Relational Reference Frame Transformation (RRFT)

• RRFT is a framework that maps the descriptions of physical systems between different reference frames using a relational and group-theoretic construction.
• It utilizes reversible and irreversible transformations via regular representations and gauge reduction, ensuring coherence in quantum reference changes.
• The approach underpins applications in quantum measurement, inertial frame shifts, and modular truncation, revealing frame-dependent entanglement and observable properties.

Relational Reference Frame Transformation (RRFT) denotes the transformation that maps a description of physical systems given relative to one reference system into a description given relative to another, without appeal to any external absolute frame. In the terminology of “Quantum reference frames for general symmetry groups,” RRFT is precisely a “change of quantum reference frame”: a group-theoretic, relational construction valid for arbitrary finite and locally compact symmetry groups $G$, with reversible transformations characterized by unitarity and regular representations, and irreversible transformations described as channels when the reference system is coarse-grained or imperfect (Hamette et al., 2020). In later work, the same notion is reformulated as gauge reduction, quantum coordinate change, or quantum gauge transformation in perspective-neutral and operator-algebraic settings (Vanrietvelde et al., 2018, Ahmad et al., 2024, Thiemann, 4 Mar 2026).

1. Relational definition and kinematical setting

The starting point is the principle of relational physics: given $n$ systems, states are defined to be relative to one of the systems. A state relative to system $i$ is a description of the other $n-1$ systems relative to $i$. In the notation of the group-theoretic framework, $\ket{\psi}_B^A$ means “state of system $B$ relative to system $A$,” while the reference system itself is assigned a trivial zero-state relative to itself, $\ket{0}_A^A\otimes\ket{\psi}_B^A$. Once a symmetry group is introduced, this $\ket{0}$ is identified with the identity element of $n$0 (Hamette et al., 2020).

For reversible frame changes, the relevant configuration space $n$1 is assumed to admit a free and transitive action of a symmetry group $n$2: for any $n$3, there is a unique $n$4 such that $n$5. This makes $n$6 a $n$7-torsor, so that points of $n$8 and coordinate systems on $n$9 can both be identified with group elements. A physical reference frame is then a physical system whose configuration space is $i$0, and the relative configuration of system $i$1 with respect to system $i$2 is a group element $i$3 defined classically by

$i$4

For an $i$5-particle classical configuration $i$6, the relational description relative to system $i$7 is

$i$8

The central RRFT problem is then: given a state $i$9, what is the corresponding state $n-1$0 relative to another system $n-1$1 (Hamette et al., 2020)?

This relational formulation has a direct classical analogue in constrained systems. In the perspective-neutral approach to the $n-1$2-body problem, the constraint surface classically and the gauge-invariant Hilbert space quantum mechanically contain all frame choices at once, while a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. In that language, RRFT is a gauge transformation, and the resulting maps are “quantum coordinate changes” (Vanrietvelde et al., 2018). A closely related general field-theoretic formulation treats a relational reference frame as a choice of reference fields $n-1$3 together with gauge-fixing conditions $n-1$4, so that RRFT becomes the canonical transformation between the reduced phase spaces defined by two such choices (Thiemann, 4 Mar 2026).

2. Group-theoretic structure and the reversible RRFT operator

Because $n-1$5, the group acts on itself by left and right multiplication. The two commuting actions are the left action

$n-1$6

and the right action

$n-1$7

In the relational interpretation, the left action is active, while the right action is passive and implements change of coordinates. On an $n-1$8-tuple $n-1$9,

$i$0

The classical change of reference frame from system $i$1 to system $i$2 is therefore

$i$3

a passive transformation generated by the relative group element $i$4 (Hamette et al., 2020).

For quantum systems with configuration space $i$5, the Hilbert space is $i$6 for locally compact groups or $i$7 for finite groups. The left and right regular representations act on basis kets $i$8 as

$i$9

or equivalently on wavefunctions $\ket{\psi}_B^A$0 by

$\ket{\psi}_B^A$1

Here $\ket{\psi}_B^A$2 encodes active transformations and $\ket{\psi}_B^A$3 passive transformations on the same space (Hamette et al., 2020).

The quantum requirement that frame changes respect superposition is expressed as the principle of coherent change of reference system: if $\ket{\psi}_B^A$4 and $\ket{\psi}_B^A$5, then

$\ket{\psi}_B^A$6

For $\ket{\psi}_B^A$7 identical $\ket{\psi}_B^A$8 systems, the coherent change of frame from system $\ket{\psi}_B^A$9 to system $B$0 is implemented by

$B$1

with $B$2. Operationally, this operator conditions on the state of the new frame, applies the corresponding passive transformation to the other systems, and swaps the old and new frame labels. It is unitary, satisfies $B$3, and is transitive in the sense that $B$4 (Hamette et al., 2020).

For $B$5, the construction reduces to the known position-shift transformation used in earlier QRF work, and for Galilean transformations it reproduces the previously known operators for translations and boosts. The same formalism also gives the $B$6 case for a particle on a circle (Hamette et al., 2020).

3. Unitarity, regular representations, and admissible reference systems

A central result of the group-theoretic theory is the Unitarity Theorem. Consider $B$7 identical systems with Hilbert spaces $B$8, an injective encoding $B$9, and two unitary representations $A$0 such that

$A$1

Suppose one demands an operator $A$2 that implements the classical frame change on product states and acts coherently on their superpositions. Then $A$3 is unitary if and only if the states $A$4 form an orthonormal basis, or orthonormal subset of a basis, and $A$5 are the left and right regular representations acting on that basis (Hamette et al., 2020).

This theorem singles out $A$6 and $A$7 equipped with regular representations as the structures that support reversible RRFT. Encodings of group elements into smaller Hilbert spaces generally fail. The example given in the paper is a qubit or rebit encoding such as $A$8 for $A$9: in that case one cannot define a linear, probability-preserving RRFT that maps classical configurations to one another, and the resulting map is non-linear (Hamette et al., 2020). A plausible implication is that “being a reference frame” is not merely a matter of carrying some representation of the symmetry group; it requires the regular-representation structure that makes all relative configurations coherently accessible.

The framework also extends to mixed collections of systems. If some subsystems are perfect $\ket{0}_A^A\otimes\ket{\psi}_B^A$0 reference frames and others are ordinary systems with Hilbert space $\ket{0}_A^A\otimes\ket{\psi}_B^A$1, injection $\ket{0}_A^A\otimes\ket{\psi}_B^A$2, and representations $\ket{0}_A^A\otimes\ket{\psi}_B^A$3 satisfying

$\ket{0}_A^A\otimes\ket{\psi}_B^A$4

then the RRFT from frame $\ket{0}_A^A\otimes\ket{\psi}_B^A$5 to frame $\ket{0}_A^A\otimes\ket{\psi}_B^A$6 is

$\ket{0}_A^A\otimes\ket{\psi}_B^A$7

This allows one to describe ordinary systems entangled with a reference frame that is itself in a superposition over group elements (Hamette et al., 2020).

The same issue reappears in later studies of spin systems. “A relational approach to quantum reference frames for spins” derives $\ket{0}_A^A\otimes\ket{\psi}_B^A$8 as the symmetry group of transformations preserving fidelities between equal-sized subsystems, and on projective Hilbert space this reduces to an $\ket{0}_A^A\otimes\ket{\psi}_B^A$9 action. In that setting, collective transformations $\ket{0}$0 preserve the internal properties of the spin system, while the space of states with identical internal properties can be larger than a single group orbit (Pienaar, 2016). This suggests that RRFT may sometimes be characterized by invariants first and by transformation groups only secondarily.

4. Irreversible RRFT and imperfect reference frames

The reversible construction depends on the torsor condition $\ket{0}$1. When a reference system resolves only a subgroup or quotient structure, RRFT becomes irreversible. The general setting considered is a symmetry group that decomposes as

$\ket{0}$2

with $\ket{0}$3 normal. “Large” systems have configuration space $\ket{0}$4, while “small” or imperfect frames have configuration space $\ket{0}$5 (Hamette et al., 2020).

Classically, one fixes a representative $\ket{0}$6 and defines an embedding

$\ket{0}$7

together with a truncation map

$\ket{0}$8

This coarse-grains the $\ket{0}$9-degree of freedom. Because many $n$00-configurations map to the same $n$01-configuration, changes of frame using only $n$02-data are irreversible. The paper gives two explicit examples: modular truncation of translations, $n$03, which models a ruler with finite resolution by mapping positions modulo a cell length $n$04 to an integer label $n$05; and projection from $n$06 to $n$07, using $n$08, which loses the out-of-plane coordinate (Hamette et al., 2020).

At the quantum level, the truncation map becomes

$n$09

The irreversible RRFT from a $n$10-frame $n$11 to an $n$12-frame $n$13 is then

$n$14

where $n$15 is the regular unitary change-of-frame operator on $n$16 systems. In the modular truncation example, $n$17 maps $n$18 to $n$19 with $n$20, and the combined frame change

$n$21

is a completely positive trace-preserving map rather than a unitary (Hamette et al., 2020).

The consistency theorem established for this setting states that truncating and then changing frame is not equivalent to changing frame in $n$22 and then truncating. This means that coarse descriptions do not merely forget an absolute sector; they also lose part of the relational information among the high-resolution systems themselves (Hamette et al., 2020). This suggests that irreversibility in RRFT is structurally tied to limited frame resolution rather than only to environmental decoherence.

5. Observables, entanglement, and interpretational applications

In the relational formalism, observables transform by conjugation. If

$n$23

describes systems $n$24 relative to system $n$25, then the same observable described relative to system $n$26 is

$n$27

Relational invariants are those observables commuting with all frame changes $n$28; they depend only on group differences such as $n$29 and not on absolute coordinates (Hamette et al., 2020).

One direct consequence is that superposition and entanglement are frame-dependent. A product state in one frame can become entangled in another if the reference frame is in a superposition relative to the original frame (Hamette et al., 2020). The same point reappears in the measurement setting: “Switching Quantum Reference Frames for Quantum Measurement” shows that von Neumann Process 2 can be embedded into the perspective-neutral framework, but the projection operation in measurement must be performed after redundancy reduction. In that framework, post-measurement states and even entanglement patterns are related by explicit QRF transformations, while outcome probabilities are preserved across frames (Yang, 2019).

The Wigner’s friend application in the $n$30 case is especially explicit. The group is $n$31, and the friend’s measurement of a system $n$32 in the basis $n$33 leads, from Wigner’s perspective, to

$n$34

Applying the RRFT $n$35 yields the friend’s perspective, where the system is definitely correlated with the friend, while Wigner becomes entangled with friend and system. The analysis recovers the conclusion that “the friend is perfectly correlated with the outcome,” in line with relational quantum mechanics, but it also shows that the state Wigner infers the friend “sees” is not the same as the state the friend actually assigns after a definite outcome is recorded. The consistency assumption $n$36 in Frauchiger–Renner is therefore nontrivial and is not automatically enforced by unitary RRFT (Hamette et al., 2020).

A related foundational claim appears in the spin-only approach of (Pienaar, 2016): when internal properties are identified with fidelities between equal-sized subsystems, a single spin in a superposition relative to a spin magnet can be physically equivalent, in the absence of an external frame, to a macroscopic superposition of the magnet relative to the spin. This does not directly follow from the group-theoretic $n$37 formalism, but it illustrates the same theme: what counts as “macroscopic” or “entangled” can depend on which subsystem is taken as reference.

6. Broader formulations, relativistic variants, and later developments

The perspective-neutral $n$38-body framework makes RRFT into a gauge transformation between reduced descriptions. For the 3-body problem, the quantum transformation from frame $n$39 to frame $n$40 is

$n$41

where each $n$42-map performs quantum symmetry reduction from the Dirac-quantized physical Hilbert space to a frame-adapted reduced Hilbert space. In this formulation, RRFTs are local “quantum coordinate changes,” and the absence of globally valid gauge fixings implies the absence of globally valid relational perspectives (Vanrietvelde et al., 2018).

The dynamical theory of inertial QRF transformations identifies a Lie algebra of canonical transformations acting on the phase space of the systems comprising the reference frames. These transformations close a group structure defined by a Lie algebra different from the usual Galilei algebra, and the standard Galilei group is recovered by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations (Ballesteros et al., 2020). This suggests that RRFT can be viewed not only as a family of ad hoc unitary maps, but as a deformation of ordinary frame symmetry induced by the frame’s own quantum degrees of freedom.

In operator-algebraic language, “Relational Quantum Geometry” identifies extended phase space, crossed products, and QRFs as manifestations of a single geometric structure. A single QRF corresponds to a crossed product von Neumann algebra or trivial quantum principal bundle, while systems containing multiple QRFs are organized as a quantum orbifold or equivalently a $n$43-framed algebra. In that framework, RRFTs are quantum gauge transformations: within one chart they are frame-preserving modifications of the trivialization, and between charts they are frame-switching maps on overlaps, satisfying cocycle conditions (Ahmad et al., 2024). A plausible implication is that RRFT is naturally a transition-function concept, not only a tensor-product unitary on a fixed Hilbert factorization.

A further generalization appears in the non-perturbative gauge-theoretic treatment of relational observables. There, a relational reference frame is a pair $n$44 consisting of reference fields and gauge-fixing conditions, and the classical RRFT from $n$45 to $n$46 at times $n$47 is the canonical transformation

$n$48

In that setting, RRFT maps between relational observables or true degrees of freedom defined by different choices of reference fields, and the physical Hamiltonians in different frames are related nontrivially rather than by simple pullback (Thiemann, 4 Mar 2026).

Operational and information-theoretic developments make the same point in more concrete terms. In the finite Abelian circuit framework, QRF transformations are implemented by

$n$49

and a local gate $n$50 in one frame transforms into

$n$51

so that symmetry-commuting gates remain local, character-sector gates acquire only frame-dependent phases, and generic gates become controlled entangling operations. This yields a frame-dependent entangling-gate count and a relational circuit complexity (Wani et al., 14 Dec 2025). In the three-qubit $n$52 model, RRFT acts as a lossless converter between local coherence and concurrence, preserving the invariant sum $n$53, and the same circuits were implemented on IBM Quantum hardware (Wani et al., 14 Dec 2025).

Relativistic and field-theoretic variants extend the relational idea beyond nonrelativistic particle mechanics. A single-particle analysis of Lorentz-related inertial frames derives the transformation

$n$54

with $n$55 fixed by probability invariance, as an RRFT preserving Born probabilities between observer-relative descriptions (Li, 2022). In quantum field theory, a displacement operator in one frame can be transformed into another via Bogolyubov coefficients, revealing distortions of phase information, modal structure, and amplitude between inertial and non-inertial frames (Onoe et al., 2019). And in curved-spacetime communication without a shared frame, correlations between two identical fields define an invariant operator $n$56 that commutes with any symmetric Bogolyubov transformation, so that information can be encoded in relational observables immune to unknown frame changes (Chȩcińska et al., 2014).

Across these formulations, RRFT is not merely a relabeling of coordinates. It is a concrete transformation between observer-relative descriptions, defined either as a unitary, a canonical map, or a quantum gauge transformation, whose form is fixed by relational observables, symmetry generators, and the chosen notion of admissible reference system.