* Tibshirani . (2011) Tibshirani.2011Tibshirani, RJ., Price, A. Taylor, J.  2011\.  A statistician plays darts A statistician plays darts. Journal of the Royal Statistical Society Series A: Statistics in Society1741213-226.
  * Yerkes  Dodson (1908) Yerkes.1908Yerkes, RM. Dodson, JD.  1908\.  The relation of strength of stimulus to rapidity of habit-formation The relation of strength of stimulus to rapidity of habit-formation. Journal of Comparative Neurology and Psychology185459-482.

##  Appendix A Appendix

###  A.1 Full descriptives

Table 7: Full descriptive statistics Panel A: Outcomes
---
Favorite wins |  | 0.665 |
Favorite 3-darts average (performance) |  | 102.254 | (9.057)
Favorite 3-darts average in first half |  | 101.687 | (11.089)
Favorite 3-darts average in second half |  | 102.892 | (11.454)
Underdog 3-darts average (performance) |  | 97.595 | (8.879)
Underdog 3-darts average in first half |  | 97.213 | (10.891)
Underdog 3-darts average in second half |  | 98.016 | (11.369)
Mean 3-darts average (performance) |  | 99.924 | (6.859)
Mean 3-darts average in first half |  | 99.450 | (8.273)
Mean 3-darts average in second half |  | 100.454 | (8.477)
Contest length (proportion of legs needed) |  | 0.790 | (0.159)
Number of max. performance (Score = 180) in contest |  | 5.921 | (4.186)
Panel B: Treatments
Ability ratio (favorite to underdog) |  | 1.055 | (0.049)
Ability difference (favorite minus underdog) |  | 4.834 | (3.997)
Odds ability difference |  | 0.350 | (0.225)
Expected ability (of opponent in next stage) |  | 94.695 | (3.963)
Underdog starts contest |  | 0.481 |
Panel C: Covariates
Favorite ability |  | 94.624 | (3.867)
Underdog ability |  | 89.791 | (4.600)
Opponent is known |  | 0.588 |
Favorite starts contest |  | 0.519 |
Favorite world ranking |  | 0.017 | (0.081)
Underdog world ranking |  | 0.046 | (0.129)
Favorite years playing darts (experience) |  | 21.333 | (8.921)
Underdog years playing darts (experience) |  | 19.766 | (10.493)
Favorite performs at home event |  | 0.034 |
Underdog performs at home event |  | 0.041 |
Prize money (standardized) |  | 0.217 | (0.330)
Notes: Mean and standard deviations (in parentheses; for non-binary
variables).

###  A.2 Additional results

Figure 6: Favorite win, future opponent

Notes: Non-parametric kernel regression. Based on Table 11, Column 2.
The blue line shows the expected winning probability for different
levels of the expected ability of the next opponent. Dashed lines
show the 90% confidence intervals.

###  A.3 Full result

####  A.3.1 Results with regard to current opponents

Table 8: First and second half performance – full table | Favorite |  | Underdog
---|---|---|---
| performance in |  | performance in
|  first half | second half |  |  first half | second half
Ability ratio1 | 12.303*** | -2.035 |  | -13.349*** | -17.198***
| (3.492) | (3.541) |  | (4.088) | (3.939)
Favorite starts first leg | 0.152 | 0.297 |  | -0.436 | -0.921***
| (0.312) | (0.367) |  | (0.285) | (0.322)
Favorite world ranking | -5.086 | 3.171 |  | -3.056* | 1.205
| (3.677) | (4.116) |  | (1.763) | (2.137)
Underdog world ranking | -0.747 | 0.231 |  | -3.122* | -3.350
| (1.296) | (1.394) |  | (1.781) | (2.096)
Favorite experience2 | 2.379** | 2.310* |  | 0.026 | 0.027
| (0.970) | (1.183) |  | (0.017) | (0.019)
Underdog experience | 0.019 | -0.004 |  | 3.676*** | 0.717
| (0.013) | (0.017) |  | (1.323) | (1.474)
Favorite at home event | -0.249 | 1.177 |  | 0.999 | 1.106
| (0.712) | (0.933) |  | (0.750) | (0.872)
Underdog at home event | -0.210 | -0.310 |  | -0.129 | 0.760
| (0.794) | (0.780) |  | (0.940) | (1.043)
Stage FE | x | x |  | x | x
Tournament-by-Year FE | x | x |  | x | x
Individual FE | x | x |  | x | x
N | 4776 | 4776 |  | 4776 | 4776
Notes: *, **, and *** represents statistical significance at the 10 %, 5 %,
and 1 %, respectively.
1 Ratio is favorite / underdog, ability measured as 3-darts average over the
past two years.
2 Experience is measured as number of years playing darts. FE = fixed effects.

####  A.3.2 Contest outcomes

Table 9: Contest outcomes – full table |  Favorite |  |  Legs
---|---|---|---
|  win |  | needed
| (1) | (2) | (3) |  | (4) | (5)
Ability ratio1 | 1.420*** | 1.423*** | 1.445*** |  | -0.346*** | -0.346***
| (0.156) | (0.159) | (0.143) |  | (0.053) | (0.053)
Favorite world ranking | 0.155 | 0.152 | -0.097 |  | -0.003 | -0.002
| (0.110) | (0.113) | (0.079) |  | (0.052) | (0.052)
Underdog world ranking | 0.174*** | 0.171*** | 0.160*** |  | 0.016 | 0.015
| (0.049) | (0.049) | (0.045) |  | (0.017) | (0.016)
Favorite starts first leg |  | 0.075*** | 0.081*** |  |  | -0.009**
|  | (0.017) | (0.014) |  |  | (0.005)
Favorite experience2 |  | -0.042 | -0.002** |  |  | 0.004
|  | (0.102) | (0.001) |  |  | (0.025)
Underdog experience |  | 0.001 | 0.001 |  |  | -0.000
|  | (0.001) | (0.001) |  |  | (0.000)
Favorite at home event |  | 0.024 | 0.035 |  |  | 0.016*
|  | (0.034) | (0.035) |  |  | (0.009)
Underdog at home event |  | 0.017 | 0.013 |  |  | -0.007
|  | (0.031) | (0.029) |  |  | (0.011)
Favorite ability |  |  | 0.012*** |  |  |
|  |  | (0.002) |  |  |
Stage FE | x | x |  |  | x | x
Tournament-by-Year FE | x | x |  |  | x | x
Individual FE | x | x |  |  | x | x
N | 4776 | 4776 | 4776 |  | 4776 | 4776
Notes: *, **, and *** represents statistical significance at the 10 %, 5 %,
and 1 %, respectively.
1 Ratio is favorite / underdog, ability measured as 3-darts average over the
past 2 years.
2 Experience is measured as number of years playing darts. FE = fixed effects.
In Column 2, both underdog and favorite fixed effects are used.

####  A.3.3 Full result table for head start

Table 10: Underdog head start – full table |  Favorite |  | Performance
---|---|---|---
| win |  | Favorite |  | Underdog |  | Average
Underdog head start | -0.075*** |  | -0.228 |  | 0.688*** |  | 0.277
| (0.017) |  | (0.280) |  | (0.202) |  | (0.169)
Ability ratio | 1.423*** |  | 5.738** |  | -15.075*** |  | -4.506
| (0.159) |  | (2.569) |  | (3.006) |  | (2.790)
Favorite rank | 0.152 |  | -0.920 |  | -1.078 |  | -1.593
| (0.113) |  | (1.672) |  | (1.257) |  | (1.476)
Underdog rank | 0.171*** |  | -0.238 |  | -3.297** |  | -1.236
| (0.049) |  | (1.000) |  | (1.547) |  | (1.042)
Favorite experience | -0.042 |  | 2.373*** |  | 0.028** |  | 1.025
| (0.102) |  | (0.689) |  | (0.013) |  | (0.839)
Underdog experience | 0.001 |  | 0.008 |  | 2.257** |  | 0.977***
| (0.001) |  | (0.012) |  | (1.049) |  | (0.320)
Favorite home | 0.024 |  | 0.417 |  | 1.062** |  | 0.838*
| (0.034) |  | (0.601) |  | (0.501) |  | (0.443)
Underdog home | 0.017 |  | -0.265 |  | 0.338 |  | 0.006
| (0.031) |  | (0.626) |  | (0.814) |  | (0.508)
|  |  |  |  |  |  |
Stage FE | x |  | x |  | x |  | x
Tournament-by-Year FE | x |  | x |  | x |  | x
Individual FE | x |  | x |  | x |  | x
Notes: *, **, and *** represents statistical significance at the 10 %, 5 %,
and 1 %, respectively.
Standard errors are clustered at the individual level or robust for average
performance.
favorite and underdog FE for average performance. FE = fixed effects. Linear
regression.

####  A.3.4 Results with regard to future opponents

Table 11: Future contest opponents – full table |  Favorite |  | Performance
---|---|---|---
| win |  | Favorite |  | Underdog |  | Average
| (1) | (2) |  | (3) |  | (4) |  | (5)
Expected ability1 | -0.004* | -0.003* |  | -0.063* |  | 0.066 |  | -0.019
| (0.002) | (0.002) |  | (0.032) |  | (0.043) |  | (0.028)
Favorite world ranking | 0.111 | -0.208** |  | -1.932 |  | -1.186 |  | -2.796**
| (0.117) | (0.104) |  | (1.764) |  | (1.377) |  | (1.243)
Underdog world ranking | 0.251*** | 0.219** |  | 0.142 |  | -3.194** |  | -1.223
| (0.052) | (0.048) |  | (1.065) |  | (1.556) |  | (1.231)
Favorite starts first leg | 0.078*** | 0.083*** |  | 0.237 |  | -0.603*** |  | -0.206
| (0.017) | (0.014) |  | (0.288) |  | (0.217) |  | (0.211)
Favorite experience2 | -0.021 | -0.002 |  | 3.370*** |  | 0.030** |  | 1.777*
| (0.100) | (0.001) |  | (0.816) |  | (0.013) |  | (0.921)
Underdog experience | 0.000 | 0.001 |  | 0.011 |  | 2.693** |  | 0.979***
| (0.001) | (0.001) |  | (0.012) |  | (1.154) |  | (0.336)
Favorite at home event | 0.026 | 0.022 |  | 0.296 |  | 1.115** |  | 0.752
| (0.037) | (0.036) |  | (0.602) |  | (0.555) |  | (0.550)
Underdog at home event | 0.029 | 0.018 |  | 0.024 |  | 0.235 |  | 0.010
| (0.035) | (0.033) |  | (0.647) |  | (0.846) |  | (0.673)
Stage FE | x |  |  | x |  | x |  | x
Tournament-by-Year FE | x |  |  | x |  | x |  | x
Individual FE | x |  |  | x |  | x |  | x
N | 4776 | 4776 |  | 4776 |  | 4776 |  | 4776
Notes: *, **, and *** represents statistical significance at the 10 %, 5 %,
and 1 %, respectively.
1 If the next opponent is known, this is their ability, otherwise it is the
ability of the
favorite of the contest that will determine the next opponent. 2 Experience is
measured
as number of years playing darts. FE = fixed effects.
Table 12: Future contest opponents – 2SLS first stage Opponent known before | -2.031
---|---
| (0.108)
| [-18.75]
Stage FE | x
Tournament-by-Year FE | x
N | 4448
Notes: Standard error (clustered on the individual level) in
parenthesis. T-value in brackets. FE = fixed effects.

###  A.4 Robustness

In Section 5.1, we showed that the effect of increased ability heterogeneity
on performance differs when focusing on higher-ability or lower-ability
contestants. This section aims to confirm the consistency of this result
across various specifications.

A potential concern regarding our results is the definition of the treatment
variable. Section 5.1 defines the treatment as the ability ratio between the
higher-ability contestant and the lower-ability contestant based on their past
performances. There are two potential concerns: 1) Dependence on the
functional form of the treatment variable and 2) Measurement error in the
treatment variable. To address these concerns, we conduct three robustness
checks. First, in Column 1 of Table 13, we demonstrate that our results remain
unchanged when we measure heterogeneity using a different functional form,
i.e., the difference in the logarithm of the past performances between
contestants instead of the ability ratio.

Second, our treatment variable could have a potential measurement error, given
that the contestant’s ability is approximated by the performance over the past
two years. Thus, we redefine our treatment variable based on bookmakers’ odds,
which are determined just before the contest starts.232323However, this
approach has a caveat. The bookmakers’ odds are recorded after the starting
contestants are known, meaning that the advantage of starting the contest is
already incorporated into the odds Goller (2023). Consequently, there is some
misspecification in identifying the ex-ante higher-ability contestant and the
ex-ante ability ratio. Additionally, since the bookmakers’ prediction model is
unknown, it is possible that the ability ratio itself was used to formulate
the prediction. Therefore, we do not use the bookmaker odds in our preferred
specification. The results presented in Column 2 of Table 13 align with our
main specifications. Third, we present a robustness check in which we remove
the highest 1% of the treatment values, which helps to address potential
outliers in the data. We present the results in Column 3 of Table 13. Removing
these observations slightly increases the point estimates for the higher-
ability contestants.

Our identification strategy relies on the tournament draw, which generates
"quasi-random" variation in opponent ability given a contestant’s own ability.
However, a potential concern arises regarding the adequacy of controlling for
individual ability using individual fixed effects, as contestant ability might
vary over time. To address this concern, we implement several robustness
checks. First, we incorporate individual-by-year fixed effects as a control
variable instead of individual fixed effects. By doing so, we account for
potential changes in individual ability over time. The robustness check in
Column 5 of Table 13 demonstrates that our main findings remain robust even
when employing individual-by-year fixed effects. Furthermore, we conduct an
additional robustness check, in which we directly control for ability by
including information on past ability over the last two years as a covariate
(Table 13, Column 6).

Another concern about our identification strategy is that while the assignment
of contestants via the tournament draw generates "quasi-random" variation in
the opponents’ ability and thus contestant heterogeneity in the first stage of
the tournament, the heterogeneity among contestants in the upcoming stages is
to some degree, but potentially not entirely exogenous. To address this
concern, we show in Column 4 of Table 13 that our results are unchanged when
we drop the last two stages of each tournament in our sample. We also show a
robustness check in which we add the ability of the expected (or realized)
next opponent as an additional control variable to our model. Our results are
not affected by this extension (Table 13, Column 7).

A final concern may relate to our outcome variable. In our main analysis, we
use the 3-darts average over the first nine darts as our performance measure.
We chose this measure for the following reasons: 1) We can observe the
performance in the same task for the higher-ability and lower-ability
contestants 2) The dominant strategy within the first nine darts is to score
the maximum number of points possible, and there are no strategic
considerations. Strategic choices solely arise as soon as a contestant ’sets’
the remaining number of points to be scored to a certain value by a throw that
is not aimed at the maximum point yield. However, even though only a very
small number of legs end with nine darts, there exists the possibility to
finish a leg with exactly nine darts or that the ninth dart is already
strategically played to ease the "double out". Therefore, we conduct a
robustness check in which we redefine our outcome variable. In Column 8 of
Table 13, we show results using the 3-darts average over the first six darts
(two moves) as an alternative outcome variable. Our results are robust to this
alternative definition of the outcome variable.

Lastly, we analyze alternative outcome variables in Table 14. These variables
include the number of times an individual scores more than 100, 140, and
precisely 180 points with three darts per leg. Professional darts commonly use
these measures to evaluate contestants’ performance, with 180 points being the
highest attainable score. By examining these alternative metrics, we verify
the consistency of our main findings across different performance indicators.
Table 14 confirms our main findings. The lower-ability contestant experiences
substantial adverse effects across all outcome variables. Conversely, the
higher-ability contestant has positive effects in all specifications, with
statistical significance observed only in the case of scoring more than 100
points per move.

We conclude that our main results are robust to different outcomes, treatment
definitions, and alternative ways of controlling the individuals’ ability.

Table 13: Robustness checks | Treatment |  | Sample |  | Controls |  | Outcome
---|---|---|---|---|---|---|---
| (1) | (2) |  | (3) | (4) |  | (5) | (6) | (7) |  | (8)
|  log spec.1 |  odds spec.1 |  | w/o outliers | w/o (s.)finals |  | ind x yr FE |  ability2 | \+ future abil. |  | 6 darts av.
Panel A: Performance in contest
A.1 Favorite’s performance
|  |  |  |  |  |  |  |  |  |  |
Ability ratio | 6.392** | 0.297*** |  | 7.145** | 5.571** |  | 5.584* | 3.934 | 6.451** |  | 7.663***
| (2.843) | (0.045) |  | (2.859) | (2.739) |  | (3.291) | (2.464) | (2.742) |  | (2.919)
|  |  |  |  |  |  |  |  |  |  |
A.2 Underdog’s performance
|  |  |  |  |  |  |  |  |  |  |
Ability ratio | -16.351*** | -0.209*** |  | -13.662*** | -15.141*** |  | -11.156** | -9.725** | -14.737*** |  | -15.263***
| (3.279) | (0.059) |  | (3.211) | (3.058) |  | (5.397) | (3.737) | (3.240) |  | (3.574)
|  |  |  |  |  |  |  |  |  |  |
Panel B: Performance in first half of contest
B.1 Favorite’s performance
|  |  |  |  |  |  |  |  |  |  |
Ability ratio | 13.686*** | 0.377*** |  | 14.908*** | 11.877*** |  | 14.412*** | 10.967*** | 13.085*** |  | 15.103***
| (3.869) | (0.077) |  | (3.775) | (3.618) |  | (4.180) | (3.508) | (3.668) |  | (4.339)
|  |  |  |  |  |  |  |  |  |  |
B.2 Underdog’s performance
|  |  |  |  |  |  |  |  |  |  |
Ability ratio | -14.674*** | -0.277*** |  | -13.230*** | -13.257*** |  | -13.441** | -10.751** | -12.319*** |  | -10.931***
| (4.674) | (0.078) |  | (4.514) | (4.184) |  | (6.659) | (4.881) | (4.427) |  | (4.704)
|  |  |  |  |  |  |  |  |  |  |
N | 4776 | 4769 |  | 4728 | 4433 |  | 4776 | 4776 | 4448 |  | 4776
Notes: *, **, and *** represents statistical significance at the 10 %, 5 %,
and 1 %, respectively. Linear Regression. All regressions include stage,
tournament,
and individual fixed effects (FE), and all covariates; unless otherwise noted.
Standard errors are clustered at the individual level. 1 The treatment is
specified as differences instead of ratio. 2 Controlled for ability instead of
Individual FE.
Table 14: Results for within contest performance, 3-darts scores per leg |  $>$100 per leg |  |  $>$140 per leg |  | 180 per leg
---|---|---|---|---|---
| (1) |  | (2) |  | (3)
Panel A: Favorite’s performance
|  |  |  |  |
Ability ratio | 0.333*** |  | 0.111 |  | 0.045
| (0.111) |  | (0.077) |  | (0.056)
|  |  |  |  |
Panel B: Underdog’s performance
|  |  |  |  |
Ability ratio | -0.578*** |  | -0.394*** |  | -0.162***
| (0.115) |  | (0.099) |  | (0.053)
|  |  |  |  |
Panel C: Mean performance
|  |  |  |  |
Ability ratio | -0.172 |  | -0.142* |  | -0.094*
| (0.109) |  | (0.083) |  | (0.050)
|  |  |  |  |
Stage FE | x |  | x |  | x
Tournament-by-Year FE | x |  | x |  | x
Individual FE | x |  | x |  | x
All Covariates | x |  | x |  | x
N | 4776 |  | 4776 |  | 4776
Notes: *, **, and *** represents statistical significance at the 10 %, 5 %,
and 1 %, respectively.
Linear Regression. Standard errors are clustered at the individual level. The
outcome variables
count the number of times an individual scores more than 100 (col. 1), more
than 140 (col. 2)
and exactly 180 (col. 3) with three darts in a leg. FE = fixed effects.


# GUIDELINES ON WRITING A GOOD PAPER FOR THE
PROCEEDINGS OF THE WINTER SIMULATION CONFERENCE

James R. Wilson
North Carolina State University
Edward P. Fitts Department of Industrial and Systems Engineering
Raleigh, NC 27695-7906, USA

## ABSTRACT

As an aid to authors who seek to improve the clarity and readability of their
papers in the Proceedings of the Winter Simulation Conference, this paper
summarizes some useful guidelines on technical writing, including current
references on each topic that is discussed.

##  1 OUTLINE OF KEY CONSIDERATIONS

Writing a clear, readable exposition of complex technical work is at least as
difficult as doing the work in the first place. Given below is an outline of
key considerations to bear in mind during all stages of writing a paper that
will be reviewed for possible presentation at the Winter Simulation Conference
(WSC) as well as publication in the Proceedings of the Winter Simulation
Conference. For questions about these guidelines, please send e-mail to
<EMAIL_ADDRESS>or contact the proceedings editors.

  1. I.

Organizing the paper (what to do before beginning to write)

    1. A.

Analyze the situation—that is, the problem, the solution, and the target
audience.

      1. 1.

Formulate the objectives of the paper.

      2. 2.

Specify the scope of the paper’s coverage of the subject and the results to be
discussed. Orient the paper toward the theme of your session as indicated
either by the title of your session or by the instructions of your session
chair. Also take into account the general focus of the track containing your
paper, which could be tutorials, case studies, vendors, methodologies, domain-
specific applications, or general applications.

      3. 3.

Identify the target audience and determine the background knowledge that you
can assume for this particular group of people. Introductory tutorials are
generally attended by newcomers who are interested in the basics of
simulation. Advanced tutorials are designed to provide more experienced
professionals with a thorough discussion of special topics of much current
interest; and some special-focus sessions in this track are designed to
provide experts with an overview of recent fundamental advances in simulation
theory. Methodology sessions are attended by professionals who have at least
an undergraduate-level background in computer simulation techniques. In the
case studies and applications tracks, session attendees are generally familiar
with the area covered by their session. Vendor sessions may contain both new
and experienced users of the relevant software products.

      4. 4.

Formulate the most logical sequence for presenting the information specified
in item 2 to the readers identified in item 3. For a discussion of effective
aids in organizing your paper (specifically, brainstorming, clustering, issue
trees, and outlining), see chapter 3 of Matthews and Matthews (2014). In
structuring your presentation, keep the following points in mind.

        1. a.

Introductory and advanced tutorials should have an educational perspective.
Within the advanced tutorials track, special-focus sessions should synthesize
the latest research results in a unified treatment of a given topic.

        2. b.

Methodology contributions should provide state-of-the-art information on
proven techniques for designing, building, and analyzing simulation models.

        3. c.

Application papers should relate directly to the practice of simulation, and
they should emphasize lessons of transferable value.

    2. B.

Make outlines to organize your thoughts and then to plan both the written and
oral presentations of your work. For excellent discussions of the construction
and use of various types of outlines, see the following: chapter 1 of Menzel,
Jones, and Boyd (1961); the sections titled “Develop an issue tree to assess
presentation balance” and “Outline to develop the paper’s framework” in
chapter 3 of Matthews and Matthews (2014); and chapter 3 of Pearsall and Cook
(2010).

      1. 1.

The introductory paragraph(s)

        1. a.

State the precise subject of the paper immediately.

        2. b.

State the problem to be solved.

        3. c.

Summarize briefly the main results and conclusions.

        4. d.

Tell the reader how the paper is organized.

      2. 2.

The main body of the paper

        1. a.

Include enough detail in the main body of the paper so that the reader can
understand what you did and how you did it; however, you should avoid lengthy
discussions of technical details that are not of general interest to your
audience.

        2. b.

Include a brief section covering notation, background information, and key
assumptions if it is awkward to incorporate these items into the introductory
paragraph(s).

        3. c.

Include sections on theoretical and experimental methods as required. For an
application paper, you should discuss the development of the simulation
model—including input data acquisition as well as design, verification,
validation, and actual use of the final simulation model. For a methodological
or theoretical paper that requires substantial mathematical development, see
Halmos (1970), Higham (1998), pages 1–8 of Knuth, Larrabee, and Roberts
(1989), Krantz (1997, 2001), or Swanson (1999). For standard mathematical
notation used in engineering and the sciences, see ISO 80000-2: Quantities and
Units—Part 2: Mathematical Signs and Symbols to Be Used in the Natural
Sciences and Technology (ISO 2009) and Scheinerman (2011).

        4. d.

Plan the results section to achieve the most effective mix of text, figures,
and tables in the presentation of the findings. The definitive reference on
the design of tables and figures is Tufte (2001).

      3. 3.

The concluding paragraph(s)

        1. a.

Explain how the theoretical and experimental results relate to the original
problem. State why these results are important.

        2. b.

Summarize any unresolved issues that should be the subject of future work.

        3. c.

State the final conclusions explicitly in plain language.

  2. II.

Writing the paper

    1. A.

Prepare an abstract that is concise, complete in itself, and intelligible to a
general reader in the field of simulation. The abstract may not exceed 150
words, and it should not contain any references or mathematical symbols.

      1. 1.

Summarize the objectives of the paper.

      2. 2.

Summarize the results and conclusions.

      3. 3.

State the basic principles underlying any new theoretical or experimental
methods that are developed in the paper.

      4. 4.

For complete instructions on the preparation of scientific abstracts, see
Guidelines for Abstracts (NISO 2010), pages 91–93 of Carter (1987), page 5 of
the AIP Style Manual (AIP 1990), or chapter 9 of Gastel and Day (2016).

    2. B.

Write the rest of the paper as though you were talking to a group of
interested colleagues about your work.

      1. 1.

Strive for accuracy and clarity above all else.

      2. 2.

In writing the introduction, you should remember the following maxim:

> The opening paragraph should be your best paragraph, and its opening
> sentence should be your best sentence. (Knuth, Larrabee, and Roberts 1989,
> 5)

You cannot achieve such an ambitious goal on the first try; instead as you add
new sections to the paper, you should review and revise all sections written
so far. For more on the spiral plan of writing, see pages 131–133 of Halmos
(1970).

        1. a.

Like the abstract, the introduction should be accessible to general readers in
the field of simulation.

        2. b.

For methodology papers and advanced tutorials, substantially more advanced
background may be assumed in the sections following the introduction.

      3. 3.

In constructing each sentence, place old and new information in the respective
positions where readers generally expect to find such information. For an
excellent discussion of the principles of scientific writing based on reader
expectations, see Gopen and Swan (1990) and Williams and Bizup (2014, 2017).

        1. a.

Place in the topic position (that is, at the beginning of the sentence) the
old information linking backward to the previous discussion.

        2. b.

Place in the stress position (that is, at the end of the sentence) the new
information you want to emphasize.

        3. c.

Place the subject of the sentence in the topic position, and follow the
subject with the verb as soon as possible.

        4. d.

Express the action of each sentence in its verb.

      4. 4.

Make the paragraph the unit of composition.

        1. a.

Begin each paragraph with a sentence that summarizes the topic to be discussed
or with a sentence that helps the transition from the previous paragraph.

        2. b.

Provide a context for the discussion before asking the reader to consider new
information.

        3. c.

Avoid paragraphs of extreme length—that is, one-sentence paragraphs and those
exceeding 200 words.

        4. d.

Place the important conclusions in the stress position at the end of the
paragraph.

      5. 5.

Allocate space to a topic in proportion to its relative importance.

      6. 6.

For methodology papers, emphasize the concepts of general applicability that
underlie the solution procedure rather than the technical details that are
specific to the problem at hand. Supply only the technical details and data
that are essential to the development.

      7. 7.

For application papers, emphasize the new insights into the problem that you
gained from designing, building, and using the simulation model.

      8. 8.

Use standard technical terms correctly.

        1. a.

For standard usage of mathematical terms, see James and James (1992) and
Borowski and Borwein (2002). For example, a nonsquare matrix cannot be called
“orthogonal” even if any two distinct columns of that matrix are orthogonal
vectors.

        2. b.

For standard usage of statistical terms, see Dodge (2003), Porkess (2005), and
Upton and Cook (2014). For example, the probability density function of a
continuous random variable cannot be called a “probability mass function.”

        3. c.

For standard usage of computer terms, see The Free On-Line Dictionary of
Computing (Howe 1993) and Dictionary of Algorithms and Data Structures (Black
1998).

        4. d.

For standard usage of industrial engineering terms, see Industrial Engineering
Terminology (IISE 2000). For example, the time that a workpiece spends in a
manufacturing cell may be called “cycle time” or “flow time” but not
“throughput time.”

      9. 9.

Avoid illogical or potentially offensive sexist language. See Miller and Swift
(2001) for a commonsense approach to this issue.

      10. 10.

Strictly avoid the following—

        1. a.

religious, ethnic, or political references;

        2. b.

personal attacks;

        3. c.

excessive claims about the value or general applicability of your work; and

        4. d.

pointed criticism of the work of other people.

Such language has no place in scientific discourse under any circumstances,
and it will not be tolerated by the proceedings editors. With respect to
vendor sessions, items c and d immediately above require authors to avoid
invidious comparisons of their products with competing products.

      11. 11.

In writing the final section of the paper containing conclusions and
recommendations for future work, you should keep in mind the following maxim:

> The mark of a good summary is revelation: “Remember this, reader? And that?
> Well, here’s how they fit together.” (van Leunen 1992, 116)

    3. C.

For each table, compose a caption that briefly summarizes the content of the
table. Comment explicitly in the text on the significance of the numbers in
the table; do not force the reader to guess at your conclusions. See sections
3.46–3.85 of The Chicago Manual of Style (University of Chicago Press 2010) or
chapter 16 of Gastel and Day (2016) for a comprehensive discussion of how to
handle tables.

    4. D.

For each figure, compose a caption (or legend) that explains every detail in
the figure—every curve, point, and symbol. See the AIP Style Manual (AIP 1990)
or chapters 17 and 18 of Gastel and Day (2016) for excellent examples.

    5. E.

Revise and rewrite until the truth and clarity of every sentence are
unquestionable.

      1. 1.

For questions about the rules of English grammar and usage, see Bernstein
(1965), Butterfield (2015), Fowler and Aaron (2016), Garner (2016), Hale
(2013), O’Conner (2009), Strunk and White (2000), the Oxford English
Dictionary (Simpson and Weiner 1989), and Webster’s Third New International
Dictionary of the English Language, Unabridged (Gove 1993).

      2. 2.

For those who use English as a second language, particularly helpful
references are Booth (1993), Fowler and Aaron (2016), Huckin and Olsen (1991),
and Yang (1995).

      3. 3.

For guidelines on how to edit your own writing effectively, see Cook (1985).

      4. 4.

For a comprehensive discussion of all aspects of scientific writing, see Alley
(1996) and Gastel and Day (2016).

    6. F.

Prepare a complete and accurate set of references that gives adequate credit
to the prior work upon which your paper is based.

      1. 1.

The author-date system of documentation is required for all papers appearing
in the Proceedings of the Winter Simulation Conference. Chapter 15 of The
Chicago Manual of Style (University of Chicago Press 2010) provides
comprehensive, up-to-date information on this citation system.

      2. 2.

In preparing your list of references, you should strive for completeness,
accuracy, and consistency. Using the information provided in your list of
references, the interested reader should be able to locate each source of
information cited in your paper.

      3. 3.

For complete instructions on citing electronic sources, see sections
14.4–14.13 of The Chicago Manual of Style (University of Chicago Press 2010).
For example, sections 14.5 and 14.6 contain basic information on uniform
resource locators (URLs) and Digital Object Identifiers (DOIs), respectively;
and section 14.12 provides useful rules for breaking a URL or a DOI across two
or more lines either in the text or in the list of references. Many specific
examples of citations for various types of electronic sources can be found
throughout chapters 14 and 15 of The Chicago Manual of Style (University of
Chicago Press 2010).

      4. 4.

The final electronic version of your paper—that is, the portable document
format (PDF) file ultimately produced from the Word or LaTeX source file of
your paper—may include external hyperlinks referring to some of the electronic
sources cited in the paper that are accessible online.

        1. a.

If an external hyperlink is live, then it is colored blue; and when viewing
the PDF file of your paper on a computer, the reader may select (click) that
hyperlink for immediate online access to the cited material. More
specifically, selecting (clicking) a live external hyperlink will activate the
reader’s web browser so that, if all goes well, the cited source of
information will be displayed in the web browser. A live external hyperlink
may also be used to activate the reader’s e-mail software for sending a
message to a specific e-mail address; for example, see the hyperlink given in
the first paragraph of this document.

        2. b.

If an external hyperlink is not live, then it is colored black; and such a
hyperlink merely displays the URL or DOI of the cited material without
providing a mechanism for immediate online access to that material.

        3. c.

If you use external hyperlinks in your paper, then you must ensure that the
text displayed for each external hyperlink is correct and complete so that a
reader who has only a hard copy of the paper can still access the cited
material by (carefully) typing the relevant displayed text of the hyperlink
into the address bar of a web browser or e-mail program. Remember that your
responsibility for the accuracy and completeness of each hyperlink in your
paper parallels your responsibility for the accuracy and completeness of each
conventional citation of a nonelectronic source—neither the editors nor the
publisher of the proceedings can verify any of this information for you.

    7. G.

See Wilson (2002) for a discussion of the following ethical and “strategic”
considerations in writing a scientific paper that will be considered for
publication in a peer-reviewed journal or conference proceedings such as the
Proceedings of the Winter Simulation Conference:

      1. 1.

achieving a consensus among collaborators on who should be a coauthor of the
paper;

      2. 2.

achieving a consensus among coauthors on the order of authorship in the
paper’s byline; and

      3. 3.

writing the paper so as to anticipate and answer key questions that will be
asked by the paper’s referees and readers.

  3. III.

Achieving a natural and effective style

    1. A.

Alfred North Whitehead memorably expressed the gist of the matter of writing
style:

> Finally, there should grow the most austere of all mental qualities; I mean
> the sense for style. It is an aesthetic sense, based on admiration for the
> direct attainment of a foreseen end, simply and without waste. Style in art,
> style in literature, style in science, style in logic, style in practical
> execution have fundamentally the same aesthetic qualities, namely attainment
> and restraint. The love of a subject in itself and for itself, where it is
> not the sleepy pleasure of pacing a mental quarter-deck, is the love of
> style as manifested in that study.
>
> Here we are brought back to the position from which we started, the utility
> of education. Style, in its finest sense, is the last acquirement of the
> educated mind; it is also the most useful. It pervades the whole being. The
> administrator with a sense for style hates waste; the engineer with a sense
> for style economises his material; the artisan with a sense for style
> prefers good work. Style is the ultimate morality of mind. (Whitehead 1929,
> 12)

Kurt Vonnegut made the following equally trenchant observation on writing
style.

> Find a subject you care about and which you in your heart feel others should
> care about. It is this genuine caring, and not your games with language,
> which will be the most compelling and seductive element in your style.
> (Vonnegut 1985, 34)

Strunk and White (2000), Williams and Bizup (2014, 2017), and Zinsser (2006)
are excellent references on achieving a natural and effective writing style.

    2. B.

Contrast the following descriptions of an experiment in optics:

      1. 1.

I procured a triangular glass prism, to try therewith the celebrated phenomena
of colors. And for that purpose, having darkened my laboratory, and made a
small hole in my window shade, to let in a convenient quantity of the sun’s
light, I placed my prism at the entrance, that the light might be thereby
refracted to the opposite wall. It was at first a very pleasing diversion to
view the vivid and intense colors produced thereby.

      2. 2.

For the purpose of investigating the celebrated phenomena of chromatic
refrangibility, a triangular glass prism was procured. After darkening the
laboratory and making a small aperture in an otherwise opaque window covering
in order to ensure that the optimum quantity of visible electromagnetic
radiation (VER) would be admitted from solar sources, the prism was placed in
front of the aperture for the purpose of reflecting the VER to the wall on the
opposite side of the room. It was found initially that due to the vivid and
intense colors which were produced by this experimental apparatus, the overall
effect was aesthetically satisfactory when viewed by the eye.

The most striking difference between these two accounts of the experiment is
the impersonal tone of the second version. According to version 2, literally
nobody performed the experiment. Attempting to avoid the first person, the
author of version 2 adopted the third person; this in turn forced the author
to use passive verbs. As Menzel, Jones, and Boyd (1961, 79) point out,
“Passive verbs increase the probability of mistakes in grammar; they start
long trains of prepositional phrases; they foster circumlocution; and they
encourage vagueness.” Notice the dangling constructions in the second sentence
of version 2. Version 1 was written by Isaac Newton (1672, 3076). Even though
it was written over 340 years ago, Newton’s prose is remarkable for its
clarity and readability.

    3. C.

To achieve a natural and effective writing style, you should adhere to the
following principles that are elaborated in chapter 5 of Menzel, Jones, and
Boyd (1961):

      1. 1.

Write simply.

      2. 2.

Use the active voice.

      3. 3.

Use plain English words rather than nonstandard technical jargon or foreign
phrases.

      4. 4.

Use standard technical terms correctly.

      5. 5.

Avoid long sentences and extremely long (or short) paragraphs.

      6. 6.

Avoid slavish adherence to any set of rules for technical writing, including
the rules enumerated here.

      7. 7.

Remember that the main objective is to communicate your ideas clearly to your
audience.

##  2 SUMMARY

In writing a paper for publication in the Proceedings of the Winter Simulation
Conference, the author should keep in mind the key considerations outlined in
this paper. Questions and suggestions for improvement of this document are
welcome.

## ACKNOWLEDGMENTS

These guidelines are based on a similar document prepared by James O.
Henriksen, Stephen D. Roberts, and James R. Wilson for the Proceedings of the
1986 Winter Simulation Conference.

## REFERENCES

  * AIP (American Institute of Physics). 1990. AIP Style Manual. 4th ed. New York: AIP. http://kmh-lanl.hansonhub.com/AIP˙Style˙4thed.pdf.

  * Alley, M. 1996. The Craft of Scientific Writing. 3rd ed. New York: Springer.

  * Bernstein, T. M. 1965. The Careful Writer: A Modern Guide to English Usage. New York: Atheneum.

  * Black, P. E., ed. 1998. Dictionary of Algorithms and Data Structures. Washington, DC: National Institute of Standards and Technology. http://www.nist.gov/dads.

  * Booth, V. 1993. Communicating in Science: Writing a Scientific Paper and Speaking at Scientific Meetings. 2nd ed. Cambridge: Cambridge University Press.

  * Borowski, E. J., and J. M. Borwein. 2002. Collins Web-linked Dictionary of Mathematics. 2nd ed. Glasgow: HarperCollins.

  * Butterfield, J. 2015. Fowler’s Dictionary of Modern English Usage. 4th ed. New York: Oxford University Press.

  * Carter, S. P. 1987. Writing for Your Peers: The Primary Journal Paper. New York: Praeger.

  * Cook, C. K. 1985. Line by Line: How to Improve Your Own Writing. Boston: Houghton Mifflin Co.

  * Dodge, Y. ed. 2003. The Oxford Dictionary of Statistical Terms. 6th ed. New York: Oxford University Press.

  * Fowler, H. R., and J. E. Aaron. 2016. The Little, Brown Handbook. 13th ed. Boston: Pearson.

  * Garner, B. A. 2016. The Chicago Guide to Grammar, Usage, and Punctuation. Chicago: University of Chicago Press.

  * Gastel, B., and R. A. Day. 2016. How to Write and Publish a Scientific Paper. 8th ed. Santa Barbara, CA: Greenwood.

  * Gopen, G. D., and J. A. Swan. 1990. “The Science of Scientific Writing.” American Scientist 78(6):550–558.

  * Gove, P. B. 1993. Webster’s Third New International Dictionary of the English Language, Unabridged. Springfield, MA: Merriam-Webster. http://unabridged.merriam-webster.com.

  * Hale, C. 2013. Sin and Syntax: How to Craft Wickedly Effective Prose. New York: Three Rivers Press.

  * Halmos, P. R. 1970. “How to Write Mathematics.” L’Enseignement Mathématique 16(2):123–152. http://www4.ncsu.edu/~jwilson/files/halmos70em.pdf.

  * Higham, N. J. 1998. Handbook of Writing for the Mathematical Sciences. 2nd ed. Philadelphia: Society for Industrial and Applied Mathematics.

  * Howe, D., ed. 1993. The Free On-Line Dictionary of Computing. London: Imperial College Department of Computing. http://foldoc.org.

  * Huckin, T. N., and L. A. Olsen. 1991. Technical Writing and Professional Communication for Nonnative Speakers of English. 2nd ed. New York: McGraw-Hill.

  * IISE (Institute of Industrial and Systems Engineers). 2000. Industrial Engineering Terminology. Rev. ed. Norcross, GA: Institute of Industrial Engineers. http://www.iise.org/details.aspx?id=645.

  * ISO (International Organization for Standardization). 2009. ISO 80000-2: Quantities and Units: Part 2: Mathematical Signs and Symbols to Be Used in the Natural Sciences and Technology. Geneva: International Organization for Standardization. http://www4.ncsu.edu/~jwilson/files/mathsigns.pdf.

  * James, R. C., and G. James. 1992. Mathematics Dictionary. 5th ed. New York: Chapman & Hall.

  * Knuth, D. E., T. Larrabee, and P. M. Roberts. 1989. Mathematical Writing. Washington, DC: Mathematical Association of America. http://www4.ncsu.edu/~jwilson/files/mathwriting.pdf.

  * Krantz, S. G. 1997. A Primer of Mathematical Writing: Being a Disquisition on Having Your Ideas Recorded, Typeset, Published, Read, and Appreciated. Providence, RI: American Mathematical Society.

  * Krantz, S. G. 2001. Handbook of Typography for the Mathematical Sciences. Boca Raton, FL: Chapman and Hall/CRC.

  * Matthews, J. R., and R. W. Matthews. 2014. Successful Scientific Writing: A Step-by-Step Guide for the Biological and Medical Sciences. 4th ed. Cambridge: Cambridge University Press.

  * Menzel, D. H., H. M. Jones, and L. G. Boyd. 1961. Writing a Technical Paper. New York: McGraw-Hill Book Company.

  * Miller, C., and K. Swift. 2001. The Handbook of Nonsexist Writing. 2nd ed. San Jose, CA: iUniverse.com.

  * Newton, I. 1672. “New Theory of Light and Colors.” Philosophical Transactions 6(80):3075–3087.

  * NISO (National Information Standards Organization). 2010\. ANSI/NISO Z39.14-1997 (R2009): Guidelines for Abstracts. Bethesda, MD: NISO Press. http://www.oregon.gov/ODOT/Programs/ResearchDocuments/Abstract-Guidelines.pdf.

  * O’Conner, P. T. 2009. Woe Is I: The Grammarphobe’s Guide to Better English in Plain English. 3rd ed. New York: Riverhead Books.

  * Pearsall, T. E., and K. C. Cook. 2010. The Elements of Technical Writing. 3rd ed. New York: Longman.

  * Porkess, R., ed. 2005. Collins Web-linked Dictionary of Statistics. 2nd ed. Glasgow: HarperCollins.

  * Scheinerman, E. R. 2011. Mathematical Notation: A Guide for Engineers and Scientists. Seattle: CreateSpace. https://www.ams.jhu.edu/ers/books/mathematical-notation.

  * Simpson, J. A., and E. S. C. Weiner, eds. 1989. Oxford English Dictionary. 2nd ed. Oxford, UK: Oxford University Press. http://www.oed.com.

  * Strunk, W., Jr., and E. B. White. 2000. The Elements of Style. 4th ed. Boston: Allyn and Bacon.

  * Swanson, E. 1999. Mathematics into Type. Updated edition by A. O’Sean and A. Schleyer. Providence, RI: American Mathematical Society.

  * University of Chicago Press. 2010. The Chicago Manual of Style. 16th ed. Chicago: University of Chicago Press. http://www.chicagomanualofstyle.org.

  * Tufte, E. R. 2001. The Visual Display of Quantitative Information. 2nd ed. Cheshire, CT: Graphics Press.

  * Upton, G., and I. Cook. 2014. A Dictionary of Statistics. 3rd ed. Oxford: Oxford University Press.

  * van Leunen, M.-C. 1992. A Handbook for Scholars. Rev. ed. New York: Oxford University Press.

  * Vonnegut, K. 1985. “How to Write with Style.” In How to Use the Power of the Printed Word, edited by B. S. Fuess, Jr., 33–38. Garden City, NY: Anchor Press/Doubleday. http://www.novelr.com/2008/08/16/vonnegut-how-to-write-with-style.

  * Whitehead, A. N. 1929. “The Aims of Education.” In The Aims of Education and Other Essays, 12. New York: The Free Press.

  * Williams, J. M., and J. Bizup. 2014. Style: The Basics of Clarity and Grace. 5th ed. Boston: Longman.

  * Williams, J. M., and J. Bizup. 2017. Style: Lessons in Clarity and Grace. 12th ed. Boston: Pearson.

  * Wilson, J. R. 2002. “Responsible Authorship and Peer Review.” Science and Engineering Ethics 8(2):155–174. http://www4.ncsu.edu/~jwilson/files/wilson02see.pdf.

  * Yang, J. T. 1995. An Outline of Scientific Writing: For Researchers with English as a Foreign Language. Singapore: World Scientific.

  * Zinsser, W. 2006. On Writing Well: The Classic Guide to Writing Nonfiction. 7th ed. New York: HarperCollins Publishers.

## AUTHOR BIOGRAPHY

JAMES R. WILSON is a professor in the Edward P. Fitts Department of Industrial
and Systems Engineering at North Carolina State University. His current
research interests are focused on probabilistic and statistical issues in the
design and analysis of simulation experiments. He has held the following
editorial positions: departmental editor of Management Science (1988–1996);
area editor of ACM Transactions on Modeling and Computer Simulation
(1997–2002); guest editor of a special issue of IIE Transactions honoring Alan
Pritsker (1999–2001); and Editor-in-Chief of ACM Transactions on Modeling and
Computer Simulation (2004–2010). He served The Institute of Management
Sciences College on Simulation (now the INFORMS Simulation Society) as
secretary-treasurer (1984–1986), vice president (1986–1988), and president
(1988–1990). His activities in the Winter Simulation Conference (WSC) include
service as proceedings editor (1986), associate program chair (1991), and
program chair (1992). During the period 1997–2004, he was a member of the WSC
Board of Directors corepresenting the INFORMS Simulation Society; and he
served as secretary (2001), vice chair (2002), and chair (2003). During the
period 2006–2009, he was a trustee of the WSC Foundation, serving as secretary
(2006), vice-president (2007), and president (2008). He is a member of ACM,
ASA, ASEE and SCS; and he is a fellow of IIE and INFORMS. His e-mail address
is<EMAIL_ADDRESS>and his web address is http://www.ise.ncsu.edu/jwilson.


$\left\\{\sigma_{i}\right\\}_{i=1}^{3}=\left\\{\left(\frac{\hbar}{2}\right)^{-1}\hat{s}_{i}\right\\}_{i=1}^{3}$.
However, unlike the canonical Pauli matrices,
$\left\\{\left(\frac{\hbar+\beta}{2}\right)^{-1}\hat{S}_{i}\right\\}_{i=1}^{3}$
depend explicitly on both quantisation scales, $\hbar$ and $\beta$.

It is straightforward to show that the simultaneous eigenvectors of
$\hat{S}^{2}$ and $\hat{S}_{i}$, for all three operators in Eqs. (4.3), have
eigenvalues

$\displaystyle\left\\{\frac{(\hbar+\beta)}{2},\frac{(\hbar+\beta)}{2},-\frac{(\hbar+\beta)}{2},-\frac{(\hbar+\beta)}{2}\right\\}\,.$
(4.54)

For $\hat{S}_{z}$, these correspond to the (un-normalised) eigenvectors

$\displaystyle\left\\{(1,0,0,0),\left(0,\frac{i\hbar}{\sqrt{\hbar\beta}},1,0\right),\left(0,-\frac{i\beta}{\sqrt{\hbar\beta}},1,0\right),(0,0,0,1)\right\\}\,,$

respectively. The normalised eigenvectors of $\hat{S}_{z}$ may then be written
as

$\displaystyle\Big{|}\frac{3(\hbar+\beta)^{2}}{4},\left(\frac{\hbar+\beta}{2}\right)_{z}\Big{\rangle}=(1,0,0,0)=\ket{\uparrow_{z}}_{1}\ket{\uparrow_{z}}_{2}\,,$
$\displaystyle\Big{|}\frac{3(\hbar+\beta)^{2}}{4},-\left(\frac{\hbar+\beta}{2}\right)_{z}\Big{\rangle}=(0,0,0,1)=\ket{\downarrow_{z}}_{1}\ket{\downarrow_{z}}_{2}\,,$
(4.55)

and

$\displaystyle\Big{|}\frac{3(\hbar+\beta)^{2}}{4},\left(\frac{\hbar+\beta}{2}\right)_{z}\Big{\rangle}_{\delta}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\delta}}(0,1,-i\sqrt{\delta},0)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\delta}}(\ket{\uparrow_{z}}_{1}\ket{\downarrow_{z}}_{2}-i\sqrt{\delta}\ket{\downarrow_{z}}_{1}\ket{\uparrow_{z}}_{2})\,,$
$\displaystyle\Big{|}\frac{3(\hbar+\beta)^{2}}{4},-\left(\frac{\hbar+\beta}{2}\right)_{z}\Big{\rangle}_{\delta}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\delta}}(0,-i\sqrt{\delta},1,0)$ (4.56)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\delta}}(\ket{\downarrow_{z}}_{1}\ket{\uparrow_{z}}_{2}-i\sqrt{\delta}\ket{\uparrow_{z}}_{1}\ket{\downarrow_{z}}_{2})\,,$

where we have introduced the dimensionless parameter

$\displaystyle\delta=\hbar/\beta\simeq 10^{-61}\,.$ (4.57)

The single-electron-plus-smeared-background system has four spin states, as
opposed to the two spin states of electrons in the fixed background of
canonical QM. However, the operators $\hat{S}^{2}$ and $\hat{S}_{i}$ that act
on the composite system have only two distinct sets of eigenvalues,
$\left\\{3(\hbar+\beta)^{2}/4,\pm(\hbar+\beta)/2\right\\}$, and each set has a
$2$-fold degeneracy. For $\hat{S}_{z}$, each eigenvalue corresponds to one
separable state and one state in which the $z$-spins of the electron and the
background are entangled. The eigenvectors
$\ket{\frac{3(\hbar+\beta)^{2}}{4},\frac{(\hbar+\beta)}{2}}$ and
$\ket{\frac{3(\hbar+\beta)^{2}}{4},\frac{(\hbar+\beta)}{2}}_{\delta}$
correspond to spin ‘up’ states, according to their measured eigenvalues,
whereas $\ket{\frac{3(\hbar+\beta)^{2}}{4},-\frac{(\hbar+\beta)}{2}}$ and
$\ket{\frac{3(\hbar+\beta)^{2}}{4},-\frac{(\hbar+\beta)}{2}}_{\delta}$
correspond to spin ‘down’ states.

For the unentangled states,
$\ket{\frac{3(\hbar+\beta)^{2}}{4},\pm\frac{(\hbar+\beta)}{2}}$, the $z$-spins
of the matter and geometry components of the tensor product smeared state,
$\ket{\psi}$ and $\ket{g}$, are aligned. The spin up state is characterised by
the individual values $\left\\{+\hbar/2,+\beta/2\right\\}$ and the spin down
state is characterised by the values $\left\\{-\hbar/2,-\beta/2\right\\}$.
However, for the entangled eigenvectors,
$\ket{\frac{3(\hbar+\beta)^{2}}{4},\pm\frac{(\hbar+\beta)}{2}}_{\delta}$,
there is no simple relation between the matter and geometry components of the
total quantum state. Remarkably, the entangled eigenstates (4.3) have the same
eigenvalues as the simple separable states (4.3).

We also note that, in the absence the interaction term $\mathbb{S}_{i}$, the
eigenvalues of the composite operator
$\hat{\mathcal{S}}_{i}+\hat{\mathcal{S}}^{\prime}_{i}$ are
$\left\\{(\hbar+\beta)/2,(\hbar-\beta)/2,-(\hbar-\beta)/2,-(\hbar+\beta)/2\right\\}$.
These correspond to the eigenvectors
$\left\\{\ket{\uparrow_{i}}_{1}\ket{\uparrow_{i}}_{2},\ket{\uparrow_{i}}_{1}\ket{\downarrow_{i}}_{2},\ket{\downarrow_{i}}_{1}\ket{\uparrow_{i}}_{2},\ket{\downarrow_{i}}_{1}\ket{\downarrow_{i}}_{2}\right\\}$,
respectively, which in the limit $\beta\rightarrow\hbar$ yield the familiar
basis vectors of a canonical two-particle state [65]. Thus, the introduction
of $\mathbb{S}_{i}$ not only restores the ${\rm su}(2)$ algebra of the
composite matter-plus-geometry system, in the presence of two quantisation
scales, $\hbar$ and $\beta\neq\hbar$, but also alters two of the four spin
eigenstates of the decoupled sectors, while leaving the remaining two
unchanged. This, in turn, shifts the corresponding eigenvalues by just the
right amount to introduce a $2$-fold degeneracy in the measured values of
$\hat{S}^{2}$ and $\hat{S}_{i}$.

A priori, there was no reason for us to anticipate that the additional terms
required to restore the ${\rm su}(2)$ algebra for the generalised operators
$\left\\{\hat{S}_{i}\right\\}_{i=1}^{3}$, i.e., those involving
$\mathbb{S}_{i}$ in the subalgebra (4.48a)-(4.48e), would simultaneously
introduce such a degeneracy. However, if had this not occurred, the doubling
of the spin degrees of freedom would, in principle, have been directly
detectable by performing simultaneous measurements of $\hat{S}^{2}$ and
$\hat{S}_{i}$. This would have caused severe problems for the smeared space
model, at least philosophically, even if the mathematical formalism remained
consistent. It is not hard to see why. In the non-spin part of the model, the
doubling of the canonical degrees of freedom is detectable only indirectly,
via the additional statistical fluctuations it induces in the measured values
of position, momentum and angular momentum, etc. These generate the GURs
derived in Secs. 3.1-3.3 and 4.2, which are consistent with our general
assumptions about the physical measurement scheme. We repeat that we assume a
scheme in which measurements are performed only on material bodies in space
[45, 47]. Therefore, we do not have direct physical access to the quantum
degrees of freedom of the background, which can be detected only indirectly
via their influence on quantum particles.

In the first formalism of the model, given in Sec. 3, this is expressed by
tracing out, or, equivalently, integrating out the degrees of freedom in the
first subspace of the tensor product Hilbert space, as in Eq. (3.7). However,
in the second formalism, on which the treatment of angular momentum is based,
the mathematical structure that renders only half of the doubled phase space
directly measurable is more complicated. The treatment of spin, given here, is
based on this second formalism, but our inability to define finite-dimensional
analogues of the canonical QM operators, $\hat{\alpha}^{i}\sim\hat{x}^{i}$,
$\hat{\alpha}^{\prime i}\sim\hat{x}^{\prime i}$,
$\hat{\beta}_{j}\sim\hat{p}_{j}$ and
$\hat{\beta}^{\prime}_{j}\sim\hat{p}^{\prime}_{j}$, means that there is no
clear analogue of either structure in the finite-dimensional case. Simply
tracing out half of the doubled spin degrees of freedom would require us to
make an arbitrary choice, namely, as to which two of the four possible spin
states we should regard as physical.

Remarkably, the algebra (4.48a)-(4.48e) saves us from this dilemma, just as it
saves the ${\rm su}(2)$ algebra of the composite state in the two-scale
quantisation scheme. The resulting model of generalised spin measurements is
mathematically consistent, and is also consistent with the physical
assumptions underlying the smeared space model as a whole, despite the
doubling of the number of dimensions in the spin Hilbert space. In this case,
the doubling is real, since
$\mathcal{H}_{2}\otimes\mathcal{H}_{2}\cong\mathcal{H}_{4}$, unlike the
infinite-dimensional case in which
$\mathcal{H}\otimes\mathcal{H}\cong\mathcal{H}$ [15].

Finally, we may write down the GURs implied by the generalised spin algebra
(4.48a)-(4.48e). By analogy with Eq. (4.34), $(\Delta_{\Psi}S_{i})^{2}$ takes
the form

$\displaystyle(\Delta_{\Psi}S_{i})^{2}$ $\displaystyle=$
$\displaystyle(\Delta_{\Psi}\mathcal{S}_{i})^{2}+(\Delta_{\Psi}\mathcal{S}^{\prime}_{i})^{2}+(\Delta_{\Psi}\mathbb{S}_{i})^{2}$
(4.58) $\displaystyle+$ $\displaystyle{\rm
cov}(\hat{\mathcal{S}}_{i},\hat{\mathbb{S}}_{i})+{\rm
cov}(\hat{\mathbb{S}}_{i},\hat{\mathcal{S}}_{i})$ $\displaystyle+$
$\displaystyle{\rm
cov}(\hat{\mathcal{S}}^{\prime}_{i},\hat{\mathbb{S}}_{i})+{\rm
cov}(\hat{\mathbb{S}}_{i},\hat{\mathcal{S}}^{\prime}_{i})\,.$

Multiplying by the equivalent expression for $(\Delta_{\Psi}S_{j})^{2}$, we
obtain the GUR for spin measurements in smeared space. Again, it is beyond the
scope of this work to investigate the consequences of this relation in detail,
but we note that it is of the general form

$\displaystyle(\Delta_{\Psi}S_{i})^{2}(\Delta_{\Psi}S_{j})^{2}\geq\dots\geq\left(\frac{\hbar+\beta}{2}\right)^{2}|(\epsilon_{ij}{}^{k})^{2}\braket{\hat{S}_{k}}_{\Psi}^{2}|\,,$
(4.59)

where the leading contribution to the terms in the middle is
$(\Delta_{\Psi}\mathcal{S}_{i})^{2}(\Delta_{\Psi}\mathcal{S}_{j})^{2}\geq(\hbar/2)^{2}|(\epsilon_{ij}{}^{k})^{2}\braket{\hat{\mathcal{S}}_{k}}_{\Psi}^{2}|$.
This is equivalent to the uncertainty relation for spin measurements in
canonical QM. The additional terms are non-canonical and depend on the ratio
of the dark energy density to the Planck density, which determines the value
of the geometry quantisation scale, $\beta$ [47, 48].

###  4.4 Relation to the theory of quantum reference frames

In Sec. 4.1 we introduced a useful unitary transformation, $\hat{U}_{\beta}$
(4.1). The action of $\hat{U}_{\beta}$ symmetrises the rigged bases of the
‘extended’ Hilbert space, to which the smeared state $\ket{\Psi}$ belongs,
such that $\ket{{\bf x},{\bf x^{\prime}}}\mapsto\ket{{\bf x},{\bf
x^{\prime}-x}}$ (4.2) and $\ket{{\bf p\,p^{\prime}}}\mapsto\ket{{\bf p},{\bf
p^{\prime}-p}}$ (4.3). In the first set of bases $\ket{\Psi}$ is non-separable
so that matter and geometry appear entangled, as in Eqs. (3.6) and (3.13).
This is consistent with the hypothesis advanced in [37]. However, in general,
entanglement is a basis-dependent property of quantum states [13, 35]. In the
symmetrised bases the smeared state is separable,
$\ket{\Psi}=\ket{\psi}\otimes\ket{g}$ (4.9), so that matter and geometry are
no longer entangled.

It follows that, if $\hat{U}_{\beta}$ represents a viable physical
transformation of the system, then the entanglement of matter and geometry in
the smeared space model is frame dependent. This is consistent with recently
obtained results in the theory of quantum reference frame (QRF)
transformations [23]. In fact, there exists a formal similarity between Eq.
(4.9) and the separable pre-QRF state considered in [23], and between Eq.
(3.6) and the entangled post-QRF state considered therein. The unitary
operator $\hat{U}_{\beta}$ (4.1) is formally analogous to the operator that
switches between QRFs in this formalism [47]. We recall that, in [23], a QRF
is defined as a superposition of classical reference frames. This is intended
to represent the realistic ‘reference frame’ defined by an observer that is
embodied as a quantum system.

The considerations above suggest a link between QRFs and GURs. Roughly
speaking, we may imagine the entangled matter-plus-geometry state of the first
formalism, Eq. (3.6), as the state of a QM particle ‘seen’ by an observer
embodied as a quantum spatial point, $\ket{{\bf x}}\otimes\ket{g_{{\bf x}}}$
(3.3). This explains the primary difference between the states and
transformations considered in the smeared space model and their counterparts
in the QRF theory [23, 47]. It may be verified that Eqs. (3.6), (3.13) and
(4.1) reduce to their counterparts in [23] in the limit
$\hbar\rightarrow\beta$. This is consistent with the fact that the QRF
formalism describes observers embodied as quantum systems on a fixed classical
background whereas the smeared space formalism treats the background itself as
a quantum mechanical object [47]. 888In [47], it was argued that the operator
$\hat{U}_{\beta}$ does not represent a physically accessible transformation of
the composite matter-plus-geometry system. However, even if the entanglement
of matter and geometry is a frame-independent feature of the theory, the
formal similarity between the smeared space formalism and the formalism of QRF
transformations, under the exchange $\beta\rightarrow\hbar$, still holds. This
strongly suggests that our interpretation of the smeared state (3.6), given
above, remains valid. It may be hoped that future work can either confirm or
disprove this [49].

However, on reflection, there is clearly something missing from both
formalisms. Specifically, the QRF formalism [23] allows us to describe the
results of physical experiments, as seen by observers that are embodied as
canonical quantum systems in a fixed classical background geometry. By
contrast, the smeared space formalism allows us to describe the results of
experiments, as seen by observers embodied as quantum spatial points [45, 47].
In any viable theory of quantum gravity, we expect realistic observers to be
embodied as material quantum systems living on, or ‘in’, a quantum background
geometry. This suggests that the two formalisms could be combined to give a
deeper picture of this scenario.

In principle, the combined theory should allow us to describe superpositions
of uniformly accelerated reference frames (i.e., uniform gravitational
fields), in which each Planck-sized volume of space is subject to additional
quantum fluctuations, as expected from generic arguments in phenomenological
quantum gravity [14, 22, 33, 58]. At present, the QRF formalism is able to
describe the former, but not the latter, whereas the smeared space formalism
describes the latter, but not the former. It may be hoped that future research
will yield further insights into the nature of both QRFs and GURs, and the
possible connections between them [49].

##  5 Avoiding the conventional wisdom

In this section, we consider possible objections to the smearing procedure
presented in [45, 47]. Two problems come immediately to mind. Both are
potentially valid and must be addressed head on, if the model is to survive as
a viable competitor to the standard modified commutator theories. Due to
limitations of space, our discussion is brief, and the problems cannot be
considered as fully resolved. Nonethless, there are encouraging signs that
they may be overcome within the existing formalism, and it may be hoped that
future research will be able to settle these issues one way or another.

The first problem is that there exist well known no go theorems for multiple
quantisation constants (see, for example, [69] and references therein). In
Sec. 5.1, we consider whether the existing theorems apply to the smeared
states defined in Secs. 3.1-3.2 and 4.1, and argue that they do not. The key
point is that the smeared space formalism treats matter and geometry
asymmetrically. This is expressed, mathematically, via the modified de Broglie
relation (3.23), in which ${\bf k}$ and ${\bf k^{\prime}}$ represent
independent degrees of freedom, but ${\bf p^{\prime}}$ remains the only
observable momentum. Therefore, quantising the composite matter-plus-geometry
system is not equivalent to quantising the state of two, distinguishable,
material particles. Because of this, the standard theorems cannot be applied,
directly, to states in the smeared space model.

The second problem is that, as every quantum gravity researcher knows,
canonical quantisation of the ‘metric’ yields a spin-2 representation of the
inhomogeneous Lorentz group [59, 80]. At first sight, this is immediately at
odds with our claim, in Sec. 4.3, that the fundamental quanta of space are
fermions. In Sec. 5.1, we present a critical view of the canonical
quantisation procedure [59] and argue that the standard interpretation applies
only to the first order perturbation of the metric. Crucially, the leading
order term, which represents the flat background space whose symmetries are
defined by the Poincaré group [21], remains purely classical in the usual
quantisation of linearised gravity [59]. We emphasise that the new degrees of
freedom introduced in the smeared space formalism correspond to the non-
relativistic limit of this flat piece, yielding superpositions of Euclidean
geometries [45, 47]. Therefore, our proposal does not contradict existing
results about the spin-2 nature of gravitons, which are represented by the
quantised perturbation.

###  5.1 Evading no go theorems for multiple quantisation constants

Well known no go theorems forbid the canonical quantisation of different
material particle species using different quantisation constants [6, 24, 69].
Given two distinguishable particles, A and B, whose degrees of freedom are
labelled by primed and unprimed variables, respectively, it may be shown that
imposing the canonical-type de Broglie relations

$\displaystyle{\bf p}=\hbar{\bf k}\,,\quad{\bf p^{\prime}}=\hbar^{\prime}{\bf
k^{\prime}}\,,$ (5.1)

with $\hbar\neq\hbar^{\prime}$, leads to fundamental inconsistencies [69].

In the limit $\hbar^{\prime}\rightarrow\hbar$, the relations (5.1) reduce to
the standard ones, ${\bf p}=\hbar{\bf k}$ and ${\bf p^{\prime}}=\hbar{\bf
k^{\prime}}$. We simply note that, in the limit $\beta\rightarrow\hbar$, the
quantisation conditions of the smeared space formalism, ${\bf p}=\hbar{\bf k}$
and ${\bf p^{\prime}-p}=\beta({\bf k^{\prime}-k})$ (3.17), which combine to
give ${\bf p^{\prime}}=\hbar{\bf k}+\beta({\bf k^{\prime}-k})$ (3.23), also
reduce to the standard relations. However, crucially, setting
$\beta=\hbar^{\prime}\neq\hbar$ does not recover Eqs. (5.1). For this reason,
the usual no go theorems presented in [6, 24, 69], and in related literature,
are not directly applicable to states in the smeared space model. The
existence of such states, given by Eqs. (3.6), (3.13) and (4.9), is not
expressly forbidden by these theorems.

Nonetheless, it is of course concievable that even slight modifications of the
existing theorems could be used to demonstrate inconsistencies in the smearing
procedure, represented by Eqs. (3.3) and (3.17). This possibility must be
thoroughly investigated, before the proposal of a second quantisation scale
for space can be taken seriously [48]. This work is of the utmost importance
for the future development of the smeared space model. If inconsistencies can
be explicitly demonstrated, it must be abandoned as a physical theory.

###  5.2 Isn’t spacetime spin-$2$? Gravitons versus quanta of space

The standard argument for the spin-$2$ nature of spacetime is that, since the
classical metric is a symmetric two-index tensor field, its canonical
quantisation yields a spin-$2$ representation of the inhomogeneous Lorentz
group in the tangent space associated with each spacetime point. This argument
is very robust as it forms part of a general scheme that may be used to
construct the dynamical equations for particles of any spin [27, 36] including
photons, spin-$1/2$ fermions, and even the composite spin-$3/2$ particles
predicted by the Standard Model [19, 79]. However, on closer inspection, a
linguistic sleight of hand has been applied with the use of the word ‘metric’.

In the standard approach the full metric of general relatively, $g_{\mu\nu}$,
is expanded to first order:

$\displaystyle g_{\mu\nu}\simeq\eta_{\mu\nu}+h_{\mu\nu}\,.$ (5.2)

Thereafter, $\eta_{\mu\nu}$ and $h_{\mu\nu}$ are regarded as dynamical and
non-dynamical pieces, respectively. This is expressed by the fact that
$\eta^{\mu\nu}=(\eta_{\mu\nu})^{-1}$ and $\eta_{\mu\nu}$ are used to raise and
lower indices on $h_{\mu\nu}$ and $h^{\mu}{}_{\nu}$, etc., as well as on
fields of any other type, corresponding to various forms of matter.

The metric perturbation is therefore defined as a field on flat spacetime
[59]. This enables canonical quantisation techniques, which are valid for
fields in Minkowski space, to be applied to $h_{\mu\nu}$. These are not
applicable to $g_{\mu\nu}$ [16]. The resulting equations of motion, known as
the Pauli-Fierz equations, describe the dynamics of spin-$2$ particles, aka
gravitons [59]:

$\displaystyle\Box h_{\mu\nu}=0\,,\quad\partial_{\nu}h_{\mu\nu}=0\,,\quad
h^{\mu}{}_{\mu}=0\,.$ (5.3)

Hence, only the quantized perturbation of the metric yields a spin-2
representation of the inhomogeneous Lorentz group, which is a subgroup of the
Poincaré group that defines the symmetries of flat Minkowski space on which it
‘lives’ [21, 36]. The Minkowski piece of the expansion (5.2) remains non-
dynamical and completely classical.

In the non-relativistic limit a similar split yields a classical Euclidean
background in which the perturbation $h_{00}$ is associated with the Newtonian
gravitational potential and the $h_{ij}$ components are neglected [32]. We
stress that the fermionic quanta of space predicted by our model are
associated with the flat Euclidean piece of the metric, which remains
classical and non-dynamical in the standard approach. By contrast, we endow
the Euclidean background with a quantum genuine state vector in the Hilbert
space of the theory [45, 47]. This forms part of the composite matter-plus-
geometry state that obeys a modified Schrödinger equation (see [45] for
details).

Though many aspects of the smeared space theory remain unclear, and have yet
to be explored, the qualitative picture that emerges is of fermionic spacetime
quanta exchanging virtual gravitons, in addition to the usual graviton-matter
and graviton-graviton interactions [45, 47]. This is perhaps not so crazy. We
recall that in the Standard Model of particle physics the fundamental
constituents of matter are fermions, which exchange virtual bosons as force-
mediators [19, 63]. Here, we treat gravitons as the fundamental quanta of
curvature, which give rise to the quantised gravitational force, but model the
underlying spacetime fabric as a web of fermions. This has the major advantage
of allowing us to answer the following question, which cannot be addressed
within the standard paradigm of linearised quantum gravity: of what is the
quantum space composed when it is flat?

##  6 Discussion

We have presented a new approach to GURs, based on the ‘smearing’ of classical
points into coherent superpositions of point-like quantum states. The
mathematical formalism of the model was presented in two forms. The first
incorporates generalised position and momentum measurements, which naturally
generate the EGUP (3.32)-(3.33), while the second allows us to define
generalised operators for angular momentum and spin, giving rise to GURs for
these observables. The key advantage of the new approach is that it generates
GURs without violating the symmetry algebras of canonical QM. This allows it
to evade the well known problems associated with modified commutation
relations, which are assumed to form the mathematical basis of GURs in the
bulk of the existing literature [33, 39, 57, 77, 76].

However, this phenomenological success comes at a price and the model implies
two radical conclusions for which there is, at present, no empirical support.
The first is that space, and hence gravity, must be quantised at a different
scale to matter, $\beta\simeq\hbar\times 10^{-61}$ (3.16) [45, 48]. The second
is that the fundamental quanta of the space-time fabric are fermions, with
spin $\pm\beta/2$, (4.3)-(4.3) [47]. To conclude, we give a critical appraisal
of the current status of the model, highlighting what is missing from the
present formalism, and what open questions need still to be addressed.

###  6.1 Current status of the model

A valid criticism of the current formalism is that the smearing scales for
position and momentum, $\Delta_{g}x^{\prime i}\simeq l_{\rm Pl}\simeq
10^{-33}\,{\rm cm}$ and $\Delta_{g}p^{\prime}_{i}\simeq m_{\rm dS}c\simeq
10^{-56}\,{\rm g\,.\,cm\,s^{-1}}$, are put into the theory by hand. That is,
although the wave-point uncertainty relation (3.21) bounds the uncertainties
associated with a quantum spatial ‘point’ such that $\Delta_{g}x^{\prime
i}\Delta_{g}p^{\prime}_{i}\gtrsim(3/2)\beta$, where $\beta\simeq l_{\rm
Pl}m_{\rm dS}c\simeq\hbar\times 10^{-61}$ (3.16), this does not bound the
values of $\Delta_{g}x^{\prime i}$ or $\Delta_{g}p^{\prime}_{i}$ individually.

Because (3.21) is formally analogous to the HUP (1.4), under the exchanges
$g\leftrightarrow\psi$ and $\beta\leftrightarrow\hbar$, it is consistent with
the limits $\Delta_{g}x^{\prime i}\rightarrow 0$,
$\Delta_{g}p^{\prime}_{i}\rightarrow\infty$ and $\Delta_{g}x^{\prime
i}\rightarrow\infty$, $\Delta_{g}p^{\prime}_{i}\rightarrow 0$, respectively,
so long as $\beta$ remains finite. However, in the former, the minimum
momentum uncertainty of a particle in the smeared background space also tends
to infinity, whereas, in the latter, the minimum position uncertainty diverges
[45]. Only by assuming the fixed values, given above, can the formalism give
rise to the EGUP expected from model-independent gedanken experiments
(3.32)-(3.33) [45, 47].

The non-dynamical nature of the smearing functions, $g({\bf x^{\prime}-x})$
and $\tilde{g}_{\beta}({\bf p^{\prime}-p})$, is therefore introduced as a
hypothesis of the model. At its present level of development, this seems
unavoidable. Nonetheless, it would certainly be preferable to derive these
conditions from a more fundamental dynamical theory. Though speculative, we
conjecture that an appropriate generalisation of the Pauli exclusion principle
(PEP), incorporating the spin-spin interaction between matter and geometry,
may enable us to achieve this. The rough idea is that the fermionic degeneracy
pressure between delocalised spatial ‘points’ is precisely what keeps them
delocalised, ensuring that a unique function $\braket{{\bf
x^{\prime}}}{g_{{\bf x}}}=g({\bf x^{\prime}-x})$ exists for each value of
${\bf x}$.

Whether or not this speculation turns out to be true, it is clear that the
canonical PEP must be generalised in the context of the smeared space model
[47]. In fact, virtually all the predictions of canonical QM could (and
should) be generalised to incorporate the effects of the smeared background.
These include all the more recent developments of the theory, conventionally
grouped under the umbrella of quantum information theory (QIT) [13]. There is
lots of work still to do, even in the non-relativistic limit. Furthermore, we
must now think seriously about why, and how, the theory should be extended
into the relativistic regime, and about how to introduce gravity into the
model in a more fundamental way. In the final section, 6.2, we briefly
consider these issues, and outline various prospects for future work.

###  6.2 Future work

Ultimately, we would like to construct a smeared version of the Standard Model
of particle physics, including all known interactions of the electromagentic,
weak, and strong nuclear forces [19]. Outstanding problems for this research
program include the questions of how to ‘smear’ time, how to smear general
gauge symmetries, and how to formulate the corresponding path integral picture
[45, 47]. However, even if we were able to overcome each of these problems,
our model still would not contain a fundamental description of gravity.

This is ironic, for a model that was originally motivated by gedanken
experiments in quantum gravity phenomenology [1, 8, 10, 51, 60, 70].
Nonetheless, it is consistent with the simplest picture of Newtonian gravity,
in which the gravitational potential is viewed as a scalar field in flat
Euclidean space [32, 45, 47]. To incorporate gravity on a more fundamental
level, two options immediately suggest themselves. The first is to complete
the research program outlined above by extending the smearing procedure to
flat Minkowski space and the Standard Model of particles physics. Armed with a
superposition of flat spacetimes, we may then consider how to construct
superpositions of curved spacetime geometries [45]. The second is to
incorporate Newtonian gravity into the model in a more fundamental way. In
principle, this may be done by smearing the Newton-Cartan formalism, in which
non-relativistic gravity emerges from the curvature of bona fide Riemmanian
geometries [5, 28].

Thankfully, these approaches are not mutually exclusive, and we may pursue
both simultaneously. In fact, this is a good strategy. If gravity is to be
viewed as spacetime curvature then the current smeared space theory
corresponds, at best, to the $G\rightarrow 0$, $c\rightarrow\infty$ limit of
the complete quantum gravity theory. By contrast, the generalisations proposed
above correspond to different limits, namely, $G\rightarrow 0$ with finite
positive $c$, and $c\rightarrow\infty$ with finite positive $G$, respectively.
Constructing each of these limits, explicitly, gives us two possible routes by
which to attack the fundamental problem of relativistic quantum gravity [45].

Finally, the model should be probed, at every stage of its development, for
mathematical inconsistencies. (For example, those associated with multiple
quantisation constants, as discussed in Sec. 5.1.) If possible, experimental
schemes should also be derived to test it empirically. Though it seems
unlikely that table-top experiments could ever be sensitive enough to directly
probe the smeared geometry, considering the extraordinarily small value of
$\beta$, this cannot be discounted a priori. In addition, we may explore the
phenomenological implications of the model for present day cosmology and the
history of the universe [45, 47]. This may prove to be an especially fruitful
research direction, given the intimate connection between $\beta$ and the
observed vacuum energy, $\rho_{\Lambda}$ (3.22) [48].

## Acknowledgements

I am extremely grateful to Marek Miller for helpful comments, suggestions, and
discussions. My thanks also to Piero Nicolini, for sharing his understanding
of nonlocal geometry models, and to Michael Hall, for bringing several
references to my attention. This work was supported by the Natural Science
Foundation of Guangdong Province, grant no. 008120251030.

## References

  * [1] Adler, R.J.; Santiago, D.I. On gravity and the uncertainty principle, Mod. Phys. Lett. A 14, 1371 (1999).
  * [2] Aghanim, N.; et al. [Planck Collaboration]. Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)].
  * [3] Amelino-Camelia, G. Planck-scale soccer-ball problem: a case of mistaken identity, Entropy 19, no.8, 400 (2017).
  * [4] Antoniou, I. and Perivolaropoulos, L. Constraints on spatially oscillating sub-mm forces from the Stanford Optically Levitated Microsphere Experiment data, Phys. Rev. D 96, no. 10, 104002 (2017).
  * [5] Banerjee, R.; Mukherjee, P. Galilean gauge theory from Poincare gauge theory, Phys. Rev. D 98, no. 12, 124021 (2018).
  * [6] Barcelo, C.; Carballo-Rubio, R.; Garay, L.J.; Gomez-Escalante, R. Hybrid classical-quantum formulations ask for hybrid notions, Phys. Rev. A 86, 042120 (2012).
  * [7] Binz, E.; Pods, S. The Geometry of Heisenberg Groups, American Mathematical Society, Providence, RI, USA, (2008).
  * [8] Bambi, C.; Urban, F.R. Natural extension of the Generalised Uncertainty Principle, Class. Quant. Grav. 25, 095006 (2008).
  * [9] Betoule, M.; et al. [SDSS Collaboration]. Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples, Astron. Astrophys. 568, A22 (2014).
  * [10] Bolen, B.; Cavaglia, M. (Anti-)de Sitter black hole thermodynamics and the generalized uncertainty principle, Gen. Rel. Grav. 37, 1255-1262 (2005).
  * [11] Bosso, P.; Das, S. Generalized Uncertainty Principle and Angular Momentum, Annals Phys. 383, 416-438 (2017).
  * [12] Burikham, P.; Cheamsawat, K.; Harko, T.; Lake, M.J. The minimum mass of a spherically symmetric object in $D$-dimensions, and its implications for the mass hierarchy problem, Eur. Phys. J. C 75, no.9, 442 (2015).
  * [13] Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, U.K. (2000).
  * [14] Crowell, L.B. Quantum Fluctuations of Spacetime, World Scientific: Hackensack, NJ, USA (2005).
  * [15] Debnath, L.; Mikusinski, P. Introduction to Hilbert Spaces with Applications, Elsevier, San Diego, CA, USA (2005).
  * [16] DeWitt-Morette, C. The pursuit of quantum gravity: Memoirs of Bryce DeWitt from 1946 to 2004, Springer, Heidelberg, Germany (2011).
  * [17] Diósi, L. Gravitation and quantum mechanical localization of macro objects, Phys. Lett. A 105, 199-202 (1984).
  * [18] Dirac, P.A.M. The Principle of Quantum Mechanics, 4th ed., Oxford University Press, Oxford, U.K. (1958).
  * [19] Donoghue, J.; Golowich, E.; Holstein, B.R. Dynamics of the standard model, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 2, 1-540 (1992).
  * [20] Frankel, T. The Geometry of Physics: An Introduction, Cambridge University Press, Cambridge, U.K. (1997).
  * [21] French, A.P. Special Relativity; W. W. Norton $\&$ Company, New York, NY, USA (1968).
  * [22] Garay, L.J. Quantum gravity and minimum length, Int. J. Mod. Phys. A 10, 145-166 (1995).
  * [23] Giacomini, F.; Castro-Ruiz, E.; Brukner, C. Quantum mechanics and the covariance of physical laws in quantum reference frames, Nature Commun. 10, no.1, 494 (2019).
  * [24] Gil, V.; Salcedo, L. Canonical bracket in quantum-classical hybrid systems, Phys. Rev. A 95, no.1, 012137 (2017).
  * [25] Girdhar, P.; Doherty, A.C. Viewing Generalised Uncertainty Principles through Quantum Noise, New J. Phys. 22, no.9, 093073 (2020).
  * [26] Gnatenko, K.P. ; Tkachuk, V.M. The soccer-ball problem in quantum space, Task Quarterly 23, no.4, 361-450 (2019).
  * [27] Gutowski, J.B. Symmetry and Particle Physics (2007). Available online: http://personal.maths.surrey.ac.uk/st/jg0033/Resources/lectnotes(master).pdf
  * [28] Hansen, D.; Hartong, J.; Obers, N.A. Action Principle for Newtonian Gravity, Phys. Rev. Lett. 122, no. 6, 061106 (2019).
  * [29] Hashiba, J. Dark Energy from Eternal Pair-production of Fermions, [arXiv:1808.06517 [hep-ph]] (2018).
  * [30] Heisenberg, W.K., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43, 172 (1927).
  * [31] Heisenberg, W.K. The physical principles of the quantum theory, Dover, New York, USA (1930).
  * [32] Hobson, M.P.; Efstathiou, G.P.; Lasenby, A.N. General Relativity: An Introduction for Physicists, Cambridge University Press, Cambridge, U.K. (2006).
  * [33] Hossenfelder, S. Minimal Length Scale Scenarios for Quantum Gravity, Living Rev. Rel. 16, 2 (2013).
  * [34] Hossenfelder, S. The Soccer-Ball Problem, SIGMA 10, 074 (2014).
  * [35] Isham, C.J. Lectures on Quantum Theory: Mathematical and Structural Foundations, Imperial College Press, London, U.K. (1995).
  * [36] Jones, H.F. Groups, Representations, and Physics, IOP Publishing, London, U.K. (1998).
  * [37] Kay, B.S. The matter-gravity entanglement hypothesis, Found. Phys. 48, no.5, 542-557 (2018).
  * [38] Kempf, A. Uncertainty relation in quantum mechanics with quantum group symmetry, J. Math. Phys. 35, 4483-4496 (1994).
  * [39] Kempf, A.; Mangano, G.; Mann, R.B. Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52, 1108-1118 (1995).
  * [40] Kempf, A. On quantum field theory with nonzero minimal uncertainties in positions and momenta, J. Math. Phys. 38, 1347-1372 (1997).
  * [41] Kelvin; Onggadinata, K.; Lake, M.J.; Paterek, T. Dark energy effects in the Schrödinger-Newton approach, Phys. Rev. D 101, no.6, 063028 (2020).
  * [42] Krishak, A.; Desai, S. Model Comparison tests of modified gravity from the Eöt-Wash experiment, J. Cosmol. Astropart. Phys. 07, 006 (2020).
  * [43] Lake, M.J. Is there a connection between “dark” and “light” physics? J. Phys. Conf. Ser. 883, no.1, 012001 (2017).
  * [44] Lake, M.J. Minimum length uncertainty relations in the presence of dark energy. Galaxies 7, no.1, 11 (2019).
  * [45] Lake, M.J.; Miller, M.; Ganardi, R.F.; Liu, Z.; Liang, S.D.; Paterek, T. Generalised uncertainty relations from superpositions of geometries. Class. Quant. Grav. 36, no.15, 155012 (2019).
  * [46] Lake, M.J. A Solution to the Soccer Ball Problem for Generalized Uncertainty Relations. Ukrainian J. Phys. _64_ , 1036 (2019).
  * [47] Lake, M.J.; Miller, M.; Liang, S.-D. Generalised uncertainty relations for angular momentum and spin in quantum geometry, Universe 6, no.4, 56 (2020).
  * [48] Lake, M.J. Why space could be quantised on a different scale to matter, SciPost Phys. Proc. 4, 014 (2021).
  * [49] Lake, M.J.: Miller, M. Quantum reference frames, revisited (in preparation).
  * [50] Landau, L.D.; Lifshitz, E.M. Mechanics, Volume 1 of Course of Theoretical Physics; Pergamon Oxford, U.K. (1960).
  * [51] M. Maggiore, A Generalized uncertainty principle in quantum gravity, Phys. Lett. B 304, 65 (1993).
  * [52] Marletto, C.; Vedral, V. Why we need to quantise everything, including gravity, npj Quantum Inf 3, 29 (2017).
  * [53] Messiah, A. Quantum Mechanics; Dover Publications, New York, NY, USA (1999).
  * [54] Moller, C. The energy-momentum complex in general relativity and related problems, Colloq. Int. CNRS 91, 15-29 (1962).
  * [55] Nakahara, M. Geometry, Topology and Physics; Taylor & Francis, Boca Raton, FL, USA (2003).
  * [56] Nicolini, P.; Niedner, B. Hausdorff dimension of a particle path in a quantum manifold, Phys. Rev. D 83, 024017 (2011).
  * [57] Nicolini, P. Nonlocal and generalized uncertainty principle black holes, [arXiv:1202.2102 [hep-th]] (2012).
  * [58] Padmanabhan, T. Planck length as the lower bound to all physical length scales. Gen. Rel. Grav. 17, 215-221 (1985).
  * [59] Pauli, W.; Fierz, M. Zur theorie der emission langwelliger lichtquanten. _Il Nuovo Cimento_ _15_ , 167–188 (1938) .
  * [60] Park, M.I. The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length, Phys. Lett. B 659, 698-702 (2008).
  * [61] Perivolaropoulos, L. Submillimeter spatial oscillations of Newton’s constant: Theoretical models and laboratory tests, Phys. Rev. D 95, no. 8, 084050 (2017).
  * [62] Perlmutter, S. et al., Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae, Astrophys. J. 517, 565 (1999).
  * [63] Peskin, M.E.; Schroeder, D.V. An Introduction to Quantum Field Theory, Perseus, Reading, MA, USA (1995).
  * [64] Pinsky, M.A. Introduction to Fourier analysis and wavelets, American Mathematical Society, Providence, RI, USA (2008).
  * [65] Rae, A.I.M. Quantum Mechanics, 4th ed., Taylor & Francis: London, U.K. (2002).
  * [66] Riess, A.G et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J. 116, 1009 (1998).
  * [67] Robertson, H.P. The Uncertainty Principle, Phys. Rev. 34, 163 (1929).
  * [68] Rosenfeld, L. On quantization of fields, Nucl. Phys. 40, 353 (1963).
  * [69] Sahoo, D. Mixing quantum and classical mechanics and uniqueness of Planck’s constant Journal of Physics A: Mathematical and General, Volume 37, Number 3 (2004).
  * [70] Scardigli, F. Generalized uncertainty principle in quantum gravity from micro - black hole Gedanken experiment, Phys. Lett. B 1999, 452, 39.
  * [71] Schrödinger, E. Zum Heisenbergschen Unschärfeprinzip, Sitzungberichten der Preussischen Akademie der Wissenschaften (Phys.-Math. Klasse), 19, ss. 296-323 (1930).
  * [72] E. Schrödinger, About Heisenberg uncertainty relation, Bulg. J. Phys. 26, 193 (1999) [Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 19, 296 (1930)].
  * [73] Spallucci, E.; Smailagic, A.; Nicolini, P. Trace Anomaly in Quantum Spacetime Manifold. Phys. Rev. D 73, 084004 (2006)
  * [74] Spradlin, M.; Strominger, A.; Volovich, A. Les Houches Lectures on de Sitter Space, [arXiv:hep-th/0110007 [hep-th]] (2001).
  * [75] Sprenger, M.; Nicolini, P.; Bleicher, M. Physics on Smallest Scales - An Introduction to Minimal Length Phenomenology, Eur. J. Phys. 33, 853-862 (2012).
  * [76] Tawfik, A.N.; Diab, A.M. Generalized Uncertainty Principle: Approaches and Applications, Int. J. Mod. Phys. D 23, no.12, 1430025 (2014).
  * [77] Tawfik, A.N.; Diab, A.M. Review on Generalized Uncertainty Principle, Rept. Prog. Phys. 78, 126001 (2015).
  * [78] Trautman, A. Clifford algebras and their representations. In Encyclopedia of Mathematical Physics; Francoise, J.-P., Naber, G.L., Tsou, S.T., Eds., Elsevier, Oxford, U.K. (2006).
  * [79] Wheeler, N. Spin Matrices for Arbitrary Spin (2000). Available online: https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/
Miscellaneous%20Essays/Angular%20Momentum,%20Spin/D3.%20Spin%20Matrices.pdf

  * [80] Wigner, E. P. On Unitary Representations of the Inhomogeneous Lorentz Group, Annals Math. 40, 149-204 (1939).