A ubiquitous feature of quantum mechanical theories is the existence of states of superposition. This is expected to be no different for a quantum gravity theory. Guided by this consideration and others we consider a framework in which classical reference frames may be in superposition relative to one another. Mirroring standard quantum mechanics we introduce a complex-valued wavefunctional, which takes as input the transformations between the coordinates, Ψ[x(x′)], with the interpretation that an interaction between the reference frames may select a particular transformation with probability distribution given by the Born rule — P[x(x′)] = probability distribution functional ≡ |Ψ[x(x′)]|2. The cases of two and three reference frames in superposition are considered explicitly. It is shown that the set of transformations is closed. A rule for transforming wavefunctions from one system to another system in superposition is proposed and consistency with the Schrodinger equation is demonstrated.

1.
J.
Foo
,
R. B.
Mann
, and
M.
Zych
, “
Schrödinger’s cat for de Sitter spacetime
,”
Classical Quantum Gravity
38
,
115010
(
2021
).
2.
R.
Jensen
, “
Can the universe be represented by a superposition of spacetime manifolds?
,”
Phys. Procedia
20
,
47
62
(
2011
).
3.
F.
Giacomini
and
Č.
Brukner
, “
Quantum superposition of spacetimes obeys Einstein’s equivalence principle
,”
AVS Quantum Sci.
4
,
015601
(
2022
).
4.
F.
Giacomini
and
Č.
Brukner
, “
Einstein’s equivalence principle for superpositions of gravitational fields
,” arXiv:2012.13754 (
2020
).
5.
M.
Christodoulou
and
C.
Rovelli
, “
On the possibility of laboratory evidence for quantum superposition of geometries
,”
Phys. Lett. B
792
,
64
68
(
2019
).
6.
C.
Anastopoulos
and
B.-L.
Hu
, “
Quantum superposition of two gravitational cat states
,”
Class. Quantum Gravity
37
,
235012
(
2020
).
7.
C.
Rovelli
, “
Quantum reference systems
,”
Class. Quantum Gravity
8
,
317
(
1991
).
8.
X.
Wu
and
H.
Zhang
, “
Chaotic dynamics in a superposed Weyl spacetime
,”
Astrophys. J.
652
,
1466
1474
(
2006
).
9.
K.
Crowther
,
Effective Spacetime
(
Springer
,
2018
).
10.
Y.
Aharonov
and
T.
Kaufherr
, “
Quantum frames of reference
,”
Phys. Rev. D
30
,
368
385
(
1984
).
11.
Y.
Aharonov
and
L.
Susskind
, “
Charge superselection rule
,”
Phys. Rev.
155
,
1428
(
1967
).
12.
S. D.
Bartlett
,
T.
Rudolph
, and
R. W.
Spekkens
, “
Reference frames, superselection rules, and quantum information
,”
Rev. Mod. Phys.
79
,
555
(
2007
).
13.
M. C.
Palmer
,
F.
Girelli
, and
S. D.
Bartlett
, “
Changing quantum reference frames
,”
Phys. Rev. A
89
,
052121
(
2014
).
14.
F.
Giacomini
,
E.
Castro-Ruiz
, and
Č.
Brukner
, “
Quantum mechanics and the covariance of physical laws in quantum reference frames
,”
Nat. Commun.
10
,
494
(
2019
).
15.
A.
Belenchia
,
R. M.
Wald
,
F.
Giacomini
,
E.
Castro-Ruiz
,
Č.
Brukner
, and
M.
Aspelmeyer
, “
Quantum superposition of massive objects and the quantization of gravity
,”
Phys. Rev. D
98
,
126009
(
2018
).
16.
A.-C.
de la Hamette
and
T. D.
Galley
,
Quantum
4
,
367
(
2020
).
17.
A.
Vanrietvelde
,
P. A.
Hoehn
,
F.
Giacomini
, and
E.
Castro-Ruiz
, “
A change of perspective: Switching quantum reference frames via a perspective-neutral framework
,”
Quantum
4
,
225
(
2020
).
18.
M. C.
Palmer
,
F.
Girelli
, and
S. D.
Bartlett
, “
Changing quantum reference frames
,”
Phys. Rev. A
89
,
052121
(
2014
).
19.
L. F.
Streiter
,
F.
Giacomini
, and
Č.
Brukner
, “
Relativistic bell test within quantum reference frames
,”
Phys. Rev. Lett.
126
,
230403
(
2021
).
20.
D. M.
Greenberger
, “
The inconsistency of the usual galilean transformation in quantum mechanics and how to fix it
,”
Z. Naturforsch. A
56
,
67
75
(
2001
).
21.
M.
Mikusch
,
L. C.
Barbado
, and
Č.
Brukner
, “
Transformation of spin in quantum reference frames
,”
Phys. Rev. Res.
3
,
043138
(
2021
).
22.
A.
Ballesteros
,
F.
Giacomini
, and
G.
Gubitosi
, “
The group structure of dynamical transformations between quantum reference frames
,”
Quantum
5
,
470
(
2021
).
23.
Y.
Aharonov
and
T.
Kaufherr
, “
Quantum frames of reference
,”
Phys. Rev. D
30
,
368
385
(
1984
).
24.
R. M.
Angelo
,
N.
Brunner
,
S.
Popescu
,
A. J.
Short
, and
P.
Skrzypczyk
, “
Physics within a quantum reference frame
,”
J. Phys. A: Math. Theor.
44
,
145304
(
2011
).
25.
R.
Angelo
and
A.
Ribeiro
, “
Kinematics and dynamics in noninertial quantum frames of reference
,”
J. Phys. A: Math. Theor.
45
,
465306
(
2012
).
26.
J.
Pienaar
, “
A relational approach to quantum reference frames for spins
,” arXiv:1601.07320 (
2016
).
27.
A. R.
Smith
,
M.
Piani
, and
R. B.
Mann
, “
Quantum reference frames associated with noncompact groups: The case of translations and boosts and the role of mass
,”
Phys. Rev. A
94
,
012333
(
2016
).
28.
H.
Everett
III
, “‘
Relative state’ formulation of quantum mechanics
,”
Rev. Mod. Phys.
29
,
454
(
1957
).
29.
C.
Rovelli
, “
Relational quantum mechanics
,”
Int. J. Theor. Phys.
35
,
1637
1678
(
1996
).
30.
C.
Rovelli
, “
Relational quantum mechanics
,” in
Quo Vadis Quantum Mechanics?
(
Springer
,
2005
), pp.
113
120
.
31.
R. P.
Feynman
,
A. R.
Hibbs
, and
D. F.
Styer
,
Quantum Mechanics and Path Integrals
(
Courier Corporation
,
2010
).
32.
L.
Hörmander
,
The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis
(
Springer
,
2015
).
33.
D.
Bohm
, “
A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I
,”
Phys. Rev.
85
,
166
(
1952
).
34.
H.
Everett
, “
The theory of the universal wave function
,” in
The Many-Worlds Interpretation of Quantum Mechanics
(
Princeton University Press
,
2015
), pp.
1
140
.
35.
D.
Albert
and
B.
Loewer
, “
Interpreting the many worlds interpretation
,”
Synthese
77
,
195
213
(
1988
).
36.
L.
Vaidman
, “
Probability in the many-worlds interpretation of quantum mechanics
,” in
Probability in Physics
(
Springer
,
2012
), pp.
299
311
.
37.
R. A.
Healey
, “
How many worlds?
,”
Nous
18
,
591
616
(
1984
).
38.
M.
Born
, “
Zur quantenmechanik der Stoßvorgänge
,”
Z. Phys.
37
,
863
867
(
1926
).
39.
E.
Schrödinger
, “
Die gegenwärtige situation in der quantenmechanik
,”
Naturwissenschaften
23
,
844
849
(
1935
).
40.
A.
Sommerfeld
and
W.
Heisenberg
, “
Die intensität der mehrfachlinien und ihrer zeemankomponenten
,”
Z. Phys.
11
,
131
154
(
1922
).
41.
E.
Schrödinger
,
Collected Papers on Wave Mechanics
(
American Mathematical Society
,
2003
), Vol.
302
.
You do not currently have access to this content.