Relativistic deformed kinematics from locality conditions in a generalized spacetime

Rela i is ic de o med kinema ics om locali y condi ions
in a gene alized space ime
J. M. Ca mona ,*J. L. Co ´es ,†and J. J. Relancio ‡
Depa amen o de Física Teó ica and Cen o de As opa ículas y Física de Al as Ene gías (CAPA),
Uni e sidad de Za agoza, Za agoza 50009, Spain
(Recei ed 21 Janua y 2020; accep ed 7 Feb ua y 2020; published 28 Feb ua y 2020)
We show how a de o med composi ion law o ou -momen a can be used o de ine, a he classical le el,
a modi ied no ion o space ime o a sys em o wo pa icles h ough he c ossing o wo ldlines in pa icle
in e ac ions. We p esen a de i a ion o a gene ic ela i is ic iso opic de o med kinema ics and discuss he
complemen a i y and ela ions wi h o he de i a ions based on κ-Poinca ´e Hop algeb a o on he geome y
o a maximally symme ic momen um space.
DOI: 10.1103/PhysRe D.101.044057
I. INTRODUCTION
Special- ela i is ic (SR) kinema ics is a consequence
o he no ion o space ime in Eins ein’s SR heo y. In a
quan um heo y o g a i y (QG), a quan um no ion o
space ime will eplace he classical no ion which leads o
SR kinema ics. A e 100 yea s o sea ching o his heo y,
we s ill do no ha e a good es able candida e o QG,
pa ially due o he di icul y in inding obse able e ec s
o he heo y. This has led o he sea ch o al e na i es o
he pu ely (unsuccess ul) heo e ical app oaches, opening a
ecen new app oach known as quan um g a i y phenom-
enology [1–5]. Many wo ks wi hin his new app oach a e
based on he na u al expec a ion ha he quan um s uc u e
o space ime will mani es h ough a modi ica ion o he
SR kinema ics. The consis ency wi h e y p ecise es s o
Lo en z in a iance [6–9] equi es his modi ica ion o be
pa ame ized by a new ene gy scale (Λ) such ha , o
obse a ions a ene gies much smalle han his new scale,
he e ec s o he modi ica ion o he SR kinema ics a e e y
small. We will e e o his si ua ion as a de o ma ion o SR
kinema ics (DK). The kinema ics o a p ocess ( ansi ion
be ween an ini ial s a e and a inal s a e o ee pa icles)
is de ined by he exp ession o he ene gy o each pa icle
in e ms o i s momen um (dispe sion ela ion) and by
he conse a ion o he o al ene gy and momen um in he
ansi ion, which is de e mined by he exp ession o he
o al ene gy and momen um o a sys em o ee pa icles
in e ms o he ene gies and momen a o he pa icles
(composi ion law). A DK will be de ined by a de o med
dispe sion ela ion (DDR) and a de o med composi ion
law (DCL).
A possible pa h o ealize he p e ious ideas is o conside
he gene aliza ion o Lie algeb as as he ma hema ical
amewo k o implemen con inuous symme ies in a
classical space ime when one in oduces a noncommu a-
i i y in space ime as a i s s ep o he ansi ion o a
quan um space ime. This leads o he o mula ion o Hop
algeb as, whose main new ing edien is a coalgeb a s uc u e
[10]. An example which has played a e y impo an ole
in a emp s o explo e de o ma ions o he SR kinema ics is
he κ-Poinca ´e Hop algeb a [11], which is based on a
de o ma ion o he Poinca ´e Lie algeb a and a noncommu-
a i e space ime whose coo dina es de ine a (spa ially
iso opic) Lie algeb a (κ-Minkowski space ime). The
Casimi o he de o med Poinca ´e algeb a de ines a DDR
and he cop oduc s o he ansla ion gene a o s (momen um
ope a o s) de ine a DCL. One can in his way associa e a DK
o he κ-Poinca ´e Hop algeb a [12]. In ac , he symme y
s uc u e o he Hop algeb a amewo k ansla es in o a
ela i is ic de o med kinema ics (RDK), i.e., a kinema ics
in a ian unde new Lo en z ans o ma ions connec ing
di e en ine ial obse e s. The de o ma ion mani es s as a
modi ica ion o he Lo en z ans o ma ions o a one-
pa icle s a e (de e mined om he de o ma ion o he
Poinca ´e algeb a) and a (non i ial) modi ica ion o he
Lo en z ans o ma ion o a wo-pa icle sys em (de e mined
om he non i ial cop oduc o he Lo en z gene a o s).
The idea o conside a ela i is ic heo y wi h a second
in a ian (a leng h l), on op o he eloci y c, was mo i a ed
by he appea ance o a minimal leng h [13,14] in di e en
app oaches o QG. This led o conside ing a DDR wi h a new
scale appea ing as a cu o on he ene gy o momen um
* elancio@uniza .es
†jca mona@uniza .es
‡co es@uniza .es
Published by he Ame ican Physical Socie y unde he e ms o
he C ea i e Commons A ibu ion 4.0 In e na ional license.
Fu he dis ibu ion o his wo k mus main ain a ibu ion o
he au ho (s) and he published a icle’s i le, jou nal ci a ion,
and DOI. Funded by SCOAP3.
PHYSICAL REVIEW D 101, 044057 (2020)
2470-0010=2020=101(4)=044057(15) 044057-1 Published by he Ame ican Physical Socie y
[15] as examples o a doubly special ela i i y (DSR).
The nonlinea i y o he Lo en z ans o ma ions which lea e
he DDR in a ian implies a nonlinea i y o he composi ion
law which should be de e mined by he in a iance unde
Lo en z ans o ma ions o he conse a ion o he o al
ene gy and momen um de ined by he nonlinea composi-
ion law. The s udy o hese examples led o iden i ying hei
ela ion wi h he κ-Poinca ´e kinema ics de e mined in he
Hop algeb a amewo k [16].
Mo e ecen ly, a new app oach o a de o ma ion o SR
kinema ics was in oduced based on a model o he
in e ac ion o pa icles de ined by a DCL. The c ossing
o wo ldlines which cha ac e izes he in e ac ion o pa -
icles in he case (SR) o a linea composi ion law no longe
happens due o he de o ma ion o he composi ion law.
The locali y o he in e ac ion ( o any obse e ) is los ;
only he obse e whose o igin is on he in e ac ion sees a
c ossing o wo ldlines. The locali y o in e ac ions in SR is
eplaced by a ela i e locali y [17]. The DCL could be used
o de ine a connec ion in momen um space and, oge he
wi h he iden i ica ion o a DDR om he dis ance be ween
he o igin and a poin in momen um space, one has an
in e p e a ion o a DK based on he geome y o momen um
space [18].
In Re . [19] i was shown ha i one conside s
a maximally symme ic momen um space and chooses
coo dina es in momen um space such ha he me ic is
spa ially iso opic, one can de ine a DDR om he dis ance
be ween he o igin and a poin in momen um space
calcula ed wi h he me ic, and also a DCL om he
isome ies o he me ic which do no lea e he o igin
in a ian ( ansla ions in momen um space). One can show
ha he de o med kinema ics de ined by he me ic is a
RDK. This gi es an al e na i e simple ela ion be ween a
de o med kinema ics and a geome y in momen um space
( he scale o de o ma ion is ela ed o he cu a u e in
momen um space). I also allows (in con as wi h he
ela ion be ween he geome y o momen um space and
he kinema ics based on ela i e locali y) o implemen he
ela i is ic in a iance in a simple way. The Lo en z in a i-
ance o he DDR is a di ec consequence o he iden i-
ica ion o he Lo en z ans o ma ion o a one-pa icle s a e
wi h he isome ies o he me ic which lea e he o igin
in a ian . The Lo en z in a iance o he conse a ion law
de ined by he DCL can be unde s ood wi hin he geome ic
amewo k h ough he iden i ica ion o he DCL and he
Lo en z ans o ma ions as isome ies [19].
A di e en pe spec i e o a de o ma ion o SR kinema ics
based on he Bo n geome y o a doubled phase space has
led o eplacing he classical model o a ee ela i is ic
pa icle by a me apa icle model [20]. Lo en z symme y is in
his case ealized in a di e en way as a g oup o ans-
o ma ions ha lea e he cons ain s which de ine he model
in a ian . The modi ied dispe sion ela ion is iden i ied om
he poles o he momen um in eg al ep esen a ion o he
me apa icle quan um p opaga o ins ead o di ec ly consid-
e ing he cons ain in he classical ac ion. The loss o
absolu e locali y associa ed wi h he modi ied ene gy-
momen um conse a ion law which de ines he in e ac ion
o pa icles in he classical model appea s in he model o
me apa icles as due o he di e en no ion o space ime o
di e en me apa icles wi h di e en alues o he doubled
momen um a iables. The ex ension o he me apa icle
model o include in e ac ions is an open p oblem.
In his wo k we a e going o ollow a di e en pa h in he
s udy o a DK. The idea is o ake he classical model o he
in e ac ion o pa icles de ined by a DCL and y o go om
he loss o locali y in he space ime whose coo dina es a e
he canonical coo dina es o a phase space oge he wi h he
ou -momen um coo dina es, o a new se o space- ime
coo dina es in phase space such ha all he pa icles ha e
he same coo dina es a he in e ac ion. The in e ac ion
de ined by a DCL is hen local in a gene alized wo-pa icle
space ime de ined as a non i ial subspace o he wo-
pa icle phase space. In Re . [21], an ansa z o he new
space- ime coo dina es o each o he pa icles, de ined as a
linea combina ion o hei space- ime coo dina es wi h
coe icien s depending on he momen a o bo h pa icles,
was in oduced. The locali y o he in e ac ion in he new
space ime leads o a sys em o di e en ial equa ions
ela ing he unc ions o momen a which de ine he new
space- ime coo dina es and he DCL. When one assumes
ha he new space- ime coo dina es o one o he pa icles
do no depend on he ou -momen um o he o he pa icle1
and ha hey a e jus a ep esen a ion o κ-Minkowski
noncommu a i e space ime, hen he equa ions de i ed
om he locali y o he in e ac ion can be used o de e mine
he DCL. I one uses he ep esen a ion o κ-Minkowski
space ime which ep oduces he phase-space s uc u e o
he κ-Poinca ´e Hop algeb a in he bic ossp oduc basis, he
co esponding DCL de e mined by locali y u ns ou o be
he one co esponding o he κ-Poinca ´e kinema ics. This
esul shows ha κ-Poinca ´e ela i is ic kinema ics can be
seen as an example o a de o med kinema ics compa ible
wi h he possibili y o iden i y a new space ime whe e
in e ac ions a e local. The au ho s o Re . [22] a i ed a
he same conclusion om a ela ed pe spec i e: by ex end-
ing he model o he in e ac ion o pa icles in 2þ1
dimensions o 3þ1dimensions and implemen ing he
igidi y o ansla ions (which is one way o eph ase he
equi emen o locali y o in e ac ions), one can ep oduce
he κ-Poinca ´e ela i is ic kinema ics.
In he p esen pape we go a s ep u he in he ela ion-
ship be ween a ela i is ic de o med kinema ics and he
de ini ion o gene alized space- ime coo dina es whe e
1This can only be he case o one o he pa icles; o he wise,
he new space- ime coo dina es a e commu a i e and hen one
can always ind a change o momen um a iables ha leads o SR
kinema ics [21].
CARMONA, CORT ´
ES, and RELANCIO PHYS. REV. D 101, 044057 (2020)
044057-2
in e ac ions a e local. While Re . [21] did no include any
es ic ion on he DCL and he noncommu a i e space ime
ha implemen s locali y, he e we conside a di e en ansa z
o implemen ing locali y2: he new space- ime coo dina es
o each pa icle a e linea combina ions o he space- ime
coo dina es o bo h pa icles, bu he coe icien s o he
space- ime coo dina es o each pa icle depend only on i s
momen um. This de ini ion o new space- ime coo dina es
in he phase space o he wo-pa icle sys em may be seen
as a mo e na u al p esc ip ion han he one made in
Re . [21], whe e he gene alized space- ime coo dina es
depend on bo h momen a bu do no mix he space- ime
coo dina es o he wo pa icles. Mo eo e , he new ansa z
imposes a s ong condi ion on he DCL, so ha no e e y
composi ion law can lead o local in e ac ions.
The s uc u e o he pape is as ollows. In Sec. II we
de ine a gene alized wo-pa icle sys em space ime which
implemen s he locali y o in e ac ions wi h a DCL. In he
new space- ime coo dina es, one has a sum o wo con-
ibu ions, each in ol ing he phase-space coo dina es o
one o he pa icles. As we show in Sec. III, he new sys em
o equa ions ela ing he unc ions o one ou -momen um
which de ine he new space- ime coo dina es and he
de i a i es o he DCL can in his case be used o di ec ly
de e mine he DCL when one makes he ansa z ha he
de o med composi ion law con ains only e ms p opo -
ional o he in e se o he scale o de o ma ion Λ(DCL1).
We will la e show ha such a locali y-compa ible DCL1
co esponds o he κ-Poinca ´e composi ion law in a basis
ha is di e en om he bic ossp oduc basis (which is he
mos widely used in κ-Poinca ´e s udies). Once we ha e
ob ained he composi ion law, we s udy he noncommu a-
i i y o he one-pa icle and wo-pa icle space imes
de ined by he locali y o in e ac ions.
In Sec. IV we de e mine he co esponding DDR which,
oge he wi h a DCL1 compa ible wi h locali y, de ines a
RDK, and he nonlinea Lo en z ans o ma ions o he
one-pa icle and wo-pa icle sys ems. This p o ides a new
way o de i e a RDK based on he physical p inciple o
locali y o in e ac ions, which is an al e na i e o he mo e
o mal de i a ions o a RDK based on κ-Poinca ´e Hop
algeb a o on he geome y o a maximally symme ic
momen um space. As we will see in Sec. V, he new
de i a ion o he ela i is ic de o med kinema ics based on
locali y (o on he geome y o a maximally symme ic
momen um space) no only ep oduces he esul s based on
he κ-Poinca ´e Hop algeb a, bu also iden i ies an al e -
na i e in which he new ene gy scale o he de o ma ion
does no appea as a maximum ene gy. Then, in Sec. VI we
s udy he ole o associa i i y in he de ini ion o a RDK
and conclude ha an associa i e DCL1 (which co esponds
o κ-Poinca ´e in a ce ain basis) is he only ela i is ic
iso opic gene aliza ion o SR kinema ics compa ible wi h
locali y.
We end in Sec. VII wi h a summa y and p ospec s o
u he wo k.
II. SPACETIME FROM LOCALITY
We conside he classical model o he in e ac ion o wo
pa icles wi h a de o med kinema ics de ined by he ac ion
S¼Z0
−∞
dτX
i¼1;2
½xμ
−ðiÞðτÞ_
p−ðiÞ
μðτÞ
þN−ðiÞðτÞ½Cðp−ðiÞðτÞÞ −m2
−ðiÞ
þZ∞
0
dτX
j¼1;2
½xμ
þðjÞðτÞ_
pþðjÞ
μðτÞ
þNþðjÞðτÞ½CðpþðjÞðτÞÞ −m2
þðjÞ
þξμ½Pþ
μð0Þ−P−
μð0Þ;ð1Þ
whe e _
a≐ðda=dτÞis he de i a i e o he a iable awi h
espec o he pa ame e τalong he ajec o y o he
pa icle, x−ðiÞ(xþðjÞ) a e he space- ime coo dina es o
he in-s a e (ou -s a e) pa icles, p−ðiÞ(pþðjÞ) a e hei ou -
momen a, m−ðiÞ(mþðjÞ) a e hei masses, P−(Pþ) is he
o al ou -momen um o he in-s a e (ou -s a e) de ining
he DCL, CðkÞis a unc ion o a ou -momen um kde ining
he DDR, ξμa e Lag ange mul iplie s ha implemen
he ene gy-momen um conse a ion in he in e ac ion,
and N−ðiÞ(NþðjÞ) a e Lag ange mul iplie s ha implemen
he dispe sion ela ion o in-s a e (ou -s a e) pa icles.
The a ia ional p inciple applied o he ac ion (1) ixes
he end (s a ing) space- ime coo dina es o he ajec o ies
o he in-s a e (ou -s a e) pa icles,
xμ
−ðiÞð0Þ¼ξν∂P−
ν
∂p−ðiÞ
μ
ð0Þ;x
μ
þðjÞð0Þ¼ξν∂Pþ
ν
∂pþðjÞ
μ
ð0Þ:ð2Þ
When he o al ou -momen um is jus he sum o
he ou -momen a o he pa icles, one has xμ
−ðiÞð0Þ¼
xμ
þðjÞð0Þ¼ξμand he wo ldlines o he ou pa icles c oss
a he poin wi h coo dina es ξμ(local in e ac ion). When
one has a DCL, he locali y o he in e ac ion is los .
We now ask he ques ion whe he i is possible o iden i y
new space- ime coo dina es in he phase space o he wo
pa icles (we conside ei he he wo pa icles in he in-s a e
o ou -s a e and hen omi he index −,þ),
˜
xα
ð1Þ¼xμ
ð1Þφα
μðpð1ÞÞþxμ
ð2Þφð2Þα
ð1Þμðpð2ÞÞ;
˜
xα
ð2Þ¼xμ
ð2Þφα
μðpð2ÞÞþxμ
ð1Þφð1Þα
ð2Þμðpð1ÞÞ;ð3Þ
2An ansa z is equi ed in o de o gi e a physical con en o he
equi emen o locali y o in e ac ions. O he wise, i would
always be possible o ind a change o a iables in he wo-
pa icle phase space such ha he composi ion law educes o he
sum o ou -momen a. In his case, one would be jus conside ing
SR in a ancy choice o coo dina es.
RELATIVISTIC DEFORMED KINEMATICS FROM LOCALITY …PHYS. REV. D 101, 044057 (2020)
044057-3
such ha he in e ac ion is local in he new space ime
[˜
xα
ð1Þð0Þ¼˜
xα
ð2Þð0Þ]. We assume ha φð2Þα
ð1Þμð0Þ¼φð1Þα
ð2Þμð0Þ¼
0so ha when one o he wo momen a is ze o he sys em
o wo pa icles educes o one pa icle wi h new space- ime
coo dina es ˜
xα¼xμφα
μðpÞ. One also has φα
μð0Þ¼δα
μso ha
he new space- ime coo dina es coincide wi h he coo di-
na es xin he limi p→0.
Locali y in he gene alized space ime equi es inding a
se o unc ions φα
μðkÞ,φð2Þα
ð1ÞμðkÞ, and φð1Þα
ð2ÞμðkÞsa is ying he
se o equa ions
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
φα
νðpð1ÞÞþ
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
φð2Þα
ð1Þνðpð2ÞÞ
¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
φα
νðpð2ÞÞþ
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
φð1Þα
ð2Þνðpð1ÞÞ;
ð4Þ
whe e we use he no a ion
Pμ¼ðpð1Þ⊕pð2ÞÞμð5Þ
o he componen s o he o al ou -momen um (P)o a
sys em o wo pa icles wi h ou -momen a pð1Þand pð2Þ.
We will e e o ⊕as he DCL.
Equa ion (4) is jus he condi ion ha he wo ldlines o
he wo pa icles in he in-s a e (o ou -s a e) c oss a a
poin . Bu he ou -momen um o he wo pa icles in he in-
s a e and ou -s a e a e cons ained by he conse a ion o
he o al ou -momen um,
ðp−ð1Þ⊕p−ð2ÞÞμ¼ðpþð1Þ⊕pþð2ÞÞμ:ð6Þ
Then, he c ossing o he wo ldlines o he ou pa icles a a
poin equi es he le -hand and igh -hand sides o Eq. (4)
o depend on he wo ou -momen a only h ough he
combina ion ðpð1Þ⊕pð2ÞÞ. When one uses he condi ions
φð2Þα
ð1Þμð0Þ¼φð1Þα
ð2Þμð0Þ¼0, one concludes ha in ac bo h
sides o Eq. (4) should be equal o φα
μðpð1Þ⊕pð2ÞÞ.3
When one akes he limi pð1Þ→0o pð2Þ→0in he
locali y equa ions, one has
φð2Þα
ð1Þμðpð2ÞÞ¼φα
μðpð2ÞÞ−lim
k→0
∂ðk⊕pð2ÞÞμ
∂kα
;
φð1Þα
ð2Þμðpð1ÞÞ¼φα
μðpð1ÞÞ−lim
k→0
∂ðpð1Þ⊕kÞμ
∂kα
ð7Þ
o he unc ions ha de ine he mixing o he phase spaces o
he wo pa icles in he gene alized space- ime coo dina es.
When hese exp essions o he mixing unc ions φð2Þ
ð1Þ,
φð1Þ
ð2Þa e plugged in o he locali y equa ions, one inds
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
lim
k→0
∂ðk⊕pð2ÞÞν
∂kα
¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
lim
k→0
∂ðpð1Þ⊕kÞν
∂kα
¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
φα
νðpð1ÞÞ
þ
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
φα
νðpð2ÞÞ−φα
μðpð1Þ⊕pð2ÞÞ:ð8Þ
The i s equali y is a se o equa ions ha a DCL (⊕) has o
sa is y in o de o be able o ha e a gene alized space ime
(whose coo dina es a e a sum o wo e ms, each in ol ing
he phase-space coo dina es o a pa icle) whe e in e -
ac ions a e local. The second equali y is a se o ela ions
be ween he DCL (⊕) and he unc ions φα
μwhich de ine
he new space- ime coo dina es o a one-pa icle sys em.
We in oduce he ela i e coo dina e
˜
xα
ð12Þ≐˜
xα
ð1Þ−˜
xα
ð2Þ
¼xμ
ð1Þ½φα
μðpð1ÞÞ−φð2Þα
ð1Þμðpð1ÞÞ
−xμ
ð2Þ½φα
μðpð2ÞÞ−φð1Þα
ð2Þμðpð2ÞÞ
¼xμ
ð1Þlim
k→0
∂ðpð1Þ⊕kÞμ
∂kα
−xμ
ð2Þlim
k→0
∂ðk⊕pð2ÞÞμ
∂kα
:
ð9Þ
The e ec o an in ini esimal ans o ma ion wi h pa am-
e e s ϵμgene a ed by he o al ou -momen um ( ansla ion)
on he ela i e coo dina e is
δ˜
xα
ð12Þ¼ϵμ ˜
xα
ð12Þ;ðpð1Þ⊕pð2ÞÞμg
¼ϵμ−
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
lim
k→0
∂ðpð1Þ⊕kÞν
∂kα
þ
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
lim
k→0
∂ðk⊕pð2ÞÞν
∂kα:ð10Þ
We see hen ha he sys em o equa ions ha a DCL (⊕)
has o sa is y in o de o ind gene alized space- ime
coo dina es wi h a local in e ac ion is jus he condi ion
o he in a iance o he ela i e coo dina e unde ans-
la ions. I one obse e sees a c ossing o wo ldlines
[˜
xα
ð12Þð0Þ¼0], ano he obse e ela ed by a ansla ion
also sees a c ossing o wo ldlines.
3Away o see his is o conside he si ua ion in which he pa icles
in he in-s a e ha e ou -momen a p−ð1Þ
μ¼ðpþð1Þ⊕pþð2ÞÞμ,
p−ð2Þ
μ¼0.
CARMONA, CORT ´
ES, and RELANCIO PHYS. REV. D 101, 044057 (2020)
044057-4
One can also conside he e ec o an in ini esimal ans o ma ion wi h pa ame e s ϵαgene a ed by he ela i e
gene alized space- ime coo dina es ˜
xα
ð12Þon he momen a pð1Þ,pð2Þ. One has
δpð1Þ
μ¼ϵα pð1Þ
μ;˜
xα
ð12Þg¼ϵαlim
k→0
∂ðpð1Þ⊕kÞμ
∂kα
¼½ðpð1Þ⊕ϵÞ−pð1Þμ;
δpð2Þ
μ¼ϵα pð2Þ
μ;˜
xα
ð12Þg¼−ϵαlim
k→0
∂ðk⊕pð2ÞÞμ
∂kα
¼−½ðϵ⊕pð2ÞÞ−pð2Þμ;ð11Þ
and hen
δðpð1Þ⊕pð2ÞÞ¼δpð1Þ⊕pð2Þþpð1Þ⊕δpð2Þ¼ðpð1Þ⊕ϵÞ⊕pð2Þ−pð1Þ⊕ðϵ⊕pð2ÞÞ:ð12Þ
Bu he in a iance o he ela i e coo dina e unde he ans o ma ion gene a ed by he o al ou -momen um implies he
in a iance o he o al ou -momen um unde he ans o ma ion gene a ed by he ela i e coo dina e. Then, om Eq. (12)
one has
ðpð1Þ⊕ϵÞ⊕pð2Þ¼pð1Þ⊕ðϵ⊕pð2ÞÞ:ð13Þ
An al e na i e, mo e di ec way o de i e his esul is based on he iden i ies
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
lim
k→0
∂ðk⊕pð2ÞÞν
∂kα
¼lim
k→0
∂ðpð1Þ⊕ðk⊕pð2ÞÞÞμ
∂ðk⊕pð2ÞÞν
∂ðk⊕pð2ÞÞν
∂kα
¼lim
k→0
∂ðpð1Þ⊕ðk⊕pð2ÞÞÞμ
∂kα
;
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
lim
k→0
∂ðpð1Þ⊕kÞν
∂kα
¼lim
k→0
∂ððpð1Þ⊕kÞ⊕pð2ÞÞμ
∂ðpð1Þ⊕kÞν
∂ðpð1Þ⊕kÞν
∂kα
¼lim
k→0
∂ððpð1Þ⊕kÞ⊕pð2ÞÞμ
∂kα
:ð14Þ
Then, he i s equali y in Eq. (8) leads o
lim
k→0
∂ðpð1Þ⊕ðk⊕pð2ÞÞÞμ
∂kα
¼lim
k→0
∂ððpð1Þ⊕kÞ⊕pð2ÞÞμ
∂kα
;ð15Þ
which is equi alen o Eq. (13).
I one makes he choice φð2Þα
ð1Þμðpð2ÞÞ¼0in Eq. (7),4one has
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
φα
νðpð1ÞÞ¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
lim
k→0
∂ðk⊕pð1ÞÞν
∂kα
¼lim
k→0∂ððk⊕pð1ÞÞ⊕pð2ÞÞμ
∂ðk⊕pð1ÞÞν
∂ðk⊕pð1ÞÞν
∂kα
¼lim
k→0
∂ððk⊕pð1ÞÞ⊕pð2ÞÞμ
∂kα
;
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
φα
νðpð2ÞÞ¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
lim
k→0
∂ðk⊕pð2ÞÞν
∂kα
¼lim
k→0∂ðpð1Þ⊕ðk⊕pð2ÞÞμ
∂ðk⊕pð2ÞÞν
∂ðk⊕pð2ÞÞν
∂kα
¼lim
k→0
∂ðpð1Þ⊕ðk⊕pð2ÞÞμ
∂kα
;
φα
μðpð1Þ⊕pð2ÞÞ¼lim
k→0
∂ðk⊕ðpð1Þ⊕pð2ÞÞÞμ
∂kα
:ð16Þ
4The same a gumen can be made i one makes he al e na i e choice φð1Þα
ð2Þμðpð1ÞÞ¼0.
RELATIVISTIC DEFORMED KINEMATICS FROM LOCALITY …PHYS. REV. D 101, 044057 (2020)
044057-5

Then he ela ions o compa ibili y wi h locali y (8) can
be w i en as
lim
k→0
∂ðpð1Þ⊕ðk⊕pð2ÞÞÞμ
∂kα
¼lim
k→0
∂ððpð1Þ⊕kÞ⊕pð2ÞÞμ
∂kα
¼lim
k→0
∂ððk⊕pð1ÞÞ⊕pð2ÞÞμ
∂kα
þlim
k→0
∂ðpð1Þ⊕ðk⊕pð2ÞÞÞμ
∂kα
−lim
k→0
∂ðk⊕ðpð1Þ⊕pð2ÞÞÞμ
∂kα
:
ð17Þ
This makes mani es ha any associa i e DCL is
compa ible wi h locali y.
III. FIRST-ORDER DEFORMED COMPOSITION
LAW OF FOUR-MOMENTA (DCL1)
We conside a DCL ha is linea as a unc ion o he
ou -momen um o each pa icle. Dimensional a gumen s
lead o he gene al o m o such a de o med composi ion
law (DCL1)
ðpð1Þ⊕pð2ÞÞμ¼pð1Þ
μþpð2Þ
μþcνρ
μ
Λpð1Þ
νpð2Þ
ρ;ð18Þ
whe e cνρ
μa e a bi a y dimensionless coe icien s. Le us
see i such a DCL can sa is y he es ic ions om he
locali y o in e ac ions. One has
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
¼δν
μþcρν
μ
Λpð1Þ
ρ;
lim
k→0
∂ðk⊕pð2ÞÞν
∂kα
¼δα
νþcασ
ν
Λpð2Þ
σ;
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
¼δν
μþcνσ
μ
Λpð2Þ
σ;
lim
k→0
∂ðpð1Þ⊕kÞν
∂kα
¼δα
νþcρα
ν
Λpð1Þ
ρ;ð19Þ
and
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
lim
k→0
∂ðk⊕pð2ÞÞν
∂kα
¼δα
μþcρα
μ
Λpð1Þ
ρþcασ
μ
Λpð2Þ
σþcρν
μcασ
ν
Λ2pð1Þ
ρpð2Þ
σ;
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
lim
k→0
∂ðpð1Þ⊕kÞν
∂kα
¼δα
μþcρα
μ
Λpð1Þ
ρþcασ
μ
Λpð2Þ
σþcνσ
μcρα
ν
Λ2pð1Þ
ρpð2Þ
σ:ð20Þ
A DCL1 is compa ible wi h locali y i he dimensionless
coe icien s sa is y he sys em o equa ions
cρν
μcασ
ν¼cνσ
μcρα
ν:ð21Þ
These a e jus he condi ions ha he coe icien s cνρ
μo a
DCL1 ha e o sa is y in o de o be associa i e. This esul
can be unde s ood since Eq. (13) implies associa i i y o
a DCL1.
The gene al o m o an iso opic DCL1 has coe icien s
cνρ
μ¼c1δν
μnρþc2δρ
μnνþc3ηνρnμþc4nμnνnρþc5ϵμνρσnσ;
ð22Þ
whe e nμ¼ð1;0;0;0Þand cia e a bi a y cons an s.
Compa ibili y wi h locali y leads o ou possible cases
o he DCL1:
cνρ
μ¼δρ
μnν;c
νρ
μ¼δν
μnρ;
cνρ
μ¼δν
μnρþδρ
μnν−nμnνnρ;
cνρ
μ¼ηνρnμ−nμnνnρ:ð23Þ
In he las wo cases, co esponding o a symme ic
composi ion law, i is possible o ind a change o he
choice o ou -momen um a iables [k0
μ¼ μðkÞ] such ha
he composi ion in he new a iables educes o he addi ion
o momen a [ðp0⊕0q0Þμ≐ðp⊕qÞ0
μ¼p0
μþq0
μ].5Then,
hey do no co espond o a de o ma ion o SR based on a
de o med composi ion law.
In he emaining wo cases, one has a nonsymme ic
composi ion law (in ac , he wo cases a e ela ed by an
exchange o he ou -momen a in he composi ion law). A
change o ou -momen um a iables applied o an addi i e
composi ion law will always p oduce a symme ic compo-
si ion law; he e o e, he wo cases o a nonsymme ic
composi ion law a e eal de o ma ions o SR. The explici
o m o he locali y-compa ible DCL1 (o , o sho , “local”
DCL1) is6
ðpð1Þ⊕pð2ÞÞ0¼pð1Þ
0þpð2Þ
0þϵpð1Þ
0pð2Þ
0
Λ;
ðpð1Þ⊕pð2ÞÞi¼pð1Þ
iþpð2Þ
iþϵpð1Þ
0pð2Þ
i
Λ;ð24Þ
whe e ϵ¼1is an o e all sign o he modi ica ion in he
composi ion law and an a bi a y cons an can be eab-
so bed in o he de ini ion o he scale Λ. We will see in
5Fo he i s one, he unc ion is 0ðkÞ¼Λlogð1þk0=ΛÞ,
iðkÞ¼ki=ð1þk0=ΛÞ, while o he las one 0ðkÞ¼k0−

k2=ð2ΛÞ, iðkÞ¼ki.
6The e is ano he DCL1 ob ained by exchanging he ou -
momen um a iables.
CARMONA, CORT ´
ES, and RELANCIO PHYS. REV. D 101, 044057 (2020)
044057-6
Sec. V ha his composi ion law co esponds in ac o κ-
Poinca ´e kinema ics.
When ϵ¼−1, one has
1−ðpð1Þ⊕pð2ÞÞ0
Λ¼1−pð1Þ
0
Λ1−pð2Þ
0
Λ;ð25Þ
so ha he scale Λplays he ole o a cu o in he ene gy.
This is he eason why his choice o sign ep oduces he
DCL in he con ex o DSR, as we will see la e . The o he
choice o sign ϵ¼þ1co esponds o a de o ma ion whe e
he scale Λis no a maximum o he ene gy and hus goes
beyond he amewo k o DSR.
I we go back o he exp ession o he ela i e gene -
alized space- ime coo dina es (9) and use he explici o m
o he local DCL1 in Eq. (24), we ind
lim
k→0
∂ðpð1Þ⊕kÞ0
∂k0
¼1þϵpð1Þ
0
Λ;
lim
k→0
∂ðpð1Þ⊕kÞ0
∂ki
¼lim
k→0
∂ðpð1Þ⊕kÞi
∂k0
¼0;
lim
k→0
∂ðpð1Þ⊕kÞi
∂kj
¼δj
i1þϵpð1Þ
0
Λ;
lim
k→0
∂ðk⊕pð2ÞÞ0
∂k0
¼1þϵpð2Þ
0
Λ;
lim
k→0
∂ðk⊕pð2ÞÞ0
∂ki
¼0;
lim
k→0
∂ðk⊕pð2ÞÞi
∂k0
¼ϵpð2Þ
i
Λ;
lim
k→0
∂ðk⊕pð2ÞÞi
∂kj
¼δj
i;ð26Þ
and hen
˜
x0
ð12Þ¼x0
ð1Þð1þϵpð1Þ
0=ΛÞ−x0
ð2Þð1þϵpð2Þ
0=ΛÞ−xj
ð2Þϵpð2Þ
j=Λ;
˜
xi
ð12Þ¼xi
ð1Þð1þϵpð1Þ
0=ΛÞ−xi
ð2Þ:ð27Þ
F om hese exp essions o he ela i e space- ime coo -
dina es, we ha e
˜
xi
ð12Þ;˜
x0
ð12Þg¼ xi
ð1Þð1þϵpð1Þ
0=ΛÞ;x
0
ð1Þð1þϵpð1Þ
0=ΛÞg
þ xi
ð2Þ;x
j
ð2Þϵpð2Þ
j=Λg
¼ðϵ=ΛÞ½xi
ð1Þð1þϵpð1Þ
0=ΛÞ−xi
ð2Þ
¼ðϵ=ΛÞ˜
xi
ð12Þ:ð28Þ
Then we see ha he ela i e space- ime coo dina es o he
wo-pa icle sys em a e he coo dina es o a (noncommu-
a i e) κ-Minkowski space ime wi h κ¼ϵ=Λ.
I we wan o de e mine he gene alized space- ime
coo dina es o he wo-pa icle sys em (no jus he ela i e
coo dina es), we ha e o sol e, using he explici o m o
he local DCL1, he sys em o equa ions in Eq. (8) o he
unc ions φα
μðpÞwhich de ine he gene alized space- ime
coo dina es o a one-pa icle sys em. One has di e en
solu ions and hen di e en choices o gene alized space-
ime coo dina es wi h a c ossing o wo ldlines. In o de o
ha e a well-de ined space ime de ined by locali y, one has
o include an addi ional equi emen .
The exp ession o he de o med composi ion law in
Eq. (24),
ðpð1Þ⊕pð2ÞÞμ¼pð1Þ
μþð1þϵpð1Þ
0=ΛÞpð2Þ
μ;ð29Þ
is a sum o pð1Þ
μ(independen o pð2Þ) and a e m p opo -
ional o pð2Þ
μdepending on pð1Þ. This sugges s conside ing
gene alized space- ime coo dina es ˜
xμ
ð1Þdepending on he
phase-space coo dina es (xð1Þ;p
ð1Þ), while he ˜
xμ
ð2Þdepend
on he phase-space coo dina es o bo h pa icles
(xð1Þ;p
ð1Þ;x
ð2Þ;p
ð2Þ), as he addi ional equi emen o de i e
he gene alized space- ime coo dina es o he wo-pa icle
sys em. In his case, one has
φð2Þα
ð1Þμðpð2ÞÞ¼0→φα
μðpð1ÞÞ¼lim
k→0
∂ðk⊕pð1ÞÞμ
∂kα
;ð30Þ
and
φα
μðpð1Þ⊕pð2ÞÞ¼lim
k→0
∂ðk⊕ðpð1Þ⊕pð2ÞÞÞμ
∂kα
¼lim
k→0
∂ððk⊕pð1ÞÞ⊕pð2ÞÞμ
∂kα
¼lim
k→0∂ððk⊕pð1ÞÞ⊕pð2ÞÞμ
∂ðk⊕pð1ÞÞν
∂ðk⊕pð1ÞÞν
∂kα
¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
lim
k→0
∂ðk⊕pð1ÞÞν
∂kα
;ð31Þ
whe e we ha e made use o he associa i i y o he local
DCL1. Bu hen [using Eqs. (30) and (31)], one has
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
φα
νðpð1ÞÞþ
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
φα
νðpð2ÞÞ
−φα
μðpð1Þ⊕pð2ÞÞ¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
lim
k→0
∂ðk⊕pð2ÞÞν
∂kα
;
ð32Þ
which is he ela ion be ween φα
μand he DCL (⊕) which
esul s om he equi emen o ha ing a c ossing o
RELATIVISTIC DEFORMED KINEMATICS FROM LOCALITY …PHYS. REV. D 101, 044057 (2020)
044057-7
wo ldlines [Eq. (8)]. This is na u al, since we al eady saw in
Eqs. (16)–(17) ha an associa i e DCL sa is ies he locali y
equa ions (8) wi h he choice φð2Þα
ð1Þμðpð2ÞÞ¼0. The unc-
ions de ining he gene alized space- ime coo dina es o a
one-pa icle sys em a e
φα
μðpÞ¼lim
k→0
∂ðk⊕pÞμ
∂kα
;ð33Þ
and hen, using he composi ion o ou -momen a in
Eq. (24),weha e
˜
x0¼xμlim
k→0
∂ðk⊕pÞμ
∂k0
¼x0ð1þϵp0=ΛÞþxjϵpj=Λ;
˜
xi¼xμlim
k→0
∂ðk⊕pÞμ
∂ki
¼xi;ð34Þ
and
˜
xi;˜
x0g¼ xi;x
jϵpj=Λg¼−ðϵ=ΛÞxi¼−ðϵ=ΛÞ˜
xi:ð35Þ
The space- ime coo dina es o a one-pa icle sys em a e
also he coo dina es o a (noncommu a i e) κ-Minkowski
space ime wi h κ¼−ðϵ=ΛÞ.
I one conside s he second case o a de o med
composi ion law quad a ic in momen a and compa ible
wi h he implemen a ion o locali y,
ðpð1Þ⊕pð2ÞÞμ¼ð1þϵpð2Þ
0=ΛÞpð1Þ
μþpð2Þ
μ;ð36Þ
one now has a sum o pð2Þ
μ(independen o pð1Þ) and a e m
p opo ional o pð1Þ
μdepending on pð2Þ. Then, one can
conside gene alized space- ime coo dina es ˜
xμ
ð2Þdepending
on he phase-space coo dina es (xð2Þ;p
ð2Þ), while he ˜
xμ
ð1Þ
depend on he phase-space coo dina es o bo h pa icles
(xð1Þ;p
ð1Þ;x
ð2Þ;p
ð2Þ) as he addi ional equi emen . In his
case one has
φð1Þα
ð2Þμðpð1ÞÞ¼0→φα
μðpð1ÞÞ¼lim
k→0
∂ðpð1Þ⊕kÞμ
∂kα
:ð37Þ
The unc ions de ining he gene alized space- ime coo -
dina es o a one-pa icle sys em a e
φα
μðpÞ¼lim
k→0
∂ðp⊕kÞμ
∂kα
;ð38Þ
and hen, using he composi ion o ou -momen a in
Eq. (36),weha e
˜
x0¼xμlim
k→0
∂ðp⊕kÞμ
∂k0
¼x0ð1þϵp0=ΛÞþxjϵpj=Λ;
˜
xi¼xμlim
k→0
∂ðp⊕kÞμ
∂ki
¼xi:ð39Þ
The exp essions o he gene alized space- ime coo dina es
o he one-pa icle sys em in e ms o he canonical phase-
space coo dina es a e he same in he wo cases.
IV. LOCAL DCL1 AS A RELATIVISTIC
KINEMATICS
Un il now, we ha e discussed one o he ing edien s in a
de o ma ion o SR kinema ics: he modi ica ion o he
composi ion law o he ou -momen um and i s ela ion o
he locali y o in e ac ions. We now conside he compa -
ibili y o he conse a ion o he o al ou -momen um in an
in e ac ion wi h Lo en z in a iance. We ha e o conside a
nonlinea implemen a ion o Lo en z ans o ma ions in he
wo-pa icle sys em, which will be de ined by he exp es-
sion o he six gene a o s Jαβ as unc ions o he wo-
pa icle phase-space coo dina es,
Jαβ ¼xμ
ð1ÞJαβ
ð1Þμðpð1Þ;p
ð2ÞÞþxμ
ð2ÞJαβ
ð2Þμðpð1Þ;p
ð2ÞÞ:ð40Þ
The ac ion o Lo en z ans o ma ions on he wo-pa icle
sys em is gi en by
pð1Þ
μ;Jαβg¼Jαβ
ð1Þμðpð1Þ;p
ð2ÞÞ;
xμ
ð1Þ;Jαβg¼−xν
ð1Þ
∂Jαβ
ð1Þνðpð1Þ;p
ð2ÞÞ
∂pð1Þ
μ
−xν
ð2Þ
∂Jαβ
ð2Þνðpð1Þ;p
ð2ÞÞ
∂pð1Þ
μ
;
pð2Þ
μ;Jαβg¼Jαβ
ð2Þμðpð1Þ;p
ð2ÞÞ;
xμ
ð2Þ;Jαβg¼−xν
ð1Þ
∂Jαβ
ð1Þνðpð1Þ;p
ð2ÞÞ
∂pð2Þ
μ
−xν
ð2Þ
∂Jαβ
ð2Þνðpð1Þ;p
ð2ÞÞ
∂pð2Þ
μ
:ð41Þ
In he one-pa icle sys em, he gene a o s o Lo en z
ans o ma ions will be gi en in e ms o he phase-space
coo dina es by
Jαβ ¼xμJαβ
μðpÞ;ð42Þ
and one has
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ES, and RELANCIO PHYS. REV. D 101, 044057 (2020)
044057-8
pμ;Jαβg¼Jαβ
μðpÞ; xμ;Jαβg¼−xν∂Jαβ
νðpÞ
∂pμ
:
ð43Þ
The iden i ica ion o he one-pa icle sys em wi h a wo-
pa icle sys em when one o he ou -momen a is ze o leads
o he ela ions
Jαβ
ð1Þμðpð1Þ;0Þ¼Jαβ
μðpð1ÞÞ;Jαβ
ð1Þμð0;p
ð2ÞÞ¼0;
Jαβ
ð2Þμðpð1Þ;0Þ¼0;Jαβ
ð2Þμð0;p
ð2ÞÞ¼Jαβ
μðpð2ÞÞ:ð44Þ
The compa ibili y o he conse a ion o he o al ou -
momen um wi h Lo en z in a iance equi es ha
ðpð1Þ⊕pð2ÞÞμ;Jαβg¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
pð1Þ
ν;Jαβg
þ
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
pð2Þ
ν;Jαβg;
ð45Þ
whe e on he le -hand side one has he gene a o s o
Lo en z ans o ma ions in a one-pa icle sys em and on he
igh -hand side he gene a o s in he wo-pa icle sys em.
Then, he conse a ion law o he ou -momen um will be
Lo en z in a ian i one can ind a solu ion o he sys em o
equa ions
Jαβ
μðpð1Þ⊕pð2ÞÞ¼
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
Jαβ
ð1Þνðpð1Þ;p
ð2ÞÞ
þ
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
Jαβ
ð2Þνðpð1Þ;p
ð2ÞÞ
ð46Þ
o he unc ions o one o wo ou -momen a ha de ine
he nonlinea ac ion o he Lo en z ans o ma ions on he
ou -momen um o a pa icle o on he ou -momen a o a
sys em o wo pa icles.
In o de o de e mine he Lo en z ans o ma ion o he
wo-pa icle sys em, one also needs an addi ional equi e-
men as in he case o he gene alized space- ime coo -
dina es. The iden i ica ion o gene alized space- ime
coo dina es wi h a mixing o phase-space coo dina es only
on he coo dina es o one o he pa icles (˜
xð2Þ) sugges s
conside ing a Lo en z ans o ma ion whe e only he ans-
o ma ion o one o he ou -momen a (pð2Þ) depends on he
ou -momen um o bo h pa icles. One has in his case
Jαβ
ð1Þμðpð1Þ;p
ð2ÞÞ¼Jαβ
μðpð1ÞÞ;ð47Þ
and he sys em o equa ions o he Lo en z in a iance o
he conse a ion law becomes
∂ðpð1Þ⊕pð2ÞÞμ
∂pð2Þ
ν
Jαβ
ð2Þνðpð1Þ;p
ð2ÞÞ
¼Jαβ
μðpð1Þ⊕pð2ÞÞ−
∂ðpð1Þ⊕pð2ÞÞμ
∂pð1Þ
ν
Jαβ
νðpð1ÞÞ:ð48Þ
This is a sys em o equa ions allowing o de e mine, gi en
he composi ion law o he ou -momen um (⊕), he
Lo en z ans o ma ion o he wo-pa icle sys em om
he Lo en z ans o ma ion o a one-pa icle sys em.
One possibili y o ix he Lo en z ans o ma ion o a
one-pa icle sys em is o equi e ha he Lo en z gene -
a o s, oge he wi h he gene alized space- ime coo dina es
˜
xα, gene a e a de o med en-dimensional Lie algeb a in
co espondence wi h he Poinca ´e algeb a gene a ed by he
space- ime coo dina es and he Lo en z gene a o s in SR.
One inds (see he Appendix)
Jij
0ðpÞ¼0;Jij
kðpÞ¼δj
kpi−δi
kpj;
J0j
0ðpÞ¼−pjð1þϵp0=ΛÞ;
J0j
k¼δj
k½−p0−ϵp2
0=2Λþðϵ=ΛÞ½ 
p2=2−pjpk:ð49Þ
The e is no e ec o he de o ma ion on he ans o ma ion
unde o a ions as a consequence o he iso opy o he
de o med composi ion law in Eq. (24).
F om Eq. (48), and using he local DCL1 (24) and he
Lo en z ans o ma ion o he one-pa icle sys em in
Eq. (49), we ind o he Lo en z ans o ma ion o he
pa icle wi h phase-space coo dina es (xð2Þ;p
ð2Þ) in he wo-
pa icle sys em
J0i
ð2Þ0ðpð1Þ;p
ð2ÞÞ¼ð1þϵpð1Þ
0=ΛÞJ0i
0ðpð2ÞÞ;
J0i
ð2Þjðpð1Þ;p
ð2ÞÞ¼ð1þϵpð1Þ
0=ΛÞJ0i
jðpð2ÞÞ
þðϵ=ΛÞðpð1Þ
jpð2Þ
i−δi
j

pð1Þ·
pð2ÞÞ;
Jij
ð2Þ0ðpð1Þ;p
ð2ÞÞ¼Jij
0ðpð2ÞÞ;
Jij
ð2Þkðpð1Þ;p
ð2ÞÞ¼Jij
kðpð2ÞÞ:ð50Þ
Once we ha e de e mined a Lo en z ans o ma ion o
he one- and wo-pa icle sys ems compa ible wi h he
in a iance o he conse a ion law o he o al ou -
momen um, one can de e mine he DDR, de ined by a
unc ion CðpÞwhich is Lo en z in a ian , i.e., such ha
CðpÞ;Jαβg¼∂CðpÞ
∂pμ
Jαβ
μðpÞ¼0:ð51Þ
When one adds he equi emen ha in he limi
ðp2
0=Λ2Þ→0,ð
p2=Λ2Þ→0 he unc ion CðpÞ educes
o p2
0−
p2, so ha one eco e s he dispe sion ela ion
o SR in he low-ene gy limi , he esul is
RELATIVISTIC DEFORMED KINEMATICS FROM LOCALITY …PHYS. REV. D 101, 044057 (2020)
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