Special relativity at eleven parameters (SR11) defines a frame by a body: a frame has a mass M , a momentum P , a spin and a position, the transformation to it is the boost L(P ) built from its momentum, and the composition law of the theory says how a second body looks from it. A frame is therefore a physical system with a quantum state, which is what the programme of quantum reference frames has been building; what the eleven add is a kinematics in which the frame body's mass appears explicitly as the scale of its rapidity and fixes its resolution. This paper takes the composition law of the eleven, with the frame body in a superposition of momenta, as the transformation law between quantum reference frames. The transformation is the composition law controlled by the frame's momentum; it carries the momentum of the body it describes linearly, and all its non-linearity sits in the Wigner rotation, which is the only place where the frame's momentum and the body's spin are correlated: the Wigner rotation is the entangler of spin with momentum, and the reduced spin of a body relative to a frame in a superposition of momenta has the purity , the result of Giacomini, Castro-Ruiz and Brukner with the frame's mass as the scale of its rapidity. The one number of the mixing is that mass. A frame body localised to ∆x has a transverse rapidity spread of at least ℏ/(2M c ∆x), so that the product of a frame's position and rapidity resolutions is bounded below by half its reduced Compton length, and the spin purity of a body of rapidity θ seen from it is 1 2 (1 + e -s 2 ) with s = (σ P /(M c)) tanh(θ/2), the small-angle Wigner rotation; the floor ∆x ≳ ℏ/(M c) below which the frame body is no longer one body is the length c 2 /a max on which the acceleration bound of SR11 is built, one scale read once as a resolution and once as a ceiling. Because the transformation is diagonal in the frame's momentum, the frame's pointer basis is its momentum, and it decoheres, when the bodies it frames are traced out, by the overlap of their relative momenta and spins; the thermal blur of a frame defined by a body in a Jüttner ensemble, computed in the thermal monograph, is that decoherence already done, 0.997, 0.992, 0.977, 0.964 at k B T /(mc 2 ) = 0.05, 0.135, 0.5, 1, and a frame defined by a conserved charge does not decohere at all. A clock is a body: its time, its energy and its dilatation charge close on an sl(2, R), the energy-time sector of the dilatation block, so that a body of the eleven is a Page-Wootters clock whose reading is a self-adjoint operator, Pauli's obstruction not applying to a commutator that is an operator; the price is that the reading is positive and its flow quadratic, x 0 -τ D + τ 2 p 0 when the clock is driven by its own energy, at the initial rate of its dilatation charge, which is Planck's constant in the Heisenberg contraction, so that the clock is covariant on neither the full line nor the full flow; and the compact conformal Hamiltonian, whose spectrum is E 0 + n, gives the clock a period, the time circle of the reverse paradigm; de Alfaro, Fubini and Furlan's conformal quantum mechanics is its model. The eleven are then placed among the kinematics that put the Planck scale into relativity, Snyder's and Yang's quantised spacetimes, doubly special relativity, κ-Poincaré, relative locality and triply special relativity, and among the quantum-reference-frame formalisms: they agree with the former on a bounded momentum, in the compact sector, and differ by the acceleration bound, the spin as one of the primary quantities, the reciprocity as an automorphism, and the linear frame composition and undeformed addition of momenta of the ordinary reading; they agree with 1 the latter that a frame is a quantum system and make explicit, in its mass, a parameter those formalisms carry and do not use. What the theory does not do is interpret: its quantum step is the Poisson bracket of the sixteen made a commutator, and it takes no position on measurement.
Special relativity at eleven parameters (SR11) defines a frame by a body: a frame has a mass M , a momentum P , a spin and a position, the transformation to it is the boost L(P ) built from its momentum, and the composition law of the theory says how a second body looks from it. A frame is therefore a physical system with a quantum state, which is what the programme of quantum reference frames has been building; what the eleven add is a kinematics in which the frame body's mass appears explicitly as the scale of its rapidity and fixes its resolution. This paper takes the composition law of the eleven, with the frame body in a superposition of momenta, as the transformation law between quantum reference frames. The transformation is the composition law controlled by the frame's momentum; it carries the momentum of the body it describes linearly, and all its non-linearity sits in the Wigner rotation, which is the only place where the frame's momentum and the body's spin are correlated: the Wigner rotation is the entangler of spin with momentum, and the reduced spin of a body relative to a frame in a superposition of momenta has the purity , the result of Giacomini, Castro-Ruiz and Brukner with the frame's mass as the scale of its rapidity. The one number of the mixing is that mass. A frame body localised to ∆x has a transverse rapidity spread of at least ℏ/(2M c ∆x), so that the product of a frame's position and rapidity resolutions is bounded below by half its reduced Compton length, and the spin purity of a body of rapidity θ seen from it is 1 2 (1 + e -s 2 ) with s = (σ P /(M c)) tanh(θ/2), the small-angle Wigner rotation; the floor ∆x ≳ ℏ/(M c) below which the frame body is no longer one body is the length c 2 /a max on which the acceleration bound of SR11 is built, one scale read once as a resolution and once as a ceiling. Because the transformation is diagonal in the frame's momentum, the frame's pointer basis is its momentum, and it decoheres, when the bodies it frames are traced out, by the overlap of their relative momenta and spins; the thermal blur of a frame defined by a body in a Jüttner ensemble, computed in the thermal monograph, is that decoherence already done, 0.997, 0.992, 0.977, 0.964 at k B T /(mc 2 ) = 0.05, 0.135, 0.5, 1, and a frame defined by a conserved charge does not decohere at all. A clock is a body: its time, its energy and its dilatation charge close on an sl(2, R), the energy-time sector of the dilatation block, so that a body of the eleven is a Page-Wootters clock whose reading is a self-adjoint operator, Pauli's obstruction not applying to a commutator that is an operator; the price is that the reading is positive and its flow quadratic, x 0 -τ D + τ 2 p 0 when the clock is driven by its own energy, at the initial rate of its dilatation charge, which is Planck's constant in the Heisenberg contraction, so that the clock is covariant on neither the full line nor the full flow; and the compact conformal Hamiltonian, whose spectrum is E 0 + n, gives the clock a period, the time circle of the reverse paradigm; de Alfaro, Fubini and Furlan's conformal quantum mechanics is its model. The eleven are then placed among the kinematics that put the Planck scale into relativity, Snyder's and Yang's quantised spacetimes, doubly special relativity, κ-Poincaré, relative locality and triply special relativity, and among the quantum-reference-frame formalisms: they agree with the former on a bounded momentum, in the compact sector, and differ by the acceleration bound, the spin as one of the primary quantities, the reciprocity as an automorphism, and the linear frame composition and undeformed addition of momenta of the ordinary reading; they agree with 1 the latter that a frame is a quantum system and make explicit, in its mass, a parameter those formalisms carry and do not use. What the theory does not do is interpret: its quantum step is the Poisson bracket of the sixteen made a commutator, and it takes no position on measurement.
At eleven parameters a frame is a body with a momentum. The quantum step of the theory makes that momentum an operator, and a frame body in a superposition of momenta is a quantum reference frame. This letter asks what using such a frame does to it, in a model with one assumption. An ideal meter comoving with the frame body reads the canonical spins of test bodies, and nothing else. Relative to the frame each spin is turned by the Wigner rotation of the composition law, by an angle that depends on the frame's own rapidity. So every spin the frame serves is a weak measurement of the frame's velocity. The quantum Fisher information per spin is the square of the derivative of the Wigner angle, in closed form. For a frame at rest it is tanh 2 (θt/2), θt the test body's rapidity, and it is bounded by 1/ cosh 2 θF . With polarised spins and the records kept, n spins sharpen the frame's rapidity, to the van Trees bound 1/(σ -2 + n F ) within one percent. With the records discarded, or with unpolarised spins, the frame's momentum coherence decays by the overlap of the two rotated spin states per spin. For a Gaussian frame its purity is 1/ 1 + N F σ 2 , equal to the purity of the spins relative to the frame. The origin of the frame is the conjugate reference. Each discarded spin kicks the frame's position by ±ℏφ ′ /2M , a random walk, and the kept records widen it by the Heisenberg bound. The velocity reference is kept or sharpened while the origin degrades, the two being the conjugate blocks of the eleven. The frame of an entangled pair with a sharp total momentum has nothing to learn and no origin to lose. The sixteen of a body being the operators of two qubits, a boost is a coupling that is not a gate, and its cost in a simulation is stated. The statements are checked numerically.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
Special relativity seems to be associated with linking physical relations as seen in one frame with those as seen in a frame moving at a speed relative to such a frame. A constant speed is often used for the moving frame. This suggests that classically there is a special frame for a particle with rest mass called a rest frame. In such a case, one simply introduces a parameter for rest mass and does not worry about the complicated motion and interactions which actually create the rest mass. It seems that even quantum mechanics, which considers a particle to be moving as one does not know its location, one uses the idea of a potential V(x) which in principle is associated with a very large rest mass, presumably at rest as no variables of such a rest mass appear in the Klein-Gordon or Dirac equations. It seems that one uses a preferred frame when solving relativistic quantum mechanical equations. Furthermore, given V(x) and a bound state with exp(-i E bound t), E applies to the overall bound system, not to a possible exp(ipx) constituent. The special relativistic invariant equation seems to exist at the system level, like with the rest mass and one may apparently calculate E in the special rest frame of the large mass and hence find a rest mass of the bound system, i..e large rest mass*cc + mo cc + E, mo is the rest mass of the single bound particle. As a result of the use of the special rest mass frame for the V(x) producing large rest mass, it seems that one does not have to worry about special relativistic constraints on the interaction with V(x), as only one frame is involved. In other words, a bound calculation is about finding a new rest mass for an overall new system. Then this system may be used in Lorentz boosts. One simplifies by working in the rest frame of the mass creating V(x). Perhaps surprisingly then, we note that exp(-iEt+ipx) for a free particle is Lorentz invariant and in fact must be Lorentz invariant for the following reason. Imagine that one wishes to create a probability for free particles which is linked with conservation of momentum. Any p must have the same real value weight because for free particles there is no bias for one p or the other. There is, however, a bias for a p1 and p2. It should have the same probability as a p3 and p4, if p1+p2 = p3+p4. (We use one dimension for simplicity.) As a result, exp(ip) seems to be a probability which respects these conditions. The problem is that one may have p=m1v1=m2v2 hence (E1,p) and (E2,p). With respect to momentum, it is irrelevant if one particle has E1 and the other E2, but not if one does a Lorentz boost. Then, in one case: p→ g(v) p + g(v) v E1 and in the second, p→ g(v) p + g(v)vE2. Here g(v) =1/sqrt(1-vv/cc). In other words, p is the same for each particle in one frame and not in the other which violates conservation of momentum because in frame 1 both particles can complete the same conservation of momentum equation, but in the second frame this is not the case, because each has a different momentum. Thus, exp(ip) cannot hold. One requires a linear form in p which is Lorentz invariant, i.e. exp(-iEt+ipx), we argue (and have done so in previous notes). We see now that momentum probability is linked with space in exp(ipx). The idea seems to be that for free colliding particles there is no preferred frame and the probabilities used must reflect this Lorentz invariance. As a result, we argue that a free particle probability in E,p arises because of Lorentz invariance, but then apply it to a V(x) problem and no longer worry about Lorentz invariance at the interaction level. V(x) with no associated rest mass variables assumes a rest frame for the V(x) producing mass. A Lorentz boost which then invalidates exp(ip), would only apply to the total rest mass of the bound particle system which has no overall momentum. It seems that one does apply the Lorentz transformation to the interaction of exp(ipx)’s with V(x). One is only interested in the overall bound system as a single entity for special relativity. As a result, one may only impose momentum considerations as V(x) is a spatial function and one must link it with exp(ipx). In fact, one may write V(x) = Sum over k Vk exp(ikx) to associate it with momentum hits linked with Vk energies, which already moves away from the idea of special relativity as Vk and k are not linked as a 4-vector. The idea of exp(ip1x)exp(i k1x) = exp(ip2x)exp(ik2x), however, seems to still hold (as it does for free particles), linking these two values (i.e. Vk1 and Vk2) with (p1+k)(p1+k) exp(i (p1+k)x). Lorentz invariance for a particle with rest mass seems to be linked with the idea that one ultimately has a rest frame for the particle. One then establishes a relation between the rest mass in one frame and its new E and p as seen by another frame. One does not, however, need to impose Lorentz invariance conditions on the interaction which creates the rest mass it seems. For example, exp(-iEt+ipx) is the Lorentz form used to describe a free particle, but exp(ipx) is the only portion used to describe its interaction with V(x) in the rest frame of the large mass creating V(x). Furthermore, exp(-iEt+ipx) ensures conservation of energy and p for a free particle, but in an interaction case, that does not happen as the potential may absorb energy.
In Part I we suggested that free particle quantum mechanics seems to follow from special relativity. In this note, we focus on the idea of a rest frame defining a rest mass. In such a frame momentum is zero i.e. one has the 4-vector (0,mo)=(p,E). (x,t), however, is also a 4-vector which is usually assigned to a point in space time. In Part I, however, we considered (x,t)=(0,to) associating p=0 and x=0 and alternatively mo and to. Velocity is usually defined as dx/dt =v, but using the Lorentz transformation (1) (with v set to -v), (x,t)=(0,to) transforms to x’/t’ = v. The relationship between x,p and t,E is further manifested through the Lorentz invariant -Et+px in which E scales t and p, x. This we suggest breaks space into pieces proportional to 1/p and time (independently) into pieces proportional to 1/E We next consider particles “created” at different x points. In other words (0,mo) applies, but the particles are at different x positions. In order to keep the relationship p=0,x=0, one must have an x origin at all particle positions. Then spatial scaling of 1/p does not line up unless the particles are n/p (n=integer) apart. Otherwise the scaling in space is unaligned. For different p, scaling in space is unaligned already with (0,mo) and (0, to). Thus for two particles with the same p, but created at different x values at different times and arriving at x1, t1, if there is no alignment, special relativity suggests these two are distinguishable, while Newtonian mechanics does not distinguish. This distinguishability, however, depends on a physical mechanism i.e. a wavelength. Thus, even without considering different p (momentum) values, two identical momenta should be distinguishable if they are not aligned in terms of p scaling even for the same rest mass mo. This we argue is strictly a result of special relativity and is support for the exp(ipx) form of quantum free particle mechanics.
Abstract Quantum reference frame as in the example of observing a quantum particle from another has been a topic of much recent interest. Quantum spatial translations, quantum rotations, and quantum Lorentz boosts in the sense have been studied to some extent. The article aims at using a consistent formulation of all that to give a full picture of what would be the symmetry of Special Quantum Relativity as special relativity as seen from quantum particles. We first look at a direct formulation of the mathematical quantum analog of any classical one-parameter Lie group of such transformations suggested by our Heisenberg picture thinking of quantum mechanics. A quantum rotation and a quantum Lorentz boost are presented. Analysis of the compositions of two such transformations shows unnatural results with various problems, especially with their commutators. From a physical point of view, we argue for a modified formulation. A quantum reference frame transformation, as the description of firstly the spacetime position of an object, idealized as a particle, as seen from another has to involve a full set of four-coordinate quantum quantities instead of one in a single coordinate only. Unlike the classical case, we illustrate that the analog of a change by the zero classical coordinate values has nontrivial effects in the transformation. A consistent full picture of the symmetry is then obtained. Results that could be interpreted essentially as the quantum analogs of the translation of a single coordinate observable, including angular ones for a rotation or a boost, can be retrieved from that but only as approximations. Under the traditional perspective of relativity symmetry being about spacetime reference frames, our task is then completed. We further look at the quantum version of Heisenberg–Weyl symmetry and perspectives on its relation with the subject matter and argue for the necessity of our background noncommutative geometric perspective for a conceptual consistent story.
In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum j and introduce the transformations between them by upgrading the Euler angles occurring in classical SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large j, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.