# Device-Robust Acoustic Scene Classification Based on Two-Stage
Categorization And Data Augmentation

###### Abstract

In this technical report, we present a joint effort of four groups, namely GT,
USTC, Tencent, and UKE, to tackle Task 1 - Acoustic Scene Classification (ASC)
in the DCASE 2020 Challenge. Task 1 comprises two different sub-tasks: (i)
Task 1a focuses on ASC of audio signals recorded with multiple (real and
simulated) devices into ten different fine-grained classes, and (ii) Task 1b
concerns with classification of data into three higher-level classes using
low-complexity solutions. For Task 1a, we propose a novel two-stage ASC system
leveraging upon ad-hoc score combination of two convolutional neural networks
(CNNs), classifying the acoustic input according to three classes, and then
ten classes, respectively. Four different CNN-based architectures are explored
to implement the two-stage classifiers, and several data augmentation
techniques are also investigated. For Task 1b, we leverage upon a quantization
method to reduce the complexity of two of our top-accuracy three-classes CNN-
based architectures. On Task 1a development data set, an ASC accuracy of 76.9%
is attained using our best single classifier and data augmentation. An
accuracy of 81.9% is then attained by a final model fusion of our two-stage
ASC classifiers. On Task 1b development data set, we achieve an accuracy of
96.7% with a model size smaller than 500KB 111Code available:
https://github.com/MihawkHu/DCASE2020_task1.

Index Terms—  acoustic scene classification, data augmentation, convolutional
neural network, device mismatch, quantization

##  1 Introduction

Acoustic scene classification (ASC) is concerned with correctly identifying
real-world sounds into a set of given environment classes, such as metro
station, street traffic, or public square. An acoustic scene sound contains
much information and rich content that makes accurate scene prediction
difficult, and thereby an intriguing research problem. ASC has thus been an
attractive research field for decades, and the Detection and Classification of
Acoustic Scenes and Events (DCASE) challenge [1, 2, 3, 4] provides the
benchmark data and a competitive platform to promote sound scene research and
analyses.

In DCASE 2020, there are two different sub-tasks of Task 1. Task 1a focuses on
the robustness problem of ASC system. Its goal is to promote research to solve
the device mismatch issue, which is a common case in ASC applications. The key
aim is to design a device-invariant system, which can classify ten scene
audios recorded by different devices well, without leveraging any device
information in the evaluation stage. Task 1b focuses on the model size of the
ASC system. The goal is to build a three-class classifier occupying no more
than 500KB.

We describe our submitted systems for the two sub-tasks of DCASE 2020. For
Task 1a, we build a two-stage acoustic scene classification system, which
includes a three-class classifier and a ten-class classifier. The final
predicted class is based on the score fusion of these two classifiers. Four
different convolutional neural network (CNN) based models are used in our two-
stage classifier. Moreover, several data augmentation strategies are adopted
to reduce the device dependency of our models. Model ensemble of 4 CNN-based
systems provided a significant boost of the ASC performance. For task 1b, we
build a small-size model at first, and then use quantization method to
compress the well-trained model. A model can be compressed to $1/8$ of the
original size by this way. In our experiments, the ensemble of two smaller
models can get better evaluation results than a single model, when they have
the same level model size.

##  2 Acoustic Scene Classification System

###  2.1 Two-Stage Classification Procedure

For Task 1a, we build a two-stage ASC system, which includes two different
classifiers and outputs the class of the input audio scene choosing among ten
classes. As shown in figure 1, the first classifier is a three-class
classifier, and it classifies an input audio scene into one of three main
classes, including in-door, out-door, and transportation. The second
classifier is a ten-class classifier, which classifies a given input audio
scene into one of ten basic scene classes, including airport, shopping mall,
metro station, pedestrian street, public square, street traffic, tram, bus,
metro, park. For each input audio, its final scene class is chosen by score
fusion of those two classifiers. If we let $\mathds{C}^{1}$ and
$\mathds{C}^{2}$ denote the set of three main classes and ten classes,
respectively, and let $F^{1}$ and $F^{2}$ indicate the output of the first and
second classifier, respectively. The final predicted class $Class(x)$ for the
input $x$ is:

$Class(x)=\underset{q,(p\in\mathds{C}^{1},q\in\mathds{C}^{2},p\supset
q)}{\operatorname{argmax}}\ F^{1}_{p}(x)*F^{2}_{q}(x),$

where $p\supset q$ means that $p$ can be though of a super set of $q$. For
example, transportation class is the super set for bus, tram, and metro
classes. Therefore, the probability of an input audio clip to be from the
public square scene is equal to the product of the probability of out-door
place given by $F^{1}_{p}(x)$ and the probability of public square given by
$F^{2}_{q}(x)$.

Figure 1: The proposed two-stage acoustic scene classification system for Task
1a.

###  2.2 CNN-based Classifiers

Five CNN based architectures, which differ one from another for specific
details concerning the usage of (i) time and frequency pooling (sub-sampling)
operations, (ii) independent frequency sub-band analysis, (iii) shortcut
connections (i.e., residual mapping functions), and (iv) number of
convolutional layers, are used:

  * •

FCNN (fully convolutional neural network): FCNN is a VGG [5]-like fully
convolutional network based on 9 stacked convolutional layers with small
kernel sizes. Each convolutional layer is followed by a Batch Normalization
operation and ReLU activation layer. Dropout is also used in the convolutional
layers from five to the last in order to alleviate over-fitting issues. A
$2\times 2$ max-pooling layer is appended after the second, fourth, and eighth
ReLu-based layers. Channel attention is applied to each output channel of the
last convolutional layer, followed by a global pooling layer. Finally, a
10-way softmax layer is used to generate the final classification decision. In
Task 1b, we use an FCNN with a similar architecture but much fewer parameters,
and we refer to that model as small-FCNN.

  * •

fsFCNN (frequency sub-sampling controlled fully convolutional neural network):
Through our experiments on DCASE 2020 Task 1a, we noticed that reducing max-
pooling in the frequency axis helped contrasting overfitting issues; thereby,
we deployed a neural architecture very similar to the above-mentioned FCNN but
having 11 stacked convolutional layers. Moreover, $2\times 2$ max-pooling
layer is appended only after the second and fourth ReLU-based layers. A
$1\times 2$ max-pooling layer instead follows the sixth, and eighth ReLU-based
layers. Channel attention is applied to each output channel of the last
convolutional layer, followed by a global pooling layer. Finally, a 10-way
softmax layer is used to generate the final classification result.

  * •

fsFCNN-s (frequency sub-sampling controlled fully convolutional neural network
with split frequency bands): Each input feature map is first split (’-s’ in
the model name) into two sub-feature maps along a frequency dimension. If
there are $N$ frequency bins, frequency bins between 0 and $(N/2)-1$ are
processed by an fsFCNN, and frequency bins between $N/2$ and $N$ would be
independently processed by another fcFCNN. The processing will happen in
parallel up to the ReLU-based layer. Then the two processed streams would be
concatenated and processed by two further convolutional layers. Finally, a
global pooling layer and 10-way softmax is used to get the final scene
classification decision.

  * •

Resnet (17-layer residual network): Resnet model is a residual network [6]. We
use the network structure proposed in [7], which has 17 convolutional layers.
There is no frequency sub-sampling throughout the whole network. Each input
feature map is divided into two sub-feature maps along the frequency
dimension. To be specific, if we have $N$ frequency bins, the first $N/2$ and
the second half are processed by two parallel stacked convolutional layers.
Thus, we have a two-stage model structure. Like FCNN, a global pooling layer
and 10-way softmax are used to get the final utterance level prediction
results. Different from the structure in [7], in our final submission, we
double the filter number of each convolutional layer, and we denote it to
Resnet-d.

  * •

Mobnet (MobileNet-v2): Mobnet is based on MobileNet-v2 [8]. The key feature of
Mobnet is its low complexities despite the high accuracies that can be
attained, as demonstrated in [8]. Mobnet uses lightweight depth-wise
convolutions to process features in the intermediate expansion layer. We
leveraged a relatively small-size Mobnet to tackle Task 1b.

Table 1: Experimental results on Task 1a. ’Scale’ means scaling input data to [0,1]. ’Spec-aug’ means performing a mini-batch level spectrum augment method. ’5*Aug’ means using extra five kinds of augmented data, including spectrum correction, pitch shift, speed change, add random noise, and mix audios. ’Two-stage’ means using a two-stage classification system. All ’Y’ in the table means we used that method. Idx. | System | Scale | Spec-aug | 5*Aug |  | Reverb
---
+DRC
Two-stage |  | A
---
acc. %
| B&C
---
acc. %
| s1-s3
---
acc. %
| s4-s6
---
acc. %
| val
---
loss
| Avg
---
acc. %
(0) |  Baseline [4] | - | - | - | - | - | 70.6 | 61.6 | 53.3 | 44.3 | 1.37 | 54.1
(1) | Resnet | - | - | - | - | - | 78.5 | 72.6 | 68.8 | 68.1 | 1.28 | 70.2
(2) | Resnet | Y | - | - | - | - | 78.8 | 72.1 | 69.3 | 69.5 | 1.30 | 71.0
(3) | Resnet | Y | Y | - | - | - | 80.3 | 73.5 | 71.4 | 67.7 | 1.26 | 71.6
(4) | Resnet-d | Y | Y | - | - | - | 79.7 | 74.8 | 71.9 | 70.7 | 1.28 | 73.0
(5) | Resnet-d | Y | Y | Y | Y | - | 83.0 | 76.1 | 73.6 | 71.0 | 1.23 | 74.6
(6) | FCNN | Y | Y | Y | - | - | 88.5 | 79.8 | 74.6 | 70.6 | 1.08 | 76.0
(7) | FCNN | Y | Y | Y | Y | - | 87.3 | 79.5 | 75.7 | 73.0 | 1.07 | 76.9
(8) | fsFCNN | Y | Y | Y | Y | - | 83.9 | 78.6 | 75.4 | 72.8 | 1.13 | 76.2
(9) | (7)+(8) | Y | Y | Y | Y | - | 87.6 | 79.8 | 76.4 | 73.8 | 1.12 | 77.5
(10) | (5)+(7)+(8) | Y | Y | Y | Y | - | 87.0 | 81.5 | 78.0 | 76.9 | 1.22 | 79.4
(11) | (7)+(8) | Y | Y | Y | Y | Y | 89.1 | 82.4 | 78.5 | 77.5 | 0.84 | 80.2
(12) | (5)+(7)+(8) | Y | Y | Y | Y | Y | 87.9 | 84.1 | 80.4 | 79.9 | 0.94 | 81.9

##  3 Data Augmentation Strategy

A key element of our submission is the use of several data augmentation
strategies. The first four strategies discussed below do not generate extra
training data; whereas, the remaining schemes generate extra training data. In
Task 1a, all the following data augmentation methods except channel confusion
were used. In Task 1b, mixup, data augmentation schemes are limited to channel
confusion, and spectrum augment.

  * •

Mixup: It was proposed in [9] and is often adopted to train ASC models. We use
mixup with alpha equal to $0.4$. Mixup is performed at a mini-batch level: Two
data batches, along with corresponding labels, are randomly mixed in each
training step.

  * •

Random cropping: It was proposed in [7]. During the training stage, the input
data is randomly cropped into a fixed-length along the time axis. In our
experiments, the input data with the size of $423\times 128\times 3$ is
cropped into $400\times 128\times 3$ input feature map. Due to quantization
constraints on dynamic ranging, random cropping is not applied to Task 1b.

  * •

Channel confusion: It is used in Task 1b only. Two channels in input data are
randomly swapped.

  * •

Spectrum augmentation: It was proposed in [10] and showed a significant boost
of the automatic speech recognition performance. In our implementation, we
carried out spectrum augmentation over each input feature map. We applied it
at a mini-batch level. For a batch data in the training step, each feature map
is randomly masked in both time and frequency axes. With respect to [10]
terminology, we set the parameter of time and frequency mask to 10% of their
dimensions, respectively.

  * •

Spectrum correction: It was proposed in [11] and demonstrated moderate device
adaptation properties. However, spectrum correction aims at transforming a
given input spectrum to that of a reference, possibly ideal, device. Different
from the original idea, we here employ spectrum correction as a data
augmentation technique. To this end, we had to modify the original procedure
as follows: (i) we create a reference device spectrum, by averaging the
spectrum from all training devices except that from device A; (ii) we correct
the spectrum of each training waveform collected with device A to obtain extra
data.

  * •

Reverberation + Dynamic Range Compression (DRC): It’s inspired to the
procedure used by the organizers to generated simulated devices, namely
s1-s11. In fact, s1-s11 data is generated by adding reverberation followed by
DRC to audio collected with device A.

  * •

Pitch shift: For each training waveform, we randomly shift the pitch based on
the uniform distribution.

  * •

Speed change: For each training waveform, we randomly change the audio speed
based on the uniform distribution. If the output waveform is longer than the
original one, extra samples are dropped from the end. If shorter, padding is
applied till attaining the same input length.

  * •

Random noise: For each training waveform, random Gaussian noise is added.

  * •

Mix audios: We randomly mix two audios from the same acoustic scene class.
It’s device-independent; this data augmentation scheme might help simulate a
new ”device.”

##  4 Quantization for Model Compression

In Task 1b, the main goal is to keep the system size within 500 Kilobytes
(KB). A post-training quantization method, which is provided by Tensorflow2
[12], is used to compress our neural models. Quantization not only reduces the
model size but also improves hardware accelerator latency with little
degradation in final classification accuracy. In our submission, we used
dynamic range quantization (DRQ). In DRQ, neural weights are quantized from
floating-point to integer having a 8-bit precision. Leveraging DRQ, we thus
transferred our neural architectures from a 32-bit TensorFlow format to a
8-bit TensorFlow-lite format, which compresses the model size to about $1/8$
of its original size. According to our experimental evidence, such a
compression method resulted in a minor ASC classification drop.

##  5 Experimental Setup & Results

###  5.1 Feature Extraction

DCASE2020 ASC audio clips have a fixed-length of 10 seconds. Log-mel filter
bank (LMFB) features were used in our experiments as audio features. The input
audio waveform is analyzed with a $2048$ SFFT points, a window size of $2048$
samples, and a frameshift of $1024$ samples. The librosa [13] library is used
to generate LMFBs, and the HTK formula definition for the Mel scale is
adopted. Due to different sampling rates for Tasks 1a and 1b, the final
spectrogram has 431-time bins in Task 1a and 469-time bins in Task 1b, but the
number of frequency bins is 128 in both tasks. Log-mel deltas and delta-deltas
without padding were also computed, which reduced the number of time samples
to 423 for Task 1a, and 461 for Tasks 1b. The final input tensor size is thus
$423\times 128\times 3$ for Task 1a, and $461\times 128\times 6$ for Task 1b.
Before feeding the speech feature tensors into CNN classifier, we scaled each
feature value into [0,1].

###  5.2 Model Training

For the train-test split, we adopt the official recommended way to split the
development material. For Task 1a, there are 13965 train audio clips, and 2970
test audio clips. The training set includes audio from devices A, B, C, and
s1-s3. The test set covers data from those six devices and extra data from
unseen devices s4, s5, and s6. Device A data dominates the training set, which
has over 10K utterances. In the test set, the number of waveforms from each
device is the same. For Task 1b, there are 9185 train waveforms and 4185 test
waveforms. All audio clips are from device A. Stochastic gradient descent
(SGD) with a cosine-decay-restart learning rate scheduler is used to train our
models. The maximum and minimum learning rates are 0.1 and 1e-5, respectively.
In our final submission, all development data is used. And due to there is no
validation data, we use the average output of models when learning rate hits
around the minimum number. Keras [14] is used to implement all our CNN-based
models.

###  5.3 Results on Task 1a

In Table 1, we report only some of the evaluation results collected on Task 1a
due to space constraints. In the training set, device A data accounts for
around 75%, and device B, C, s1-s3 accounts for around 5%, respectively. Thus,
based on the device information of data, we have divided the test set into
four different subsets, which represent real source data (device A), real
target data (device B & C), the target seen simulated data (device s1-s3), and
target unseen simulated data (device s4-s6).

For comparison, model (0) gives the accuracy of the baseline ASC system
provided by the DCASE2020 organizers. For systems (1) to (3), we investigate
the effect of scaling and spectrum augmentation. Scaling features to [0,1]
give a 0.8% absolute improvement, and that gain is mainly from device s1-s6.
Spectrum augmentation boosts the ASC accuracy from 71.0% to 71.6%, yet the
accuracy on unseen devices drops significantly. Model (4) doubled Resnet
parameters, and that leads to a significant enhancement of the ASC accuracy.
Leveraging all data augmentation schemes, we can further improve ASC accuracy
up to 74.6%, which is the best result using the Resnet based models discussed
earlier.

Systems (6) to (8) are based on the FCNN architecture. The fsFCNN-s model is
not evaluated on this train-test setup, but it’s used in our final submission.
Moving from Resnet-d to FCNN and leveraging all available training data
(original and augmented), ASC accuracy goes from 74.6% to 76.9%, which
represents the best result with a stand-alone FCNN-based architecture.
Comparing models (6) and (7) allows us to appreciate the effect of the
reverberation + DRC data augmentation. Indeed, it provides a large improvement
in s1-s6 data, especially on unseen s4-s6. That is expected since s1-s6 data
is generated by adding reverberation and applying DRC on device A data. Models
(7), (8), and fsFCNN-split have very similar structures, yet when comparing
classification results per testing utterance and among those three models, we
observed over 20% difference in the final decision taken by the three models.
Therefore, we keep all of those three models for the model ensemble stage.

Model ensemble is indeed known to boost the ASC accuracy, and we use a simple
non-weighted average score fusion as system combination scheme. From the
results corresponding to systems (9) to (10), we can see that a two-model
combination outperforms any stand-alone systems. By increasing the number of
models in the ensemble, we can attain an enhancement of the ASC accuracy from
77.5% (two models) to 78.1% (three models). Finally, after model ensemble, a
three-class classifier is integrated through a score fusion approach
leveraging the two-stage fusion scheme discussed in Section 2. From results
referring to systems (11) and (12), we can argue that the proposed two-stage
fusion strategy significantly improves ASC accuracy. In a two-model ensemble,
the ASC accuracy increases from 77.5% to 80.2%; and further increases from
79.4% to 81.9% with a three-model ensemble. For the sake of completeness, it
should be mentioned that the accuracy of the three-class classification system
is 93.2%.

For our final system submissions, all of the development data is used. Due to
resource and time limitations, some of our final submissions are not tested on
the above-mentioned train-test split. Our four submissions include: 1) average
ensemble of Resnet-d, Resnet-d with attention, fsFCNN-s, and fsFCNN with
attention; 2) average ensemble of Resnet-d, FCNN, fsFCNN, and fsFCNN-s trained
by different data strategies; 3) average ensemble of all models in 1) and 2);
and 4) average ensemble of all models in 1) and 2) except Resnet based models.

###  5.4 Results on Task 1b

Table 2 shows our evaluation results on Task 1b. Mobnet and small-fcnn models
can attain an ASC accuracy of 95.2% and 96.4%, respectively. The model size
can be reduced to about $1/8$ of its original size using quantization. For
Mobnet, it is observed a 0.4% performance drop. For small-fcnn, the
performance drop is only equalt to 0.1%. We thus decide to carry out model
ensemble, but we further reduce mobnet and small fcnn parameters before system
combination to keep the final combined model size under 500 KB. Therefore,
systems in rows 3 and 4 in Table 2 are slightly different from those in row 1
and 2. In Table 2, we present the results by logistic regression based model
ensemble. As expected, model ensemble achieves better accuracy than a stand-
alone system even after quantization.

For our final submission, all of the development data is used. Among our four
submissions, the first is from a single small-FCNN model, and the other three
are from the ensemble of Mobnet and small-FCNN, including average ensemble of
Mobnet and small-FCNN, logistic regression based ensemble of Mobnet and small-
FCNN, and two small-FCNN models.

Table 2: Experimental results on Task 1b.  Model |  | Original
---
acc. % (size)
| Quantization
---
acc. % (size)
Mobnet | 95.2 (3.2M) | 94.8 (411K)
small-FCNN | 96.4 (2.8M) | 96.3 (357K)
| Mobnet
---
\+ small-FCNN-v1
96.8 (1.8M+1.9M) | 96.7 (490K)
| small-FCNN-v1
---
\+ small-FCNN-v2
96.5 (1.9M+2.1M) | 96.3 (499K)

##  6 Discussion & Conclusion

Although we achieve over 80% and 95% ASC accuracy on Task 1a and Task 1b,
respectively, there is yet a large prediction difference between systems with
very similar performance. For example, the prediction overlap of FCNN and
fsFCNN is only 77%, which implied that there exists some audio clips are
differently predicted by different systems with similar performance. By
listening inspection, we realize that several audio clips are difficult to
classify even for a human listener because some acoustic scenes are fuzzy by
design. To bright ideas: voices of people in an indoor area can either be from
a shopping mall or an airport, because the same micro-scene, i.e., a small
shop, can be in either a shopping mall or an airport.

In this technical report, we have described our submitted systems to DCASE
2020 Tasks 1a and 1b challenges. For Task 1a, a two-stage classification
system is designed, which includes both 3-class classifiers and 10-class
classifiers. The final predicted class is determined by the score fusion of
these two classifiers. Different fully CNN based classifiers and data
augmentation techniques are investigated in our experiments. For Task 1b,
small-size FCNN and mobnet are used as a classifier. The quantization method
is performed to compress the ASC model to be less than 500KB. From our
evaluation of development data, we achieve an accuracy of 81.9% on Task 1a,
and 96.7% on Task 1b.

## References

  * [1] A. Mesaros, T. Heittola, E. Benetos, P. Foster, M. Lagrange, T. Virtanen, and M. D. Plumbley, “Detection and classification of acoustic scenes and events: Outcome of the DCASE 2016 challenge,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 26, no. 2, pp. 379–393, 2018.
  * [2] A. Mesaros, T. Heittola, A. Diment, B. Elizalde, A. Shah, E. Vincent, B. Raj, and T. Virtanen, “DCASE 2017 challenge setup: Tasks, datasets and baseline system,” in _Proceedings of the Detection and Classification of Acoustic Scenes and Events 2017 Workshop (DCASE2017)_ , November 2017, pp. 85–92.
  * [3] A. Mesaros, T. Heittola, and T. Virtanen, “A multi-device dataset for urban acoustic scene classification,” in _Proceedings of the Detection and Classification of Acoustic Scenes and Events 2018 Workshop (DCASE2018)_ , November 2018, pp. 9–13.
  * [4] T. Heittola, A. Mesaros, and T. Virtanen, “Acoustic scene classification in dcase 2020 challenge: generalization across devices and low complexity solutions,” in _Proceedings of the Detection and Classification of Acoustic Scenes and Events 2020 Workshop (DCASE2020)_ , 2020, submitted. [Online]. Available: https://arxiv.org/abs/2005.14623
  * [5] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _arXiv preprint arXiv:1409.1556_ , 2014.
  * [6] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.
  * [7] M. D. McDonnell and W. Gao, “Acoustic scene classification using deep residual networks with late fusion of separated high and low frequency paths,” in _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 141–145.
  * [8] M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, and L.-C. Chen, “Mobilenetv2: Inverted residuals and linear bottlenecks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 4510–4520.
  * [9] H. Zhang, M. Cisse, Y. N. Dauphin, and D. Lopez-Paz, “mixup: Beyond empirical risk minimization,” _arXiv preprint arXiv:1710.09412_ , 2017.
  * [10] D. S. Park, W. Chan, Y. Zhang, C.-C. Chiu, B. Zoph, E. D. Cubuk, and Q. V. Le, “Specaugment: A simple data augmentation method for automatic speech recognition,” _arXiv preprint arXiv:1904.08779_ , 2019.
  * [11] T. Nguyen, F. Pernkopf, and M. Kosmider, “Acoustic scene classification for mismatched recording devices using heated-up softmax and spectrum correction,” in _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 126–130.
  * [12] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “TensorFlow: Large-scale machine learning on heterogeneous systems,” 2015, software available from tensorflow.org. [Online]. Available: https://www.tensorflow.org/
  * [13] B. McFee, C. Raffel, D. Liang, D. P. Ellis, M. McVicar, E. Battenberg, and O. Nieto, “librosa: Audio and music signal analysis in python,” in _Proceedings of the 14th python in science conference_ , vol. 8, 2015.
  * [14] F. Chollet _et al._ , “Keras,” https://keras.io, 2015.


# Interference with non-interacting photons and a special type of detector

Ioannis Contopoulos<EMAIL_ADDRESS>Research Center for Astronomy
and Applied Mathematics, Academy of Athens, Athens 11527, Greece

###### Abstract

We present a statistical model of non-interacting individual classical
particles that may lead to a microscopic implementation of quantum mechanics.
The model requires the action of a special type of detector that detects and
records individual particles. According to our model, microscopic particles
are classical entities that follow classical trajectories without nonlocal
interactions between them. As long as they remain undetected, particles carry
the information of a phase equal to an action integral along their trajectory.
At the point of their detection, our special type of detector collects the
phases from all particles reaching it, adds them up over time as complex
numbers, and divides them by the square root of their number. The detector
records a detection each time the square of the magnitude of the ensuing
vector crosses an integer value. We test our model with numerical simulations
of a double-slit experiment, and obtain an interference pattern analogous to
the one observed in an actual physical experiment.

Statements and Declarations: The datasets generated during during the current
study are available from the corresponding author on reasonable request. There
are no ethical issues nor external funding associated with this work. On
behalf of all authors, the corresponding author states that there is no
conflict of interest.

computer simulation, quantum theory, quantum interpretations

††preprint: APS/123-QED

##  I Introduction

Quantum mechanics presents a conceptual conundrum that led to a multitude of
interpretations, all of which are, in our opinion, unsatisfactory 1 ; 2 ; 3 ;
7 . Individual measurements detect individual particles, but the event-by-
event collection of such measurements shows wave-like interference patterns
which, quoting Feynman, are ‘impossible, absolutely impossible to explain in
any classical way and have in them the heart of quantum mechanics’ (Feynman
Lectures on Physics 4 ). We will argue, as did Jin et al. (2010 5 ) a few
years ago, that it is wrong to think that waves are the only possible physical
cause of interference.

We will first show that it is possible to replace the collective description
of a wave with a sequence of non-interacting individual particles if each
particle carries a phase that evolves along the particle trajectory. Not only
that. In order to observe wave effects, we need the contribution of a
particular type of classical detector, namely one that records the phases of
the detected individual particles. In § II we derive the operating
characteristics of such a detector in the case of photons, without any
reference to Planck’s constant. In § III we show how we can obtain an
interference pattern from the collection of a sequence of such non-interacting
individual photons. This will further elucidate the role of our detector. In §
IV we try to generalize our conclusions for massive particles with mass $m$,
velocity ${\rm v}$. We will argue that the classical description of the
trajectories of microscopic particles with energy $\epsilon$, and potential
energy $U$ is missing something fundamental about their motion. This may be
that particles undergo stochastic interactions with a background, or with the
detectors that result in their measurement. We summarize our conclusions in §
V. The novel result of our work is that, if physical measurement operates
according to the characteristics of the special type of detector described in
§ II, the interference pattern is a-posteriori collective information that is
obtained from the collection of a growing number of particle detections, and
is not known by each individual particle.

##  II Collection of the phases of non-interacting photons

Our eventual goal is to obtain a physical interpretation of quantum mechanics
based on individual non-interacting particles that behave classically.
Although we will eventually consider classical particles like electrons, it
will help our discussion to begin by considering individual photons. Let us
discuss first what information individual photons may carry that can lead to
the construction of event-by-event interference patterns at a detector.

According to classical electromagnetism, an electromagnetic wave in vacuum is
characterized by its electric field, its direction, and its frequency
$\nu\equiv c/\lambda$ ($c$ is the speed of light, and $\lambda$ is the
wavelength of the radiation). These are the most fundamental quantities from
which one can calculate the Poynting flux of energy and the electromagnetic
energy density. Quantum mechanics offers a different picture of the
electromagnetic wave in terms of individual photons. If we were to reconcile
the wave and particle pictures without invoquing non-local interactions
between photons, the flux of energy may be viewed as the flow of $N$ photons
per unit time each carrying energy $\epsilon$ at the speed of light along the
direction of the Poynting vector. We will now argue that non-interacting
photons cannot carry the information of the electric field which is
fundamental for the manifestation of wave behavior (e.g. refraction,
diffraction, interference, etc.). The reason is that the collection of $N$
indistinguishable photons carries $N$ times the energy $\epsilon$ of one
photon, but the amplitude of the electric field of the corresponding classical
wave is proportional to only $\sqrt{N}$ times the square root of $\epsilon$.
Taking the square root of $N$ is obviously a collective operation that cannot
be carried as information by non-interacting individual particles.

We will now propose one particular way to count the number of photons in a
detector. Firstly, we will assume that individual photons carry an evolving
phase $\phi$, something like an ‘internal clock’. For photons of wavelength
$\lambda$ we propose the natural expression

$\displaystyle\phi$ $\displaystyle\equiv$ $\displaystyle 2\pi\frac{{\rm
l}}{\lambda}\ .$ (1)

Here, ${\rm l}$ is the length of the photons’ trajectory from their point of
origin to the finite region of spacetime where and when their detection takes
place. Secondly, we will assume a special type of detector that collects the
phases of the $N$ photons that reach it at a finite region around the
spacetime position $({\bf x};t)$, and forms the expression

$\overline{\Psi}({\bf
x};t)\equiv\sum\nolimits_{k=1}^{K}\frac{1}{\sqrt{N_{k}}}\sum\nolimits_{j=1}^{N_{k}}{\rm
e}^{2\pi{\rm l}_{k}/\lambda}$ (2)

Hereafter, $\overline{(\ldots)}$ will denote quantities that are calculated
after several repetions of an experiment. $k=1$ to $K$ denotes the $K$
possible different trajectories that photons may follow to reach the spacetime
position $({\bf x};t)$, as e.g. in an interference experiment, and $l_{k}$ are
the lengths of every such trajectory. $N_{1}+\ldots+N_{K}=N$ is the total
number of photons collected by the detector at spacetime position $({\bf
x};t)$. We will discuss the physical significance of eq. (2) below. Finally,
the number of photons recorded by the detector in that particular region of
spacetime around $({\bf x};t)$ is assumed to be equal to

$\overline{N}\equiv|\overline{\Psi}|^{2}({\bf
x};t)=\left|\sum\nolimits_{k=1}^{K}\frac{1}{\sqrt{N_{k}}}\sum\nolimits_{j=1}^{N_{k}}{\rm
e}^{2\pi i{\rm l}_{k}/\lambda}\right|^{2}\ .$ (3)

In general, the number $\overline{N}$ of recorded photons is not equal to the
number $N$ of collected ones. This is the main ansatz that characterizes our
process of photon collection and recording:

> We assume that the only way to count photons is via eq. (3).

In other words, $N$ photons are collected at a particular position in
spacetime, but the detector says that $\overline{N}\neq N$ have been recorded.
The important thing to emphasize here is that $\overline{\Psi}$ is not known
a-priori, and is certainly not ‘known’ by each individual photon.
$\overline{\Psi}$ is determined at the detector after the collection of a
large number of photons from subsequent repetitions of the experiment.

We will now see how eq. (2) allows us to generate an interference pattern with
individually collected non-interacting photons.

##  III Event-by-event emergence of photon interference

Let us consider a double-slit experiment where individual photons of energy
$\epsilon$ emerge horizontally from a certain source, cross a screen with two
parallel perpendicular slits separated by a distance $D$, and then hit a
horizontal detector some distance $l\gg D$ behind the screen. The detector
consists of an array of pixels where individual photons are collected and
their phases are processed according to eq. (2). The detector then triggers
detections (i.e. records photons) every time $|\overline{\Psi}|^{2}$ crosses
an integer value. The process repeats itself over and over again.

Figure 1: Schematic of a double slit experiment with $f$ individual non-
interacting photons emitted horizontally per unit time from each slit. $l,L\gg
D$. Notice that there is no requirement for a common ‘coherent’ origin of
these photons. The only requirement is that their phases start counting from
the time they emerge from the source. Figure 2: Gradual buildup of the double-
slit interference pattern for 100, 1,000, 10,000 and 100,000 photon detections
respectively. Shown are the distributions of photons along the 28 $100\mu$m
wide segments of the detector. In this experiment we chose detection pixels of
width $\delta x=1\mu{\rm m}\sim\lambda$. Solid line: the analytical functional
form of eq. (6). Note that the peaks drop very slowly with distance $x$
because photons are assumed to move on the horizontal plane, and therefore,
their number density drops inversely to the distance from the slits, not the
distance squared. $R^{2}>0.9$ after about 4,000 photon detections. Figure 3:
Similar to figure 2 but for detection pixels of width $\delta x=0.2\mu{\rm
m}<\lambda$. The interference pattern also develops clearly but more slowly
than in figure 2. Figure 4: Similar to figure 2 but for detection pixels of
width $\delta x=10\mu{\rm m}\gg\lambda$. As expected, however long we may
wait, the interference pattern will never be recovered.

Let us assume that photons emerge at a rate of $f$ photons per unit time from
each slit at the speed of light distributed isotropically on the horizontal
plane, and that each detection position has a width $\delta x$ at positions
$x$ from the detector midpoint (see figure 1). In reality, the detection
position is a finite region of spacetime and not just one point. In other
words, in order for our detector to process phases according to eq. (2), we
need to consider a finite photon collecting area, and not just one point. As
we will see in the following subsection, in order to obtain an interference
pattern from the collection of a reasonable number of photons, the detection
pixel width $\delta x$ must be neither too small, neither too large.

We also assume that there are no other external perturbations acting on the
photons, thus all photons move along straight lines during their flight from
the source to their eventual detection at the detector. Each pixel receives on
average $fl\delta
x/\pi\cdot[1/((x-\frac{D}{2})^{2}+l^{2})+1/((x+\frac{D}{2})^{2}+l^{2})]$
photons per unit time, which for $D\ll l$ becomes approximately equal to
$2fl\delta x/(\pi(x^{2}+l^{2}))$, i.e. each pixel receives about $N=fl\delta
x/(\pi(x^{2}+l^{2}))$ photons per unit time from each slit. According to eq.
(2), $K=2$ (i.e. there are 2 independent paths to reach each pixel), and the
function $\overline{\Psi}$ recorded by the pixel at position $x$ over an
integration time $t_{\rm int}$ is equal to

$\displaystyle\overline{\Psi}(x;t_{\rm int})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{N_{1}}}\sum\nolimits_{j=1}^{N_{1}}{\rm e}^{2\pi
i(L+l_{1j})/\lambda}$ (4)
$\displaystyle+\frac{1}{\sqrt{N_{2}}}\sum\nolimits_{\mu=1}^{N_{2}}{\rm
e}^{2\pi i(L+l_{2\mu})/\lambda}\ .$

Here, $N_{1}$ and $N_{2}$ are the numbers of photons that passed through slits
No 1 and 2 respectively (they are both approximately equal to $N=fl\delta x\
t_{\rm int}/(\pi(x^{2}+l^{2}))$), and $L+l_{1j},L+l_{2\mu}$ are the respective
lengths of the two trajectories from the source to the detector pixel. From
this point on we can only proceed with a Monte Carlo numerical simulation (see
subsection III.A). We can proceed analytically only if we assume that we have
roughly an equal number $N$ of $j$ and $\mu$ summations. In that case, eq. (4)
yields

$\displaystyle|\overline{\Psi}|^{2}(x;t_{\rm int})$ $\displaystyle\approx$
$\displaystyle\frac{1}{N}\left|{\rm e}^{2\pi
i(L+l_{1})/\lambda}\sum\nolimits_{j=1}^{N}\left(1+{\rm e}^{2\pi
i(l_{2}-l_{1})/\lambda}\right)\right|^{2}$ (5) $\displaystyle=$
$\displaystyle\frac{1}{N}\left(N\left|1+{\rm e}^{2\pi
i(l_{2}-l_{1})/\lambda}\right|\right)^{2}$ $\displaystyle=$ $\displaystyle
2N[1+\cos(2\pi(l_{1}-l_{2})/\lambda)]$ $\displaystyle=$ $\displaystyle
4N\cos^{2}\left(\frac{\pi xD}{2\lambda\sqrt{x^{2}+l^{2}}}\right)$
$\displaystyle\approx$ $\displaystyle\frac{4fl\delta x\ t_{\rm
int}}{\pi(x^{2}+l^{2})}\ \cos^{2}\left(\frac{\pi
xD}{2\lambda\sqrt{x^{2}+l^{2}}}\right)$

when $D\ll l$. Notice that our process of photon detection differs from that
of Jin et al. (2010)Jin . In the next subsection we will perform a numerical
experiment to test our result.

We must emphasize a very important point in eq. (4). The summations over
$N_{1}$ and $N_{2}$ are performed over photons that reached the detector after
crossing slit No 1 and 2 respectively. However,

> the detector ‘knows’ the direction each photon came from

and processes separately photons reaching it having crossed slit No1 from
those reaching it having crossed slit No 2. Otherwise, if it added the
exponentials with the phases of all the photons that reach it and then divided
by the square root of their total number, namely $\sqrt{2N}$, the
normalization in front of eq. (5) would have been one half of the correct
value.

Eq. (5) yields the following variation of intensity with distance $x$ from the
detector midpoint

$\displaystyle\frac{{\rm d}I(x)}{{\rm d}x}$ $\displaystyle=$
$\displaystyle\frac{\epsilon|\overline{\Psi}|^{2}(x;t_{\rm int})}{\delta x\
t_{\rm int}}$ (6) $\displaystyle=$
$\displaystyle\frac{4fl\epsilon}{\pi(x^{2}+l^{2})}\cos^{2}\left(\frac{\pi
xD}{\lambda\sqrt{x^{2}+l^{2}}}\right)$

when $D\ll l$. Notice that if we integrate eq. (6) over all $x$’s we obtain
the total energy per unit time crossing the screen with the two slits, namely

$I=\int_{-\infty}^{+\infty}\frac{{\rm d}I(x)}{{\rm d}x}\ {\rm d}x=2f\epsilon\
.$ (7)

We thus confirm that, despite the collection and processing of phases and the
emergence of an interference pattern, the total number of collected photons is
equal to the total number of photons that passed through the two slits. Had we
performed the summation in eq. (4) differently (first adding all the
exponentials, and then dividing by $\sqrt{2N}$) we would be missing one half
of the photons.

The derivation in eq. (5) is correct only in the limit of an infinite number
of detection positions $x$, and an infinite number of collected photons. In
reality, each detection position has a finite size $\delta x$, thus the
collected phases contain some variability on the order of

$\delta\phi\sim 2\pi\frac{\delta x}{\lambda}\ .$ (8)

The larger $\delta x$, the more the phases are mixed, resulting in a worse
interference pattern. The smaller $\delta x$, the less the phases are mixed,
resulting in a clearer interference pattern. At the same time, however, the
detector needs to collect a larger and larger number of photons to populate
all detection sites, thus the interference pattern takes longer to appear. As
we will see the optimal conditions for the generation of an interference
pattern that closely mimicks an actual physical experiment with the minimum
number of photon detections require $\delta x\sim\lambda$. In other words, the
detection size must be ‘tuned’ to the wavelength of the photon that it is
going to measure!

###  III.1 Numerical interference experiment analogous to a physical
experiment

Let us now perform a numerical simulation of an idealized photon double-slit
experiment similar to the one performed by Kolenderski et al. (2014 8 ). In
that experiment, they recorded individual photons as they were gradually
building the interference pattern one by one. They also compared the outcome
of their quantum mechanical experiment with the model of Jin et al. (2010 5 )
and found that, although it too was able to reproduce the interference
pattern, it did so about 10 times slower than the physical experiment.

We too consider a source of coherent light with wavelength $\lambda=842\ {\rm
nm}$. The detector consists of 28 segments of width $\Delta x=100\ \mu{\rm m}$
aligned around $x=0$ at a distance $l=3\ {\rm mm}$ away from a screen with 2
narrow slits a distance $D=5\lambda$ apart ($\lambda,D\ll l$). This particular
distance between the screen and the detector was chosen so as to observe 4
interference peaks within the 28 segments of the detector, as in the actual
experiment. A schematic representation of the setup between the screen and the
detector is shown in figure 1. We are also given the information that the
pixels have a 5% detection efficiency in that particular wavelength range.
Photons are collected at each of the 28 detector segments. In reality,
however, photons are collected at detection positions (pixels) of width
$\delta x\ll\Delta x$ within each segment. As we will see below, we were able
to reproduce the results of the actual physical experiment for $\delta
x\sim\lambda$. If $\delta x\gg\lambda$, the phases of individual photons that
arrive at each segment are mixed so dramatically that no interference pattern
is ever obtained.

We perform the numerical experiment as follows. We send photons that move
randomly through either one of the two slits, emerge at random angles behind
the screen, and are collected in one of the $28\times(\Delta x/\delta x)\gg
28$ detection areas (pixels) of width $\delta x$. The phases of the collected
particles are processed as in eq. (2) at each detection position. When
$|\Psi|^{2}$ becomes equal to or larger than 1 at a certain position, the
first particle is detected at that position. The next threshold is $2$ which
when crossed, a second particle is detected, and so on and so forth. We stop
the experiment when a total number of 100, 1,000, 10,000 and 100,000
detections are recorded. We then distribute these detections among the 28
detector segments because this is the only information available about how
particle detections are recorded in the actual Kolenderski et al. (2014)
experiment with which we want to compare our results.

In figures 2-4 we depict the statistics of our numerical experiment for
$\delta x=1,0.2$ and $10\mu{\rm m}$ respectively. These show how the
interference pattern builds-up for particular values of the detection position
width. We clearly see that when $\delta x\gg\lambda$, the interference pattern
is destroyed. We found that, when $\delta x\sim\lambda$, the interference
pattern is reproduced with a coefficient of determination $R^{2}\approx 0.9$
after about 4,000 detector triggerings. This is 20 times higher than the
corresponding number of detector triggerings in the actual experiment of
Kolenderski et al. (2014). The discrepancy, however, may not be as high as it
seems. In Kolenderski et al. (2014) it is stated that, out of the $2\times
10^{6}$ photons per second generated by their source, only 72,000 photons per
second reach their detector, whereas detector triggerings take place at a rate
of only $200/(115\ \mbox{ms})=1,700$ triggerings per second. This would imply
that each of the 28 pixels of their detector collects on average roughly
$72,000/1,700/28=1.5$ photons for every triggering it makes. This, however, is
not consistent with the claimed pixel photon detection efficiency of 5%. This
information leads us to believe that their detector collected 20 times more
photons, i.e. their experiment reached a coefficient of determination
$R^{2}\approx 0.9$ after collecting about 4,000 particles. This value is
consistent with our result, and corresponds to their stated 200 detector
triggerings. Notice, however, that in order to record 4,000 triggerings in our
numerical experiment with $\delta x=1\mu$m, our detector collected 3 times
more photons. Bearing further testing and comparison with actual experiments,
we would like now to discuss how the above may be generalized for massive
particles.

##  IV Towards the wavefunction of massive particles

The success of the above numerical simulation in reproducing an actual
physical interference pattern leads us to discuss how our model may be
generalized for massive particles, i.e. how one may generate an interference
pattern from an ensemble of individual non-interacting massive particles. It
is interesting that, for massless photons, there was no mention whatsoever of
the Planck constant $h$, except for the fact that the energy $\epsilon$ of a
photon is equal to $hc/\lambda$, where $c$ is the speed of light. This result
is very promising because we knew about the wave character of electromagnetic
radiation well before the discovery of quantum mechanical interference. The
new result that quantum mechanics brought in the discussion is that
electromagnetic radiation consists of a sequence of individual photons. We
have shown above that, for our proposed special type of detector, individual
non-interacting photons too manifest wave characteristics without invoking
quantum mechanics.

All the above discussion is directly generalizable for massive particles of
mass $m$ and velocity ${\rm d}{\bf x}/{\rm d}t$ if we replace the photon
wavelength $\lambda$ with the de Broglie wavelength $\lambda_{\rm de\
Broglie}\equiv\hbar/(m|{\rm d}{\bf x}/{\rm d}t|)$, and the phase of the
massive particle with

$\phi\equiv\int\frac{2\pi{\rm d}{\rm l}}{\lambda_{\rm de\
Broglie}}=\int\frac{m}{\hbar}\frac{{\rm d}{\bf x}}{{\rm d}t}\cdot{\rm d}{\bf
x}\ .$ (9)

Here, the ${\rm d}{\bf x}$ integration is taken along the particle trajectory.
The reason we need an integration is because the magnitude of the particle
velocity $|{\rm d}{\bf x}/{\rm d}t|$ is in general not constant, and thus the
de Broglie wavelength may vary along the particle trajectory. On the contrary,
a photon’s velocity always remains fixed, and the above integration yields
$2\pi{\rm l}/\lambda$ (eq. 1). Eqs. (2) and (3) are directly generalizable as

$\overline{\Psi}({\bf
x};t)\equiv\sum\nolimits_{k=1}^{K}\frac{1}{\sqrt{N_{k}}}\sum\nolimits_{j=1}^{N_{k}}{\rm
e}^{i\int_{k}m\frac{{\rm d}{\bf x}_{j}}{{\rm d}t}\cdot{\rm d}{\bf x}/\hbar}\
,$ (10) $\overline{N}\equiv|\overline{\Psi}|^{2}({\bf
x};t)=\left|\sum\nolimits_{k=1}^{K}\frac{1}{\sqrt{N_{k}}}\sum\nolimits_{j=1}^{N_{k}}{\rm
e}^{i\int_{k}m\frac{{\rm d}{\bf x}_{j}}{{\rm d}t}\cdot{\rm d}{\bf
x}/\hbar}\right|^{2}\ .$ (11)

Once again, the integrals $\int_{k}$ are calculated along the $K$ possible
different trajectories that particles may follow to reach the spacetime
position $({\bf x};t)$, $N_{1}+\ldots+N_{K}=N$ is the total number of
particles collected by the detector, and in general $\overline{N}\neq N$. From
the above discussion, it seems that the de Broglie wavelength may be equally
fundamental as Planck’s constant $h$.

We soon realized that, in certain physical systems, massive particles cannot
follow classical trajectories with conserved energy $\epsilon$ and at the same
time be distributed according to a solution of the Schrodinger equation.
Quantum tunneling is one such example. Another example is the hydrogen atom
where an electron say at its ground state with enery $\epsilon_{1}$ can be
found at all distances away from the nucleus (the corresponding wave function
extends all the way to infinity), although this is impossible for a classical
particle with negative energy $\epsilon_{1}$ that is under the action of the
Coulomb potential. Obviously, we are missing something important.

Nelson (1966) first proposed that microscopic particles may follow
frictionless Brownian motion. According to his proposition, particles of mass
$m$ interact with a background with diffusion coefficiant $\nu$ and no
friction. This stochastic motion introduces randomness in the determination of
the phase of a particle along its trajectory. Nelson proposed the following
choice of wavefunction,

$\Psi_{\rm Nelson}=\sqrt{n}\ {\rm e}^{i{\cal S}}\equiv\sqrt{n}\ {\rm e}^{i\int
m{\rm{\bf v}}\cdot{\rm d}{\bf x}/\hbar}\ ,$ (12)

where, ${\rm{\bf v}}$ is the so-called particle current velocity, and $n$ is
the density of detected particles at a particular spacetime position. Let us
backtrack a little and present to the reader a simplified formulation of
Brownian motion similar to the one used by Nelson. Let ${\bf x}(t)$ be a
stochastic process that describes the particle position with time $t$ and
satisfies

${\rm d}{\bf x}(t)={\rm{\bf v}}\ {\rm d}t+{\rm d}{\bf w}(t)\ ,$ (13)

where ${\bf w}$ is a Wiener process, and ${\rm{\bf v}}$ is the mean particle
velocity. The ${\rm d}{\bf w}$ are independent Gaussian with mean $\langle{\rm
d}{\bf w}_{i}(t)\rangle=0$, and $\langle{\rm d}{\bf w}_{i}(t)\cdot{\rm d}{\bf
w}_{j}(t)\rangle=2\nu\delta_{ij}{\rm d}t$. In a stochastic process, ${\bf
x}(t)$ is not differentiable, thus we need to take proper care of time
derivatives. In an average sense, one may integrate the following set of
equations,

$\displaystyle\langle\frac{{\rm d}{\rm{\bf x}}}{{\rm d}t}\rangle$
$\displaystyle=$ $\displaystyle{\bf v}$ (14)
$\displaystyle\frac{\partial{\rm{\bf v}}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{{\bf F}}{m}-({\bf v}\cdot\nabla){\bf v}+({\bf
u}\cdot\nabla){\bf u}+\nu\Delta{\bf u}$ (15)
$\displaystyle\frac{\partial{\rm{\bf u}}}{\partial t}$ $\displaystyle=$
$\displaystyle-\nu\nabla(\nabla\cdot{\bf v})-\nabla({\bf v}\cdot{\bf u})\ ,$
(16)

and obtain the distributions of ${\rm{\bf v}}$ and $n$ from them. Here, ${\bf
F}\equiv-\nabla V$ is the external force acting on the particle, and ${\bf
u}\equiv\nu\nabla\ln n$ is the so-called osmotic velocity. The final
assumptions are that ${\bf v}$ is a gradient, namely

${\bf v}\equiv\frac{\hbar}{m}\nabla{\cal S}\ ,$ (17)

and that the diffusion coefficient is equal to

$\nu=\frac{\hbar}{2m}\ .$ (18)

Under the above assumptions, Nelson was able to show that the expression for
$\Psi_{\rm Nelson}$ in eq. (12) satisfies Schrodinger’s equation.

The new element brought into the discussion by our model of non-noninteracting
particles is that $\Psi_{\rm Nelson}$ may somehow be related to our function
$\overline{\Psi}$. $\overline{\Psi}$ is not carried by each individual
particle, but is calculated by a detector at a certain position of the
experiment. According to eqs. (10) and (13),

$\overline{\Psi}=\sum\nolimits_{k=1}^{K}\frac{{\rm e}^{i\int_{k}m{\rm{\bf
v}_{k}}\cdot{\rm d}{\bf
x}/\hbar}}{\sqrt{N_{k}}}\sum\nolimits_{j=1}^{N_{k}}{\rm
e}^{i\int_{k}m\frac{{\rm d}{\bf w}_{j}}{{\rm d}t}\cdot{\rm d}{\bf x}/\hbar}\
,$ (19)

where again, the first summation is over all the different discrete
trajectories that particles may follow to reach the detector at spacetime
position $({\bf x};t)$. According to eq. (11), $|\Psi|^{2}$ will be equal to
the number $\overline{N}$ of detected particles at a particular detection
region of spacetime. If we ignore for the moment the stochastic perturbations
${\rm d}{\bf w}_{j}$, $\overline{\Psi}$ becomes

$\overline{\Psi}=\sum\nolimits_{k=1}^{K}\sqrt{N_{k}}\ {\rm e}^{i\int_{k}m{\bf
v}_{k}\cdot{\rm d}{\bf x}/\hbar}\ .$ (20)

When we divide by the square root of the volume of space $\delta{\cal V}$
where the detection takes place we obtain

$\displaystyle\Psi_{\rm Nelson}$ $\displaystyle\equiv$
$\displaystyle\frac{\overline{\Psi}}{\sqrt{\delta{\cal V}}}\equiv\sqrt{n}\
{\rm e}^{i\int m{\bf v}\cdot{\rm d}{\bf x}/\hbar}$ (21) $\displaystyle=$
$\displaystyle\sum\nolimits_{k=1}^{K}\sqrt{n_{k}}\ {\rm e}^{i\int_{k}m{\bf
v}_{k}\cdot{\rm d}{\bf x}/\hbar}\ ,$

where $n_{k}\equiv N_{k}/\delta{\cal V}$ are the partial densities of
particles that reached the detection point following path $k$, and the
integrations in the last term are along the directions of the particle current
velocities ${\bf v}_{k}$. As we emphasized over and over again, in general
$\sum\nolimits_{k}n_{k}\neq n\equiv\overline{N}/\delta{\cal V}$. Notice that
${\bf v}$ as defined in eq. (17) is the velocity of particles in Bohmian
mechanics 6 . In the case of multiple particle trajectories, however, our
model of non-interacting particles assumes that individual particles do not
flow along ${\bf v}$, but follow their own classical trajectories with average
velocities ${\bf v}_{k}\neq{\bf v}$.

Nelson’s derivation of the Schrodinger equation requires the action of a
Wiener process ${\bf w}$ with diffusion coefficient $\nu=\hbar/2m$, and the
integration of eqs. (14-16). We here propose that the wavefunction may be
constructed as

$\Psi_{\rm Nelson}\equiv\sqrt{n}\ {\rm e}^{i\int m{\rm{\bf v}}\cdot{\rm d}{\bf
x}/\hbar}\equiv\frac{\overline{\Psi}}{\sqrt{\delta{\cal V}}}$
$=\sum\nolimits_{k=1}^{K}\sqrt{n_{k}}\ {\rm e}^{i\int_{k}m{\rm{\bf
v}_{k}}\cdot{\rm d}{\bf x}/\hbar}\left(\frac{\sum\nolimits_{j=1}^{N_{k}}{\rm
e}^{i\int_{k}m\frac{{\rm d}{\bf w}_{j}}{{\rm d}t}\cdot{\rm d}{\bf
x}/\hbar}}{N_{k}}\right)$ (22)

where the integrations are performed along the independent stochastic
trajectories that individual particles follow from their position of origin to
the position of their eventual detection after a large number of repetitions
of the experiment. We can only get a hint here about how the random phases in
the above expressions behave after a large number of repetitions of the
experiment. Their average value is equal to

$\displaystyle\langle\int\frac{m}{\hbar}\frac{{\rm d}{\bf w}_{j}}{{\rm
d}t}\cdot{\rm d}{\bf x}\rangle$ $\displaystyle\approx$
$\displaystyle\sum\nolimits\frac{m}{\hbar\ {\rm d}t}\ \langle{\rm d}{\bf
w}_{j}\cdot{\rm d}{\bf w}_{j}\rangle$ (23) $\displaystyle=$
$\displaystyle\sum\nolimits\frac{m\ 2\nu}{\hbar}=\sum\nolimits 1={\cal M}\ ,$

where ${\cal M}$ is the number of stochastic kicks that each particle
experiences along its trajectory. In other words, particles collected at the
same time instance $t$ in eq. (22), all of them experience the same number
${\cal M}$ of stochastic kicks, and therefore, the term in parentheses in eq.
(22) is expected to yield only an unimportant extra phase from the
contribution of these stochastic kicks. This result remains to be shown with
an actual numerical experiment that will be performed in a followup
publication.

In a world where all measurements are performed according to our model of a
special type of detector, an interference pattern is not something that
characterizes each individual particle. Particles have no a-priori knowledge
of their distribution in space and time during a large number of repetitions
of the experiment. This is in contrast to other non-local particle theories
like Bohmian mechanics 6 . Particles carry their own phases, follow classical
trajectories that are acted upon by external forces, and are stochastically
perturbed according to eq. (13) everytime they interact with a background, or
every time we decide to make a measurement on them. This is all there is to
it. The particles do not know anything a-priori about their distribution $n$.
In order to determine their distribution after several repetitions of the
experiment, we need to place detectors at all positions in space and time and
collect detections over and over again. The role of the detector is to collect
the phases of the particles that reach it, and process them according eq. (2).
Then we can gradually build a spatial density distribution $n$ that is
proportional to the square of the amplitude of the function $\overline{\Psi}$.

We cannot emphasize more this result. In our special world, individual
particles are not associated with any wave. Wave-like behavior emerges from
classical orbits of non-interacting particles perturbed by a large number of
stochastic interactions. This is how $n_{k}$ and ${\bf v}_{k}$ are determined
and from them $n$, ${\bf v}$ and $\Psi_{\rm Nelson}$ in eq. (21), not the
other way around by first solving Schrodinger’s equation.

##  V A speculative new interpretation of quantum mechanics

As we will see in a followup publication, our individual particle model holds
great promise in reproducing the quantum wavefunction from local first
principles. This leads us to propose a new interpretation of non-relativistic
quantum mechanics according to which particles characterized by their mass
$m$, velocity ${\rm d}{\rm{\bf x}}/{\rm d}t$, and phase $\phi$ follow
classical Brownian trajectories. When these particles start to move, they
receive some kind of ‘kick’ that ‘sets-off’ an ‘internal clock’ that records
the evolution of their phase according to eq. (9). In other words, it is the
transfer of energy and momentum to a particle that also triggers its internal
phase. An analogy may be made with a soccer ball which, when kicked, travels
in space but also oscillates internally. The same thing will happen also every
time the particle interacts stochastically with a detector.

When particles reach a detector, the detector records their phases and
processes them according to eq. (10). Particles exist and behave classically,
but manifest themselves only when they interact with a detector, or
equivalently their macroscopic classical environment. The detector records a
detection when the square of $\overline{\Psi}$ obtained from several
repetitions of the experiment reaches a certain detection threshold. In other
words, individual particles exist and travel along stochastic classical
trajectories, it is only when we decide to detect them that we observe wave-
like characteristics. Nature behaves classically, and wave-like behavior
manifests itself only in the eyes of the detector. This interpretation is
intriguing and we plan to continue investigating it.

#### Acknowledgements:

We acknowledge extensive corrections by Pr. George Contopoulos which greatly
improved the presentation of our ideas.

## References

  * (1) Ball, P., Nature, 12198 (2013)
  * (2) Basi, A. et al., Reviews of Modern Physics, 85, 471 (2013)
  * (3) Bell, J., Physics World, 3, 33 (1990)
  * (4) Feynman, R. P., Leighton, R. B., Sands, M., ‘The Feynman Lectures on Physics. Volume III, Addison-Wesley, Reading (1965)
  * (5) Jin, F., Yuan, S., De Raedt, H., Michielsen, K., Miyashita, S., Journal of the Physical Society of Japan, 79, 074401 (2010)
  * (6) Goldstein, S., in Stanford Encyclopedia of Philosophy, Zalt. E. N. (ed.), Stanford Univ. (2017)
  * (7) Harrigan, N. R., Spekkens, R. W., Foundations of Physics, 40, 125 (2010)
  * (8) Kolenderski, P. et al., Scientific Reports, 4, 4685 (2014)
  * (9) Nelson, E., Physical Review, 150, 1079 (1966)
  * (10) Jin et al. (2010) propose to process the phases of individual photon detections as $\overline{\Psi}_{N}\equiv\gamma\overline{\Psi}_{N-1}+(1-\gamma){\rm e}^{i\phi_{N}}=(1-\gamma)\sum\nolimits_{j=1}^{N}\gamma^{N-j}{\rm e}^{i\phi_{j}}$, where $0<\gamma<1$ is one parameter of their model. This procedure differs from our eq. (2). In their model, $0<|\overline{\Psi}|_{N}<1$, whereas in ours it is of order $\sqrt{N}$, i.e. it grows indefinitely as more and more photons are collected in subsequent realizations of the experiment. In order to trigger their detector, they compare $|\overline{\Psi}|^{2}_{N}$ with a random number $r_{N}$ uniformly distributed betwen 0 and 1. Although both detector models generate wave-like interference patterns, we believe that our approach is more physical. This is why we believe there is potential to use it as a new interpretation of quantum mechanical measurement.