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Quantum Reference Frames for Lorentz Symmetry
Luca Apadula, Esteban Castro-Ruiz, Časlav Brukner
TL;DR
QRF transformations lacked a formulation for Lorentz symmetry. The paper develops a spacetime-based relativistic framework and quantum Lorentz transformations between relativistic QRFs. It obtains superpositions of time dilations and length contractions, while noting limitations involving free dynamics and particle-based probabilities.
Problem
A formulation of QRF transformations for Lorentz symmetry was lacking despite prior non-relativistic and partial relativistic developments.
Method
The paper treats space and time symmetrically in spacetime states and uses coherent twirling to define transformations between relativistic QRFs.
Results
The framework yields quantum Lorentz transformations and superpositions of special-relativistic time dilations and length contractions, with wave-packet widths linked to length-contraction superpositions.
Takeaways & Limitations
The work extends Einstein’s operational approach to spacetime to the quantum domain through relativistic QRF descriptions.
Takeaways & Limitations
The construction assumes perspectival free dynamics, and its particle-based probability formula does not eliminate propagation between spacelike-separated regions; a field-theoretic extension is left for future work.
Abstract
from arXiv · showhide
Since their first introduction, Quantum Reference Frame (QRF) transformations have been extensively discussed, generalising the covariance of physical laws to the quantum domain. Despite important progress, a formulation of QRF transformations for Lorentz symmetry is still lacking. The present work aims to fill this gap. We first introduce a reformulation of relativistic quantum mechanics independent of any notion of preferred temporal slicing. Based on this, we define transformations that switch between the perspectives of different relativistic QRFs. We introduce a notion of ''quantum Lorentz transformations'' and ''superposition of Lorentz boosts'', acting on the external degrees of freedom of a quantum particle. We analyse two effects, superposition of time dilations and superposition of length contractions, that arise only if the reference frames exhibit both relativistic and quantum-mechanical features. Finally, we discuss how the effects could be observed by measuring the wave-packet extensions from relativistic QRFs.
1 Introduction
The paper extends quantum reference-frame transformations to Lorentz symmetry, addressing a gap left by primarily non-relativistic and partial relativistic treatments. It uses spacetime states without preferred temporal slicing to define quantum Lorentz transformations and predicts superpositions of relativistic effects.
- Prior QRF work mainly treated Galilean symmetry and non-relativistic or post-Newtonian settings.
- Existing relativistic extensions addressed quantum Wigner rotations for internal spin or spacetime translations, but not a full Lorentz-symmetry formulation for QRFs.
- The paper uses spacetime states that treat space and time symmetrically, avoiding any preferred temporal slicing because Lorentz transformations mix them.
- Coherent twirling defines maps between relativistic QRFs, interpreted as quantum Lorentz transformations.
- These transformations produce superpositions of special-relativistic time dilations and length contractions, including states on superpositions of spacetime slices.
2 Spacetime states and Probability
The paper formulates relativistic quantum states over spacetime regions rather than preferred time slices, establishes Lorentz-covariant probabilities, and describes how boosts transform state supports and simultaneity surfaces.
- Relativistic quantum states are represented in 1+1-dimensional spacetime with positive-energy relativistic momentum states and symmetric treatment of space and time.
- A spacetime function f localizes a state through its support and extends initial conditions to arbitrary spacetime regions, including preparations unsharp in time.
- The spacetime-state inner product reduces to the standard Klein–Gordon product on an arbitrary spatial slicing and is Lorentz invariant.
- 2.1 Probability: Complete observation tests use POVM elements whose probabilities are well defined and bounded, with completeness ensured by the resolution of identity.
- 2.1 Probability: The particle-based probability formula permits propagation between spacelike-separated regions, motivating a future quantum-field-theoretic extension.
- 2.2 State transformation under Lorentz boost: Lorentz boosts are represented unitarily and transform spacetime-state supports and simultaneity surfaces between inertial observers.
3 Relativistic Quantum Reference Frame Transformations
The paper formulates relativistic quantum reference-frame transformations using spacetime states and quantum-controlled Lorentz boosts, then derives superpositions of relativistic time dilation and length contraction.
- Relativistic Quantum Reference Frame Transformations: The construction removes redundant global Lorentz information by group-averaging states into perspective-neutral relational states.The authors impose invariance of the state vector under the 1+1-dimensional Lorentz group.
- 3.1.1 Superposition of boosts: A particle can undergo a quantum superposition of distinct Lorentz boosts, producing spacetime and energy-momentum correlations unavailable from a classical boost.The transformed state admits passive and active descriptions: superposed coordinates of fixed supports or an entangled state in one frame.
- Relativistic Quantum Reference Frame Transformations: The perspective transformation is not a symmetry of free dynamics because Lorentz-group averaging does not commute with the free evolution, and the time operator becomes entangling.The free Hamiltonian is therefore not invariant under the quantum Lorentz boosts introduced here.
- 3.1.2 Superposition of spacetime slices: A simultaneity surface in one frame becomes a superposition of tilted hypersurfaces for a quantum reference frame moving in superposed velocities.The particle state becomes correlated with the sharp velocity branches of the other reference frame.
- 3.2 Superposition of special-relativistic time dilations: A time interval becomes a superposition of two relativistically dilated intervals when the reference frame has two Lorentz-boost branches.The branches correspond to special-relativistic dilation of the same rest-frame interval for two boost values.
- 3.3 Superposition of special-relativistic length contractions: The same formalism describes a superposition of length contractions and superposed wave-packet widths for a relativistic quantum reference frame.The spacetime-distance operator remains invariant, so each quantum inertial observer measures the same spacetime distance.
- Relativistic Quantum Reference Frame Transformations: A quantum-controlled Lorentz coordinate transformation changes perspectives between relativistic quantum reference frames.The transformation is controlled by the reference frame’s momentum and acts on external degrees of freedom.
4 Conclusions
The paper develops Lorentz-covariant quantum reference frames using a spacetime-symmetric formulation and identifies superpositions of relativistic time dilation and length contraction. Its applicability is limited by known single-particle relativistic quantum-mechanics difficulties, while the framework suggests extensions toward broader covariance.
- The paper formulates QRF Lorentz covariance, addressing a gap left by prior relativistic QRF work focused on internal degrees of freedom or spacetime translations.
- Relativistic quantum mechanics is reformulated without preferred temporal slicing by treating space and time symmetrically in a 1 + 1-dimensional spacetime.
- The framework analyzes quantum superpositions of time dilations, length contractions, and invariant spatiotemporal distances for reference frames in superpositions of momenta.
- The approach is constrained by single-particle relativistic quantum mechanics, where initially localized particles may propagate superluminally, although a suitable limit restores relativistic causality.
- The work uses Lorentz boosts in a fixed background as a step toward extending general covariance to QRFs and eventually the full diffeomorphism symmetry group.
A Superposition of the wave packet extensions
The appendix constructs a superposition of Gaussian wave packets defined on different simultaneity surfaces and shows that changing to a moving QRF produces Lorentz-contracted widths.
- A superposition of two Gaussian packets is prepared on distinct tilted simultaneity surfaces with slopes α_i = −tanh ω_i.
- The joint state contains Gaussian states defined on the tilted hypersurfaces, with a width parameter characterizing each wave packet.
- Applying the perspective-changing map ˆS_C→A transforms the state from observer C’s description to reference frame A’s description.
- Figure 8 contrasts tilted hypersurfaces viewed from the ground with simultaneity surfaces viewed from the moving ship, where contracted Gaussian widths witness length contraction.
- In A’s perspective, both Gaussian branches lie on a single simultaneity surface at t = 0, while their widths are transformed.
- The transformed branches have widths contracted by γ(ω_i)^−1 < 1, realizing a quantum superposition of special-relativistic length contractions.
A.0.1 Probing superposition in the non-relativistic regime
The paper probes the transformed wave-packet state through position measurements in a non-relativistic regime, retaining relativistic corrections through second order in rapidity.
- Because the relativistic position operator is not well defined, the analysis uses the non-relativistic limit to study position measurements.
- The transformation is expanded for |ω_i| ≪ 1, retaining relativistic corrections up to second order in the rapidities.
- From A’s perspective, the state of particle B and reference frame C is described using spatial and temporal wave-packet standard deviations σ_x and σ_t.
- The perspective-changing map ˆS_C→A is applied before taking the small-rapidity expansion of the transformed state.
- The protocol measures B’s spacetime location conditional on a postselected result for C, using a POVM element for B.
B Perspectival states
This appendix provides intermediate derivation steps for obtaining perspectival states from the earlier state-transformation equations.
- The appendix shows intermediate steps in deriving perspectival states from Eq. (29) to Eq. (34).
C Superposition of boosts
The appendix provides the full derivation of the state in Eq. (39) under the conditions specified in Eqs. (37) and (38), including the spacetime function associated with relativistic QRF C.
- C Superposition of boosts: The state in Eq. (39) is derived under the conditions expressed in Eqs. (37) and (38).
- C Superposition of boosts: The derivation identifies a spacetime function of relativistic QRF C whose Fourier transform is a delta function centered in ω_i for i = 1, 2.
D Superposition of simultaneity surfaces
The appendix gives the detailed derivation of Eq. (47), defining states for A and B and introducing quantities that determine time dilation or contraction depending on mass ratios.
- D Superposition of simultaneity surfaces: The states for A and B used in Eq. (47) are given in Eqs. (45) and (46).
- D Superposition of simultaneity surfaces: The quantities t′_C := m_A m_C t_A and t_C := m_A m_C t determine either a time dilation or a contraction, depending on the ratio of masses.
- D Superposition of simultaneity surfaces: The construction defines a spacetime function describing a relativistic particle with momentum Λω_i k_C, located on the simultaneity surface labelled by t_C.