color perception[END_REF]. As mentioned in the introduction and explained in [START_REF] Provenzi | Geometry of color perception. part 1: A homogeneous color space[END_REF], one of the key arguments of H. Resnikoff is the existence of a transitive action of the group denoted GL(P) on the space P of perceived colors. This group is supposed to be composed of all linear changes of background illumination. In what preceds, we make use of the action of P SL(2, R) on H(2, R), see (96). But the matrices A(ρ, ζ 0 ) of P SL(2, R) that are used are also symmetric, due to the fact that σ 1 and σ 2 are symmetric. Actually, the action (96) can be also viewed as the action:

X -→ AXA , (103)
of the Jordan algebra H(2, R) on itself. This is precisely the action:

Q(A) : X -→ (2L(A) 2 -L(A 2 ))X , (104)
of the quadratic representation of A on X [START_REF] Faraut | Analysis on Symmetric Cones[END_REF]. But once again, the matrices X that we consider are σ 0 , σ 1 and σ 2 . The matrices σ 1 and σ 2 are not elements of the positive cone H + (2, R). It appears in consequence that the homogeneity of H + (2, R) is not so important in our approach. Instead of postulating the existence of a group of linear changes of background illumination, we have shown that the quantum description that we propose naturally leads to consider boost maps as illumination changes. These illumination changes are isometries of the Klein disk K 1/2 .

Conclusion

We have shown that the space P of perceived colors can be interpreted as the effect space of a real quantum system, more precisely a rebit. The state space S of this system is isometric to the Klein disk K of constant negative curvature. The chromatic state space of perceived colors is a Bloch disk of real dimension 2 that coincides with the Hering disk given by the color opponency mechanism. This quantum description creates a deep connection between the works of H. Yilmaz [START_REF] Yilmaz | Color vision and a new approach to general perception[END_REF], [START_REF] Yilmaz | On color perception[END_REF] and H.L. Resnikoff [START_REF] Resnikoff | Differential geometry and color perception[END_REF]. In particular, it allows to recover in a natural way the following result of H. L. Resnikoff: the surface of constant magnitude are isometric to the Klein disk of constant negative curvature equal to -1 [START_REF] Resnikoff | Differential geometry and color perception[END_REF]. Finally, one may envisage to characterize the individual color vision from a convex subset of the effect space E(S) and/or a convex subset of the state space S endowed with the Hilbert metric. Further investigations will be devoted to the properties of this metric space in connection with MacAdam ellipses [START_REF] Macadam | Visual sensitivities to color differences in daylight[END_REF].

  It is important to notice that the four state density matrices ρ(1, 0), ρ(1, π), ρ(1, π/2) and ρ(1, 3π/2) correspond to two pairs of state vectors (|u 1 , |d 1 ), (|u 2 , |d 2 ), the state vectors |u i and |d i , for i = 1, 2, being linked by the "up and down" Pauli-like matrix σ i .

Acknowledgements

The author would like to thank E. Provenzi and C. Choquet for helpful comments on earlier versions of this article.

Funding

This work was partially supported by the french CNRS through the project GOALVISION and by the french region Nouvelle Aquitaine through the project RECOGER.

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