# Spatial curvature with the Alcock-Paczyński effect

Luca Amendola  Institute of Theoretical Physics, Philosophenweg 16, Heidelberg
University, 69120, Heidelberg, Germany  Marco Marinucci  Dipartimento di
Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova, via
Marzolo 8, I-35131, Padova, Italy  INFN, Sezione di Padova, Via Marzolo 8,
I-35131, Padova, Italy  Miguel Quartin  Instituto de Física, Universidade
Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil  Observatório
do Valongo, Universidade Federal do Rio de Janeiro, 20080-090, Rio de Janeiro,
RJ, Brazil  PPGCosmo, Universidade Federal do Espírito Santo, 29075-910,
Vitória, ES, Brazil

###### Abstract

We propose a methodology to measure the cosmological spatial curvature by
employing the deviation from statistical isotropy due to the Alcock-Paczyński
effect of large scale galaxy clustering. This approach has a higher degree of
model independence than most other proposed methods, being independent of
calibration of standard candles, rulers, or clocks, of the power spectrum
shape (and thus also of the pre-recombination physics), of the galaxy bias, of
the theory of gravity, of the dark energy model and of the background
cosmology in general. We find that a combined DESI-Euclid galaxy survey can
achieve $\Delta\Omega_{k0}=0.057$ at 1$\sigma$ C.L. in the redshift range
$z<2$ by combining power-spectrum and bispectrum measurements.

##  I Introduction

The spatial curvature $\Omega_{k0}$ has been one of the most investigated
cosmological parameters over the last decades. It is a standard degree of
freedom of the Friedman-Lemaître-Robertson-Walker metric, with a very
important role in our understanding of the universe. The possibility of a non-
flat universe thus continues to captivate both researchers and laymen.

The first strong observational constraints on flatness came from the
measurements of the first peak of the Cosmic Microwave Background (CMB)
angular power spectrum [1]. Since then, high-resolution maps of the CMB have
continued to tighten these constraints and the current best one comes from the
Planck satellite. There is, however, a strong debate on which are the current
most reliable measurements. The combination of temperature, polarization and
lensing yields $\Omega_{k0}=-0.0106\pm 0.0065$ [2], consistent with flatness.
But the CMB lensing itself is too large to fit the standard $\Lambda$CDM model
[3, 4]. Dropping lensing, one gets $-0.095<\Omega_{k0}<-0.007$ at 99% CL [5],
favoring a closed universe. This disagreement highlights the fact that CMB
measurements are always performed within a cosmological paradigm which
requires assuming a specific model for both the early and late universes.

Curvature can also be measured through its effects on the late-time universe.
In particular, it affects measurements of the luminosity ($D_{L}$) and angular
diameter ($D_{A}$) distances, of the expansion rate ($H(z)$) and of both weak
and strong lensing. A large number of works analyzed combinations of these
observables to constrain curvature independently from the CMB. Several of
these made use of the so-called cosmic chronometers (CC) to infer $H(z)$ and
constrained $\Omega_{k0}$ by combining $H(z)$ with supernova distances (SN)
[6, 7, 8, 9, 10, 11, 12], with BAO [13, 14, 10, 12], or with lensing [15]. The
obtained uncertainties on $\Omega_{k0}$ are around $\sim 0.1-0.2$. However, CC
are based on modelling passively evolving galaxies, and their accuracy level
is still under debate [16]. Without CC, constraints on $\Omega_{k0}$ were also
obtained with precision $\sim 0.5-0.9$ combining supernova distances and
lensing [17]. A promising avenue to avoid CC relies on measurements of the
large-scale structure (LSS) alone, since radial and transversal correlations
allow measurements of both $D_{A}(z)$ and $H(z)$. The recent DESI 2024 results
using BAO alone obtained $\Omega_{k0}=0.065^{+0.068}_{-0.078}$
($0.087^{+0.100}_{-0.085}$) assuming the $\Lambda$CDM (o$w_{0}w_{a}$CDM) model
[18].

Forecasts on $\Omega_{k0}$ have also been performed using weak-lensing from
Euclid or LSST [19]; intensity mapping [20]; supernovae and BAO [11]; standard
sirens [21, 22] and the clustering of standard candles [23, 22].

One important recent concern in the field has been to push for model-
independent measurements. Non-parametric fits have been employed to mitigate
late-time modelling using Gaussian Processes [7, 24, 11, 12], polynomial fits
[9] or smoothing techniques [25]. Another option is to use directly the
measurements of $D_{A}$ and $H$ in different redshift bins [6, 26]. Here we
follow the latter approach.

This late-time model-independence has the advantage of being robust with
respect to uncertainties related to dark energy, which is important since
there are hints of a tension with late-universe data (e.g. [27, 18]). We
remark that model-independence however can also be extended to the early
universe, as we will discuss below, which makes results also robust against
non-standard early universe physics. In fact, the current Hubble tension has
sparked interest in more exotic early universe scenarios as a possible
explanation [28]. Extending model-independence to the early universe means we
do not have to assume that $P(k)$ has the $\Lambda$CDM shape, or is
parametrized by a restricted set of parameters, for instance the Alcock-
Paczyński (AP) parameters $\alpha_{\parallel},\alpha_{\perp}$, plus the growth
rate $f$ and the normalization $\sigma_{8}$ as in, e.g., [29] where, moreover,
the non-linear corrections were evaluated only for the $\Lambda$CDM model.
Thus, to the best of our knowledge, no work so far has investigated the
possibility of measuring the spatial curvature in the same model-independent
way that we propose in this paper.

In this work we employ the FreePower method [30, 31, 32]. FreePower extracts
cosmological information in a purely geometrical way through the AP effect by
binning the spectra in $k$-wavebands. The AP effect depends only on the
dimensionless expansion rate $E$ and the dimensionless comoving angular
diameter distance $L_{A}$

$E(z)\equiv H(z)/H_{0}\,,\quad L_{A}(z)\equiv H_{0}D_{A}(z)\,.$ (1)

Instead of choosing a particular cosmological model, the method leaves free to
vary, in each redshift bin, the functions $f$, $E$ and $L_{A}$ together with
all necessary nuisance parameters. We take as data the one-loop power spectrum
and tree-level bispectrum of galaxy clustering. Our strategy is to use only
the AP effect to constrain both $E$ and $L_{A}$ while ensuring that the
clustering correlators are written down in the most model-independent way and
all relevant parameters are marginalized over. We adopt for the non-linear
correlators the general expressions derived in [33], which is based on general
considerations of symmetry rather than on specific models. Our basic
parameters are then 25 values of the linear $P(k_{i})$ in the $k$ interval
$0.01-0.25\,h/$Mpc, plus, for each redshift bin, $f$, $E$ and $L_{A}$, seven
bias and bootstrap parameters, a smoothing velocity dispersion and a
counterterm parameter, and finally three shot noises. In total, we have 15
parameters for each redshift bin plus 25 $k$-band parameters. We adopt two
cut-off schemes: a more “aggressive” one, which is our default scheme, in
which we take $k_{\rm max}=0.25\,h/$Mpc for the power spectrum and $k_{\rm
max}^{\rm B}=0.1\,h/$Mpc for the bispectrum; and a “conservative” one, in
which the two cut-offs are $0.20\,h/$Mpc and $0.08\,h/$Mpc, respectively. We
assume that $f$ does not depend on $k$ in the interval here considered. This
is not a fundamental limitation, as we have shown in [31], but it is a safe
approximation in many models (e.g., massive neutrinos induce a variation of
$f$ with $k$ of less than 1% in the viable range, see e.g. [34]). More details
in Ref. [32].

| $z$ | $V$ | $10^{3}\,n_{g}$ | $b_{1}$
---|---|---|---|---
|  |  [Gpc$/h]^{3}\\!\\!$ | $[h/{\rm Mpc}]^{3}$ |
| 0.1 | 0.263 | 118. | 1.41
| 0.3 | 1.53 | 11.9 | 1.57
DESI | 0.5 | 3.33 | 1.14 | 1.74
| 0.7 | 5.15 | 1.07 | 1.15
| 0.9 | 7.22 | 1.54 | 1.26
| 1.1 | 8.61 | 0.891 | 1.34
| 1.3 | 9.66 | 0.521 | 1.42
Euclid | 1.5 | 10.4 | 0.274 | 1.5
| 1.7 | 11. | 0.152 | 1.58
| 1.9 | 11.3 | 0.0899 | 1.66
Table 1:  Our forecast specifications and fiducials, based on DESI and Euclid
forecasts for the full surveys. We use DESI BGS for $z<0.6$, DESI ELG for
$0.6<z<0.8$ and Euclid ELG for $z>0.8$. For $z<0.6$ we take values for the
bias parameters from the low-z BOSS results [35], while for $z>0.6$ we use the
models in [36].

Our approach addresses therefore both the issue of improving accuracy (being
more model-independent than other approaches) and improving precision
(employing the information in the one-loop spectrum and in the bispectrum).
Another crucial advantage of the FreePower approach is that we can derive
constraints directly on the dimensionless variables $E$ and $L_{A}$, and thus
directly on $\Omega_{k0}$. This is in contrast with using the popular
combination CC and SN or standard sirens, which constrains only the quantity
$\Omega_{k0}$$H_{0}^{2}$, and thus requires either an extra probe to constrain
$H_{0}$ or an extrapolation of the $H(z)$ data to $z\rightarrow 0$ to break
the degeneracy.

##  II Spatial curvature constraints

We applied the FreePower method to produce Fisher matrix forecasts for a joint
DESI and Euclid dataset. The DESI survey [37, 38] is a ground telescope which
will produce a spectroscopic map covering 14000 deg2 of the sky, covering the
range $z=(0-1.6)$ with a combination of BGS, LRG and ELG galaxies [39]. The
Euclid survey is a space telescope, launched in 2023, that will map 15000 deg2
of the sky [40], covering the range $z=[0.8-2.0]$. We adopted redshift bins of
width $\Delta z=0.2$ centered on the redshifts listed in the tables, and
assume negligible cross-bin correlations. We used DESI specifications (only
for BGS and low-redshift ELG in order to be conservative) for the bins with
$z\leq 0.8$ and Euclid for $0.8\leq z\leq 2.0$. The main details of the
surveys and the fiducial parameters are displayed in Table 1. This is similar
to what was considered in [41]; the main difference with respect to [32] is
the inclusion of the low-$z$ DESI bins.

As mentioned above, the AP effect distorts the wavenumber $k$ and the cosine
angle $\mu$ in a way that depends only on $h\equiv E/E_{r}$ and $l\equiv
L_{A}/L_{A,r}$, where the subscript $r$ refers to the (arbitrary) reference
cosmological value adopted to convert distances and angles into $k,\mu$, such
that $\mu=\mu_{r}h/\alpha$ and $k=\alpha k_{r}$, where [42, 43, 44]

$\alpha\,=\,l^{-1}\sqrt{\mu_{r}^{2}(h^{2}l^{2}-1)+1}\,.$ (2)

Once we marginalize over all the other parameters, we see that we can measure
$h(z),l(z)$ down to 2–3% in several redshift bins in the aggressive case, as
we show in Table 2. The marginalized Fisher matrix for $h,l$ is the main input
for the next section.

$z$ | $\Delta f/f$ | $\Delta h/h$ | $\Delta l/l$
---|---|---|---
0.1 | 0.072 | 0.038 | 0.024
0.3 | 0.052 | 0.023 | 0.015
0.5 | 0.044 | 0.019 | 0.013
0.7 | 0.034 | 0.022 | 0.018
0.9 | 0.031 | 0.019 | 0.016
1.1 | 0.03 | 0.018 | 0.015
1.3 | 0.032 | 0.019 | 0.015
1.5 | 0.039 | 0.021 | 0.017
1.7 | 0.049 | 0.025 | 0.02
1.9 | 0.063 | 0.031 | 0.026
Table 2: Fully marginalized relative error forecast for all redshift bins,
(see [32] for a full description). Here and in then rest of this work we
report 1$\sigma$ uncertainties.

The spatial curvature is related to $E(z)=H(z)/H_{0}$ and to the dimensionless
comoving angular diameter distance $L_{A}(z)$ by a relation that in our $h,l$
variables reads

$\Omega_{k0}=\frac{\big{(}h(\partial_{z}l)E_{r}L_{A,r}+hl\big{)}^{2}-1}{l^{2}L_{A,r}^{2}}\,,$
(3)

where $\partial_{z}$ means derivative with respect to $z$. Notice that in
terms of $E(z)$ and $L_{A}(z)$, $\Omega_{k0}$ is independent of $H_{0}$, as we
discussed previously. Also, the expression above is valid for both open and
closed curvature. Therefore, once we measure $E,L_{A}$, we can also measure
$\Omega_{k0}$.

We need then to propagate the constraints on $h,l$ from each bin, including
their correlation, to $\Omega_{k0}$. Since $\Omega_{k0}$ depends in a non-
linear way on the variables $h,l$, we choose to propagate the errors
numerically. We generate $10^{5}$ random values of $h,l$ in $n_{B}$ redshift
bins $i=1,...,n_{B}$ from a Gaussian multivariate distribution with means 1 in
each bin and covariance given by the inverse of our marginalized Fisher matrix
for $h,l$. Then we discretize Eq. (3)

$\Omega_{k,i}=\frac{\left[h_{i}\frac{(l_{i+1}-l_{i-1})}{2\Delta
z}E_{r}(z_{i})L_{A,r}(z_{i})+h_{i}l_{i}\right]^{2}-1}{l_{i}^{2}L_{A,r}(z_{i})^{2}}\,,$
(4)

where $\Delta z$ is the bin size, and from every set of $h_{i},l_{i}$ with
$i=2,..n_{B}-1$ we produce a value of $\Omega_{k,i}$. The $j-$th value
corresponding to the $i$-th bins is denoted as $\Omega_{k,(i,j)}$. These
values are correlated. Then we estimate the $(n_{B}-2)\times(n_{B}-2)$
covariance matrix of $\Omega_{k,(i,j)}$:

$C_{\Omega_{k,(i,n)}}=\big{\langle}\Omega_{k,(i,j)}\Omega_{k,(n,m)}\big{\rangle}_{j,m}\,.$
(5)

The errors on $\Omega_{k0}$ for each bin are in Table 3, while in Fig. 1 we
show the distribution for some redshift bins. Since the distribution is well
approximated by a Gaussian, we can safely interpret the errors in the table in
the usual Gaussian way, i.e. as 68$\%$ confidence regions. The variance of
$\Omega_{k0}$ is obtained by projecting the $(n_{B}-2)\times(n_{B}-2)$ Fisher
matrix $F=C^{-1}_{\Omega_{k0}}$ onto a single $\Omega_{k0}$. The result is
simply

$\sigma^{2}_{\Omega_{k0}}=\bigg{(}\sum_{i,j}F_{ij}\bigg{)}^{-1}\,.$ (6)

Finally, we obtain

$\sigma_{\Omega_{k0}}=0.057$ (7)

at 68$\%$ for the aggressive specifications, and $\sigma_{\Omega_{k0}}=0.075$
for the conservative ones. If only the power spectrum is employed, then we get
0.094. If one artificially takes the limit of infinite galaxy number density,
then the cosmic-variance limited value of 0.033 can be reached. These results,
and the comparison with $\Lambda$CDM, are in Table 4. Let us remark that these
results are not prior-dominated, that is, the priors for each parameter have
been chosen to be much wider than the final constraints.

$z$ | FreePower | FreePower |  $\Lambda$CDM
---|---|---|---
|  | CV limit | $+\Omega_{k0}$
0.3 | 0.769 | 0.735 | 0.250
0.5 | 0.365 | 0.325 | 0.140
0.7 | 0.244 | 0.207 | 0.110
0.9 | 0.240 | 0.198 | 0.097
1.1 | 0.205 | 0.161 | 0.090
1.3 | 0.218 | 0.143 | 0.089
1.5 | 0.263 | 0.134 | 0.095
1.7 | 0.351 | 0.129 | 0.110
combined | 0.0572 | 0.0335 | 0.033
Table 3: Forecast uncertainties on $\Omega_{k0}$ for each bin (and combined).
We also show the constraints for the standard full-shape approach assuming
$\Lambda$CDM.  Figure 1:  Numerical distribution of $\Omega_{k0}$ for two
redshift bins. The dashed curves are Gaussian fits. The continuous thicker red
curve is the likelihood when combining all bins.

How do these numbers compare with other methods with some degree of model
independence? Forecasts for Euclid data using the standard full-shape
approach, which assumes a parametrized shape of $P(k)$, and assuming linear
theory was valid up to (an optimistic) $k_{\rm max}=0.20h$/Mpc were performed
in [6]. They found $\sigma$($\Omega_{k0}$) around $0.1-0.2$ in 13 redshift
bins, which if combined would result in $\sigma(\Omega_{k0})=0.033$. A
forecast for Euclid + DESI was performed in [11] using the radial BAO scale
combined with Nancy Roman SN, resulting in $\sigma(\Omega_{k0})=0.026$, but
this assumes the BAO scale does not evolve. Forecasts for 21cm intensity
mapping for HIRAX combining with the CMB distance scale were also performed in
[20], resulting in $\sigma(\Omega_{k0})=0.0085$ for an agnostic binned $w(z)$
dark-energy model. Of course, assuming both an early and late-time model
allows tighter constraints. For instance, the same HIRAX+CMB constraints
shrink to $\sigma(\Omega_{k0})=0.0028$ assuming $w$CDM. Other methods also
become very precise. Assuming the $\Lambda$CDM model, using the clustering of
Einstein Telescope bright sirens and DESI BGS, $\sigma(\Omega_{k0})=0.018$ was
forecast by [22], while combining upcoming CMB with Euclid BAO and weak-
lensing could yield $\sigma(\Omega_{k0})=0.0018$ (degrading to $0.0088$ for
the $w$CDM model) [19].

method |  combined $\sigma_{\Omega_{k0}}$
---|---
FreePower P+B | $0.057$
FreePower conservative P+B | 0.075
FreePower CV limit P+B | 0.033
FreePower only P | 0.094
$\Lambda$CDM full shape P+B | 0.033
$\Lambda$CDM full shape only P | 0.037
$\Lambda$CDM full shape P+B+CMB | 0.0021
Table 4: Results combining all redshift bins. Note that CMB can only be added
by considering a model for both early and late times. P stands for using the
power spectrum alone while P+B adds also the bispectrum. CV limit denotes the
cosmic variance limit, i.e. $n_{g}\rightarrow\infty$.

We emphasize that, in contrast with our approach, all these constraints have
been obtained either assuming specific parametrizations, or the reliability,
accuracy and correct calibration in general of standard candles and clocks.
For CC in particular, this requires assuming the reliability and robustness of
stellar population synthesis models, which form the basis of the method, and
that all CC systematic effects can be kept under control.

Finally, in Fig. 2, we show how the uncertainty on $\Omega_{k0}$ decreases
with an increasing power spectrum cut-off (keeping the bispectrum cut-off at
$0.1\,h/$Mpc). The sensitivity to $k_{\rm max}$ is relatively weak.

##  III Discussion

We presented a methodology to measure the late-time cosmic spatial curvature
that is independent of calibration of standard candles, clocks, or rulers, of
the cosmological background, and of the power spectrum shape and growth. This
model-independent approach makes use of the statistical isotropy of the
Universe embedded in the linear and non-linear power spectrum and bispectrum
of galaxy clustering. We find that a combination of the DESI and Euclid
surveys can constrain $\Omega_{k0}$ to within 0.057, a level competitive with
several other less model-independent methods.

Figure 2:  Scaling of $\sigma_{\Omega_{k0}}$ versus $k_{\rm max}$ for the
spectrum (keeping fixed to $k_{\rm max}=0.10h/$Mpc the bispectrum value). The
bigger yellow dot represents our reference value. The method has only a weak
sensitivity to $k_{\rm max}$ values.

One can further improve these constraints in a number of ways, e.g. by adding
other redshift bins or larger sky areas, or combining different tracers of
structure.

One can also consider external constraints on $H,D$ from standard candles or
cosmic chronometers. We tested adding strong external priors for either
distance or expansion constraints. We find that FreePower benefits the most
from the former: external distance data could improve precision by a factor of
almost three. However this comes at the cost of assuming these independently
measured distances are free of biases and systematic effects in general.

## Acknowledgements

We thank Massimo Pietroni for several discussions. LA acknowledges support
from DFG project 456622116. MM acknowledges support by the MIUR PRIN Bando
2022 - grant 20228RMX4A. MQ is supported by the Brazilian research agencies
FAPERJ, CNPq and CAPES. Many integrals of the FreePower method have been
performed using the CUBA routines [45] by T. Hahn (http://feynarts.de/cuba).

*

##  Appendix A Constraints on $\Omega_{k0}$ assuming $\Lambda$CDM

The results shown in this letter are obtained using the FreePower method where
the linear power spectrum is not fixed by cosmology, while standard LSS
analyses usually assume a specific model, $\Lambda$CDM, [35, 46, 47, 48, 49,
50, 51], or its generalizations, e.g. see [52, 53, 54]. We perform a Fisher
analysis using the same specifications and fiducial values listed in Table 1
adopting $\Lambda$CDM plus a non-zero $\Omega_{k0}$. We use the code PyBird
[55] (https://github.com/pierrexyz/pybird), re-adapted to match the biasing
scheme of FreePower. We vary simultaneously the cosmology, the bias and the
small scales parameters.

It has been observed that, at the scales considered, the one-loop power
spectrum and the tree-level bispectrum are not very sensitive to some of the
EFT parameters, such as the non-linear bias(es), the counterterms and the shot
noise, that are usually fixed or marginalized, see [35, 50]. For this reason
we fix the third order bias, the next-to-next-to-leading order counterterm and
the scale dependent shot noise [55], while we leave free to vary all the other
bias parameters (a total of seven) in each redshift bin. We consider the
monopole and quadrupole of the two statistics, neglecting the correlation
among different redshift bins as this effect has been shown to be negligible
[56].

The results of this $\Lambda$CDM forecast are reported in Tables 3–4. Adding
the bispectrum does not significantly improve the constraints on the curvature
parameter: with $P$ alone we obtain $\sigma_{\Omega_{k0}}=0.037\;(0.073)$ at
68$\%$ (95$\%$) CL while with $P+B$ we have
$\sigma_{\Omega_{k0}}=0.033\;(0.065)$ at 68$\%$ (95$\%$) CL. These results are
in line with the analysis performed in [57], and analogous analyses have shown
that the bispectrum is essential to constrain the non-linear biases but adds
little information about cosmology compared to the $P$ only case [48].
Furthermore, if we include Planck CMB data [2] we obtain
$\sigma_{\Omega_{k0}}=0.0021$, ten times better than LSS alone. This results
illustrate how, assuming a particular model, the combination of LSS and CMB
data is capable of breaking important degeneracies among the cosmological
parameters.

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# Bell-type inequalities for systems of relativistic vector bosons

Alan J. Barr  Department of Physics, Keble Road, University of Oxford, OX1 3RH
and
Merton College, Merton Street, Oxford, OX1 4JD  Paweł Caban
<EMAIL_ADDRESS>(corresponding author) Jakub Rembieliński  Department
of Theoretical Physics, University of Łódź
Pomorska 149/153, PL-90-236 Łódź, Poland

###### Abstract

We perform a detailed analysis of the possible violation of various Bell-type
inequalities for systems of vector boson-antiboson pairs. Considering the
general case of an overall scalar state of the bipartite system, we identify
two distinct classes of such states, and determine the joint probabilities of
spin measurement outcomes for each them. We calculate the expectation values
of the CHSH, Mermin and CGLMP inequalities and find that while the generalised
CHSH inequality is not expected to be violated for any of the scalar states,
in the case of the Mermin and CGLMP inequalities the situation is different –
these inequalities can be violated in certain scalar states while they cannot
be violated in others. Moreover, the degree of violation depends on the
relative speed of the two particles.

##  1 Introduction

Quantum mechanics predicts that the results of measurements exhibit
correlations that differ radically from those of classical physics. Einstein,
Podolsky and Rosen were sufficiently disturbed by the apparent lack of realism
in quantum measurements, in particular those corresponding to non-commuting
operators, that they doubted the completeness of the theory [1]. In response,
Bell [2] considered the predictions of theories that are both local (such the
physical influences cannot travel faster than the speed of light), and
realistic (having physical properties that are independent of observation). He
showed that, under certain assumptions, one could perform experimental tests
that could distinguish between the predictions of quantum theory and those of
such local realistic theories. His method was based on the observation that
quantum mechanical predictions for particular correlated expectation values
disobey a mathematical inequality – a so-called ‘Bell Inequality’ – that local
realistic theories must satisfy.

Subsequent experimental tests of Bell-like inequalities have been performed a
variety of physics systems using e.g. photons [3, 4], ions [5],
superconducting systems [6] and solid-state systems [7]. More recent examples
include so-called ‘loop-hole-free’ tests [8, 9, 10], in which special
attention was paid to issues of causality, efficiency and freedom of choice of
experimental settings. Bell tests have also been performed experimentally on
pairs photons with orbital angular momentum [11], where any measurement of
each of the two subsystems results in one of three possible outcomes.

Violation of Bell inequalities by quantum-mechanical probabilities is usually
discussed within the framework of non-relativistic quantum mechanics. However,
the description of EPR experiments with relativistic particles should be
performed in a relativistic setting. Quantum entanglement and Bell
inequalities in such a setting have been considered in the literature starting
from Czachor’s paper [12]. Since then a large number of papers on the subject
have been published, see, e.g., [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30] and references therein. It is worth taking note
that the description of EPR experiments in a relativistic framework is
hindered by theoretical and interpretational difficulties. One of the most
important problems is related to the appropriate definition of a relativistic
spin operator which is rooted in the non-existence of the Lorentz-covariant
position operator in relativistic quantum mechanics [31]. We discuss these
issues in more detail in Sec. 3.

In Ref. [21], the EPR correlations in the spins of a pair of identical
relativistic spin-1 bosons were considered and the expectation values of
various correlators calculated analytically. More recently a proposal was made
[32] for testing Bell inequalities using a pair of spin-1 $W^{+}$ and $W^{-}$
bosons resulting from the decay of a spin-0 Higgs boson111Further analysis of
the prospects for Bell violation measurements in bipartite systems of weak
vector bosons at high-energy colliders have been described subsequently in
refs [33, 34, 35, 36].. Several distinctive features of these recently-
proposed measurements make them particularly interesting both in terms of
their experimental setup and the interpretation of the results. Firstly, the
extremely short (sub-fm) sub-nuclear length scales of the bipartite systems
under test are many orders of magnitude shorter than any existing
measurements, making this a novel and unexplored regime. Secondly at least one
$W$ must be off-mass-shell if produced in a Higgs boson decay, since the mass
of the Higgs boson is less than the sum of the $W^{+}$ and $W^{-}$ bosons.
This opens the possibility of performing Bell-inequality tests using rather
virtual particles, again a regime in which we are not aware of existing tests.
Finally, the use of “self-measuring” quantum spin states – which exploit the
chiral nature of the weak force to measure the bosons’ spins from the emitted
direction of their daughter leptons – challenges the assumption of the
experimentalist’s freedom-of-choice that is typically made in Bell tests.
These features provide motivation for increasing our understanding Bell-
inequality violation in systems of vector bosons, however we do not rely on
them in what follows.

##  2 Scalar states of two vector bosons

We are interested in states containing two relativistic vector bosons, which
we label $a$ corresponding to the particle, and $b$ corresponding to its
antiparticle. We consider here states that have sharp momentum, and label the
four momentum of the particle $k$ and that of the antiparticle $p$. The
simplifying assumption of states with sharp momentum allows the study of the
relativistic effects without the additional complications associated with
finite-width effects.

The method of constructing the corresponding single-particle (and single-anti-
particle) states is presented for completeness in Appendices B and C, which
themselves further develop the formalism described in Ref. [21]. The result is
that a covariant boson-antiboson state corresponding to this situation takes
the following form

$e_{\lambda}^{\mu}(k)e_{\sigma}^{\nu}(p)|(k,\lambda);(p,\sigma)\rangle\equiv
e_{\lambda}^{\mu}(k)e_{\sigma}^{\nu}(p)a^{\dagger}_{\lambda}(k)b^{\dagger}_{\sigma}(p)|0\rangle,$
(1)

where the meanings of the amplitudes $e(k)$ and the creation operators
$a^{\dagger},b^{\dagger}$ are as described in Appendix B.

Here we consider a general scalar state in the following form

$|\alpha(k,p)\rangle=g_{\mu\nu}(k,p)e_{\lambda}^{\mu}(k)e_{\sigma}^{\nu}(p)|(k,\lambda);(p,\sigma)\rangle,$
(2)

where

$g_{\mu\nu}(k,p)=\eta_{\mu\nu}+\tfrac{c}{(kp)}\big{(}k_{\mu}p_{\nu}+p_{\mu}k_{\nu}\big{)},\quad
c\in{\mathbb{R}}.$ (3)

We note that transversality condition (60) for amplitudes $e(k)$ reduces the
second term in the bracket in (3) to the $p_{\mu}k_{\nu}$ only. Choosing the
above parametrization we exclude from our considerations the state
$p_{\mu}k_{\nu}e_{\lambda}^{\mu}(k)e_{\sigma}^{\nu}(p)|(k,\lambda);(p,\sigma)\rangle$.
However, this state is separable.

The scalar state defined in Eq. (2) is normalized as follows

$\langle\alpha(k,p)|\alpha(k,p)\rangle=4k^{0}p^{0}(\delta^{3}(\textbf{0}))^{2}A(k,p),$
(4)

with

$A(k,p)=2+\Big{[}c\tfrac{m^{2}}{(kp)}-\tfrac{(kp)}{m^{2}}(1+c)\Big{]}^{2}.$
(5)

Eq. (2) defines a whole variety of scalar states, two of which are
distinguished.

The first one, $|\psi(k,p)\rangle$, corresponds to the choice $c_{\psi}=0$:

$|\psi(k,p)\rangle=\eta_{\mu\nu}e_{\lambda}^{\mu}(k)e_{\sigma}^{\nu}(p)|(k,\lambda);(p,\sigma)\rangle.$
(6)

It is the simplest and the most natural scalar state. In [21] we considered
the Einstein–Podolsky-Rosen type experiment with two bosons in the state
$|\psi(k,p)\rangle$. The normalization factor $A(k,p)$ of the state
$|\psi(k,p)\rangle$ has the simple form

$A^{\psi}(k,p)=2+\tfrac{(kp)^{2}}{m^{4}}.$ (7)

The second interesting scalar state, $|\xi(k,p)\rangle$, corresponds to the
choice $c_{\xi}=-1$:

$|\xi(k,p)\rangle=\Big{(}\eta_{\mu\nu}-\tfrac{1}{(kp)}p_{\mu}k_{\nu}\Big{)}e_{\lambda}^{\mu}(k)e_{\sigma}^{\nu}(p)|(k,\lambda);(p,\sigma)\rangle.$
(8)

The state $|\xi(k,p)\rangle$ has the property that in the massless limit it
converges to a scalar two-photon state. This question was discussed in detail
in [37]. Notice also that the normalization factor $A(k,p)$ of the state
$|\xi(k,p)\rangle$ also takes the simple form

$A^{\xi}(k,p)=2+\tfrac{m^{4}}{(kp)^{2}}.$ (9)

##  3 Spin operator for a relativistic particle

When we want to calculate explicitly correlation functions, we need to
introduce the spin operator for relativistic massive particles. The choice of
such an operator is not a trivial problem. We know that in the carrier space
of a unitary representation of the Poincaré group there exists a well-defined
square of the spin operator

$\hat{\vec{S}}^{2}=s(s+1)I=-\frac{W^{\mu}W_{\mu}}{m^{2}},$ (10)

where $s$ denotes spin of a particle and

$\hat{W}^{\mu}=\tfrac{1}{2}\epsilon^{\mu\nu\gamma\delta}\hat{P}_{\nu}\hat{J}_{\gamma\delta}$
(11)

is the Pauli-Lubanski four-vector, $\hat{P}_{\nu}$ is the four-momentum
operator, $\hat{J}_{\mu\nu}$ denote the generators of the Lorentz group such
that $U(\Lambda)=\exp(i\omega^{\mu\nu}\hat{J}_{\mu\nu})$, and we assume
$\epsilon^{0123}=1$. On the other hand, spin can be defined as a difference
between total angular momentum and the orbital angular momentum
$\hat{\vec{L}}=\hat{\vec{Q}}\times\hat{\vec{P}}$:

$\hat{\vec{S}}=\hat{\vec{J}}-\hat{\vec{Q}}\times\hat{\vec{P}}.$ (12)

Total angular momentum $\hat{\vec{J}}$ is well defined as the generator of the
rotations, $\hat{J}^{i}=\varepsilon^{ijk}\hat{J}^{jk}$, and the momentum
operator $\hat{\vec{P}}$ is also well defined (compare Eq. (53)), but there
does not exist a generally accepted position operator $\hat{\vec{Q}}$[31].
Different choices of the position operator $\hat{\vec{Q}}$ lead to different
spin operators. The most popular position operator was introduced by Newton
and Wigner[38]

$\hat{\vec{Q}}_{NW}=-\frac{1}{2}\Big{[}\frac{1}{\hat{P}^{0}}\hat{\textbf{K}}+\hat{\textbf{K}}\frac{1}{\hat{P}^{0}}\Big{]}-\frac{\hat{\textbf{P}}\times\hat{\textbf{W}}}{m\hat{P}^{0}(m+\hat{P}^{0})},$
(13)

where $\hat{\vec{K}}$ denotes the boost generators,
$\textbf{K}^{i}=\textbf{J}^{0i}$. The Newton-Wigner position operator
possesses many desirable properties: it is a vector with commuting, self-
adjoint components and it is defined for arbitrary spin. Unfortunately,
$\hat{\vec{Q}}_{NW}$ does not transform in a manifestly covariant way under
Lorentz boosts.

The spin operator related to the Newton-Wigner position operator is equal to

$\hat{\textbf{S}}=\frac{1}{m}\Big{(}\hat{\textbf{W}}-\hat{W}^{0}\frac{\hat{\vec{P}}}{\hat{P}^{0}+m}\Big{)}.$
(14)

This operator has several desirable features. Components of this operator
satisfy the standard su(2) Lie algebra commutation relations. Moreover, the
square of the operator (14) in a unitary irreducible representation of the
Poincaré group is equal to (10). What more, the operator (14) is the only
axial vector which is a linear function of the Pauli-Lubanski four-vector
components[39]. Finally, as was shown in [40], the operator $\hat{\textbf{S}}$
has an elegant transformation formula
$\hat{\textbf{S}}^{\prime}=R(\Lambda,\hat{P})\hat{\textbf{S}}$ under Lorentz
group action, where $R(\Lambda,\hat{P})$ is the corresponding Wigner rotation
while $\hat{P}$ is the four-momentum operator. That is why in our opinion the
spin operator (14) is the most appropriate one and we chose it for our
calculations (compare [41]).

The influence of the Newton-Wigner localization of a particle inside a
detector during spin measurement on relativistic quantum correlations, but in
a case of a fermion pair, was considered in [26]. In Appendix D we have
shortly recalled these results adapting them to a system of vector bosons. We
concluded there that for real, massive particles there is no problem with
localization of a particle inside the detector in situation in which it is
directly detected. The extent to which such arguments might also be relevant
to measurements of gauge boson spin via the kinematics of their chiral decays,
as was used e.g. in [32], is a more involved question, and lies beyond the
scope of the present paper.

We note that other spin operators have been also used in the description of
relativistic EPR experiments, the most popular one is the operator used by
Czachor[12]. This operator is related with the so-called center of mass
position operator which has non-commuting components. For more exhaustive
discussion of the problem of choice of the proper relativistic spin operator
see, e.g., [42, 19, 20, 41, 43, 44, 45, 46, 47, 48].

The spin operator $\hat{\textbf{S}}$ (14) acts on one-particle states
according to

$\hat{\textbf{S}}|k,\sigma\rangle=\textbf{S}_{\lambda\sigma}|k,\lambda\rangle,$
(15)

where $S^{i}$ are standard spin-1 matrices (compare, e.g., [49]):

$\displaystyle S^{1}=\tfrac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\\\ \end{pmatrix},\quad S^{2}=\tfrac{i}{\sqrt{2}}\begin{pmatrix}0&-1&0\\\
1&0&-1\\\ 0&1&0\\\ \end{pmatrix},$ (16) $\displaystyle
S^{3}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&0&-1\\\ \end{pmatrix}.$ (17)

An operator which acts like a spin on particles whose momenta belong to some
definite region $\Omega$ in momentum space and gives 0 otherwise has the form

$\hat{\textbf{S}}^{a}_{\Omega}=\int_{\Omega}\tfrac{d^{3}\textbf{k}}{2k^{0}}a^{\dagger}(k)\textbf{S}a(k),$
(18)

where $a(k)=\big{(}a_{+1}(k),a_{0}(k),a_{-1}(k)\big{)}^{T}$. A similar
operator but acting on antiparticles has the form

$\hat{\textbf{S}}^{b}_{\Omega}=\int_{\Omega}\tfrac{d^{3}\textbf{k}}{2k^{0}}b^{\dagger}(k)\textbf{S}b(k).$
(19)

We have

$\hat{\textbf{S}}^{a}_{\Omega}|(k,\lambda)_{a};(p,\sigma)_{b}\rangle=\chi_{\Omega}(k)\textbf{S}_{\lambda^{\prime}\lambda}|(k,\lambda^{\prime})_{a};(p,\sigma)_{b}\rangle$
(20)

and

$\hat{\textbf{S}}^{b}_{\Omega}|(k,\lambda)_{a};(p,\sigma)_{b}\rangle=\chi_{\Omega}(p)\textbf{S}_{\sigma^{\prime}\sigma}|(k,\lambda)_{a};(p,\sigma^{\prime})_{b}\rangle,$
(21)

where $\chi_{\Omega}$ denotes the characteristic function of the set $\Omega$:
$\chi_{\Omega}(k)=1$ for $k\in\Omega$ and $\chi_{\Omega}(k)=0$ for
$k\not\in\Omega$. The spectral decomposition of the operator
$\boldsymbol{\omega}\cdot\hat{\textbf{S}}^{a}_{\Omega}$ can be written as

$\boldsymbol{\omega}\cdot\hat{\textbf{S}}^{a}_{\Omega}=1\cdot\Pi^{a+}_{\Omega\boldsymbol{\omega}}+(-1)\cdot\Pi^{a-}_{\Omega\boldsymbol{\omega}}+0\cdot\Pi^{a0}_{\Omega\boldsymbol{\omega}},$
(22)

where the projectors act in the following way:

$\Pi^{a\pm}_{\Omega\boldsymbol{\omega}}|(k,\lambda)_{a};(p,\sigma)_{b}\rangle=\\\
\tfrac{1}{2}\chi_{\Omega}(k)\Big{(}(\boldsymbol{\omega}\cdot\textbf{S})^{2}_{\lambda^{\prime}\lambda}\pm(\boldsymbol{\omega}\cdot\textbf{S})_{\lambda^{\prime}\lambda}\Big{)}|(k,\lambda^{\prime})_{a};(p,\sigma)_{b}\rangle,$
(23)
$\Pi^{a0}_{\Omega\boldsymbol{\omega}}|(k,\lambda)_{a};(p,\sigma)_{b}\rangle=\\\
\chi_{\Omega}(k)\Big{(}\delta_{\lambda^{\prime}\lambda}-(\boldsymbol{\omega}\cdot\textbf{S})^{2}_{\lambda^{\prime}\lambda}\Big{)}|(k,\lambda^{\prime})_{a};(p,\sigma)_{b}\rangle.$
(24)

Analogous formulas can easily be found for
$\boldsymbol{\omega}\cdot\hat{\textbf{S}}^{b}_{\Omega}$.

##  4 Probabilities

Now, let two distant observers, Alice and Bob, be at rest with respect to a
given inertial frame and share a pair of bosons in the scalar state
$|\alpha(k,p)\rangle$ (Eq. (2)). The probability that Alice obtains $\sigma$
and Bob $\lambda$ ($\lambda,\sigma\in\\{-1,0,1\\}$), when measuring spin
projections on the directions a and b, respectively, are given by the formula

$P_{\sigma\lambda}=\frac{\langle\alpha(k,p)|\hat{\Pi}_{A\textbf{a}}^{\sigma}\hat{\Pi}_{B\textbf{b}}^{\lambda}|\alpha(k,p)\rangle}{\langle\alpha(k,p)|\alpha(k,p)\rangle}.$
(25)

Here we assume that Alice (Bob) can register only particles (antiparticles)
whose momenta belong to the region $A$ ($B$) in the momentum space and that
they use the spin operator given in Eq. (14). Projectors in the above formula
are defined in Eqs. (23,24). Further, assuming that Alice can measure only
bosons with four-momentum $k$ and Bob those with four-momentum $p$ we find

$P_{\pm\pm}=\frac{1}{4A(k,p)}\operatorname{Tr}\\{M(k,\textbf{a})g(k,p)M(p,\textbf{b})g(k,p)^{T}\\\
-N(k,\textbf{a})g(k,p)N(p,\textbf{b})g(k,p)^{T}\\},$ (26a)
$P_{\pm\mp}=\frac{1}{4A(k,p)}\operatorname{Tr}\\{M(k,\textbf{a})g(k,p)M(p,\textbf{b})g(k,p)^{T}\\\
+N(k,\textbf{a})g(k,p)N(p,\textbf{b})g(k,p)^{T}\\},$ (26b)
$P_{0\pm}=\frac{1}{2A(k,p)}\operatorname{Tr}\\{T(k,\textbf{a})g(k,p)M(p,\textbf{b})g(k,p)^{T}\\},$
(26c) $P_{\pm
0}=\frac{1}{2A(k,p)}\operatorname{Tr}\\{M(k,\textbf{a})g(k,p)T(p,\textbf{b})g(k,p)^{T}\\},$
(26d)
$P_{00}=\frac{1}{A(k,p)}\operatorname{Tr}\\{T(k,\textbf{a})g(k,p)T(p,\textbf{b})g(k,p)^{T}\\},$
(26e)

where

$g(k,p)=[g_{\mu\nu}(k,p)]$ (27)

and where we have introduced the following notation

$M(q,\boldsymbol{\omega})=[M(q,\boldsymbol{\omega})^{\mu\nu}]=e^{*\mu}_{\lambda}(q)(\boldsymbol{\omega}\cdot\textbf{S})^{2}_{\lambda\sigma}e^{\nu}_{\sigma}(q),$
(28)
$N(q,\boldsymbol{\omega})=[N(q,\boldsymbol{\omega})^{\mu\nu}]=e^{*\mu}_{\lambda}(q)(\boldsymbol{\omega}\cdot\textbf{S})_{\lambda\sigma}e^{\nu}_{\sigma}(q),$
(29)
$T(q,\boldsymbol{\omega})=[T(q,\boldsymbol{\omega})^{\mu\nu}]=e^{*\mu}_{\lambda}(q)\big{(}\delta_{\lambda\sigma}-(\boldsymbol{\omega}\cdot\textbf{S})^{2}_{\lambda\sigma}\big{)}e^{\nu}_{\sigma}(q).$
(30)

With the help of the above formulas one can find $P_{\lambda\sigma}$ for
arbitrary $k$ and $p$. However, the resulting formulas are complicated. Thus,
here we restrict ourselves to the situation when Alice and Bob’s frame
coincides with the center of mass frame of the boson pair
($p=k^{\pi}=(k^{0},-\textbf{k})$). Let us introduce the notation

$x=\tfrac{\textbf{k}^{2}}{m^{2}},\quad\textbf{n}=\tfrac{\textbf{k}}{|\textbf{k}|}.$
(31)

Using this notation we have in the center of mass frame

$A(k,k^{\pi})=2+\tfrac{[4x(x+1)(c+1)+1]^{2}}{(2x+1)^{2}}$ (32)

and

$P_{\pm\pm}=\frac{1}{4A(k,k^{\pi})}\Big{\\{}1+4x(x+1)\big{(}1+2c\tfrac{x}{2x+1}\big{)}\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}\big{[}1-(\textbf{a}\cdot\textbf{n})^{2}-(\textbf{b}\cdot\textbf{n})^{2}\big{]}+\Big{[}(\textbf{a}\cdot\textbf{b})+2x\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\Big{]}^{2}\\\
-2\Big{[}(\textbf{a}\cdot\textbf{b})+2x\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}\big{[}(\textbf{a}\cdot\textbf{b})-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}\Big{]}\Big{\\}},$
(33a)
$P_{\pm\mp}=\frac{1}{4A(k,k^{\pi})}\Big{\\{}1+4x(x+1)\big{(}1+2c\tfrac{x}{2x+1}\big{)}\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}\big{[}1-(\textbf{a}\cdot\textbf{n})^{2}-(\textbf{b}\cdot\textbf{n})^{2}\big{]}+\Big{[}(\textbf{a}\cdot\textbf{b})+2x\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\Big{]}^{2}\\\
+2\Big{[}(\textbf{a}\cdot\textbf{b})+2x\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}\big{[}(\textbf{a}\cdot\textbf{b})-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}\Big{]}\Big{\\}},$
(33b)
$P_{0\pm}=\frac{1}{2A(k,k^{\pi})}\Big{\\{}1+4x(x+1)\big{(}1+2c\tfrac{x}{2x+1}\big{)}\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}(\textbf{a}\cdot\textbf{n})^{2}-\Big{[}(\textbf{a}\cdot\textbf{b})+2x\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\Big{]}^{2}\Big{\\}},$
(33c) $P_{\pm
0}=\frac{1}{2A(k,k^{\pi})}\Big{\\{}1+4x(x+1)\big{(}1+2c\tfrac{x}{2x+1}\big{)}\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}(\textbf{b}\cdot\textbf{n})^{2}-\Big{[}(\textbf{a}\cdot\textbf{b})+2x\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\Big{]}^{2}\Big{\\}},$
(33d)
$P_{00}=\frac{1}{A(k,k^{\pi})}\Big{[}(\textbf{a}\cdot\textbf{b})+2x\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\Big{]}^{2}.$
(33e)

The correlation function defined as

$\displaystyle C(\textbf{a},k;\textbf{b},k^{\pi})$
$\displaystyle=\sum_{\lambda,\sigma=-1,0,1}\lambda\sigma
P_{\lambda\sigma}=2(P_{\pm\pm}-P_{\pm\mp})$ (34)

is equal to

$C(\textbf{a},k;\textbf{b},k^{\pi})=-\tfrac{2}{A(k,k^{\pi})}\Big{[}(\textbf{a}\cdot\textbf{b})\\\
+2x\big{(}1+2c\tfrac{x+1}{2x+1}\big{)}\Big{(}(\textbf{a}\cdot\textbf{b})-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\Big{)}\Big{]}.$
(35)

Let us now consider the nonrelativistic and ultrarelativistic limits of the
above probabilities.

###  4.0.1 Nonrelativistic limit

In the nonrelativistic limit ($x\to 0$) we obtain

$P_{\pm\pm}(x\to 0)=\tfrac{1}{12}[1-(\textbf{a}\cdot\textbf{b})]^{2},$ (36a)
$P_{\pm\mp}(x\to 0)=\tfrac{1}{12}[1+(\textbf{a}\cdot\textbf{b})]^{2},$ (36b)
$P_{0\pm}(x\to 0)=P_{\pm 0}(x\to
0)=\tfrac{1}{6}[1-(\textbf{a}\cdot\textbf{b})^{2}],$ (36c) $P_{00}(x\to
0)=\tfrac{1}{3}(\textbf{a}\cdot\textbf{b})^{2}.$ (36d)

These probabilities are the same as those calculated in the framework of
nonrelativistic quantum mechanics in the singlet spin-1 state

$|\Psi^{\mathrm{nonrel}}_{\mathrm{singlet}}\rangle=\tfrac{1}{\sqrt{3}}\big{(}|1\rangle|-1\rangle-|0\rangle|0\rangle+|-1\rangle|1\rangle\big{)}.$
(37)

###  4.0.2 Ultrarelativistic limit

In the ultrarelativistic limit ($x\to\infty$) we have to distinguish two
separate cases: $c=-1$, corresponding to the state $|\xi\rangle$, and
$c\not=-1$ which contains the state $|\psi\rangle$.

In the case $c\not=-1$ we get

$P^{c\not=-1}_{\pm\pm}(x\to\infty)=P^{c\not=-1}_{\pm\mp}(x\to\infty)\\\
=\tfrac{1}{4}\big{[}1-(\textbf{a}\cdot\textbf{n})^{2}\big{]}\big{[}1-(\textbf{b}\cdot\textbf{n})^{2}\big{]},$
(38a)
$P^{c\not=-1}_{0\pm}(x\to\infty)=\tfrac{1}{2}(\textbf{a}\cdot\textbf{n})^{2}\big{[}1-(\textbf{b}\cdot\textbf{n})^{2}\big{]},$
(38b) $P^{c\not=-1}_{\pm
0}(x\to\infty)=\tfrac{1}{2}(\textbf{b}\cdot\textbf{n})^{2}\big{[}1-(\textbf{a}\cdot\textbf{n})^{2}\big{]},$
(38c)
$P^{c\not=-1}_{00}(x\to\infty)=(\textbf{a}\cdot\textbf{n})^{2}(\textbf{b}\cdot\textbf{n})^{2}.$
(38d)

The correlation function in this case vanishes

$C^{c\not=-1}(\textbf{a},k;\textbf{b},k^{\pi},x\to\infty)=0.$ (39)

The case $c\not=-1$ includes the state $|\psi(k,k^{\pi})\rangle$. The
probabilities in the ultrarelativistic limit in this state were given in Eq.
(60) in Ref. [21] and they coincide with the above formulas.

The case $c=-1$ corresponds to the state $|\xi(k,k^{\pi})\rangle$ and for this
state we obtain

$P^{\xi}_{\pm\pm}(x\to\infty)=\tfrac{1}{8}\big{\\{}\big{[}(\textbf{a}\cdot\textbf{b})-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\\\
+\big{[}(\textbf{a}\cdot\textbf{n})-(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(40a)
$P^{\xi}_{\pm\mp}(x\to\infty)=\tfrac{1}{8}\big{\\{}\big{[}(\textbf{a}\cdot\textbf{b})-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\\\
+\big{[}(\textbf{a}\cdot\textbf{n})+(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(40b)
$P^{\xi}_{0\mp}(x\to\infty)=\tfrac{1}{4}\big{\\{}1-(\textbf{a}\cdot\textbf{n})^{2}\\\
-\big{[}(\textbf{a}\cdot\textbf{b})-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(40c) $P^{\xi}_{\mp
0}(x\to\infty)=\tfrac{1}{4}\big{\\{}1-(\textbf{b}\cdot\textbf{n})^{2}\\\
-\big{[}(\textbf{a}\cdot\textbf{b})-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(40d)
$P^{\xi}_{00}(x\to\infty)=\tfrac{1}{2}\big{[}(\textbf{a}\cdot\textbf{b})-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2},$
(40e)

while the correlation function is equal to

$C^{\xi}(\textbf{a},k;\textbf{b},k^{\pi},x\to\infty)=-(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n}).$
(41)

It is interesting to observe that the state $|\xi(k,k^{\pi})\rangle$ is
distinguished by the property that for this state only the correlation
function does not vanish in the ultrarelativistic limit.

In the following we concentrate on the most interesting (and natural) states
$|\psi\rangle$ and $|\xi\rangle$.

###  4.0.3 Probabilities in the state $|\psi(k,k^{\pi})\rangle$

We have calculated these probabilities in our previous paper [21], they are
given by Eq. (58) from [21]. For convenience we recall them here

$P^{\psi}_{\pm\pm}=\tfrac{1}{4[(2x+1)^{2}+2]}\big{\\{}(2x+1)^{2}-2(2x+1)(\textbf{a}\cdot\textbf{b})\\\
+4x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})-4x(x+1)\big{[}(\textbf{a}\cdot\textbf{n})^{2}+(\textbf{b}\cdot\textbf{n})^{2}\big{]}\\\
+\big{[}(\textbf{a}\cdot\textbf{b})+2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(42a)
$P^{\psi}_{\pm\mp}=\tfrac{1}{4[(2x+1)^{2}+2]}\big{\\{}(2x+1)^{2}+2(2x+1)(\textbf{a}\cdot\textbf{b})\\\
-4x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})-4x(x+1)\big{[}(\textbf{a}\cdot\textbf{n})^{2}+(\textbf{b}\cdot\textbf{n})^{2}\big{]}\\\
+\big{[}(\textbf{a}\cdot\textbf{b})+2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(42b)
$P^{\psi}_{0\mp}=\tfrac{1}{2[(2x+1)^{2}+2]}\big{\\{}1+4x(x+1)(\textbf{a}\cdot\textbf{n})^{2}\\\
-\big{[}(\textbf{a}\cdot\textbf{b})+2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(42c) $P^{\psi}_{\mp
0}=\tfrac{1}{2[(2x+1)^{2}+2]}\big{\\{}1+4x(x+1)(\textbf{b}\cdot\textbf{n})^{2}\\\
-\big{[}(\textbf{a}\cdot\textbf{b})+2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(42d)
$P^{\psi}_{00}=\tfrac{1}{(2x+1)^{2}+2}\big{[}(\textbf{a}\cdot\textbf{b})+2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}.$
(42e)

The correlation function for the state $|\psi(k,k^{\pi})\rangle$ was given in
Eq. (59) in [21] and reads

$C^{\psi}(\textbf{a},k;\textbf{b},k^{\pi})=\tfrac{-2}{(2x+1)^{2}+2}\big{[}(2x+1)(\textbf{a}\cdot\textbf{b})\\\
-2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}.$ (43)

###  4.0.4 Probabilities in the state $|\xi(k,k^{\pi})\rangle$

In this case we insert into Eq. (33) the following values:

$c_{\xi}=-1,\quad A^{\xi}(k,k^{\pi})=2+\tfrac{1}{(2x+1)^{2}}.$ (44)

The resulting probabilities have the following form

$P^{\xi}_{\pm\pm}=\tfrac{1}{4[2(2x+1)^{2}+1]}\big{\\{}\big{[}(2x+1)(\textbf{a}\cdot\textbf{b})-2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\\\
+(2x+1)^{2}-4x(x+1)\big{[}1-(\textbf{a}\cdot\textbf{n})^{2}-(\textbf{b}\cdot\textbf{n})^{2}\big{]}\\\
-2(2x+1)\big{[}(\textbf{a}\cdot\textbf{b})+2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}\big{\\}},$
(45a)
$P^{\xi}_{\pm\mp}=\tfrac{1}{4[2(2x+1)^{2}+1]}\big{\\{}\big{[}(2x+1)(\textbf{a}\cdot\textbf{b})-2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\\\
+(2x+1)^{2}-4x(x+1)\big{[}1-(\textbf{a}\cdot\textbf{n})^{2}-(\textbf{b}\cdot\textbf{n})^{2}\big{]}\\\
+2(2x+1)\big{[}(\textbf{a}\cdot\textbf{b})+2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}\big{\\}},$
(45b)
$P^{\xi}_{0\pm}=\tfrac{1}{2[2(2x+1)^{2}+1]}\big{\\{}1+4x(x+1)\big{[}1-(\textbf{a}\cdot\textbf{n})^{2}\big{]}\\\
-\big{[}(2x+1)(\textbf{a}\cdot\textbf{b})-2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(45c) $P^{\xi}_{\pm
0}=\tfrac{1}{2[2(2x+1)^{2}+1]}\big{\\{}1+4x(x+1)\big{[}1-(\textbf{b}\cdot\textbf{n})^{2}\big{]}\\\
-\big{[}(2x+1)(\textbf{a}\cdot\textbf{b})-2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}\big{\\}},$
(45d)
$P^{\xi}_{00}=\tfrac{1}{2(2x+1)^{2}+1}\big{[}(2x+1)(\textbf{a}\cdot\textbf{b})-2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}^{2}.$
(45e)

The correlation function (35) in the state $|\xi(k,k^{\pi})\rangle$ reads

$C^{\xi}(\textbf{a},k;\textbf{b},k^{\pi})=-\tfrac{2(2x+1)}{2(2x+1)^{2}+1}\big{[}(\textbf{a}\cdot\textbf{b})+2x(\textbf{a}\cdot\textbf{n})(\textbf{b}\cdot\textbf{n})\big{]}.$
(46)

##  5 Bell-type inequalities

Now we are in a position to discuss the violation of Bell-type inequalities in
a system of two vector bosons. We restrict our considerations to the situation
when Alice and Bob are in the center–of–mass frame of a boson pair. We
consider here three inequalities: the Clauser–Horne–Shimony–Holt (CHSH)
inequality [50], the Mermin inequality [51] and the
Collins–Gisin–Linden–Massar–Popescu (CGLMP) inequality [52].

#### CHSH inequality.

The generalized CHSH inequality can be written in the form

$|C(\textbf{a},\textbf{b})-C(\textbf{a},\textbf{d})|+|C(\textbf{c},\textbf{b})+C(\textbf{c},\textbf{d})|\leq
2,$ (47)

where $C(\textbf{a},\textbf{b})$ is the correlation function of spin
projections on the directions a and b. The CHSH inequality is optimal for
detecting quantum nonlocality in a system of two qubits. However, in a system
of two spin 1 particles in a singlet state in nonrelativistic quantum
mechanics the CHSH inequality cannot be violated. In our paper [21] we have
considered the violation of the CHSH inequality in the state
$|\psi(k,k^{\pi})\rangle$, we have shown that this inequality is not violated
in $|\psi(k,k^{\pi})\rangle$. Our further numerical simulations also show that
the CHSH inequality cannot be violated in the state $|\xi(k,k^{\pi})\rangle$,
either.

#### Mermin inequality.

The Mermin inequality for spin 1 particles reads

$C(\textbf{a},\textbf{b})+C(\textbf{b},\textbf{c})+C(\textbf{c},\textbf{a})\leq
1.$ (48)

As it was shown in [51] this inequality should be satisfied in any local,
realistic theory. This inequality cannot be violated in nonrelativistic
quantum mechanics. However, as we have shown in [21], relativistic vector
bosons in the state $|\psi(k,k^{\pi})\rangle$ can violate the Mermin
inequality. We find that bosons in the state $|\xi(k,k^{\pi})\rangle$ also can
violate the Mermin inequality. In Figs. 1,2 we have compared the violation of
the Mermin inequality in the states $|\psi(k,k^{\pi})\rangle$ and
$|\xi(k,k^{\pi})\rangle$ in different configurations.

Figure 1: Comparison of the violation of the Mermin inequality in the state
$|\xi(k,k^{\pi})\rangle$ (blue, dashed line) and in the state
$|\psi(k,k^{\pi})\rangle$ (green, dotted line). The configuration of particles
momenta and measurements directions is the following: $\textbf{n}=(0,0,1)$,
$\textbf{w}=(\cos\phi_{w}\sin\theta_{w},\sin\phi_{w}\sin\theta_{w},\cos\theta_{w})$,
$\textbf{w}\in\\{\textbf{a},\textbf{b},\textbf{c}\\}$ and $\theta_{a}=1.593$,
$\phi_{a}=3.236$, $\theta_{b}=1.564$, $\phi_{b}=1.150$, $\theta_{c}=1.514$,
$\phi_{c}=5.322$. Figure 2: Comparison of the violation of the Mermin
inequality in the state $|\xi(k,k^{\pi})\rangle$ (blue, dashed line) and in
the state $|\psi(k,k^{\pi})\rangle$ (green, dotted line). The configuration of
particles momenta and measurements directions is the following:
$\textbf{n}=(0,0,1)$,
$\textbf{w}=(\cos\phi_{w}\sin\theta_{w},\sin\phi_{w}\sin\theta_{w},\cos\theta_{w})$,
$\textbf{w}\in\\{\textbf{a},\textbf{b},\textbf{c}\\}$ and $\theta_{a}=1.891$,
$\phi_{a}=3.820$, $\theta_{b}=2.589$, $\phi_{b}=0.653$, $\theta_{c}=0.220$,
$\phi_{c}=0.716$.

#### CGLMP inequality.

The CGLMP inequality is an optimal inequality for detecting quantum
nonlocality in a system of two qudits. For two qubits it reduces to the CHSH
inequality. We consider here spin one bosons therefore we present the CGLMP
inequality for two qutrits. We assume that Alice can perform two possible
measurements $A_{1}$ or $A_{2}$, and Bob can perform measurements $B_{1}$ or
$B_{2}$. Each of these measurements can have three outcomes: 0,1,2. Denoting
by $P(A_{i}=B_{j}+k)$ the probability that the outcomes $A_{i}$ and $B_{j}$
differ by $k$ modulo 3, i.e.,
$P(A_{i}=B_{j}+k)=\sum_{l=0}^{l=2}P(A_{i}=l,B_{j}=l+k\mod 3)$, and defining

$\mathcal{I}_{3}=\big{[}P(A_{1}=B_{1})+P(B_{1}=A_{2}+1)\\\
+P(A_{2}=B_{2})+P(B_{2}=A_{1})\big{]}\\\
-\big{[}P(A_{1}=B_{1}-1)+P(B_{1}=A_{2})\\\
+P(A_{2}=B_{2}-1)+P(B_{2}=A_{1}-1)\big{]},$ (49)

the CGLMP inequality can be written in the form

$\mathcal{I}_{3}\leq 2.$ (50)

Identifying spin projection values $-1,0,1$ with outcomes $0,1,2$ in the
following way

$-1\leftrightarrow 0,\quad 0\leftrightarrow 1,\quad 1\leftrightarrow 2,$ (51)

and measurements $A_{1}$, $B_{1}$, $A_{2}$, $B_{2}$ with spin projections on
a, b, c, d, respectively, the $\mathcal{I}_{3}$ takes the form

$\mathcal{I}_{3}=C(\textbf{a},\textbf{b})+C(\textbf{c},\textbf{d})+C(\textbf{a},\textbf{d})-C(\textbf{c},\textbf{b})\\\
+P_{+-}(\textbf{a},\textbf{b})+P_{+-}(\textbf{c},\textbf{d})+P_{-+}(\textbf{a},\textbf{d})-P_{+-}(\textbf{c},\textbf{b})\\\
+P_{00}(\textbf{a},\textbf{b})+P_{00}(\textbf{c},\textbf{d})+P_{00}(\textbf{a},\textbf{d})-P_{00}(\textbf{c},\textbf{b})\\\
-\big{[}P_{0-}(\textbf{a},\textbf{b})+P_{0-}(\textbf{c},\textbf{d})+P_{-0}(\textbf{a},\textbf{d})-P_{0-}(\textbf{c},\textbf{b})\\\
+P_{+0}(\textbf{a},\textbf{b})+P_{+0}(\textbf{c},\textbf{d})+P_{0+}(\textbf{a},\textbf{d})-P_{+0}(\textbf{c},\textbf{b})\big{]}.$
(52)

The probabilities and correlation functions in Eq. (52) are given in Eqs. (33)
and (35). We have found that bosons can violate the CGLMP inequality either in
the state $|\psi(k,k^{\pi})\rangle$ or in the state $|\xi(k,k^{\pi})\rangle$.
In Figs. 3, 4, 5 we have compared the violation of the CGLMP inequality for
the states $|\psi(k,k^{\pi})\rangle$ and $|\xi(k,k^{\pi})\rangle$ in different
configurations. In all these figures the point $x=0$ corresponds to the
nonrelativistic case.

Figure 3: Comparison of the violation of the CGLMP inequality in the state
$|\xi(k,k^{\pi})\rangle$ (blue, dashed line) and in the state
$|\psi(k,k^{\pi})\rangle$ (green, dotted line). The configuration of particles
momenta and measurements directions is the following: $\textbf{n}=(0,0,1)$,
$\textbf{w}=(\cos\phi_{w}\sin\theta_{w},\sin\phi_{w}\sin\theta_{w},\cos\theta_{w})$,
$\textbf{w}\in\\{\textbf{a},\textbf{b},\textbf{c},\textbf{d}\\}$ and
$\theta_{a}=2.667$, $\phi_{a}=4.109$, $\theta_{b}=0.924$, $\phi_{b}=0.974$,
$\theta_{c}=2.699$, $\phi_{c}=1.005$, $\theta_{d}=0$, $\phi_{d}=0$. Figure 4:
Comparison of the violation of the CGLMP inequality in the state
$|\xi(k,k^{\pi})\rangle$ (blue, dashed line) and in the state
$|\psi(k,k^{\pi})\rangle$ (green, dotted line). The configuration of particles
momenta and measurements directions is the following: $\textbf{n}=(0,0,1)$,
$\textbf{w}=(\cos\phi_{w}\sin\theta_{w},\sin\phi_{w}\sin\theta_{w},\cos\theta_{w})$,
$\textbf{w}\in\\{\textbf{a},\textbf{b},\textbf{c},\textbf{d}\\}$ and
$\theta_{a}=3.141$, $\phi_{a}=0$, $\theta_{b}=0$, $\phi_{b}=0$,
$\theta_{c}=0.836$, $\phi_{c}=5.044$, $\theta_{d}=2.754$, $\phi_{d}=1.897$.
Figure 5: Comparison of the violation of the CGLMP inequality in the state
$|\xi(k,k^{\pi})\rangle$ (blue, dashed line) and in the state
$|\psi(k,k^{\pi})\rangle$ (green, dotted line). The configuration of particles
momenta and measurements directions is the following: $\textbf{n}=(0,0,1)$,
$\textbf{w}=(\cos\phi_{w}\sin\theta_{w},\sin\phi_{w}\sin\theta_{w},\cos\theta_{w})$,
$\textbf{w}\in\\{\textbf{a},\textbf{b},\textbf{c},\textbf{d}\\}$ and
$\theta_{a}=2.532$, $\phi_{a}=3.141$, $\theta_{b}=1.213$, $\phi_{b}=0$,
$\theta_{c}=2.378$, $\phi_{c}=1.363$, $\theta_{d}=0$, $\phi_{d}=0$.

##  6 Conclusions

The recent paper [32] suggested that it might be possible to experimentally
test the violation of Bell inequalities with a pair of $W^{+}W^{-}$ bosons.
Motivated by this paper and by our previous theoretical works [21, 37] we have
discussed the violation of the Mermin and CGLMP inequalities in a system of
two relativistic vector bosons in a scalar state. We have derived formulas for
probabilities in the EPR-type experiment in the general scalar state (2),
assuming that Alice and Bob measure spin projections on given directions.
These probabilities depend on spin projection direction and bosons momenta. We
have considered in detail the situation when Alice and Bob are at rest in the
center of mass frame of the boson pair. In this case we have explicitly
calculated probabilities in two states of particular interest: the simplest
nonseparable state $|\psi\rangle$ (6) and the state which in the massless
limit converges to the scalar two-photon state $|\xi\rangle$ (8). We have
shown that both the Mermin and CGLMP inequalities can be violated in both
states $|\psi\rangle$ and $|\xi\rangle$ and that the degree of violation
depends on bosons momenta.

Large violations of the CGLMP inequality are predicted even for boson pairs at
rest. For small boosts of the vector bosons, both the Mermin, and more
particularly the CGLMP inequalities show sufficiently large violations that
they might well be measurable in practical experiments. For the CGLMP
inequality, our analytical results, here calculated in the narrow-width
approximation, and with observers at rest in the centre-of-mass-frame of the
boson pair, support the conclusion found using numerical results in
simulations of $H\to WW^{*}$ decays in Ref. [32] under different assumptions.

Potential applications of these results extend beyond the case of Higgs boson
decays. Other example applications might include relativistic hadronic,
nuclear, atomic or molecular systems.

AJB is grateful to Christopher Hays and to Alexander Karlberg for helpful
discussions. AJB is supported through STFC grants ST/R002444/1 and
ST/S000933/1, and by Merton College, Oxford. PC and JR are supported by
University of Lodz under the IDUB project.

##  Appendix A Massive representations of the Poincaré group for $s=1$

For self-consistency we recall here basic facts related to massive spin 1
representations of the Poincaré group. For more details see, e.g, [53]. Let
$\mathcal{H}$ be the carrier space of the irreducible massive representation
of the Poincaré group for $s=1$. $\mathcal{H}$ is spanned by the four-momentum
operator eigenvectors $|k,\sigma\rangle$

$\hat{P}^{\mu}|k,\sigma\rangle=k^{\mu}|k,\sigma\rangle,$ (53)

$k^{2}=m^{2}$, $m$ is the mass of the particle, and $\sigma$ its spin
component along $z$ axis, $\sigma=-1,0,1$. We use the following Lorentz-
covariant normalization

$\langle
k,\sigma|k^{\prime},\sigma^{\prime}\rangle=2k^{0}\delta^{3}(\textbf{k}-\textbf{k}^{\prime})\delta_{\sigma\sigma^{\prime}}.$
(54)

The vectors $|k,\sigma\rangle$ can be generated from the standard vector
$|\tilde{k},\sigma\rangle$, where $\tilde{k}=m(1,0,0,0)$ is the four-momentum
of the particle in its rest frame. We have
$|k,\sigma\rangle=U(L_{k})|\tilde{k},\sigma\rangle$, where Lorentz boost
$L_{k}$ is defined by relations $k=L_{k}\tilde{k}$,
$L_{\tilde{k}}={\mathbb{I}}$.

By means of Wigner procedure we get

$U(\Lambda)|k,\sigma\rangle=\mathcal{D}_{\lambda\sigma}(R(\Lambda,k))|\Lambda
k,\lambda\rangle,$ (55)

where the Wigner rotation $R(\Lambda,k)$ is defined as
$R(\Lambda,k)=L_{\Lambda k}^{-1}\Lambda L_{k}$, and for $s=1$ the
representation $\mathcal{D}(R)$ is unitary equivalent to $R$ by

$\mathcal{D}(R)=VRV^{{\dagger}},\qquad V^{\dagger}V={\mathbb{I}},$ (56)

and the explicit form of the matrix $V$ is the following:

$V=\frac{1}{\sqrt{2}}\begin{pmatrix}-1&i&0\\\ 0&0&\sqrt{2}\\\ 1&i&0\\\
\end{pmatrix}.$ (57)

##  Appendix B Boson field

In order to describe two types of vector bosons, particle and antiparticle
(e.g. $W^{+}$ and $W^{-}$), we consider the free field operator
$\hat{\phi}^{\mu}(x)$ with the following momentum expansion:

$\hat{\phi}^{\mu}(x)=(2\pi)^{(-3/2)}\int\tfrac{d^{3}\textbf{k}}{2\omega_{k}}\big{[}e^{ikx}e^{\mu}_{\sigma}(k)a^{\dagger}_{\sigma}(k)\\\
+e^{-ikx}e^{*\mu}_{\sigma}(k)b_{\sigma}(k)\big{]},$ (58)

where $\omega_{k}=\sqrt{\textbf{k}^{2}+m^{2}}$ and $m$ is a mass of a particle
(and antiparticle). $a^{\dagger}_{\sigma}(k)$, $a_{\sigma}(k)$ and
$b^{\dagger}_{\sigma}(k)$, $b_{\sigma}(k)$ are creation and annihilation
operators of a particle and antiparticle, respectively.
$a^{\dagger}_{\sigma}(k)$ ($b^{\dagger}_{\sigma}(k)$) creates particle
(antiparticle) with four-momentum $k$ and spin component along $x$ axis equal
to $\sigma$. They fulfill the standard canonical commutation relations

$[a_{\sigma}(k),a^{\dagger}_{\sigma^{\prime}}(k^{\prime})]=[b_{\sigma}(k),b^{\dagger}_{\sigma^{\prime}}(k^{\prime})]=2k^{0}\delta(\textbf{k}-\textbf{k}^{\prime})\delta_{\sigma\sigma^{\prime}}$
(59)

and all the other commutators vanish. The Klein-Gordon equation and Lorentz
transversality condition imply

$k^{2}=m^{2},\quad k_{\mu}e^{\mu}_{\sigma}(k)=0.$ (60)

The one-particle and one-antiparticle states

$|k,\lambda\rangle_{a}=a^{\dagger}_{\lambda}(k)|0\rangle,\quad|p,\sigma\rangle_{b}=b^{\dagger}_{\sigma}(p)|0\rangle,$
(61)

where $|0\rangle$ is a Lorentz-invariant vacuum, $\langle 0|0\rangle=1$,
should transform like a basis states of a carrier space of the irreducible,
massive representation of the Poincaré group for $s=1$ considered in Appendix
A. This condition allows us to determine the explicit form of amplitudes
$e^{\mu}_{\sigma}(k)$. The derivation is the same as in [21], therefore we
give here only the results:

$e(k)=[e^{\mu}_{\sigma}(k)]=\begin{pmatrix}\tfrac{\textbf{k}^{T}}{m}\\\
{\mathbb{I}}+\tfrac{\textbf{k}\otimes\textbf{k}^{T}}{m(m+k^{0})}\end{pmatrix}V^{T},$
(62)

where $V$ is given in Eq. (57). Moreover, one can show that
$e^{\mu}_{\sigma}(k)$ fulfills the following relations:

$\displaystyle e^{*}(k)=e(k)VV^{T},$ (63) $\displaystyle
e^{*\mu}_{\sigma}(k)e^{\nu}_{\sigma}(k)=-\eta^{\mu\nu}+\tfrac{k^{\mu}k^{\nu}}{m^{2}},$
(64) $\displaystyle
e^{*\mu}_{\sigma}(k)e_{\mu\lambda}(k)=-\delta_{\sigma\lambda},$ (65)
$\displaystyle
e^{\mu}_{\sigma}(k)e_{\mu\lambda}(k)=-(VV^{T})_{\sigma\lambda}.$ (66)

##  Appendix C Boson states transforming covariantly under Lorentz
transformations

In [21] it was shown that with the help of amplitudes $e^{\mu}_{\sigma}(k)$
one can construct states transforming in the explicitly covariant manner under
Lorentz transformations. In our case covariant particle/anti-particle states
have the form

$|(\mu,k)\rangle_{a/b}=e^{\mu}_{\sigma}(k)|k,\sigma\rangle_{a/b},$ (67)

and under Lorentz transformations transform according to

$U(\Lambda)|(\mu,k)\rangle_{a/b}=(\Lambda^{-1})^{\mu}_{\nu}|(\nu,\Lambda
k)\rangle_{a/b}.$ (68)

Two-particle state describing boson with four-momentum $k$ and spin projection
$\lambda$ and antiboson with four-momentum $p$ and spin projection $\sigma$
has the form

$|(k,\lambda)_{a};(p,\sigma)_{b}\rangle=a^{\dagger}_{\sigma}(k)b^{\dagger}_{\sigma}(p)|0\rangle,$
(69)

and consequently a boson–antiboson covariant state reads

$e^{\mu}_{\lambda}(k)e^{\nu}_{\sigma}(p)|(k,\lambda)_{a};(p,\sigma)_{b}\rangle.$
(70)

##  Appendix D Localization inside detectors

Eigenvectors of the Newton–Wigner operator have the following form[26]

$|\textbf{x},\sigma\rangle=\frac{1}{(2\pi)^{3/2}}\int\frac{d^{3}\textbf{p}}{2p^{0}}\sqrt{2p^{0}}e^{-i\textbf{p}\cdot\textbf{x}}|p,\sigma\rangle,$
(71)

where $p^{0}=\sqrt{\textbf{p}^{2}+m^{2}}$. The problem of localization of a
particle inside a detector during spin measurement in the context of
relativistic quantum correlations of a fermion pair was also discussed in
[26]. Here we briefly present that paper’s conclusions, adapting them to a
system of vector bosons. The central element in such a discussion is the
projector on a region $D$ in the coordinate space

$\displaystyle\hat{\Pi}_{D}$
$\displaystyle=\int_{D}d^{3}\textbf{x}\sum_{\sigma=-1,0,1}|\textbf{x},\sigma\rangle\langle\textbf{x},\sigma|$
(72)
$\displaystyle=\iint\limits_{{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}}\frac{d^{3}\textbf{p}}{\sqrt{2p^{0}}}\frac{d^{3}\textbf{p}^{\prime}}{\sqrt{2{p^{\prime}}^{0}}}\Delta_{D}(\textbf{p}^{\prime}-\textbf{p})\sum_{\sigma}|p^{\prime},\sigma\rangle\langle
p,\sigma|,$ (73)

where $|\textbf{x},\sigma\rangle$ is defined in (71) and

$\displaystyle\Delta_{D}(\textbf{p}^{\prime}-\textbf{p})$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int_{D}d^{3}\textbf{x}e^{-i(\textbf{p}^{\prime}-\textbf{p})\cdot\textbf{x}}$
(74)
$\displaystyle=\frac{1}{(2\pi\hbar)^{3}}\int_{D}d^{3}\textbf{x}e^{-i\frac{(\textbf{p}^{\prime}-\textbf{p})\cdot\textbf{x}}{mc\lambda}}.$
(75)

To facilitate further discussion, in the second line we have explicitly
expressed $\Delta_{D}$ in the standard units, $c$ is the velocity of light and
$\lambda=\frac{\lambda_{C}}{2\pi}=\frac{\hbar}{mc}$ is the particle Compton
wavelength divided by $2\pi$. To evaluate quantitatively the influence of
localization let us choose $D$ as the cube located in the center of the
coordinate frame with edges of the length $l$ parallel to coordinate axis. In
this case $\Delta_{\textsf{Cube}}$ can be calculated and we obtain

$\Delta_{\textsf{Cube}}(\textbf{p}^{\prime}-\textbf{p})=\Big{(}\frac{\lambda}{\hbar}\Big{)}^{3}\prod_{j=1}^{3}\Big{[}\frac{1}{\pi}\frac{l}{2\lambda}\mathrm{sinc}\,\Big{(}\frac{p_{j}^{\prime}-p_{j}}{mc}\frac{l}{2\lambda}\Big{)}\Big{]}.$
(76)

Using the formula

$\lim\limits_{\tau\to\infty}\frac{\tau}{\pi}\mathrm{sinc}\,(\tau x)=\delta(x)$
(77)

we observe that for $l\to\infty$:
$\Delta_{\textsf{Cube}}(\textbf{p}^{\prime}-\textbf{p})\to\delta^{3}(\textbf{p}^{\prime}-\textbf{p})$
and consequently

$\hat{\Pi}_{D}\to\hat{\Pi}_{{\mathbb{R}}^{3}}=I.$ (78)

Therefore, there is no problem with localization of a quantum particle
provided that its Compton wavelength is sufficiently small in comparison with
a detecting element size $l$, i.e., when the condition $\lambda\ll l$ holds.

For muons and electrons at rest the corresponding Compton wavelengths take the
following approximate values: $\lambda_{\mu}=1.87\times 10^{-15}$ m and
$\lambda_{e}=3.86\times 10^{-13}$ m. Consequently, assuming a realistic size
of particle detector pixels $l=10^{-6}$ m, the scaling factors
$\tau=l/(2\lambda)$ take the values $\tau_{\mu}=0.27\times 10^{9}$ and
$\tau_{e}=0.13\times 10^{7}$, respectively. Moreover, for relativistic
particles we should use $m=\gamma m_{\textsf{rest}}$, where $\gamma$ is the
particle Lorentz factor, so $\lambda=\gamma^{-1}\lambda_{\textsf{rest}}$ and
in effect the Compton wavelength decreases so the scaling factor increases.
Thus, for real, massive particles
$\Delta_{\textsf{Cube}}(\textbf{p}^{\prime}-\textbf{p})$ is very close to
$\delta^{3}(\textbf{p}^{\prime}-\textbf{p})$ and, indeed, there is no problem
with localization of a particle inside the detector in situation in which it
is directly detected.

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