P Albrodt

M T Jamal

A K Hansen

O B Jensen

² G Blume

K Paschke

P Crump

P Georges

And G Lucas-Leclin
email: gaelle.lucas-leclin@institutoptique.fr

Coherent combining of high brightness tapered amplifiers for efficient non-linear conversion

We report on coherent beam combining of three high brightness tapered amplifiers seeded by a single-frequency laser at λ = 976 nm in a simple architecture with efficientlycooled emitters. The maximal combined power of 12.9 W is achieved at a combining efficiency of > 65 % limited by the intrinsic beam quality of the amplifiers. The coherent combining cleans up the spatial profile, as the power content in the central lobe increases up to 86 %. This highbrightness infrared beam is converted into the visible by second harmonic generation, and results in a high non-linear conversion efficiency of 4.5 %/W and a maximum power over 2 W at 488 nm, limited by thermal effects in the PPLN.

Introduction

High-power high brightness diode lasers are in strong demand for many applications and benefit from their efficiency, reliability and compactness [START_REF] Bachmann | High Power Diode Lasers: Technology and Applications[END_REF]. However, the power limit of diode lasers is much lower in the visible spectral range than in the near infrared and not all wavelengths are directly accessible with diode lasers. Nonlinear frequency conversion, like second harmonic generation (SHG) is an important technology for the development of high power laser sources in the visible spectral range. Efficient SHG requires a good beam quality, a narrow spectral linewidth and a high input power [START_REF] Boyd | Nonlinear Optics[END_REF]. High conversion efficiencies are commonly achieved with diode pumped solid state and fiber lasers [START_REF] Miller | 42%-efficient singlepass cw second-harmonic generation in periodically poled lithium niobate[END_REF][START_REF] Kumar | High-power, single-frequency, continuous-wave secondharmonic-generation of ytterbium fiber laser in PPKTP and MgO:sPPLT[END_REF], however diode laser based visible light sources have become increasingly attractive for biomedical applications [START_REF] Müller | Diode laser based light sources for biomedical applications[END_REF] to fulfill the need for compact and efficient visible laser sources. Different high-brightness diode laser architectures are used for blue-green light generation [START_REF] Jechow | Blue-green light generation using high brilliance edge emitting diode lasers[END_REF]. Among those, tapered lasers (TPL) and tapered amplifiers (TPA) are the most promising architecture since they combine high power and acceptable beam quality in one device.

TPL and TPA devices consist of a single mode ridge waveguide (RW) followed by flared gain-guided amplifier section. Spectrally stabilized TPL are available in the visible (VIS) and near-infrared (NIR) spectral range and reach output powers up to 15 W per device [START_REF] Sumpf | Spectrally stabilized high-power high-brightness DBR-tapered lasers in the VIS and NIR range[END_REF] and can be converted to multiple watts of VIS light [START_REF] Jensen | Efficient generation of 3.5W laser light at 515nm by frequency doubling a single-frequency high power DBR tapered diode laser[END_REF] by making use of a cascade of two SHG crystals [START_REF] Hansen | Concept for power scaling second harmonic generation using a cascade of nonlinear crystals[END_REF][START_REF] Kumar | High-efficiency, multicrystal, single-pass, continuouswave second harmonic generation[END_REF]. Further power scaling can be achieved using an increased NIR pump laser power enabled by research on highly efficient high power diode lasers [START_REF] Crump | Efficient High-Power Laser Diodes[END_REF]. Power can also be scaled by beam combining techniques. Incoherent spectral beam combining of tapered diode lasers is a straightforward concept for this power scaling [START_REF] Jensen | Power Scaling of Nonlinear Frequency Converted Tapered Diode Lasers for Biophotonics[END_REF], however the ultimate limitation is the available output power per wavelength channel. Coherent beam combining (CBC) of light from several devices is the only way to achieve much higher powers within a single beam that retains excellent spatial and spectral beam quality [START_REF] Fan | Laser beam combining for high-power, high-radiance sources[END_REF].

We describe in this paper the CBC of three high-brightness tapered amplifiers in a simple optical setup. Furthermore, we demonstrate the importance of CBC for the development of high power visible light sources by demonstrating a significant improvement of the achievable nonlinear conversion efficiency for single pass SHG in a bulk periodically poled lithium niobate (PPLN) crystal.

Coherent beam combining

CBC is the superposition of multiple laser beams by constructive interference. It requires a proper and stable phase relationship of the sampled gain medium. Different approaches have been demonstrated: active phase locking of amplifiers seeded by a single frequency laser split into several beams or passive phase locking of emitters in an extended cavity [START_REF] Goodno | Engineering of Coherently Combined, High-Power Laser Systems[END_REF]. Impressive results for CBC using a limited number of high power fiber amplifiers (P > 100W per channel) show the potential of power scaling by CBC reaching kW level powers [START_REF] Klenke | Coherent Beam Combination of Ultrafast Fiber Lasers[END_REF][START_REF] Redmond | Diffractive coherent combining of a 2.5 kW fiber laser array into a 1.9 kW Gaussian beam[END_REF]. CBC with diode lasers and amplifiers is still limited to lower power levels but has been demonstrated in various configurations. CBC of arrays of diode lasers in an external cavity [START_REF] Huang | Coherent combination of slab-coupled optical waveguide lasers[END_REF][START_REF] Montoya | External cavity beam combining of 21 semiconductor lasers using SPGD[END_REF][START_REF] Liu | Coherent addition of high power broad-area laser diodes with a compact VBG V-shaped external Talbot cavity[END_REF] has been demonstrated but did not lead to a significant increase of the available NIR power since the combined power was limited to a few watts. CBC in MOPA configuration has been demonstrated for larger arrays of amplifiers [START_REF] Redmond | Active coherent beam combining of diode lasers[END_REF][START_REF] Levy | Coherent array of 900 semiconductor laser amplifiers[END_REF] and led to a significant increase of the available NIR power of diode laser architectures reaching about 40 W for CBC of 47 Slab Coupled Optical Waveguide Amplifiers (SCOWA) [START_REF] Creedon | High efficiency coherent beam combining of semiconductor optical amplifiers[END_REF].

Our approach to make CBC of diode lasers more attractive for future applications is to make use of high-brightness tapered amplifiers in order to reach similar power levels but with a reduced number of elements. Tapered laser diodes are a promising building block for CBC architectures as they can deliver high power in a close to diffraction limited beam out of one monolithic device. However, one important drawback to tapered devices is the slightly degraded beam quality at high powers with about 70 % power-content in the diffraction limited central lobe while the rest of the power is distributed in side lobes. TPAs are nevertheless our preferred choice for power scaling by CBC to simplify the optical setup by using only a limited number of high-power amplifiers. At the same time as demonstrated in this study, CBC is one way to reduce the relative power content in the side lobes since CBC with non-perfect Gaussian beams favors the diffraction-limited power content leading to an improved beam quality [START_REF] Schimmel | Rear-side resonator architecture for the passive coherent combining of high-brightness laser diodes[END_REF]. In our previous studies, an output power of 11.5 W was reached by CBC of a monolithically integrated array of five TPAs, but it was limited by thermal effects caused by emitter to emitter heating of the laser bar [START_REF] Schimmel | Coherent beam combining architectures for high power tapered laser arrays[END_REF]. For this reason, we make use of separate individual amplifiers that can be efficiently cooled in order to be able to operate them at higher currents and achieve a higher combined power per element [START_REF] Albrodt | Coherent combining of high brightness tapered lasers in master oscillator power amplifier configuration[END_REF]. Additionally one can develop a simplified optical setup with standard optical elements, as splitting and combining the beam is simpler.

Experimental setup

We used 6 mm long TPAs mounted p-side up on CuW-heat spreaders and C-mounts. We made use of amplifiers identical to lasers previously described in [START_REF] Fiebig | 12 W high-brightness single-frequency DBR tapered diode laser[END_REF] but with a 2 mm long RW instead of the internal grating. Two amplifiers had a 5 µm wide single-mode RW entry section and the third amplifier a 4.5 µm wide entry section, limited by the availability of the amplifiers. Currents in RW (Irw) and tapered section (Itp) can be controlled independently, allowing decoupled control of the phase (via Irw) and the power (via Itp) in each amplifier. The amplifiers are saturated for ridge currents Irw > 300 mA. We use small variations of Irw above this limit in order to control the phase with limited impact on the optical power. The maximum power extracted from the amplifiers reaches 6.5 W at 10 A and is limited by thermal rollover due to poor thermal resistance of the c-mount packaging. The power content in the central lobe is >70 %. A detailed analysis of the amplifiers used is given in [START_REF] Albrodt | Coherent combining of high brightness tapered lasers in master oscillator power amplifier configuration[END_REF] showing that the phase noise of the amplifier is dominated by low frequencies (f < 10 Hz).

The CBC interferometer used in this work is sketched in Fig. 1 and is based on three arms. The seed laser, a narrow linewidth DFB laser at 976 nm (Pmax = 100 mW), was isolated by a double stage optical isolator with an isolation > 50 dB. The beam was split and recombined using non-polarizing 50:50 beam splitters (BS). The beam was coupled into the ridge waveguides (RW) using aspheric lenses (L2: f = 8 mm, NA = 0.5) and recollimated in the fast axis (FA) by an aspheric lens (L3: f = 2.75 mm, NA = 0.55) and in the slow axis by a cylindrical lens (L4: f = 19 mm). We note that the injection powers into the amplifiers were slightly different due to the asymmetry of the setup and the used 50:50 beam splitter coatings. However, this did not affect the performance of the setup since the input powers into all amplifiers (Pin ≥10 mW) were high enough to saturate them. The beams of amplifiers A1 and A2 interfered on a 50:50 beam combiner (BS1). The combined beam was then directed to the second 50:50 beam combiner (BS2) where it interfered with the beam from A3. One path at each element of combination, representing the combining losses B1 and B2, was absorbed by a beam block. We note that the use of two 50:50 beam combiners leads to unbalanced intensities on the second combiner resulting in minor losses smaller than 3 % in the combining setup [START_REF] Schimmel | Coherent beam combining architectures for high power tapered laser arrays[END_REF]. The power of the combined beam is measured by a fast photodiode and a control loop is used to maximize and stabilize the output power by active feedback on the ridge currents of A2 and A3 sequentially. We used a standard hill climbing algorithm for the phase control, which is characterized in details in [START_REF] Albrodt | Coherent combining of high brightness tapered lasers in master oscillator power amplifier configuration[END_REF].

The output of the CBC interferometer was used for single pass SHG, which will be described later in section 3.

Combined power

We measured the coherently combined power PCBC behind the combining elements at different currents into the tapered section, corresponding to the superposition of two and three beams, respectively (see Fig. 2). The positions of the cylindrical lenses for SA-collimation (Fig. 1) were adapted at each operating point to correct the astigmatism and to ensure a good beam overlap on the beam combiners. The combining efficiency is defined by the ratio of the coherently combined power over the sum of the extracted optical powers, which is equivalent to

  CBC CBC B B P P P P     12
for the superposition of A1+2+3 neglecting the losses at the optical elements. As shown in Fig. 2 we reached a maximum power of 12.9 W at Itp = 10 A. This is higher than the 11.5 W previously achieved by CBC of 5 TPAs on a minibar limited by thermal effects [START_REF] Schimmel | Coherent beam combining architectures for high power tapered laser arrays[END_REF]. Though the design of amplifiers in [START_REF] Schimmel | Coherent beam combining architectures for high power tapered laser arrays[END_REF] was slightly different to those used in this work, it still appears that individual mounting of emitters allows the operation of the elements at higher currents since they can be cooled more efficiently. Measurements of combination of only the first two amplifiers (A1+2) show a higher combining efficiency (ranging from 85 % at 2 A to 74 % at 10 A). The combining efficiency of the total setup was > 65 %. The combining efficiency decreases for higher currents in the tapered section caused by the degradation of the individual beam quality of each emitter, which will be discussed in the following subsection.

Measurements of the spectrum showed that the spectral properties of the DFB seed laser (976 nm) were maintained. The side mode suppression ratio was better than 40 dB. The full width at half maximum (FWHM) was smaller than 20 pm, limited by the resolution of the optical spectrum analyzer. Fig. 2. Combined power at output and total combining efficiency as a function of the currents in the tapered section. The ridge currents were actively controlled in the range of 250 to 400 mA, the heatsink temperature was T = 20 °C. The astigmatism of the amplifiers was corrected at each operating current.

Beam quality and combining efficiency

Since the beam quality of the TPA is not diffraction limited in the slow axis (SA), one has to investigate the achievable combining efficiency together with the resulting beam quality of the combined beam. Two beams with a slightly different profile interfere at each combining element in the CBC interferometer (Fig. 1). The combining efficiency η at each combining step can be written as

  '( , ) ( , ) ( , ) , ( , ) ( , )
x y I x y I x y dxdy I x y I x y dxdy

     12 12 (1)
with the local combining efficiency (2)

( , ) ( , ) ( , ) ( , ) ( , ) ( , )
defined by the overlap of the incident spatial amplitude and phase profiles I1,2(x,y) and φ1,2 (x,y) respectively [START_REF] Goodno | Perturbative analysis of coherent combining efficiency with mismatched lasers[END_REF].

One can consider the beam of a tapered laser as a sum of the fundamental mode, corresponding to the central lobe, and emission at high angle ("side lobes"). The central lobes of each beam are very similar and interfere constructively with a high combining efficiency. But the high-order modes exhibit a significantly higher beam mismatch and get therefore partially filtered during the CBC. Furthermore, amplified spontaneous emission is incoherent to the seed source and is therefore weakened by 50 % at each beam combiner. Consequently, CBC of beams from tapered lasers leads to a clean-up of the spatial beam profile where the central lobe is maintained and the high angle non diffraction-limited emission is suppressed.

The beam profile measurements shown in Fig. 3(a)-3(c) illustrate this effect. The shown beam profiles correspond to the beam waist after the focusing lens with a 1/e 2 waist diameter of about 95 µm. One can clearly see that the side lobes in SA get filtered by the two CBC steps. The power content in the central lobe gets increased from 71 % for one individual amplifier to 81 % for CBC of A1+2 and 86 % for CBC of A1+2+3. As a result the beam quality factor is reduced from . M   5 for the final beam. Consequently the experimental combining efficiency of our setup is mostly limited by the power losses induced by this beam clean-up, which removes the undesired high-angle side lobes of the amplifier beams. In order to estimate the combining efficiency of the central lobes alone, we used intensity and wavefront measurements of the individual beams at maximum power. The combining efficiency is then calculated following Eqs. ( 1) and ( 2). We applied a strong clipping criterion at 1/e 2 intensity level to the experimental data as a rough estimation for the central lobe and analyzed the overlap of the intensity and phase profiles within this area. We identified a mismatch of the intensity profile in SA caused by slightly different widths of the ridge section as mentioned earlier in section 2.1. Furthermore small differences in the collimation of each beam lead to a mismatch of the intensity profile in FA. Both effects reduce the achievable combining efficiency by 7-10 % at each combining step. Furthermore the wavefront measurements were used to evaluate combining losses linked to differences in the phase profile in between the beams. Those differences are small and had a limited impact smaller than 3 % on the achievable combining efficiency of the central lobe.

Altogether, the calculated combining efficiency is η = 91 % for the first CBC step (A1+2) and η = 87 % for the second step (A1+2+3), resulting in an overall combining efficiency of 82 % for the central lobe power content at the maximum current of 10 A. Our lower experimental combining efficiency of >65 % considering the whole beam is thus clearly related to the power content in the fundamental mode of the individual amplifier beams.

Second harmonic Generation

The increase of the brightness in the NIR by CBC of laser diodes can be useful for many applications requiring high power as well as good spectral and spatial beam quality. Especially for CW single-pass SHG, the conversion efficiency is strongly influenced by the focusing conditions in the nonlinear crystal, as described by Boyd and Kleinman [START_REF] Boyd | Parametric interaction of focused Gaussian light beams[END_REF]. Furthermore, the spatial quality of the pump beam directly impacts the achievable conversion efficiency [START_REF] Uebernickel | Study of the properties of the SHG with diode lasers[END_REF].

We demonstrate in the following the improvement of single pass SHG efficiency resulting from the scaled brightness by coherent combining, and the subsequent increase of the visible power.

Description of the experiment

The output of the CBC-interferometer was used for single pass SHG as shown in Fig. 1. In order to test the nonlinear conversion efficiency at different levels of brightness, we did modify the CBC setup slightly in order to use the beam from either a single amplifier (A1) or from the CBC of two (A1+2) or three power amplifiers (A1+2+3) as the input pump beam for the SHG. This was done by replacing the relevant combining elements with highly reflective mirrors. During the SHG experiments all three amplifiers were operated at constant current into the tapered section (Itp = 9 A) and the power used for SHG was adjusted by turning the polarization before the first polarizing beam splitter (PBS). The optical isolation between the CBC interferometer and the SHG experiment was >25 dB. After reshaping optics (L5: f = 50 mm and L6: f = 100 mm) for beam size adjustment and astigmatism correction, another half-wave plate is used to align the polarization of the NIR beam with the crystallographic Z-axis of the nonlinear crystal which is parallel to the thickness of the crystal in this case. The NIR beam was then focused with a lens (L7: f = 150 mm) into the 40 mm long periodically poled MgO:LiNbO3 (PPLN). The 1/e 2 waist diameter of the focused beam inside the crystal was measured to be ~ 95 µm, which was experimentally verified to be the optimum focusing condition at the highest input power. The focusing conditions were identical for all pump source configurations. The PPLN crystal was mounted into a temperature-controlled closed-top oven. The temperature was measured with a temperature sensor in the oven and corresponds to an average temperature. It was optimized at each pump power level. The spectral components were separated by a dichroic mirror (DM).

Experimental results

The SHG power was measured for three source configurations: a single amplifier (A1), CBC of two amplifiers (A1+2) and CBC of three amplifiers (A1+2+3). Results are shown in Fig. 4. We fitted the pump depletion approximation [START_REF] Risk | Compact Blue-Green Lasers[END_REF] tanh (

),

P P P       2 2 (3)
where η is the nonlinear conversion efficiency, to the experimental values for pump powers < 6 W. The nonlinear conversion efficiencies in case A1, A1+2 and A1+2+3 obtained by numerical fitting are 2.6 %/W, 3.7 %/W and 4.5 %/W, respectively. This corresponds to an increase of up to 73 % enhanced by the beam clean-up and the scaled brightness. We attribute the deviation of measured SHG power from the theoretical fit for Pω ≥ 6 W to thermal dephasing due to localized heating caused by SH absorption [START_REF] Hansen | Concept for power scaling second harmonic generation using a cascade of nonlinear crystals[END_REF]. This limited the maximum SHG output power to 2.09 W for 9.2 W of NIR input power (conversion efficiency >22 %). However it is so far the highest power achieved at λ =488 nm by single-pass SHG of a diode-laser-based system, thanks to the simultaneous increase of power and brightness provided by the coherent combining architecture. A further investigation regarding the effects of thermal dephasing on the SHG power was performed by mechanical low frequency chopping (~0.7 Hz, 50 % duty cycle) of the NIR beam at different pump power levels. Fig. 5 (a) shows a photodiode measurement of the SHG power and chopped pump beam with 7 W peak pump power. The results show a clear degradation in the SH power level during each pulse. This degradation in the SHG power indicates the onset of thermal dephasing. This effect was observed for pump powers higher than 6 W. The peak power of SHG can be considered as the SHG power level prior to adverse effects of thermal dephasing. When the peak power of SHG is plotted against the fundamental input power for Pω > 6 W, it followed the numerical fit nicely as shown in Fig. 5 (b). This supports our conclusion that further increase of the SH power was limited by thermal effects in the PPLN crystal and not by the available pump power or the pump beam quality. The crystal temperature acceptance bandwidth measured at P=1 W (blue curve) and P= 7 W (red curve) is shown in Fig. 6. The FWHM temperature bandwidth was about 0.6°C. The down shift of 0.4°C in the crystal phase matching temperature also indicates the overheating of the crystal due to SH absorption at high pump powers [START_REF] Kumar | High-power, single-frequency, continuous-wave secondharmonic-generation of ytterbium fiber laser in PPKTP and MgO:sPPLT[END_REF]. These thermal effects can be reduced by using nonlinear crystals with better thermal handling in a cascade scheme as described in [START_REF] Hansen | Concept for power scaling second harmonic generation using a cascade of nonlinear crystals[END_REF][START_REF] Kumar | High-efficiency, multicrystal, single-pass, continuouswave second harmonic generation[END_REF].

Conclusion and perspectives

Coherent beam combining is an important approach for power scaling of diode laser systems when high spatial quality and narrow linewidth are required. In this work, we used a simple architecture and only three amplifiers allowing to combine 12.9 W in one single beam. The beam quality was significantly increased by a beam cleanup inherent to the CBC process, resulting in a beam quality factor of and 86 % power content in the central lobe. Those results compare favorably with previous demonstrations of CBC with diode lasers [START_REF] Huang | Coherent combination of slab-coupled optical waveguide lasers[END_REF][START_REF] Montoya | External cavity beam combining of 21 semiconductor lasers using SPGD[END_REF][START_REF] Liu | Coherent addition of high power broad-area laser diodes with a compact VBG V-shaped external Talbot cavity[END_REF][START_REF] Redmond | Active coherent beam combining of diode lasers[END_REF][START_REF] Levy | Coherent array of 900 semiconductor laser amplifiers[END_REF][START_REF] Creedon | High efficiency coherent beam combining of semiconductor optical amplifiers[END_REF][START_REF] Schimmel | Rear-side resonator architecture for the passive coherent combining of high-brightness laser diodes[END_REF][START_REF] Schimmel | Coherent beam combining architectures for high power tapered laser arrays[END_REF], regarding both the combined power per amplifier and the simplicity of the setup. Furthermore, the multi-arm MOPA configuration used here could easily be scaled to a larger number of amplifiers, should a larger NIR power be required.

The coherently combined beam was used for single pass nonlinear frequency conversion in a PPLN crystal. As compared to direct SHG of single tapered devices, the improved beam quality of the combined beam in the infrared increases significantly the nonlinear conversion efficiency. A maximum SH power of 2.09 W at 488 nm was reached, limited by thermal dephasing of the PPLN, as confirmed using a pulsed pump beam. Further increase of the output power in the visible spectral range would firstly require the use of different nonlinear crystals in order to deal with the thermal roll-over of the conversion efficiency.

The CBC architecture demonstrated here relies only on off-the-shelf optical elements combined with state of art TPAs. As tapered amplifiers are now available at a wide range of wavelengths, from to the red to the NIR [START_REF] Sumpf | Spectrally stabilized high-power high-brightness DBR-tapered lasers in the VIS and NIR range[END_REF], our setup can easily be adapted for other applications. Besides its use for nonlinear frequency conversion towards the visible, our approach would also be effective to develop powerful pump sources for high-brightness pumping of fibre or bulk-crystal amplifiers. The output power could even be further increased by dense spectral beam combining of several similar CBC interferometers operating at slightly different wavelengths, as commonly used with single emitters and bars [START_REF] Witte | kW-class direct diode laser for sheet metal cutting based on DWDM of pump modules by use of ultra-steep dielectric filters[END_REF] offering a potential path to future CBC-based direct diode high brightness material processing systems.

Fig. 1 .

 1 Fig. 1. Experimental setup of the CBC interferometer and the single pass SHG in a MgO:PPLN bulk crystal; BS: 50:50 beam splitters; PBS: polarizing beam splitters.

Fig. 3 .

 3 Fig. 3. Beam profiles at waist for (a) one individual amplifier A2, (b) CBC of two amplifiers A1+2 and (c) the final CBC step of all three amplifiers A1+2+3. Measurement for Itp = 10 A, T = 20 °C and actively controlled ridge currents in the range of 250 to 400 mA. The FA is in vertical and the SA is in horizontal direction. The given central lobe power content was calculated by fitting a Gaussian beam profile to the central lobe of the beam and calculating the power content within the fitted profile.

Fig. 4 .

 4 Fig.4. SHG output power (P2ω) vs. fundamental pump power (Pω) with corresponding numerical fits using pump depletion approximation.

Fig. 5 .

 5 Fig. 5. (a) Photodiode measurements of the SHG output power with chopped pump beam. (b) SHG output power in CW and SHG peak power in QCW (50 % duty cycle) operation. The used pump beam was generated by CBC of A1+2.

Fig. 6 .

 6 Fig. 6. Normalized SHG power vs. crystal temperature at P=1 W (blue curve) and P= 7 W (red curve) under CW operation. The measured spectral FWHM linewidth of the generated blue-green light (488 nm) was less than 20 pm limited by the resolution of the optical spectrum analyzer. The beam quality of SHG was close to diffraction-limited ( . M   2 4 12). The improved beam quality of the SHG beam compared to the fundamental input beam ( . M  

4 1 1 2 5

Acknowledgment

Portions of this work were presented at the 8 th EPS-QEOD Europhoton conference in 2018, TuM1.4. The authors are grateful to F. Moron for technical assistance.

Funding

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 721766.


Vincent Samy

Ko Ayusawa

Yusuke Yoshiyasu

Ryusuke Sagawa

Eiichi Yoshida

Musculoskeletal Estimation Using Inertial Measurement Units and Single Video Image

We address the problem of estimating the physical burden of a human body. This translates to monitor and estimate muscle tension and joint reaction forces of a musculoskeletal model in real-time. The system should minimize the discomfort generating by any sensors that needs to be fixed on the user. Our system combines a 3D pose estimation from vision and IMU sensors. We aim to minimize the number of IMU fixed to the subject while compensating the remaining lack of information with vision.

I. INTRODUCTION

Not all factories are automatized and humans will keep working in this environment. One of the main reason is that robots have still problems to adapt to the human environment. In opposite, humans navigate easily in any environment, however, several studies [START_REF] Duthey | Background paper 6.24 low back pain," Priority medicines for Europe and the world[END_REF] [START_REF] Punnett | Estimating the global burden of low back pain attributable to combined occupational exposures[END_REF] have shown that factory workers face muscular fatigue and physical health difficulties on the long term. One of the focus of this paper is low back (lumbar) pain which is one of the main problem encountered by workers. The National Institute of Occupational Safety and Health (NIOSH) has set a maximum limit load on the inter-vertebral discs [START_REF] Waters | Revised niosh equation for the design and evaluation of manual lifting tasks[END_REF] which is often used to qualify a back pain risk. Monitoring in real-time this load is a complex operation, first because its computation is not trivial and second because the used system must not be cumbersome for the worker. The latter excludes intrusive methods like [START_REF] Wilke | Intradiscal pressure together with anthropometric data-a data set for the validation of models[END_REF] which provide a better precision results but are impracticable. The former goes with estimations (and thus imprecision), and with some time constraints since all the computations need to be done in a short amount of time. Although several off-line estimations exist [START_REF] Imamura | Risk estimation for intervertebral disc pressure through musculoskeletal joint reaction force simulation[END_REF] [START_REF] Samy | Real-time musculoskeletal visualization of muscle tension and joint reaction forces[END_REF], the focus will be made on on-line computation while reducing cumbersomeness.

In our previous work [START_REF] Samy | Real-time musculoskeletal visualization of muscle tension and joint reaction forces[END_REF], we developed a framework of visualizing the physical burden of human body during movements by using the human musculoskeletal model [START_REF] Nakamura | Somatosensory computation for man-machine interface from motion-capture data and musculoskeletal human model[END_REF]. The system can realize the real-time estimation of the several information like joint angles, joint torques, muscle tensions, and joint reaction forces [START_REF] Imamura | Risk estimation for intervertebral disc pressure through musculoskeletal joint reaction force simulation[END_REF]. The main objective is to support factory workers by monitoring the risk of physical health problems like low back pains. However, the system requires a large amount of Inertial Measurement Unit (IMU) sensors in order to obtain the accurate results, which reduces the workers' comfort during their tasks. Recently, Ohashi et al. [START_REF] Ohashi | Video motion capture from the part confidence maps of multi-camera images by spatiotemporal filtering using the human skeletal model[END_REF] presented an algorithm based on 4 camera that computes Human pose by using a spatiotemporal filter. They could reach a precision of less than 3cm in average. While the precision is remarkable, the system confines the subject in a space where all cameras need to see him. IMU-based systems also exist, Marcard et al. [START_REF] Marcard | Sparse inertial poser: Automatic 3d human pose estimation from sparse imus[END_REF] suggest a method that merges a statistical body model that includes anthropometric constraints with 6 IMU sensors. Malleson et al. [START_REF] Malleson | Real-time full-body motion capture from video and imus[END_REF] proposed a method to get the 3D human pose estimation from multiple cameras and from 6 to 13 IMUs in real-time. Methods as in [START_REF] Trumble | Total capture: 3d human pose estimation fusing video and inertial sensors[END_REF] uses the deep neural network to train both video images and the data of IMU sensors. This approaches have a benefit of integrating several inputs of data resources easily. The video cameras can be introduced in a factory and will provide supplementary information about human movement. On the other hand, they are not always available during whole period of working. Therefore, the pose estimation requires the flexibility of changing the inputs of data resources according to situations. In addition, the straightforward pose estimation often lacks the human musculoskeletal feasibility like joint or muscletendon constraints.

In this paper, we propose a new framework of musculoskeletal estimation by utilizing the vision-based pose estimation technique with deep learning [START_REF] Yoshiyasu | Skeleton transformer networks: 3d human pose and skinned mesh from single rgb image[END_REF]. Therefore, our method is based on the motion optimization with the musculoskeletal model to integrate the pose information obtained from IMU sensors and vision images. The musculoskeletal optimization can provide both the human feasibility and the flexibility about data integration. Based on the fast robotics kinematics computation [START_REF] Murai | Musculoskeletal-see-through mirror: Computational modeling and algorithm for whole-body muscle activity visualization in real time[END_REF], [START_REF] Ayusawa | Fast inverse dynamics algorithm with decomposed computation of gradient for wire-driven multi-body systems and its application to estimation of human muscle tensions[END_REF], [START_REF]Fast inverse kinematics algorithm for large dof system with decomposed computation of gradient and its application to musculoskeletal model[END_REF], our method also realize the real-time computation of the detailed human musculoskeletal model [START_REF] Nakamura | Somatosensory computation for man-machine interface from motion-capture data and musculoskeletal human model[END_REF], By introducing the vision-based pose estimation, the number of IMU sensors can be reduced while keeping the accuracy of musculoskeletal estimation. Our method does not use the data of IMU sensors in the training of the deep neural network, which also enables the flexible change of the placement of IMU sensors according to applications. This paper also show several experimental validations about the accuracy by testing the several configurations of sensor placements.

II. METHOD

We propose a system that estimates musculoskeletal activities from different set of sensors. Fig. 1: Process from sensors outputs to wire tension and joint reaction forces computation. It takes as input IMUs and an image. XSens SDK provide IMUs' orientation S E Ro and SkeletonNet provides joint positions m r Rv , both are respectively transform in world coordinates to get S E 0 and m r 0 . Then, they are sent to the IK along model constraints q t . Output of the IK q is derived and filtered to get q and q. Finally, data are sent to the ID along wire constraints f max to output the wire tension f , the joint torque τ j and the joint reaction force τ c .

A. Overview

The flow of the system (see Fig. 1) works like this: i) get sensors' data, ii) Transform data into the musculoskeletal model reference frame, iii) compute the Inverse Kinematics (IK), and iv) feed the Inverse Dynamics (ID) with IK results.

We used two sets of sensors. For body orientation, 12 IMU sensors1 are attached to wrists, shoulders, ankles, thighs, head, torso and back, and for body position, a webcam2 visualizes the scene. We use the XSens provided SDK to collect IMU outputs and SkeletonNet [START_REF] Yoshiyasu | Skeleton transformer networks: 3d human pose and skinned mesh from single rgb image[END_REF] system to get the 3D pose estimation.

SkeletonNet uses a single camera with depth and deep learning methods to return joint position in the world frame. The system uses a two-step regressor (see Fig. 2) that first performs a bone rotation regression then a cross heatmap regression. Note that the regressor used here is less complex than the one presented in [START_REF] Yoshiyasu | Skeleton transformer networks: 3d human pose and skinned mesh from single rgb image[END_REF].

Although IMU sensors don't need any model to work, the 3D pose estimator is based on a simplified human model composed of 16 points that correspond to the ankles, knees, hips edges, hip center, neck, shoulders, elbows, hands, lower and upper head positions. This model is rather simple and thus, is different from the one in the IK and ID. The IK/ID model is composed of 14 joints, which correspond to 47 degrees of freedom, 314 muscles, 6 tendons and 34 cartilages.

B. Musculoskeletal variables estimation

To synchronize XSens with the IK model, and SkeletonNet with the IK model, we added an initialization step which allow us to get correct transformations from measurement to model. At the initialization step, the subject is asked to match an upright configuration with arms along the body Fig. 4. From this configuration we get both Ro E i S the rotation of the i-th sensor from its current frame F S to its reference frame F Ro and m r k Rv the k-th marker position from its reference frame F Rv to the marker position m. For XSens, we need to know the rotation from the IMU reference frame and the orientation of the arm it is attached to. Let's have G E b the Given rotation from the body orientation to the attached sensor. This value is decided beforehand and is chosen so that the IMU is easy to place on the body (like a flat surface). Provided that the frame F G and the frame F S should match, we compute the transformation

Ro E i 0 = Ro E i S G E i b b E i 0 (1)
with b E i 0 the orientation of body i in the world frame. We also compute the offset from F G to F S with

S E i G = S E i Ro Ro E i 0 0 E i G . (2)
In the same manner, we define

b r i m = b r i 0 -0 E Rv m r i Rv ( 3
)
where 0 E Rv is the rotation form SkeletonNet reference frame to the world frame and b r i 0 the position of body i in the world frame. Fig. 2: SkeletonNet system that outputs joint positions using a two-step regressor. Now that data are transformed into the same frame, we can compute the IK:

min q ω q q i t -q i 2 + ω r t r i 0 -b r i 0 2 + ω E t E i 0 -b E i 0 2 (4)
with q i the generalized coordinates of angle i and q i t its target, b r i 0 the position of body b in the world frame and t r i 0 its target, b E i 0 the orientation of body b in the world frame and t E i 0 its target, ω k , k = {q, r, E} the weight. The vision system provides markers as position markers relative to the camera and IMU sensors provide body orientation. When the subject performs a whole body rotation, the IK has difficulties to rotate the root body at the same amount. This is because the pelvis (e.g. the root body), has no parent joint (it is freely attached to the world). To prevent this, we add a special task which target a specific joint angle value for all joints that link a body to the pelvis. We also add tasks that acts as joint limit constraints. These values are set at the initialization step and are not changed afterwards.

Last point is the synchronization of data. SkeletonNet returns data every 16ms in average but varies from 14ms up to 22ms for worst cases. XSens returns values every 16ms. We decided to have a loop that runs every 30ms. The loop first get data from both SkeletonNet and XSens (using protective methods like mutex to ensure data validity). Then it computes the IK. The IK can either end because a limit of time is reached or if converged. The derivative of IK's result are computed and filtered and then sent to the ID.

The optimization system to compute the inverse dynamics is [START_REF] Imamura | Risk estimation for intervertebral disc pressure through musculoskeletal joint reaction force simulation[END_REF]:

min f ,τj ,τc Z f (f ) + Z j (f , τ j ) + Z c (f , τ c ) s.t. -f max ≤ f ≤ 0 (5)
with

Z f = f T W f f Z k = (τ k -J T k f ) T W k (τ k -J T k f ), k = j, c, (6)
where f ∈ R Nw is the wire tension, τ j ∈ R Ndof is the joint torques, τ c ∈ R Nc is the joint reaction forces, J j ∈ R Nw×Ndof is the Jacobian matrix that maps the joint torques to the wire tension, J c ∈ R Nw×Nc is the Jacobian matrix that maps the joint reaction forces to the wire tension, and W f , W j and W c are weighting matrix.

III. EXPERIMENT

We conducted experiments in order to compare the results of estimated physical quantities among several configurations of input data resources. Though our framework utilizes IMUs and a single video camera, the optical motion capture system (Motion Analysis Corp.) was also used in order to be compared.

Our aim is not only to reduce the number of IMUs but also to know which IMUs should be used. To compare different IMU configuration, we decided to record a complete dataset with all systems. As in Fig. 3, the subject wears a suit with 34 motion capture markers and 12 IMUs. The vision system is composed of one camera (see Fig. 4) set in front of the subject. All data are streamed to the main computer that record data altogether so that all are synchronized. Although it is possible to compute wire tension and joint reaction forces in a dynamic way, to simplify the comparison, we decided to limit the computation of the ID at the static level, leaving q = 0 and q = 0. During the recording, the subject is asked to perform several movement with different facings (facing the camera or not). Some of the movement includes step, throw, lift, crouch, sit, radio gymnastic, etc.

IV. RESULTS

Among all different motion we will show some results on 4 different level. Fig. 5a and 5b is about the joint angle (xaxis) of the left shoulder during lifts. Fig. 6a and 6b compares joint angle (y-axis) of the lumbar vertebra 5 during steps and throws. Fig. 7a and7b shows joint reaction force (yaxis) of the lumbar vertebra 5 during crouches. And finally Fig. 8a and 8b represents data of the rectus femoris muscle during crouches and sit. For each motion (step, crouch, etc), the subject performs it facing, siding and backing on to the camera. All the computation process has been made in realtime at a rate of 33Hz on a single computer (Intel Xeon Gold 6134 CPU @3.20 GHz and NVIDIA Quadro P200).

Here, we compare different set of IMU disposition on the body along the vision system. In the following graph we used Ext for 5 IMUs on chest, wrists and ankles, Int for 5 IMUs on chest, shoulders and thighs, Mix5 for 5 IMUs on chest, wrists and thighs, and Mix6 the same as Mix5 with one more IMU on the lumbar vertebra 1.

Fig. 5a shows that the vision system does not yet have a precision good enough to run on its own, although it sometimes can perform as good as a full IMU set, it can not maintain the precision. This is due to the occlusion that might happen and the difficulties for the estimator to get the 3D pose from dynamic motion that generates blurry images. To prevent and alert potential back pain we need to focus on lumbar forces. We now consider a joint belonging to the trunk and that is used to compute these forces. In that case, Fig. 6a, the vision system can not see how the person is bending since it only return the hip and chest position. Of course, having a sensor close to the lumbar vertebra (line Mix6 in Fig. 6b 1 ) provides a better solution than others.

We can see in Fig. 8a and 8b the impacts of these errors on the lumbar vertebra joint reaction forces. Vision system can not output any correct data while the XSens system can at least provide intensity spikes. Mixing these data with IMUs 1 The results are subject to small time delay that increase slightly through time. return Fig. 7b. It shows that adding a sixth IMU on the lumbar vertebra also greatly increase the system accuracy in term of joint reaction force. Note that the Ext provides better results than other methods but it can not catch sudden spike intensities.

Finally, we also looked at a muscle tension, the rectus femoris, that links the pelvis to the femur Fig. 8a and8b. Again, regarding the subject posture, the vision system sometimes fails matching the 3D pose. Here, the Ext disposition gives the worst results, it is also the only one that does not have IMU on its thighs.

Globally using vision allows to reduce the number of IMUs while keeping accuracy. For case of the spine, it is necessary to attach several IMUs on it. Another solution would be to have a better model for the vision. For dynamic computation, it is better to have IMUs close to muscle tension/joint reaction force you want to have results on.

V. CONCLUSION

We proposed a new framework of musculoskeletal estimation by using IMUs and a single video camera. The motion optimization integrates the input data resources with keeping the accuracy of musculoskeletal estimation. By integrating the vision-based pose estimation, we showed that the number of IMU sensors attached to a subject can be reduced. The consistency between two time instances is also preserved thanks to the IMUs sensors, while the only vision-based pose estimation often loses the consistency. The method can provide a flexible IMU disposition so that it can be adjusted depending on which body segment we want to analyze accurately; for example, IMUs on the trunk would allow the accurate estimation of the joint reaction forces of the lumber-vertebra. Therefore, our method is expected to be used to monitor the back pain risk of a individual subject in an actual working environment.

VI. FUTURE WORK

Here, we discuss a way to reduce the number of IMU without loosing too much precision. A next job would be to use multiple camera and to be robust with lost signals (out-of-range IMU or occlusion).

The benefits of using vision is twofold, it is less cumbersome for the subject and, in the future, we can exploit it to get the global pose and even detect contacts with the environment. And to better exploit vision capabilities we do need to update our vision system to have a more robust system. For the sake of accuracy, we also want to include dynamic consistency inside the optimization process so that wire tension and joint reaction forces would return better results.

In a further future, we would like to use the software on site, to monitor workers. Doing so, we hope that we will be able to reduced the amount of physical pains and improve the workers' work life.

Fig. 3 :

 3 Fig. 3: Sensor positioning. The suit is composed of 34 motion capture markers and 12 IMUs.

Fig. 4 :

 4 Fig. 4: Experiment initialization. Red circle shows motion capture cameras, blue circle shows the camera used for the vision, green for the recording.

  Fig. 5b compares different IMUs disposition. It turns out that the best disposition in this case is the Int. It almost always follow the MoCap value which is consider of the closest to reality.

  X-axis of spherical joint of the left shoulder during lifts motion for MoCap, IMU and vision methods alone. X-axis of spherical joint of the left shoulder during lifts motion for MoCap and several mix methods combining vision and IMUs.

Fig. 5 :

 5 Fig. 5: Joint angle visualization of the left shoulder during lifts motion.

  (a) Y-axis of spherical joint of the lumbar vertebra 5 during steps and throws motion for MoCap, IMU and vision methods alone. (b) Y-axis of spherical joint of the lumbar vertebra 5 during steps and throws motion for MoCap and several mix methods combining vision and IMUs.

Fig. 6 :

 6 Fig. 6: Joint angle visualization of the lumbar vertebra 5 during steps and throws motion.

  Y-axis of joint reaction force of the lumbar vertebra 5 during crouches motion for MoCap, IMUs and vision methods alone. Y-axis of joint reaction force of the lumbar vertebra 5 during crouches motion for MoCap and several mix methods combining vision and IMUs.

Fig. 7 :

 7 Fig. 7: Joint reaction force of the lumbar vertebra 5 during crouches motion.

  Rectus femoris muscle force during crouches and sit motion for MoCap, IMUs and vision methods alone. Rectus femoris muscle force during crouches and sit motion for MoCap and several mix methods combining visions and IMUs.

Fig. 8 :

 8 Fig. 8: Rectus femoris muscle force during crouches and sit motion.

XSens awinda series: https://www.xsens.com/products/ mtw-awinda/

Logitech HD C615: https://www.logitech.com/en-us/ product/hd-webcam-c615


Richard David Gill
email: gill@math.leidenuniv.nl

Does Geometric Algebra provide a loophole to Bell's Theorem?

Keywords: Bell's theorem; geometric algebra, Clifford algebra, quantum information

, exposing conceptual and algebraic errors. Since 2015, when

Introduction

In 2007, Joy Christian surprised the world with his announcement that Bell's theorem was incorrect, because, according to Christian, Bell had unnecessarily restricted the co-domain of the measurement outcomes to be the traditional real numbers. Christian's first publication on arXiv spawned a host of rebuttals and these in turn spawned rebuttals of rebuttals by Christian. Moreover, Christian also published many sequels to his first paper, most of which became chapters of his book [START_REF] Christian | Disproof of Bell's Theorem: Illuminating the Illusion of Entanglement[END_REF], second edition; the first edition came out in 2012.

His original paper [START_REF] Christian | Disproof of Bell's theorem by Clifford algebra valued local variables[END_REF] took the Clifford algebra C 3,0 (R) as outcome space. This is "the" basic geometric algebra for the geometry of R 3 , but geometric algebra goes far beyond this. Its roots are indeed with Clifford in the nineteenth century but the emphasis on geometry and the discovery of new geometric features is due to the pioneering work of David Hestenes who likes to promote geometric algebra as the language of physics, due to its seamless combination of algebra and geometry. The reader is referred to the standard texts [START_REF] Doran | Geometric Algebra for Physicists[END_REF] and [START_REF] Dorst | Geometric Algebra for Computer Science[END_REF]; the former focussing on applications in physics, the latter on applications in computer science, especially computer graphics. The wikipedia pages on geometric algebra, and on Clifford algebra, are two splendid mines of information, but the connections between the two sometimes hard to decode; notations and terminology are not always consistent.

David Hestenes was interested in marrying quantum theory and geometric algebra. A number of authors, foremost among them Doran and Lasenby, explored the possibility of rewriting quantum information theory in the language of geometric algebra. This research programme culminated in the previously mentioned textbook [START_REF] Doran | Geometric Algebra for Physicists[END_REF] which contains two whole chapters carefully elucidating this approach. However, it did not catch on, for reasons which I will attempt to explain at the end of this paper.

It seems that many of Christian's critics were not familiar enough with geometric algebra in order to work through Christian's papers in detail. Thus the criticism of his work was often based on general structural features of the model. One of the few authors who did go to the trouble to work through the mathematical details was Florin Moldoveanu. But this led to new problems: the mathematical details of many of Christian's papers seem to be different: the target is a moving target. So Christian claimed again and again that the critics had simply misunderstood him; he came up with more and more elaborate versions of his theory, whereby he also claimed to have countered all the objections which had been raised.

It seems to this author that it is better to focus on the shortest and most self-contained presentations by Christian since these are the ones which a newcomer has the best chance of actually "understanding" totally, in the sense of being able to work through the mathematics, from beginning to end. In this way he or she can also get a feeling for whether the accompanying words (the English text) actually reflect the mathematics. Because of unfamiliarity with technical aspects and the sheer volume of knowledge which is encapsulated under the heading "geometric algebra", readers tend to follow the words, and just trust that the (impressive looking) formulas match the words.

In order to work through the mathematics, some knowledge of the basics of geometric algebra is needed. That is difficult for the novice to come by, and almost everyone is a novice, despite the popularising efforts of David Hestenes and others, and the popularity of geometric algebra as a computational tool in computer graphics. Wikipedia has a page on "Clifford algebra" and another page on "geometric algebra" but it is hard to find out what is the connection between the two. There are different notations around and there is a plethora of different products: geometric product, wedge product, dot product, outer product. The standard introductory texts on geometric algebra start with the familiar 3D vector products (dot product, cross product) and only come to an abstract or general definition of geometric product after several chapters of preliminaries. The algebraic literature jumps straight into abstract Clifford algebras, without geometry, and wastes little time on that one particular Clifford algebra which happens to have so many connections with the geometry of three dimensional Euclidean space, and with the mathematics of the qubit.

Interestingly, in recent years David Hestenes also has ardently championed mathematical modelling as the study of stand-alone mathematical realities. I think the real problem with Christian's works is that the mathematics does not match the words; the mismatch is so great that one cannot actually identify a well-defined mathematical world behind the verbal description and corresponding to the mathematical formulas. Since his model assumptions do not correspond to a well-defined mathematical world, it is not possible to hold an objective discussion about properties of the model. As a mathematical statistician, the present author is particularly aware of the distinction between mathematical model and reality -one has to admit that a mathematical model used in some statistical application, e.g., in psychology or economics, is really "just" a model, in the sense of being a tiny and oversimplified representation of the scientific phenomenon of interest. All models are wrong, but some are useful. In physics however, the distinction between model and reality is not so commonly made. Physicists use mathematics as a language, the language of nature, to describe reality. They do not use it as a tool for creating artificial toy realities. For a mathematician, a mathematical model has its own reality in the world of mathematics.

The present author already published (on arXiv) a mathematical analysis, [START_REF] Gill | Simple refutation of Joy Christian's simple refutation of Bell's simple theorem[END_REF], of one of Christian's shortest papers: the so-called "one page paper", [START_REF] Christian | Disproof of Bell's theorem[END_REF], which moreover contains the substance of the first chapter of Christian's book. In the present paper he analyses in the same spirit the paper [START_REF] Christian | Disproof of Bell's theorem by Clifford algebra valued local variables[END_REF], which was the foundation or starting shot in Christian's project. The present note will lay bare the same fundamental issues which are present in all these works. In fact, the model is not precisely the same in every publication, but constantly changes.

We will find out that [START_REF] Christian | Disproof of Bell's theorem by Clifford algebra valued local variables[END_REF] model is not complete: the central feature, a definition of local measurement functions, is omitted. However there is little choice in how to fill the gap, and that is done in the 2011 version (though the author does not state explicitly that he is actually adding something to the earlier version of the model).

What is common to both these short versions of Christian's theory is a basic conceptual error and a simple algebraic (sign) error. I think it is important to identify these with maximal clarity and that is why it is best to focus on the shortest and most self-contained presentations of the theory. One sees that the model is based on a dream which evaporates in the cold light of day.

If one only uses mathematics as an intuitive language in which to describe physical insights, if a particular mathematical formalism does not match physical intuition, then the mathematics is at fault: not the intuition. However if there is no stand-alone mathematical model behind the dream. . . then it remains a dream which cannot be shared with other scientists. I shall return to both projects: Doran and Lasenby's project to geometrise quantum information, and Christian's to disprove Bell; at the end of this paper. First it is necessary to set up some basic theory of geometric algebra. What is the Clifford algebra C 3,0 (R) and what does it have to do with Euclidean (three-dimensional) geometry? What does it have to do with the quantum information theory of a system of qubits? We will actually start very close to a single qubit by discussing an alternative way of looking at the two-by-two complex matrices. Since each of the four complex entries of such a matrix has a real and an imaginary part, the whole matrix is determined by 8 real numbers. Restricting attention to the addition and subtraction of these matrices, and scalar multiplication (the scalars in question being reals, we are looking at an eight-dimensional real vector space. We will study the interaction between the real vector space structure and the operation of matrix multiplication, together giving us what the pure mathematicians call a real algebra.

Geometric Algebra

I will first discuss the real algebra of the two-by-two complex matrices. We can add such matrices and obtain a new one; we can multiply two such matrices and obtain a new one. The zero matrix acts as a zero and the identity matrix acts as a unit. We can multiply a two-by-two complex matrix by a real number. This is what we call scalar multiplication. We can therefore take real linear combinations. Matrix multiplication is associative so we have a real unital associative algebra. Unital just means: with a unit; real means that the scalars in scalar multiplication are reals, and therefore, as a vector space (i.e., restricting attention to addition and scalar multiplication), our algebra is a vector space over the real numbers.

The four complex number entries in a two-by-two complex matrix can be separated into their real and imaginary parts. In this way, each two-by-two complex matrix can be expressed as a real linear combination of eight "basis" matrices: the matrices having either +1 or i in just one position, and zeros in the other three positions. Obviously, if a real linear combination of those 8 matrices is zero, then all eight coefficients are zero. The vector-space dimension of our space is therefore eight.

I will now specify an alternative vector-space basis of our space. Define the Pauli spin matrices

σ x = 0 1 1 0 , σ y = 0 -i i 0 , σ z = 1 0 0 -1
and the identity

I = 1 0 0 1 and define = I, e 1 = σ x , e 2 = σ y , e 3 = σ z , β 1 = iσ x , β 2 = iσ y , β 3 = iσ z , M = iI.
It is not difficult to check that this eight-tuple is also a vector space basis. Note the following:

e 2 1 = e 2 2 = e 2 3 = , e 1 e 2 = -e 2 e 1 = -β 3 , e 2 e 3 = -e 3 e 2 = -β 2 , e 3 e 1 = -e 1 e 3 = -β 1 ,
e 1 e 2 e 3 = M.

These relations show us that the two-by-two complex matrices, thought of as an algebra over the reals, are isomorphic to C 3,0 (R), or if you prefer, form a representation of this algebra. By definition, C 3,0 (R), is the associative unital algebra over the reals generated from , e 1 , e 2 and e 3 , with three of e 1 , e 2 and e 3 squaring to and none of them squaring to -, which is the meaning of the "3" and the "0" : three squares are positive, zero are negative. Moreover, the three basis elements e 1 , e 2 and e 3 anti-commute. The largest possible algebra over the real numbers which can be created with these rules has vector space dimension 2 3+0 = 8 and as a real vector space, a basis can be taken to be , e 1 , e 2 , e 3 , e 1 e 2 = -β 3 , e 1 e 3 = β 2 , e 2 e 3 = -β 1 , e 1 e 2 e 3 = M. The algebra is associative (like all Clifford algebras), but not commutative.

This description does not quite tell you the "official" definition of Clifford algebra in general, but is sufficient for our purposes. We are interested in just one particular Clifford algebra, C 3,0 (R),

which is the Clifford algebra of three dimensional real geometry. It is called the (or a) geometric algebra and the product is called the geometric product, for reasons which will soon become clear.

I wrote for the identity matrix and will later also write for the zero matrix, as one step towards actually indentifying these with the scalars 1 and 0. However, for the time being, we should remember that we are not (at the outset) talking about numbers, but about elements of a complex matrix algebra over the reals. Complex matrices can be added, multiplied, and multiplied by scalars (reals).

Within this algebra, various famous and familiar structures are embedded.

To start with something famous, we can identify the quaternions. Look at just the four elements , β 1 , β 2 , β 3 . Consider all real linear combinations of these four. Notice that

β 2 1 = β 2 2 = β 2 3 = -, β 1 β 2 = -β 2 β 1 = -β 3 , β 2 β 3 = -β 3 β 2 = -β 1 , β 3 β 1 = -β 1 β 3 = -β 2 .
This is close to the conventional multiplication table of the quaternionic roots of minus one: the only thing that is wrong is the last minus sign in each of the last three rows. However, the sign can be fixed in many ways: for instance, by taking an odd permutation of (β 1 , β 2 , β 3 ), or an odd number of sign changes of elements of (β 1 , β 2 , β 3 ). In particular, changing the signs of all three does the job. But also the triple e 1 e 2 = -β 3 , e 1 e 3 = β 2 , e 2 e 3 = -β 1 (the order is crucial) does the job too. Notice that we reversed the order of the three β j s, which is an odd permutation, and changed two signs, an even number. We ended up with e 1 e 2 , e 1 e 3 , e 2 e 3 in the conventional lexicographic ordering.

Switching some signs or permuting the order does not change the algebra generated by , β 1 , β 2 , β 3 . These four elements do generate a sub-algebra of C 3,0 (R) which is isomorphic to the quaternions. The quaternion sub-algebra is called the "even sub-algebra" of our Clifford algebra since it is built from just the products of even numbers of e 1 , e 2 and e 3 .

Obviously, M commutes with everything, and obviously

M 2 = -. Also Me 1 = β 1 , Me 2 = β 2 , Me 3 = β 3 and, in duality with this, e 1 = -Mβ 1 , e 2 = -Mβ 2 , e 3 = -Mβ 3 .
Thus any element of C 3,0 (R) can be expressed in the form p + Mq where p and q are quaternions, M commutes with the quaternions, and M is itself another square root of minus one. However one must be careful with parametrisation and be aware of "left-handed" and "right-handed" ways to define and work with the quaternions.

Notice that (e i )( + e i ) = (the zero two-by-two matrix). So the algebra C 3,0 (R)

possesses zero divisors (in fact, very many!), and hence not all of its elements have inverses.

The real reason for calling this algebra a geometric algebra comes from locating real 3D space within it, and also recognising geometric operations and further geometric structures with the algebra. To start with, look at the linear span of e 1 , e 2 and e 3 . This can be identified with R 3 : from now on, real 3D vectors are real linear combinations of e 1 , e 2 and e 3 . So let us look at two elements a, b of R 3 , thus simultaneously elements of the linear span of e 1 , e 2 and e 3 . In the latter incarnation (i.e., as elements of C 3,0 (R)), we can multiply them; what do we find? The answer is easily seen to be the following:

ab = (a • b) + M (a × b).
Here, a • b stands for the (ordinary) inner product of two real (three dimensional) vectors, hence a real number, or a scalar; a × b stands for the (ordinary) cross product of two real three dimensional vectors, hence a real vector. As such, it is a real linear combination of e 1 , e 2 and e 3 . Multiplying by M gives us the same real linear combination of β 1 , β 2 , β 3 .

Thus the geometric algebra C 3,0 (R) contains as a linear subspace the real vectors of R 3 ; the geometric product of two such elements encodes both their vector dot product and their vector cross product. The dot product and the cross product of real vectors can both be recovered from the geometric product, since these parts of the geometric product live in parts of the eight dimensional real linear space C 3,0 (R) spanned by disjoint sets of basis elements. The real linear subspace generated by β 1 , β 2 , β 3 is called the set of bivectors, just as the real linear subspace generated by e 1 , e 2 , e 3 is identified with the set of vectors. The real linear subspace generated by the single element is identified with the real numbers or scalars and in particular, we identify the scalar 1 and the element , the scalar 0 and the element . The real linear subspace generated by the single element M is called the set of trivectors, also known as pseudo-scalars.

Bivectors can be thought of as directed planes. Going further still, rotation of vectors can be represented in terms of bivectors and so-called rotors. I will come back to this point later.

So we have seen that C 3,0 (R) is a beautiful object, containing within itself the quaternions, the complex numbers, the two-by-two complex matrices, three dimensional real vectors; and its product, sometimes called "geometric product", contains within it both the inner and outer product of real vectors. Moreover, everything comes in dual pairs: the vectors and the bivectors;

the quaternions and the product of the quaternions with M, the scalars and the pseudo-scalars.

Every element of the algebra can be split into four components: scalar, vector, bivector and pseudo-scalar. The algebra is called graded. The parts we just named constitute the zeroth, first, second and third grades of the algebra. The product of elements of grades r and s belong to the grade r + s modulo 4.

The fact that C 3,0 (R) has as a representation the algebra of complex two-by-two matrices over the reals is, on the whole, not much more than a side-note in the literature on geometric algebra. In the theory of Clifford algebra, it is just one tiny piece in the theory of classification of Clifford algebras in general. It seems to this author that this representation should help very much to "anchor" the theory for those coming from quantum theory and quantum foundations.

However, it must be pointed out that the standard textbook on geometric algebra for physicists, [START_REF] Doran | Geometric Algebra for Physicists[END_REF], has two whole chapters on quantum mechanics, the first containing a geometric algebraic representation of the quantum theory of one spin half particle, the second doing the same for two entangled spin half particles. This even contains a carefully worked out analysis of the singlet state, including derivation of the famous EPR-B correlationcos(a • b).

The reader is referred to Chapters 8 and 9 of [START_REF] Doran | Geometric Algebra for Physicists[END_REF], whose endnotes contain many, many further references. An important point to notice is that prior to Christian's work, there already was a large literature devoted to expressing quantum theory in terms of geometric algebra. Though it seems that this literature did not gain a place in mainstream quantum theory, one of Christian's aims had already been fulfilled: geometric algebra had already been put forward as a vehicle for re-writing quantum theory with geometry instead of Hilbert space at the forefront. I will later try to explain why, I believe, Doran and Lasenby's project to geometrise quantum information theory never caught on.

Remarks on computation

Before continuing, I would like to make some remarks on computation. In order to test formulas, or in order to simulate models, it is convenient to have access to computer languages which can "do" computer algebra.

First of all, the fact that C 3,0 (R) can be identified with the two-by-two complex matrices means that one can implement geometric algebra as soon as one can "do" complex matrices. One needs to figure out how to get the eight coordinates out of the matrix. There are easy formulas but the result would be clumsy and involve a lot of programming at a low level. The relation with the quaternions is more promising: we can represent any element of C 3,0 (R) as a pair of quaternions, and the real and the imaginary parts of the two component quaternions immediately give us access to the scalar and bivector, respectively trivector (also called "pseudo scalar") and vector parts of the element of the algebra. For those who like to programme in the statistical language R it is good to know that there is an R package called "onion" which implements both the quaternions and the octonions.

Here a small digression starts. There exists a huge menagerie of somewhat related mathematical objects, and sometimes notations and nomenclature are confused and confusing. For instance, the octonions are another eight dimensional algebra which however is not associative.

There are weird and wonderful and deep connections between these various objects and in particular, concerning the sequence 1, 2, 4, 8, which is where a particular construction taking us from the reals to the complex numbers to the quaternions and finally to the octonions stops. But this is not part of our story. Another Clifford algebra C 0,3 (R) is also related to pairs of quaternions, but also not part of our story. End of digression.

Nicest of all is to use a computer system for doing real geometric algebra, and for this purpose I highly recommend the programme GAViewer which accompanies the book [START_REF] Dorst | Geometric Algebra for Computer Science[END_REF]. It can be obtained from the authors' own book website http://www. geometricalgebra.net. It not only does geometric algebra computations, it also visualises them, i.e., connects to the associated geometry. Moreover the book and the programme are a nice starting point for two higher dimensional geometric algebras, of dimension 16 and 32 respectively, which encode more geometric objects (for instance, circles and affine subspaces), and more geometric operations, with the help of the further dimensions of the algebra.

Christian's first model

After several pages of general introduction, [START_REF] Christian | Disproof of Bell's theorem by Clifford algebra valued local variables[END_REF] gives a very brief specification of his (allegedly) local realist model for the singlet (or EPR-B) correlations, obtained through the device of taking the co-domain of Bell's measurement functions to be elements of the geometric algebra C 3,0 (R) rather than the conventional (one dimensional) real line.

Conventionally, a local hidden variables model for the singlet correlations consists of the following ingredients. First of all, there is a hidden variable λ which is an element of some arbitrary space over which there is a probability distribution referred to in physicist's language sometimes as ρ(λ)dλ, sometimes as dρ(λ). This hidden variable is often thought to reside in the two particles sent to the two measurement devices in the two wings of the experiment, and therefore to come from the source; but one can also think of λ as an assemblage of classical variables in the source and in both particles and in both measurement devices which together determine the outcomes of measurement at the two locations. Any "local stochastic" hidden variables model can also be re-written as a deterministic local hidden variables model. This rewriting (thinking of random variables as simply deterministic functions of some lower level random elements) might not correspond to physical intuition but as a mathematical device it is a legitimate and powerful simplifying agent.

Secondly, we have two functions A(a, λ) and B(b, λ) which take the values ±1 only, and which denote the measurement outcomes at Alice's and Bob's locations, when Alice uses measurement setting a and Bob uses measurement setting b. Here, a and b are 3D spatial directions conventionally represented by unit vectors in R 3 . The set of unit vectors is of course also known as the unit sphere S 2 .

Bell's theorem states that there do not exist functions A and B and a probability distribution ρ, on any space of possible λ whatever, such that

A(a, λ)B(b, λ)dρ(λ) = -a • b
for all a and b in S 2 .

Christian claims to have a counter-example and the first step in his claim is that Bell "unphysically" restricted the co-domain of the functions A and B to be the real line. Now this is a curious line to take: we are supposed to assume that A and B take values in the two-point set {-1, +1}. In fact, the correlation between A and B in such a context is merely the probability of getting equal (binary) outcomes minus the probability of getting different (binary) outcomes. In other words: Bell's theorem is about measurements which can only take on two different values, and it is merely by convention that we associate those values with the numbers -1 and +1. We could just as well have called them "spin up" and "spin down". In the language of probability theory, we can identify λ with the element ω of an underlying probability space, and we have two However, let us bear with Christian, and allow that the functions A and B might just as well be thought of as taking values in a geometric algebra . . . as long as we insist that they each only take on two different values.

Christian used the symbol n to denote an arbitrary unit vector (element of S 2 ) and in formulas ( 15) and ( 16) makes the following bold suggestion:

A n = B n = µ • n ∼ = ±1 ∈ S 2
where µ • n has been previously defined to be ±Mn (I use the symbol M instead of Christian's I). Christian talks about the dot here standing for a "projection" referring, presumably, to the duality between vectors and bivectors. Christian sees µ as his local hidden variable, giving us the story that space itself picks at random a "handedness" for the trivector M = e 1 e 2 e 3 , thought of as a directed volume element. This story is odd; after all, the "handedness" of e 1 e 2 e 3 is merely the expression of the sign of the evenness of the permutation 123. Of course, the multiplication rule of geometric algebra, bringing up the cross product does again involve a handedness convention: but this is nothing to do with physics, it is only to do with mathematical convention, i.e., with book-keeping.

But anyway, within Christian's "model" as we have it so far, we can just as well define λ to be a completely random element of {-1, +1}, and then define µ = λM. The resulting probability distribution of µ is the same; we have merely changed some names, without changing the model.

So now we have the model

A(a, λ) = λMa, B(b, λ) = λMb which says that the two measurement functions have outcomes in the set of pure (unit length) bivectors. Now, those two sets are both isomorphic to S 2 , and that is presumably the meaning intended by Christian when using the congruency symbol ∼ =: our measurements can be thought Let us take a look at each mis-step separately. Regarding the first (incomplete specification), there are actually only two options (without introducing completely new elements into the story), since Christian already essentially told us that the two values ±1 of (my) λ are equally likely. Without loss of generality, his model (just for these two measurement directions) becomes either

A(a, λ) = B(b, λ) = λ = ±1 or -A(a, λ) = B(b, λ) = λ = ±1
and hence his correlation is either +1 or -1, respectively.

Regarding the second mis-step, Christian proceeds to compute the geometric product (µ • a)(µ • b) (later, he averages this over the possible values of µ). Now as we have seen this is

equal to (λMa)(λMb) = λ 2 M 2 ab = -a • b -M(a × b) and therefore certainly not equal to -a • b -λM(a × b) = -a • b -µ(a × b).

Christian's second model

I next would like to take the reader to Christian's one page paper, [START_REF] Christian | Disproof of Bell's theorem[END_REF], simultaneously the main material of the first chapter of his book [START_REF] Christian | Disproof of Bell's Theorem: Illuminating the Illusion of Entanglement[END_REF].

It seems clear to me that by 2011, Christian himself has realised that his "model" of 2007 was incomplete: there was no explicit definition of the measurement functions. So now he does come up with a model, and the model is astoundingly simple . . . it is identical to my second model:

A(a, λ) = -B(b, λ) = λ = ±1.
However, he still needs to get the singlet correlation from this, and for that purpose, he daringly redefines correlation, by noting the following: associated with the unit vectors a and b are the unit bivectors (in my notation) Ma and Mb. As purely imaginary elements of the bivector algebra or quaternions, these are square roots of minus one, and we write

A(a, λ) = (Ma)(-λMa) = λ B(b, λ) = (λMb)(Mb) = -λ
where λ is a "fair coin toss", i.e., equal to ±1 with equal probabilities 1 2 .

Now the cunning device of representing these two random variables as products of fixed bivectors and random bivectors allows Christian to propose a generalised Pearson bivector correlation between A and B by dividing the mean value of the product, by the non random "scale quantities" Ma and Mb. In other words, Ma and Mb are thought of as standard deviations. The Clifford algebra valued measurement outcome ±1 is thought of as having a standard deviation which is a bivectorial root of -1.

Since the geometric product is non commutative, one must be careful about the order of these operations (and whether we divide on the left or the right), but I will follow Christian in taking what seems natural choices: "a" always on the left, "b" on the right.

Unfortunately, since AB = -1 with probability one, the Christian-Pearson correlation should be -

(Ma) -1 (Mb) -1 = -(Ma)(Mb) = -a • b -M(a × b)
, just as before. However, just as in the 2007 paper, Christian succeeds again in 2011 in getting a sign wrong, so as to "erase" the unwanted bivector cross-product term from the "correlation". In Gill (2012) I have analysed where he went wrong and put forward an explanation of how this mistake could occur (ambiguous notation and inaccurate terminology, driven of course by powerful wishful thinking). It should be noted that he also hides the sign error somewhat deeper in complex computations, looking at the average of a large number of realisations and using the law of large numbers, rather than just computing the expectation "theoretically".

I will discuss the other versions of Christian's theory at the end of this paper. He has elaborated more or less the same "theory" with the same repertoire of conceptual and algebraic errors in various papers with increasing levels of complexity. This work shows a remarkable degree of dedication and persistence, and erudition too, as more and more complex mathematical constructions are brought into play. However the flaws in the foundation are unchanged and one of them (as has been said by many before) is unredeemable: Bell's theorem is about probabilities of different combinations of pairs of binary outcomes. A local hidden variable theory has to explain the correlations between binary outcomes which are predicted by quantum mechanics, following standard, minimal, interpretations: quantum mechanics predicts the probability distribution of measurement outcomes. Experimenters (in this field) count discrete events and estimate probabilities with relative frequencies.

Geometrizing quantum information theory

I assume that the reader is already familiar with the standard "Bloch sphere" picture of quantum states, quantum operations, and quantum measurements for a single qubit. This means that the first of Doran and Lasenby's two chapters on quantum information hold few surprises. Quantum states can be represented as points in the Bloch ball, thus as real 3D vectors of length less than or equal to one. Pure states lie on the surface of the ball, mixed states in the interior. Unitary evolutions correspond to rotations of the Bloch sphere and these are represented by right and left multiplication with a unitary matrix and its Hermitian conjugate. In geometric algebra, rotation of a 3D real vector is expressed using an object called a "rotor" and the geometric idea is that we do not rotate around a vector, but in a plane. Oriented planes are represented by bivectors.

One curious point is that Doran and Lasenby also attempt to map the vector (wave function)

representation of a pure state into the geometric algebra framework. Since the density matrix is a completely adequate representation of any state, pure or mixed, this seems to me to be superfluous.

Why not agree that a pure state is represented by its density matrix, hence a point on the unit sphere, hence a real vector of unit length? They do achieve their aim by thinking of state vectors as being the result of a 3D rotation from a fixed reference direction, hence pure states are represented by rotors which involve a choice of reference direction.

All in all, one could argue that rephrasing the mathematics of a single qubit in terms of real geometric algebra demystifies the usual approach: no more complex numbers, just real 3D geometry. Doran and Lasenby hint several times in their book and other publications that their approach also has the promise of delivering an unconventional interpretation of quantum mechanics, but they never appear to follow this up.

Having all the mathematical machinery of the complex two-by-two matrices at their disposal, it is essentially a straightforward matter to now also extend the discussion to several (entangled) qubits. Indeed, their book contains a reworking of the computation of the singlet correlations for the special case of a maximally entangled pair of qubits.

The approach is simply to take the (outer) product of as many copies of C 3,0 (R) as we have qubits. Thus no new geometry is involved and in fact we move out of the realm of Clifford algebras: a product of several Clifford algebras is not a Clifford algebra itself. Certainly it is the case that the outer product of two copies of C 3,0 (R) contains, as an algebra, everything that we need relative to the usual Hilbert space approach. However it contains more than we need. The problem is that in the Hilbert space approach, the tensor product of the identity with i times the identity is equal to the tensor product of i times the identity with the identity. But taking the outer product of two Clifford algebras defines two distinct elements: ⊗ M and M ⊗ .

Thus we should form the basic space for a composite system not just by taking a product of spaces for the subsystems but also by adding a further equality ⊗ M = M ⊗ .

Notice that the two-by-two complex matrices have real dimension eight, while the four-by-four complex matrices have real dimension 32. However, 8 × 8 = 64. Thus if we believe the usual Hilbert space story for the mathematics of two qubits, then states, operators, measurements, . . . can all be represented in a real algebra of dimension 32. Adding an extra equality ⊗ M = M ⊗ reduces the dimension of C 3,0 (R) ⊗2 from 64 to 32 and now we do have exactly the algebra which we need in order to carry the "usual" Hilbert space based theory.

In fact, Doran and Lasenby, taking their cue from their representation of pure states as rotors, representing a rotation from a reference direction to the direction of the state, construct a different reduction by imposing a different equality. Their point is that geometrically, we need to align the reference directions of the two subsystems. This leads to an algebraically more complicated "base space", with perhaps better geometric interpretability, than the obvious algebraic reduction, which comes down to identifying the complex root of minus one, i, as being "the same" in the two Hilbert spaces of the two subsystems: mathematically that is a natural step to take, but what does it mean physically?

The proof of the pudding should be in the eating. On the one hand, everything one can do with the Clifford algebra approach can also be done in the Hilbert space approach: after fixing the dimension issue, one has two mathematically isomorphic systems, and the choice between the two is merely a matter of "picking a coordinate system". So far, it seems that the Clifford algebra approach did not pay off in terms of easier computations or geometric insight, when we look at systems of several qubits. On the other hand, the geometric insight for a single qubit was already available and has become completely familiar. The geometric algebra approach did not lead to new interpretational insights in quantum theory. Finally, it does not extend in an easy way to higher-dimensional systems: where is the geometric algebra of the qutrit?

However I hope that this paper will encourage others to take a fresh look for themselves. which is not a division algebra at all.

Further publications

He writes "the corresponding algebraic representation space ... is nothing but the eight-dimensional even sub-algebra of the 2 4 = 16-dimensional Clifford algebra C 4,0 ". This space is well known to be isomorphic to C 0,3 . The wikipedia pages on Clifford Algebras are very clear on this, and as far as I can see, well sourced and highly accurate. One can take as basis for the 8 dimensional real vector space C 0,3 the scalar 1, three anti-commuting vectors e i , three bivectors v i , and the pseudo-scalar M = e 1 e 2 e 3 . The algebra multiplication is associative and unitary (there exists a multiplicative unit, 1). The pseudo-scalar M squares to -1. Scalar and pseudo-scalar commute with everything. The three basis vectors e i , by definition, square to -1. The three basis bivectors v i = Me i square to +1. Take any unit bivector v. It satisfies v 2 = 1 hence v 2 -1 = (v -1)(v + 1) = 0. If the space could be given a norm such that the norm of a product is the product of the norms, we would have v -1 . v + 1 = 0 hence either v -1 = 0 or v + 1 = 0 (or both), implying that either v -1 = 0 or v + 1 = 0 (or both), implying that v = 1 or v = -1, neither of which are true.

Recall that a normed division algebra is an algebra that is also a normed vector space and such that the norm of a product is the product of the norms; a division algebra is an algebra such that if a product of two elements equals zero, then at least one of the two elements concerned must be zero. A splendid resource on the Hurwitz theorem and the octonions is John Baez' celebrated 2002 article in the Bulletin of the American Mathematical Society reproduced on his web pages http://math.ucr.edu/home/baez/octonions/. Full proofs are given but already the key definitions and main theorems are clearly stated in the first couple of pages.

There do exist some exotic objects called "exotic spheres": differentiable manifolds that are homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one. They were first discovered by Milnor in 1956 for the 7-sphere, and have been further

classified by him and many other researchers. It is unknown if exotic 4-spheres exist. There are no other exotic spheres in dimensions 1 up to 6; from then on, they only exist in some dimensions, and much is still unknown about them.

Christian claims that his model depends on the Hopf fibration, a wonderful way to look at the 3-sphere. He claims that a uniform distribution on S 3 relative to the Hopf fibration differs from the usual uniform distribution on S 3 . But this makes no sense. "Uniform distributions" are defined relative to some symmetries, and the Hopf fibration does not introduce new symmetries to S 3 beyond the familiar ones of "flatland", i.e., of Euclidean geometry.

  families of random variables A a and B b , taking values in a two point set, without loss of generality the set {-1, +1}, and Bell's claim is: it is impossible to have Prob(A a = B b ) -Prob(A a = B b ) = -a • b for all a and b.

  of as taking values in S 2 . At the same time, each measurement takes on only one of two different values (given the measurement direction) hence we can also claim congruency with the two point set {-1, +1} = {±1}. But of course, these are two different congruencies! And they still need to be sorted out. What is mapped to what, exactly . . . This is where things go badly wrong. On the one hand, the model is not yet specified, if Christian does not tell us how, exactly, he means to map the set of two possible values ±Ma onto {±1} and how he means to map the set of two possible values ±Mb onto {±1}. On the other hand, Christian proceeds to compute a correlation (in the physicist's sense: the expectation value of a product) between bivector valued outcomes instead of between {±1} valued outcomes. No model, wrong correlation.

  This paper mysteriously vanished from the web-site of the once famous journal AoP after it had been accepted but before the contents page of the next issue was definitive. On the web https://ieeexplore.ieee.org/document/8836453 IEEE Access, arXiv:1405.2355 This is actually a revised version of the disappearing IJTP paper Local Causality in a Friedmann-Robertson-Walker Spacetime which I discussed earlier. It combines elements from all Eight-dimensional Octonion-like but Associative Normed Division Algebra.In this latest paper, which Christian says is Submitted (to a pure mathematics journal), Christian claims he has found a counter-example to Hurwitz's theorem that the only division algebras are R, C, H, O of dimensions 1, 2, 4 and 8. His "counter-example" is the real Clifford Algebra C 0,3

	previous works.
	6.5. The Pure Mathematics paper, 2018
	arXiv:1806.02392
	6.1. The International Journal of Physics paper, 2015
	Macroscopic Observability of Spinorial Sign Changes under 2π Rotations.
	https://link.springer.com/article/10.1007%2Fs10773-014-2412-2
	I wrote a paper pointing out various extraordinary errors and entitled Macroscopic
	Unobservability of Spinorial Sign Changes. It was accepted and published in IJTP,
	https://link.springer.com/article/10.1007%2Fs10773-015-2657-4. Christian posted a
	rebuttal on arXiv but this was not published in IJTP, though no doubt incorporated in later
	papers.
	6.2. The Annals of Physics paper, 2015
	Local Causality in a Friedmann-Robertson-Walker Spacetime.
	https://arxiv.org/pdf/1405.2355v1.pdf

page one can read "This article was erroneously included in this issue. We apologize for any inconvenience this may cause." The author had not been informed. Quite amazingly, this paper includes a computer simulation, allegedly of the author's model but actually a simulation of Pearle's famous detection loophole model of 1970. The author does not himself read or write computer programs, but has some supporters who write software for him.

Since it is impossible to implement his own model while still getting the right answer, they are forced to modify Christian's own definitions or even to simply write code to implement already existing models earlier ihvented by others, of which there are many. I will elsewhere publish further details about my own re-discovery, correction, and implementation of Pearle's model.

The RSOS paper, 2018

Quantum correlations are weaved by the spinors of the Euclidean primitives.

https://royalsocietypublishing.org/doi/full/10.1098/rsos.180526 RSOS (Royal Society -Open Source), arXiv:1806.02392

The author claims that Bell made an extraordinary error in the proof of his theorem. Namely, it is claimed that the use of expressions involving simultaneously measurement functions evaluated at several different measurement settings cannot possibly have any physical meaning and hence that proofs which depend on the evaluation of such expresssions are physically meaningless and hence invalid.

The point, of course, is that, under local realism, the local hidden variable λ is supposed to exist, and various functions thereof are also supposed, by the assumption of local realism, to exist.

Hence one can mathematically consider any combinations of those functions one likes, and if that leads to observable consequences, for instance, bounds on correlations which can be observed in experiments, then those bounds must be valid, if local realism is true.

Note that when I use the word "exists", I speak as a mathematician, and I am discussing mathematical models. For me, local realism means the existence of a mathematical model which reproduces the predictions of quantum mechanics (or approximately does so). It is up to physicists and to philosophers to thrash out what they mean when they use words like local, real, or exist.

The paper yet again proposes a local hidden variables model in which the hidden variable λ is a fair coin toss, taking the values ±1. It follows that the pair A(a, λ), B(b, λ) can only take the values (-1, -1), (-1, +1), (+1, -1) or (1, +1). Their product can only take the values ±1 and the only probabilities around are 1 2 , 1 2 . This means that the expectation value of a product A(a, λ)B(b, λ), whatever the values of the settings a, b, can only be -1, 0 or +1. I learnt this argument from Heine Rasmussen.

The proof that the singlet correlations are produced by this model is more sophisticated than ever; and the paper actually claims to deal with all quantum correlations, not just the EPR-B correlations. The usual errors can be located in the proofs, together with a new one: a limit is computed as s 1 → a and s 2 → b, while imposing the "physical" constraint s 1 = -s 2 . The notation is non-standard and not explained, but not surprisingly, this doesn't make much sense unless

There is also a proof of an astonishing pure mathematical result in this paper, which has been separated off and submitted as a separate publication to a pure mathematics journal. I will deal with that at the end of this section. The paper has not yet appeared in print.

The IEEE Access paper, 2018

Bell's Theorem Versus Local Realism in a Quaternionic Model of Physical Space.