# Image Generators with Conditionally-Independent Pixel Synthesis

I. Anokhin1,2, K. Demochkin1,2, T. Khakhulin1,2, G. Sterkin1, V. Lempitsky1,2,
D. Korzhenkov1

1Samsung AI Center, Moscow
2Skolkovo Institute of Science and Technology, Moscow

###### Abstract

Existing image generator networks rely heavily on spatial convolutions and,
optionally, self-attention blocks in order to gradually synthesize images in a
coarse-to-fine manner. Here, we present a new architecture for image
generators, where the color value at each pixel is computed independently
given the value of a random latent vector and the coordinate of that pixel. No
spatial convolutions or similar operations that propagate information across
pixels are involved during the synthesis. We analyze the modeling capabilities
of such generators when trained in an adversarial fashion, and observe the new
generators to achieve similar generation quality to state-of-the-art
convolutional generators. We also investigate several interesting properties
unique to the new architecture.

Figure 1: Samples from our generators trained on several challenging datasets
(LSUN Churches, FFHQ, Landscapes, Satellite-Buildings, Satellite-Landscapes)
at resolution $256\times 256$. The images are generated without spatial
convolutions, upsampling, or self-attention operations. No interaction between
pixels takes place during inference.

##  1 Introduction

State-of-the-art in unconditional image generation is achieved using large-
scale convolutional generators trained in an adversarial fashion [10, 11, 1].
While lots of nuances and ideas have contributed to the state-of-the-art
recently, for many years since the introduction of DCGAN [22] such generators
are based around spatial convolutional layers, also occasionally using the
spatial self-attention blocks [32]. Spatial convolutions are also invariably
present in other popular generative architectures for images, including
autoencoders [14], autoregressive generators [30], or flow models [3, 13].
Thus, it may seem that spatial convolutions (or at least spatial self-
attention) is an unavoidable building block for state-of-the-art image
generators.

Recently, a number of works have shown that individual images or collections
of images of the same scene can be encoded/synthesized using rather different
deep architectures (deep multi-layer perceptrons) of a special kind [20, 26].
Such architectures are not using spatial convolutions or spatial self-
attention and yet are able to reproduce images rather well. They are, however,
restricted to individual scenes. In this work, we investigate whether deep
generators for unconditional image class synthesis can be built using similar
architectural ideas, and, more importantly, whether the quality of such
generators can be pushed to state-of-the-art.

Perhaps surprisingly, we come up with a positive answer (Fig. 1), at least for
the medium image resolution (of $256\times{}256$). We have thus designed and
trained deep generative architectures for diverse classes of images that
achieve similar quality of generation to state-of-the-art convolutional
generator StyleGANv2 [11], even surpassing this quality for some datasets.
Crucially, our generators are not using any form of spatial convolutions or
spatial attention in their pathway. Instead, they use coordinate encodings of
individual pixels, as well as sidewise multiplicative conditioning (weight
modulation) on random vectors. Aside from such conditioning, the color of each
pixel in our architecture is predicted independently (hence we call our image
generator architecture Conditionally-Independent Pixel Synthesis (CIPS)
generators).

In addition to suggesting this class of image generators and comparing its
quality with state-of-the-art convolutional generators, we also investigate
the extra flexibility that is permitted by the independent processing of
pixels. This includes easy extention of synthesis to non-trivial topologies
(e.g. cylindrical panoramas), for which the extension of spatial convolutions
is known to be non-trivial [17, 2]. Furthermore, the fact that pixels are
synthesized independently within our generators, allows sequential synthesis
for memory-constrained computing architectures. It enables our model to both
improve the quality of photos and generate more pixel values in a specific
areas of image (i.e. to perform foveated synthesis).

##  2 Related Work

Feeding pixel coordinates as an additional input to the neural network
previously was successfully used in the widely known CoordConv technique [18]
to introduce the spatial-relational bias. Recently, the same idea was employed
by the COCO-GAN [17] to generate images by parts or create ”looped” images
like spherical panoramas. However, those models still used standard
convolutions as the main synthesis operation. The synthesis process for
neighboring pixels in such architectures is therefore not independent.

To the best of our knowledge, the problem of regressing a given image from
pixel coordinates with a perceptron (that calculates each pixel’s value
independently) started from creating compositional patterns with an
evolutionary approach [28]. Those patterns, appealing for digital artists,
were also treated as kind of differentiable image parametrization [21].
However, this approach was not capable of producing photorealistic hi-res
outputs (e.g., see the demo [8]).

Some machine learning blogs reported experiments with GANs, where the
generator was a perceptron that took a random vector and pixel coordinates as
an input, and returned that pixel’s value as an output [5, 6]. The described
model was successfully trained on MNIST, but has not been scaled to more
complex image data.

Scene-representation networks [27] and later the neural radiance fields (NeRF)
networks [20] have demonstrated how 3D content of individual scenes can be
encoded with surprising accuracy using deep perceptron networks. Following
this realization, systems [26] and [29] considered the usage of periodic
activation functions and so-called Fourier features to encode the pixel (or
voxel) coordinates, fed to the multi-layer perceptron. In particular, the
ability to encode high-resolution individual images in this way was
demonstrated. All these works however have not considered the task of learning
image generators, which we address here.

The very recent (and independent) Generative Radiance Fields (GRAF) system
[25] showed promising results at embedding the NeRF generator into an image
generator for 3D aware image synthesis. Results for such 3D aware synthesis
(still limited in diversity and resolution) for certain have been
demonstrated. Here, we do not consider 3D-aware synthesis and instead
investigate whether perceptron-based architectures can achieve high 2D image
synthesis quality.

##  3 Method

Our generator network synthesizes images of a fixed resolution $H\times W$ and
has the multi-layer perceptron-type architecture $G$ (see Fig. 2). In more
detail, the synthesis of each pixel takes a random vector
$\mathbf{z}\in\mathcal{Z}$ shared across all pixels, as well the pixel
coordinates $\left(x,y\right)\in\left\\{0\dots
W-1\right\\}\times\left\\{0\dots H-1\right\\}$ as input. It then returns the
RGB value $\mathbf{c}\in\left[0,1\right]^{3}$ of that pixel
$G:\left(x,y,\mathbf{z}\right)\mapsto\mathbf{c}$. Therefore, to compute the
whole output image $I$, the generator $G$ is evaluated at at each pair
$\left(x,y\right)$ of the coordinate grid, while keeping the random part
$\mathbf{z}$ fixed:

$\displaystyle
I=\left\\{G\left(x,y;\mathbf{z}\right)\mid\left(x,y\right)\in\texttt{mgrid}\left(H,W\right)\right\\},$
(1)

where

$\texttt{mgrid}\left(H,W\right)=\\{\left(x,y\right)\mid 0\leq x<W,\,0\leq
y<H\\}$

is a set of integer pixel coordinates.

Following [10], a mapping network $M$ (also a perceptron) turns $\mathbf{z}$
into a _style_ vector $\mathbf{w}\in\mathcal{W}$,
$M:\mathbf{z}\mapsto\mathbf{w}$, and all the stochasticity in the generating
process comes from this style component.

We then follow the StyleGANv2 [11] approach of injecting the style
$\mathbf{w}$ into the process of generation via weight modulation. To make the
paper self-contained, we describe the procedure in brief here.

Any modulated fully-connected (ModFC) layer of our generator (see Fig. 2) can
be written in the form $\mathbf{\psi}=\hat{B}\mathbf{\phi}+\mathbf{b}$, where
$\mathbf{\phi}\in\mathbb{R}^{n}$ is an input, $\hat{B}$ is a learnable weight
matrix $B\in\mathbb{R}^{m\times n}$ modulated with the style $\mathbf{w}$,
$\mathbf{b}\in\mathbb{R}^{m}$ is a learnable bias, and
$\mathbf{\psi}\in\mathbb{R}^{m}$ is an output. The modulation takes place as
follows: at first, the style vector $\mathbf{w}$ is mapped with a small net
(referred to as A in Fig. 2) to a scale vector $\mathbf{s}\in\mathbb{R}^{n}$
Then, the $\left(i,j\right)$-th entry of $\hat{B}$ is computed as

$\hat{B}_{ij}=\frac{s_{j}B_{ij}}{\sqrt{\epsilon+\sum\limits_{k=1}^{n}\left(s_{k}B_{ik}\right)^{2}}},$
(2)

where $\epsilon$ is a small constant. After this linear mapping, a LeakyReLU
function is applied to $\mathbf{\psi}$.

Finally, in our default configuration we add skip connections for every two
layers from intermediate feature maps to RGB values and sum the contributions
of RGB outputs corresponding to different layers. These skip connections
naturally add values corresponding to the same pixel, and do not introduce
interactions between pixels.

We note that the independence of the pixel generation process, makes our model
parallelizable at inference time and, additionally, provides flexibility in
the latent space $z$. E.g., as we show below, in some modified variants of
synthesis, each pixel can be computed with a different noise vector $z$,
though gradual variation in $z$ is needed to achieve consistently looking
images.

Figure 2: The Conditionally-Independent Pixel Synthesis (CIPS) generator
architecture. Top: the generation pipeline, in which the coordinates
$\left(x,y\right)$ of each pixel are encoded (yellow) and processed by a
fully-connected (FC) network with weights, modulated with a latent vector
$\mathbf{w}$, shared for all pixels. The network returns the RGB value of that
pixel. Bottom: The architecture of a modulated fully-connected layer (ModFC).
Note: our default configuration also includes skip connections to the output
(not shown here).

###  3.1 Positional encoding

The architecture described above needs an important modification in order to
achieve the state-of-the-art synthesis quality. Recently two slightly
different versions of positional encoding for coordinate-based multi-layer
perceptrons (MLP), producing images, were described in literature. Firstly,
SIREN [26] proposed a perceptron with a principled weight initialization and
sine as an activation function, used throughout all the layers. Secondly, the
Fourier features, introduced in [29], employed a periodic activation function
in the very first layer only. In our experiments, we apply a somewhat in-
between scheme: the sine function is used to obtain Fourier embedding
$e_{fo}$, while other layers use a standard LeakyReLU function:

$e_{fo}\left(x,y\right)=\sin\left[B_{fo}\left(x^{\prime},y^{\prime}\right)^{T}\right],$
(3)

where $x^{\prime}=\frac{2x}{W-1}-1$ and $y^{\prime}=\frac{2y}{H-1}-1$ are
pixel coordinates, uniformly mapped to the range $\left[-1,1\right]$ and the
weight matrix $B_{fo}\in\mathbb{R}^{2\times n}$ is learnable, like in SIREN
paper.

However, only Fourier positional encoding usage turned out insufficient to
produce plausible images. In particular, we have found out that the outputs of
the synthesis tend to have multiple wave-like artifacts. Therefore, we also
train a separate vector $e_{co}^{\left(x,y\right)}$ for each spatial position
and call them coordinate embeddings. They represent $H\times W$ learnable
vectors in total. For comparison of these two embedding from the spectral
point of view, see Sec. 4.5. The full positional encoding $e\left(x,y\right)$
is a concatenation of Fourier features and a coordinate embedding

$e\left(x,y\right)=\text{concat}\left[e_{fo}\left(x,y\right),e_{co}^{\left(x,y\right)}\right]$
(4)

and serve as an input for the next perceptron layer:
$G\left(x,y;\mathbf{z}\right)=G^{\prime}\left(e\left(x,y\right);M\left(\mathbf{z}\right)\right)$.

##  4 Experiments

###  4.1 Architecture details

In our experiments, both Fourier features and coordinate embeddings had the
dimension of 512. The generator had 14 modulated fully-connected layers of
width 512 We use leaky ReLU activation with the slope $0.2$. We implement our
experiments on top of the public
code111https://github.com/rosinality/stylegan2-pytorch for StyleGANv2. Our
model is trained with a standard non-saturating logistic GAN loss with $R_{1}$
penalty [19] applied to the discriminator $D$. The discriminator has a
residual architecture, described in [11] (we have deliberately kept the
discriminator architecture intact). Networks were trained by Adam optimizer
[12] with learning rate $2\times 10^{-3}$ and hyperparameters: $\beta_{0}=0$,
$\beta_{1}=0.99$, $\epsilon=10^{-8}$.

###  4.2 Evaluation

We now evaluate CIPS generators and their variations on a range of datasets.
For the sake of efficiency, most evaluations are restricted to $256\times 256$
resolution. The following datasets were considered:

  * •

The Flickr Faces-HQ (FFHQ) [10] dataset contains 70,000 high quality well-
aligned, mostly near frontal human faces. This dataset is the most regular in
terms of geometric alignement and the StyleGAN variants are known to perform
very well in this setting.

  * •

The LSUN Churches [31] contains 126,000 outdoor photographs of churches of
rather diverse architectural style. The dataset is regular, yet images all
share upright orientation.

  * •

The Landscapes dataset contains 60,000 manually collected landscape photos
from the Flickr website.

  * •

The Satellite-Buildings222https://www.crowdai.org/challenges/mapping-challenge
dataset contains 280,741 images of $300\times 300$ pixels (which we crop to
$256\times 256$ resolution and randomly rotate). This dataset has large size,
and is approximately aligned in terms of scale, yet lacks consistent
orientation.

  * •

Finally, the Satellite-Landscapes333https://earthview.withgoogle.com/ contains
a smaller curated collection of 2,608 images of $512\times 512$ resolution of
satellite images depicting various impressive landscapes found on Google Earth
(which we crop to $256\times 256$ resolution). This is the most “textural”
dataset, that lacks consistent scale or orientation.

| StyleGANv2 | CIPS (ours)
---|---|---
FFHQ | 3.83 | 4.38
LSUN Churches | 3.86 | 2.92
Landscapes | 2.36 | 3.61
Satellite-Buildings | 73.67 | 69.67
Satellite-Landscapes | 51.54 | 48.47
Table 1: FID on multiple datasets at resolution of $256^{2}$ for CIPS-skips model. Note that CIPS is of comparable quality with state-of-the-art StyleGANv2, and better on Churches. The value for CIPS model on FFHQ differs from the one reported in Tab. 3 as we trained this model for more time and with larger batch size.  Model | Precision | Recall
---|---|---
StyleGANv2 | 0.609 | 0.513
CIPS | 0.613 | 0.493
Table 2: Precision & Recall measured on FFHQ at $256^{2}$. The resulting
quality of our model is better in terms of precision (corresponds to
plausibility of images) and worse in recall (this points to the greater number
of dropped modes).

For evaluation, we relied on commonly used metrics for image generation:
Frechet Inception Distance (FID) [7] as well as more recently introduced
generative Precision and Recall measures [23, 15].

Our main evaluation is thus against the state-of-the-art StyleGANv2 generator
[11]. We took FID value for LSUN Churches directly from the original paper
[11] and trained StyleGANv2 on other datasets in authors’ setup but without
style-mixing and path regularization of generator – as noted in the original
paper, these changes do not influence the FID metric. The results of this key
comparison are presented in Tab. 1 and 2. Neither of the two variants of the
generator dominates the other, with StyleGANv2 achieving lower (better) FID
score on FFHQ and Landscapes, while CIPS generator achieving lower score on
LSUN Churches and both Satellite datasets.

###  4.3 Ablations

CIPS | “base” | “No embed (NE)” | “No Fourier” | Main | “Residual” | Sine
---|---|---|---|---|---|---
Fourier Features | + | + | – | + | + | –
Coordinate Embeddings | + | – | + | + | + | +
Residual blocks | – | – | – | – | + | –
Skip connections | – | – | – | + | – | –
Sine Activation | – | – | – | – | – | +
FID | 6.71 | 12.71 | 10.18 | 6.31 | 6.52 | 10.0
Table 3: Effects of the modifications of CIPS generator on the FFHQ dataset in
terms of Frechet Inception Distance (FID) score. Each column corresponds to a
certain configuration, while rows correspond to present/missing features. The
simultaneous usage of Fourier features and coordinate embeddings is necessary
for a good FID score. Also, both residual connections and cumulative skip
connections (default configuration) to the output outperform the plain
multilayer perceptron.  Figure 3: Image corresponding to the mean style vector
in the space of $\mathcal{W}$ for CIPS (left) and CIPS-NE (no embeddings)
generators (right). Left image has more plausible details like hair which
confirms the results in Tab. 3.

We then evaluate the importance of different parts of our model by its
ablation on the FFHQ dataset (Tab. 3). We thus consider removing Fourier
features, coordinate embeddings (config referred to as _CIPS-NE_) and replace
LeakyReLU activation with sine function in all layers. We also compare the
variants with residual connections (we follow StyleGANv2 [11] implementation
adjusting variance of residual blocks with the division by $\sqrt{2}$) with
our main choice of cumulative projections to RGB. Additionally, the “base”
configuration without skip connections and residual connections is considered.
In this comparison, all models were trained for 300K iterations with batch
size of 16.

As the results show, coordinate embeddings, residual blocks and cumulative
projection to RGB significantly improve the quality of the model. The removal
of coordinate embeddings most severely worsens the FID value, and affects the
quality of generated images (Fig. 3). We further investigate the importance of
coordinate embeddings for the CIPS model below.

###  4.4 Influence of positional encodings

|
---|---
(a) Fourier features | (b) Coordinate embeddings
Figure 4: The spectrum magnitude for our two kinds of positional encoding (color scale is equal for both plots). The output of coordinate embeddings clearly has more higher frequencies. |
---|---
(a) Fourier features | (b) Coordinate embeddings
Figure 5: PCA plot (3 components) for two kinds of positional encoding of CIPS-base. Coordinate embeddings contain not just more fine-grained details, but also key points of the averaged face.  |  |
---|---|---
Figure 6: Influence of different types of positional encoding on the resulting
image. Left: original image. Center: coordinate embeddings zeroed out (the
image contains no fine-grained details). Right: Fourier features zeroed out
(only high-frequency details are present).

To analyze the difference between Fourier features $e_{fo}$ and coordinate
embeddings $e_{co}$, we plotted the spectrum of these codes for the generator
CIPS-base, trained on FFHQ. As shown in Fig. 4, Fourier encoding generally
carries low-frequency components, whereas coordinate embeddings resemble more
high-frequency details. The Principal Component Analysis (PCA) of the two
encodings supports the same conclusion (Fig. 5) The possible explanation is
simple: coordinate embeddings are trained independently for each pixel, while
$e_{fo}\left(x,y\right)$ is a learned function of the coordinates. However,
the next layers of the network could transform the positional codes and, for
example, finally produce more fine-grained details, relying on Fourier
features. To demonstrate that this is not the case, we conducted the following
experiment. We have zeroed out either the output of Fourier features or
coordinate embeddings and showed the obtained images in Fig. 6. One can notice
that the information about the facial hair’s details as well as the forelock
is located in the coordinate embeddings. This proves that it is the coordinate
embeddings that are the key to high-frequency details of the resulting image.

###  4.5 Spectral analysis of generated images

(a) Magnitude spectrum. Our models produce less artifacts in high frequency
components (note the grid-like pattern in StyleGANv2). Two CIPS models are
difficult to distinguish between (better zoom in).

(b) Azimuthal integration over Fourier power spectrum. The curve of StyleGANv2
has heavy distortions in most high frequency components. Surprisingly, CIPS-NE
demonstrates a more realistic and smooth tail than CIPS-base, while being
worse in terms of FID.

Figure 7: Spectral analysis for models trained on FFHQ at resolution of
$256^{2}$. All results are averaged across 5000 samples. We demonstrate that
CIPS-NE is most similar to real images.

Recently, [4] observed that the common convolutional upsampling operations can
lead to the inability to learn the spectral distribution of real images, in
spite of any generator architecture. In contrast, CIPS operates explicitly
with the coordinate grid and has no upscaling modules, which should to
improved reproduction of the spectrum. Indeed, we compare the spectrum of our
models (CIPS-“base” without residual and skip connections; CIPS-NE) to
StyleGANv2 and demonstrate that CIPS generators design has the advantage in
the spectral domain.

The analysis of magnitude spectra for produced images is given in Fig. 7(a).
The spectrum of StyleGANv2 has artifacts in high-frequency regions, not
present in both CIPS generators under consideration. Following prior works
[4], we also use the azimuthal integration (AI) over the Fourier power
spectrum (Fig. 7(b)). It is worth noting that AI statistics of CIPS-NE are
very close to the ones of real images. However, adding the coordinate
embeddings degrades a realistic spectrum while improving the quality in terms
of FID (Tab. 3).

We note that the introduction of skip connections in fact makes the spectra
less similar to those of natural images.

###  4.6 Interpolation

We conclude the experimental part with the demonstration of the flexibility of
CIPS. As well as many other generators, CIPS generators have the ability to
interpolate between latent vectors with meaningful morphing (Fig. 10). As
expected, the change between the extreme images occurs smoothly and allows for
the use of this property, in a similar vein as in the original works (e.g.
[11]).

Figure 8:  Images generated using foveated synthesis. In each case, the CIPS
generator was sampled on a 2D Gaussian distribution concentrated in the center
of an image (standard deviation = $0.4*$image size). Left to right: sampled
pattern covers 5% of all pixels, 25%, 50%, 100% (full coordinate grid).
Missing color values have been filled via bicubic interpolation.  Figure 9:
Left: the generated image of resolution $256\times{}256$, upscaled with
Lanczos upsampling scheme [16] to $1024\times{}1024$. Right: the image,
synthesized by CIPS, trained at resolution of $256\times{}256$ on the
coordinate grid of resolution $1024\times{}1024$. Note the
sharpness/plausibility of the eyelid and the more proper shape of the pupil.
Figure 10: Latent linear morphing between two sampled images – the left-most
and right-most ones.

###  4.7 Foveated rendering and interpolation

One of the inspiring applications of our per-pixel generator is the foveated
synthesis. The foveated synthesis ability can be beneficial for computer
graphics and other applications, as well as mimics human visual system. In
foveated synthesis, an irregular grid of coordinates is sampled first, more
dense in the area, where the gaze is assumed to be directed to, and more
sparse outside of that region. After that, CIPS is evaluated on this grid (its
size is less than the full resolution), and color for missing pixels of the
image is filled using interpolation. The demonstration of this method is
provided in Fig. 8.

Alongside the foveated rendering, we are also able to interpolate the image
beyond the training resolution by simply sampling denser grids. Here we use a
model, trained on images of $256\times{}256$ resolution to process a grid of
$1024\times{}1024$ pixels and compare it with upsampling the results of
upsampling the image synthesized at the $256\times{}256$ resolution with the
Lanczos filter [16]. As Fig. 9 suggests, more plausible details are obtained
with denser synthesis than with Lanczos filter.

###  4.8 Panorama synthesis

(a)

(b)

(c)

Figure 11: Panorama blending. We linearly blend two upper images from CISP
generator trained on the Landscapes dataset with a cylindrical coordinate
system. The resulting image contains elements from both original panoramas:
land and water integrated naturally.

As CIPS is built upon a coordinate grid, it can relatively easily use non-
Cartesian grids. To show this, we thus adopt a cylindrical system to produce
landscape panoramas. The training setup is as follows. We uniformly sample a
crop $256\times 256$ from the cylindrical coordinate grid and train the
generator to produce images using these coordinate crops as inputs. A similar
idea was also explored in [17]. We note, however, that during training we do
not use any real panoramas in contrast to other coordinate-based COCO-GAN
model [17]. Fig. 11(a) and 11(b) provide examples of panorama samples obtained
with the resulting model.

As each pixel is generated from its coordinates and style vector only, our
architecture admits pixel-wise style interpolation (Fig. 11(c)). In these
examples, the style vector blends between the central part (the style of Fig.
11(a)) and the outer part (the style of 11(b)).

###  4.9 Typical artifacts

Finally, we show the typical artifacts that keep recurring in the results of
CIPS generators (Fig. 12). We attribute the wavy texture (in hair) and
repeated lines pattern (in buildings) to the periodic nature of sine
activation function within the Fourier features. Also we note that sometimes
CIPS produces a realistic image with a small part of the image being
inconsistent with the rest and out of the domain. Our belief is that this
behaviour is caused by the LeakyReLU activation function that divides the
coordinate grid into parts. For each part, CIPS effectively applies its own
inverse discrete Fourier transform. As CIPS generators do not use any
upsampling or other pixel coordination, it is harder for the generator to
safeguard against such behaviour.

Figure 12: Examples of the most common kinds of artifacts on different
datasets. They are best described as wavy textures on hair, background, and
glowing blobs. see text for the discussion.

##  5 Conclusion

We presented a new generator model called CIPS, a high-quality architecture
with conditionally independent pixel synthesis, such that the color value is
computed using only random noise and coordinate position.

Our key insight is that the proposed architecture without spatial
convolutions, attention or upsampling operations has the ability to compete in
the model market and obtain decent quality in terms of FID and precision &
recall; such results have not been presented earlier for perceptron-based
models. Furthermore, in the spectral domain outputs of CIPS are harder to
discriminate from real images. Interestingly, CIPS-NE modification is weaker
in terms of plausibility, yet has a more realistic spectrum.

Direct usage of a coordinate grid allows us to work with more complex
structures, such as cylindrical panoramas, just by replacing the underlying
coordinate system.

In summary, our generator demonstrates quality on par with state-of-the-art
model StyleGANv2; moreover, it has applications in various diverse scenarios.
We have shown that the considered model could be successfully applied to
foveated rendering and super-resolution problems in their generative
interpretations. Future development of our approach assumes researching these
problems in their image-to-image formulations.

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##  Appendix A Architecture details

Figure 13: The diagram of the CIPS generator (default version). Modification | # parameters (mln)
---|---
CIPS-“base” | 43.8
CIPS-NE | 10.2
CIPS-Default | 45.9
StyleGANv2 | 30.0
Table 4: The number of parameters for different version of the CIPS generator.
For reference, the number of parameters within the StyleGANv2 generator is
also given.

In this section we provide additional information about the default version of
our CIPS generator (Fig. 13). In total, its backbone contains 15 fully
connected layers. The first layer projects concatenated coordinate embeddings
and Fourier features into the joint space with the dimension of 512. Next, the
following layer pattern is repeated seven times. The representation is put
through two modulated fully-connected layers and a projection to RGB color
space is computed. The projections coming from the seven iterations are summed
together to create the final image. The number of parameters for the different
modifications of the CIPS generator discussed in the paper are given in Tab.
4.

##  Appendix B Coordinate embeddings

We also run the Principle Components Analysis (PCA) for coordinate embeddings
of models trained on Landscapes and LSUN-Churches images (similar pattern for
the FFHQ dataset is shown in the main paper). Fig. 14 provides the
visualisation for the three main components. Note, that as these datasets are
as aligned as FFHQ, there is considerably less spatial structural information
in the learned embeddings.

|  |
---|---|---
Figure 14: Visualisation of three main principal components of coordinate
embeddings for CIPS models, trained on Landscapes (left) and LSUN-Churches
(center). As these datasets are not as aligned as the face dataset, there is
less recognizable structure in the learned coordinate embeddings. The bottom
horizontal structure in the LSUN-Churches case is likely due to frequent
watermark pattern in the dataset (a sample from the model with such watermark
is shown on the right).

##  Appendix C Patch-based generation

Dataset | Patch size | FID
---|---|---
FFHQ | 256 | 4.38
128 | 9.08
64 | 11.79
LSUN-Churches | 256 | 2.92
128 | 7.08
64 | 11.53
Table 5: Frechet Inception Distance (FID) values for CIPS models trained on
patches of varying receptive field and fixed resolution ($64\times{}64$ and
$128\times{}128$). The results for patch-based training are worse than the
default training procedure, in which the discriminator observes the full
$256\times 256$ image.

To show one benefit of coordinate-based approach, we demonstrate the results
of memory-constrained training, where the discriminator observes patches at
lower resolution than the full image (inspired by the GRAF system [25]). Since
pixel generation is conditionally-independent, at each iteration only low-
resolution patches need to be generated. Thus, only the following $K\times K$
patch is synthesized and submitted to the discriminator:

$\displaystyle
P_{K,\sigma}\left(u,v\right)=\\{G\left(u+i\sigma,v+j\sigma;\mathbf{z}\right)\mid\left(i,j\right)\in\texttt{mgrid}\left(K,K\right)\\},$

where $0\leq u<W-\left(K-1\right)\sigma$ and $0\leq
v<H-\left(K-1\right)\sigma$ are the coordinates of the corner pixel of the
patch. For $\sigma=1$ this produces dense patch, while for $\sigma>1$ dilated
patch with increased receptive field is obtained. Applying this patch sampling
to real images before putting them into the discriminator may be thought of as
an example of a differentiable augmentation, the usefulness of which was
recently proved by [9, 34].

Tab. 5 reports the quality (FID) for CIPS generators trained on patches of
sizes $64\times{}64$ and $128\times{}128$, while the resolution of full images
equals $256\times{}256$. Fig. 15 shows the outputs of models, trained with the
patch-based pipeline. In our experiments, training with smaller size of
patches degrades the overall quality of resulting samples.

LSUN-Churches

FFHQ

Figure 15: Samples from CIPS generators learned with memory-constrained patch-
based training. Within every grid, the top row contains images from models
trained with patches of size $128\times 128$ and the bottom row represents
outputs from training on $64\times 64$ patches. While the samples obtained
with such memory-constrained training are meaningful, their quality and
diversity are worse compared to standard training.

##  Appendix D Additional samples

In Fig. 16, we provide additional samples from CIPS generators trained on
different datasets. We also demonstrate more samples of cylindrical panoramas
in Fig. 17.

Although we do not apply mixing regularization [10] at train time, our model
is still capable of layer-wise combination of latent variables at various
depth (see Fig. 19). The examples suggest that similarly to StyleGAN,
different layers of CIPS control different aspects of images.

##  Appendix E Nearest neighbors

To assess the generalization ability of CIPS architecture, we also show the
samples from the model trained on the FFHQ face dataset alongside the most
similar faces from the train dataset. To mine the most similar faces, we
extract faces using the MTCNN model [33], and then compute their embeddings
using FaceNet [24] (the public implementation of these
models444https://github.com/timesler/facenet-pytorch was used). Fig. 18 shows
five nearest neighbors (w.r.t. FaceNet descriptors) for each samples. The
samples generated by the model are clearly not duplicates of the training
images.

LSUN-Churches

Landscapes

Satellite-Landscapes

Satellite-Buildings

Figure 16: Samples from CIPS generators trained on various datasets. The top
row of every grid shows real samples, and the remaining rows contain samples
from the models. The samples from CIPS generators are plausible and diverse.

Figure 17: Additional samples of cylindrical panoramas, generated by the CIPS
model trained on the Landscapes dataset. The training data contains standard
landscape photographs from the Flickr website. No panoramas are provided to
the model during training.

Figure 18: Nearest neighbors for generated faces. Within each row, we show a
sample from the model on the left. The remaining columns contain real images
that are closest to the respective sample in terms of the FaceNet [24]
descriptor. The visualization suggests that the CIPS model generalizes well
beyond memorization of the training dataset.  Figure 19: Layer-wise style
mixing. The two leftmost columns contain source images A and B. In the
rightmost three columns, we replace the latent code $\mathbf{w}$ of A with the
latent code $\mathbf{w}$ of B at layers (left to right): 6-8, 3-5, 1-2. The
visualization suggests that layers 1-2 control the pose and the shape of the
head, the middle layers (3-5) control finer geometry such as the shape of
eyes, eyebrows and nose, and the final layers (6-8) controls the skin color
and the textures. Interestingly, this CIPS model was trained without layerwise
mixing, and therefore such decomposition likely arises from the architectural
prior.


On Bell's Paradox

Bart Jongejan

Cæciliavej 31, 2500 Valby, Denmark

Electronic address<EMAIL_ADDRESS>

The correlations between the outcomes of pairs of spin component measurements
on particles in the singlet state exist and can be observed independently of
any a priori given frame of reference. We can even construct a frame of
reference that is based on these correlations. This observation-based frame of
reference is equivalent to the customary a priori given frame of reference of
the laboratory when describing real Bohm-Aharonov experiments.

J.S. Bell has argued that local hidden parameter theories that reproduce the
predictions of Quantum Mechanics cannot exist, but the counterfactual
reasoning leading to Bell's conclusion is physically meaningless if the frame
of reference that is based on spin component measurements is accepted as the
backdrop for Bohm-Aharonov experiments.

The refutal of Bell's proof opens up for the construction of a viable hidden
parameter theory. A model of a spin $\hbar$/2 particle in terms of a non-flat
metric of space-time is shown to allow the predictions of quantum mechanics in
the Bohm-Aharonov experiment to be reproduced, without introducing non-
locality.

PACS numbers: 03.65.Bz, 02.40.Ky, 04.20.Gz

I.HIDDEN VARIABLES

A.The Einstein, Podolsky and Rosen Gedanken experiment

Einstein, Podolsky and Rosen (EPR) [1] argued that quantum mechanics is an
incomplete theory. Their reasoning was that two non-commuting quantities, such
as momentum and position of one of the particles, can be determined by
measuring one of the quantities directly and by deriving the value of the
other quantity from the outcome of the measurement of that quantity on the
other particle in the pair. Bohr [2] opposed to this idea.

Bohm and Aharonov [3] devised a version of the EPR Gedanken experiment that
has been the focus of much theoretical and experimental work. In their
experiment, the measurements are done on pairs of spin $\hbar$/2 particles
that are prepared in the singlet state, which is a quantum state that does not
hold any information about the directions of the spins of the individual
particles.

In the Bohm-Aharonov experiment the particles fly apart toward two widely
separated observation posts, where they traverse the gaps of Stern-Gerlach
magnets. During such a traversal, due to a coupling between the particle’s
intrinsic spin and the longitudinal gradient of the magnetic field, the
particle’s path is bent either away from or toward the pole where the magnetic
field is strongest. Each particle finally hits one of two detectors, depending
on which route it took. One of the detectors only records particles that had
spin up $\left(\uparrow\right)$, while the other only records those with spin
down $\left(\downarrow\right)$, “up” and “down” being defined relative to the
Stern-Gerlach magnet. The detectors are fixed to their respective Stern-
Gerlach magnets, so that the directions that are “up” and “down” rotate
together with the freely orientable Stern-Gerlach magnets.

B.Can counterfactual considerations complete the description of physical
reality?

Employing arguments of the type that EPR used to show that quantum mechanics
cannot be complete, J.S. Bell [4] showed that any theory that reproduces the
predictions made by quantum mechanics and yet is more complete than quantum
mechanics necessarily postulates instantaneous action at a distance. In other
words, the kind of theories that Einstein envisaged as successors to quantum
mechanics would be difficult to reconcile with relativity theory, which
champions locality and does not allow any signal to travel faster than light.

Bell’s proof is dependent on counterfactual propositions. A counterfactual
proposition assigns a determinate value to a quantity that could have been
directly observable, but is not, typically because another, incommensurable
quantity is measured. In the context of Bohm-Aharonov experiments,
counterfactual propositions are justified by first noting that the measurement
on the twin is as good as the measurement on the particle itself, because the
inner states of the particles must be fully correlated in order to preserve
the isotropy of the quantum state of the pair. The next step in the
justification is to show that the measurement on the twin particle is not even
necessary. Knowing that no signal can bridge spatially separated regions of
space time, whether the measurement on the twin is performed or not, cannot
have impact on the validity of the assignment of some value to the outcome of
a counterfactual measurement.

To give teeth to Bell's counterfactual propositions, one also needs to ascribe
definiteness to the angle between the counterfactual orientation of a Stern
Gerlach magnet and the (factual or counterfactual) orientation of the other
magnet. It is this angle that determines the statistical correlations between
the outcomes. Without this angle no comparison with the quantum mechanical
predictions can be made. Normally the definiteness and well-definedness of the
angle between the instruments, even in the case of counterfactual set-ups, is
not disputed, but in the following it is shown that the seeming clarity of the
experimental set-up is deceiving.

Let us hold Bell’s counterfactual reasoning against the background of
Riemannian geometry, or, more specifically, relativity theory. Only geometric
relations, such as angles and distances, between things that are local to each
other in space and in time bear physical meaning in this setting. Geometric
relations over long spatio-temporal distances, such as the angle between the
directions of observation in the Bohm-Aharonov experiment, do not have an
immediate physical meaning.

If it is not assumed that space-time is flat, then only an operational
definition can lend physical meaning to these relations. What then is the
angle between a factual set-up of a Stern-Gerlach magnet at one of the two
observation posts and a counterfactual set-up of the Stern-Gerlach magnet at
the other observation post? Of course, because one of the two set-ups is not
effectuated, there is no direct means of measuring this angle. We can only
base our answer on interpolation, together with a smoothness assumption that
coordinates the interpolated but counterfactual observation with real ones.
Normally, interpolation depends on a smoothness assumption that is innocuous,
because the missing, interpolated values may some day be replaced by outcomes
of real experiments. In such cases, the made interpolations are, in principle,
falsifiable and therefore acceptable. In Bell’s proof, the interpolations are
not falsifiable, because the measuring apparatus cannot possibly have two
orientations at the same time. One will never be able to come up with
experimental results that replace outcomes of counterfactual measurements of
spin components. The deeply rooted non-falsifiability of counterfactual
statements undermines Bell’s conclusion. We can only accept Bell’s proof if we
assume that the space time backdrop is smooth and constant enough to allow us
to interpolate between measurements, but this assumption excludes from
consideration any theory that denies that space-time is like that.

For another recent development playing with the idea that the backdrop of
physical experiments needs an operational justification see e.g. [5] and [6].

C.The Bohm-Aharonov experiment without flat-space preconception.

Bell's proof hinges on the postulate that space time is flat, but this
postulate may be false. This is the main theme of this paper and we will dwell
on it a little more, because understanding the epistemological restrictions
that relativity imposes is essential for appreciating the approach towards
hidden variables that is presented later.

First think of the Bohm-Aharonov experiment as a set-up consisting of two
observation posts connected by the floor of a laboratory or something else
that we may regard as rigid. Each observation post consists of a Stern-Gerlach
magnet that is freely orientable in its mounting into any of a large number of
directions, or lines of observation. On each Stern-Gerlach magnet sits a
pointer that is aligned with the axis of the magnetic field. Also fixed to a
Stern-Gerlach magnet are an "up" and a "down" particle detector. The
orientation of a Stern-Gerlach magnet is identifiable, for example by the
color of a mark on the mounting that the pointer is pointing at. There may be
many differently colored marks, each identifying a unique orientation of the
magnet. (If one wishes so one can define the mountings of the instrument to
include far away stars, which then can be used as the marks for that post.)

Suppose also that we have a calibrated scale on each mounting so that we have
the option to read off the angular coordinates of the pointer of the Stern-
Gerlach magnet. The rigid connection between the two posts (or rather: between
the mountings), together with conventional means of doing geodesy (measuring
rods, light beams, gyroscopes, etc.) gives us ample means to define a
reference frame in which both measuring instruments have definite positions
and orientations. Also counterfactual orientations can be tracked, because the
rigid frame fixes all thinkable orientations. This is the normal, "robust"
experimental context of Bell's proof.

Now remove all unnecessary equipment: the measuring rods, light beams,
gyroscopes, scales and even the laboratory floor. Would that make any
difference? We did not do away the mountings of the instruments and are
therefore still able to identify each orientation by reading off the color of
the mark that a Stern-Gerlach magnet is pointing at. Can we reconstruct the
experiment with this basic equipment?

From each pair of measurements we obtain a data-triplet: the color of the mark
that the left magnet was pointing at, idem for the right magnet and finally
the outcome of the detectors, which arbitrarily may be defined to be "S" (for
"same") if both "up" or both "down" detectors were hit and "N" (for "not
same") if one "up" and one "down" detector was triggered. Our logbook will
have just three columns; the experiment has only three degrees of freedom.

After doing a long series of such experiments, with the magnets having been
oriented in many combinations of directions many times, the log of outcomes
will enable us to assign statistical probabilities for measuring the same spin
component to each pair of orientations of the Stern-Gerlach magnets. We might
for example observe that "yellow" left and "blue" right have a 73% chance of
resulting in the value "S". There is a conceptual difference with the full
blown experimental set-up, though: we have no prior knowledge of the geometric
relations between the Stern-Gerlach magnets; we can only pairwise
statistically relate marks on the mountings with each other. We do not even
know the angle between two orientations of the same magnet!

Now we will try to organize the three columns of obtained data by mapping the
colored marks that identify directions of observations onto points on a sphere
in such a way that exactly one point is assigned to each mark. This mapping
must map the marks on both surroundings, however far separated from each
other, onto a single sphere that does not exist physically, but only as a
mathematical tool. Underlying the mapping is the working hypothesis that there
is a functional relation between the statistical probability to measure the
same component and the angle, as measured on this sphere, between the
orientations of the Stern-Gerlach magnets. We could simply assume a linear
dependency: a 100% probability and a 0 % probability correspond to angles of
$180{{}^{\circ}}$ and of $0{{}^{\circ}}$ between the magnets, a 10%
probability would correspond to $18{{}^{\circ}}$, and so on. Such a relation
would be appropriate if the spinning particles were macroscopic objects with
observationally well defined "north" and "south" hemispheres. However, if we
apply this relation to our data then the obtained angles force us to map the
same mark onto several points, which is not what we wanted.

Eventually, we would find out that we can establish a 1-1 mapping by defining
the angle between the orientations in such a way that one of the following
relations becomes true:

$P\left(S\right)=probability\,to\,measure\,same\,value=$
$=\sin^{2}\left(angle\,between\,orientations\over 2\right)$ $None$

or

$P\left(S\right)=probability\,to\,measure\,same\,value=$
$=\cos^{2}\left(angle\,between\,orientations\over 2\right).$ $None$

Now, not only would we have learned how to define angles between the two
measuring instruments when directed towards two given colored marks, we would
also have a consistent way to figure out the angles between the colored marks
at one and the same post. That means that we would have regained the angular
relations that we did not assume as given a priori. That, in turn, would
enable us to deliver exactly the same kind of experimental report as someone
who had a rigid frame to connect the measuring posts and rods, light beams,
gyroscopes and scales to measure the geometry. We could also verify the
contingent fact that this statistical way of determining geometric relations
is perfectly consistent with the other, more conventional means. But of
course, there is a difference: as little physical sense as it makes to ascribe
temperature and pressure (or the mean kinetic energy and momentum per
particle) to a single particle in a gas, as little sense would it make to
ascribe the statistically defined angle to a pair of really made measurements
that contributed to the very determination of the angle. A fortiori, we cannot
draw firm conclusions from an argument that hinges on a counterfactual
measurement controlled by the statistically defined angle. There is no
observational basis for assigning a value to such an angle, because the value,
as defined operationally in the above way, is of statistical character and
therefore not applicable to any pair of orientations in any specific pair of
measurements, but only in the long run of many measurement event pairs. In
addition, the assignment of a value of the angle, operationally defined as
above, to pairs of orientations that are not both effectuated, can only be
based on an arbitrary convention and therefore renders any argument that is
based on such angles unconvincing.

If the presented way of constructing a frame of reference is so feeble that it
does not allow us to assign values to angles between counterfactual setups,
why then should we not stick with the conventional means using rods,
gyroscopes and so on? The reason is that our method is not any more feeble
than conventional means! Unless conventional methods mysteriously gain
strength somewhere in the transition from the quantum to the classical regime,
any angle that can be measured by conventional means can also be measured
using a sufficiently large ensemble of spin component observations on an
equally large number of pairs of spin $\hbar$/2 particles, to any desired
accuracy. However, the proposed method, which is obviously based on quantum
phenomena, has the additional advantage that it very clearly delineates the
domain of applicability of the method; a domain of applicability that cannot
be surpassed by conventional methods, unless the aforementioned mysterious
powers opened a back door for the conventional methods to do measurements of
angles that are out of reach for the proposed method.

Whereas proofs like Bell's are hard pressed because of the lack of an
observational basis for the assumed frame of reference, a realist point of
view is not obstructed in the same way. It is sensible to imagine a
counterfactual set-up of a measuring apparatus that is oriented towards a
particular colored mark and with a definite value of the spin component in
that direction, but we must keep in mind that the whereabouts of the
instrument's orientation relative to other (factual or counterfactual)
orientations is unknown. Underlying Bell's and similar proofs is a concept of
realism that is far more encompassing than necessary. Things that derive the
status of being part of reality by force of real observations, such as the
angle between directions of observation and even systems of coordinates in
general, cannot be idealized to an existence detached from these observations
without introducing trouble in some corners of our theoretical picture of the
world.

II.MATHEMATICAL CONSTRAINTS ON HIDDEN VARIABLE THEORIES OF SPIN $\hbar$/2
PARTICLES

A.What makes an aspirant hidden variable theory?

We have seen that the Bohm-Aharonov experiment has just three relevant degrees
of freedom: the color of the left mark, the color of the right mark and the
combination of the outcomes. We will now discuss hidden variable theories that
also exhibit three degrees of freedom and hope to find one that can be made to
correspond to the Bohm-Aharonov experiment and that explains the statistical
correlations found in the Bohm-Aharonov experiment (which are assumed to be
accurately predicted by quantum mechanics).

The contemplated hidden variable theories all have one aspect in common: not
only the Stern-Gerlach magnets have definite orientations, also the particle
itself has one, which is the axis of rotational symmetry or "spin axis". The
three variables that specify any configuration of the three directions are (a)
the angle between the left measuring apparatus and the spin axis, (b) the
angle between the right measuring apparatus and the spin axis and (c) the
angle between the left measuring apparatus and the right measuring apparatus.
None of these variables are precisely measurable, but each corresponds to
observed data: if the left "up" detector is triggered, then the angle between
the left measuring apparatus and the spin axis is less than $90{{}^{\circ}}$.
If the "down" detector is triggered, the angle is somewhere between
$90{{}^{\circ}}$ and $180{{}^{\circ}}$. Likewise for the right detector. The
angle between the measuring apparatuses is taken to be the angle that was
statistically derived from the outcomes of a long series of Bohm-Aharonov
measurements, using Eq. (1a).

The main point made by Bell was that no local hidden variable theory is able
to reproduce the predictions of quantum mechanics. The requirement that our
aspirant hidden variable theories are local puts three constraints on the
statistical distribution of the angles between the measuring apparatuses and
between each measuring apparatus and the particle's spin axis. These will be
discussed now.

Constraint 1.  The orientations of the measuring instruments are unrelated.

Each measuring apparatus is oriented in a way that does not depend on the
orientation of the other instrument, not even statistically. If the movements
of the measuring apparatuses A and B were restricted to a plane, then this
constraint would translate to a uniform distribution of angles $\angle AB$ in
the range $0\leq\angle AB\leq\pi$. We do, however, assume that the instruments
are freely orientable in space. In that case, the probability that the angle
between A and B is $\angle AB$ is proportional to $\sin\angle AB$.

Whereas the angle is not uniformly distributed, its cosine is. So we require a
uniform distribution $\rho\left(Z_{AB}\right)$ of the inner product

$\rho\left(Z_{AB}\right)=const.,Z_{AB}\equiv{\tenrm-cos}\angle AB=-{\tenit
a}.{\tenit b}$ $None$

in the range $-1\leq Z_{AB}\leq 1$. The minus sign is arbitrarily introduced
to compensate for the circumstance that the particles' spins are anti-parallel
in Eq. (1a).

Normalization requires $\int^{1}_{-1}\rho\left(Z_{AB}\right)dZ_{AB}=1$, so
that

$\rho\left(Z_{AB}\right)={\mathchoice{\raise 2.15277pt\hbox{$\scriptstyle
1$}\\!/\\!_{2}}{\raise 2.15277pt\hbox{$\scriptstyle 1$}\\!/\\!_{2}}{\raise
0.86108pt\hbox{$\scriptscriptstyle 1$}\\!/\\!_{2}}{\raise
0.86108pt\hbox{$\scriptscriptstyle 1$}\\!/\\!_{2}}}$ $None$

We ask that during each measurement the measuring apparatus points at a
randomly chosen point of its own surroundings. This does not automatically
imply an isotropic distribution of the orientations with respect to each
other: one could imagine that each observation post's orientations, taken
separately, would survive a "randomness test", but that the orientations were
not distributed isotropically with respect to each other. That situation could
arise if the observers used the same sequence of random numbers to prepare the
instruments for each pair of measurements. We assume that such correlation
does not occur, because that seems to be the only assumption that is
compatible with the principles of locality, causality and free will.

Constraint 2. (Locality condition.) The orientation of one magnet does not
influence the result of a spin component measurement obtained with the other.

Suppose that someone came up with a hidden variable theory of a spin $\hbar$/2
particle. In order to test the claim that it reproduces the predictions of QM
in Bohm-Aharonov experiments, we had to subject the theory to a Gedanken
experiment in which a great number of spin-component measurements were
randomly chosen. How would the randomly chosen orientations of a measuring
instrument be distributed with respect to the particle's axis of rotational
symmetry? As we have no means of observing this distribution, we postulate
one. In the absence of any reason to assume a non-isotropic distribution, we
assume the isotropic distribution. Call the inner product of the orientations
of the instruments and the direction of the particle (denoted by unit vectors)
$Z_{A}$ and $Z_{B}$ (for instrument A and instrument B). Require that

$\hbox{$\rho\left(Z_{A}\right)=\rho\left(Z_{B}\right)={\mathchoice{\raise
2.15277pt\hbox{$\scriptstyle 1$}\\!/\\!_{2}}{\raise
2.15277pt\hbox{$\scriptstyle 1$}\\!/\\!_{2}}{\raise
0.86108pt\hbox{$\scriptscriptstyle 1$}\\!/\\!_{2}}{\raise
0.86108pt\hbox{$\scriptscriptstyle 1$}\\!/\\!_{2}}}$},$ $None$

using the same reasoning that leads to Eq. (3).

The sign of $Z_{A}$ determines the outcome of the measurement made with the
magnet at observation post A. The locality condition is fulfilled if $Z_{A}$
is not dependent on the orientation of instrument B. The angle between the
instruments can be taken to represent this orientation, in which case the
locality condition is fulfilled if

$\rho\left(Z_{A}\,|\,Z_{AB}\right)=\rho\left(Z_{A}\right)$

which is equivalent to

$\rho\left(Z_{A},Z_{AB}\right)\equiv\rho\left(Z_{A}\,|\,Z_{AB}\right)\rho\left(Z_{AB}\right)=\rho\left(Z_{A}\right)\rho\left(Z_{AB}\right).$
$None$

Alternatively, we can take the (hidden) angle between instrument B and the
spin axis as representing the orientation, so, to be on the safe side, we also
require that

$\rho\left(Z_{A},Z_{B}\right)=\rho\left(Z_{A}\right)\rho\left(Z_{B}\right).$
$None$

As we are used to specify orientations with two mutually independent angles,
it is tempting to require that $Z_{A}$ is independent of both of $Z_{AB}$ and
$Z_{B}$:

$\rho\left(Z_{A},Z_{B},Z_{AB}\right)=\rho\left(Z_{A}\right)\rho\left(Z_{B}\right)\rho\left(Z_{AB}\right).$
$None$

However, below we will see that this conflicts with the next constraint and
even with the statistics of classical spinning particles.

Constraint 3. The theory reproduces the predictions of quantum mechanics

The correlation between the outcomes of measurements on flown-apart particles
with counter parallel spin axis conforms exactly to the predictions of QM (see
also Eqs. (1a) and (2)):

$P\left(S\right)\equiv\sin^{2}{\angle AB\over 2}={1+Z_{AB}\over 2}$ $None$
$P\left(N\right)\equiv\cos^{2}{\angle AB\over 2}={1-Z_{AB}\over 2}$ $None$
$P\left(\uparrow\uparrow\right)=P\left(\downarrow\downarrow\right)={P\left(S\right)\over
2}$
$P\left(\uparrow\downarrow\right)=P\left(\downarrow\uparrow\right)={P\left(N\right)\over
2}.$ $None$

B.The joint distribution $\rho\left({\tenit Z}_{\tenit A},{\tenit Z}_{\tenit
B},{\tenit Z}_{{\tenit A}{\tenit B}}\right)$ that explains the quantum
mechanical predictions.

We assume that in a hidden variable theory, a full specification of a
measurement of two spin components in the Bohm-Aharonov experiment requires
three angles, or their cosines. We must now investigate whether there are
distributions $\rho\left(Z_{A},Z_{B},Z_{AB}\right)$ of these three quantities
that fulfill all three constraints. For example, the first constraint requires
that

$\int^{1}_{-1}\int^{1}_{-1}\rho\left(Z_{A},Z_{B},Z_{AB}\right)dZ_{A}dZ_{B}=\rho\left(Z_{AB}\right)=1/2,$
$None$

and the second constraint, the locality condition, requires that

$\int^{1}_{-1}\mathop{}\rho\left(Z_{A},Z_{B},Z_{AB}\right)dZ_{B}=\rho\left({Z_{A},Z}_{AB}\right)=1/4.$
$None$

Finally, according to quantum mechanics we must find that

$\int^{1}_{0}\int^{1}_{0}\rho\left(Z_{A},Z_{B},Z_{AB}\right)dZ_{A}dZ_{B}\over\int^{1}_{-1}\int^{1}_{-1}\rho\left(Z_{A},Z_{B},Z_{AB}\right)dZ_{A}dZ_{B}$
$=P\left(\uparrow\uparrow\right)={1+Z_{AB}\over 4}.$ $None$

The distribution in Eq. (5) fully acknowledges the freedom of the
experimenters to vary the angle between the instruments ($Z_{AB}$) and it
guarantees the isotropic distribution of the axis of the model relative to the
lines of observation ($Z_{A}$ and $Z_{B}$). Yet this distribution is not
realistic at all, because it does not restrict the angles between the
instruments ($\arccos{-Z}_{AB}$) and the angles between the instruments and
the axis of the model ($\arccos Z_{A}$ and $\arccos Z_{B}$). These three
angles cannot be completely independent: for example can the angle between the
measuring instruments not exceed the sum of the angles between the instruments
and the axis of the model.

If two of the angles already are fixed to any values between $0$ and $\pi$
(any two of $\\{\arccos Z_{AB}$,$\arccos Z_{A}$,$\arccos Z_{B}\\}$, call them
$\alpha_{1}$ and $\alpha_{2}$), then we have the following constraint on the
third angle $\alpha_{3}$:

$\left|\alpha_{1}-\alpha_{2}\right|\leq\alpha_{3}\leq\alpha_{1}+\alpha_{2}.$
$None$

That means that whereas any two of the three angles are independent of each
other, there exists a mutual dependency between the three angles, even in a
classical context without any hint at non-locality. For example, the uniform
distribution of three vectors ${\tenit a},{\tenit b}$ and $\tenit c$ over all
directions illustrates this dependency. In an Euclidean frame of reference,
the joint density of the three inner products $s=Z_{A}={\tenit a}.{\tenit
c},\,t=Z_{B}={\tenit b}.{\tenit c},\,$$u=Z_{AB}={\tenit a}.{\tenit b}$ is

$\eqalignno{\rho\left(s,t,u\right)={}&{\left(8\pi\sqrt{1+2stu-s^{2}-t^{2}-u^{2}}\right)}^{-1}\cr\vskip
2.0pt\cr\hfil\qquad\qquad\qquad\qquad\left(1+2stu-s^{2}-t^{2}-u^{2}>0\right)\cr\hfil&(11)\cr={}&0\qquad(1+2stu-s^{2}-t^{2}-u^{2}\leq
0).\cr}$

Notice the discontinuity in this, the classical, density distribution.

It is well known that the above classical distribution is not able to fulfill
Constraint 3. By employing a discrete numerical method we can find
distributions that come arbitrarily close to the fulfilment of Constraint 3.
(See [7] for a description and some illustrations of the results). All these
distributions are characterized by regions of almost emptiness and steep
climbs to high values of probability density in narrow bands. The following
distribution, which is a limiting case of these numerical approximations, uses
Dirac delta functions and fulfills all three constraints:

$\rho\left(Z_{A},Z_{B},Z_{AB}\right)=1/8\left[\delta\left(Z_{A}+Z_{B}+Z_{AB}+1\right)\right.$
$+\,\delta\left(Z_{A}-Z_{B}-Z_{AB}+1\right)$
$+\,\delta\left(Z_{A}+Z_{B}-Z_{AB}-1\right)$
$\left.+\,\delta\left(Z_{A}-Z_{B}+Z_{AB}-1\right)\right]$ $None$

This distribution indicates that one continuous degree of freedom is replaced
by a discrete one: the density is only non-zero on the surface of a
tetrahedron spanned by four of the eight corners of the configuration cube.
For any pair of values of, say, $Z_{A}$ and $Z_{AB}$, in the range $[-1,1]$,
only two values of the third variable, $Z_{B}$ in this example, also in the
range $[-1,1]$, make the joint density different from zero.

A classical configuration of three independent vectors is specified by three
numbers, which are the lengths of the sides of a triangle on the unit sphere.
They cannot live within less than the two dimensions of this sphere. On the
other hand, from (12) we can learn that configurations which are compatible
with QM require only two numbers and a sign, the third number being a function
of either the sum or the difference of the other two, depending on the sign.

Perhaps somewhat unexpectedly, our search for a probability distribution for
the angle between the instruments and the angles between each instrument and
the (hidden) spin axis did not merely result in a non-classical distribution,
but also in a qualitative characterization of any hidden variable theory with
hopes to fulfill all three constraints: the theory must endow a model of a
spin $\hbar$/2 particle with a degeneracy that replaces one continuous degree
of freedom with a two-valued one. We will now look at a theory that
accomplishes this feat.

III.EXAMPLE: A HIDDEN VARIABLE MODEL BASED ON NON-FLAT SPACE-TIME STRUCTURE

A.Overview

In the foregoing, a weakness in Bell's counterfactual reasoning was exposed by
peeling away unwarranted and mostly silent assumptions that are underlying his
proof, until we were left with the nitty gritty observational data. Then we
postulated, in spite of Bell's conclusion, that the direction of a particle's
spin is really existing, although hidden from observation. We found that the
QM-compatible configurations of three vectors, denoting the orientations of
the measuring instruments and a candidate model's hidden variable, seem to
live within one dimension less than classically expected.

The next step is the construction of a model with a characteristic that we
rightly can call the spin direction of the model. We postulate that the stuff
that the particle is made of is space time itself. We will check that the full
specification of a geodetic path herein - tentatively representing the world
line of a measuring apparatus during a measurement - exhibits the same lack of
one degree of freedom.

Such a postulated space-time structure is in part based on guesswork, in part
on esthetic rules, such as simplicity and symmetry. The real structure of
space time is perhaps unknowable, but we may hit upon a theory of the
structure of space time that survives the observations that we perform to test
it.

The presented model tries only to explain a very limited set of phenomena,
namely the correlation between two spin component measurements. We have not
tried very hard to incorporate and explain other phenomena. Thus, a simple
thing like the spatial distance between the spinning particles is not
expressed very well by the proposed model, nor their relative movements. In
fact, the model explains two widely different phenomena without making a
distinction, which shows that in any case the differences between these
phenomena are not expressed in the model. The first phenomenon is the
correlation between the measurements of spin components on two different
particles that together form a system in the singlet state. The second
phenomenon, that is explained equally well, is the passage of a single
spinning particle first through one, then through a second Stern-Gerlach
magnet at some distance from the first, that is inclined with respect to the
first.

B.Metric and geodesic equations

Consider a metric $g_{ik}$

$ds^{2}=\cos^{2}\vartheta\,dt^{2}-dr^{2}-r^{2}d\vartheta^{2}-r^{2}\sin^{2}\vartheta\,d\varphi^{2}$
$+\,2r\sin^{2}\vartheta\,d\varphi dt.$ $None$

This space-time has axial symmetry due to the last term in the metric ground
form, $+2r\sin^{2}\vartheta\,d\varphi d\vartheta$. This term specifies the
model's spin direction relative to the chosen frame of reference; by changing
the sign of this term or by rotating $180^{\circ}$ along an axis perpendicular
to the $\vartheta=0$ axis we obtain a model with opposite spin direction.

In order to investigate how free test-particles move in this space-time we
have to solve the equations of motion

${d^{2}x_{i}\over ds^{2}}=-\Gamma^{i}_{jk}{dx_{j}\over ds}{dx_{k}\over ds}.$
$None$

With the shorthand notation $U^{r}=r^{\prime}={dr\over ds}$,
$U^{r^{\prime}}=r^{\prime\prime}={d^{2}r\over ds^{2}}$ etc. the geodesic
equations become

$\eqalignno{U^{t^{\prime}}={}&{-\,U^{r}\sin\vartheta\left[\sin\vartheta(U^{t}-rU^{\varphi})\right]\over
r}&(15a)\cr\vskip 2.0pt\cr\hfil\cr
U^{r^{\prime}}={}&{(rU^{\vartheta})^{2}-({rU^{\varphi}\sin}\vartheta)[\sin\vartheta(U^{t}-rU^{\varphi})]\over
r}\cr\vskip 2.0pt\cr\hfil&(15b)\cr
U^{\vartheta^{\prime}}={}&{-\,2\,U^{r}(rU^{\vartheta})+{\cos\vartheta\over\sin\vartheta}[\sin\vartheta(U^{t}-rU^{\varphi})]^{2}\over
r^{2}}\cr\vskip 2.0pt\cr\hfil&(15c)\cr U^{\varphi^{\prime}}={}&{1\over
r^{2}\sin\vartheta}\cr\vskip
2.0pt\cr{}\hfil&\times\bigl{\\{}-U^{r}(rU^{\varphi}\sin\vartheta)\cr}$
$+\,U^{r}\cos^{2}\vartheta\left[\sin\vartheta(U^{t}-rU^{\varphi})\right]$
$+\,2{\cos\vartheta\over\sin\vartheta}(rU^{\vartheta})\left[\sin\vartheta(U^{t}-rU^{\varphi})\right]\bigr{\\}}.$
$None$

These equations have the following solutions:

$U^{t}=P+{X\over r}$ $None$ $U^{\varphi}={P-{X{\tenrm cotg}^{2}\vartheta\over
r}\over r}$ $None$
${\left(U^{\vartheta}\right)}^{2}={A-{X^{2}\over\sin^{2}\vartheta}\over
r^{4}}$ $None$ ${\left(U^{r}\right)}^{2}=-\,{A-X^{2}\over r^{2}}+{2PX\over
r}+P^{2}-W,$ $None$

where $P,X,A$ and $W$ are constants. We can also write

$P=\cos^{2}\vartheta\,U^{t}+r\sin^{2}\vartheta\,U^{\varphi}$ $None$
$X=r\sin^{2}\vartheta\left(U^{t}-rU^{\varphi}\right)$ $None$
$A={\left(r^{2}U^{\vartheta}\right)}^{2}+{\left[r\sin\vartheta\left(U^{t}-rU^{\varphi}\right)\right]}^{2}$
$None$
$W={\left(\cos\vartheta\,U^{t}\right)}^{2}-{\left(U^{r}\right)}^{2}-{\left(rU^{\vartheta}\right)}^{2}-{\left(r\sin\vartheta\,U^{\varphi}\right)}^{2}$
$+\,2r\sin^{2}\vartheta\,U^{t}U^{\varphi}$ $None$

$W$ is simply $g_{ij}U^{i}U^{j}$, the length of the vector $\tenit U$ squared.
Time-like geodesics have $W>0$, space-like geodesics have $W<0$ and light-like
geodesics have $W=0$. We will restrict $W$ to the values -1, 0 and 1. This
restriction removes an arbitrary scaling factor.

C.Comparison with paths in central force fields

If we only look at time-like geodesics in the direction from past to future
(the paths that test particles follow, $W=1,\,U^{t}>0$), then only three
numbers $P,X$ and $A$ are needed to fully specify a geodesic. What are the
consequences of this paucity of orbit-fixing numbers? Let us compare a
geodesic in a central gravitational field and a geodesic in this model-space
time. Four constants are needed to specify the orbit of a freely falling test
particle A with respect to a massive body, such as the orbit of a planet
around a star. The shape of the orbit or eccentricity \- whether the orbit is
a circle, an ellipse, a parabola or a hyperbola - provides one number. The
size of the orbit - e.g. the distance of closest approach to the central
massive body \- provides another number. The orientation of the orbital plane,
defined as the unit vector normal to the orbital plane, requires the
specification of two angular coordinates. That adds two more degrees of
freedom and brings the total number of constants to four.

For the time-like geodesics in our model-space-time the situation is
different. If we assume that two numbers are needed to specify "shape" and
"size", then only one number is left to specify the "orbital plane". The
quotes indicate that we cannot be sure that it makes sense to talk about
shape, size and orbital plane. We will later have to look at that.

If one number fixes an orbital plane then obviously the orbital orientation
has only one degree of freedom. Given one orbital plane, then another orbital
plane (of test particle B, say) can be specified with reference to the given
orbital plane by providing the difference of the "orbital plane numbers". In
fact, there is an ambivalence in such a specification because of the sign of
the difference. This sign cannot be specified without breaking the symmetry
between the two planes: we have to assume that either plane can play the role
of reference plane and the expressions should not depend on this choice in any
arbitrary way.

We have seen the same "directional degeneracy" before: the distribution of
values $Z_{A},Z_{B},Z_{AB}$ that reproduces quantum mechanics is such that
given one $Z$-value, the other $Z$-values can be specified with reference to
this single $Z$-value. For example, if $Z_{A}$ is given then $Z_{B}$ is
specified, up to a bivalent choice, by giving just one more number: $Z_{AB}$
(see 7a-d).

D.How the constants define shape and orientation of the geodesic

We will now look more closely at the constants of the motion and see whether
it really is the case that there is only one number to specify the orbital
plane.

$A,P,X$ and $W$ are real numbers that only to some degree can be chosen
freely. The expression for ${\left(U^{\vartheta}\right)}^{2}$ indicates that
only values $\left|X\right|\leq\sqrt{A}$ can lead to geodesics. It also shows
that such geodesics are restricted to points where
$\sin\vartheta\geq{\left|X\right|\over\sqrt{A}}$. Geodesics with values of
$\left|X\right|$ close to $\sqrt{A}$ are everywhere close to the equatorial
plane $\vartheta={\pi\over 2}$, while a value of $\left|X\right|\over\sqrt{A}$
close to zero allows the geodesic to approach the poles very closely.
Therefore we define the "tilt of the orbital plane" as

$S={X\over\sqrt{A}}\qquad\left(-1\leq S\leq 1\right).$ $None$

If $S=-1$ or $S=1$ then $\sin\vartheta=1$; such geodesics are equatorial. All
other geodesics are wavering north and south (and through) the equatorial
plane and their orbital plane, if such a mathematical object can be defined,
is tilted with respect to the equatorial plane. The maximum angular distance
from the equatorial plane is reached when $\left|U^{\vartheta}\right|=0$,
which is when $\sin\vartheta=\left|S\right|$. We call $S$ the "tilt of the
orbital plane with respect to the equatorial plane", because of the analogy to
the tilt of a planetary orbit with respect to the Solar system's ecliptic
plane, which also is equal to the angle where the planet reaches its greatest
angular distance from the ecliptic plane. While we already have seen that the
tilt of a planetary orbit is only one of two constants that define the
orientation of the orbital plane, we still have to see whether there is such a
second constant in the case of our model-space-time geodesics.

We can learn much about the orbital shape and size from investigating the
radial velocity. If $A\neq X^{2}$ then ${\left(U^{r}\right)}^{2}$ is a second
order function of $1/r$ and otherwise it is a first order function of $1/r$.
We will now look at the constraints on $P,X,W$ and $A$ that ensure that

${\left(U^{r}\right)}^{2}=-\,{A-X^{2}\over r^{2}}+{2PX\over r}+P^{2}-W=0$
$None$

has positive solutions of $r$.

We have:

${1\over r}|_{U^{r}=0}={PX\pm\sqrt{AP^{2}-AW+X^{2}W}\over A-X^{2}}.$ $None$

The condition that there be solutions is that

$P^{2}\geq W\left(1-{X^{2}\over A}\right).$ $None$

The factor in parentheses is non-negative, because $A\geq X^{2}$. If $W\leq
0$, then the relation is fulfilled for all values of $P$. For positive $W$ the
above relation puts a lower bound on the absolute value of $P$.

Because the radial parameter $r$ is non-negative, we are only interested in
non-negative solutions of $1/r$. If there is one positive solution, then the
trajectory is unbound: the positive solution is the point of closest approach,
but there is no point of greatest radial distance. If the other solution is
zero, then the trajectory is just barely unbound and has the status of a
parabolic trajectory in a central force field. If the other solution is
negative, then the trajectory is "hyperbolic". If there are two positive
solutions, then the trajectory is bound, like the elliptic trajectories in a
central field with a $1/r$ potential. If the two positive solutions are equal,
then the trajectory has constant radius, i.e. it is comparable with circular
orbits.

The close analogy between classical orbits in central force fields and
geodesics in our space-time model is concisely expressed by

${P^{2}-W\over 2}={{\left(U^{r}\right)}^{2}+{\left(U^{\perp}\right)}^{2}\over
2}-{XP\over r}-{X^{2}\over{2r}^{2}},$ $None$

where

${\left(U^{\perp}\right)}^{2}={\left(rU^{\vartheta}\right)}^{2}+\sin^{2}\vartheta(U^{t}-rU^{\varphi})^{2}\qquad$
${}^{\prime\prime}square\,of\,tangential\,velocity^{\prime\prime}.$ $None$

Eq. (22a) unmistakably has the signature of an energy, having a kinetic
contribution depending on the squares of the radial and tangential velocities
and a potential contribution in two parts, one of which is due to a long range
$1/r$ potential that can be attractive or repelling, like a Coulomb potential,
while the other is due to a short range attractive $1/r^{2}$ potential. The
$1/r$ potential gives rise to orbits having circular, elliptic, parabolic or
hyperbolic shapes, while the $1/r^{2}$ potential adds a precession of the
pericentrum to the movement, like the shift of the perihelion of Mercury that
also is caused by a $1/r^{2}$ term.

The energy expression [Eq. (22a)] contains two independent constants, $P$ and
$X$, that together define shape and size of an orbit in the same way that
shape and size of planetary orbits are defined by the mass of the sun and the
energy per kilogram of planetary mass, where $X$ plays the role of the solar
mass and $\left(P^{2}-W\right)/2$ the role of energy per unit of planetary
mass.

Now we have "used" three constants to specify the tilt of the orbital plane
[Eq. (18)], the size of the orbit [Eq. (20)] and the shape of the orbit [Eq.
(22a)] and there is no constant to completely specify the orientation of the
orbital plane. This situation is explained by the fact that the orbital plane
has to rotate to ensure that a geodesic test particle does not leave the
orbital plane. The pace $\Omega={d\psi\over dt}$ with which the intersection
line $\left\\{\vartheta=\pi/2,\varphi=\psi\left(t\right)\right\\}$ of orbital
plane and the equatorial plane rotates depends on the instantaneous radial
distance $r$ of the test particle:

$\Omega=1/r.$ $None$

The movement of the test particle for the case where the orbit has constant
$r$ ("circular" orbit) is accurately modeled by the movement of a point on the
rim of a coin that is set to spin on its side on a table. The coin may start
almost upright, slowly falling due to frictional dissipation and decreasing
its tilt with respect to the surface of the table until it lies down flat on
the table. In the model space-time, of course, there is no friction and the
initial tilt remains the same forever. While the coin is wobbling on the
table, it rolls on its surface, which means that points on the rim not only
take part in the rotation of the plane of the coin, but also in a rotation
around the axis that is perpendicular to the coin, moving up and down and
around in a complex dance.

The movement becomes even more complex if the radial distance is not constant,
but it can still be understood easily if one imagines that the movement takes
place in a tilted orbital plane that rotates. Figures 1-2 (stereograms) depict
two unbound geodesics, only differing by the sign of the $X$-parameter, from
these two perspectives.

E.Spin component measurements

In a Stern-Gerlach experiment the direction in which a spinning particle is
deflected depends on the direction of the force

$F_{z}=-\nabla\left(-\mu.B\right)=\mu_{z}{\partial B_{z}\,\over\partial z},$
$None$

where $\mu$ is the magnetic moment of the particle associated with the spin of
the particle and $B$ is the magnetic field of the magnet, having a gradient
$\partial B_{z}/\partial z$. As seen from a spin-1/2 particle's point of view,
depending on the sign of $\mu_{z}$, the pole where the magnetic field is
strongest is either squeezed away from or attracted towards the particle as
the particle passes through the diverging field: the two situations are not
symmetric. Neither are the (unbound) geodesics of two test particles that only
differ by the sign of $XP$, see Figs. 1 and 2. According to Eq. (22a) the
$1/r$ potential gives rise to either a repelling or an attractive force,
depending on the sign of $XP$. This analogy is the reason why we tentatively
want to say that the sign of $X$ corresponds to the outcome of a spin
component measurement. ($P$'s sign defines the direction of time, see Eq.
17a.) Thus, according to Eq. (18), the outcome of a spin component measurement
depends on whether the orbital plane tilts less or more than $90{{}^{\circ}}$
with respect to the equatorial plane. Or, in other words, if the normal vector
of the orbital plane points into the "upper" demi-sphere, then the outcome is
"up", if the normal points "down", then the outcome is "down".

F.Generalization

The presented model has two gross characteristics that make it a candidate
hidden variable theory. The first characteristic is the lack of asymptotic
flatness for large values of the radial parameter $r$. There is no scale at
which the model returns to classicality. In a Bohm-Aharonov experiment on a
cosmological scale with Stern Gerlach instruments each the size of a galaxy
the same results will follow, even though the "distance" of the particle under
observation to the nearest tangible part of the instrument may measure several
light years. This is in fine agreement with three accepted physical facts. The
first fact is that some particles with spin, especially electrons, don't seem
to have a characteristic size. The second fact is that the Coulomb force has
an infinite range. Finally, Quantum Mechanics does not declare a scale of size
above which the theory gives up or reverts to "classical" predictions. Of
course, the lack of asymptotic flatness has its downsides too. The most severe
is that a single particle fills the whole universe. Perhaps other, more
realistic models lacking asymptotic flatness can be constructed or do already
exist.

An important treat of the presented model is that flatness is emergent.
Another important characteristic of the model is that orbital planes of test
particles (which are thought of as representing measuring instruments) are
determined by a single parameter only. Precession wipes out the other.
Presumably, all theories that describe elementary particles as vortices in
space, exhibit this feature, and are, if they also lack asymptotic flatness,
candidates for being hidden variable theories.

As an aside, much existing work in relativity centers on making a quantum
theoretical "update" of the original theory. Although some adaptations to the
relativistic framework may be necessary, the present article tries to show
that, after all, relativity should be at the basis of the interpretation of
quantum physics. Some popular concepts introduced in the wake of quantum
theory, most notably "entanglement", should not be regarded as fundamental but
merely as emergent, just as the perceived flatness of space-time.

IV.CONCLUSION

If we accept that we cannot know for sure whether space-time is flat at the
scale (in space and time) at which the Bohm-Aharonov experiment is performed,
then we cannot evade the conclusion that Bell's and similar proofs, which are
all based on counterfactual statements, do not apply. Non-local angles and
other geometric relations over some distance involving counterfactual set-ups
are not measurable and have, even from a realist point of view, no definite
values, because non-local geometric relations in curved space-time require
operational definitions, which are of course not applicable to counterfactual
set-ups.

A proper description of the Bohm-Aharonov experiment amalgamates the
indefiniteness of the traditional quantum description with the realist point
of view of the traditional classical description. Such a description is in the
spirit of relativity theory and makes plausible that, contrary to common
opinion, the philosophical foundation of relativity theory is also fundamental
to the proper interpretation of quantum mechanics as a statistical theory in a
non-flat and largely unknown playground.

An analysis of the results of the Bohm-Aharonov experiment (which are assumed
to agree with the predictions of quantum mechanics) indicates that a hidden
variable can be introduced to explain the results, but that the configuration
of three directions (the orientations of the measuring instruments and the
direction of the hidden variable) has one degree of freedom less than expected
classically. Classically, given two of the three angles that identify a
configuration, the third angle can be chosen freely from a continuous
spectrum. On the other hand, configurations that are compatible with the
predictions of QM restrict the third angle to a bivalent choice. This
restriction is not severe and does not introduce non-locality by itself: even
classically the third angle is restricted.

A model based on the simple, even naive assumption that spin has to do with
space time structure with rotational symmetry in one direction, exhibits
geodesic movements with fewer orbit defining constants than are needed for a
Keplerian orbit of a test particle in a central force field. Of the three
constants, only one constant defines the orientation of the orbital plane.
Relative to an orbital plane, any other orbital plane can, up to a bivalent
choice, be specified with a single number. Thus the model has exactly the
required property to ensure that the predictions of quantum mechanics can be
reproduced.

[1] A. Einstein, B. Podolsky & N. Rosen, Phys. Rev. 47, 777 (1935)

[2] N. Bohr, Phys. Rev, 48, 696 (1935)

[3] D. Bohm and Y. Aharonov, Phys. Rev. 108, 1070 (1957)

[4] J.S. Bell, Physics 1, 195 (1964)

[5] R. Breslav, G.N. Parfionov, R.R.Zapatrin, http://xxx.lanl.gov/abs/quant-
ph/9903011

[6] Ioannis Raptis and Roman R.Zapatrin, http://xxx.lanl.gov/abs/gr-qc/9904079

[7] Bart Jongejan, http://xxx.lanl.gov/abs/quant-ph/0007010

Figure 1. Geodesic with attractive $1/r$ potential

Figure 2. Geodesic with repelling $1/r$ potential