Abstract
Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.
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1 Introduction
1.1 The Classical Spacetime
An operational definition of the classical spacetime of special and general relativity is based on the use of freely moving/falling point-like (test) particles and light rays [1], and to measure the distance between these particles idealized standard (purely classical geometric [2, 3] or atomic quantum [4]) clocks, mirrors and light rays are used. Thus, in particular, the classical spacetime structure is defined by classical point mechanical and geometrical optical notions. (Classical field theoretical concepts, e.g. the energy-momentum tensor, are used only in the introduction of Einstein’s field equations. For a strict axiomatic treatise of the ideas of [1], see [5,6,7].)
Historically, however, the well known structure of the Galilei and Minkowski spacetime was read off from the structure of the Galilei–Newton mechanics and Maxwell’s electrodynamics, respectively. In particular, the Galilei and Poincaré symmetries were recognized first as the symmetries of the equations of motion in the theories describing these particular material physical systems. The notion of spacetime with these symmetries was introduced only later as a useful notion to specify the spatio-temporal location of phenomena happening with the concrete physical systems in a convenient and transparent way.
Although the classical Galilei or Minkowski spacetime is defined to be the set of the idealized (viz. structureless, i.e. pointlike and instantaneous) classical events (abstracted from the realistic events happening with the existing macroscopic physical object, e.g. from collisions of billiard balls), this notion turned out to be very useful even in quantum physics [8, 9]. In particular, many of the quantum field theoretical calculations became much more transparent when special relativistic spacetime concepts, e.g. Lorentz covariance, are used.
1.2 The Problem and a Possible Resolution
However, the great successes of quantum theory should not hide the absurdity that, e.g. in quantum mechanics, the independent variables in the argument of the wave function of an electron are defined e.g. by collisions of classical, macroscopic billiard balls; or, in quantum electrodynamics, the causal structure that the propagators are based on are defined by light rays in the geometrical optical approximation of classical electrodynamics. The resolution of this contradiction is that the a priori ‘spacetime’, used as a background in the quantum physical calculations, and the operationally defined ‘physical spacetime’ in which we arrange the events (e.g. in a scattering process) are not the same. The former is a geometrical representation of certain aspects of the internal structure of the quantum systems, viz. that it is the dual of the momentum space defined via the Fourier transformFootnote 1, but the latter is the set of the actual eventsFootnote 2 that happen in the Universe.
The theoretical significance of the clarification of the origin of the structures of the physical spacetime is twofold: first, this might yield a deeper understanding how the classical physical world that we see emerges from quantum theory; and, second, if gravity is the non-triviality of the geometry of the operationally defined physical spacetime (as we understand it according to general relativity), then these investigations may, ultimately, reveal how gravity, and, in particular, its universal nature, is rooted in the quantum world. Thus, in a conceptually coherent approach, the spacetime should be re-defined by purely quantum physical concepts, without using directly any classical physical structure, and, in particular, any a priori notion of spacetime.
As a first step to derive the classical spacetime geometry from purely quantum concepts, already in 1966, Penrose suggested the spin network model of spacetime [13]. Using combinatorial techniques, he showed that, in the classical limit, the geometry of directions in Euclidean 3-space (i.e. the conformal structure of \({\mathbb {R}}^3\)) can be recovered from this model [13,14,15,16]. Later, this result was derived using the more familiar formalism of quantum mechanics, and it became known as the Spin Geometry Theorem [17].
In our previous paper [18], we re-derived the above result in an even simpler and more direct way in the algebraic formulation of quantum mechanics, in which the system was, in fact, the formal union of a large number of SU(2)-invariant elementary quantum mechanical systems. Also, in an analogous way but extending the symmetry group from SU(2) to the quantum mechanical Euclidean group E(3) (and using geometric methods and complex techniques developed in general relativity to work out the classical limit), we recovered the metric (rather than only the conformal) structure of the Euclidean 3-space [19]. The next step in this program would be the derivation of the metric of the flat Minkowski space. The aim of the present paper is to show that the metric structure of the Minkowski space can indeed be recovered in this way by extending the symmetry group further into the quantum mechanical Poincaré group E(1, 3). (By the quantum mechanical Poincaré group E(1, 3) we mean the semidirect product of the group of translations and \(SL(2,{\mathbb {C}})\), i.e. the universal covering of the connected component of the classical Poincaré group, which is the isometry group of the Minkowski space.)
1.3 The Strategy
The main idea is that, instead of the classical test particles in the operational definition of the physical spacetime, we should use E(1, 3)-invariant elementary quantum mechanical systems, and the various geometrical structures of the spacetime should be introduced by the observables of (rather than measurements on) these quantum systems. Then, in the spacetime structure that emerges in this way, the intrinsic quantum nature of the elementary systems is manifested in a nontrivial way, and the various geometrical structures of the classical spacetime should emerge from these observables in the classical limit.
Since the general strategy of the present approach has only been sketched earlier [18, 19], here we summarize its key points in a more coherent and a bit more detailed way:
First, following [13,14,15,16], our approach is also positivistic as it does not depend on any a priori notion of spacetime. However, it is a bit ‘more Machian’ than that of [13,14,15,16] in the sense that the quantum systems are not intended to be used to define any notion like ‘direction’ or ‘position’. We define only relative orientations by some sort of ‘empirical angles’ and relative positions by ‘empirical distances’ between elementary subsystems of a large composite system, even though the ‘directions’ and ‘positions’ themselves are not defined at all. The ‘empirical geometry’ of the physical spacetime should ultimately be synthesized from these ‘empirical geometrical quantities’. (For the idea how to introduce the geometry from empirically given distances between pairs of points, see e.g. [3].)
Second, since at the fundamental level no instruments (e.g. mirrors or clocks) exist, the notion of physical spacetime cannot be based on concrete experimental/measuring processes. Rather, it should be based on the use of observables of the quantum systems. The various geometrical quantities are built exclusively from the basic quantum observables in the analogous way how the corresponding classical geometrical quantities are built from the basic classical observables. (Hence, this aspect of the present approach is perhaps a bit ‘Platonist’ rather than positivistic, and not ‘instrumentalist’ at all.)
Third, according to the abstract, algebraic formulation of quantum theory, a quantum system is thought to be specified completely if the algebra of its observables and its representation are given. In particular, elementary quantum mechanical systems with symmetry group G are specified by the (\(C^*\)-closure of the) universal enveloping algebra \({{\mathcal {A}}}\) of the Lie algebra of G in which the observables are the self-adjoint elements, and the states belong to the carrier space of a unitary, irreducible representation of G on some complex, separable Hilbert space \({{\mathcal {H}}}\). This representation yields a representation of \({{\mathcal {A}}}\), too. The self-adjoint generators of the action of G will be called the basic quantum observables. (For the Poincaré group, this notion of elementary quantum mechanical systems was introduced by Newton and Wigner [20].) It is this symmetry, attributed to the elementary quantum physical systems, that can be expected to emerge in some way as the symmetry of the resulting physical spacetime; just like the Galilei and Poincaré symmetries of the Galilei and Minkowski spacetimes that came, respectively, from the Galilei–Newton mechanics and Maxwell’s electrodynamics (see the second paragraph above).
Moreover, since general relativity is the counterpart of the quantum theory as a general framework rather than the theory of any particular interaction (as it is manifested in the universality of free fall via the Galilei–Eötvös experiments), the resulting spacetime structure should also share this universality. Therefore, the quantum observables that the construction is based on must depend only on the kinematical framework of the quantum systems, but they may not depend e.g. on any particular Hamiltonian. It is these elementary systems that replace the classical point particles in the operational definition of the classical physical spacetime.
Finally, to recover the classical spacetime structure from the above model an appropriate notion of classical limit is needed. Although there are attempts to represent the classical limit of a quantum system by special but otherwise completely regular states of the quantum system (e.g. by considering the classical theory to be the \(\hbar \rightarrow 0\) limit of the quantum theory [21,22,23], or by some version of the coherent states, see e.g. [24, 25] and the references therein to a sample from the extended literature), here, following Wigner [26], Ch. 27, we adopt a much weaker notion of classical limit: we do not require the states of the classical system to be among the states of the quantum system. This limit is defined through a (not necessarily convergent) sequence of quantum states, and only the expectation value of the basic quantum observables that is required to tend to their large classical value, formally to infinite, in such a way that the corresponding standard deviations tend to zero relative to the expectation values. Here, the classical theory is not expected to be a limit of the quantum theory in any sense, and hence the states of the classical theory need not to be states of the quantum system either.
In the above notion of elementary G-invariant quantum mechanical systems not only the algebra \({{\mathcal {A}}}\) of observables, but the unitary, irreducible representation, labeled by the value of the Casimir invariants, is also fixed. The definition of the distance operator and the empirical distance will be based on this notion in subsections 4.1 and 4.2, respectively. However, in the definition of the classical limit we should consider a sequence of states whose elements belong to different irreducible representations. Thus, we should also consider the more general G-invariant quantum mechanical systems with the same algebra \({{\mathcal {A}}}\) of observables, but the representation is the direct sum/integral of all its unitary irreducible representations. Strictly speaking, it is this more general notion of E(1, 3)-invariant quantum mechanical systems that we should use in our Theorem in subsection 4.3.
1.4 The Key Ingredients and the Main Result
One of the key notions of the present paper, viz. the empirical distance, is based on the observation that the intrinsic spin part of the angular momentum of a composite system is the sum of the spin of the constituent subsystems and their relative orbital angular momenta; and, in the traditional formulation of quantum mechanics, the latter is an expression of the distance between the centre-of-mass (world)lines of the subsystems and their momenta. This distance could in fact be recovered as the classical limit of the expectation value of expressions of the abstract basic quantum observables of E(3)-invariant elementary quantum mechanical systems [19]; and, as we will see, this can indeed be done in the present case, too. (The idea that the distance could be inferred somehow from the angular momentum has already been raised by Penrose in [13].)
The other key ingredient is the appropriate mathematical form of the idea of the classical limit of Poincaré-invariant quantum systems. This is introduced in two steps, generalizing Wigner’s notion of the classical limit of SU(2)-invariant quantum systems first to E(3)-invariant ones (as we did it in [19]), and then, in the present paper, to E(1, 3)-invariant systems with positive rest mass. These notions are based on another notions, viz. on the canonical angular momentum states in the SU(2)-invariant case, on special centre-of-mass states in the E(3)-invariant case, and on special co-moving, centre-of-mass states in the present case. The latter are the states in which the expectation value of the centre-of-mass vector operator is vanishing, the expectation value of the energy-momentum vector operator has only the time component, and the Pauli–Lubanski spin has only a single spatial component.
In the present paper, we found an explicit expression for the empirical distance in terms of the 4-momenta and angular momenta, and also the mathematical form of the states defining the classical limit of Poincaré-invariant quantum systems for which our main result, formulated mathematically in the Theorem in subsection 4.3, is proven: we show that for any finite number of timelike straight lines in Minkowski space we can always find E(1, 3)-invariant quantum mechanical systems such that the empirical distance between them reduces in the classical limit just to the classical Lorentzian distance between them. Therefore, the metric structure of the Minkowski space can be recovered from the abstract E(1, 3)-invariant quantum mechanical systems. Also, this result of the present investigations confirms that it is the world lines of the freely moving particles (rather than the points) that should be considered to be the elementary objects in spacetime, just in accordance with the basic idea of [1]. The actual notion of the classical limit of E(1, 3)-invariant quantum systems as well as their co-moving, centre-of-mass states may have significance in other problems in relativistic quantum theory, independently of the present context.
Nevertheless, although the strategy of the present investigations is the same and simple as that in the E(3)-invariant case, technically the calculations in the present E(1, 3)-invariant case is considerably longer and more complicated; but they are mostly only routine ones.
1.5 The Structure of the Paper
In the next section, we summarize the key properties of the Poincaré-invariant classical mechanical systems, express the distance between the world lines of any two of them (with positive rest mass) by basic classical observables, and show that the knowledge of this distance between any two timelike straight lines is equivalent to the knowledge of the metric structure of the Minkowski space. In Sect. 3, the E(1, 3)-invariant elementary quantum mechanical systems are defined, and their co-moving, centre-of-mass states as well as the mathematical notion of their classical limit are introduced. Section 4 is devoted to composite systems consisting of two (and, in principle, any finite number of) E(1, 3)-invariant elementary quantum mechanical systems. The empirical distance between the elementary subsystems is introduced and its classical limit is calculated here. The key result of the paper is summarized in a theorem also in this section. The main part of the paper concludes with some final remarks in Sect. 5.
Since the representation theory of the Poincaré group, at least in the form that we use in the present paper, is not very well known in the relativity and differential geometry communities; and also the geometric and complex techniques and methods developed in general relativity are almost completely unknown outside the relativity community, for the sake of completeness we summarize the key elements of these ideas and methods in the appendices.
The signature of the Lorentzian metric is chosen to be \((+,-,-,-)\). Small Latin indices, say \(a,b,c,...,h=0,...,3\) and \(i,j,k,...=1,2,3\), are concrete tensor indices, referring to a fixed orthonormal basis in the momentum space; and the capital Latin indices, e.g. \(A,B,C,...=0,1\), are concrete spinor name indices with respect to a fixed spin frame associated with this orthonormal vector basis. We do not use abstract indices. Round/square brackets around indices denote symmetrization/anti-symmetrization. We use the units in which \(c=1\), but Planck’s constant \(\hbar \) is kept. Our standard references to the spinorial and complex techniques are [27, 28], but a more concise summary of them is given in the appendices of [24, 25].
2 Poincaré Invariant Elementary Classical Mechanical Systems
2.1 The Definition of the Elementary Systems
A classical mechanical system will be called a Poincaré invariant elementary system (or, for brevity, a single particle) if its states can be characterized completely by the energy-momentum and angular momentum, \(p^a\) and \(J^{ab}\), respectively; and under a Lorentz transformation they transform as a Lorentzian 4-vector and anti-symmetric tensor, respectively, while under a translation with \(\xi ^a\in {\mathbb {R}}^4\) they change as \((p^a,J^{ab})\mapsto ({\tilde{p}}^a,{\tilde{J}}^{ab}):=(p^a,J^{ab}+2 \xi ^{[a}p^{b]})\). \(p^a\) and \(J^{ab}\) are elements of the dual space of the commutative ideal and the so(1, 3) Lie sub-algebra of the Poincaré algebra e(1, 3), respectively. The commutative ideal is endowed with the Lorentzian metric \(\eta _{ab}:={\textrm{diag}}(1,-1,-1,-1)\). This metric identifies the space of translation generators with the momentum space, which becomes a Minkowski vector space \({\mathbb {R}}^{1,3}=({\mathbb {R}}^4, \eta _{ab})\); and it determines a metric on the whole tensor algebra over it. We lower and raise freely the small Latin indices a, b, c, ... by \(\eta _{ab}\) and its inverse, \(\eta ^{ab}\), respectively. Also, this yields the Lie brackets on the space of the basic observables,
which are just those of the Poincaré Lie algebra e(1, 3). In addition to the conditions on the basic observables above, we assume that \(p^a\) is non-zero, non-spacelike and future pointing (i.e. its timelike component, \(p^0\), is strictly positive) with respect to \(\eta _{ab}\) and a fixed time orientation.
The rest mass, the Pauli–Lubanski spin and the centre-of-mass vectors are defined, respectively, according to
Here \(\varepsilon _{abcd}\) is the natural volume 4-form on \({\mathbb {R}}^{1,3}\) determined by \(\eta _{ab}\). \(\mu ^2\) and \(S_aS^a\) are invariant with respect to Poincaré transformations; which invariance follows from (2.1)-(2.2), too: \(\mu ^2\) and \(S_aS^a\) are Casimir invariants. From the definitions it follows that \(S_ap^a=0\) and \(M_ap^a=0\), and that the identity
holds. Since \(p^a\) is non-spacelike, \(\mu ^2\ge 0\) holds; and since \(S_ap^a =0\), \(S^a\) is spacelike or null: \(S_aS^a\le 0\). Since \(S_a\) and \(p^a\) are invariant with respect to translations while \(M_a\mapsto {\tilde{M}}_a=M_a+ (\mu ^2\eta _{ab}-p_ap_b)\xi ^b\), for \(\mu >0\) this identity is interpreted as the decomposition of the total angular momentum into the sum of its spin and orbital parts. (Strictly speaking, if \(\mu >0\), then the dimensionally correct Pauli–Lubanski spin and centre-of-mass vectors are \(1/\mu \)-times and \(1/\mu ^2\)-times the expressions above, respectively.)
If \(\mu =0\), then (2.4) does not provide a decomposition of \(J^{ab}\) into its spin and orbital parts. In this case, contracting (2.4) with \(S^a\), we find that \(S_aM^a=0\) holds. If \(M^a\) is null, then by \(M_ap^a=0\) it follows that \(M_a=kp_a\) for some \(k\in {\mathbb {R}}\), which by (2.4) implies \(S_a=\chi p_a\) for some \(\chi \in {\mathbb {R}}\). In a similar way, if \(S^a\) is null, then by \(S_ap^a=0\) it follows that \(S_a= \chi p_a\), and hence, also by (2.4), \(M_a=kp_a\) follows. Therefore, either both \(M_a\) and \(S_a\) are spacelike and orthogonal to each other or both are proportional to the null 4-momentum \(p_a\). In the latter case, the factor of proportionality \(\chi \) turns out to be a Casimir invariant and it might be called the ‘classical helicity’ of the system. k is invariant with respect to Lorentz transformations, but, under a translation, it changes according to \(k\mapsto k-p_a\xi ^a\).
In [29], Penrose and MacCallum clarified under what conditions can we find a translation \(\xi ^a\) resulting \({\tilde{M}}_a=0\). The summary of their results are as follows. If \(\mu >0\), then \(S_a\) and \(M_a\) are spacelike or zero, and \(0={\tilde{M}}_a=M_a+(\mu ^2\eta _{ab}-p_ap_b)\xi ^b\) can always be solved for \(\xi ^a\). The solutions form a 1-parameter family: \(\xi ^a=-M^a/\mu ^2+up^a/\mu \), \(u\in {\mathbb {R}}\). Hence, as it is interpreted in Minkowski space in [29], an elementary Poincaré-invariant system with \(\mu >0\) can always be ‘localized’ to a timelike straight line, viz. to \(\gamma ^a(u):=M^a/\mu ^2+up^a/\mu \). The timelike straight line \(\gamma ^a\), associated with the Poincaré-invariant elementary classical system in this way, will be called the centre-of-mass line of the system.
If \(\mu =0\), then \({\tilde{M}}_a=0\) can be solved for \(\xi ^a\) precisely when \(M_a=kp_a\), which, as we noted above, is equivalent to \(S_a=\chi p_a\). Thus, in particular, if \(\mu =0\) and the Pauli–Lubanski spin vector is spacelike, then no translation \(\xi ^a\) yielding \({\tilde{M}}_a =0\) exists.
If \(\mu =0\), \(S_a=\chi p_a\) and \(M_a=kp_a\), then the translations \(\xi ^a\) that yield \({\tilde{M}}_a=0\) satisfy \(p_a\xi ^a=k\), and hence these translations form a null hyperplane with \(p^a\) as its null normal, rather than only a single null straight line with tangent \(p^a\). This set of translations can be reduced further in a natural way to form only a 1-parameter family if \(\chi =0\) also holds. Therefore, an elementary system with \(\mu =0\) and \(S _a=\chi p_a\) with \(\chi \not =0\) can be ‘localized’ only to a null hyperplane \({{\mathcal {N}}}\), and it can be ‘localized’ further in a natural way to a null straight line \(\gamma ^a\) precisely when \(\chi =0\). This result is analogous to the result that a relativistic quantum mechanical particle with zero rest mass can be localized (with respect to the Newton–Wigner position operator [20]) only when its helicity is vanishing [30,31,32].
Thus, to summarize, in the zero rest mass case (2.4) does not provide a decomposition of the total angular momentum into its spin and orbital parts, and such elementary systems would have well defined centre-of-mass lines only if their helicity were zero. But since in the present paper we intend to recover the metric structure of the Minkowski space from the orbital part of the angular momentum by expressing the distance between the centre-of-mass straight lines in terms of the spins (as in the E(3)-invariant case in [19]), in (2.4) we must assume that the rest mass \(\mu \) is strictly positive. The elementary systems with zero rest mass do not seem to provide an appropriate framework for the present project. Thus, in the rest of the paper, we concentrate only on the \(\mu >0\) case.
The (future) mass shell \({{\mathcal {M}}}^+_\mu :=\{p^a\in {\mathbb {R}}^{1,3}\vert \,\, p^0>0,\, p^ap^b\eta _{ab}=\mu ^2>0\}\) is a spacelike hypersurface in the momentum space \({\mathbb {R}}^{1,3}\). (We discuss the geometry of \({{\mathcal {M}}} ^+_\mu \) further in Appendix A.1.) Its cotangent bundle, \(T^* {{\mathcal {M}}}^+_\mu \), is homeomorphic to the manifold of the future directed timelike straight lines \(\gamma ^a\): the unit tangent of \(\gamma ^a\) is \(p^a/\mu \), while a point on \(\gamma ^a\) can be chosen to be the intersection point of \(\gamma ^a\) and the spacelike hyperplane through the origin of \({\mathbb {R}}^{1,3}\) with normal proportional to \(p^a\). This point of intersection is given by the spatial vector \(M^a/\mu ^2\).
Clearly, if \(\gamma ^a\) is any timelike straight line, then this can always be obtained from the special one \(\gamma ^a_0(u):=u\,\delta ^a_0\), \(u\in {\mathbb {R}}\), by an appropriate Lorentz boost \(\Lambda ^a{}_b\) and a translation \(\xi ^a\): \(\gamma ^a(u)=\Lambda ^a{}_b\gamma ^b_0(u)+\xi ^a\). The ambiguity in \(\xi ^a\) is that we can add to it any term proportional with \(p^a\). Thus, the Cartesian frame in \({\mathbb {R}}^{1,3}\) is a special co-moving, centre-of-mass frame for the system whose centre-of-mass world line is \(\gamma ^a_0\); and whose centre-of-mass vector is zero, its energy-momentum has only time component, and the Pauli–Lubanski spin vector points e.g. in the z-direction.
2.2 Classical Two-Particle Systems and the Empirical Distance
In this subsection, we show that the Lorentzian distance between any two non-parallel timelike straight lines, considered to be the centre-of-mass lines of Poincaré-invariant elementary classical mechanical systems with positive rest masses, can be expressed in terms of Poincaré-invariant observables of the two-particle system and its constituent elementary subsystems.
Let \((p^a_{\textbf{i}},J^{ab}_{\textbf{i}})\), \({\textbf{i}}=1,2\), characterize two Poincaré-invariant elementary classical mechanical systems, and let us form their formal union characterized by \(p^a:=p^a_1+p^a_2\) and \(J^{ab}:=J^{ab}_1+ J^{ab}_2\). Then the rest mass, the Pauli–Lubanski spin and the centre-of-mass vectors of the composite system are defined in terms of \(p^a\) and \(J^{ab}\) according to the general rules. As a consequence of the definitions, \(S^a_{12}:=S^a-S^a_1-S^a_2=\frac{1}{2}\varepsilon ^a{}_{bcd}(J^{bc}_1p^d_2+J^{bc}_2p^d_1)\); and, in a similar way, \(P^2_{12}:=\frac{1}{2}(\eta _{ab}p^ap^b-\mu ^2_1-\mu ^2_2)= \eta _{ab}p^a_1p^b_2\). Since both \(p^a_1\) and \(p^a_2\) are future pointing and timelike, \(P^2_{12}\ge \mu _1\mu _2>0\) holds, in which the equality holds precisely when \(p^a_1\) and \(p^a_2\) are parallel with each other. As we will see, \(S^a_{12}\) and \(P^2_{12}\) play fundamental role in the subsequent analyses because they characterize the relationship between the two elementary subsystems in the composite system.
Since both \(p^a_1\) and \(p^a_2\) are timelike, the definitions and equation (2.4) yield
If the two 4-momenta are not parallel, then \(P^4_{12}>\mu ^2_1\mu ^2_2\), and the last term on the right is not identically zero. Then (2.5) can be solved for \(M^a_1/\mu ^2_1-M^a_2/\mu ^2_2\):
where \(u_1,u_2\in {\mathbb {R}}\) are arbitrary. Hence, although its component in the timelike 2-plane spanned by \(p^a_1\) and \(p^a_2\) is ambiguous, its component in the orthogonal spacelike 2-plane, viz.
is well defined. Here
is the projection to the spacelike 2-plane orthogonal to \(p^a_1\) and \(p^a_2\).
\(d^a_{12}\) is a translation invariant Lorentzian spacelike 4-vector. It points from a well defined point \(\nu _{21}\) of the centre-of-mass line of the second system to a point \(\nu _{12}\) of the the centre-of-mass line of the first system, and it is orthogonal to these two straight lines. Thus, this is the relative position vector of the first system with respect to the second at a particular instant. Its length, defined by \(d_{12}:=\sqrt{-\eta _{ab}d^a_{12}d^b_{12}}\) and whose physical dimension is, indeed, cm, is just the spatial Lorentzian distance between the points \(\nu _{21}\) and \(\nu _{12}\) in the spatial 2-plane orthogonal to the timelike 2-plane spanned by \(p^a_1\) and \(p^a_2\). Note that it is given exclusively by Poincaré invariant observables of the composite system and its constituent subsystems; and it is analogous to the empirical distance between the Euclidean-invariant elementary classical mechanical systems [19]. Thus, we call it the empirical distance between the two straight lines. \(d^a_{12}\) is zero precisely when the two centre-of-mass lines intersect each other; and it is invariant with respect to the rescalings \((p^a_1,J^{ab}_1)\mapsto (\alpha p ^a_1,\alpha J^{ab}_1)\) and \((p^a_2,J^{ab}_2)\mapsto (\beta p^a_2,\beta J^{ab} _2)\) for any \(\alpha ,\beta >0\).
Let us represent the timelike straight lines as \(\gamma ^a_{\textbf{i}}(u)=\Lambda ^a_{\textbf{i}}{}_b\gamma ^b_0(u)+\xi ^a_{\textbf{i}}\), \({\textbf{i}}=1,2\), i.e. by a pair \((\Lambda ^a_{\textbf{i}}{}_b,\xi ^a_{\textbf{i}})\) of Poincaré transformations (in fact, by a pair of Lorentz boosts and translations) and the straight line \(\gamma ^a_0\). Then with the choice \(\gamma ^a_0(u):=u\,\delta ^a_0\) (as at the end of the previous subsection) the 4-momenta are given by \(p^a_1=\mu _1\Lambda ^a_1{}_0\) and \(p^a_2=\mu _2\Lambda ^a_2{}_0\), and hence the explicit form of the (non-negative) square of the spatial Lorentzian distance between \(\gamma ^a _1\) and \(\gamma ^a_2\) is
This is invariant with respect to the rotations about \(\gamma ^a_0\) in the Lorentz transformations \(\Lambda ^a_{\textbf{i}}{}_b\), it is free of the ambiguities \(\xi ^a_{\textbf{i}}\mapsto \xi ^a_{\textbf{i}}+up^a_{\textbf{i}}\) in the translations, and it is independent of the rest masses \(\mu _1\) and \(\mu _2\). The expression \((d_{12})^2=-\eta _{ab}d^a_{12}d^b_{12}\) above is an alternative form of (2.9), given by classical observables of a Poincaré-invariant composite system and its elementary subsystems. As we will see in subsection 4.3, \((D_{12})^2\) can be recovered in the classical limit of the square of the empirical distance between E(1, 3)-invariant elementary quantum mechanical systems, where the latter is built from the basic quantum observables analogously to how the alternative expression \((d_{12})^2\) is built form the classical observables.
If the two 4-momenta, \(p^a_1\) and \(p^a_2\), are parallel, then the above strategy to recover the distance between the corresponding centre-of-mass lines does not work. Nevertheless, following an argumentation analogous to that in [19], one can show that this distance can be recovered as the limit of distances between elementary systems with non-parallel 4-momenta; but the points analogous to \(\nu _{21}\) and \(\nu _{12}\) above are not well defined. Therefore, it is enough to consider timelike straight lines with non-parallel tangents.
2.3 The Minkowski Metric from Empirical Distances
In Minkowski space the metric determines the Lorentzian distance between any two (e.g. non-parallel) timelike straight lines, and now we show that the converse is also true: if the distance between any two non-parallel timelike straight lines is known, then the Lorentzian distance function on the Minkowski space is completely determined.
Let the points \(x^a_1,x^a_2\in {\mathbb {R}}^{1,3}\) be spacelike separated. Then there is a uniquely determined timelike 3-plane through \(x^a_1\), and another timelike 3-plane through \(x^a_2\) which are orthogonal to the direction of \(x^a_1-x^a_2\). Then let us choose any timelike straight line \(\gamma ^a_1\) through \(x^a_1\) lying in the timelike 3-plane above; and, in a similar way, let \(\gamma ^a_2\) be any timelike straight line through \(x^a_2\) in the corresponding timelike 3-plane through \(x^a_2\). Clearly, \(\gamma ^a_1\) and \(\gamma ^a_2\) can be chosen to be non-parallel. Then the Lorentzian distance between the points \(x^a_1\) and \(x^a_2\) is just the empirical distance between \(\gamma ^a_1\) and \(\gamma ^a_2\); i.e. in this case the Lorentzian distance is realized directly by the empirical distance.
Next, let us suppose that the points \(x^a_1\) and \(x^a_2\) are timelike separated and \(x^a_1\) is in the chronological future of \(x^a_2\). These two points determine uniquely a timelike straight line \(\gamma ^a\) through them, and the point with the coordinates \(x^a:=(x^a_1+x^a_2)/2\) is on this straight line. Let \(T^2:=\eta _{ab}(x^a_1-x^a_2)(x^b_1-x^b_2)\), the square of the Lorentzian distance between \(x^a_1\) and \(x^a_2\). Then let \(\Sigma \) be the spacelike hyperplane through \(x^a\) whose timelike normal is proportional to \(x^a_1-x^a_2\), i.e. that \(\Sigma \) is the ‘instantaneous spacelike 3-space’ with respect to \(\gamma ^a\) through \(x^a\), and let \({\tilde{x}}^a\) be any point in \(\Sigma \) whose spatial distance from \(x^a\) in \(\Sigma \) is T/2. Finally, let \({\tilde{\gamma }}^a\) be any timelike straight line through \({\tilde{x}}^a\) which is orthogonal to the spacelike straight line through \(x^a\) and \({\tilde{x}}^a\). Then the empirical distance between the timelike straight lines \(\gamma ^a\) and \({\tilde{\gamma }}^a\) is just T/2. Hence, there are timelike straight lines such that the empirical distance between them reproduces the Lorentzian distance between any two timelike separated points.
To summarize, considering the timelike straight lines of the Minkowski space \({\mathbb {R}}^{1,3}\) to be centre-of-mass lines of elementary, Poincaré-invariant classical mechanical systems with positive rest mass, the Lorentzian distance between these straight lines can be expressed in terms of Poincaré-invariant classical observables, or at least the limit of such observables. Since the Lorentzian distance between any two points of the Minkowski space can be written as the Lorentzian distance between appropriate timelike straight lines, we conclude that the metric structure of the Minkowski space can be determined by Poincaré-invariant classical observables of Poincaré-invariant elementary systems. It is this characterization of the Minkowski metric by observables of elementary classical physical systems that makes it possible to derive the Minkowski metric from observables of quantum systems, too, in the classical limit.
3 Poincaré Invariant Elementary Quantum Mechanical Systems
3.1 The Definition of the Elementary Systems
A Poincaré-invariant elementary quantum mechanical system is defined to be a system whose states belong to the representation space of a unitary, irreducible representation of the quantum mechanical Poincaré group E(1, 3); and, in this representation, the momentum and angular momentum tensor operators, \({\textbf{p}}^a\) and \({\textbf{J}} ^{ab}\), are the self-adjoint generators of the translations and boost-rotations (see [20]). In what follows, the letters in boldface will denote quantum mechanical operators. Such a representation is characterized by a fixed value \(\mu ^2\ge 0\) and w, respectively, of the two Casimir operators
Here \({\textbf{S}}_a:=\frac{1}{2}\varepsilon _{abcd}{\textbf{J}}^{bc}{\textbf{p}}^d\) is the Pauli–Lubanski spin operator. The commutators of \({\textbf{p}}_a\) and \({\textbf{J}} _{ab}\) can be obtained formally from (2.1)-(2.2) with the \(p_a\mapsto {\textbf{p}}_a\), \(J_{ab}\mapsto -({\textrm{i}}/\hbar ){\textbf{J}}_{ab}\) substitution:
Moreover, under the action of the translations with \(\xi ^a\), these transform as \({\textbf{p}}^a\mapsto \tilde{\textbf{p}}^a={\textbf{p}}^a\) and \({\textbf{J}}^{ab}\mapsto \tilde{\textbf{J}}^{ab}={\textbf{J}}^{ab}+2\xi ^{[a}{\textbf{p}}^{b]}\), while, under the action of \(SL (2,{\mathbb {C}})\), they transform as Lorentzian vector and anti-symmetric tensor operators. For the sake of brevity, we call such a system a single particle, though, as Newton and Wigner stress [20], such a system is not necessarily ‘elementary’ in the usual sense that it does not have any internal structure: it may have such a structure, but, in the given physical context, it is considered to be irrelevant.
All the operators above are formally self-adjoint. However, \({\textbf{M}}_a:= {\textbf{J}}_{ab}{\textbf{p}}^b\) is not self-adjoint, because \({\textbf{M}}_a^\dagger ={\textbf{p}}^b{\textbf{J}}_{ab}={\textbf{M}}_a+[{\textbf{p}}^b,{\textbf{J}}_{ab}]={\textbf{M}}_a-3{\textrm{i}} \hbar {\textbf{p}}_a\). Thus we form \({\textbf{C}}_a:=\frac{1}{2}({\textbf{M}}_a+{\textbf{M}} ^\dagger _a)={\textbf{M}}_a-\frac{3}{2}{\textrm{i}}\hbar {\textbf{p}}_a\), which is formally self-adjoint by definition, and we consider this to be the centre-of-mass operator. The commutators of these operators can be derived from those for \({\textbf{p}}_a\) and \({\textbf{J}}_{ab}\) above. In particular, in any unitary, irreducible representation labeled by \(\mu \) they are
where \({\textbf{I}}\) is the identity operator. As a consequence of the definitions, the analog of (2.4) holds for the operators too:
Under translations by \(\xi ^a\) the operators \({\textbf{p}}^a\) and \({\textbf{S}}_a\) do not change, but \({\textbf{C}}_a\mapsto \tilde{\textbf{C}}_a={\textbf{C}}_a+\xi ^b(\mu ^2\eta _{ab} -{\textbf{p}}_a{\textbf{p}}_b)\).
Several different forms of unitary, irreducible representations of E(1, 3) are known. Our approach is based on the use of square integrable completely symmetric spinor fields with index 2s, say \(\phi ^{A_1...A_{2s}}\), on the future mass-shell \({{\mathcal {M}}}^+_\mu \) with rest mass \(\mu \) in the classical momentum space. Here, s and \(\mu \) are determined by the two Casimir operators. The key points of this representation are sketched in [8, 9], and we summarize them in Appendix A.2: in this representation the operators \({\textbf{p}}^a\) are multiplication, while \({\textbf{J}} _{ab}\) are differential operators acting on the spinor fields, and they are given explicitly by equations (A.18) and (A.19), respectively.
However, to prove our main result, it will be enough to consider only states \(\phi \), or in the bra-ket notation \(\vert \phi \rangle \), given by spinor fields \(\phi ^{A_1...A_{2s}}\) that are obtained by E(1, 3)-transformations from special states \(\vert \psi \rangle \), specified by special spinor fields \(\psi ^{A_1...A_{2s}}\), which are some sort of centre-of-mass states. It is well known (see e.g. [27, 28]) that, in general, any totally symmetric spinor field with 2s indices, say \(\psi ^{A_1...A_{2s}}\), can be completely specified by \(2s+1\) complex scalars, \(\psi _r\), \(r=0,1,...,2s\). The centre-of-mass states, as we will see in the next subsection, are those that can be specified by two such scalars, viz. \(\psi _{2\,s}\) and \(\psi _0\), which, in addition, are holomorphic and anti-holomorphic, respectively, with respect to a certain complex structure. Thus, in the calculation of the empirical distance, we use the form of the quantum mechanical operators that fits naturally to this representation of the quantum states. This Newman–Penrose (NP) form of the operators is determined in Appendix A.3. The technical, geometric background to the Appendices A.2 and A.3 is summarized in Appendix A.1.
3.2 The Co-moving, Centre-of-Mass States
According to subsection 2.1, every state of a Poincaré-invariant elementary classical mechanical system with positive rest mass can be obtained by an appropriate Poincaré transformation from the special state in which \(M^a=0\), \(p^a=\mu \delta ^a_0\) and \(S^a=S\delta ^a_3\), where \(\mu >0\) and \(S\ge 0\) are fixed by the two Casimir invariants. (Thus, this state may be called a ‘special co-moving, centre-of-mass state’.)
In the case of E(3)-invariant elementary quantum mechanical systems, the centre-of-mass states were introduced as the states that minimized the expectation value of the square of the centre-of-mass vector operator, and the expectation value of the Euclidean centre-of-mass vector operator itself turned, in fact, out to be zero in all these states [19]. These states are given by spinor fields of the form \(\psi _{A_1...A_{2\,s}} =({}_{-s} Y_{s,m}/p)o_{A_1}\cdots o_{A_{2\,s}}\) or \((-)^{2\,s}({}_sY_{s,m}/p)\) \(\iota _{A_1}\cdots \iota _{A_{2s}}\) on the 2-sphere \({{\mathcal {S}}}_p\) of radius p (the analog of the ‘mass shell’), where p is the magnitude of the linear momentum, \({}_{\pm s} Y_{s,m}\), \(m=-s,-s+1,...,s\), are special spin weighted spherical harmonics on \({{\mathcal {S}}}_p\), and \(\{o^A,\iota ^A\}\) is a Newman–Penrose normalized spinor basis adapted to \({{\mathcal {S}}}_p\). The value of p and s is fixed by the two Casimir operators of E(3). (For the notations and the technical background, see Appendix A.1 and A.2.)
Unfortunately, in the actual E(1, 3)-invariant case, the mass shell is non-compact and the metric \(\eta _{ab}\) by means of which the square of \({\textbf{C}}^a\) is defined is indefinite, yielding that the analysis done in [19] cannot be successfully repeated in the present case. Nevertheless, in this subsection, we specify certain special states, \(\vert \psi _{s,m}\rangle \), motivated by the centre-of-mass states of the E(3)-invariant systems above, and we justify a posteriori that these are, in fact, analogous to the special co-moving, centre-of-mass states of the E(1, 3)-invariant elementary classical mechanical systems, viz. that \(\langle \psi _{s,m}\vert {\textbf{C}}^a\vert \psi _{s,m}\rangle =0\), \(\langle \psi _{s,m}\vert {\textbf{p}}^a\vert \psi _{s,m}\rangle \sim \delta ^a_0\) and \(\langle \psi _{s,m}\vert {\textbf{S}}^a\vert \psi _{s,m}\rangle \sim \delta ^a_3\) hold in all of these states. The factor of proportionality in the last two expressions will also be fixed here, but the justification of that choice will be given only in the next subsection.
Since \(E(3)\subset E(1,3)\), the structure of the centre-of-mass states in the E(3)-invariant case suggests to consider the special states \(\vert \psi ^\pm _{s,m}\rangle \) given, respectively, by the completely symmetric spinor fields
on \({{\mathcal {M}}}^+_\mu \). Here f is a real valued function of p satisfying the normalization condition \(1=\langle \psi ^\pm _{s,m}\vert \psi ^\pm _{s,m}\rangle :=\langle \psi ^\pm _{A_1...A_{2\,s}}\vert \psi ^\pm _{A_1...A_{2\,s}}\rangle = \int ^\infty _0f^2(p)^2(p^0)^{-1}{\textrm{d}}p\) (see equation (A.2)). For \(s>0\), \(\psi ^+_{A_1...A_{2\,s}}\) and \(\psi ^-_{A_1...A_{2s}}\) are orthogonal to each other with respect to the \(L_2\) scalar product (A.17), but they coincide for \(s=0\). The normalization condition requires the fall-off \(f=o(1/p)\), i.e. that f should fall-off slightly faster than 1/p for large p. However, the various basic quantum observables have well defined action on these states only if f has some faster fall-off (see Appendix A.2). Thus, temporarily, we require the faster fall-off \(f=o(p^{-3/2})\), but the ultimate fall-off that we choose for the function f at the end of this subsection will be the much faster exponential decay.
However, the states \(\vert \psi ^\pm _{s,m}\rangle \) in themselves cannot be expected to be co-moving, centre-of-mass states of E(1, 3)-invariant elementary systems. In fact, while in the E(1, 3)-invariant case what the Pauli–Lubanski spin operator determines is the non-negative s, in the E(3)-invariant case the analogous ‘spin’ operator (which is, in fact, a helicity operator) gives the same s but with either sign ±. Thus the sign of the ‘spin’ in the centre-of-mass states of the E(3)-invariant systems is also fixed by the Casimir invariants, and hence these states have some definite chirality. Such a chirality is not provided by the Casimir invariants of the E(1, 3)-invariant systems. Therefore, the co-moving centre-of-mass states of these systems can be expected to be only those combinations of the states \(\vert \psi ^\pm _{s,m}\rangle \) that are symmetric in the two chiralities. We will see that this expectation is correct and is justified by the explicit calculations of the expectation values of the basic quantum observables.
First, let us compute the expectation value of the centre-of-mass operator. Recalling that by (A.9) the only non-zero component \(\psi _r\) of the spinor fields of the form \(\psi ^+_{A_1...A_{2s}}\) is \(\psi _{2s}\), by (A.15) equation (A.30) reduces to
(To avoid potential confusion, we inserted a comma between the two indices of the Kronecker delta.) Using the explicit form of the components of \(v^a\) and \(m^a\) given in Appendix A.1, equations (A.5), (A.10), (A.17) and the fact that the spherical harmonics have unit \(L_2\)-norm on the unit sphere, from (3.8) we obtain
In the last step we used that, for large p, \(f=o(p^{-3/2})\). Also, for \(a =i=1,2,3\), (3.8) gives
where, according to Appendix A.4, \(\langle .\,\vert .\,\rangle _1\) denotes the \(L_2\) scalar product on the unit 2-sphere. Here, to get the first expression, we formed a total 
-derivative and used the explicit form of \(v^i\) and equation (A.10) given in Appendix A.1; then we used 
and, finally, we used equations (A.34), (A.37) and (A.40), we formed a total derivative and used the fall-off \(f=o(p^{-3/2})\). In a similar way, one can show that
Therefore, \(\vert \psi ^\pm _{s,m}\rangle \) and \((\vert \psi ^+_{s,m}\rangle +\vert \psi ^-_{s,m}\rangle )/\sqrt{2}\) could be centre-of-mass states.
Using (A.24), similar calculations yield that
Hence, \(\vert \psi ^\pm _{s,m}\rangle \) are not co-moving states, i.e. \(\langle \psi ^\pm _{s,m}\vert {\textbf{p}}^a\vert \psi ^\pm _{s,m}\rangle \) are not proportional to \(\delta ^a_0\), but their sum, \((\vert \psi ^+_{s,m}\rangle + \vert \psi ^-_{s,m}\rangle )/\sqrt{2}\), are. In an analogous way, using (A.25), we obtain
Hence, the expectation value 4-vectors \(\langle \psi ^\pm _{s,m}\vert {\textbf{S}}^a \vert \psi ^\pm _{s,m}\rangle \) are not proportional to \(\delta ^a_3\), but \(\langle \psi ^+_{s,m}+\psi ^-_{s,m}\vert {\textbf{S}}^a\vert \psi ^+_{s,m}+\psi ^-_{s,m}\rangle \) are.
Therefore, to summarize the moral of equations (3.9)-(3.18), the states \(\vert \psi _{s,m}\rangle \), given by the spinor fields
on \({{\mathcal {M}}}^+_\mu \) for \(2\,s=0,1,2,...\), \(m=-s,-s+1,...,s\) and for some real functions f of p with the \(f=o(p^{-3/2})\) fall-off and normalization condition \(\int ^\infty _0f^2(p)^2(p^0)^{-1}{\textrm{d}}p=1\), are special states of E(1, 3)-invariant elementary quantum mechanical systems with positive rest mass \(\mu \) and spin s in which the expectation value of \({\textbf{C}}^a\), \({\textbf{p}}^a\) and \({\textbf{S}}^a\) is zero, proportional to \(\delta ^a_0\) and to \(\delta ^a_3\), respectively. In particular, \(\langle \psi _{s,m}\vert {\textbf{p}}^a \vert \psi _{s,m}\rangle \eta _{ab}\langle \psi _{s,m}\vert {\textbf{S}}^b\vert \psi _{s,m} \rangle =0\) holds. Note that \(\langle \psi _{s,m}\vert \psi _{s,m}\rangle =1+\delta _{s,0}\), i.e. these states are normalized only for \(s>0\).
Later, we will need to know the action of \({\textbf{J}}^{ab}\) on these special states, as well as their expectation values. In particular, for the action of \({\textbf{J}}^{ab}\) on \(\vert \psi ^+_{s,m}\rangle \) (A.31) gives
and (A.31) gives a similar expression for \(({\textbf{J}}^{ab}\psi ^-_{s,m}) _r\) too. Then, using 
(A.47), (A.48) and the techniques of the calculations above we find that
and hence that
Therefore, while the expectation value of \({\textbf{p}}^a\) and \({\textbf{S}}^a\) in the special co-moving, centre-of-mass states still depends on the specific choice for the function f, the expectation value of \({\textbf{J}}^{ab}\) does not. To have a definite value of \(\langle \psi _{s,m}\vert {\textbf{p}}^a\vert \psi _{s,m}\rangle \) and \(\langle \psi _{s,m}\vert {\textbf{S}}^a\vert \psi _{s,m}\rangle \) too, we should specify f.
Our choice for f is the square root of the Gaussian distribution function on the hyperboloidal \({{\mathcal {M}}}^+_\mu \), which is concentrated on the point \(p^a=(\mu ,0,0,0)\). Namely, with the formal substitution \(x^i:=p^i/\mu \) and for any \(\epsilon >0\), the normalization condition for the standard three-dimensional Gaussian distribution function \(g_\epsilon =g_\epsilon (x^i)\) on \({\mathbb {R}}^3\) can be written as
where we used (A.2). Comparing this equation with the normalization condition for f, we find that
could be a possible choice. Clearly, this satisfies even the strongest fall-off condition. In what follows, we use this \(f_\epsilon \) in \(\vert \psi _{s,m}\rangle \). We justify this choice in the next subsection, though many other choices could be equally good. Thus, in contrast to the centre-of-mass states of the E(3)-invariant systems, the present co-moving, centre-of-mass states are not canonically determined by the physical system itself: they depend on our choice for f, too.
With this particular choice for f, the expectation values of \({\textbf{p}}^a\) and \({\textbf{S}}^a\) are
Since
for this integral we have the estimates
and hence we find that, in the \(\epsilon \rightarrow 0\) limit, this integral tends to \(\mu \). Here we used the known expression
for the definite integrals, where the Gamma function is known to be \(\Gamma (k+1/2)=(2k)!\sqrt{\pi }/(4^kk!)\) and \(\Gamma (k+1)=k!\) for \(k= 0,1,2,...\). Therefore, for \(s>1/2\), the expectation values (3.25) and (3.26) tend in the \(\epsilon \rightarrow 0\) limit to \(\mu \delta ^a_0\) and \(\hbar \mu ms(s+1)^{-1}\delta ^a_3\), respectively,.
The sequence of states by means of which the classical limit of E(1, 3)-invariant systems is defined in the next subsection will be based on \(\vert \psi _{s,m}\rangle \), given by (3.19) with (3.24), but in which we should link \(\epsilon \) to s.
3.3 The Definition of the Classical Limit
Intuitively, the classical limit of a quantum system with basic observables \({\textbf{O}}_\alpha \), \(\alpha =1,2,...\), can be expected to be defined by a sequence of normalized states, \(\vert \phi _k\rangle \), \(k\in {\mathbb {N}}\), in which the expectation values \(\langle \phi _k\vert {\textbf{O}}_\alpha \vert \phi _k \rangle \) tend in the \(k\rightarrow \infty \) limit to their large classical value, formally to infinity, such that the rate of growth of the standard deviations \(\Delta _{\phi _k}{\textbf{O}}_\alpha \), given by \((\Delta _\phi {\textbf{O}}_\alpha )^2=\langle \phi \vert {\textbf{O}}_\alpha ^2\vert \phi \rangle -(\langle \phi \vert {\textbf{O}} _\alpha \vert \phi \rangle )^2\), is definitely smaller than the typical rate of growth of the expectation values. Thus, in this limit, the standard deviations should tend to zero relative to the typical expectation values.
In this subsection, we construct such a sequence of states explicitly for E(1, 3)-invariant systems, which states are based on the co-moving, centre-of-mass states introduced in the previous subsection.
The notion of classical limit of SU(2)-invariant quantum mechanical systems was introduced by Wigner [26] in the canonical angular momentum basis. This notion was extended in a natural way in [19] to E(3)-invariant quantum mechanical systems. This limit was defined by a sequence of states belonging to unitary irreducible representations of E(3) such that, in each of these states the quantum number \(\vert m \vert \) took its maximal value, and the two Casimir invariants, the magnitude of the linear momentum and of the spin (more precisely, of the helicity), viz. p and s, respectively, tended to infinity in the same order, say with s. In this limit, the expectation value of the basic observables \({\textbf{p}}^i\) and \({\textbf{J}}^i\) also grow typically with s, but the standard deviations grow only with \(\sqrt{s}\), which is slower than that of the typical expectation values.
Since \(E(3)\subset E(1,3)\), the classical limit of E(1, 3)-invariant quantum mechanical systems should be compatible with this notion. Thus, we will say that a sequence of special co-moving, centre-of-mass states defines a classical limit if \(\vert m\vert =s\rightarrow \infty \) and \(\mu \rightarrow \infty \) such that, in this limit, \(\mu =Ms+O(1/s)\) for some positive constant M. Hence, the Casimir invariants s and \(\mu \) are required to tend to their large classical value in the same orderFootnote 3. Then by (3.23), (3.25) and (3.27) the expectation value of the basic observables \({\textbf{p}}^a\) and \({\textbf{J}}^{ab}\) of the E(1, 3)-invariant system also diverge, typically with s.
To see whether or not this limit is physically well established, we should check that the rate of growth of the standard deviation of the basic observables in this limit is less than that of the expectation values. In this subsection, we show that this is the case provided the parameter \(\epsilon \) of the Gaussian distribution function \(f^2_\epsilon \) tends to zero in an appropriate way as \(s\rightarrow \infty \). Thus the parameter \(\epsilon \) should be linked to s.
A straightforward calculation yields that the square of the standard deviation of the various components of \({\textbf{p}}^a\) in the special co-moving, centre-of-mass states \(\vert \psi _{s,m}\rangle \) is given by
where we used (3.25), (3.28), (3.29), (A.32), (A.42) and (A.43). Thus, although in the \(\vert m\vert =s\rightarrow \infty \) limit \(\Delta _{\psi _{s,m}}{\textbf{p}}^1\) and \(\Delta _{\psi _{s,m}}{\textbf{p}}^2\) diverge like \(\sqrt{s}\) even with finite \(\epsilon \), the rate of growth of \(\Delta _{\psi _{s,m}}{\textbf{p}}^3\) is s unless \(\epsilon \) tends to zero, say as \(\epsilon =1/s^\alpha \) for some \(\alpha >0\). Under this condition, by (3.30), \(\Delta _{\psi _{s,m}} {\textbf{p}}^0\) grows also not faster than \(s^{1-\alpha }\).
Using the same techniques, in particular \(\langle \psi _{s,m}\vert ({\textbf{J}} ^{ab})^2\vert \psi _{s,m}\rangle =\langle {\textbf{J}}^{ab}\psi _{s,m}\vert {\textbf{J}}^{ab} \psi _{s,m}\rangle \) and equations (3.19), (3.20) and (A.10), \(\langle \psi _{s,m}\vert ({\textbf{J}}^{ab})^2\vert \psi _{s,m} \rangle \) can be calculated in a straightforward way. Since by (A.10) the terms of the form \(\langle \psi ^-_{s,m}\vert ({\textbf{J}} ^{ab})^2\vert \psi ^+_{s,m}\rangle \) are non-zero only for \(s\le 1\), and since basically we are interested only in the \(s\rightarrow \infty \) limit, we need to calculate only \(\langle \psi ^\pm _{s,m}\vert ({\textbf{J}}^{ab})^2\vert \psi ^\pm _{s,m} \rangle \).
In particular, since by (A.3) and (A.4) \(m^i{\bar{m}} ^j-m^j{\bar{m}}^i={\textrm{i}}\delta ^{ik}\delta ^{jl}\varepsilon _{klr}p^r/p\) and \(p^im^j- p^jm^i={\textrm{i}}\delta ^{ik}\delta ^{jl}\varepsilon _{klr}pm^r\) hold, using 
equations (A.15), (A.16) and the normalization condition for f, by integration by parts we obtain
Here \(\varepsilon _{ijk}\) is the three dimensional anti-symmetric Levi-Civita symbol. Using the explicit form (A.44), (A.45), (A.47) and (A.48) of the various matrix elements and equation (3.23), finally we obtain that, for \(s>1\),
These expressions are precisely those obtained in the case of SU(2) and E(3)-invariant systems, these are independent of the specific form of the function f, and they have the correct and expected \(\sqrt{s}\) growth for the standard deviation of \({\textbf{J}}^{ij}\) in the \(\vert m\vert =s\rightarrow \infty \) limit.
In a similar way, by (3.23), (3.24), (3.29) and (A.47), for \(s>1\) we obtain
Then by (A.42) and (A.43) this yields
and
Thus, for finite \(\epsilon \) the variances \((\Delta _{\psi _{s,m}}{\textbf{J}}^{0i}) ^2\) grow like s in the \(s\rightarrow \infty \) limit even if \(m\not =s\). However, as we concluded from (3.30) and (3.32), we had to assume that \(\epsilon \rightarrow 0\) when \(s\rightarrow \infty \). Hence, with the assumption \(\epsilon =s^{-\alpha }\) for some \(\alpha >0\), by (3.35) and (3.36) even in the \(m=s\) case, \((\Delta _{\psi _{s,m}}{\textbf{J}}^{01})^2\) and \((\Delta _{\psi _{s,m}}{\textbf{J}}^{02})^2\) grow like \(s^{1+2\alpha }\) and \((\Delta _{\psi _{s,m}} {\textbf{J}}^{03})^2\) grows like \(s^{2\alpha }\) as \(s\rightarrow \infty \). Comparing these rates with those obtained for \((\Delta _{\psi _{s,m}}{\textbf{p}}^a)^2\) above, the optimal choice for \(\epsilon \) seems to be \(\epsilon =1/\root 4 \of {s}\), yielding \(s\sqrt{s}\) or \(\sqrt{s}\) divergences for the variances. Therefore, the sequence \(\vert \psi _{s,s}\rangle \) of states for \(2s=3,4,...\) with \(\epsilon =1/\root 4 \of {s}\) can be used to define the classical limit of E(1, 3)-invariant quantum systems.
Clearly, the sequence of states of the form \({\textbf{U}}_{(A,\xi )}\vert \psi _{s,s} \rangle \) defines the same kind of classical limit, where \({\textbf{U}}_{(A,\xi )}\) is the unitary operator representing the Poincaré transformation \((A^A{}_B, \xi ^a)\in E(1,3)\). The general, large classical values \(p^a\) and \(J^{ab}\) of \({\textbf{p}}^a\) and \({\textbf{J}}^{ab}\), respectively, can be recovered by sequences of states of this form. The states that we use in the proof of our main result will have this structure.
4 Two-Particle Systems
4.1 The Relative Position and Distance Operators
On the tensor product space \({{\mathcal {H}}}_1\otimes {{\mathcal {H}}}_2\) of two irreducible representation spaces, labeled, respectively, by \((\mu _1,s_1)\) and \((\mu _2, s_2)\), equation (3.6) yields
Let the 4-momentum and Pauli–Lubanski spin operators of the composite system be denoted by \({\textbf{p}}^a\) and \({\textbf{S}}^a\), respectively, and let us introduce the notations
Here \({\textbf{I}}_{\textbf{i}}\), \({\textbf{i}}=1,2\), denote the identity operators on the respective Hilbert spaces. The operators \({\textbf{P}}^2_{12}\) and \({\textbf{S}}^a_{12}\) characterize the relationship between the two subsystems in the composite system. For later use, it seems useful to calculate the commutator of these with some other operators. By the second of (3.4), it follows that \([{\textbf{S}}^a_1\otimes {\textbf{I}}_2,{\textbf{P}}^2_{12}]=[{\textbf{I}}_1\otimes {\textbf{S}} ^a_2,{\textbf{P}}^2_{12}]=0\); and, by the second of (3.2), it follows that \([{\textbf{S}}^a_{12},{\textbf{p}}^b_1\otimes {\textbf{I}}_2]=-[{\textbf{S}}^a_{12},{\textbf{I}}_1 \otimes {\textbf{p}}^b_2]=-{\textrm{i}}\hbar \varepsilon ^{ab}{}_{cd}{\textbf{p}}^c_1\otimes {\textbf{p}}^d_2\). The latter implies that \([{\textbf{S}}^a_{12},{\textbf{P}}^2_{12}]=0\).
Using (4.1)-(4.2) and the commutators (3.4)-(3.5), from (4.4) we obtain
In the second step, we used that in the actual representation \({\textbf{p}}^a_1\), \({\textbf{p}}^a_2\) and \({\textbf{P}}^2_{12}\) are multiplication operators (see Appendix A.2). Multiplying it (from the left) by \(\varepsilon ^e{}_{cda}p^c_1 p^d_2\) and recalling that \((\varepsilon ^a{}_{cde}p^c_1p^d_2)(\varepsilon ^e{} _{fgb}p^f_1p^g_2)=-(P^4_{12}-\mu ^2_1\mu ^2_2)\Pi ^a_b\), where \(\Pi ^a_b\) is the projection to the spacelike 2-plane orthogonal to \(p^a_1\) and \(p^a_2\) (see equation (2.9)), we obtain
Since \({\textbf{S}}^e_{12}\) does not commute with \(\varepsilon ^a{}_{bcd}p^b_1p^c_2\), the operator \({\textbf{d}}^a_{12}\) cannot be expected to be self-adjoint. In fact,
Here we used that \({\textbf{S}}^e_1\) and \({\textbf{S}}^e_2\) commute with both \(\varepsilon ^a{}_{bcd}p^b_1p^c_2\) and \(1/(P^4_{12}-\mu ^2_1\mu ^2_2)\), and \({\textbf{S}}^e_{12}\) commutes with \(1/(P^4_{12}-\mu ^2_1\mu ^2_2)\), which can be verified directly using e.g. expression (A.21) of the centre-of-mass operator. Thus the self-adjoint part of \({\textbf{d}}^a_{12}\) can be written into the form
\({\textbf{D}}^a_{12}\) can be interpreted as the ‘relative position operator’ of the first elementary system with respect to the second. However, the ‘correction term’ in (4.7) that makes \({\textbf{d}}^a_{12}\) self-adjoint is rather trivial: since \(p_{1a}{\textbf{D}}^a_{12}=-p_{2a}{\textbf{D}}^a_{12}={\textrm{i}} \hbar {\textbf{I}}_1\otimes {\textbf{I}_2}\), its only non-zero component is proportional to \(p^a_1-p^a_2\), which is a universal expression: \((p_{1a}-p_{a2}){\textbf{D}}^a _{12}=2{\textrm{i}}\hbar {\textbf{I}}_1\otimes {\textbf{I}}_2\). This \({\textbf{D}}^a_{12}\) is the quantum mechanical analog of (2.7).
Nevertheless, despite the non-self-adjointness of \({\textbf{d}}^a_{12}\), its square, \(\eta _{ab}{\textbf{d}}^a_{12}{\textbf{d}}^b_{12}\), is self-adjoint. To see this, in \((\eta _{ab}{\textbf{d}}^a_{12}{\textbf{d}}^b_{12})^\dagger =\eta _{ab} ({\textbf{d}}^a_{12})^\dagger ({\textbf{d}}^b_{12})^\dagger \) it is enough to use (4.5)–(4.6) and the commutators above to obtain that it is, in fact, \(\eta _{ab}{\textbf{d}}^a_{12}{\textbf{d}}^b_{12}\). Its manifestly self-adjoint form is given explicitly by
Here, to derive the expression in the second line from the first expression in (4.5) we used \([{\textbf{C}}^a_1\otimes {\textbf{I}}_2, \Pi ^b_a]\Pi _{bc}=[{\textbf{I}}_1\otimes {\textbf{C}}^a_2,\Pi ^b_a]\Pi _{bc}=0\); while to get the third line we used (2.9) and that \({\textbf{S}}^a_{12}\) commutes with \(1/(P^4_{12}-\mu ^2_1\mu ^2_2)\). Thus, the operator for the square of the distance between the two constituent, elementary subsystems in the composite system can be defined by either \(\eta _{ab}{\textbf{d}}^a _{12}{\textbf{d}}^b_{12}\) or \(\eta _{ab}{\textbf{D}}^a_{12}{\textbf{D}}^b_{12}\); and they differ only in a term of order \(\hbar ^2\). (4.8) is a well defined operator, given by the centre-of-mass and the 4-momentum operators, or by \({\textbf{S}}^a_{12}\), \({\textbf{P}}^2_{12}\) and the 4-momentum and spin operators of the constituent systems in an E(1, 3)-invariant manner.
However, in contrast to the classical expression (2.9) (see also (2.8)), \(P^4_{12}\) in the denominator is not constant: this is \(P^4_{12}=(\eta _{ab}p^a_1p^b_2)^2\), an expression of the variables \(p^a_1\) and \(p^a_2\) of the wave functions representing the states of the subsystems. Hence, it could be difficult to evaluate the expectation value or to calculate the standard deviation of (4.8). Nevertheless, the role of the factor \(1/(P^4_{12}-\mu ^2_1\mu ^2_2)\) in (4.8) is only the normalization of the middle factor \(\varepsilon _{acde}p^c_1p^d_2\, \varepsilon ^e{}_{ghb}p^g_1p^h_2\) to be the projection \(\Pi _{ab}\). Hence, its role is analogous to the magnitude of the angular momentum 3-vectors in the definition of the empirical angle \(\delta _{ab}J^a_1J^b_2/\vert J_1\vert \vert J_2\vert \) in SU(2)-invariant systems [18], and to the similar normalization factor in the definition of the empirical distance in E(3)-invariant systems [19]. Thus, although \(\eta _{ab}{\textbf{d}} ^a_{12}{\textbf{d}}^b_{12}\) might yield a well defined operator for the distance between the subsystems, and certainly it would be worth studying this, in the present paper we follow the strategy of [18] for the SU(2) and of [19] for the E(3)-invariant systems, and we calculate only the ‘empirical distance’ based on (4.8) and introduced in the next subsection. As we will see, this approximation of the expectation value of the operator (4.8) is enough to prove our key result in subsection 4.3.
4.2 The Empirical Distance
Let \({{\mathcal {S}}}_1\) and \({{\mathcal {S}}}_2\) be two E(1, 3)-invariant elementary quantum mechanical systems, characterized by the Casimir invariants \((\mu _1, s_1)\) and \((\mu _2,s_2)\), respectively. Then, motivated by (4.8) and introducing the notation
their empirical distance in their state \(\phi _1\) and \(\phi _2\), respectively, is defined to be the square root of
Note that this is not the expectation value of the operator (4.8), it is the quotient of the expectation value of that’s numerator and of that’s denominator. Since, however, the denominator \(1/(P^4_{12}-\mu ^2_1\mu ^2_2)\) in (4.8) is a multiplication rather than a differential operator, it is translation invariant. Hence, \(d^2_{12}\) above can be expected not to deviate from the expectation value of (4.8) in the leading order in the classical limit. Since \(d^2_{12}\) is not the expectation value of some operator, its standard deviation is not defined. Nevertheless, as we will see in subsection 4.3.2, its uncertainty can be defined.
Clearly, just as in the SU(2) and E(3)-invariant cases, this empirical geometrical quantity can be defined between any two constituent elementary subsystems of any composite system consisting of any finite number of subsystems, say \({{\mathcal {S}}}_1,...,{{\mathcal {S}}}_N\), in any of its state (not only in pure tensor product states of a bipartite system). In fact, in this more general case the Hilbert space of the pure states of the composite system is the tensor product \({{\mathcal {H}}}:={{\mathcal {H}}}_1\otimes \cdots \otimes {{\mathcal {H}}}_N\) of the Hilbert spaces of the constituent elementary subsystems, a general state is a density operator \(\rho :{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}\), and the square of the empirical distance between the \({\textbf{i}}\)’th and the \({\textbf{j}}\)’th subsystems, \({\textbf{i}},{\textbf{j}}=1,...,N\), in this state is defined by
where \(\Sigma _{\textbf{i}\textbf{j}}\) and \({\textbf{P}}^4_{\textbf{i}\textbf{j}}\) are the obvious generalizations of \(\Sigma \) and \({\textbf{P}}^4_{12}\), respectively. If, however, \(\rho \) represents the tensor product \(\phi :=\phi _1\otimes \cdots \otimes \phi _N\) of pure states of the elementary subsystems, then (4.11) reduces to an expression of the form (4.10) and the result depends only on the states \(\phi _{\textbf{i}}\) and \(\phi _{\textbf{j}}\), and will be independent of the states of the other subsystems. Since the states in the proof of our main result will be chosen to be such tensor product states, it is enough to carry out the subsequent analysis for a bipartite system.
Unfortunately, the calculation of the empirical distance is considerably more complicated than in the E(3)-invariant case: while in the latter the relative position vector is proportional to the uniquely determined direction orthogonal to the two linear momenta \(p^i_1\) and \(p^i_2\), \(i= 1,2,3\), and hence the (signed) distance between the two subsystems is already given by the component of the relative position vector orthogonal to \(p^i_1\) and \(p^i_2\), in the present E(1, 3)-invariant case the relative position vector has two components orthogonal to \(p^a_1\) and \(p^a_2\). Thus, to get the distance between the subsystems in the E(3)-invariant case it was enough to consider the relative position vector itself, in the E(1, 3)-invariant case we must calculate the square of the relative position vector. This yields much more terms to evaluate in the calculation of the expectation values and variances.
Since our aim is to recover the metric structure of the classical Minkowski space, we calculate these expectation values directly in the classical limit without evaluating them in general quantum states. This will be done in the next subsection.
4.3 The Classical Limit
The main result of the paper is the following theorem:
Theorem
Let \(\gamma ^a_1,\cdots ,\gamma ^a_N\) be timelike straight lines in the Minkowski space \({\mathbb {R}}^{1,3}\) such that no two of them are parallel. Then there are E(1, 3)-invariant quantum mechanical systems \({{\mathcal {S}}}_1,...,{{\mathcal {S}}}_N\) and a sequence of their pure quantum states \(\phi _{1k},...,\phi _{Nk}\), \(k\in {\mathbb {N}}\), respectively, such that, in the \(k\rightarrow \infty \) limit, the empirical distances \(d_{\textbf{i}\textbf{j}}\), calculated in the states \(\phi _{1k}\otimes \cdots \otimes \phi _{Nk}\), tend with asymptotically vanishing uncertainty to the classical Lorentzian distances \(D_{\textbf{i}\textbf{j}}\) between the straight lines \(\gamma ^a_{\textbf{i}}\) and \(\gamma ^a_{\textbf{j}}\), given by (2.9), for any \({\textbf{i},\textbf{j}}=1,...,N\).
We split the proof of this theorem into two parts: the statement on the classical limit of the empirical distance will be proven in subsection 4.3.1, while that on the uncertainty in subsection 4.3.2.
4.3.1 The Classical Limit of the Empirical Distance
As we noted in connection with the general form (4.11) of the empirical distance, if the state \(\rho \) is the pure tensor product state \(\phi _{1k}\otimes \cdots \otimes \phi _{Nk}\) as in the present case, then \(d_{\textbf{i}\textbf{j}}\) depends only on the states \(\phi _{\textbf{i}}\) and \(\phi _{\textbf{j}}\) of \({{\mathcal {S}}}_{\textbf{i}}\) and \({{\mathcal {S}}}_{\textbf{j}}\), respectively, and independent of the other factors in this tensor product. Hence, it is enough to prove the Theorem for \(N=2\), i.e. when \({\textbf{i}}=1,2\). Then the operator in the numerator of (4.10) can be written as
while that in the denominator is
We calculate the leading order terms in the expectation value of these operators in the states \(\phi _1\otimes \phi _2\) specified below.
Let us represent the timelike straight lines \(\gamma ^a_{\textbf{i}}\) by the Lorentz boosts \(\Lambda ^a_{\textbf{i}}{}_b\), the translations \(\xi ^a_{\textbf{i}}\) and a single timelike straight line \(\gamma ^a_0\) just as we did it at the end of subsection 2.1: \(\gamma ^a_{\textbf{i}}(u)=\Lambda ^a_{\textbf{i}}{}_b\gamma ^b_0(u)+\xi ^a_{\textbf{i}}\); and let us consider the states \(\vert \phi _{\textbf{i}}\rangle =\exp ({\textrm{i}}p_e\xi ^e_{\textbf{i}}/\hbar ){\textbf{U}}_{\textbf{i}}\vert \psi _{s_{\textbf{i}},s_{\textbf{i}}}\rangle \) of the E(1, 3)-invariant elementary quantum mechanical system \({{\mathcal {S}}} _{\textbf{i}}\). Here \({\textbf{U}}_{\textbf{i}}={\textbf{U}}_{(A_{\textbf{i}},0)}\), the unitary operator representing one of the two \(SL(2,{\mathbb {C}})\) matrices \(\pm A^A_{\textbf{i}}{}_B\) corresponding to the Lorentz boost \(\Lambda ^a_{\textbf{i}}{}_b\). Thus, the states \(\vert \phi _{\textbf{i}}\rangle \) depend on \(s_{\textbf{i}}\). However, to reduce the number of indices, we do not write out s on the states, and we simply write e.g. \(\vert \psi _{\textbf{i}}\rangle =\vert \psi _{s_{\textbf{i}},s_{\textbf{i}}}\rangle \).
Next we evaluate the factors in the tensor products in the operators \({\textbf{A}}\) and \({\textbf{B}}\). Using (A.20) and (A.21), for the first term in (4.12) we obtain that
Therefore, we should determine the expectation values \(\langle \psi _{s,m} \vert {\textbf{p}}^a{\textbf{p}}^b\vert \psi _{s,m}\rangle \), \(\langle \psi _{s,m}\vert {\textbf{p}}^a{\textbf{p}}^b{\textbf{p}}^e{\textbf{J}}^c{}_e\vert \psi _{s,m}\rangle \), \(\langle \psi _{s,m}\vert {\textbf{J}}^a{}_e{\textbf{p}}^e{\textbf{p}}^b{\textbf{p}}^c\vert \psi _{s,m} \rangle \) and \(\langle \psi _{s,m}\vert {\textbf{J}}^a{}_e{\textbf{p}}^e{\textbf{p}}^b{\textbf{p}}^c {\textbf{p}}^f{\textbf{J}}^d{}_f\vert \psi _{s,m}\rangle \), at least for large \(m=s\).
By (3.24), (3.25), (3.29), (A.42) and (A.43) it is easy to calculate \(\langle \psi _{s,m}\vert {\textbf{p}}^a{\textbf{p}}^b\vert \psi _{s,m}\rangle \): for \(s>0\) it is
This yields that, for \(s_1,s_2>0\), the first term on the right hand side of (4.14) is
Note that, also for \(s_1,s_2>0\), (4.15) gives the asymptotic form of
and hence of the operator \({\textbf{B}}\) given by (4.13), too:
Apart from the coefficient and the asymptotically vanishing terms between the brackets, this is just the denominator in the second line of the classical expression (2.9) for the Lorentzian distance between \(\gamma ^a_1\) and \(\gamma ^a_2\).
Since \(\langle \psi _{s,m}\vert {\textbf{J}}^a{}_e{\textbf{p}}^e{\textbf{p}}^b{\textbf{p}}^c\vert \psi _{s,m}\rangle =\overline{\langle \psi _{s,m}\vert {\textbf{p}}^c{\textbf{p}}^b{\textbf{p}}^e {\textbf{J}}^a{}_e\vert \psi _{s,m}\rangle }\), it is enough to calculate only one of these. Using (A.31) and (A.10), for \(s>1/2\) we obtain
Then by integration by parts and using \(m^i{\bar{m}}^j+{\bar{m}}^im^j=\delta ^{ij} -p^ip^j/(p)^2\), \(m^i{\bar{m}}^j-m^j{\bar{m}}^i={\textrm{i}}\delta ^{ik}\delta ^{jl} \varepsilon _{klr}p^r/p\) (see Appendix A.4) and equations (A.47) and (A.48), we obtain
Finally, using (3.24) and evaluating the integrals by (3.29), we find that
where \(C^{abc}\) is a constant. Therefore, the second and third lines on the right hand side of (4.14) have the structure
where C is a constant. We will not need the exact form of this term.
Using the same techniques (i.e. forming total derivatives, using the explicit form of the various components of \(v^a\), the product \(m^i{\bar{m}} ^j\) expressed by \(m^i\), \({\bar{m}}^i\) and \(p^i\), the explicit form of f, the integral formula (3.29), the various matrix elements given in Appendix A.4, etc), the fourth term in (4.14) can be calculated in a straightforward way. However, this computation is considerably longer than the previous ones and we do not need its explicit form. We need to show only that the rate of the divergence of this term is smaller than the rate of the divergence of the denominator of (4.17). Next we show that this is the case.
The last term on the right hand side of (4.14) can be written into the form
in which the second expectation value is of the form \(\mu ^2_2(\delta ^{h_2}_0 \delta ^{g_2}_0+O(1/\sqrt{s}))\) by (4.15). Since also by (4.15) the order of the denominator in (4.10) is of \(\mu ^2_1\mu ^2_2\), we should show only that the first expectation value in this expression tends to infinity more slowly than \(\mu ^6_1\), and hence this does not give any contribution to the numerator of (4.10) in the classical limit.
Since by (3.24)
(A.15), (A.16) and (A.31) yield immediately that
and, in a similar way, that
Using \(\varepsilon _{aecd}p^cm^d=-{\textrm{i}}\mu (v_am_e-v_em_a)\) and \(\varepsilon _{aecd}p^cv^d=-{\textrm{i}}\mu (m_a{\bar{m}}_e-m_e{\bar{m}}_a)\) (see Appendix A.1) and equations (A.10) and (A.17), (4.21) yields that
Substituting the explicit form of \(v^a\) here (see Appendix A.1), the integrals of the resulting expression on the unit sphere \({{\mathcal {S}}}_1\) can be evaluated using 
equations (A.15) and (A.16) and the matrix elements given in Appendix A.4. The evaluation of the integrals with respect to p can be based on the integral
where the integral on the right is a finite, bounded function of \(\epsilon \). Thus, its behaviour in the classical limit depends only on the coefficient \((\epsilon \mu )^{n-2}\mu ^{m+1}\). Using these, for the order of the leading term of the 00-component of (4.23) is \(\mu ^4s\), the order of the leading term of the 0i-components is \(\mu ^4s\root 4 \of {s}\), and that of the ij-components is \(\mu ^4s\sqrt{s}\). Using (4.22), we obtain similar orders. Finally, by (A.10) the ‘cross-term’ \(\langle \varepsilon ^{he}{}_{gb}({\textbf{p}}^g{\textbf{p}}^{f_1}{\textbf{J}}^b{}_{f_1})\psi ^+_{s,m}\vert \varepsilon ^d{}_{eac}({\textbf{p}}^a{\textbf{p}}^{f_2}{\textbf{J}}^c{}_{f_2})\psi ^-_{s,m} \rangle \) is zero for \(s>1\). Therefore, since \(\psi _{s,m}=(\psi ^+_{s,m}+\psi ^- _{s,m})/\sqrt{2}\), we obtain that the leading term in \(\langle \varepsilon ^{he} {}_{gb}({\textbf{p}}^g{\textbf{p}}^{f_1}{\textbf{J}}^b{}_{f_1})\psi _1\vert \varepsilon ^d{}_{eac} ({\textbf{p}}^a{\textbf{p}}^{f_2}{\textbf{J}}^c{}_{f_2})\psi _1\rangle \) is of order \(\mu ^4s \sqrt{s}\), and hence, for \(s>1\), the fourth term on the right hand side of (4.14) has the structure
where \(C'\) is a constant.
Since the first and fourth terms in (4.12) have the same structure after interchanging the indices 1 and 2, the asymptotic forms (4.16), (4.19) and (4.25) immediately yield the asymptotic form of the expectation value of the first and the last terms in (4.12):
where \(C_1\), \(C_2\), \(C_3\) and \(C_4\) are constants.
Using (A.20) and (A.21), the expectation value of the second term in (4.12) can be written as
Using (4.15) and (4.18), we obtain the asymptotic form of the second term in (4.14) immediately. But since the third term in (4.14) has the same structure; and also the second and third terms in (4.12) have the same structure, finally we have that the asymptotic form of these two terms in (4.12) is
where \(C_5\), \(C_6\) and \(C_7\) are constants.
The sum of (4.26) and (4.28) gives the expectation value of \({\textbf{A}}\), i.e. the numerator of (4.10), in the state \(\phi :=\phi _1\otimes \phi _2\) for large \(s_1\) and \(s_2\), which is
where \(c_1\), \(c_2\), \(c_3\) and \(c_4\) are constants. Comparing this with the numerator of (2.9) and recalling the form (4.17) of the expectation value of \({\textbf{B}}\) in the same state, we conclude that, in the \(s_1=s_2\rightarrow \infty \) limit, the empirical distance \(d_{12}\) tends to the classical Lorentzian distance \(D_{12}\); and, apart from an additive constant, the index k in the Theorem can be chosen to be \(2s_1=2s_2\).
4.3.2 The Uncertainty
As we already stressed at the end of subsection 4.1 and at the beginning of subsection 4.2, the square of the empirical distance (4.10) is the quotient of two expectation values, \(\langle \phi _1 \otimes \phi _2\vert {\textbf{A}}\vert \phi _1\otimes \phi _2\rangle \) and \(\langle \phi _1 \otimes \phi _2\vert {\textbf{B}}\vert \phi _1\otimes \phi _2\rangle \), rather than the expectation value of a single operator, say of \(\eta _{ab}{\textbf{d}}^a_{12}{\textbf{d}} ^b_{12}\) given by (4.8). Thus, its standard deviation is not defined. However, borrowing the idea from experimental physics how the error of a quantity, built from experimental data, is defined, in [19] we introduced the notion of uncertainty of the E(3)-invariant empirical distance in the state \(\phi _1\otimes \phi _2\). Namely, if \(Q=Q(q^1,...,q^n)\) is a differentiable function and in a series of experiments we obtain the mean values \({\bar{q}}^\alpha \) and errors \(\delta q^\alpha \) of the quantities \(q^\alpha \), \(\alpha =1,...,n\), then the mean value and error of Q are defined to be \({\bar{Q}}:=Q({\bar{q}}^1,...,{\bar{q}}^n)\) and
respectively.
Motivated by this formula, we define the uncertainty of the square of the empirical distance (4.10) in the state \(\phi :=\phi _1\otimes \phi _2\) by
where \(\Delta _\phi {\textbf{A}}\) and \(\Delta _\phi {\textbf{B}}\) are the standard deviations in the state \(\phi \). We show that both terms between the brackets tend to zero in the classical limit. Since in this limit \(d^2 _{12}\) is bounded, this implies that the uncertainty \(\delta _\phi {d^2_{12}}\) tends to zero.
It is easy to show that the second term between the brackets is zero in this limit. In fact, the variance of \({\textbf{B}}\) in the state \(\phi :=\phi _1 \otimes \phi _2\) is
where, in the last step, we used (4.17). Since by (A.46) we have that
we conclude that the second term on the right in (4.30) tends to zero if \(s_1,s_2\rightarrow \infty \).
To show that the first term also tends to zero in this limit, we should show that, in the leading order, the expectation value \(\langle \phi \vert {\textbf{A}}^2\vert \phi \rangle \) is just \((\langle \phi \vert {\textbf{A}}\vert \phi \rangle )^2\) and all the remaining terms grow slower than \(\mu ^4_1\mu ^4_2\). By (4.12) the expectation value \(\langle \phi \vert {\textbf{A}}^2\vert \phi \rangle \) can be written as
Thus, these terms are quartic expressions of ‘units’ of the form \(\varepsilon _{acde}{\textbf{p}}^c{\textbf{p}}^f{\textbf{J}}^d{}_f\) between the round brackets. Note that, by the second commutator in (3.2), the order of the factors in such a ‘unit’ is irrelevant; moreover, \([{\textbf{p}}_b, \varepsilon _{acde}{\textbf{p}}^c{\textbf{p}}^h{\textbf{J}}^d{}_h]={\textrm{i}}\hbar \mu ^2 \varepsilon _{abce}{\textbf{p}}^c\) holds.
As in subsection 4.3.1, let the states be of the form \(\vert \phi _{\textbf{i}}\rangle =\exp ({\textrm{i}}p_e\xi ^e_{\textbf{i}}/\hbar )\vert \chi _{\textbf{i}}\rangle :=\exp ({\textrm{i}}p_e\xi ^e_{\textbf{i}}/\hbar )\) \({\textbf{U}}_{\textbf{i}}\vert \psi _{\textbf{i}}\rangle \). Then the action of a ‘unit’ on such a state is
Hence, each of the 16 terms on the right hand side of (4.31) will be the sum of 16 terms. One of these 16 terms is a quartic expression of the translation(s) \(\xi ^a\), there are terms that are proportional to the first, others to the second and some to the third powers of the translation(s); and also there is one term which does not contain any \(\xi ^a\).
The 16 terms being quartic in the translation(s) can be written as
Analogously to (4.15), it is a straightforward calculation to show that, for,
Using this, expression (4.32) can be written into the form
which is precisely \((\langle \phi \vert {\textbf{A}}\vert \phi \rangle )^2\). We show that the rate of growth of all the remaining terms in (4.31) is less than that of \(\mu ^4_1\mu ^4_2\).
Next, let us consider the terms in (4.31) that are cubic in the translation(s). There is only one ‘unit’ in the expectation values of such a term. For example, one of the four such terms coming from the first term on the right of (4.31) is
By (4.33) the last factor grows in the classical limit as \(\mu ^4_2\), and hence we should show only that the first expectation value grows more slowly than \(\mu ^6_1\).
Since by (A.20) and (A.21) this expectation value can be written as
and \(\psi _{s,m}=(\psi ^+_{s,m}+\psi ^-_{s,m})/\sqrt{2}\), it is enough to determine the order of the leading terms in the expectation values \(\varepsilon ^{ef} {}_{gb}\langle \psi ^\pm _{s,m}\vert {\textbf{p}}^{d_1}{\textbf{p}}^{h_1}{\textbf{p}}^{d_2}({\textbf{p}}^g {\textbf{p}}^c{\textbf{J}}^b{}_c)\vert \psi ^\pm _{s,m}\rangle \). (For large enough s the ‘cross terms’ \(\varepsilon ^{ef}{}_{gb}\langle \psi ^-_{s,m}\vert {\textbf{p}}^{d_1} {\textbf{p}}^{h_1}{\textbf{p}}^{d_2}({\textbf{p}}^g{\textbf{p}}^c{\textbf{J}}^b{}_c)\vert \psi ^+_{s,m} \rangle \) are vanishing.) Substituting (4.21) here, we obtain
Integrating by parts, using the explicit form of the components of \(v^e\), the expressions \(v^i{\bar{m}}^j-v^j{\bar{m}}^i=-{\textrm{i}}p^0\eta ^{ik}\eta ^{jl}\varepsilon _{klr}{\bar{m}}^r/\mu \) (Appendix A.1) and \(m^i{\bar{m}}^j-m^j{\bar{m}}^i= {\textrm{i}}\eta ^{ik}\eta ^{jl}\varepsilon _{klr}p^r/p\) (Appendix A.4), the various matrix elements in Appendix A.4 and equation (4.24), we can determine the order of its leading term. For \(d_1 =h_1=d_2=0\) it is \(\mu ^5\root 4 \of {s}\), for \(d_1=d_2=0\), \(h_1=k\) it is \(\mu ^5\), for \(d_1=k\), \(d_2=l\), \(h_1=0\) it is \(\mu ^5/\root 4 \of {s}\), and for \(d_1=k\), \(d_2=l\), \(h_1=m\) it is \(\mu ^5/\sqrt{s}\).
Repeating the above analysis with \(\psi ^-_{s,m}\) and using (4.22), we obtain the same leading orders. Finally, using the commutator \([{\textbf{p}} _b,\varepsilon _{acde}{\textbf{p}}^c{\textbf{p}}^h{\textbf{J}}^d{}_h]={\textrm{i}}\hbar \mu ^2 \varepsilon _{abce}{\textbf{p}}^c\), one can show that the position of the ‘unit’ in (4.33) is irrelevant: the difference of any two such configurations give an even slowly growing term. This shows that any term in (4.31) that is cubic in the translation(s) grow not faster than \(\mu ^3_1\mu ^4_2\root 4 \of {s_1}\) or \(\mu ^4_1\mu ^3_2\root 4 \of {s_2}\).
The proof that the terms being quadratic or linear in or independent of the translation(s) in (4.31) grow more slowly than \(\mu ^4_1 \mu ^4_2\) is similar.
5 Final Remarks
In the present investigations, the basic notion is a composite quantum system that is the union of E(1, 3)-invariant elementary quantum mechanical systems. Apart from their own intrinsic properties, fixed by their abstract algebra of observables and their representations, no additional, extra structures, in particular, no notion of 3-space or spacetime, is used. We found that, in addition to the familiar observables representing the usual physical quantities in quantum theory, one can construct observables that mimic geometrical notions, e.g. angles between ‘directions’, or distances between ‘centre-of-mass (world)lines’. Using these, genuine ‘quantum geometrical structures’, defined by the quantum system itself, can be introduced. The significance of these structures is that in the classical limit these structures reproduce the well known, but a priori given geometrical structures of the Minkowski space. Thus, the latter, like the geometrical structures in the case of the Euclidean 3-space [18, 19], might be considered to be coming from quantum theory. This is the main message of the present paper.
Nevertheless, different quantum observables may be used to define one kind of quantum geometry. For example, in the E(3)-invariant case, the linear momenta, the angular momenta, and the relative position vectors could be used to define some ‘quantum conformal structure’, which are all different, though all three can be used to derive the one and the same conformal structure of the classical 3-space in the classical limit. In a similar way, in E(1, 3)-invariant systems, the ‘quantum conformal structure’ can be defined using the (timelike) 4-momenta, the (spacelike) Pauli–Lubanski spin, or the (spacelike) relative position vectors. The resulting ‘quantum conformal structures’ are expected to yield the same conformal structure of the Minkowski space in the classical limit, though they are not expected to be equivalent at the genuine quantum level. Thus, at the quantum level, there might be no uniquely defined geometric structure of a given kind: that may depend on the observables that that is based on.
In the present (as well as in our previous [18, 19]) investigation(s) the observables representing geometric quantities were evaluated in pure tensor product states of the constituent elementary subsystems of the large composite system. Thus, the subsystems were assumed to be independent. The fact that non-trivial geometric quantities, e.g. the relative distance between them, could be derived is due to the structure of these geometric observables: all these are joint, or rather ‘entangled’ observables. In particular, the empirical distance between the E(1, 3)-invariant elementary system ‘1’ and elementary system ‘2’ is built from the quantum operators \({\textbf{P}}^2_{12}:=\eta _{ab}{\textbf{p}}^a_1\otimes {\textbf{p}} ^b_2\) and \({\textbf{S}}^a_{12}:=\frac{1}{2}\varepsilon ^a{}_{bcd}({\textbf{J}}^{bc}_1 \otimes {\textbf{p}}^d_2+{\textbf{p}}^d_1\otimes {\textbf{J}}^{bc}_2)\). Since the observables are ‘entangled’ in this sense, the states do not need to be entangled to get well defined, non-trivial results, e.g. actually the relative distance.
Notes
Since the momentum space is a Lorentzian vector space with the physically distinguished origin, the group of its isometries is only the homogeneous Lorentz (rather than the Poincaré) group, and its algebraic dual is also a vector space. However, its dual space becomes an affine vector space (i.e. without any distinguished point as its origin) if the dual is defined via the Fourier transform of the wave functions \(\phi \) on the momentum space in which not only a constant, but even a special \(p^a\)-dependent phase ambiguity is allowed, i.e. when \(\phi \) and \({\tilde{\phi }}\), defined pointwise by \({\tilde{\phi }}:p^a\mapsto \exp ( {\textrm{i}}p_e\xi ^e/\hbar )\phi (p^a)\), are considered to be equivalent for any \(\xi ^e\in {\mathbb {R}}^4\) (see e.g. [10], and Appendix A.2 of the present paper). This makes the dual space, i.e. the resulting Minkowski coordinate space, a homogeneous space with the larger Poincaré group as its group of isometries. Thus, in the present framework, while the Lorentzian symmetries of the Minkowski coordinate space come from the isometries of the momentum space, the translational symmetries have a different root; and this latter could be considered to be of quantum mechanical origin.
For example, the mass and spin of the electron in cgs units are \(\mu _e=9.1 \times 10^{-29}\,gram\) and \(s_e=\hbar /2=3.3\times 10^{-28}gram\,cm^2\,sec^{-1}\), respectively. Thus, if the classical scale is that in which the relevant quantities are of (or maybe a few orders less than the) order one in these units, then both the spin and the mass should be enlarged roughly by the same factor. Hence, accepting that enlarging the spin by a factor of \(10^{28}\) can be approximated in the given circumstances by tending with s to infinity, then it seems reasonable to approximate enlarging the mass by the factor \(10^{29}\) in the same way. In the \(c=1\) units (that we use in the present paper), \(\mu _e\) remains the same but \(s_e=1.1\times 10^{-38}gram\,cm^3\). This raises the possibility that in the classical limit it is only the spin s that should tend to infinity, but the rest mass \(\mu \) only to a large but finite value. Another possibility is that both s and \(\mu \) tend to infinity, but in different orders, e.g. when \(\mu \) grows only like \(\sqrt{s}\). Although in these cases \(\epsilon \) would not be forced to be linked to s and it could be kept finite (see below), but these asymptotic properties would not guarantee the finiteness of the empirical distance.
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A Appendix: The Summary of the Unitary, Irreducible Representations of E(1, 3)
A Appendix: The Summary of the Unitary, Irreducible Representations of E(1, 3)
In the first of these appendices, mostly to fix the notations, we review the geometry of the mass-shell and the various structures thereon (for the general background, see [27, 28, 33]). Then we summarize the key results on the unitary, irreducible representations of the quantum mechanical Poincaré group E(1, 3) in a slightly more geometric form than it is usually given in standard textbooks (and which are only sketched e.g. in [8, 9]). In Appendix A.3, we rewrite the centre-of-mass and other relevant operators in the Newman–Penrose (NP) form that we use to calculate the empirical distant and its classical limit. Finally, in Appendix A.4, matrix elements of certain polynomials of the linear momentum \({\textbf{p}}^i\) in the basis of spin weighted spherical harmonics are presented.
1.1 A.1 The Geometry of the Mass-Shell
The future mass shell \({{\mathcal {M}}}^+_\mu \) with rest mass \(\mu >0\), defined in subsection 2.1, is homeomorphic to \({\mathbb {R}}^3\). The induced (negative definite) metric on \({{\mathcal {M}}}^+_\mu \), \(h_{ab}=\eta _{ab}-p_ap_b/\mu ^2\), is of constant curvature with curvature scalar \({{\mathcal {R}}}=-6/\mu ^2\), and its extrinsic curvature is \(\chi _{ab}=h_{ab}/\mu \). By definition, the Cartesian components \(p^a=(p^0,p^i)\), \(i=1,2,3\), of the position vectors on the mass shell satisfy \((p^0)^2=\mu ^2+\delta _{ij}p^ip^j\). \({{\mathcal {M}}}^+_\mu \) can be foliated by the 2-spheres \({{\mathcal {S}}}_p:=\{p^a\in {{\mathcal {M}}}^+_\mu \vert \,p:= \sqrt{\delta _{ij}p^ip^j}={\textrm{const}}\,\}\). Clearly, \(p\in [0,\infty )\), and for \(p=0\) the ‘sphere’ \({{\mathcal {S}}}_0\) is a single point. (To avoid confusion, the square of p will be written as \((p)^2\), rather than \(p^2\), which has already been used to denote the 2nd Cartesian component of \(p^a\).) Let \((\zeta ,{\bar{\zeta }})\) be the standard complex stereographic coordinates on these 2-spheres, defined e.g. in terms of the more familiar spherical polar coordinates by \(\zeta :=\exp ({\textrm{i}}\varphi )\cot (\theta /2)\). Then \((p,\zeta , {\bar{\zeta }})\) is an intrinsic coordinate system on \({{\mathcal {M}}}^+_\mu \); and \((\mu ,p,\zeta ,{\bar{\zeta }})\) is a coordinate system inside the future null cone of the origin in \({\mathbb {R}}^{1,3}\). In these coordinates, the line element of the intrinsic metric on \({{\mathcal {M}}}^+_\mu \) is
Thus \(\sqrt{-\det (h_{ab})}=\mu (p)^2/\sqrt{\mu ^2+(p)^2}\), and hence the natural volume element on \({{\mathcal {M}}}^+_\mu \) divided by \(\mu \) is given by
where \({\textrm{d}}{{\mathcal {S}}}_1\) is the area element on the unit sphere. This \({\textrm{d}}v_\mu \) is Lorentz-invariant, and it is well defined even on the \(\mu =0\) mass shell, i.e. on the null cone of the origin in \({\mathbb {R}}^{1,3}\). It is this volume element that will be used in the definition of the Hilbert space of the square integrable spinor fields on \({{\mathcal {M}}}^+_\mu \).
The Cartesian components of \(p^a\) expressed by these coordinates are
and \(p^a/\mu \) is the future pointing unit timelike normal of \({{\mathcal {M}}}^+_\mu \) at the point \(p^a\in {{\mathcal {M}}}^+_\mu \). Its spinor form, \(p_{AA'}:=p_a\sigma ^a _{AA'}\), multiplied by \(\sqrt{2}/\mu \) defines a positive definite Hermitean pointwise scalar product on the spinor spaces, by means of which the primed spinors can be converted to unprimed ones. Here \(\sigma ^{AA'}_a\) are the four standard \(SL(2,{\mathbb {C}})\) Pauli matrices (including \(1/\sqrt{2}\)), according to the conventions of [27], and whose indices are lowered/raised by \(\eta _{ab}\) and the symplectic spinor metrics \(\varepsilon _{AB}\), \(\varepsilon _{A'B'}\) and the corresponding inverses.
The outward directed unit normal of the 2-spheres \({{\mathcal {S}}}_p\) that is tangent to \({{\mathcal {M}}}^+_\mu \) is given by \(v^a=(p/\mu ,p^0p^i/\mu p)\). This \(v^a\) is completed to be a frame field on \({{\mathcal {M}}}^+_\mu \) by the complex null vector field \(m^a\) and its complex conjugate \({\bar{m}}^a\) such that \(m^a\) and \({\bar{m}}^a\) are tangent to the 2-spheres \({{\mathcal {S}}}_p\) and that they satisfy \(m^am_a=0\) and \(m^a{\bar{m}}_a=-1\). They can be chosen to be linked to the complex stereographic coordinates so that, in particular, the Cartesian components of \(m^a\) are
while \(v^a\) and \(m^a\), as differential operators, are given by
Thus, \(\eta _{ab}=p_ap_b/\mu ^2-v_av_b-m_a{\bar{m}}_b-{\bar{m}}_am_b\) and \(p^av^bm^c {\bar{m}}^d\varepsilon _{abcd}={\textrm{i}}\mu \) hold, where \(\varepsilon _{abcd}\) is the natural volume 4-form on \({\mathbb {R}}^{1,3}\).
We associate a spinor basis \(\{o^A,\iota ^A\}\), normalized according to \(o_A \iota ^A=1\), to the basis \(\{p^a,v^a,m^a,{\bar{m}}^a\}\) such that the spinor form of these basis vectors has the form \(p^{AA'}=\mu (o^A{\bar{o}}^{A'}+\iota ^A {\bar{\iota }}^{A'})/\sqrt{2}\), \(v^{AA'}:=v^a\sigma ^{AA'}_a=(o^A{\bar{o}}^{A'}-\iota ^A {\bar{\iota }}^{A'})/\sqrt{2}\) and \(m^{AA'}:=m^a\sigma ^{AA'}_a=o^A{\bar{\iota }}^{A'}\), respectively. Explicitly, this spinor dyad is given by
This basis is related to the Cartesian spinor basis \(O^A=\delta ^A_0\), \(I^A= \delta ^A_1\) by a general, \(p,\zeta ,{\bar{\zeta }}\)-dependent \(SL(2,{\mathbb {C}})\) transformation. In particular, this is boosted with respect to the Cartesian spinor basis, and the boost parameter is \(\sqrt{(p^0+p)/\mu }=1/ \sqrt{(p^0-p)/\mu }\). Without this the bases \(\{o^A,\iota ^A\}\) and \(\{O^A, I^A\}\) would be linked only by a \(\zeta ,{\bar{\zeta }}\)-dependent SU(2), rather than a general \(p,\zeta ,{\bar{\zeta }}\)-dependent \(SL(2,{\mathbb {C}})\) transformation. The corresponding un-boosted dyad (i.e. without the boost parameter) will be denoted by \(\{{\tilde{o}}{}^A,{\tilde{\iota }}{}^A\}\).
By (A.5) and (A.6) it is easy to check that
where \(\beta :=-(2\sqrt{2}p)^{-1}\zeta \).
If \(\phi _{A_1...A_{2s}}\) is a completely symmetric spinor field, then we form \(\phi _r:=\phi _{A_1...A_{2s}}\iota ^{A_1}\cdots \iota ^{A_r}\) \(o^{A_{r+1}}\cdots o ^{A_{2\,s}}\), \(r=0,...,2\,s\) (see [27, 28]). The spinor field itself and the pointwise Hermitean scalar product of two such spinor fields can be given in terms of these functions:
In particular, \(\phi _r=0\) for \(r=0,...,2s-1\) is equivalent to the proportionality of \(\phi _{A_1...A_{2s}}\) to \(o_{A_1}\cdots o_{A_{2\,s}}\), and \(\phi _r =0\) for \(r=1,...,2\,s\) to its proportionality to \(\iota _{A_1}\cdots \iota _{A_{2s}}\).
Recalling that a scalar \(\phi \) is said to have spin weight \(\sigma \) if, under the transformation \(\{o^A,\iota ^A\}\mapsto \{\exp ({\textrm{i}}\alpha )o^A, \exp (-{\textrm{i}}\alpha )\iota ^A\}\) of the spinor basis, it transforms as \(\phi \mapsto \exp ({\textrm{i}}\sigma \alpha )\phi \), the scalar \(\phi _r\) has spin weight \(s-r\) (see [27, 28]). Since the components of the spin vectors of the NP basis can be considered to be the contractions with the Cartesian spinors, e.g. \(o^0=-I_Ao^A\) and \(o^1=O_Ao^A\), they are spin weighted scalars, and the spin weight of \(o^A\) is 1/2, while that of \(\iota ^A\) is \(-1/2\). Hence the spin weight of \(p^{AA'}\) and of \(v^{AA'}\) is zero, while that of \(m^{AA'}\) and \({\bar{m}}^{AA'}\) is 1 and \(-1\), respectively.
Another particularly important examples for the spin weighted scalars are the spin weighted spherical harmonics, which can be defined by
where the factor of normalization is
and \(2\sigma \in {\mathbb {Z}}\), \(j=\vert \sigma \vert ,\vert \sigma \vert +1,\vert \sigma \vert +2,...\) and \(m=-j, -j+1,...,j\) (see [27]). Note that in the definition of \({}_\sigma Y_{j,m}\) above the un-boosted spinor basis \(\{{\tilde{o}}{}^A,{\tilde{\iota }}{}^A\}\) is used. Since, however, the boost parameter is constant on the \(p={\textrm{const}}\) 2-spheres, the notion of the spin weight defined with respect to \(\{o^A,\iota ^A\}\) and to \(\{{\tilde{o}}^A, {\tilde{\iota }}^A\}\) is the same. The advantage of using \(\{{\tilde{o}}^A,\tilde{\iota }^A\}\) rather than \(\{o^A,\iota ^A\}\) in the definition of \({}_\sigma Y _{j,m}\) is that the resulting spherical harmonics depend only on \(\zeta \) and \({\bar{\zeta }}\), but do not on the radial coordinate p. The spin weighted spherical harmonics form an orthonormal basis in the space of the square integrable spin weighted scalars on the unit sphere. Note also that (after correction of a misprint in equation (4.15.104) of [27]) the complex conjugate of the spin weighted spherical harmonics is \(\overline{ {}_\sigma Y_{j,m}}=(-)^{m-\sigma }{}_{-\sigma }Y_{j,-m}\).
The edth and edth-prime operators of Newman and Penrose [33], acting on spin weighted scalars with spin weight \(\sigma \) on 2-spheres of radius p, can be defined by
increases, while 
decreases the spin weight by one. These definitions yield, in particular, that
which imply that 
, 
, 
and 
. Also, the action of these operators on the spin weighted spherical harmonics is
These imply that 
. A spin weighted scalar \(\phi \) is said to be holomorphic if 
, and it is anti-holomorphic if 
. (A.15)-(A.16) imply that the space of the holomorphic scalars of weight \(\sigma \) is spanned by the harmonics \({} _\sigma Y_{\vert \sigma \vert ,m}\) with \(\sigma \le 0\) and that of the anti-holomorphic ones by the harmonics \({}_\sigma Y_{\sigma ,m}\) with \(\sigma \ge 0\). Both these spaces are \(2\vert \sigma \vert +1\) complex dimensional.
1.2 A.2 The Representation
We a priori assume that the spectrum of the operators \({\textbf{p}}^a\) form the solid future null cone in the classical momentum space \({\mathbb {R}} ^{1,3}\) (without its vertex), i.e. the set of the non-zero future pointing non-spacelike vectors. In the present paper, we consider only the case when the spectrum elements \(p^a\) are strictly timelike, i.e. when the corresponding rest mass \(\mu \) is strictly positive.
The representation space is the Hilbert space \({{\mathcal {H}}}_{\mu ,s}\) of the square integrable totally symmetric unprimed spinor fields \(\phi ^{A_1...A_{2s}}\), \(2s=0,1,2,...\), on the future mass shell \({{\mathcal {M}}}^+_\mu \) of mass \(\mu >0\), and the scalar product of \(\phi ^{A_1...A_{2s}}\) and \(\psi ^{A_1...A_{2s}}\) in \({{\mathcal {H}}}_{\mu ,s}\) is defined by
This is the integral of \((\phi _{A_1...A_{2s}},\psi _{A_1...A_{2s}})\) given by (A.10) on the future mass shell \({{\mathcal {M}}}^+_{\mu }\). This implies that \(\phi ^{A_1...A_{2s}}\) is normalizable precisely when its pointwise norm, \(\vert \phi ^{A_1...A_{2s}}\vert \), defined by the pointwise scalar product (A.10), falls off as o(1/p), i.e. \(\vert \phi ^{A_1...A_{2s}}\vert \) must fall off slightly faster than 1/p for large p. Since (A.10) defines a \({\mathbb {C}}\)-linear isomorphism between the space of the primed and the unprimed spinors, it is enough to consider only e.g. the unprimed ones.
Clearly, the rotation-boost Killing fields of \({\mathbb {R}}^{1,3}\) that are vanishing at the origin \(p^a=0\) are tangent to the mass-shell and generate the isometries of \({{\mathcal {M}}}^+_\mu \), too. Then the action of \(SL(2,{\mathbb {C}})\) on any \(\phi ^{A_1...A_{2s}}\in {{\mathcal {H}}}_{\mu ,s}\) is defined to be just the action of the corresponding isometries, while the action of the translation with \(\xi ^a\) is a multiplication of \(\phi ^{A_1...A_{2s}}\) by the phase factor \(\exp ({\textrm{i}}p_a\xi ^a/\hbar )\). Explicitly, for any \(A^A{}_B\in SL(2,{\mathbb {C}})\) and \(\xi ^a\in {\mathbb {R}}^4\), this action is given by \(({\textbf{U}}_{(A,\xi )}\phi ) ^{A_1...A_{2\,s}}(p^e):=\exp ({\textrm{i}}p_a\xi ^a/\hbar )A^{A_1}{}_{B_1}\cdots A^{A_{2\,s}}{} _{B_{2\,s}}\phi ^{B_1...B_{2\,s}}((\Lambda ^{-1})^e{}_bp^b)\). Here, the Lorentz matrix in the argument on the right is built from the \(SL(2,{\mathbb {C}})\) matrix \(A^A{}_B\) according to \(\Lambda ^a{}_b:=\sigma ^a_{AA'}A^A{}_B{\bar{A}}^{A'}{}_{B'} \sigma ^{BB'}_b\). These transformations are unitary with respect to the scalar product (A.17). In this representation, the operators \({\textbf{p}}_a\) and \({\textbf{J}}_{ab}\) are defined via 1-parameter families of E(1, 3) transformations, \(\xi ^a(u)=T^au\) and \(A^A{}_B(u)=({\textrm{Exp}} (\lambda u))^A{}_B:=\delta ^A_B+\lambda ^A{}_Bu+\frac{1}{2}\lambda ^A{}_C \lambda ^C{}_Bu^2+...\) for any \(T^a\in {\mathbb {R}}^4\) and \(\lambda ^A{}_B\in sl (2,{\mathbb {C}})\), respectively, and \(u\in {\mathbb {R}}\). Explicitly, \(({\textrm{i}} /\hbar )(T^a{\textbf{p}}_a+M^{ab}{\textbf{J}}_{ab})\phi ^{A_1...A_{2s}}:=\frac{\textrm{d}}{{\textrm{d}}u}({\textbf{U}}_{(A(u),\xi (u))}\phi )^{A_1...A_{2s}}\vert _{u=0}\) with \(2\,M^a{}_b:=-\sigma ^a_{AA'}(\lambda ^A{}_B\delta ^{A'}_{B'}+{\bar{\lambda }}^{A'}{}_{B'}\delta ^A_B)\sigma ^{BB'}_b\). The resulting operators are self-adjoint on some dense subspaces of \({{\mathcal {H}}}_{\mu ,s}\) and are given explicitly by
Thus, \({\textbf{p}}_a\) acts as a multiplication operator, while \({\textbf{J}}_{ab}\) is just \({\textrm{i}}\hbar \)-times the Lie derivative of the smooth spinor fields along the rotation-boost Killing vector \(p_a(\partial /\partial p^b)-p_b (\partial /\partial p^a)\) on \({{\mathcal {M}}}^+_\mu \). \({\textbf{p}}^a\) and \({\textbf{J}}^{ab}\) are not bounded, and hence they are defined only on some appropriate dense subspaces of \({{\mathcal {H}}}_{\mu ,s}\): to ensure the square integrability of \({\textbf{p}}_a\phi ^{A_1...A_{2s}}\), the spinor field should fall off slightly faster than \(1/(p)^2\) for large p, and \({\textbf{J}}_{ab}\phi ^{A_1...A_{2s}}\) to be well defined, the spinor field must also be smooth. It is straightforward to check that, on an appropriate common domain, these operators satisfy the commutation relations (3.2)-(3.3).
(A.18) and (A.19) yield that, under the action of E(1, 3), the operators \({\textbf{p}}^a\) and \({\textbf{J}}^{ab}\) transform according to
Thus, their transformation properties postulated in subsection 3.1 have been implemented successfully by the unitary operators \({\textbf{U}}_{(A,\xi )}\).
Equations (A.18) and (A.19) yield
By (A.18), in this representation, \({\textbf{p}}_a{\textbf{p}}^a\) is \(\mu ^2\)-times the identity operator; and by (A.22) \({\textbf{S}}_a{\textbf{S}} ^a\) is \(w=-\hbar ^2\mu ^2s(s+1)\)-times the identity operator. Therefore, \({{\mathcal {H}}}_{\mu ,s}\) is a subspace of, and, in fact, it coincides with the carrier space of the unitary, irreducible representation of E(1, 3) labeled by \(\mu \) and s.
1.3 A.3 The NP Form of Operators
By (A.18) and the definition of the spinor components \(\phi _r\), \(r=0,...,2\,s\), for the Newman–Penrose (NP) form of the energy-momentum vector operator we obtain
Using the spinor identity \(\delta ^A_B=\iota ^Ao_B-o^A\iota _B\), the spinorial expression of the vectors \(p_a\), \(v_a\), \(m_a\) and \({\bar{m}}_a\) given in Appendix A.1 and the notations therein, it is also straightforward to derive from (A.22) the NP form of the Pauli–Lubanski spin vector operator:
for \(r=0,1,...,2s\).
Contracting (A.23) with the vectors of the basis \(\{p^a,v^a,m^a, {\bar{m}}^a\}\), we obtain that
To find their components in the basis \(\{o^A,\iota ^A\}\), let us use \(\delta ^a_b=p^ap_b/\mu ^2-v^av_b-m^a{\bar{m}}_b-{\bar{m}}^am_b\), equations (A.5), (A.7), (A.8) and the definitions (A.13) of the edth operators. We obtain that, for any \(r=0,1,...,2s\),
holds. Using (3.6), the definitions yield that \(v^ap^b{\textbf{J}}_{ab} =v^a{\textbf{C}}_a\), \(m^ap^b{\textbf{J}}_{ab}=m^a{\textbf{C}}_a\), \({\bar{m}}^ap^b{\textbf{J}}_{ab}= {\bar{m}}^a{\textbf{C}}_a\), \({\textrm{i}}\mu m^av^b{\textbf{J}}_{ab}=-m^a{\textbf{S}}_a\), \({\textrm{i}} \mu {\bar{m}}^av^b{\textbf{J}}_{ab}={\bar{m}}^a{\textbf{S}}_a\) and \({\textrm{i}}\mu m^a{\bar{m}}^b {\textbf{J}}_{ab}=v^a{\textbf{S}}_a\) hold. Then by (A.25) and (A.30) these imply that
which is the NP form of the angular momentum operator \({\textbf{J}}_{ab}\).
1.4 A.4 Matrix Elements of Homogeneous Polynomials of \({\textbf{p}}^i\)
Denoting the \(L_2\) scalar product on the unit 2-sphere \({{\mathcal {S}}}_1\) by \(\langle \,.\,\vert \,.\,\rangle _1\), by the orthonormality of the spin weighted spherical harmonics we have that
In the appendix of [19] we calculated the matrix elements of \({\textbf{p}}^i\) in the basis of spin weighted spherical harmonics. There we found that the only non-zero matrix elements are
Thus, the subspaces spanned by \({}_\sigma Y_{j,m}\) with given \(\sigma \) and j are not invariant under the action of the momentum operators; and while \({\textbf{p}}^3\) does not change the index m, \({\textbf{p}}^1\pm {\textrm{i}}{\textbf{p}}^2\) increases/decreases the value of m.
Also, using (A.33)-(A.41), we found that
and all the other components of \(\langle {}_\sigma Y_{j,m}\vert {\textbf{p}}^i {\textbf{p}}^j\vert {}_\sigma Y_{j,m}\rangle \) are vanishing. Although at first sight these are singular for \(\vert \sigma \vert =0,1/2\), but they are not; and they are \((p)^2/3\) is both cases.
\(\langle {}_{\pm s}Y_{s,m}\vert {\textbf{p}}^i{\textbf{p}}^j{\textbf{p}}^k\vert {}_{\pm s}Y_{s,m} \rangle _1\) can be calculated in a similar way, and for its non-zero matrix elements we obtain
\(\langle {}_{\pm s}Y_{s,m}\vert {\textbf{p}}^i{\textbf{p}}^j{\textbf{p}}^k{\textbf{p}}^l\vert {}_{\pm s} Y_{s,m}\rangle _1\) could also be calculated in the same manner, but we need to know only that, for large \(m=s\),
holds, which can be derived directly from (A.33)-(A.41), too.
We also need to know \(\langle {}_\sigma Y_{j,m}\vert m^i{\bar{m}}^j\vert {}_\sigma Y _{j,m}\rangle _1\). Since \(\eta ^{ab}=p^ap^b/\mu ^2-v^av^b-m^a{\bar{m}}^b-{\bar{m}}^am^b\), we have that \(\delta ^{ij}=-p^ip^j/\mu ^2+v^iv^j+m^i{\bar{m}}^j+{\bar{m}}^im^j\), i.e. \(m^i{\bar{m}}^j+{\bar{m}}^im^j=\delta ^{ij}-p^ip^j/(p)^2\). Hence,
Using \(m^i{\bar{m}}^j-m^j{\bar{m}}^i={\textrm{i}}\delta ^{ik}\delta ^{jl}\varepsilon _{klr} p^r/p\), a simple consequence of (A.3) and (A.4), we obtain
where we used (A.39)-(A.41), too. Finally, the sum of (A.47) and (A.48) together with (A.42) and (A.43) give the explicit form of \(\langle {}_\sigma Y_{j,m} \vert m^i{\bar{m}}^j\vert {}_\sigma Y_{j,m}\rangle _1\).
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Szabados, L.B. Minkowski Space from Quantum Mechanics. Found Phys 54, 25 (2024). https://doi.org/10.1007/s10701-024-00753-x
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DOI: https://doi.org/10.1007/s10701-024-00753-x