A change of perspective: switching quantum reference frames via a perspective-neutral framework
Augustin Vanrietvelde1, Philipp A. Höhn1,2, Flaminia Giacomini1,2, and Esteban Castro-Ruiz1,2
1Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria 2Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzman- ngasse 5, 1090 Vienna, Austria
Treating reference frames fundamentally tional language also inspires a new inter- as quantum systems is inevitable in quan- pretation of Dirac and reduced quantized tum gravity and also in quantum foun- theories within our model as perspective- dations once considering laboratories as neutral and perspectival quantum theo- physical systems. Both fields thereby face ries, respectively, and reveals the explicit the question of how to describe physics rel- link between them. In this light, we sug- ative to quantum reference systems and gest a new take on the relation between a how the descriptions relative to different ‘quantum general covariance’ and the dif- such choices are related. Here, we ex- feomorphism symmetry in quantum grav- ploit a fruitful interplay of ideas from both ity. fields to begin developing a unifying ap- proach to transformations among quantum reference systems that ultimately aims at 1 Introduction encompassing both quantum and gravita- Reference frames are essential in our description tional physics. In particular, using a grav- of physical phenomena. Every time we measure a ity inspired symmetry principle, which en- physical quantity or describe a physical event, we forces physical observables to be relational do so with respect to a reference frame. In prac- and leads to an inherent redundancy in tice, reference frames are physical objects that are the description, we develop a perspective- sufficiently decoupled from the system we want neutral structure, which contains all frame to describe. When the reference frame does not perspectives at once and via which they influence the system of interest for all practical are changed. We show that taking the purposes, we treat it like an external entity in perspective of a specific frame amounts to our theoretical analysis. For example, in our cur- a fixing of the symmetry related redun- rent most successful theories, general relativity dancies in both the classical and quan- and quantum theory (incl. quantum field theory), tum theory and that changing perspec- such reference frames are taken as idealized clas- tive corresponds to a symmetry transfor- sical systems that are non-dynamical and neither mation. We implement this using the lan- back-react on spacetime itself, nor on other fields guage of constrained systems, which natu-
arXiv:1809.00556v4 [quant-ph] 22 Jan 2020 contained in it. rally encodes symmetries. Within a simple However, a more fundamental approach to one-dimensional model, we recover some physics should dispose of such an idealization and of the quantum frame transformations of take seriously the fact that reference frames are [1], embedding them in a perspective- always physical systems themselves and thereby neutral framework. Using them, we il- subject to interactions and the laws of physics. In lustrate how entanglement and classical- particular, if we accept the universality of quan- ity of an observed system depend on the tum theory, we have to face the question of how to quantum frame perspective. Our opera- describe physics with respect to a quantum frame of reference and, subsequently, how the descrip- Philipp A. Höhn: [email protected], corresponding author tions relative to different such quantum frames
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 1 are related to one another. Classical frame trans- ativity this is the diffeomorphism symmetry); we formations will not suffice to switch from the per- shall see a toy version of this below. spective of one quantum frame to that of another, This entails the paradigm of relational local- as already epitomized, in its most extreme, by the ization: some dynamical matter or gravitational Wigner’s friend scenario. So what will take their degrees of freedom (in the full theory these will place in a fully quantum formulation? be fields) serve as temporal or spatial reference The answer to this question will, of course, de- systems for others and these relations are invari- pend on the concrete physics at hand. But our ant under diffeomorphisms. That is, these rela- aim here and in [2–5] will be to initiate the devel- tions are invariant under the gauge symmetry of opment of a novel and systematic method for an- general relativity and thereby physically mean- swering this question, that ultimately can encom- ingful. Such gauge invariant relations are usu- pass quantum reference frames in both quantum ally referred to as relational observables. This and gravitational physics. That is, this method applies not only to classical general relativity, but shall be applicable in both fields to produce also to (background independent) quantum grav- the sought-after transformations among quantum ity approaches where one aims at quantizing all frame perspectives. dynamical degrees of freedom, while retaining (a Reference frames (or, more generally, reference quantum version of) diffeomorphism invariance. systems) indeed provide a natural arena for an in- As such, quantum reference systems and rela- terplay of quantum and gravitational physics, ap- tional observables appear ubiquitously in quan- pearing ubiquitously in both fields. Their recog- tum gravity and, given their indispensability, nition as quantum systems themselves dates back they have been studied extensively [7,8,11–21]. to at least 1967 when, in a historical coincidence, However, owing to the challenges in a field the- two seminal papers, by Aharonov and Susskind ory context, what has not been studied exten- [6] and DeWitt [7], separately brought this recog- sively is how to switch among different choices nition to center stage in the foundations of quan- of such relational quantum reference systems in tum theory and quantum gravity, respectively. quantum gravity. For systems with finitely many The ensuing study and usage of quantum refer- phase space degrees of freedom (such as quantum ence systems took, however, rather different di- cosmological models), the first systematic frame- rections in the two fields and our goal will be to work for changes of quantum reference systems unify some of these developments. was developed in [22–24] and applied to tem- In (quantum) gravity, there is even a necessity poral references called relational clocks, but re- to employ physical systems as references with re- stricted to sufficiently semiclassical states. The spect to which to describe the remaining physics. crucial feature of this framework is a ‘perspective- On account of the diffeomorphism symmetry of neutral’ quantum theory which contains all clock general relativity, which is a consequence of gen- choices at once and each clock choice corresponds eral covariance, physical systems are not local- to a gauge fixing. ized and oriented relative to some absolute spa- In quantum information, on the other hand, tiotemporal structure, but with respect to one quantum reference frames have been extensively another [8]. This is spacetime relationalism and, discussed mainly with the purpose of devising for later purpose, we note that this is closely re- communication tasks with physical systems serv- lated to Mach’s principle, which roughly states ing as detectors. In the seminal papers [6, 25] it that what is an inertial frame is not determined was shown that it is possible to overcome super- with respect to an absolute space (as in Newto- selection rules (such as the charge superselection nian physics), but by the other dynamical con- rule in Ref. [6]) via the introduction of a quan- tent of the universe [8–10]. Physics is purely rela- tum reference frame. In Ref. [26] it was shown tional: intuitively, if one moves around the entire that quantum mechanics can be consistently for- dynamical content of the universe while keeping mulated without appealing to abstract reference the relations among its constituents intact, it will frames of infinite mass. The subsequent liter- not change the physics. Mach’s principle thereby ature [27–34] has then focused on different as- implies a symmetry principle and a correspond- pects of the introduction of quantum systems as ing redundancy in the description (in general rel- reference frames, and mainly on a) the lack of
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 2 a shared reference frame, b) bounded-size refer- principle [41], stating that it is always possible to ence frames, and c) the possibility of overcoming find a quantum reference frame having a definite general superselction rules by employing quan- causal structure in the local vicinity of any point. tum reference frames (e.g., see [27] for a review). Our main ambition will be to synthesize these These approaches resort to an encoding of quan- developments in quantum gravity and quantum tum information into relational degrees of free- information into a unifying method for switch- dom. The latter are invariant under an averaging ing perspectives in the quantum theory that in- over the external symmetry group, defining deco- cludes both spatial and temporal quantum refer- herence free subsystems. A relational approach ence systems and applies in both fields. Indeed, to quantum reference frames has been consid- in the course of this work, here and in [2–5], we ered also in [35–37]. The transformation between shall show how (some of) the quantum reference two quantum reference frames is in general not frame transformations of [1] and the method of considered in this applied quantum information relational clock changes [22–24] can be accom- context, with the important exception of [30]. modated and reproduced within one framework. More foundationaly, quantum reference frames This will be achieved by adopting key ingredients have also been used to derive the Lorentz group from both sides. from operational conditions on quantum commu- In particular, we shall adopt a gravity inspired nication without presupposing a specific space- symmetry principle to develop, as proposed in time structure [38]; this exemplifies how quantum [42], a perspective-neutral super structure that information protocols can constrain the space- encodes, so to speak, all perspectives at once time structures in which they are feasible. and requires additional choices to ‘jump’ into the A suitable starting point for establishing a con- perspective of a specific frame. Technically, we nection between these efforts in quantum infor- will achieve this by availing ourselves of the tools mation and in quantum gravity is the approach to and concepts of constrained Hamiltonian systems quantum reference frames developed in [1], which [43, 44] that also play a key role in the canoni- we shall further exploit in the present paper. The cal formulation of general relativity and quantum main idea of Ref. [1] is to formulate changes be- gravity [8,19] and that were also used for the re- tween quantum reference frames in an operational lational clock changes in [22–24]. and fully relational way, without referring to any The symmetry principle will, as mentioned external or absolute entity. Within this formu- above, entail two related key features: (i) an lation, Ref. [1] investigates the extension of the inevitable redundancy in the description of the covariance of physical laws under such reference physics (gauge freedom and constraints), and (ii) frame transformations, paving the way for a for- that the physically meaningful (i.e. gauge invari- mulation of a notion of “quantum general covari- ant) information is purely relational. The inher- ance”. Such developments give concrete meaning ent redundancy will permit us to treat all pos- to the idea of describing physics from the point sible reference frames as part of a larger phys- of view of a quantum frame of reference. This ical system at once and on an equal footing; approach might be particularly relevant in the a priori no choice of frame is preferred and no context of quantum gravity, where a fixed no- frame is described externally. However, in order tion of spacetime (spacetime metric) is not avail- to make operational sense out of physical phe- able. As an indication of the concrete possibility nomena, we must make additional choices to fix of formulating physics on a non-fixed spacetime these redundancies. We will show that choosing metric, Ref. [1] develops an extension of Galileo’s a system from the perspective-neutral picture to weak equivalence principle for quantum reference serve as our reference frame is equivalent to fix- frames, which holds when the reference frame is ing these redundancies and that classically this is a system falling in a superposition of accelera- a choice of gauge. Accordingly, (at least classi- tions. This approach is in resonance with works cally) switching from the internal perspective of aiming at formulating physics on indefinite causal one frame to another will amount to a symme- structures from an observer-dependent perspec- try transformation as in [22–24]. Our approach tive [39–41]. In particular, it is closely related thereby connects with, but also extends the dis- to Hardy’s proposal for a quantum equivalence cussion in [45], where it is argued that the redun-
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 3 dancy in gauge theories is not just a mathemat- linking Dirac with a given reduced quantization ical artifact, but expresses the fact that physics constitutes a quantum symmetry reduction pro- is relational and provides the ‘handles’ through cedure, i.e. the quantum analog of phase space which systems can couple (and relate to one an- reduction. This reduction is always formulated other) in different ways.1 relative to a choice of quantum reference frame. Conversely, the operational language of quan- In this simple model, all these transformations tum foundations and, specifically, the approach will be valid globally on phase and Hilbert space, to quantum reference frames in [1] will supply so that no technical subtleties cloud our main ar- our operational interpretation of the formalism. guments and interpretation. In particular, it will inspire compelling new in- However, in generic systems, it will not al- sights into the quantization of constrained sys- ways be true that the quantum theory obtained tems. These insights will thereby be of relevance by applying the quantum symmetry reduction to for quantum gravity. Indeed, there exist two main Dirac quantization coincides with a specific re- strategies in the literature for canonically quan- duced phase space quantization, in line with the tizing constrained systems: observations in [20, 21, 51–59]. This will not be a problem for our approach, as we explain in more Reduced quantization: One solves the con- detail later: in general, we will interpret the result straints first at the classical level and only of applying the quantum symmetry reduction to quantizes non-redundant gauge invariant de- Dirac quantization, which removes redundancy in grees of freedom. that description, as the perspective of the associ- Dirac quantization: One quantizes first all ated frame. (incl. redundant and gauge) degrees of free- Furthermore, in generic systems, globally valid dom and solves the constraints in the quan- perspectives of quantum reference frames will be- tum theory. come impossible (this is analogous to the Gribov problem in gauge theories) and, accordingly, the There has been an ample discussion in the lit- transformations between them will likewise not erature as to how these two quantization strate- be of global validity. This is illustrated in the gies are related – with the general conclusion that companion paper [2], where we extend our discus- ‘constraint imposition and quantization do not sion to the three-dimensional N-body problem, commute’ – and about when one or the other and in [3,4], where it will be shown how to change should be applied [20, 21, 47–59]. Adopting the relational quantum clocks, using our new method. operational language of [1], we will shed some In particular, in contrast to [22–24], these clock new light on this discussion, both technically and changes will also be valid beyond a semiclassical conceptually. regime. In this article, we will begin with a techni- cally rather simple model, subject to a linear In connection with discussions of relative and constraint, on a finite dimensional phase space. global states in the literature, we emphasize that Within this model both quantization methods our perspective-neutral structure itself will not are necessary for a complete relational interpre- admit an immediate operational interpretation, tation. As we shall see, Dirac quantization will only the description relative to a given perspec- yield a perspective-neutral quantum theory, con- tive. Concretely, this means that there will be taining all quantum reference frame perspectives global quantum states for the entire physics at at once, while reduced quantization produces the once, namely those of the perspective-neutral quantum physics as seen by a specific frame. We Dirac quantized theory. However, there will be no will also provide the transformations that take global operational states and only states relative us from one to the other and will exploit this to to a frame (which will not include its own degrees establish switches between different quantum ref- of freedom) will admit an operational interpreta- erence frames. In particular, the transformation tion. This will be exploited in [4] to develop a novel take on the ‘wave function of the universe’ 1A complementary extension of these ideas, which does in quantum cosmology, as proposed in [42, 60], not rely on gauge fixings to define frames, has also recently and suggests a new conception of the relative been put forward for the field theory context in [46]. states of relational quantum mechanics [61, 62]
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 4 and their interrelations. which we will encounter in the course of our work. Quantum foundations and (quantum) gravity The aim of physics is to describe the phys- are usually considered independently. However, ical world, or at least a subset thereof. Usu- our results are a clear testimony to how a fruitful ally, this is done by choosing a perspective from interplay of their tools and perspectives can lead which to describe the physical situation at hand. to new conceptual and technical insights in both Abstractly, choosing a perspective is thus tanta- fields. mount to choosing a map from the physics of in- The rest of this article is organized as follows. terest to some suitable mathematical description In sec.2, we explain the interplay of perspective- thereof. More precisely, denote by Sphys the set neutral structures and internal perspectives in of possible physical situations one wishes to de- physics more carefully; a quick reader can skip scribe (and could, in principle, measure) and by it on a first reading. Subsequently, in sec.3, Sdes the set of mathematical objects used for the we introduce a toy model of N particles in one- description of these situations. Then choosing a dimensional Newtonian space in which we impose perspective defines a map a symmetry principle, namely global translation ϕ : Sphys → Sdes invariance, which will serve as a toy version of Mach’s principle. Here we show how frame per- from the actual physics to its description. spectives are related to gauge choices. In sec.4 we The important point to notice is that the actual quantize the classical model and explicitly reveal physics, encoded in Sphys, is, in fact, perspective- the conceptual and technical relation between the neutral. For instance, suppose the physical situ- Dirac and reduced quantization of our toy model, ation is that a billiard ball flies through space so which here give equivalent expectation values. Fi- that Sphys denotes the set of all its possible spa- nally, in sec.5 we analyze some of the operational tial velocities. The statement of such a physical consequences of describing physics from the point situation per se does not require the perspective of view of a quantum system. In particular, a con- of some reference frame, but it can be described crete example will illustrate the quantum frame from many different perspectives. Indeed, sup- dependence of the degree of entanglement of an pose there is an observer Alice who measures the observed system. Finally, we conclude in sec.6 (components of the) velocity of the ball in three with an outlook on further applications of our spatial directions and reads it off the scales of approach. Details have been moved into appen- her measurement device. Then Alice would usu- 3 dices. ally take Sdes to be R and ϕA associates to each physical velocity a three-dimensional vector, cor- responding to the three real numbers she reads off 2 A meta-perspective on perspectives her measurement device, thereby specifying the velocity relative to her frame of reference. This The quick reader can skip this section and proceed is a second point to notice: the choice of a map, directly to sec.3. i.e. a perspective, ϕA is (usually) associated with The purpose of this section is to motivate a choice of reference frame, which is why we have and specify more clearly what we mean by a now attached a frame label to it. Note that only perspective-neutral theory, as proposed in [42]. a concrete perspective has an immediate opera- To this end, we shall adapt the abstract lan- tional interpretation. guage introduced in [38] to explain from a very Of course, this structure is completely general. general standpoint how different perspectives can For example, Sphys could also represent the quan- fit into one framework and how one can switch tum states associated with (possibly an ensemble between them. We shall thus revisit some of) a physical system that one can try to estimate, fairly basic questions, illustrating along the way using tomography, in a laboratory. A physically how perspective-neutral structures already ap- distinguished choice for Sdes would be the appro- pear ubiquitously in all of physics. This discus- priate set of density matrices. Clearly, a concrete sion will also highlight some peculiarities, such as description ϕA of the quantum states depends on an absence of global perspectives in most systems a choice of reference frame as it involves the choice of interest, and explain within a broad context the of a Hilbert space basis that Alice associates with structure of the sought-after perspective changes certain measurement outcomes, say, of spin in her
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 5 z-direction on which another observer Bob may tures of constrained systems, incl. canonical grav- not agree. ity. In the previous two examples, while the ac- Indeed, for a system with first class constraints, tual physics is perspective-neutral, the theories we shall propose to interpret the classical con- describing it are arguably not. For example, if straint surface and the gauge invariant physi- one wrote down a standard Lagrangian for the cal Hilbert space Hphys of its Dirac quantiza- billiard ball, it would fail to be invariant under tion as the perspective-neutral physics Sphys of general time-dependent changes of coordinates in the classical and quantum theory, respectively. configuration space; it does not abide by a full Correspondingly, we shall argue that the re- symmetry principle and thereby presupposes a duced (gauge fixed) phase spaces and their re- special class of (e.g., inertial) frames with respect duced quantizations, the reduced Hilbert spaces to which it is formulated. Similarly, at least the Hred, assume the role of the descriptions Sdes of standard textbook formulation of quantum me- the classical and quantized physics, respectively. chanics implicitly assumes the frame of the ob- Hence, a perspective ϕA will define a mapping server and her measurement and preparation de- from the constraint surface/Hphys to the reduced vices at the outset. phase space/Hred and we shall only assign an By contrast, a prime example of a perspective- operational interpretation to the latter reduced neutral theory is general relativity. The Einstein- structures; these are the physics described with Hilbert action is completely independent of coor- respect to a given classical or quantum reference dinates and choices of reference frame (it is dif- frame.3 feomorphism invariant) and so the theory does For any of the above examples and theories, it not dictate the choice of perspective from which is now also clear how to switch from the perspec- to interpret and describe the physics in space- tive of, say Alice’s frame, to another, say Bob’s, time; it contains all frame perspectives at once namely through the following transformation: and on equal footing and it is up to the physi- −1 cist to pick one. For example, when considering TA→B = ϕB ◦ ϕA . (1) the dynamics in a given spacetime in general rel- Note that, while TA→B : Sdes → Sdes is a map ativity, Sphys may represent the possible physical from description to description,4 it always pro- situations happening in that spacetime.2 Given a ceeds via the perspective-neutral structure Sphys reference frame associated to some observer Alice in-between, thanks to its compositional form. (usually an orthonormal tetrad), Sdes is then nor- This is the general form of our sought-after trans- 4 mally taken to be R . Her perspective ϕA defines formations and we shall encounter it repeatedly a (usually only locally valid) coordinate descrip- throughout our work, i.e. below and in [2–5]. tion of the physics in that given spacetime, e.g., Hence, we will always switch perspectives via encoding the tangent vector corresponding to the the perspective-neutral meta-structure in both the motion of a massive object in a four-vector whose classical and the quantum theory. Notice also the components describe the velocity relative to Al- structural resemblance to coordinate changes on ice. a manifold. However, here it is more than just a In the course of our work, we shall show how coordinate transformation: it is a change of per- to embed the discussion of quantum reference spective. frames into such a meta-framework. In partic- ular, in analogy to general relativity, we will use 3There is an interesting analogy to the relation between a symmetry principle, in the form of invariant shape dynamics and general relativity [10, 63]. The two Lagrangians and (first class) constraints, to for- theories are related via a ‘linking theory’ [64] that can be regarded as the perspective-neutral theory. When re- mulate perspective-neutral theories of reference stricted to solutions admitting constant-mean-curvature frames. We shall use these theories to argue for slicings, shape dynamics and general relativity (as reduc- a novel, more general interpretation of key struc- tions of the linking theory) can be regarded as two differ- ent descriptions of the same physics.
2 4 phys By contrast, when considering the dynamics of space- Similarly, one can construct transformations TA→B := −1 time in general relativity, Sphys may represent the space of ϕB ◦ϕA : Sphys → Sphys that are actual operations on the solutions or, in the canonical formulation, the constraint physics [38]. Since we are only interested in perspectives surface (see also comments below). and their relations, such operations are not relevant here.
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 6 The transformation (1) clearly assumes the For intuition: ‘jumping’ into the perspective perspective map ϕA to be invertible somewhere. of a given reference frame defines, e.g., what it The example of general relativity above makes it means to be ‘at the origin’ in position space, clear, however, that this will in general not be fixing the translational symmetry. Conversely, possible globally; ϕA need not be defined every- starting from the assumption that one can always where on the perspective-neutral physics Sphys. ‘jump’ into a frame that is ‘at the origin’, one is In other words, in general we will find that global led to a symmetry, because our ability to ‘fix’ any perspectives on the physics (with operational in- system at the origin means its ‘absolute position’ terpretation) do not exist in most interesting sys- is not physical. tems. This will also be illustrated in the compan- The technical simplicity of the model will per- ion articles [2–4]. mit us to illustrate in sec.4 the general method In consequence, the perspective changes (1) for changing perspectives via a perspective- will generally constitute non-global transforma- neutral structure in the quantum theory and, in tions and it will become a non-trivial question particular, to derive the quantum reference frame whether (and where) Bob’s perspective ϕB can transformations constructed in [1] for the one- −1 be concatenated with the inverse ϕA of Alice’s dimensional case from first principles. In this perspective. Hence, in general it will be a non- manuscript we will thus not need to worry about trivial question too whether perspective changes technical subtleties that cloud the main argu- (1) can be concatenated and constitute a group ments and which will be studied in more com- or more general structures such as a groupoid. plicated models in [2–5]. Such questions are crucial as a lot of information about the physics resides in perspectives and their 3.1 A toy model for Mach’s principle in 1D relations. For instance, the information about a space spacetime’s geometry is encoded in the relations among its reference frames and this is also where For simplicity, we shall take the N particles to symmetries reside. be of unit mass5 and the configuration space In the following, we shall now transition from as Q = RN so that the phase space is simply 2N N a perspective-neutral structure to internal per- R . We use canonical pairs (qi, pi)i=1 as coordi- spectives and study operational consequences of nates. It turns out (see AppendixA) that a La- the ensuing transformations (1). By contrast, the grangian with global translation invariance nec- constructions in [38, 65] can be regarded as pur- essarily leads to a (primary) constraint, namely suing in the opposite direction: they start with that the center of mass momentum vanishes operational conditions on relations among inter- N nal perspectives and attempt to reconstruct a P = X p ≈ 0 , (2) perspective-neutral structure. i i=1
so that the momenta of the individual particles 3 Classical reference frame perspec- are not all independent. Note that this equation tives as gauge-fixings defines a (2N − 1)-dimensional constraint surface in phase space. The symbol ≈ denotes a weak We now construct a simple model, which incorpo- equality, i.e. an equality that only holds on this rates a toy version of Mach’s principle for N in- constraint surface. (See [43, 44] for an introduc- teracting particles in one-dimensional Newtonian tion to constrained Hamiltonian systems.) space through a global translational invariance. On the constraint surface defined by (2), the We will argue that it constitutes a perspective- Hamiltonian (following from the Lagrangian of neutral theory in which no reference frame has AppendixA) will be of the form been chosen yet, and in which physical quanti- 1 ties are relational. We will then show that going H = X p2 + V ({q − q } ). (3) 2 i i j i,j to the perspective of a particular reference frame i amounts to a gauge fixing, and, correspondingly, that switches from one frame perspective to an- 5It is straightforward to generalize the model to arbi- other are gauge transformations. trary individual particle masses and we come back to this.
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 7 Clearly, the constraint is preserved by the dynam- generator, for otherwise the evolution of all vari- ics {P,H} = 0 (where {., .} denotes the Pois- ables would be unambiguously determined, given son bracket) and so, in the terminology of Dirac, initial data, leaving no room for gauge freedom. no secondary constraints arise to enforce the con- Note that λ will get fixed below when fixing the servation of P and it is automatically first-class. gauge. It is evident that the equations of motion It is therefore a generator of gauge transforma- of Dirac observables (generated by Htot) will not tions [43, 44]. Indeed, in line with the symmetry depend on λ on the constraint surface; their dy- of the Lagrangian of AppendixA, it generates namics thus features no arbitrariness, given suit- global translations, infinitesimally given by able initial data. However, there is redundancy among the Dirac ( qi → qi + {qi,P } ε = qi + ε observables mentioned. Thanks to (2), only N −1 . (4) pi → pi + {pi,P } ε = pi of the pi are independent on the constraint sur- face. Similarly, only N − 1 of the relative dis- The physical interpretation is here (see Appendix tances are independent, as qi − qk is just the sum A for a discussion): the localizations qi(t) and of qi − qj and qj − qk. Altogether, we therefore motions q˙i(t) of the N particles with respect to have 2(N −1) independent gauge invariant phase the Newtonian background space have no physi- space functions. Indeed, given that P generates cal meaning, but are gauge dependent. Only the one-dimensional gauge orbits in its (2N − 1)- relative localization and motion of the particles dimensional constraint surface, the reduced (i.e. are physically relevant, thereby providing a toy gauge invariant) phase space [44] is 2(N − 1)- model for Mach’s principle. Thanks to the sym- dimensional for this model. metry, physics is here relational.6 We propose to interpret what we have de- Given the gauge symmetry, we need to find scribed thus far as a perspective-neutral super physical quantities that are gauge invariant and theory. Using this structure, we derived the gauge thus do not depend on the localization and mo- invariant degrees of freedom, but we have not de- tion relative to the Newtonian background space. scribed them from the perspective of, e.g., any Technically, these are phase space functions O, of the N particles, each of which could serve as a which Poisson-commute with the gauge genera- physical reference system. That is to say, we have tor on the constraint surface {O,P } ≈ 0 (i.e., not chosen any reference frame from which to de- are invariant under the gauge flow generated by scribe the physics. The perspective-neutral super P ) and are called Dirac observables. In this sim- structure contains, so to speak, all perspectives at ple model, there are obvious examples: For in- once and thereby does not by itself admit an im- N stance, all N momenta pi and all 2 relative mediate operational interpretation. Instead, we distances qi − qj, i, j = 1,...,N are Dirac ob- shall now argue that choosing the internal per- servables. Clearly, H is also a Dirac observable spective of a reference system on the physics is and the total Hamiltonian (sum of a gauge in- tantamount to choosing a particular gauge fix- variant Hamiltonian plus a linear combination of ing. In particular, the perspective-neutral struc- gauge generators [43, 44]) thereby reads ture tells us there is a redundancy among the ba- sic Dirac observables, but it does not by itself 1 H = X p2 + V ({q − q }) + λP , (5) choose which of the Dirac observables to consider tot 2 i i j i as the redundant ones. Gauge fixing will take care of this. where λ is a Lagrange multiplier, namely an ar- bitrary function of time which encodes the gauge freedom (eq. (58) in the Lagrangian formulation 3.2 Choosing an internal perspective = choos- in AppendixA) in the canonical equations of mo- ing a gauge tion. Intuitively, it is clear that an arbitrary func- We shall now reduce the phase space, getting rid tion of time will have to appear in the evolution of gauge freedom altogether and working only 6In this simple model, only the spatial physics is rela- with the physical quantities written in a partic- tional, while we have kept the absolute Newtonian time as ular gauge. Suppose we want to describe the physical. One can also make the temporal physics of such physics from the internal perspective of particle N particle models relational, see e.g. [9, 10, 66, 67]. A. We are free to define A as the origin from
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 8 which to measure distances in coordinates, im- being a new gauge-fixed phase space, we have to posing (emphasized through the symbol !) specify the bracket structure on it that is inher- ited from the original phase space R2N . For con- ! χ = qA = 0 , (6) strained systems, this amounts to replacing the Poisson bracket with the Dirac bracket [43, 44]. which is a global gauge fixing, as {χ, P } = 17 In the present model it simply reads: and technically this implies that the constraints become second class. {F,G}D = {F,G}−{F,P }{χ, G}+{F, χ}{P,G} , This gauge choice indeed corresponds to ‘tak- (11) ing the point of view of A’, since now all relative for any phase space functions F,G, where {., .} distances between A and the other N −1 particles denotes the usual Poisson bracket. The Dirac (these are a complete set of independent configu- brackets of our basic phase-space variables are ration Dirac observables) simply become then:
qi − qA 7→ qi , i 6= A. (7) {qA, pA}D = 0 , {qi, pj}D = δij , ∀ i, j 6= A. Accordingly, we can consistently interpret the qi (12) as position measurements of the remaining parti- Hence, this reduction simply discards parti- cles relative to particle A. (The relative distances cle A’s position and momentum from among the among the remaining particles are clearly redun- physical degrees of freedom and we pick the re- dant information.) maining ones as coordinates of the reduced phase It is clear that we should also solve the redun- space. We thus end up with a theory for N − 1 dancy among the basic momenta for pA, particles – as seen by A. The corresponding re- X duced Hamiltonian can be computed from (5) and pA ≈ − pi , (8) (2,6): i6=A so that all the N − 1 pi6=A become the indepen- Hred = X p2 + X p p + V ({q } ) . (13) dent momentum variables. Note that pA is not A i i j i i6=A i6=A i6=j proportional to q˙A alone (see (61) in Appendix i,j6=A A) so that the fact that generally now pA 6= 0 does not mean the motion of A is not fixed. This Hamiltonian is of a somewhat non- In fact, we have to ensure that defining A as standard form: the usual 1/2 factor in the kinetic the origin is consistent at all times. This fixes the energy is not present and there are couplings be- Lagrange multiplier λ. Indeed, the equations of tween the pi’s. However, it encodes the relational motion are physics correctly. Indeed, restricting ourselves to the N = 3 case for clarity, the equations of mo- ∂Htot q˙i = ≈ pi + λ , (9a) tion give the accelerations (writing ∂i := ∂/∂qi): ∂pi ∂H ∂V q¨ = −2∂ V − ∂ V, (14a) p˙ = − tot ≈ − , (9b) B B C i ∂q ∂q i i q¨C = −2∂C V − ∂BV. (14b) ! so that the conservation of (6), namely q˙A = 0, Recall that the variables qB and qC encode the imposes relative positions of B and C with respect to A λ = −pA , (10) in the reduced phase space. Thus, if we take for thereby fixing any arbitrariness in the equations example (14a), the factor 2 in ∂BV stems from of motion. Inserting (10) in (9) gives us the dy- the fact that the effect of, for instance, an attrac- namics of all particles in the chosen gauge, i.e. ‘as tive force between A and B has to be counted seen by A’. twice, as it both pulls B towards A and A to- We noted above that the reduced phase space wards B. As for the presence of a ∂C V , it is due is 2(N −1)-dimensional and it is clear that it is co- to the fact that, even in the absence of an interac- tion between A and B, an interaction between A ordinatized by the (qi, pi) where i 6= A. However, and C will affect the position of A, and thus the 7This implies that χ = 0 intersects every P -generated position of B relative to it. These considerations gauge orbit once and only once. generalize to arbitrary N.
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 9 As an aside, it is interesting to also look at what defined through P , by C. PBC|A canonically em- Hamiltonian (13) becomes had we permitted the beds into C as the intersection C ∩ GBC|A where particles to have differing masses mi in (57): GBC|A is the gauge fixing surface defined by the gauge χ = 0. This defines an embedding map
1 1 1 ιBC|A : PBC|A ,→ C . (16) Hred = X + p2 A 2 m m i i6=A i A whose image is C ∩ GBC|A. It is important here p p that P is equipped with the interpretation as + X i j + V ({q } ) (15) BC|A m i i6=A the physics seen by A to avoid ambiguities in the i6=j A i,j6=A embedding map. Indeed, abstractly, the reduced phase space is the space of gauge orbits (i.e., ev-
In the limit mA → ∞,(15) becomes the usual ery gauge orbit corresponds to one physical state) Hamiltonian, in agreement with the fact that a and thus simply the quotient Pred = C/ ∼, where reference system with infinite mass can be used ∼ is the equivalence relation that identifies points as an inertial frame. This limit also recovers the in the same orbit. This abstract Pred can be inter- standard situation in quantum mechanics, where preted as the perspective-neutral phase space: it the description is given with respect to a classical is gauge invariant and coordinatized by Dirac ob- reference frame. servables (which really are functions on the set of In summary, we interpret the reduced phase in orbits). It is also isomorphic to every (globally) a particular gauge as the physics described rel- gauge fixed reduced phase space. Without fur- ative to a reference frame, which corresponds to ther information, the embedding of the abstract that gauge. reduced phase space Pred into C would be highly ambiguous. However, here it is the physical in- terpretation of the gauge fixed PBC|A that singles 3.3 Switching internal perspectives out its embedding. Note that C ∩ GBC|A indeed defines a 2(N − 1)-dimensional hypersurface in It is clear that going from one reference frame to the original phase space R2N . another amounts to a finite gauge transformation Similarly, one can define a ‘projection’ on the constraint surface (and a corresponding swap of which Dirac observables to treat as re- πBC|A : C ∩ GBC|A → PBC|A , (17) dundant, achieved simply through an exchange so that πBC|A ◦ ιBC|A = IdPBC|A . This projection of A, C labels). That is, in order to switch per- essentially drops all redundant embedding infor- spective, we have to go back to the perspective- mation. The same construction can, of course, neutral structure of the original phase space, into also be carried out for the reduced phase space which the reduced phase space embeds. We shall PAB|C in C perspective, by simply exchanging A only be schematic here as the situation is geo- and C labels. In particular, GAB|C is now defined metrically evident (see fig.1); the details of the by qC = 0. following discussion can be found in AppendixB. In order to switch from A to C perspective, we The quick reader may skip the following para- now need the gauge transformation αA→C , gen- graphs and proceed directly to sec.4 . erated by the constraint P , that takes us from Denote the reduced phase space in A perspec- one embedding C ∩ GBC|A to the other C ∩ GAB|C . tive by PBC|A. As discussed above its canonical In AppendixB, we show that this defines a map coordinates are (qi, pi)i6=A. Next, denote the con- SA→C : PBC|A → PAB|C that produces the ex- straint surface in the original phase space R2N , pected result
0 0 0 0 (qB, pB, qC , pC ) 7→ qA = −qC , pA = −pB − pC , qB = qB − qC , pB = pB (18) and satisfies the following commutative diagram, where ζC denotes the invertible map that associates to each orbit in C its intersection point with GAB|C (and similarly for ζA):
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 10 C-perspective (gauge fixing surface) GAB|C
q = 0 GBC|A C
qA = 0 gauge orbits
C perspective-neutral P = 0 A-perspective constraint surface (gauge fixing surface)
Figure 1: Phase space geometry of classical frame perspective switches.
Pred = C/ ∼
−1 ζA ζC
αA→C C ∩ GBC|A C ∩ GAB|C
ιBC|A πAB|C
SA→C PBC|A PAB|C
Taking Sphys := C/ ∼, ϕA := πBC|A ◦ ζA and in position and velocity space. Upon transition ϕC := πAB|C ◦ ζC , this perspective change is in- to phase space, it is clear that such symmetries −1 are generated by (primary) constraints that nec- deed of the form (1), SA→C := ϕC ◦ ϕA , pro- ceeding via the perspective-neutral phase space. essarily involve momenta. As such, the gauge fix- Equivalently, we could also take Sphys := C, ing must include position information and relative ϕ˜A := πBC|A and ϕ˜C := πAB|C to write SA→C := positions are here indispensable relational observ- −1 ables, as opposed to, for example, relative mo- ϕ˜C ◦ αA→C ◦ ϕ˜A , exploiting the shortcut via the intermediate gauge transformation αA→C . (Such menta. This is also reflected in the interpretation a shortcut will be absent in the quantum theory.) of the frames and their relations. By contrast, if, This yields a similar compositional structure as as in [1], one also wanted to switch the roles of in (1) via the perspective-neutral constraint sur- the configuration and momentum basis, one ide- P face. ally would like to have a constraint Q = i qi as a symmetry generator in the one-dimensional N-body problem. However, this is a so-called 3.4 Remark on the preferred role of the posi- holonomic constraint (involving only configura- tion basis tion data) and such constraints can only arise Note that in our present construction of frame through equations of motion as secondary ones, transformations the position basis plays a special and are usually second class (thus not symmetry role, in contrast to [1]. This is a consequence generators). Hence, one would have to proceed of our symmetry principle which is formulated differently than in our construction here. at the level of the Lagrangian (in AppendixA)
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 11 4 Quantum reference frames in 1D globally valid gauge choices are generically ab- space sent, e.g., not only in Yang-Mills theories or gen- eral relativity (the Gribov problem), but also in Our task is now to translate the perspective- much simpler systems [20–24,68]. There even ex- neutral super structure and the inside perspec- ist extreme cases where a reduced quantization tives into the quantum theory. This will permit is outright impossible (without changing other us to switch between different quantum reference structure in the model), while the Dirac method frame perspectives and suggests a new interpreta- can be applied [20,21]. tion of two quantization methods for constrained Nevertheless, in our simple model both meth- systems. ods are necessary for a complete relational in- The two most commonly used strategies for terpretation. In particular, we will establish canonically quantizing constrained systems are: a new systematic quantum symmetry reduction Reduced quantization: Solve the constraints procedure of the Dirac quantized theory which (and possibly gauge fix) first at the classi- is the quantum analog of the classical phase cal level, then quantize the reduced theory. space reduction through gauge-fixing discussed in Sec. 3.2. In particular, this quantum symmetry Dirac quantization: Quantize the system first reduction procedure removes the redundancy in (incl. unphysical degrees of freedom), then the description and is always relative to a choice solve the constraints in the quantum theory. of quantum reference frame: it picks out the ref- erence frame degrees of freedom as the redundant There is a general debate in the literature about ones because we do not want to describe the ref- the relation between these two methods and, in erence frame relative to itself. particular, about when one or the other is the correct method to be employed. In our simple model, this procedure can, in In the context of the Guillemin-Sternberg con- fact, be interpreted as the ‘quantization’ of the jecture, the two methods have been shown to be phase space symmetry reduction in the sense that equivalent for the case of compact phase spaces symmetry reduction and quantization ‘commute’. and compact symmetry groups acting on them That is, first quantizing according to the Dirac [47, 48] (see also [49] for an attempt at a gen- method, then performing the new quantum sym- eralization). It follows from [50, 52, 55] that for metry reduction procedure yields a quantum the- the more interesting case of non-compact phase ory, which coincides with the quantization of the spaces, such as the cotangent bundle R2N used classical symmetry reduced phase space. In our in the present model, an equivalence of the two model, this quantum symmetry reduction pro- methods can also be sometimes established. This cedure will thus map the Dirac quantized the- happens provided certain factor ordering choices ory bijectively into the reduced quantized theory. are made, the gauge transformations take the This permits us to interpret the reduced quan- form of point transformations, as in (4), and tum theory (in a specific gauge) as the descrip- there are no global obstructions to either sepa- tion of the quantum physics relative to a (quan- rating gauge from gauge-invariant degrees of free- tum) reference frame, while the Dirac quantum dom or fixing a gauge. We note that constraints theory will assume the role of the perspective- which are linear in the momenta, as in our model, neutral quantum theory via which internal per- always generate point transformations. How- spectives will be changed. The new quantum ever, the two methods are known to yield uni- symmetry reduction maps will thereby be the key tarily inequivalent results in more general set- to the quantum reference frame switches. Given tings [20,21, 51–59]. that the Dirac quantized theory constitutes the Usually the Dirac method is invoked because perspective-neutral super structure, it does not classically solving constraints and fixing a gauge by itself admit an immediate operational inter- can become arbitrarily complicated. For in- pretation (recall the discussion in sec.2). stance, in general relativity constructing the re- Of course, our model is essentially as simple a duced phase space is tantamount to also solving constrained system as it gets: it features a single the dynamics of the theory, which for the full linear constraint, global gauge fixing conditions theory seems a hopeless endeavor. Furthermore, and accordingly, we have globally valid inside per-
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 12 spectives in both the classical and quantum the- and arbitrary reduced states can be represented ory. In generic models, not only do global chal- as follows: lenges (such as the Gribov problem) arise, which Z may inhibit the existence of globally valid inside |ψiBC|A = dpB dpC ψBC|A(pB, pC ) |pB, pC i . perspectives [2], nor will it always be true that (21) symmetry reduction and quantization commute. The quantum theory obtained by applying the The corresponding reduced Hilbert space will be new quantum symmetry reduction procedure to denoted by HBC|A, see fig.2. It is clear that the Dirac quantized theory will not always coin- generalizing to arbitrary N (or differing masses) cide with the quantization of a classical reduced poses no efforts. phase space, due to the well-known inequivalence In line with the classical case, we interpret this of Dirac and reduced quantization in more gen- reduced quantum theory as the description of the eral situations [20, 21, 51–59]. However, this will quantum dynamics of the remaining particles as not be a problem for our interpretation and we seen from the quantum reference frame of parti- shall comment on this further in the conclusions, cle A. Yet, as we moved to the reference frame Sec.6. of A (that is, fixed the gauge accordingly) before Since we will be dealing with a number of dis- quantizing, we have washed out the perspective- tinct Hilbert spaces through the two methods, we neutral information and the reduced quantum de- have organized the various steps of the construc- scription alone lacks structure to tell us how to tion and their relation in fig.2 for visualization. switch from this frame to the perspective of an- other one (for instance, B), while staying in the quantum theory. To switch perspectives within 4.1 Reduced quantization – quantum theory quantum theory, we need to relate the reduced from an internal perspective descriptions to the Dirac method, which quan- We begin with the reduced quantization of the tizes the classical perspective-neutral structure. model from the previous section, as we aim to recover it subsequently from the Dirac quantiza- 4.2 Dirac quantization – the perspective- tion. That is, we return to the gauge correspond- neutral quantum theory ing to, say, A’s internal perspective and simply quantize the reduced phase space of sec. 3.2. This We now quantize first, then solve the constraints. This requires two distinct Hilbert spaces, see fig. is standard and amounts to promoting the qi, pi, 2. i 6= A, to operators and the Dirac brackets (12) N to commutators9 First we promote all canonical pairs (qi, pi)i=1 (i.e., incl. physically redundant and gauge degrees of freedom), coordinatizing the original phase [ˆqi, pˆj] = i δij, [ˆqi, qˆj] = [ˆpi, pˆj] = 0 , i, j 6= A, space T ∗Q' R2N of sec. 3.1, to operators and the (19) Poisson brackets to commutators on a kinemati- on an L2(RN−1) Hilbert space. cal (or auxiliary) Hilbert space Hkin = L2(RN ). In order to simplify the equations, we restrict Next, we employ this Hilbert space to quantize to N = 3 in the sequel. For instance, the quan- the total momentum constraint (2) and solve the tized Hamiltonian (13) for 3 particles reads: latter in the quantum theory by requiring that physical states |ψiphys of our system are annihi- Hˆ := Hˆ red =p ˆ2 +p ˆ2 +p ˆ pˆ + V (ˆq , qˆ ) , BC|A A B C B C B C lated by it. Returning in the sequel to the N = 3 (20) case for simplicity, we thus impose
8 The entire diagram may also be considered commuta- ˆ phys phys ! tive in the sense that one obtains the same quantum the- P |ψi = (ˆpA +p ˆB +p ˆC ) |ψi = 0 . (22) ory in the bottom right corner, although clearly individual elements of T ∗Q cannot be associated with individual el- Physical states are zero-eigenstates of the con- ements of HBC|A. By contrast, in the red diagram every straint. kin element of H will be mapped to a unique element of One subtlety arises: Pˆ has a continuous spec- Hphys . A,BC trum around zero and so physical states are not 9 Henceforth, we shall work in units where ¯h = 1. normalized with respect to the standard inner
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 13 ∗ 6 P =qA=0 original phase space T Q' R PBC|A
Dirac quantization
ˆ kin TA,BC kin H HA,BC reduced quantization
ˆ δ(P ) δ(ˆpA)
√ ˆ phys TA,BC phys 2π AhqA=0| H HA,BC HBC|A
Figure 2: Diagram of the two quantization methods and their relation for three particles. The horizontal arrows between Hilbert spaces are all isometries. The red diagram is commutative.a The quantum symmetry reduction procedure from the perspective-neutral physical Hilbert space Hphys of the Dirac quantization to the reduced Hilbert space, say, in A-perspective HBC|A involves two steps: 1. a constraint trivialization TA,BC which transforms the constraint in such a way that it only acts on the reference frame variables; 2. the reference frame variables, having become redundant, are discarded by projecting onto the classical gauge fixing conditions. product on Hkin; they are thus not actually con- an (improper) projector onto solutions of (22): tained in the kinematical Hilbert space. Instead, we have to construct a new inner product for δ(Pˆ): Hkin → Hphys Z +∞ physical states, to turn the space of solutions to kin phys 1 isPˆ kin phys |φi 7→ |φi := ds e |φi . (22) into a proper physical Hilbert space H . 2π −∞ (But see also [69] for an alternative method using (23) a modification of the Hilbert space topology.) To this end, we employ group averaging (or re- Projecting an arbitrary state of Hkin in momen- fined algebraic quantization) [19,70,71] and define tum representation,
Z kin kin |φi = dpA dpB dpC φ (pA, pB, pC ) |pAi |pBi |pC i , a general solution becomes, depending on which particle’s momentum is solved for, Z phys |φi = dpB dpC φBC|A(pB, pC ) |−pB − pC iA |pBiB |pC iC Z = dpA dpC φAC|B(pA, pC ) |pAiA |−pA − pC iB |pC iC (24) Z = dpA dpB φAB|C (pA, pB) |pAiA |pBiB |−pA − pBiC .
where for later use we have defined It turns out (see AppendixC) that the sought- after inner product between physical states is kin φBC|A(pB, pC ) := φ (−pB − pC , pB, pC ) , phys phys kin ˆ kin kin (ψ , φ )phys := hψ| δ(P ) |φi , (26) φAC|B(pA, pC ) := φ (pA, −pA − pC , pC ) , (25) kin kin φAB|C (pA, pB) := φ (pA, pB, −pA − pB) . where h·|·i is the original inner product of H . Through Cauchy completion (and other technical All three lines in (24) give different descriptions subtleties which we shall here ignore), the space of the same physical state |φiphys and we shall of solutions to (22) can thereby be turned into a exploit this below. Note that δ(Pˆ) is an improper proper Hilbert space Hphys. projector since δ(Pˆ)2 is clearly singular. Clearly, in analogy to the classical case, ob-
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 14 servables Oˆ on Hphys must satisfy [O,ˆ Pˆ] = 0, for mute with the constraint. For example, the cen- otherwise they would map out of the space of so- ter of mass position qˆcm = 1/3 (ˆqA +q ˆB +q ˆC ) is lutions. Any such Oˆ is thus gauge invariant and a conjugate to the constraint, [ˆqcm, Pˆ] = i. Hence, quantum Dirac observable. For instance, in this physical states as zero-eigenstates of Pˆ must be simple model, the quantization of the elementary maximally spread out over qcm. But this is gauge classical Dirac observables, relative distances and invariance: to smear/average over the gauge or- momenta, are obviously quantum Dirac observ- bit. Indeed, this is precisely what the improper ables, projector (23) does. We thus have to proceed differently in the qˆ − qˆ , qˆ − qˆ , qˆ − qˆ , pˆ , pˆ , pˆ , B A C A B C A B C quantum theory, in order to map from the (27) perspective-neutral structure to the perspective as is the total Hamiltonian, which on Hphys reads of a specific reference frame, i.e. to map from the Dirac to a reduced quantum theory. In particu- ˆ 1 2 2 2 10 Htot = (ˆpA +p ˆB +p ˆC ) (28) lar, we can not fix a gauge. Instead, quantum 2 symmetry reduction proceeds as follows: + V (ˆqB − qˆA, qˆC − qˆA, qˆB − qˆC ) . Just as in the classical case, the observables 1. Pick a reference system. (27) are redundant and only define four indepen- dent Dirac observables on Hphys. Related to this, 2. Transform the quantum constraint(s) in such (24) shows that we also have a redundancy in the a way that the result only acts on the refer- description of a fixed physical state. Dirac quan- ence system variables, which thereby become tization by itself does not tell us which of the fixed. This step is called constraint trivial- Dirac observables to treat as the redundant ones. ization. We thus interpret the gauge invariant physics in Hphys as the perspective-neutral quantum theory. 3. Discard the now redundant reference system Here, we have not chosen a quantum reference degrees of freedom through a projection onto frame from which to describe the non-redundant the classical gauge fixing conditions. physics and precisely the redundancy (originating in gauge symmetry) permits us to choose from This quantum symmetry reduction procedure among a multitude of perspectives. is the quantum analog of phase space reduction through gauge fixing. In particular, the entire 4.3 From Dirac to reduced quantum theory: procedure will define a map from the gauge- recovering relative states invariant states and observables on the phys- ical Hilbert space Hphys to the corresponding Classically, solving constraints means restricting states and observables on the appropriate reduced to the constraint surface in phase space and this Hilbert space. This map can be interpreted as by itself does not lead to gauge invariance because the ‘quantum coordinate map’ taking us from the first class constraints still generate gauge flows on perspective-neutral description to the perspective the constraint surface. We have exploited this in of a reference frame. It maps from the gauge- our classical construction: choosing an internal invariant description on Hphys to what can be perspective corresponded to imposing an addi- tional gauge fixing condition to break the flow 10While gauge-fixing is thus not feasible in the opera- of the constraint (see sec. 3.2). torial Dirac quantization after imposing the constraints, In Dirac quantization, the situation is differ- in the path integral formulation the gauge-fixing happens inside the path integral, for example using the Faddeev- ent: solving the constraint in the quantum the- Popov trick for canonical gauges [72], or the Batalin- ory already implies gauge invariance. Indeed, Fradkin-Vilkovisky framework for arbitrary relativistic Hphys (‘the quantum constraint surface’) is invari- gauge systems [73–75]. In light of our work, some of the ant under the flow of the constraint since, owing different gauge choices inside the path integral may thus ˆ phys phys encode different quantum reference frame perspectives. In to (22), exp(i s P ) |φi = |φi . Intuitively, particular, expressing the gauge-invariant path integral in this difference to the classical case can be under- a suitably gauge-fixed fashion would then correspond to stood through the Heisenberg uncertainty rela- describing the dynamics relative to a choice of quantum tions: gauge dependent quantities do not com- reference frame.
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 15 understood as the quantum analog of a ‘gauge- tives. fixed’ description of this gauge-invariant formu- We illustrate the procedure by moving to the lation. In particular, this map will preserve inner perspective of particle A and recovering the cor- products, algebraic properties of observables and responding reduced quantum theory of sec. 4.1 their expectation values, despite transforming ob- (see fig.2 for illustration of the following steps). servables and states. Accordingly, when reducing Hence, the degrees of freedom corresponding to later relative to different reference frame choices, A are the redundant ones and we need to remove we will still always describe the same physical them. To this end, use (25) and write an arbi- situation, however, from different frame perspec- trary physical state (24) as
Z phys |ψi = dpB dpC ψBC|A(pB, pC ) |−pB − pC iA |pBiB |pC iC . (29)
Next, on Hkin we define the unitary transforma- actually map out of Hphys. Yet, it will define tion: an isometry to a transformed set of distribu- kin TˆA,BC = exp i qˆA(ˆpB +p ˆC ) . (30) tions on H without losing physically relevant information. That is, the end product can be Understanding physical states as distributions on considered just a new representation Hphys := Hkin, we can apply this transformation also to A,BC Tˆ (Hphys) of the physical Hilbert space. In- physical states. However, given that Tˆ does A,BC A,BC deed, we obtain not commute with Pˆ, this transformation will
Z ˆ phys |ψiA,BC := TA,BC |ψi = |p = 0iA ⊗ dpB dpC ψBC|A(pB, pC ) |pBiB |pC iC , (31)
so that we can write: where † is defined with respect to Hkin, so that |ψi = |p = 0i ⊗ |ψi . (32) ˆ A,BC A BC|A PA,BC |ψiA,BC =p ˆA |ψiA,BC = 0 (34) It is important to note that this step does not and |ψiA,BC is actually a physical state, but in a correspond to ‘gauge fixing’ to pA = 0 (there is no different representation. It is clear that, in con- gauge symmetry left and pA is in any case a Dirac trast to the classical case, there is no sense in observable). Instead, this is really a rewriting – which we can talk about additionally gauge fix- 11 a trivialization – of the constraint to system A, ing A’s position. since Crucially, observe that the information in the † PˆA,BC := TˆA,BC Pˆ (TˆA,BC ) =p ˆA , (33) A-slot of |ψiA,BC contains no relevant informa- tion about the original state (29). We may thus 11 Classically, it is also often useful to implement canon- consider A as redundant and, consequently, inter- ical transformations that trivialize constraints in the sense pret the remainder of the state |ψi prelimi- that they become new momentum variables. If the con- BC|A straints are first class then the gauge degrees of freedom narily as the quantum state of B and C relative to can be made directly conjugate to them, while the other A, corresponding to the perspective-neutral state new canonical pairs would be directly Dirac observables. |ψiphys. This is subject to further justification, For examples of this method, see, e.g., [16, 17, 44, 76–80]. but notice already that |ψi is now precisely In the present model, this would amount to the linear BC|A canonical transformation of the form of the reduced states (21). Given the redundancy of |p = 0i , it is nat- (q , p ) 7→ (q ,P ) , (q − q , p ) , (q − q , p ) . A i i i=A,B,C A B A B C A C ural to discard it altogether and consider only Upon gauge fixing qA = 0, implementing the Dirac bracket |ψiBC|A, which contains all the physical (that and dropping the redundant A-variables, this is equivalent phys to what we constructed in sec. 3.2. Here, we are imple- is, relational) information about |ψi . We menting the quantum analog of that procedure – except can achieve this – in some analogy to the Page- that it does (and can) not employ gauge fixing. Wootters construction [81] – by projecting the
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 16 factor of the reference system onto the classical the Schrödinger equation on Hphys implies the gauge fixing condition (6): Schrödinger equation on HBC|A: √ |ψi = 2π hq = 0|ψi BC|A A A,BC phys ˆ phys Z i ∂t |ψi = Htot |ψi 0 0 = dp A hp |ψi (35) ˆ A,BC ⇒ i ∂t |ψiBC|A = HBC|A |ψiBC|A . (38) Z = dpB dpC ψBC|A(pB, pC ) |pBiB |pC iC . In conjunction, it follows (see AppendixC for As shown in AppendixC, Tˆ , followed by the more detail) that expectation values of relevant A,BC phys projection (35), defines an isometry from Hphys to Dirac observables on H are identical to those of the transformed observables on the reduced the reduced Hilbert space HBC|A of sec. 4.1. That is, the procedure preserves the inner product. Hilbert space HBC|A and we do not lose any phys- Before claiming that (30) defines a cor- ically relevant information through our transfor- ˆ phys rect transformation from the perspective-neutral mation TA,BC , despite mapping out of H . In- quantum theory to the one described from A’s deed, our procedure illustrated in fig.2 exploits ˆ phys perspective, we have to check that the relevant that TA,BC (H ) is just a new, but equivalent Dirac observables from (27) transform correctly representation of the physical Hilbert space. We to those of the reduced theory. Indeed, we find thus conclude that the constraint trivialization map (30), followed by the projection (35), indeed † 12 TˆA,BC (ˆqB − qˆA)(TˆA,BC ) =q ˆB , constitutes the desired transformation from the † perspective-neutral to the quantum theory ‘seen TˆA,BC pˆB (TˆA,BC ) =p ˆB , (36) from A’s perspective’. Regardless of our model’s ˆ ˆ † TA,BC (ˆqC − qˆA)(TA,BC ) =q ˆC , simplicity, this sheds new light on both the con- † TˆA,BC pˆC (TˆA,BC ) =p ˆC . ceptual and technical relation between the Dirac and reduced quantization methods. Indeed, in phys Hence, the operator qˆB − qˆA on |ψi corre- the companion articles [2–4], we shall corroborate this with more complicated models. sponds to the operator qˆB on |ψiA,BC , and there- fore also on |ψiBC|A. In other words, the position information stored in the B-slot of |ψiBC|A is in- deed the relative position of B with respect to A, 4.4 Switching internal perspectives in the and the same goes for C. quantum theory Let us also check that the total Hamiltonian (28) transforms as desired. The Hamiltonian In the previous section, we could equally well ˆ have chosen C as the reference system, starting HA,BC for |ψiA,BC becomes (assuming V can be Taylor expanded) with the respective expressions in the last lines of each of (24, 25, 70) and repeating the same † HˆA,BC = TˆA,BC Hˆtot (TˆA,BC ) steps by switching A and C labels. It is thus 1 clear how to change from the internal perspec- = pˆ2 +p ˆ2 +p ˆ2 2 A B C tive of quantum reference frame A to that of C via the perspective-neutral Dirac quantum the- +p ˆBpˆC − pˆApˆB − pˆApˆC + V (ˆqB, qˆC ) . ory: invert the transformations to A-perspective
Yet, pˆA annihilates |ψiA,BC ; it is thus equivalent and apply the transformations to C-perspective. This is the quantum analog of the classical pro- to eliminate the terms containing it from HˆA,BC , which has then no component acting on the A- cedure in sec. 3.3. Concretely, this defines a map factor of |ψiA,BC , and which can therefore also ˆ be considered as a Hamiltonian for the relative SA→C : HBC|A → HAB|C , (39) state |ψiBC|A: 12In fact, as discussed in AppendixD, this constraint ˆ 2 2 HBC|A =p ˆB +p ˆC +p ˆBpˆC + V (ˆqB, qˆC ) . (37) trivialization map is mathematically not unique. However, this non-uniqueness only affects the irrelevant information This is precisely the Hamiltonian (20) of the re- in the redundant A slot and thus has no physical conse- duced quantum theory in A-perspective. Hence, quences.
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 17 of the form In AppendixE, we show that (i) in- ˆ Z deed SA→C |ψiBC|A = |ψiAB|C , where Sˆ := dp0 hp0| Tˆ A→C C C,AB |ψiBC|A , |ψiAB|C correspond via (24, 25) to phys ˆ † the same state |ψi , and that (ii) this × (TA,BC ) |p = 0iA ⊗ [·] Z transformation is equivalent to 0 0 = dp C hp | exp i qˆC (ˆpA +p ˆB) Sˆ = Pˆ ei qˆC pˆB , (41) A→C CA × exp − i qˆA(ˆpB +p ˆC ) |p = 0iA ⊗ [·], where Pˆ is the parity-swap operator defined (40) CA in [1], which, acting on momentum eigenstates of A yields: where the reduced state |ψiBC|A of interested has to be inserted into the empty slot [·] of the tensor ˆ factor associated to particles B and C. PCA |piC = |−piA . (42) A few comments are in place which bring us back to the classical discussion of Sec. 3.3. There Crucially, (41) is precisely the transformation be- we emphasized that it is the physical interpreta- tween quantum reference frame perspectives con- tion of the reduced description as the perspective structed in a different approach in [1] for parti- of reference frame A that singles out the other- cle systems in one-dimensional Newtonian space. wise highly ambiguous embedding of the gauge- In Ref. [1], this transformation arises as a spe- fixed reduced phase space into the perspective- cific instance of a more general class of quan- neutral constraint surface. This interpretation is, tum reference frames transformations, including of course, added information compared to the re- also a generalization of extended Galilean trans- duced description alone, but it is crucial. Note formations. The present construction permits us that this is also qualitatively analogous to what to derive the specific transformation (41) from happens in coordinate changes on a manifold. first (symmetry) principles and via an associated Suppose one picks the coordinate map associated perspective-neutral quantum structure into which to some observer in general relativity to map a all perspectives can be embedded. It also is clear spacetime neighbourhood into some coordinate that the reduced observables transform correctly description thereof. If one only kept the coor- from HBC|A to HAB|C dinate description of the neighbourhood but dis- † 0 0 carded all information about the coordinate map SA→C qˆB SA→C =q ˆB − qˆA, † 0 itself (and thus the interpretation of this coordi- SA→C qˆC SA→C = −qˆA, nate description), it would be impossible to map (43) S pˆ S† =p ˆ0 , back into the spacetime manifold and thus to con- A→C B A→C B † 0 0 sistently change from one frame perspective to SA→C pˆC SA→C = −pˆB − pˆA, another. One has to keep track of which coordi- where the primed operators represent the position nates are associated to which spacetime point in and momentum operators in the reference frame order to compare descriptions of it. of C. Note that the way the operators transform From the previous section it follows that the coincides with the results found in Ref. [1] and ‘quantum coordinate map’ from the perspective- matches the classical case of Eq. (18). neutral physical Hilbert space Hphys to the per- √ For clarity, we summarize this internal per- spective of A is ϕ := 2π hq = 0| Tˆ . Just A A A,BC spective change in the following commutative di- like in the case of classical coordinate changes, agram: we cannot discard the information about ϕA (and thereby the interpretation of the reduced theory) and must invert it to return to Hphys. Append- ing, e.g., the A tensor factor |p = 0i (rather than map’ ϕA is mathematically non-unique up to the number A to which we fix A’s momentum. The same thus holds true another state of A) to the reduced state in (40) −1 for the inverse map ϕA . However, this non-uniqueness in order to map back into the perspective-neutral is physically irrelevant as discussed in appendixD and as Hphys has to be understood in this sense.13 long as one sticks to a convention one will always relate the same reduced state |ψiBC|A with the same original phys 13AppendixD entails that the ‘quantum coordinate physical state |ψi .
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 18 Hphys ˆ † TA,BC TˆC,AB
phys phys HA,BC HC,AB
√ |p=0iA⊗(·) 2π C hq=0|
SˆA→C HBC|A HAB|C
phys Notice that setting Sphys := H as B √the perspective-neutral structure, ϕA := ˆ 2π A hq√= 0| TA,BC as A’s perspective map and kB ϕC := 2π C hq = 0| TˆC,AB as C’s perspective ˆ −1 map, we find that SA→C = ϕB ◦ ϕA is indeed of C the general form (1).
5 Some operational consequences of k A A switching perspectives in the classical and quantum theory Figure 3: In the perspective-neutral description, the three systems A, B, and C behave like two harmonic The operational consequences of the transforma- oscillators, with springs being attached to system C and tion between two quantum reference frames have A (with spring constanst kA), and to systems C and B been thoroughly analyzed in Ref. [1]. There, it (with spring constant kB). From this perspective, the Hamiltonian (both in the classical and quantum case) is was shown that entanglement and superposition 2 2 2 pA pB pC 1 2 1 H = + + + kA(qC − qA) + kB(qC − depend on the quantum reference frame, and this 2mA 2mB 2mC 2 2 2 is operational in that it can in principle be tested qB) . experimentally. In other words, a state which ap- pears as “classical” (for instance, in a coherent state) from the point of view of a certain quan- on the quantum reference frame, compatibly with tum reference frame, might appear entangled, or what has been found in [1]. Here, we additionally in a superposition state from the point of view of provide a study of the entanglement in different a different quantum reference frame. Addition- frames in a dynamical setting, i.e., by studying ally, the notion of quantum reference frame can the solutions of the equations of motion. Let us turn out to be extremely useful in concrete ap- consider a system of two harmonic oscillators, as plications. For instance, the approach in Ref. [1] seen from the perspective of C. In the reduced allows one to identify the transformation to jump theory, the Hamiltonian is into the rest frame of a quantum system, intended 2 2 2 ξA ξB (ξA + ξB) as a system moving in a superposition of veloci- HAB|C = + + ties. This operation would be impossible with a 2mA 2mB 2mC (44) standard reference frame transformation. 1 2 1 2 + kAxA + kBxB, In the following, we analyse a simple model of 2 2 two harmonic oscillators (as illustrated in fig.3), where mA, mB, mC are respectively the mass of both in the classical and in the quantum case. system A, B, and C, and kA, kB are the spring Our goal is to show how the behaviour of the constants of systems A and B respectively. Note different systems is described in two different ref- that in this section we have renamed the relative erence frames. In particular, in the quantum case coordinates in C’s reference frame as xA and xB we will recover the dependence of entanglement and the momenta in C’s reference frame as ξA and
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 19 q(t) ξB. Under the assumption that mC mA, mB, 2 the systems B and C behave as two decoupled oscillators, moving along the trajectories 1
x (t) = A cos (ω t + φ ) , xA(t) A 0 A A t (45) 2 4 6 8 10 xB(t) xB(t) = B0 cos (ωBt + φB) ,
-1 where A0,B0, φA, φB are fixed by the initial con- ditions and ω2 ≈ ki , i = A, B. i mi -2 If we now change to the reference frame of A by changing to the coordinates qC = −xA and qB = xB − xA using Eq. (18), the Hamiltonian q(t) becomes 2 2 2 2 πB πC (πB + πC ) 1 HBC|A = + + 2mB 2mC 2mA (46) qC (t) t 1 2 1 2 qB(t) + k q + k (q − q ) , 2 4 6 8 10 2 A C 2 B B C -1 where qB and qC are the new coordinates and πB,
πC the new momenta in A’s reference frame. The -2 solutions of the equations of motion, matched with the transformed initial conditions, read Figure 4: Up: xA(t) and xB(t) when A0 = B0 = 1, ωA = 1, ωB = 10, φA = 0 and φB = π/2. Down: the qB(t) = B0 cos (ωBt + φB) − A0 cos (ωAt + φA) , solutions of the equations of motion qB(t) and qC (t) in A’s reference frame. qC (t) = −A0 cos (ωAt + φA) . (47)
Note that these solutions coincide with qB(t) = q(t) xB(t) − xA(t) and qC (t) = −xA(t). The solutions of the equations of motion in the initial and final 1.0 reference frames are illustrated, for different val- 0.5 ues of the parameters, in fig.4 and in fig.5. In xA(t) t x (t) particular, we notice that, while in the reference 2 4 6 8 10 B frame C the two solutions are independent, in the new reference frame correlations arise. -0.5 In the particular case when ω = ω and when A B -1.0 the oscillators are in phase, one finds the solution qB(t) = 0. Physically, this means that if the two oscillators are perfectly in phase and oscillate at q(t) the same amplitude and frequency, from the point of view of A the system B doesn’t move. 1.0
After quantization, as we have shown in sec.4, 0.5 the Hamiltonian acting on the reduced phase qC (t) t qB(t) space is quantized as 2 4 6 8 10 -0.5 ˆ2 ˆ2 ˆ ˆ 2 ˆ ξA ξB (ξA + ξB) HAB|C = + + -1.0 2mA 2mB 2mC (48) 1 2 1 2 + kAxˆ + kBxˆ , 2 A 2 B Figure 5: Up: xA(t) and xB(t) when A0 = 0.3 B0 = 1, ωA = 10, ωB = 1, φA = 0 and φB = π/2. Down: the where the parameters and the operators have the solutions of the equations of motion qB(t) and qC (t) in same meaning as in the classical case. For sim- A’s reference frame. plicity, we assume that the system is initially pre- pared in an eigenstate of this Hamiltonian (so
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 20 that the time evolution of the state only amounts point of view of A is an entangled state. Thus, to a global phase, which we can then discard. we have mapped, via a quantum reference frame Notice that this method is general, because any transformation, a product state into an entangled other state of the Hibert space L2(R2) can be ob- state, showing the dependence of entanglement tained by linear combinations of the eigenstates on the quantum reference frame. The Hamilto- of the harmonic oscillator). nian from the viewpoint of A can easily be calcu- Under the assumptions that the derivatives of lated as n the total eigenstates of A and B Ψ (xA, xB), ˆ ˆ ˆ ˆ† n ∈ N, are of the same order, and that mC HBC|A = SC→AHAB|C SC→A mA, mB, we can consider a perturbative expan- 2 2 2 m πˆB πˆC (ˆπB +π ˆC ) sion in A(B) . To the lowest order in pertur- = + + (51) mC 2mB 2mC 2mA bation theory, we have two decoupled harmonic 1 1 p + k qˆ2 + k (ˆq − qˆ )2, oscillators with frequency ωA = kA/mA and A C B B C p 2 2 ωB = kB/mB. The eigenstate can then be split where qˆi and πˆi, i = B,C are the position and into the two eigenstates of A and B, which are momentum operator in the reduced phase space easily expressed in terms of the Hermite polyno- from the point of view of A. mials. For concreteness, we shall focus on the In order to analyse the dependence of quan- first two eigenstates tum features on the reference frame, it is conve- nient, in this particular example, to look at the 1/4 α x2 0 αi − i i Wigner function of the relative states in the two ψi (xi) = e 2 , π reference frames. In the initial reference frame (49) !1/4 2 √ 3 α x C, the Wigner function of the state of A and 1 αi − i i ψ (x ) = 2 x e 2 , i i π i B is the product of the two Wigner functions fW,i|C (xi, ξi), with i = A, B. In particular, the miωi Wigner function of the ground state ψ0(x ) of the where αi = ¯h and i = A, B. Since i the quantum reference frame transformation harmonic oscillator is is unitary, the transformed state is also an ξ2 i eigenstate of the new Hamiltonian with the 0 1 −α x2 − 2 f (x , ξ ) = e i i e h¯ αi , (52) same eigenvalue. Therefore, if in the ini- W,i|C i i π¯h tial reference frame we have |Ψ(t)iAB|C = − i (En +Em)t n m and the Wigner function of the first excited state e h¯ A B |ψ i |ψ i , where n = 0, 1, in A|C B|C ψ1(x ) is A’s reference frame this state is transformed to i ˆ |Ψ(t)i = SC→A |Ψ(t)i . Explicitly, ! BC|A AB|C 1 2ξ2 f 1 (x , ξ ) = 2α x2 + i − 1 Z W,i|C i i i i 2 − i (En +Em)t n π¯h αi¯h |Ψ(t)i = e h¯ A B dqB dqC ψ (−qC ) (53) BC|A ξ2 i −α x2 − 2 m × e i i e h¯ αi . × ψ (qB − qC ) |qBiB|A |qC iC|A . (50) When we change to the reference frame A, we We can see that the state of B and C from the get
j k fW,BC|A(qB, qC , πB, πC ) = fW,A|C (−qC , −πB − πC )fW,B|C (qB − qC , πB), (54) where j, k = 0, 1 and A, B label the initial Wigner functions of systems A and B respectively. In order to find the Wigner function of B or C, it is enough to take the marginals Z fW,B|A(qB, πB) = dqC dπC fW,BC|A(qB, qC , πB, πC ), (55) Z fW,C|A(qC , πC ) = dqB dπB fW,BC|A(qB, qC , πB, πC ). (56)
Different combinations of these Wigner functions are plotted in the figures6–10. In particular, in
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 21 0.3 0.1
0 0.2
-0.1
0.1 -0.2
-0.3 0
0 Figure 6: On the left, the Wigner function of the ground state of the Hamiltonian, fW,A|C (xA, ξA), with α = 1. 1 On the right, the Wigner function of the first excited eigenstate of the Hamiltonian, fW,A|C (xA, ξA), with α = 1. The Wigner functions for system B are analogous to those of system A, and are calculated by taking the marginals on either system A or B in the initial reference frame C.
0.08 0.08
0.06 0.06
0.04 0.04
0.02 0.02
0 0
Figure 7: In the final quantum reference frame A, the marginals of the total Wigner function representing the reduced state of system B (on the left) and C (on the right) when both A and B were initially in the ground state. In both cases, αA = 0.1. αB
0.125 0.10
0.100 0.05
0.075 0 0.050
-0.05 0.025
-0.10 0
Figure 8: In the final quantum reference frame A, the marginals of the total Wigner function representing the reduced state of system B (on the left) and C (on the right) when A was initially in the ground state and B in the first excited state. In both cases, αA = 1. αB
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 22 0.125 0.10
0.100 0.05
0.075 0
0.050
-0.05 0.025
-0.10 0
Figure 9: In the final quantum reference frame A, the marginals of the total Wigner function representing the reduced state of system B (on the left) and C (on the right) when A was initially in the first excited state and B in the ground state. In both cases, αA = 1. αB
fig.6 the Wigner functions of the ground and ex- 6 Conclusions and outlook cited state of the harmonic oscillator are illus- trated. These functions can refer to both system In this work, we have exploited a fruitful inter- A and B from the viewpoint of the initial refer- play of ideas from quantum gravity and quantum ence frame C. On the right in fig.6, the negativity foundations to begin developing a unifying ap- of the Wigner function indicates the nonclassical- proach to transformations among quantum ref- ity of the excited state. In the figures7,8,9, and erence systems – of both temporal and spatial 10 the Wigner functions of the reduced state of character – that ultimately should be applicable B (on the left) and of C (on the right) are shown in both fields. Methodologically, we have com- in the new reference frame A for different combi- bined tools and concepts from constrained sys- nations of states. In particular, fig.7 shows the tems, also inherently used in the relational clock Wigner functions of B and C from the point of changes of [22–24], with the operational approach view of A when the state of A and B from the to quantum reference frames recently put for- point of view of C was the product of the ground ward in [1]. In particular, as proposed in [42], state eigenstates in the initial reference frame. we took recourse to a gravity inspired symmetry Figures8 and9 show the Wigner functions of principle to formulate a perspective-neutral super B and C relative to A when the state of A and B structure that, so to speak, contains all perspec- from the point of view of C was the product of the tives at once and via which one can switch among ground state and the excited state. Finally, fig. 10 the individual perspectives of the different classi- shows the Wigner functions of B and C when in cal or quantum reference systems. This extends C’s reference frame the total state was the prod- the method of [22–24], equips it with a novel uct of the two excited states. Compared to the operational interpretation thanks to [1] and em- states in fig.6, the states in the reference frame beds the approach of [1] in a perspective-neutral A appear more spread out, and the characteristic framework. Our construction offers a systematic quantumness (i.e., the negativity of the Wigner method for transforming quantum reference sys- function, an indicator of quantum behaviour) is tems, with possible applications in both quantum sharply reduced. This happens because in the foundations and gravity. new reference frame the total state of B and C Using this novel perspective-neutral frame- is entangled, as can easily be seen in Eq. (50), in work, we have been able to recover one of such a way that the marginals describe a mixed the transformations between quantum reference state. frames in one-dimensional space constructed, This concludes our quantum discussion of us- among other things, in [1] through a different ap- ing a perspective-neutral structure in order to proach. Finally, we have also studied some strik- switch from one particle reference frame in one- ing operational consequences of these quantum dimensional space to another. frame switches. Specifically, we have illustrated
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 23 0.10 0.10
0.05 0.05
0 0
-0.05 -0.05
Figure 10: In the final quantum reference frame A, the marginals of the total Wigner function representing the reduced state of system B (on the left) and C (on the right) when both A and B were initially in the first excited state. In both cases, αA = 1. αB
how entanglement and classicality of a system in- usually not coincide with the quantization of a teracting with two quantum reference frames de- reduced phase space, in contrast to the simple pends on whether the perspective of one or the model discussed here. However, this is not a prob- other is chosen. lem for our perspective-neutral approach. As we showed, in our new approach classi- We propose to always interpret the quantum cally choosing the perspective of a specific frame symmetry reduced theory as the description of amounts to a choice of gauge and perspective the quantum physics of the remaining degrees of changes require a gauge transformation within freedom relative to the associated quantum ref- the perspective-neutral constraint surface. In the erence frame. The reason is that Dirac quanti- quantum theory, on the other hand, it was the re- zation is more general than reduced quantization duced quantum theories which assumed the role in the sense that it also encodes quantum fluctu- of the quantum physics as seen from a partic- ations of the reference frame degrees of freedom. ular quantum reference frame, while the Dirac By contrast, there are no such quantum fluctua- quantized theory constitutes the perspective- tions of the reference frame in reduced quantiza- neutral quantum theory, without immediate op- tion since the reference frame degrees of freedom erational interpretation, via which quantum ref- have been removed altogether prior to quantiza- erence frame perspectives have to be switched. tion. Nonetheless, the classical symmetry reduc- In particular, for our model we have clarified tion procedure, such as in Sec. 3.2, leading to the quantum symmetry reduction procedure that reduced quantization is conceptually important maps Dirac to reduced quantization. because it clarifies that we can think of our new Our quantum symmetry reduction procedure quantum symmetry reduction procedure as being relative to a choice of quantum reference frame its proper quantum analog. This supports the in- can be adapted to more complicated models, as terpretation of the quantum symmetry reduced shown in the companion articles [2–4]. We sus- theory as the ‘perspective’ of an associated quan- pect that the general steps of the procedure – tum reference frame, which is why we have also 1. choose a quantum reference frame, 2. suit- discussed reduced quantization in this article. ably trivialize the constraint with respect to this We emphasize that in more general systems choice of quantum reference frame, and 3. project another property will feature: unlike in our toy onto the classical gauge-fixing conditions – may model, globally valid gauge-fixing conditions will be adaptable to an even much more general class be absent (globally valid means that every gauge of models. Of course, due to the well-known orbit is intersected once and only once by the inequivalence of Dirac and reduced quantization gauge-fixing surface). This has the consequence in more general situations [20, 21, 51–59], the re- that internal frame perspectives, which are asso- sult of applying the quantum symmetry reduc- ciated to a choice of gauge, will not be globally tion procedure to the Dirac quantized theory will valid, neither classically nor in the quantum the-
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 24 ory. More precisely, both the classical and quan- a perspective-neutral meta-structure, similar tum symmetry reduction procedures will not be to here, may open up a new approach to the defined on the entire perspective-neutral struc- problem. Of course, this would require the ture, i.e. classically the constraint surface and in inclusion of measurement interactions into the quantum theory the physical Hilbert space. the perspective-neutral structure that lead This too is not a fundamental problem, but to ‘collapses’ in the respective internal per- rather has to be expected from the general dis- spectives. cussion in Sec.2: globally valid internal perspec- Quantum general covariance and diffeomorphism tives are special. We have seen that changing symmetry. Classical general covariance and from one quantum reference frame perspective to diffeomorphism symmetry, while intimately another has a compositional structure analogous related, are not the same concept [8]. In- to coordinate changes on a manifold. The quan- deed, within the language of sec.2, general tum symmetry reduction maps assume the role covariance refers to the operational level of ‘quantum coordinate maps’ and in analogy to of frame perspectives onto the physics and classical coordinate maps they will generically not their relations (all the laws of physics are be globally defined. We propose to nevertheless the same in every reference frame). Diffeo- use such reduction maps to describe the physics morphism symmetry, on the other hand, from the internal perspective of a dynamical ref- refers to the perspective-neutral structure erence frame wherever defined. In [2] it will be (the diffeomorphism equivalence class of a shown through the relational N-body problem in spacetime) that contains and connects all 3D that our new approach indeed remains valid these different individual frame perspectives. also in more complicated systems where global internal perspectives will be absent. As long as Our approach suggests to extend this inter- one can locally (in a phase space sense) fix a play to the quantum case and we now see gauge, one can, in principle, construct local re- how the ‘quantum general covariance’, as duced quantum descriptions (see also [22–24]). advocated in [1], in principle fits, through Furthermore, in [3,4] it will be demonstrated the language of sec.2, into a bigger pic- how our method can be employed to switch tem- ture together with the diffeomorphism sym- poral reference systems, i.e. relational quantum metry in quantum gravity [8, 19, 53]. The clocks (such as in quantum gravity and cosmol- ‘quantum general covariance’ of [1], again, ogy) where subtleties due to the quadratic nature refers to the operational level of quan- of the constraints arise. Finally, in [5], it will be tum reference frames and their relations, established that our new method is indeed equiv- which, in our new approach, is encoded in alent to that developed in [22–24] when restricted the perspectives and their corresponding re- to a semiclassical regime within which the latter duced quantum theories. The diffeomor- was formulated. phism symmetry in canonical quantum grav- None of these systems include internal degrees ity [8,19,53], on the other hand, refers to the of freedom. In forthcoming work [82], relativis- Dirac quantized theory where one attempts tic particles with spin will be incorporated into to implement the Hamiltonian and diffeo- the original quantum reference frame approach morphism constraints, which constitute the of [1] and the operational consequences of quan- (first class) Dirac hypersurface deformation tum frame transformations will be explored in algebra that generates the diffeomorphism this setting. symmetry. The corresponding diffeomor- We conclude with an outlook on some prob- phism invariant physical Hilbert space, solv- lems where our approach may inspire new per- ing these constraints, in the language of spectives: sec.2, defines the perspective-neutral meta- structure. In line with our new approach, Wigner’s friend. A paradigmatic example for the and the simplicity of the present model challenges of fitting different perspectives notwithstanding, we propose to view this lat- in quantum theory into one picture is the ter perspective-neutral quantum gravity the- Wigner friend scenario on which much has ory as the structure containing and connect- been written (e.g., see [61,83–86]). Including ing all the different quantum reference sys-
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 25 tem perspectives that one refers to when one ceived funding from the European Union’s Hori- speaks about ‘quantum general covariance’ zon 2020 research and innovation programme un- as in [1]. This will be further elaborated on der the Marie Sklodowska-Curie grant agreement in [3,4], where it will also inspire a new per- No 657661 (awarded to PH). PH also acknowl- spective on the ‘wave function of the uni- edges support through a Vienna Center for Quan- verse’. tum Science and Technology Fellowship. F.G. and E.C.R. acknowledge support from the John Relational quantum mechanics and perspectives. Templeton Foundation, Project 60609, “Quantum In his seminal paper [61] on relational quan- Causal Structures”, from the research platform tum mechanics, Rovelli suggested “... to “Testing Quantum and Gravity Interface with investigate the extent to which the noticed Single Photons” (TURIS), and the Austrian Sci- consistency between different observers’ ence Fund (FWF) through the project I-2526- descriptions, which I believe characterizes N27 and the doctoral program “Complex Quan- quantum mechanics so marvellously, could tum Systems” (CoQuS) under Project W1210- be taken as the missing input for recon- N25. This publication was made possible through structing the full formalism." Whether or the support of a grant from the John Templeton not a consistency among different observers’ Foundation. The opinions expressed in this pub- descriptions can be used in a reconstruc- lication are those of the authors and do not nec- tion of quantum theory remains an open essarily reflect the views of the John Templeton question. In fact, meanwhile, the formal- Foundation. ism has been reconstructed without it, while still being compatible with relational quantum mechanics [60, 87] (see [88] for a summary).14 However, in line with our perspective-neutral approach of sec.2, this consistency among different observer per- spectives seems to be rather a characterizing feature of physics in general.15
Acknowledgments
We are grateful to Časlav Brukner for numer- ous discussions on the topic and to Lucien Hardy for conversations over the interplay of quantum reference frames and indefinite causal structures. PH is indebted to Tim Koslowski for intensive conversations over perspective changes in quan- tum theory in the past years and, in particu- lar, for suggesting the model of [2], which ulti- mately also inspired the simplified version of this manuscript. PH would also like to thank Sylvain Carrozza, Bianca Dittrich, Henrique (aka Hein- rich) Gomes, Markus Müller, Dennis Rätzel and Wolfgang Wieland for helpful discussions. The project leading to this publication has initially re-
14There exist further operational reconstructions of quantum theory, see e.g. [89–94], which however pursue a conceptually different route than that proposed in [61]. 15See also [95, 96] for a striking recent approach to de- riving an inter-observer consistency from subjective indi- vidual perspectives using algorithmic information theory.
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 26 A Lagrangian with translational invariance
For simplicity, we shall take the N particles to be of unit mass and the configuration manifold as Q = RN . The Lagrangian on the tangent bundle T Q' R2N reads !2 1 N 1 N L = X q˙2 − X q˙ − V {q − q }N . (57) 2 i 2N i i j i,j=1 i=1 i=1
| {zcm } Ekin We have subtracted the kinetic energy of the center of mass so that only the motion relative to the latter contributes to the energy. The potential is translation invariant. In consequence, this Lagrangian is singular and features a gauge symmetry: it is invariant under global translations
(qi, q˙i) 7→ (qi + f(t), q˙i + f˙(t)), (58) where f(t) is an arbitrary function of time that does not depend on particle i. In particular, the equations of motion are underdetermined and read
∂V 1 N − =q ¨ − X q¨ , (59) ∂q i N j i j=1 so that there are only N − 1 independent equations as their sum implies
N ∂V X = 0 , (60) ∂q i=1 i which is automatically satisfied for a translation invariant potential. The physical interpretation is clear: the localizations qi(t) and motions q˙i(t) of the N particles with respect to the Newtonian background space have no physical meaning, but are gauge dependent. Only the relative localization and motion of the particles is physically relevant, thereby providing a toy model for Mach’s principle. Thanks to the symmetry, physics is here relational. This becomes especially explicit in the canonical formulation on which we shall henceforth focus. ∗ 2N The Legendre transformation to the phase space T Q' R , in coordinates (qi, q˙i) 7→ (qi, pi), where
∂L 1 N p = =q ˙ − X q˙ , (61) i ∂q˙ i N j i j=1 fails to be surjective and evidently maps onto the (2N − 1)-dimensional (primary) constraint surface defined by (2), in line with the symmetry of the Lagrangian.
B Switching internal perspectives as a gauge transformation
The embedding map of the reduced phase space in A perspective into the constraint surface reads
ιBC|A : PBC|A ,→ C X (qi6=A, pi6=A) 7→ qi6=A, pi6=A, qA = 0, pA = − pi (62) i6=A and its image is precisely C ∩ GBC|A. Conversely, we can also define a projection
πBC|A : C ∩ GBC|A → PBC|A X qi6=A, pi6=A, qA = 0, pA = − pi 7→ (qi6=A, pi6=A) , (63) i6=A
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 27 that drops all redundant information so that πBC|A ◦ ιBC|A = IdPBC|A . Clearly, the same structures can be constructed for C perspective. Now what is the gauge transformation that takes us from C ∩ GBC|A to C ∩ GAB|C , where GAB|C is s defined by qC = 0? Denote the flow on the constraint surface generated by (2) by αP , where s is the flow parameter. The gauge transformation of a phase space function F corresponds to transporting s s the argument along the flow αP ·F (x) = F (αP (x)) with x a point on the constraint surface. Explicitly, it reads ∞ sk αs · F (x) = X {F,P } (x) , (64) P k! k k=0 where {F,P }k = {... {{F,P },P },...,P } is the k-nested Poisson bracket of F with P . Using (4), these gauge transformations are easy to evaluate for the canonical variables
s s αP · qi(x) = qi(x) + s , αP · pi(x) = pi(x) . (65)
Hence, jumping from the reference frame of A to the reference frame of, say, C corresponds to the gauge transformation −qC (x) αA→C := αP , (66) i.e. to flowing with ‘parameter distance’ s = −qC (x) (where qC (x) is the actual value of the relative distance of A and C prior to the transformation), as it verifies αA→C · qC (x) = 0 and αA→C · qA(x) = −qC (x). It is clear that altogether this defines a map, depicted in the diagram of sec. 3.3,
SA→C := πAB|C ◦ αA→C ◦ ιBC|A : PBC|A → PAB|C . (67)
Taking into account the swap of non-redundant Dirac observable from qB − qA to qB − qC (and the inverse switch of redundant Dirac observable) through the A, C label exchange, it reads in coordinates:
0 0 0 0 (qB, pB, qC , pC ) 7→ qA = −qC , pA = −pB − pC , qB = qB − qC , pB = pB . (68)
C Physical inner product for Dirac quantization
The improper projector (23) δ(Pˆ) defines equivalence classes of states in Hkin that are mapped to the same solution |φiphys. One can define an inner product between |ψiphys and |φiphys by using any member |ψikin, |φikin of their respective equivalence classes:
phys phys kin kin (ψ , φ )phys := hψ| δ(Pˆ) |φi , (69) where h·|·i is the original inner product of Hkin. Since δ(Pˆ) is symmetric in Hkin, this construction is independent on which representative is chosen from each equivalence class. Through Cauchy completion (and other technical subtleties which we shall here ignore), the space of solutions to (22) can thereby be turned in a proper Hilbert space Hphys. Using (25), the physical inner product in momentum representation takes either of the following equivalent forms: Z phys phys ∗ (ψ , φ )phys = dpB dpC [ψBC|A(pB, pC )] φBC|A(pB, pC ) Z ∗ = dpA dpC [ψAC|B(pA, pC )] φAC|B(pA, pC ) (70) Z ∗ = dpA dpB [ψAB|C (pA, pB)] φAB|C (pA, pB) , i.e., essentially just drops a redundant (and singular) momentum integration.
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 28 Next, we show that the ‘Page-Wootters like’ projection (35) is consistent with the inner products. phys ˆ phys More precisely, if the inner product on the transformed set HA,BC := TA,BC (H ) is defined, in analogy to (69), as
kin ˆ (ψA,BC |φA,BC )A,BC := hTA,BC ψ | φiA,BC kin ˆ † = hψ| TA,BC |φiA,BC (71) Z ∗ = dpB dpC [ψBC|A(pB, pC )] φBC|A(pB, pC ) ,
ˆ phys phys then TA,BC indeed defines an isometry from H to HA,BC . The last line also coincides with the ˆ inner product on the reduced Hilbert space HBC|A of sec. 4.1 and so TA,BC , followed by the projection phys (35), also defines an isometry from H to HBC|A. Given the transformations (36–37), it is also clear that
phys phys † (ψ , Oˆ ψ )phys ≡ (ψA,BC |TˆA,BC Oˆ (TˆA,BC ) | ψA,BC )A,BC ˆ = hψ|BC|A OBC|A |ψiBC|A , (72) ˆ ˆ where O is a relevant Dirac observable containing B and C information and OBC|A is the corresponding reduced observable on the reduced Hilbert space HBC|A. Hence, expectation values of relevant Dirac phys observables on H coincide with those of the correctly transformed observables in HBC|A.
D Mathematical non-uniqueness of constraint trivialization
The trivialization amounts to transforming the constraint such that it acts only on the reference frame degrees of freedom and the latter become completely fixed and redundant. In the present model, given the linear structure of the constraint Pˆ, this means fixing the momentum of the chosen reference frame, e.g. of A. Hence, the trivialization (30) is unique only up to the number to which we fix A’s momentum. For example, if instead we chose
ˆ 0 TA,BC = exp i qˆA(ˆpB +p ˆC + k) , (73) where k ∈ R, we would have
† ˆ 0 ˆ ˆ 0 TA,BC P TA,BC =p ˆA − k (74) and
|ψiA,BC = |p = kiA ⊗ |ψiBC|A . (75) Yet, also in this case, does one find
ˆ 0 ˆ ˆ 0 † ˆ TA,BC Htot (TA,BC ) |ψiA,BC = |p = kiA ⊗ HBC|A |ψiBC|A , (76) and, in fact, all of the relevant structures (71, 36, 38, 72) are actually independent of the choice of k. The non-uniqueness of the transformation thereby has no physical consequences and only affects the irrelevant information in the A-slot. Up to the irrelevant number k the trivialization is unique.
E Transformation between two quantum reference frames
Here we shall prove the claim of sec. 4.4. Writing an arbitrary state in HBC|A as in (21, 35), Z |ψiBC|A = dpB dpC ψBC|A(pB, pC ) |pBiB |pC iC , (77)
Accepted in Quantum 2019-12-26, click title to verify. Published under CC-BY 4.0. 29 one finds Z ˆ 0 SA→C |ψiBC|A = dpC dpB dpC ψBC|A(pB, pC )
0 C hpC | exp i qˆC (ˆpA +p ˆB) exp − i qˆA(ˆpB +p ˆC ) |p = 0iA |pBiB |pC iC Z = dpB dpC ψBC|A(pB, pC ) |−pB − pC iA |pBiB . (78)
Recalling from (25) that ψAB|C (pA, pB) = ψBC|A(pB, −pA − pB) (79) and using the change of variables pA = −pB − pC , we obtain from (78) Z ˆ SA→C |ψiBC|A = dpA dpB ψAB|C (pA, pB) |pAiA |pBiB
= |ψiAB|C . (80) This transformation is equivalent to
i qˆC pˆB SˆA→C = PˆCA e , (81) where PˆCA is the parity-swap operator defined in [1] on position eigenstates as ˆ PCA |xiC = |−xiA . (82)
Note the similarity to the action of the gauge transformation αA→C in AppendixB. Indeed, it can be checked that on momentum eigenstates this yields
|−pB − pC i |pBi = PˆCA |pB + pC i |pBi A B C B (83) ˆ i qˆC pˆB = PCA e |pC iC |pBiB , so that, upon using again (79) and the variable redefinition, Z ˆ i qˆC pˆB |ψiAB|C = PCA e dpB dpC ψBC|A(pB, pC ) |pC iC |pBiB (84) ˆ i qˆC pˆB = PCA e |ψiBC|A .
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