Yasin Memiş
email: ymemis@anadolu.edu.tr

Comparison of Japanese and Turkish textbooks: Giving opportunities for creative reasoning in terms of proportion

Keywords: Mathematical creative reasoning, proportional reasoning, textbook analysis

Empirical research indicates that students, when solving tasks, frequently use superficial similarities to find the right algorithm. Furthermore, mathematics textbooks continue to use a teaching design that supports rote learning. This kind of approach creates difficulties in terms of promoting proportional reasoning as well as other mathematical skills. The purpose of this study is to examine the extent to which selected Grade 7 Japanese and Turkish textbooks provide opportunities to use creative mathematical reasoning, in terms of proportion topics. The results imply that proportion topics in the selected Turkish textbook mostly design opportunities for students to use algorithmic mathematical reasoning. However, Japanese textbook provides more creative mathematical reasoning opportunities.

Introduction

Proportional reasoning is a very important skill that students must acquire in middle school. The National Council of Teachers of Mathematics (NCTM) (2000) has highlighted that proportional reasoning should be "an integrative theme in the middle grades mathematics program" (p. 212). Being able to reason proportionally, on the one hand, plays a key role in the meaning of many mathematical content areas (e.g. similarity, trigonometry, probability), while at the same time constituting the basis of the concepts (e.g. speed, density) often encountered in real life [START_REF] Lesh | Proportional Reasoning[END_REF]. However, most students struggle with developing proportional reasoning, and one of the reasons for this failure is related to weak teaching-design methods. Typically, a textbook on the subject of ratio and proportion starts by showing the different ways of writing the ratio, and the proportion is defined as two equal ratios. Then, students usually focus on finding answers to similar questions in the cross-multiplication algorithm. Students frequently decide that a task involves a proportion following superficial cues such as, "the problem gives 3 numbers, and 1 other number is missing". When solving this type of proportion task, students only need to recognise the task, insert the relevant numbers, and perform the calculation (Labato & Ellis, 2010). In this process, only the result is important; students do not usually concern themselves with proportional reasoning. The correct solution to this kind of example or task used by students is insufficient to show that a degree of proportional thinking has occurred at the desired level. The repetition of similar questions will allow for more practice of multiplication and division rather than improving proportional thinking. Furthermore, the continuance of such algorithmic questions will result in the student not being sufficiently educated about the multiplicative relationship required for proportional thinking. Instead, situations in which students create their own solutions will provide opportunities for them to connect with previous information and, in particular, allow them to improve multiplicative thinking (Labato & Ellis, 2010).

Unfortunately, this kind of instructional design that uses highly algorithmic tasks is presented in the textbooks. Textbooks frequently provide questions that are similar to each other, and students are expected to engage in solving them [START_REF] Boesen | The relation between types of assessment tasks and the mathematical reasoning students use[END_REF][START_REF] Hiebert | What research says about the NCTM standards[END_REF]. Studies also show that students who decide on which algorithm to use when solving a problem with superficial evidence usually try to remember the algorithm and then try to find the result [START_REF] Mcneal | Learning not to think in a textbook-based mathematics class[END_REF][START_REF] Kamii | To teach or not to teach algorithms[END_REF]. This situation is similar to proportional reasoning in Turkish and other countries' textbooks. The research indicates that proportional reasoning tasks in various countries' textbooks are mostly low-level and contain highly familiar routine problems [START_REF] Da Ponte | Proportion in school mathematics textbooks: A comparative study[END_REF][START_REF] İncikabı | A Comparative Analysis of Ratio and Proportion Problems in Turkish and the US Middle School Mathematics Textbooks[END_REF]. Learning to reason proportionally takes effort and time, and in addition to routine problems, non-routine problems are also needed to help students develop. The algorithms presented without letting students create the ideas they need will not provide them with opportunities to improve their proportional reasoning skills (Labato & Ellis, 2010).

Textbooks play a key role in many important areas, such as teaching design and student achievement. Since mathematics textbooks are a bridge between students and teachers, it could be said that textbooks are effective tools that offer students learning opportunities [START_REF] Rezat | A model of textbook use[END_REF]. Due to this important role of the textbook, it could be considered as a mediator between the intended and the implemented curriculum (Schmidt, McKnight, & Raizen, 1997), and therefore, textbooks are important research areas in terms of mathematics education. Due to the important roles of textbooks and proportional reasoning, it is possible to find both national (Bayazıt, 2013;[START_REF] Dole | The capacity of two Australian eighth-grade textbooks for promoting proportional reasoning[END_REF] and cross-national studies [START_REF] Da Ponte | Proportion in school mathematics textbooks: A comparative study[END_REF][START_REF] İncikabı | A Comparative Analysis of Ratio and Proportion Problems in Turkish and the US Middle School Mathematics Textbooks[END_REF]. Although there are studies on how the Turkish reform curriculum handles mathematics education, studies about how to design the contents of proportional reasoning in textbooks are very limited. For this reason, this cross-national study will make an important contribution to those who prepare education programmes, as well as textbook authors.

The aim of this research is to compare the extent to which selected Grade 7 Japanese and Turkish textbooks provide opportunities for creative mathematical reasoning in terms of proportion.

The two central research questions in this investigation are as follows:

1. To what extent does the teaching design of selected Japanese and Turkish textbooks support the development of creative reasoning in the chapters on proportion? 2. What kind of reasoning is expected from students for solving the tasks presented in the textbooks?

Conceptual framework

Algorithms are important components in mathematics education; however, algorithms presented to students without preparing an adequate background prevent the development of mathematical skills. The students will use the algorithms they have learnt to solve the questions that they classify as superficial. This process, like any mathematical skill, will prevent the development of the desired level of proportional thinking (Labato & Ellis, 2010). Lithner's framework of mathematical reasoning types sets apart algorithmic and creative reasoning, and so it could present a different perspective that examines the extent to which textbooks promote the development of proportional reasoning.

In this study, Lithner's framework of mathematical reasoning types will be used [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF], which was first created as a result of empirical studies in the field [START_REF] Bergqvist | Upper secondary students task reasoning[END_REF][START_REF] Lithner | Mathematical reasoning in calculus textbook exercises[END_REF][START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF]. Throughout those studies, reasoning was defined as "the line of thought adopted to produce assertions and reach conclusions in task-solving" (Lithner, 2008, p. 257). [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] identified two main reasoning types: Imitative Reasoning (IR) and Creative Reasoning (CR). The main distinction between these two reasoning types is that IR can be identified as copying a solution process or as using an algorithm directly from a source example or from a textbook narrative, whilst CR creates a product from a creative mathematical thinking process (Figure 1). MR is characterised by the idea that "the strategy choice is founded on recalling a complete answer, the strategy implementation consists only of writing it down" (Lithner, 2008, p.258). An example of this is writing a proof step that has already been memorised. The AR sequence should function as follows: "the strategy choice is to recall a solution algorithm. The remaining reasoning parts of the strategy implementation are trivial for the reasoner, only a careless mistake can prevent an answer from being reached" (Lithner, 2008, p. 259).

CR can also be separated into two sub-categories: Local Creative Reasoning (LCR) and Global Creative Reasoning (GCR).

LCR contains major parts of IR (e.g. recalling facts and using algorithmic sub-procedures), and minor parts of CMR. This means that a LCR task solution is always based on algorithms, although the student needs to modify the algorithm locally in a "non-trivial" way. A GCR task does not have a solution that is based on an algorithm, and therefore requires CR throughout the whole tasksolving process. GCR may contain construction of an example, proof of something new, or modelling [START_REF] Bergqvist | Upper secondary students task reasoning[END_REF].

CR fulfils the following three properties:

1. "Novelty: a new (to the reasoner) reasoning sequence is created or a forgotten one is recreated". 2. "Plausibility: there are arguments supporting the strategy choice and/or strategy implementation motivating why the conclusions are true or plausible". 3. "Mathematical foundation: the arguments are anchored in intrinsic mathematical properties of the components involved in the reasoning" (Lithner, 2008, p. 266).

For the analysis of this study, textbooks will be examined as having two main parts: the tasks and examples provided to the students. In this study, a mathematical task is defined as a question that requires a solution from students, whilst an example is defined as a question that is given with an appropriate solution or solution process.

Method

Japanese and Turkish textbooks were selected for this study. The reason for this choice is primarily the success of Japanese students in international examinations, such as Trends in International Mathematics and Science Study (TIMSS). The 2015 TIMSS results saw 15-yearold Japanese students ranked 5 th , and Turkish students ranked 40 th , in terms of mathematics performance. The results of the 2015 TIMMS for Grade 8 students on the topics of "ratio, proportion and percent" emphasised the difference between the achievements of students from these countries. In response to questions within this study, 67% of Japanese students got the correct answers, compared with just 34% of Turkish students.

The textbooks that have been chosen for analysis in this study are Mathematics International Grade 7 (Fujii & Matano, 2012), published by Tokyo Shoseki, and Mathematics Textbook Grade 7 [START_REF] Bilen | Ortaokul matematik 7 ders kitabı[END_REF], published by Gizem in Turkey. These two textbooks were passed through an approval process by their respective governments for use in classrooms in these two countries. The Tokyo Shoseki series, which has been translated into English, has a big market in Japan [START_REF] Miyakawa | Comparative analysis on the nature of proof to be taught in geometry: the cases of French and Japanese lower secondary schools[END_REF].

For the purpose of this study, only the proportion chapter was examined. The proportion topic is included in the Grade 7 textbook in both countries. In the Grade 7 Turkish textbook, the chapter on proportion is presented with two subdivisions, including nine examples and ten tasks. In the Grade 7 Japanese textbook, there were only three examples and 18 tasks in the proportion chapter.

The examples and tasks were examined for analysis and were classified according to mathematical reasoning frameworks. Although the CR process is similar to the solution structure of non-routine problems, the framework is more focussed on the distinction between imitation and creation [START_REF] Lithner | Principles for designing mathematical tasks that enhance imitative and creative reasoning[END_REF]. For this reason, templates and solutions were considered. If the example/task could be solved with previous algorithms without any modification, they were classified as AR, whilst if the example/task could be solved with previous solution processes yet needed some modification to this process, they were classified as LCR. However, if the example/task did not have a solution that was based on an already-given algorithm and required CR throughout the whole solution process, it was classified as GCR.

Figure 2 shows two examples from the Grade 7 Turkish textbook in the proportion chapter. These examples were classified as AR, as they present a teaching design that simply requires replacing the appropriate numbers in the template solutions in order to do the calculations correctly.

1. You need 2 cups of rice and 3 cups of water to cook rice. In this case, how much water is needed for 1 cup of rice?

2. 12 small bottles of 50ml cologne costs 15 TL.

If you want to buy 1 small bottle of 50ml cologne, how much would it cost? There are five tasks at the end of the chapter, which students are required to solve. However, to complete these tasks, they only need to remember the previous solution process. Students need to replace the numbers in the template and perform basic calculations to obtain the solution. For this reason, the task below (in Figure 3) is classified as AR.

1) If 3 buses have 132 seats, how many seats are there in 1 bus? The task below (in Figure 4), taken from the Japanese textbook, was determined to be of the LCR type. There are no similar tasks from previous examples, and so students cannot solve them by directly using cross multiplication algorithms (a:b=c:d); instead, they need to carry out local modifications to the previous solution methods (160+4x:60+3x=2:1). The example below (in Figure 5), taken from the Japanese textbook, was determined to be GCR. This example investigates whether "3: 4 = 6: 8" is true. There is an argument given that "if a: b = m: n then a. n = b. m", and it is then proved using mathematical foundations. Finally, the example presents a generalisation, which provides the student with a totally new creative process with regards to the proofing argumentation.

Results

In this chapter, the pilot study results of the classified mathematical reasoning types in the proportion chapters in Turkish and Japanese textbooks are presented (Tables 1 and2). The first conclusion drawn from the report of the study is that the Turkish textbook features no GCR-type tasks or examples, and furthermore, ten of the tasks within the chapter do not even require any CR reasoning. These results show that the Turkish textbook only provides limited and weak opportunities for students to perform CR. In addition, the Turkish textbook's teaching style mostly shows the necessary algorithm for solving a specific task within the examples, which means that when the students are solving the tasks at the end of the chapters, these algorithms can be used directly. This type of teaching style mostly supports IR, especially AR. In contrast, within the Japanese textbook's proportion chapter, there is a total of three examples. Two of these examples are classified as AR, and the other example is classified as GCR. There are 18 tasks for the students to solve, with seven of these tasks requiring CR. This result reveals that the Japanese textbook provides more, and stronger, opportunities for CR than the Turkish textbook in the proportion chapter.

Discussion

The development of proportional thinking is a milestone for students, especially for those who take many years to progress. In this process, students will be supported by different problem-solving situations as much as possible, to support healthy development. The mathematical reasoning framework has shown that the Grade 7 Turkish textbook mostly provides opportunities for AR only in terms of learning proportion, in contrast to the Grade 7 Japanese textbook.

Algorithms have an important place in mathematics, as they are an effective solution to a general class task. There are short-term gains and benefits, such as the 'quick response to tasks'; however, AR can be a barrier to obtaining effective skills that can be used in the long term. Using algorithms in proportional reasoning can create the same problem. A teaching design based solely on using algorithms will, in the long term, allow students to solve familiar tasks that they recognise as only being superficial. The fact that students are engaged in different problem situations that require CR will lead them to understand the components of proportionality (multiplicative thinking, absolute thinking, etc.) in greater depth. Proper use of CR in the textbooks would contribute to the solution of this problem and should enable further development of students' mathematical ability. The results of this study offer useful information on content design for educators, especially textbook authors. Applying the process should provide teachers with insights into the strengths and weaknesses of the textbooks being employed and enable them to make effective decisions about the selection and use of textbooks. In future studies, the comparison of different book series from different countries will provide new data that could be used to evaluate the results of the study in a wider context.

Figure 1 :

 1 Figure 1: An overview of Lithner's (2008) mathematical reasoning framework

Figure 2 :

 2 Figure 2: Grade 7 Turkish textbook examples (p.127)

Figure 3 :

 3 Figure 3: Grade 7 Turkish textbook task (p.128)

Figure 4 :

 4 Figure 4: Grade 7 Japanese textbook task (p.112)

Figure 5 :

 5 Figure 5: Grade 7 Japanese textbook simple example (p.99)

Table 1 : Classification of tasks and examples in the Turkish textbook Table 2: Classification of tasks and examples in the Japanese textbook

 1

		Example	Task	Total
	MR	0	0	0
	AR	9	10	19
	LCR	1	0	1
	GCR	0	0	0
	IR (MR + AR)	9	10	19
	CR (LCR + GCR)	1	0	1


Meghna Nag Chowdhuri
email: nc.meghna@gmail.com

Two primary school teachers' pedagogical design capacity of using mathematics textbooks in Delhi, India

Keywords: Mathematics curriculum, primary teacher, pedagogic design capacity

India introduced a new National Curriculum framework in 2005, which included reformed goals for primary school mathematics in an attempt to make them more inclusive and student-centred. One of the efforts towards operationalizing the framework was to create new primary mathematics textbooks. In a context where textbooks are seen as the basis of all curricular activityfrom sequencing, to teaching, to examination, it becomes important to understand how such reformbased textbooks are used by teachers. Drawing from a multiple case study in the context of government-run primary schools in Delhi, this paper reports that teachers seem to primarily reject these reform-based textbooks. Further it shows that while using the textbook, teachers have varied levels of design capacities to productively interpret the textbook tasks. The findings imply that curriculum and policy makers need to develop materials that teachers are firstly willing to engage with, and also provide space for teachers' design capacity to develop.

Introduction

India, much like the rest of the world and especially the Global South, is attempting to improve teaching and learning practices in mathematics not just to increase learning outcomes, but also to ensure equity in its processes. In post-colonial India, quality and equity have remained fundamental to its Education system. These ideals are explicitly reflected in the National Curriculum Framework -2005(NCF 2005), which forefronts "processes" in mathematical activities as well as connecting mathematics to the lives of children (NCERT, 2005, p. 9). Another important feature of the Indian education system which is recurring in the literature is its 'textbook culture' [START_REF] Kumar | Political agenda of education: A study of colonialist and nationalist ideas[END_REF]. The textbook continues to be the only resource in the classroom; and for state-run schools, prescribed and made available to students free. [START_REF] Kumar | Political agenda of education: A study of colonialist and nationalist ideas[END_REF] even claims that textbooks act as "de facto curriculum" (p. 67), despite an official curriculum framework and syllabus. Thus, it was not surprising that the textbook series (NCERT Math-Magic) was chosen as the first (and only) point of intervention to operationalize the visions of NCF 2005. Yet very little is known in India about how the textbook is navigated in classrooms. In this paper, which comes from a larger PhD study, I focus on the following research question:

In what ways do teachers use reform-based textbooks in their mathematics classrooms?

Theoretical framework

Keeping the context in mind, as we explore the Anglophone literature in Mathematics education on the relationship between the teacher and the textbook (or other curricular resources), we see various developments. Different frameworks are used by researchers to make sense of these aspects based on how they conceptualise the teacher-textbook relationship as well as textbook as a resource. In this paper, I use [START_REF] Brown | Teaching by design: Understanding the intersection between teacher practice and the design of curricular innovations[END_REF] framework of "design capacity for enactment" (p. 4), to explore how teachers in the Indian context use textbooks. Central to Brown's idea is that teachers design textbook use by offloading-adapting-improvising based on the degree of responsibility they share with the textbook and their own resources. Thus, while using the textbooks, teachers may place agency primarily on the textbook (offloading), or change the activity based on their own needs (adapting), or change the course of the work altogether, making it very different from the textbooks (improvising) [START_REF] Brown | Teaching by design: Understanding the intersection between teacher practice and the design of curricular innovations[END_REF]. While there is no prescribed hierarchy to their choice of offloadingadapting-modifying (one is not better than the other), these indicate the relevance and priority given to the textbook as a resource in teaching. Furthermore, this use is a function of both textbook affordance (textbooks' own attributes) and teachers' pedagogical design capacity. The notion of pedagogical design capacity -"teacher's capacity to perceive and mobilize existing resources in order to craft instructional episodes" (Brown, 2011, p. 29) is evaluative in nature signifying teachers' capacity to use the textbook in productive ways. More recently [START_REF] Pepin | Refining teacher design capacity: Mathematics teachers' interactions with digital curriculum resources[END_REF] have further described 'teachers' design' as a "deliberate/conscious act" of "creating something new" (p. 801); thus, bringing dimensions of intentionality and genesis to the fore. They further distinguish between 'design' (during planning); and 'design-in-use' (during enactment). This exploratory paper, only focuses on teachers' 'design-in-use', as they mobilise textbooks' affordances and create lessons (without considering teachers' intentions). By doing so, I examine limited yet crucial facets of the notion of teachers' pedagogical design capacity.

Methodology

Sample: The larger study (a part of which is presented in this paper) is a multiple case study of ten teachers in four government-run municipality schools using Math-Magic textbooks. Delhi government schools, mandated to use these textbooks, were the site for the study. Importantly, these schools are low resourced both in terms of infrastructural resources, as well as teaching and learning resources. Getting permission to conduct my field work in these under-studied sites was particularly difficult. The ten primary school teachers were thus chosen based on convenience sampling, in schools which allowed me to conduct my study. However, I ensured that all the teachers had more than eight years of experience, including experience of using the textbook from the time of its introduction. Also, I focused on grades 4 and 5 teachers, where these textbooks were being used.

Data collection: Keeping the theoretical framing in mind, I focused on the three parts of the design capacity framework, and collected data on textbook affordances (the textbooks themselves), teachers' textbook use and thinking (through classroom observation and teacher interviews). While collecting classroom observation data (which is the data that I am analysing in this paper), I was given permission only to audio-record the classrooms, which was sufficient to capture the textbook use dimensions of teachers' teaching. The audio-recording was supplemented by observation notes, where I noted how the textbook was treated in the classroom (were the teachers reading it before starting the lesson, were they showing the textbook to students, did they provide time for students to read the text etc). In total, for each of ten teachers, I observed 3-4 lessons.

Data analysis: After transcribing and translating the audio recordings, the lessons were categorised into episodes which were defined on the basis of the tasks that teachers chose, so that an equivalent unit of analysis could be used to compare the textbook tasks with the teachers' chosen tasks. A total of 34 lesson observations from the ten teachers, were categorised into 152 episodes (each episode corresponding to a task from the textbook analysis). These episodes were coded based on Brown's notion of offload-adapt-improvise. However, while doing so, two differences arose. First, teachers seemed to be using tasks of their own (or with the help of some other resource material) which were completely unrelated to tasks in the textbook. Thus, a notion of 'insertion' was necessary which accounted for such episodes, as done by [START_REF] Leshota | Disaggregating a mathematics teacher's pedagogical design capacity[END_REF] in their adaptation of Brown's framework. Moreover, a second difficulty arose in defining the difference between adaptation and improvisation. One of the findings of the textbook analysis was that clearly stated goals for the tasks were missing, often leaving the tasks open to interpretation (perhaps even purposefully so) (e.g. Rampal & Subramanian, 2012). Thus, any modification to the tasks in the textbooks was coded as adapted use, without attempting to code episodes as being closer to the goals of the textbook task or not. As a result, a suitable adaptation to the 'offload-adaptation-improvisation' framework was made by identifying textbook task use in terms of 'direct use-adapted use-inserted use' which closely supported my data.

Overall findings

Teacher acceptance or rejection of the textbook: Figure 1 below shows details of the textbook use of each of the ten teachers. There are two notable results from this analysis. Firstly, we can see that four teachers are using the textbook directly for majority of their teaching, and the remaining six teachers are using their own insertions for more than half of their episodes. This is a clear indication that the textbook is not being used most of the teachers in my sample, who are rejecting them.

Direct use of textbook tasks: While analysing the episodes of 'direct textbook use', there were two ways in which teachers used tasks. One was by following each sentence in the book, (often teachers would read the text, or ask students to read it out), thus addressing every aspect of the book as a script. The second way of using which seemed more flexible was to pick and choose tasks (or elements of them) as the teacher felt suitable. These two kinds of approaches to directly using the tasks were especially exemplified by teachers Jagdeesh and Kamala, discussed in the next section. explored within each of these three types of textbook engagements. For this paper, I look at two frequent direct users and their contrasting interpretations and mobilisations of the textbook affordances; i.e. their design-in-use. Jagdeesh and Kamala, both have eight years of teaching experience and both teach grade 4 in the same school. Jagdeesh tended to follow every task given in the textbook, while Kamala was pickier in terms of what she chose to include in her class. In the following section, I analyse how two focus tasks (Figure 2, 3) were used by Jagdeesh and Kamala; as they designed two very different lessons. While Jagdeesh used the textbooks' affordances in productive ways (as afforded by the textbook), Kamala does not do so.

Contrasting cases of teacher design capacity

Focus tasks chosen for analysis and its affordances

The task in Figure 2 has two kinds of questions -'Have you heard about marathon races?' invites students to engage with the context of the problem which follows. Such questions which were posed to students encouraging them to talk about their thinking, life experiences and feelings were coded as expression tasks. The following two questions are specified tasks (fill the blanks) where students are expected to answer the pre-formulated question. It is crucial to note that the text breaks down (possibly for support) the specified tasks into two stages, first asking students to find distance covered in 10 rounds (laps), and indicate its use to find the number of rounds for the 40 km. In the second task shown below (Figure 3), we see another typical problem from the textbook which I refer to generation tasks. Here, the task is not pre-formulated, and to answer all these questions students have to bring in information from their own experiences or surrounding to answer the question. These particular aspects are viewed as task 'affordances', which are also aligned with the NCF reforms aiming to make mathematics process oriented and linked with everyday life.

Comparison of teacher use of focus tasks:

Table 1 below, shows the different approaches taken by Jagdeesh and Kamala, while attempting these tasks. To provide some context, these teachers teach all the subjects for their class and are free to allot any time for each of the subjects, including for mathematics. Thus, the amount of time for each task or subject depends on the teachers' own goals for the lesson (there are institutional time limitations due to administrative work which are not discussed in this paper).

Teacher

Expression task Specified task Generation task Jagdeesh (productive design)

Expanded discussion

Explanatory approach

Opening the task Kamala (unproductive design) Omitting versus expanding the expression task: Jagdeesh, instead of approaching the first expression task as a yes or no response, spends 10 minutes discussing different aspects of the contextwhat is a marathon, how a stadium looks like, what sports can be played in a stadium. He even encourages students to bring forth their own understandings, by listing the different sports they have heard being played in a stadium. He thus expands this expression task further creating his own probing questions supporting the task. On the other hand, Kamala omits this question all together, and only focuses on the specified tasks.

Omitted Direct approach Closing the task

Direct versus explanatory approach to specified task: In his approach to the specified tasks, as possibly intended by the task, Jagdeesh uses the first question as the first step to the solution of the second question. That is, first finding distance travelled in 10 rounds and then discussing how many 10 rounds would be needed for 40 km. Throughout these stages he focuses on a detailed explanation of the solution procedure (spending 20 minutes), by breaking down the bigger question into smaller parts, which are more easily answered by the students. For example:

T: [We have to run] 40 kilometres. Now in 10 rounds, [we covered] 4000 meters that is 4 kilometres. In the next 10 rounds, how much will it be? In the next 10, if we add more, then how much is it adding 4 and 4? 8 km. After that another 10 rounds if we add?

Thus, Jagdeesh focuses more on the process of the solution (with some support from the structure of the specified task) rather than presenting formal operations for solutionssuch as multiplication.

Kamala on the other hand only spends 10 minutes in total on the task. She introduces the stadium with circumference 400 m at the beginning of the lesson and draws it on the blackboard to help model the stadium. Consequently, she only focuses on the first questions of the given task unlike Jagdeesh's use of the both the questions to explain the process of the solution. Instead of an elaborate explanation, we see Kamala giving direct procedures.

T: Now, this is given in your book. If you take 10 rounds of this stadium, then what will you get? [] If you take one round of this stadium, then how much will it be? 400. You will multiply 400 with 10 rounds, then we will get how many metres we have run.

As we can see above, she directly tells the students that 400 needs to be 'multiplied' by 10 and consequently she writes the solution on the blackboard using the standard column algorithm form of multiplication. It is also important to note here, that Kamala was using an unofficial privately published guidebook while discussing this chapter alongside the textbook. The guidebook (see Figure 4) which acts like an answer book, provides a solution to each task in the textbook which just like her approach are also direct and formalised rather than process oriented (interestingly the guidebook even has a calculation error). Thus, instead of using the affordance within the textbook task (two questions as support for process explanation), Kamala takes a direct approach as given in the guidebook. Opening versus closing the generation task: While attempting the generation task (shown in Figure 3), we again see two very different approaches by Jagdeesh and Kamala. Jagdeesh while addressing this question (See Figure 3: How long does your assembly take?), tries to bring to fore the fact that the answer will vary based on the activity conducted during the assembly:

T: Hmm. Sometimes it is 15 minutes and sometimes it is 30 minutes. Sometimes, when there is a children's meeting, it may take longer, otherwise the time for prayer meeting is about 20 minutes which includes the national song.

This shows that the Jagdeesh is open to the possibility that generation tasks can have many answers and the aim is to keep that in mind while answering. In contrast, Kamala gives definite answers which the students have to write down.

T: Who said it is 30 minutes? It is only 20 minutes. Write down 20 minutes.

As we can see above, in Kamala's approach to the generation tasks, no openness is accommodated. Instead she renders the answer '30 minutes' given by a student wrong and goes on to give a 'correct' answer to the student. Thus, despite the question being open (depending on the student's perspective), she closes the kind of answers it can have.

If we look back at Table 1, along with the description of these episodes, we can see that while Jagdeesh seemed to interpret the key affordances within the tasks in alignment with the task's rationale; Kamala's interpretations contradict them. We can thus say that Jagdeesh seems to have a higher pedagogical design capacity with respect to these textbooks; compared to Kamala. It is important to note that here I have not included evidence from teachers' own thinking which will enhance our understanding of the dimension of 'intentionality' motivating these designs-in-use. Furthermore, this does not imply that Jagdeesh and Kamala will always teach in these ways, and thus these are not fixed labels for teachers rather helpful indicators to help situate teachers' work in relation to the textbooks.

Conclusion

In this I have discussed primary school teachers' relationship with mathematics textbooks, in the context of Delhi, a city where there is hardly any research in understanding how teachers interact with textbooks. Especially the perspective of viewing teachers as designers (and not just 'implementers'), actively participating while using the textbooks, is completely missing from the Indian policy discourse. Although the need for reforming curriculum and textbook has been articulated strongly in the last 15 years, there is a need to now put teachers at the centre of the discussion. In such a context, using this approach to study Indian teachers is a novel idea (even though in other parts of the world this idea has been developed in innovative ways). By focusing on such a framework, with some adaptation in terms of categories and unit of analysis, this paper reports two findings.

Firstly, the finding that majority of the teachers in my sample are rejecting the textbooks, is a common finding across the globe when teachers encounter reforms. Yet this finding has severe consequences in a context where officially, this is the only textbook provided to teachers, and the schools are not resourced to provide supplementary materials. As we have seen, this might mean that teachers rely on unofficial privately published guidebooks which often have contradictory pedagogical suggestions. Or teachers might rely on their own traditional notions of teaching, thus reforms miss the opportunity of using the textbook's potential of becoming 'educative material' (Davis & Krajkic, 2005).

Secondly, within the group of teachers who were primarily accepting and directly using textbook tasks, by focusing on two teachers we were able to more closely investigate teachers' design capacities as they engaged with the textbook's affordances. In case of the two teachers we clearly see two different ways of perceiving and mobilizing the resource around the tasks; ie their pedagogical design capacity. Jagdeesh is attempting to approach the tasks more openly, attempting to engage students with the contexts integrated in the text and also giving more explanatory solutions. On the other hand, Kamala does not pick up the opportunity of engaging children with the context, and tends to give direct formulaic solutions and closes tasks which are more open. While Jagdeesh seemed to interpret the key affordance within the textbook productively, Kamala does not. Recognising that teachers' design capacities are not homogenous is important in terms of not just curriculum designing but also the accompanying teacher support. Small scale initiatives are starting in science education, such as Ramadas (2017)where teachers' varied experiences with curriculum feeds back into curriculum design. Yet, these are almost absent in primary Mathematics. Developing partnerships of designing needs to be the next step for the Indian curricular context; which extends the view of teachers as designers and integrates that into curriculum designing (at national, state, school or classroom level) (Jones & Pepin, 2017). After more than 10 years of introduction of reform-based textbooks such as Math-magic, conceptualising ways of developing design capacities needs to become central.

In this paper, I have explored different designs-in-use of the teachers, yet not discussed notions of intentionality: Are teachers consciously making these design choices? What are their intentions and goals behind them? By exploring designs-in-use across the three levels of textbook use: direct useadapted use-inserted use, along with teachers' rationales for those choices, my larger study aims to develop this notion of 'pedagogical design capacity' further.

Figure 1 :

 1 Figure 1: Per cent of episodes observed categorised as direct, adapted or inserted textbook use

Figure 2 :

 2 Figure 2: Expression task and Specified task within Book 4 chapter 2

Figure 3 :

 3 Figure 3: Generation task within Book 4 chapter 4

Figure 4 :

 4 Figure 4: Page C-10, Apollo guidebook

  As discussed in my theoretical framework, the use patterns analysed above do not indicate the evaluative aspect of use; i.e teachers' pedagogical design capacities. Design capacities can be

	120,0
	60,0 80,0 100,0	0,0 27,3	63,6	86,7	61,1	61,5	57,1	6,7 6,7	50,0	6,3 6,3	26,3 5,3
	0,0 20,0 40,0	72,7	4,5 31,8	6,7 6,7	27,8 11,1	15,4 23,1	14,3 28,6	86,7	0,0 50,0	87,5	68,4
		Bhumika Afreen Kanchan Deepali Sonali Ganesh Mohini Neetu Jagdeesh Kamala
					Direct use	Adapted use	Insertion

Table 1 : Comparison of teacher use of focus tasks

 1


C S Unnikrishnan
email: unni@tifr.res.in

Quantum Noise in Balanced Di¤erential Measurements in Optics: Implication to the Wave Modes of Quantum Vacuum

Experimental tests for assessing the physical reality of the hypothetical wave modes of quantum vacuum with zero-point energy are of fundamental importance for quantum …eld theories and cosmology. Physical e¤ects like the Casimir e¤ect have alternate description in terms of retarded interaction between charged matter, due to quantum ‡uctuations of material dipoles. However, there are simple quantum optical con…gurations where the hypothetical quantum vacuum modes seem to assume an essential real role in the observable quantum noise of optical signals. I present the logical and theoretical basis of a decisive test that relies on the comparisons of balanced homodyne detection with a novel di¤erential scheme of balanced wave-front division detection, when the two real optical beams at the detectors are derived from one coherent beam as input. Both ideal and practical con…gurations of my experimental test are discussed. Results from the experiments on balanced detection, beam localization of optical beams, and atomic Bose-Einstein condensates are used to reach de…nite conclusions against the reality of the wave modes of quantum vacuum. It is shown that the entire quantum noise follows consistently from the state reduction of quantum superpositions of particle-number states at the point of detection, where the quantum measurement is completed. This is consistent with the demonstrated applications of squeezed light in interferometry and quantum metrology. This result achieves consistency between quantum noise in quantum optics and observational cosmology based on general relativity, by avoiding the wave modes of quantum vacuum with divergent zero-point energy density. Generalization from the limited sphere of quantum optics to general quantum …eld theories promises the complete solution to the problem of a divergent cosmological constant.

Quantum Optics as a mature theory is much younger than quantum mechanics and quantum …eld theories. The new foundations were …rmly established in 1963, with papers from R. Glauber [START_REF] Glauber | Photon correlations[END_REF] and E. C. G. Sudarshan [START_REF] Sudarshan | Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams[END_REF]. There are di¤erent approaches to the calculations, which have di¤erent foundational basis. One important issue is the notion of 'Quantum Vacuum'. In the quantum optical picture that quantizes electromagnetic wave modes, the vacuum mode is a wave mode at frequency with the zero-point energy h =2. In this picture, photons are just excitations of the wave mode with di¤erent amplitudes, with the wave-energy quantized in units of h . That is, the wave modes with …nite zero-point energy (ZPE) are present even if there is no detectable light. The use of real wave mode subject to quantization rules in calculations raises the question of the physical reality and observability of the hypothetical quantum vacuum modes with a nonzero ZPE. The notion is seriously troublesome because each mode has the ZPE of h =2 and there is an in…nity of such modes in space and time, implying a divergent energy density that is in con ‡ict with general relativistic (observational) cosmology. The problem of the divergent ZPE density, commonly called the problem of a divergent cosmological constant, has remained unresolved for a long time [START_REF] Weinberg | The cosmological constant problem[END_REF][START_REF] Rugh | The quantum vacuum and the cosmological constant problem[END_REF].

In contrast, there is a logically robust picture that rejects real waves underlying the quanta, pioneered and argued for by S. N. Bose, of photons as particles of light obeying Bosonic rules of collective behaviour [START_REF] Bose | Planck's law and the light-quantum hypothesis[END_REF]. Hailed by Einstein as a fundamentally new way of treating the statistical behaviour of quanta, this has no divergence problems (as Bose himself had exclaimed [START_REF] Unnikrishnan | Bose-Einstein condensates as universal quantum matter[END_REF]). The Fock number state approach formalizes this line of thought and integrates it to the underlying notion of a '…eld' [START_REF] Fock | Kon…gurationsraum und zweite Quantelung[END_REF]. The quantum state can be then described by labelling the number of photons in a particular state, as jni and forming arbitrary superpositions. The vacuum state is just the formal no-photon state j0i. The theory of quantum optics treats the vacuum state as fundamental because it serves as the basis from which n-photon Fock states are 'created'as a y n j0i. The superposition of such number states then provides all states of light. Modern quantum optics uses a hybrid view, based on quantum …eld theories, where the notion of the underlying …eld and particles as excitations is retained. This is reinforced by the usage of coherent states as a universal basis, pioneered by E. C. G. Sudarshan [START_REF] Sudarshan | Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams[END_REF]. However, while mathematically complete, this hybrid view carries the same conceptual inconsistency of the divergences.

In the rigorous 'Dirac view'of quantum mechanics, the ‡uctuations in physical quantities happens at the (repeated) measurements and there is no ‡uctuations in quantum state itself either in time or in space until a measurement in made. In contrast, the description with the quantum vacuum modes explicitly links the quantum noise with the space-time ‡uctuations of the amplitude and phase of the modes. Thus, the coherent state in the Dirac-Fock picture is the superposition of all number states at all instants, until the measurement at a detector. In the 'mode-view', in contrast, the …eld ‡uctuates in its amplitude and phase, the statistical ensemble average of which gives the familiar 'stick-and-ball'representation of the quantum coherent state and its uncertainty noise. The latter view is obviously not fully consistent with the principles of quantum mechanics, since it implies de…nite amplitude and phase at each instant, ‡uctuating. But it is extensively in use due to its convenience.

There is another conceptual issue in the description of the quantum vacuum as a physical state with average ZPE h =2, carrying ‡uctuations in amplitude and phase distributed as a Gaussian about zero. The probability for a ‡uctuation in amplitude that far exceeds the average ZPE and even h is …nite; this is in con ‡ict with requirement that no photon shall be detected in the vacuum state. Though I will not discuss this further, the di¤erence between the 'wave modes'view and the 'quanta-state'view becomes crucial in the discussion of quantum noise, its interpretation, and the exact expressions in speci…c physical situations.

The Casimir force, Lamb shift, and the spontaneous emission from excited atoms are phenomena cited in support of the physical reality of quantum vacuum modes. However, the Casimir force, the prime example, can be derived as the interaction of the quantum ‡uctuating dipoles (atoms) in the material making the two surfaces, as the retarded van der Waals force, or as the integrated Casimir-Polder force between a material boundary and an atom [START_REF] Schwinger | Casimir e¤ect in dielectrics[END_REF][START_REF] Milton | The Casimir E¤ect: Physical Manifestations of Zero-Point Energy[END_REF][START_REF] Milonni | The Quantum Vacuum: An Introduction to Quantum Electrodynamics[END_REF]. Since all real boundaries are equal to the factual presence of matter with quantum zero-point motion, and not mere static mathematical conditions, the necessity of the vacuum modes in the Casimir force cannot be insisted because that would be double-counting; either picture is mathematically consistent when invoked alone. In other words, when wave modes and their di¤erential radiation pressure are invoked to calculate the Casimir e¤ect, one has to assume passive mathematical boundaries without any atomic and electromagnetic structure subject to quantum mechanical zero-point ‡uctuations. However, since matter and its zero point ‡uctuations are the only reality that is directly observed and veri…ed, the mode picture can be seen only as a calculational tool, without physical reality to the wave modes. This argument is logically robust and it solves the problem of the divergent ZPE, since the matter density in the universe is …nite and its ZPE is negligible. Therefore, there is strong reason and motivation to explore laboratory physical situations to …nd a direct demarcating experiment that tests the physical reality or otherwise of quantum vacuum modes. The rest of the paper is devoted to the identi…cation of such experiments, results, and their interpretation. The simple physical situation involving a 'beam splitter' (BS) in interferometers and quantum optical experiments is often discussed as a reliable indication of the necessity of the real quantum vacuum mode. The two-port balanced homodyne detection (BHD) of light [START_REF] Yuen | Noise in homodyne and heterodyne detection[END_REF] exploits the fundamental peculiarity of the beam splitter that there is relative phase of exp(i ) = 1 between the two output beams of the passive device (…gure 1). BHD involves a strong phase coherent local oscillator (LO) beam of amplitude + at one port (say 'a' port) of a 50:50 beam splitter and the weak signal beam s + s at the other. The terms and s represent the ‡uctuations in the …elds. The beams emerge superposed from the two output ports and the intensity is measured in each port independently by photo-detectors. The di¤erence in the outputs of the detectors is obtained after a di¤erencing circuit, and this is the homodyne signal, which consists of only the interference terms ( + ) (s + s) between the …elds in the two input ports due to the crucial relative phase factor of at the BS [START_REF] Schumaker | Noise in homodyne detection[END_REF][START_REF] Loudon | Quantum noise in homodyne detection[END_REF][START_REF] Loudon | Quantum Theory of Light[END_REF][START_REF] Garrison | Quantum Optics[END_REF]. If we consider the situation when only the LO light is entering one port, with no light at the other, one would have expected a zero homodyne signal because the cross interference term would be identically zero. Yet, the actual homodyne output is non-zero Gaussian noise centered on mean zero. This is the reason to postulate that a physically real ‡uctuating quantum vacuum mode (s = 0; s 6 = 0) enters the open empty port of the beam splitter and mixes (interferes) with the LO to give the observed noise in the homodyne output. This interpretation was …rst stressed by C. M. Caves in the context of the quantum noise in the interferometric gravitational wave (GW) detectors based on Michelson interferometers with the balanced beam splitter [START_REF] Caves | Quantum-mechanical radiation-pressure ‡uctuations in an interferometer[END_REF][START_REF] Caves | Quantum-mechanical noise in an interferometer[END_REF][START_REF] Caves | New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states[END_REF]. The radiation pressure noise is proportional to di¤erence in the mode intensities in the two arms, c y c d y d , which can be written in terms of the interference terms of the input operators a 1 and a 2 as a y b + b y a . This is also the di¤erence signal measured in the homodyne scheme. Then, the vacuum mode through the open port of the beam splitter enters the description even if there is no input light in it. At present, this is the standard interpretation for the irreducible quantum noise in the balanced homodyne output. More over, the demonstrated use of squeezed light in GW detectors is based on this physical picture [START_REF] Aasi | Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light[END_REF][START_REF] Barsotti | Squeezed vacuum states of light for gravitational wave detectors[END_REF]. Thus, quantum optics involving a beam splitter seems to provide a ubiquitous situation and transparent proof for the physical reality of the quantum vacuum mode. However, the divergent ZPE in general relativistic cosmology remains as a genuine and troublesome physical problem.

How does one reconcile the two contradictory physical pictures, each with seemingly convincing support? To obtain a reliable answer experimentally, I devised a novel signal di¤erencing con…guration that avoids the phase factor of the beam splitter, which can then be compared with the homodyne scheme. My results are completely consistent with Fock-state quantum mechanics without the real wave modes of quantum vacuum. Then, quantum noise is explained as entirely due to the reduction of the quantum state where the quantum measurement is completed right at the detector, with no physical role for the di¤erencing operation after the intensity detectors. While resolving the con ‡ict with general relativistic cosmology, my results are in consistency with demonstrated quantum physical applications like squeezed light metrology.

II. BALANCED DIFFERENTIAL MEASUREMENTS

The simple physical situation involving a 50:50 'symmetric'beam splitter in interferometers and quantum optical experiments is conventionally interpreted as a reliable indication of the necessity of the real quantum vacuum mode. See …gure 2A. One mode of real radiation a is split into two (c; d) in amplitude by the BS and the mode intensities in the two beams after BS are c y c and d y d. There is no real optical bean in port 'b'. When detected with di¤erencing photo-detector, the quantity measured directly is

N (t) = N c N d = c y c d y d;
which can be formally written in terms of the interference terms of the input mode operators a and b as a y b + b y a . The explicit calculations are described later. This relation between the input and output beams can be traced to the fundamental feature of the BS that there is a relative phase of between the two output beams of the BS. Due to this factor, only the cross interference terms survive without cancellation in the balanced subtraction N (t) = c y c d y d . This is the signal measured in the balanced homodyne scheme. But there is no real beam in the port 'b'! However, when such an experiment is done, one does not measure 'zero' at the detector output. The average di¤erential signal h N (t)i = 0 and there is irreducible quantum noise and variance in N (t). This is the quantum noise, consistent with the (non)commutation relations a; a y = b; b y = 1. Thus, one is forced to postulate that the invisible quantum vacuum mode enters through the open port 'b'of the beam splitter, even if there is no input light in it. Only then the quantity a y b + b y a can be non-zero. At present, this is the standard interpretation for the irreducible quantum noise in the balanced homodyne output.

How reliable is this quantum optical evidence? Can one …nd an equally simple measurement scheme in quantum optics that invalidates this interpretation and disproves the physical reality of the invisible quantum vacuum mode? This is indeed possible with a simple di¤erential detection scheme where one splits the wavefront, instead of the amplitude, of the coherent beam before the detection with the di¤erential detector. I call this the balanced wavefront-division detection (BWDD). The detected quantity in BWDD is also the di¤erence in the intensity in the two detectors N (t) = a y a a 0y a where a 0 is the symmetric replica of the mode a, being split from the same wavefront with the same mode features (…gure 2B). Both parts of the wavefront have quantum noise, carried by the same quantum vacuum mode associated with the a beam. However, since the crucial phase factor is absent in the wavefront-split di¤erential scheme, the subtraction is total, with no surviving interference term. Thus, the same quantum optical calculations that attributed the quantum noise entirely to the uncanceled interference terms in BHD predict near-zero noise at the di¤erential output of BWDD. This di¤erence between the two schemes is the vital idea of the experiment. Yet, from the quantum mechanical analysis of a coherent beam as an instantaneous superposition of all particle (photon) number states, instead of as a …eld mode with noise ‡uctuations, the di¤erence signal and its noise variance should not be zero; rather, the noise in N should be p N c + N d in both BHD and BWDD, because of the completion of the quantum measurement right at the detectors. It is this di¤erence in the predicted residual noise in the di¤erential signal that helps reject the reality of the hypothetical quantum vacuum modes.

III. DIFFERENTIAL SIGNALS AND THEIR NOISE A. Direct Detection of a Coherent Beam

First, let us consider the quantum noise in a single mode of coherent radiation, detected with a photodetector. The detected intensity is related to the normal ordered number operator a y a. In terms of the coherent amplitude this is just . The variance needs the square of the number operator, N 2 = a y aa y a, which is not a normal ordered operator product. However, a y aa y a = a y a y a + 1 a = j j 4 + j j 2 . Since a y a 1 in the experiments we consider, all quantum mechanical calculations are accurately done using the complex coherent amplitude. The optical amplitude is = a + a 1 + i a 2 , where the quantum ‡uctuations in the two quadratures are designated as a 1 and a 2 [START_REF] Schumaker | Noise in homodyne detection[END_REF][START_REF] Loudon | Quantum noise in homodyne detection[END_REF]. For a coherent beam, the uncertainty principle dictates j a 1 a 2 j = 1=4. The phase of the beam is …xed such that a is real. We assume that there is no other excess amplitude noise of the optical beam (like technical ‡uctuations of the laser intensity). The detector output is

i(t) / N (t) = a y a ' = a 2 + 2a a 1 + a 2 1 + a 2 2 (1)
The quantum noise is re ‡ected in the last three terms, of which only the …rst is signi…cant in magnitude. Then,

N (t) ' a 2 + 2a a 1 ! N p N
The variance is

V (N ) = D N N 2 E ' 4a 2 a 2 1 ( 2
)
We see that the entire quantum noise is consistently described by the ‡uctuations a 1 and a 2 in the 'quantum vacuum mode'. When evaluated, V (N ) ' N , because a 1 = a 2 = 1=2 for a coherent mode, from the uncertainty relation a 1 a 2 = 1=4.

The alternate and conceptually very di¤erent interpretation of quantum noise based on a particle picture of standard quantum mechanics is also consistent. There, the coherent state of photons is a superposition of all number states jni with Poissonian weights.

j (t)i = exp( j j 2 =2) n=1 X n=0 n p n! jni
The detector detects the average N = 2 and its ‡uctuations (standard deviation) are p N . Thus V (N (t)) = N . The quantum measurement is completed at the square-law detector. In fact, it is identical in structure to a coherent state of atoms (for which there is no underlying real wave or vacuum mode with a zero-point energy). There is no real wave mode and its ‡uctuations in this picture. Then one can get ‡uctuations in the photon number on detection, with Poissonian probability, with the collapse of the state that happens (only) at the detection event.

For this particular case of the direct detection of the optical beam, either view is consistent. Our goal is to demarcate and decide between the two di¤erent views, by a decisive experiment.

B. Balanced Homodyne Detection

We refer to the …gure 2A. The …eld amplitudes corresponding to the modes are complex quantities. Since only relative phases are observable, one of the modes can be taken as real quantity. The quantum noise in each mode is however in two quadratures. So, a ! a + a 1 + i a 2 . All experiments under discussion have only one real optical beam. For the BS experiment, there are two input ports and two possible independent modes (di¤erent 'k'vectors etc.). But, since there is no real optical beam in the second port, only the hypothetical and invisible quantum vacuum modes with ‡uctuations enter the …nal expressions,

b ! b 1 + i b 2 .
The general beam splitter has the transmission T = jtj 2 , to port 'd'from port 'a', and hence re ‡ection R = jrj 2 = 1 T from port 'a' to port 'c' (or from port b to port d). Fixing explicitly one relative phase of between ports 'b'and 'd', we have real r and t. The phase of 'a' beam is chosen such that a is real. Since we are concerned with experiments on quantum vacuum, b will eventually be zero or very small. Thus, the terms second order in small quantities (b 2 ; a

N c = c y c = ra y + tb y (ra + tb) ' jr (a + a) + t (b + b)j 2 (3) (r (a + a) + t (b + b)) (r (a + a) + t (b + b)) = R a 2 + a 2 + 2a a + T b 2 + b 2 + 2b b + rt(ab + a b + b a + a b ) + rt(ba + b a + a b + b a ) ' Ra 2 + 2Ra a + p T R (ab + ab + a b + a b) (4)
= Ra 2 + 2Ra a + 2 p T R (ab 1 + a b 1 ) (5)
For the other output port 'd', inserting the relative phase factor from port b to d,

N d = d y d ' [t (a + a) r (b + b)] 2 (6)
' T a 2 + 2T a a 2 p T R (ab 1 + a b 1 ) (7)
The symmetric (50:50) BS has equal transmission and re ‡ection coe¢ cients (tt = T = 1=2 = R = rr ). Then,

N c = 1 2 a 2 + a a + ab 1 + a b 1 (8)
N d = 1 2 a 2 + a a ab 1 a b 2 (9)
Balanced homodyne detects the di¤erence signal N (t) = N c N d and its noise directly. The total quantum noise is 2 (a b 1 + a 1 b 1 + a 2 b 2 ), where the last two terms are negligible. The di¤erence signal is identical to the (cross) interference term between the two beams in the ports a and b,

N (t) = 2ab 1 + 2a b 1 (10) h N i = 2ab 1 (11)
It is remarkable that we get the interference term in the di¤erence signal despite a 'phasedestroying' square law detection at each detector. Similar 'magic' was seen in the classic case of the Hanburry Brown-Twiss intensity interferometry that prompted the new theory of correlations and quantum coherence in optics. Most importantly, the signal has no self interference terms between the mode amplitude a and its quantum vacuum ‡uctuations a. Only the cross interference term a b 1 is present. It should be noted that if the BS did not introduce the phase, all signs in N d = d y d (eq. 6) would have been positive and the di¤erence signal in equation 10 would have been identically zero in the subtraction, without any averaging.

When there is only one coherent beam at the port 'a', with only 'vacuum' at the port 'b', h N i = 0. The balanced homodyne signal has the variance

V ( N (t)) = D N N 2 E = 4a 2 b 2 1 +4 a 2 1 b 2 1 +4 a 2 2 b 2 2 +8 h a 1 b 1 a 2 b 2 i ' 4a 2 b 2 1 (12)
All other terms average to zero. The last of the four terms can give a nonzero average if there are correlations in the two quadratures of each mode. However, only the …rst term is signi…cant in magnitude. It is of order N , whereas the other terms are of order 1, totally negligible. Thus, V ( N ) ' 4a 2 b 2 1 . Therefore, the quantum noise in balanced homodyne detection is attributed entirely to the quantum noise in the 'b' mode, with the amplitude of the 'a'mode beam acting as a noise-free linear ampli…er, with gain N .

It is instructive to look at the variance of the signal at each detector. We have

V (c) = D N c N c 2 E = a 2 a 2 1 + a 2 b 2 1 + a 2 1 b 2 1 + a 2 2 b 2 2 + 2 a 1 b 1 a 2 b 2 ' a 2 a 2 1 + a 2 b 2 1 (13) V (d) = D N d N d 2 E = a 2 a 2 1 + a 2 b 2 1 + a 2 1 b 2 1 + a 2 2 b 2 2 + 2 a 1 b 1 a 2 b 2 ' a 2 a 2 1 + a 2 b 2 1 ( 14
)
Clearly the expression for the sum of the variances of the two signals,

V (c) + V (d) ' 2 (a 2 a 2 1 + a 2 b 2 1
) is not equal to the variance of the homodyne di¤erence signal (eq. 12), V ( N (t) ' 4a 2 b 2 1 . The theoretical quantum noise in the output of each detector is contributed equally by the quantum ‡uctuations a and b of the quantum vacuum modes 'a'and 'b', whereas the theoretical quantum noise in the direct di¤erence signal is entirely contributed by the ‡uctuations in the 'b'mode, b.

V (c) + V (d) ' 2 a 2 a 2 1 + a 2 b 2 1 6 = V ( N (t) ' 4a 2 b 2 1 ( 15
)
This is an important and fundamental result. For equal uncertainties in both quadratures, the numerical magnitudes 2 (a 2 a 2 1 + a 2 b 2 1 ) and 4a 2 b 2 1 are obviously indistinguishably equal. However, the result becomes crucial, with quantitatively di¤erent total noise, in experiments with squeezed light or in those situations where there are quantum correlations. For example, with b 1 squeezed by a factor exp( s),

V (c) + V (d) ' 2 a 2 a 2 1 + e 2s a 2 b 2 1 ( 16
)
whereas the variance in the homodyne signal is very di¤erent,

V ( N ) ' 4e 2s a 2 b 2 1 (17)
Hence, jV ( N )j 6 = jV (C) + V (C)j. We see right here some fundamental inconsistency and con ‡ict in the notion of real quantum vacuum modes with the idea of measurement in quantum mechanics, but I will defer a detailed discussion on that and focus on the new wavefront division experiment.

C. Balanced Wavefront-Divison Detection

We write the essential steps in the calculation for the balanced wavefront-division di¤erential detection (BWDD) schemes of …gure 3. The mode a with its vacuum noise is split into two sections, which we designate as u and d . They are the same mode with all quantum numbers matching. But we can try to distinguish the parts of the wavefront with a spatial label, up (u) and down (d) of the wavefront. The fact that there is only one beam and one mode now, from which two beams are generated by diving the wavefront, and there is no phase to introduce a negative sign in any term, re ‡ects in the di¤erential output. The detectors generate the intensity signals,

u = 1 p 2 (a + a 1 + i a 2 ) d = 1 p 2 (a + a 1 + i a 2 ) (18)
y u u ' u u = 1 2 a 2 + 2a a 1 + a 2 1 + a 2 2 (19)
y d d ' d d = 1 2 a 2 + 2a a 1 + a 2 1 + a 2 2 ( 20
)
The output of the di¤erential detector is identically zero, without any averaging.

N (t) = y u u y d d = 0 (21)
We have no freedom in this calculation if we follow the same logic as what was used in the homodyne calculation. There, the split occurs in the time domain whereas here it occurs in the spatial domain; that is the only di¤erence. There are only self interference terms of the mode amplitude with its own quantum ‡uctuations here, but there is no cross interference term with the ‡uctuations of another mode. The variance of the output signal N (t) is obviously zero, since N (t) itself is zero.

V ( N (t)) = D N N 2 E = 0 (22)
If the balance is not ideal, but in the ratio P : Q with P ' Q, then the expression modi…es to

N (t) = (P Q)a y a a y a (23) V ( N (t)) = 4(P Q) 2 a 2 a 2 1 N (24)
Therefore, the experiment can decisively determine whether a real quantum vacuum mode is the source of the quantum noise or whether it is the collapse of the instantaneous quantum superposition of the particle number states. If the di¤erence signal and its variance is near zero, then it supports the view that the quantum vacuum mode with the ZPE is the carrier of the quantum noise. On the other hand, the collapse of the particle number superposition at the detectors predicts that the variance is the sum of the variances at the individual

detectors, V ( N (t)) = V (c) + V (d) = N c + N d ,
because the quantum measurement is completed (only) at the square law photodetectors. Therefore, noise variance that scales as the intensity (average photon number) in BWDD would be reliable evidence against the real wave modes of quantum vacuum. One might think and hope that one could introduce new rules for the wave modes to save the interpretation of the zero-point wave as the source of the quantum noise. One possibility is to postulate that the ‡uctuations are cancelled in the subtraction only when the amplitude is split, but not when the wavefront is split. This would be the case if di¤erent spatial portions of the same wavefront have di¤erent ‡uctuations, even after integrating over the response time of the detectors. But, this arbitrary postulate does not solve the problem. We repeat the calculation with the new postulate. The intensity signals with di¤erent and independent quantum noise a and a 0 in the two haves of the wavefront are,

y 1 1 ' 1 1 = 1 2 a 2 + 2a a 1 + a 2 1 + a 2 2 ( 25
)
y 2 2 ' 2 2 = 1 2 a 2 + 2a a 0 1 + a 02 1 + a 02 2 (26)
After the subtraction,

N (t) = 1 1 2 2 ' a( a 1 a 0 1 ) < 2a a 1 (27)
V ( N ) = a 2 ( a 1 a 0 1 ) 2 ' a 2 a 2 1 + a 02 1 2a 2 h a 1 a 0 1 i = 2a 2 a 2 1 2a 2 h a 1 a 0 1 i < 4a 2 a 2 1 (28)
Therefore, V BW D = V BH =2 for independent quantum ‡uctuations in the two parts of the wavefront. So, this patch repair does not match the variance that we got for the homodyne scheme in magnitude, V BW D ( N (t)) < V BH ( N (t)) ' 4a 2 b 2 1 . When the ‡uctuations are fully anticorrelated ( a 1 = a 0 1 ), then h a 1 a 0 1 i = a 2 1 , but one cannot get this condition for every partition of the wavefront. If this were the case, then the total intensity would be

1 1 + 2 2 = a 2 + a ( a 1 + a 0 1 ) = a 2
, with no variance! The wavefront-division detection can provide a strong proof against the wave modes of quantum vacuum.

D. Insigni…cance of Steering Mirrors

It is convenient (and often necessary) to use one or more steering mirrors in the experiments to get the beam to the detector location etc. While it is intuitively obvious that a mirror with high re ‡ectivity does not a¤ect the concept or the quantitative expressions of the experiment, it is perhaps appropriate to prove this explicitly. (This clarifying calculation, obvious to most, is included after seeing persistent confusion among some colleagues that a single steering mirror interpreted as a beam splitter with very low transmission is essential and crucial in the wavefront division experiment to compare it with the homodyne experiment). A steering mirror of re ‡ectivity R can be interpreted as a R : T beam splitter, with T R ' 1. We note from equation 4 that the use of the mirror alters the quantum noise contribution in that beam slightly,

N c N c = 2Ra a + 2 p RT a b (29) V (N c (t)) ' 4R 2 a 2 a 2 + 4RT a 2 b 2 (30)
The di¤erential signal and its variance in the wavefront-division experiment with a steering mirror (…g. 3B), assuming 50:50 wavefront division before the mirror, is calculated next. The already divided amplitude (a + a) = p 2 becomes r (a + a) = p 2 after the R : T mirror. This is superposed with t b when we assume that the vacuum mode leaks in due to the small transmissivity T . I include also the phase for re ‡ection at the mirror. The intensity at the detector is

N c = e i r p 2 (a + a) + t b e i r p 2 (a + a) + t b ' R 2 a 2 + 2a a 1 2rt p 2 a b 1 (31)
The phase factor at re ‡ection drops out. The other half of the wavefront goes straight to the detector 'd'. So,

N d ' 1 2 a 2 + 2a a 1 N (t) = N d N c = 1 2 (1 R) a 2 + 2a a 1 + p 2RT a b 1 h N (t)i = 1 2 (1 R)a 2 (32) V ( N ) = (1 R) 2 a 2 a 2 1 + 2RT a 2 b 2 ' 2T a 2 b 2 = (T =2) V ( N BHD ) (33)
As stated before, with T < 1%, the variance in the di¤erential signal is negligible compared to the variance in the balanced homodyne signal, suppressed by the factor T =2. The phase at the steering mirror is irrelevant, unlike the crucial phase of in the homodyne BS. Similar remarks apply if a prism (total internal re ‡ection) is used for steering the beam.

IV. THE REAL SOURCE OF THE QUANTUM NOISE

Now I show that the real source of the quantum noise is the quantum state reduction of the particle superposition of the real optical beam at the detector that completes the quantum measurement, in all cases in quantum optics, without exception. The coherent state in the example of direct beam detection is

j (t)i = exp( j j 2 =2) n=1 X n=0 n p n! jni (34)
Before reaching the detector, the quantum state is a superposition of all number states, with its only time evolution re ‡ected in an overall phase. Till the state j i interacts with the square law detector, subject to an interaction Hamiltonian, the state is stable and under pure unitary evolution. The quantum measurement is completed at the square-law detector, and only at that point. This gives a value for the number of photons with probability p(n) dictated by the Poissonian weight of the state with n number of photons. It is clear that whether we make the measurement on the beam by direct detection, or on any part of the beam by splitting it in amplitude or wavefront, the state is the same, multiplied by an overall numerical factor. The relative probabilities remain the same. For an ensemble of identical quantum measurements with the detector, a sequence of such results are obtained; then we get the average N = 2 and its statistical ‡uctuations, with standard deviation p N . Thus V (N (t)) = N .

For balanced homodyne detection as well as for balanced wavefront division detection, each detector completes the quantum measurement independently and the rest of the operations are irrelevant for the quantum state. Each detector measures the average N =2 with the standard deviation p N =2. Therefore, the di¤erence signal is centred on zero, with standard deviation p N . This is the prediction for any balanced dual detector measurement from the theory that rejects the physical wave mode of quantum vacuum. Then the noise is attributed to the quantum measurement that reduces a quantum superposition of number states to a particular number state in each measurement event. It does not make any di¤erence in the expressions for the quantum noise whether one performs the individual measurements and then takes the di¤erence, or whether the di¤erential signal is directly measured. This interpretation and prediction are universally applicable in all cases, from direct detection to general unbalanced detection of light. Since the prediction for the variance of the balanced di¤erential signal is very di¤erent in the quantum state view and in the vacuum-waves view ( N vs 0), one can decisively determine the correct physical picture directly from the experiment and rule on physical reality of the wave modes of quantum vacuum.

V. EXPERIMENTAL RESULTS

A. Balanced Homodyne Measurements

Balanced homodyne measurement (…g. 2A) is the primary tool for the characterization of quantum noise in squeezed light. There are already several experimental results that are consistent with the calculations described in the section on BHD. Therefore, BHD measurements that measure the di¤erence signal and its variance are consistent with the hypothesis that a real wave mode of quantum vacuum entering the open port of the BS is the source of the entire quantum noise [START_REF] Smithey | Measurement of number-phase uncertainty relations of optical …elds[END_REF][START_REF] Leonhardt | Measuring the Quantum State of Light[END_REF]. The quantum noise variances in the individual detectors are, however, contributed by the vacuum modes entering both the input ports. The di¤erencing operation after the intensiy detectors cancels one contribution while doubling the other. Other caveats are brie ‡y mentioned in section IIB.

B. Balanced Wavefront-Division Measurements

Now we examine the hypothesis of the quantum vacuum modes against the balanced wavefront-division measurements. The experimental scheme is as indicated in …gure 3. Direct experiments can be with two matched photodetectors, at relatively high intensity of about a mW (>0.5 mA photocurrent or 10 16 photons/s) or by photon counting detectors and subsequent di¤erencing schemes (hardware or software) at much attenuated intensity of less than 10 12 W. The calculation of the di¤erential signal based on splitting the wavefront of the wave modes predicts near-zero ( N ) noise variance. In sharp contrast, the variance V ( N ) = N c + N d = N is predicted by the de…nite completion of the quantum measurement and the reduction of the state at the individual intensity detectors, without the wave modes of quantum vacuum. In the di¤erential detection experiments, both in the BHD scheme and in the BWDD scheme, there is no role of physical signi…cance for the di¤erencing operation after the intensity detectors.

The dark current of commercial balanced detectors is in the range 1 5 pA/ p Hz, below 10 MHz. For example, a NewFocus-1807 model balanced detector has a noise equivalent The quantum shot noise exceeds the detector dark noise only when the optical power is about 0.05 mW (>10 14 photons/s), corresponding to photocurrent ‡uctuations of a few pA/ p Hz. The optical quantum noise measured at 110 kHz in a wavefront division di¤erential detection experiment using the NewFocus-2007 detector, in the scheme shown in …gure 3A, is plotted in the …gure 4 (the noise spectrum is ‡at within 2 dBm/Hz in the range 100 kHz-130 kHz). The detector shot noise, without any light input, is measured to be below -126 dBm/Hz (<0.25 fW/Hz). With 0.02-0.8 mW of light in each half of the wavefront from a He-Ne laser, the light beams are balanced to get a di¤erential voltage output less than 10 mV, corresponding to a di¤erential power below 1 W. The di¤erential scheme cancels the relatively large (1% rms) laser intensity noise by a factor of nearly 10 5 . The residual noise is monitored in the 100 kHz-150 kHz bandwidth with a spectrum analyzer. The near-zero electrical (voltage) output of the di¤erential circuit is connected to the spectrum analyzer and voltmeter, to directly measure the spectral density and rms value of the residual noise variance of the di¤erential signal, V (

N (t)) = D N N 2 E / D i i 2 E
. The noise variance in BWDD scheme scales linearly with the optical power in each half of the wavefront, in spite of the subtraction of equal portions of the same wavefront, without the phase factor that is characteristic of the homodyne di¤erential measurement. Data from the balanced homodyne measurement is also shown for comparison, in the range 0.25-1 mW. Variances in both cases are similar and scale as the average optical power. The subtraction scheme does not cancel the quantum noise in the BWDD scheme, directly contradicting the standard calculation in section IIIC. This indicates that the quantum noise in each half of the optical wavefront is independent, as predicted for the quantum state reduction at the detector, ruling out the physical reality of the ‡uctuating wave modes of the quantum vacuum.

There is another class of measurements that are e¤ectively identical to balanced wavefront-division di¤erential measurements. To understand this, consider the measurement of beam parameters of the optical beam from a single coherent laser beam of any intensity. This can be done by sub-saturation imaging with a CCD camera. We note at once that the task of determining the 'centre' of the beam, or 'centroiding', involves just taking the di¤erence signal between 'two halves'on any chosen diameter and averaging (…gure 5). The sum of pixels on each half is equivalent to one detector. Since the image of the beam is two-dimensional, the division can be done in many ways. This di¤erence signal has quantum noise, exactly like the quantum noise observable in the experiment indicated in …g. [START_REF] Weinberg | The cosmological constant problem[END_REF].

Another example is the determination of centroid of the light through a slit, as in an optical lever that measures angular changes. Still another is the localization of the telescope image of stars. While the total width of the image is determined by di¤raction limit of the imaging system, the centroid error is ultimately de…ned by the quantum noise.

Centroiding is the operation of …nding out the pixel location that equally divides the total number of photons in the detector ( N (t) or i(t)) into two bins. If one repeats this, the pixel location ‡uctuates (given su¢ cient spatial resolution) even if only quantum noise is present. For our task of determining the quantum noise, the exact location of the centroid is not critical. We consider the situation in which the balance of average number of photons is better than 1%. Our target is the variations in N (t) and adjusting N (t) to exactly zero is not required. A cooled EMCCD camera (detector) has high quantum e¢ ciency and nearly single photon sensitivity. The optical beam can be expanded so that a large part of the wavefront (>98%) illuminates the sensor. The di¤erence in the total counts from pixels on the two sides of the …ducial pixel is

N (t) = n=2 X 0 N i n X n=2 N i (35)
where each N i is y ' that we calculated earlier. In all experiments dealing with wavefront division, it is necessary to make sure that the spatial (angular) ‡uctuation of the whole beam is small enough to be negligible. Since the whole beam with its quantum ‡uctuations is described by the amplitude = a + a 1 + i a 2 , each N i is proportional to (with factors for the pixel area, ampli…cation factor and a Gaussian beam factor etc.) Therefore, N (t) ' 0 and its variance is also zero, according to the equations 23 and 24. However, centroiding experiments that measure the beam pro…les of weak laser beams show the ‡uctuations in N (t) with variance

V ( N ) ' N (36)
The physical problem of locating the centroid of the optical image of an illuminated slit, as in autocollimating optical levers to measure angular changes, is similar. The angular shift is measured by tracking the centroid of the image of the slit, which in turn is the balanced di¤erencing operation on the light intensities of the two 'halves'of the image. So, a BWDD scheme is operational in such devices. Optical levers operated at their quantum sensitivity limit [START_REF] Jones | Some developments and applications of the optical lever[END_REF] already indicate that the observed noise ‡oor is in con ‡ict with the hypothesis that the quantum vacuum mode is the source of quantum noise. Take the case of the instrument with an experimentally determined sensitivity limit of < 10 10 radians/ p Hz, with total light detected at the photodiode of about 1 W , or about 10 13 photons/s [START_REF] Jones | Some developments and applications of the optical lever[END_REF]. The di¤raction limited image of the slit had an angular width of 2 10 4 radians. With p V ( N ) = p N ' 3 10 6 , the quantum limited sensitivity of such an optical lever is then > 2 10 4 =3 10 6 ' 7 10 11 radians/ p Hz. In contrast, the prediction for the quantum noise in the di¤erence signal, assuming that its source is the ‡uctuating quantum vacuum wave mode, is V ( N ) ' 0 . Since other noise sources are controlled better in the device, the experimental results of shot-noise limited sensitivity of the optical lever reject the hypothesis of real wave modes of the quantum vacuum. Instead, they support the picture of local quantum state reduction of the superposition of number states at each pixel, at the instant of square-law (intensity) detection.

It is instructive to compare results on the quantum noise in optics with the quantum noise in the direct detection of atoms in a coherent beam of atomic Bose-Einstein condensate (BEC) of metastable Helium atoms (which allows such direct detection and counting of the atoms on a delay line detector due to the nature of spontaneous ionization of the He* atoms on contact) [START_REF] Schellekens | Hanbury Brown Twiss E¤ect for Ultracold Quantum Gases[END_REF]. There is quantum zero-point energy in the dynamics of the atomic BEC, …nite and proportional to the number of atoms. Atomic BEC is in a coherent state, but it has no associated real wave modes of quantum vacuum in space. All zero-point energy of the atomic BEC is associated with the atoms and it is truly zero when there are no atom in the beam. However, the quantum noise in the di¤erence in the number of atoms on either side of a 'centroid'has the variance, V ( N ) ' N . Clearly, this quantum noise is not related to any wave modes of quantum vacuum.

VI. BALANCED HOMODYNE SCHEME AND MICHELSON INTERFEROME-TRY

The common passive optical element of signi…cance in a balanced homodyne measurement and Michelson interferometry is the symmetric (50:50) BS. The detection is entirely di¤erent though. In homodyne, there are two detectors that complete the quantum measurements and a di¤erencing circuit (hard or soft) produces the …nal di¤erence signal. In Michelson interferometry, the input light is divided into two paths by the BS and two end mirrors re ‡ect the beams back to the BS, preserving the average phase after the double-pass through BS. This is equivalent to a di¤erencing operation on the beam amplitudes. This is then detected with a single square law detector that completes the quantum measurement (…gure 6). Compared to the homodyne detection, there are signi…cant di¤erences in details of the quantum noise that are important in the application of squeezed light in sub-shot noise metrology.

The di¤erence signal in the homodyne scheme with a single beam at port 'c' is the interference term N (t) = c y c d y d = ab y + a y b ' 2a b 1 with variance 4a 2 b 2 1 . The radiation pressure noise in a Michelson interferometer has the same feature since the light pressure on each mirror is proportional to the number of photons re ‡ecting o¤ the mirror and the interferometer optical noise comes from the ‡uctuating di¤erential motion of the two mirrors; the common mode displacement does not lead to an optical intensity modulation at the detector. However, this view is simplistic. Consider a Michelson interferometer with one of the mirrors anchored, with its free motion arrested. The interferometer is still sensitive to the gravitational wave strain, albeit with a reduction in sensitivity by a factor of two. However, now the radiation pressure noise is related to N (t) = c y c c y c = a a 1 + a b 1 with variance

V ( N ) = a 2 a 2 1 + b 2 1 + 2 a 1 b 1 ' a 2 a 2 1 + b 2 1 = 1 2 V ( N ) (37)
It is not natural or logical to claim that just by holding one of the mirrors the radiation pressure noise became equal contribution of vacuum noise in both modes a and b, whereas with mirrors left free, the entire quantum noise was solely from the vacuum noise of b entering the open port. In both cases, there is no constraint on the operation point, which can continue to be the dark fringe. This becomes important in the context of squeezed light injection in the open port for reducing radiation pressure noise at low frequencies. With both mirrors free, the wave-mode picture allows the possibility of arbitrary reduction of radiation pressure noise by injecting light squeezed by the factor exp( s),

V ( N ) ' 4e 2s a 2 b 2 (38)
whereas the theory allows only limited noise reduction when one of the mirrors in held rigid,

V ( N ) ' a 2 a 2 + e 2s a 2 b 2 (39)
Currently, the advances in quantum noise reduction in gravitational wave detectors based on Michelson interferometers are based on the notion that the quantum noise in the Michelson signal is similar to the quantum noise in the homodyne signal, which is the cross interference term between the coherent amplitude (or LO) and the quantum noise in the vacuum mode entering the open port, 2a b. The quantum noise is in two quadratures, b 1 'in phase' with the LO amplitude and b 2 that is orthogonal. While the cross term in the homodyne signal is a b 1 , it is a b 2 in the Michelson interferometer signal, operated near the dark fringe. This is because the intensity ‡uctuations near the dark fringe (no light output at port 'b') are due to the ‡uctuating phase di¤erence, rather than amplitude ‡uctuations, of the light in the two arms. Therefore, one argues that replacing the rogue quantum vacuum mode with the squeezed vacuum, with smaller b 2 , at the cost of increased b 1 , reduces the noise in the relevant quadrature [START_REF] Caves | Quantum-mechanical noise in an interferometer[END_REF]. This expectation turns out to be correct, but the detailed physical reason does not involve the wave modes of the quantum vacuum and their ‡uctuations; the comparison of the BWDD scheme with the balanced homodyne scheme already rules out the reality of the wave modes of quantum vacuum. The correct interpretation, in which the quantum state reduction of the superposition of the photon number states happens at the intensity detector where the quantum measurement is completed, will be discussed in a subsequent paper. While the use of the wave modes is very convenient in calculations, just like in the calculation of the Casimir e¤ect etc., the physical reality of such modes is in con ‡ict with both laboratory experiments and cosmology. Hence a consistent and complete description in terms of photon statistics during quantum state reduction at measurement, without referring to the ‡uctuating wave modes, is necessary and possible.

VII. SUMMARY

I have examined two kinds of balanced di¤erential optical measurement involving the di¤erence of intensities recorded by two photodetectors. The …rst is the balanced amplitudedivision homodyne detection (BHD) that uses a symmetric BS and the second is the balanced wavefront-division detection (BWDD) in which the BS is absent. The two beams in each scheme are detected with separate detectors and the di¤erence signal and its quantum ‡uctuations are compared to determine the physical reality and the contribution of the hypothetical quantum vacuum mode to the factually observed quantum noise. The relevant theoretical expressions were derived from …rst principles, for each experiment. It was shown that the predictions for the variance of the di¤erence signals are very di¤erent in the BHD scheme and the BWDD scheme. In BHD, the interference from ‡uctuations in the quantum vacuum mode from one input port of the BS entirely cancels in the subtraction, while the ‡uctuations of the vacuum mode through the second port survive as the cross interference term with the carrier and manifest as the quantum noise of the homodyne signal. This conventional picture and the associated theory seem consistent with the results of the homodyne experiments.

However, the BWDD scheme provides a decisive counter-point because the interference of the ‡uctuations disappears totally in the subtraction without a beam splitter and there is no cross interference term. Therefore, if the measured quantum noise in the BWDD is very di¤erent from the prediction of vanishing residual noise, then the experiment decisively rules out the reality of the quantum vacuum mode and its role in the observed quantum noise. An experiment in which a balanced pair of photodetectors were used to measure the di¤erential intensity and its variance from splitting the wavefront showed that the quantum noise variance scales linearly with optical power, instead of cancelling in the subtraction. Several experimental schemes that are equivalent to the BWDD scheme are examined to con…rm this result. This decisively rejects the physical reality of the wave modes of quantum vacuum. Instead, the reduction of the superposition of particle number states to a speci…c but random number state in each quantum measurement is the sole consistent description and the preferred interpretational choice of the quantum noise. This is also consistent with observational cosmology and its general relativistic description, since there is no divergent ZPE density in this picture. I have derived the quantum noise in this alternate picture in all cases of optical detection. It correctly predicts equal quantum noise in both detection schemes, BHD and BWDD.

These results are corroborated by the recent reformulation of general dynamics and quantum mechanics in terms of a universal wave equation for the 'waves of action', rather than for the hypothetical matter-energy waves [START_REF] Unnikrishnan | Reconstructing quantum mechanics without foundational problems[END_REF]. This modi…cation of dynamics that promotes Hamilton's equation to an action-wave equation reproduces quantum interference, correlations and uncertainty noise, without the divergent ZPE because the action waves carry 'action' and not energy or momentum. The uncertainty principle is recast as S ~. Schrödinger equation and its wavefunction pertain to statistical averages over an ensemble of dynamical histories and not single quantum history, contrary to the prevailing understanding of quantum mechanics. The physical basis of the use of squeezed light in quantum metrology in the light of these results remains to be discussed, especially in the context of the Michelson interferometer. More results and discussion on squeezed light quantum metrology without quantum vacuum modes will be presented in a subsequent paper.

  of Steering Mirrors IV. The Real Source of the Quantum Noise V. Experimental Results A. Balanced Homodyne Measurements B. Balanced Wavefront-Division Measurements VI. Balanced Homodyne Scheme and Michelson Interferometry

FIG. 1 :

 1 FIG.1:The scheme for the balanced homodyne detection. The di¤erential signal after the di¤erencing circuit and its ‡uctuations are directly measured.

FIG. 2 :

 2 FIG. 2: The two di¤erential detection schemes to test the physical reality of the quantum vacuum modes. The dashed arrows indicate the hypothetical vacuum modes. Left: Balanced homodyne detection (BHD) with one real optical input at port 'a'. The di¤erential photodetector gives either the di¤erential current i(t) or the di¤erential photon counts N (t). Right: Balanced wavefrontdivision detection (BWDD). The beam is expanded and then the wavefront is split equally into two parts and sent to the di¤erential detector.

  2 ; b 2 a b; b b) are negligible. The beam splitter combines the two input modes to give two output modes c and d. The photon counts in port 'c'(re ‡ection of b and transmission of a) is

FIG. 3 :

 3 FIG.3: Two con…gurations of balanced wavefront-division detection. This does not involve a 50:50 BS and associated relative phase of . B) Scheme with one or more beam steering elements, like a prism or a mirror.

FIG. 4 :

 4 FIG.4:The noise variance measured in the balanced wavefront-division measurement, plotted as a function of the (balanced) optical power in each detector. Noise variance scales with the optical power (dotted line), similar to the noise in the homodyne measurement, indicating that the wave mode of quantum vacuum is not the source of the quantum noise.

FIG. 5 :

 5 FIG. 5: Direct wavefront-division di¤erential measurement of an optical beam to study quantum noise.

FIG. 6 :

 6 FIG. 6: The quantum measurement in the Michelson interferometer.

Acknowledgements

I have bene…tted from conversations on the wave modes of quantum vacuum and the Casimir e¤ect with G. Rajalakshmi, D. Suresh, R. Cowsik, N. D. Haridass, K. A. Milton, S. Reynaud, and N. Jetty. I thank N. Jetty and S. Sankaranarayanan for several discussions on optical quantum noise and its calculation and interpretation in quantum optics. P. G. Rodrigues and P. V. Sudersanan helped in setting up the experiments.