Classical perspectives on the Newton–Wigner position observable

We use the ‘mostly plus’ $(-{++}+)$ signature convention for the spacetime metric and stick, as indicated, to four dimensions. This is not to say that our analysis cannot be generalised to other dimensions. In fact, as will become clear as we proceed, many of our statements have an obvious generalisation to other, in particular higher dimensions. On the other hand, as will also become clear, there are a few constructions which would definitely look different in other dimensions, like, e.g., the use of the Pauli–Lubański ‘vector’ in section 2.5, which becomes an $(n-3)$-form in $n$ dimensions, or the classification of elementary systems.

The velocity of light will be denoted by $c$, and not set equal to $1$. Affine Minkowski spacetime will be denoted by $M$, and the corresponding vector space of ‘difference vectors’ will be denoted by $V$. The Minkowski metric will be denoted by $\eta\colon V\times V\to\mathbb{R}$. The isomorphism of $V$ with its dual space $V^{*}$ induced by $\eta$ (‘index lowering’) will be denoted by a superscript ‘flat’ symbol $\flat$, i.e. for a vector $v\in V$ the corresponding one-form is $v^{\flat}:=\eta(v,\cdot)\in V^{*}$. The inverse isomorphism (‘index raising’) will be denoted by a superscript sharp symbol $\sharp$. Note that under a Lorentz transformation $\Lambda$, $v\in V$ transforms under the defining representation, $(\Lambda,v)\mapsto\Lambda v$, whereas its image $v^{\flat}\in V^{*}$ under the $\eta$-induced isomorphism transforms under the inverse transposed, $(\Lambda,v^{\flat})\mapsto(\Lambda^{-1})^{\top}v^{\flat}=v^{\flat}\circ\Lambda^{-1}$.

We fix an orientation and a time orientation on $M$. The (homogeneous) Lorentz group, i.e. the group of linear isometries of $(V,\eta)$, will be denoted by $\mathcal{L}:=\mathsf{O}(V,\eta)$. The Poincaré group, i.e. the group of affine isometries of $(M,\eta)$, will be denoted by $\mathcal{P}$. The proper orthochronous Lorentz and Poincaré groups (i.e. the connected components of the identity) will be denoted by $\mathcal{L}_{+}^{\uparrow}$ and $\mathcal{P}_{+}^{\uparrow}$, respectively²²2Note that speaking of just orthochronous or proper Lorentz / Poincaré transformations does not make invariant sense without specifying a time direction..

We employ standard index notation for Minkowski spacetime, using lowercase Greek letters for spacetime indices. When working with respect to bases, we will, unless otherwise stated, assume them to be positively oriented and orthonormal, and we will use 0 for the timelike and lowercase Latin letters for spatial indices. We will adhere to standard practice in physics where lowering and raising of indices are done while keeping the same kernel symbol; i.e. for a vector $v\in V$ with components $v^{\mu}$, the components of the corresponding one-form $v^{\flat}\in V^{*}$ will be denoted simply by $v_{\mu}$. For the sake of notational clarity, we will sometimes denote the Minkowski inner product of two vectors $u,v\in V$ simply by

$$u\cdot v:=\eta(u,v)=u_{\mu}v^{\mu}.$$

(2.1)

We fix, once and for all, a reference point / origin $o\in M$ in (affine) Minkowski spacetime, allowing us to identify $M$ with its corresponding vector space $V$ (identifying the reference point $o\in M$ with the zero vector $0\in V$, i.e. via $M\ni x\mapsto(x-o)\in V$), which we will do most of the time. Using the reference point $o\in M$, the Poincaré group splits as a semidirect product

$$\mathcal{P}=\mathcal{L}\ltimes V$$

(2.2)

where the Lorentz group factor in this decomposition arises as the stabiliser of the reference point – i.e. a Poincaré transformation is considered a homogeneous Lorentz transformation if and only if it leaves $o$ invariant. Thus, a homogeneous Lorentz transformation $\Lambda\in\mathcal{L}$ acts on a point $x\in M\equiv V$ as $(\Lambda x)^{\mu}=\Lambda^{\mu}_{\hphantom{\mu}\nu}x^{\nu}$, and a Poincaré transformation $(\Lambda,a)\in\mathcal{P}$ acts as $((\Lambda,a)\cdot x)^{\mu}=\Lambda^{\mu}_{\hphantom{\mu}\nu}x^{\nu}+a^{\mu}$.

We will sometimes make use of the set of spacelike hyperplanes in (affine) Minkowski spacetime $M$, which we will denote by

$$\mathsf{SpHP}:=\{\Sigma\subset M:\Sigma\;\text{spacelike hyperplane}\}.$$

(2.3)

Since the image of a spacelike hyperplane under a Poincaré transformation is again a spacelike hyperplane, there is a natural action of the Poincaré group on $\mathsf{SpHP}$, which we will denote by $((\Lambda,a),\Sigma)\mapsto(\Lambda,a)\cdot\Sigma$ and spell out in more detail in equation (3.4) below.

When considering the Lie algebra $\mathfrak{p}$ of the Poincaré group (or symplectic representations thereof), we will denote the generators of translations by $P_{\mu}$ such that $a^{\mu}P_{\mu}$ is the ‘infinitesimal transformation’ corresponding to the translation by $a\in V$, and the generators of homogeneous Lorentz transformations (with respect to the chosen origin $o$) by $J_{\mu\nu}$, such that $-\frac{1}{2}\omega^{\mu\nu}J_{\mu\nu}$ is the ‘infinitesimal transformation’ corresponding to the Lorentz transformation $\exp(\omega)\in\mathcal{L}_{+}^{\uparrow}\subset\mathsf{GL}(V)$ for $\omega\in\mathfrak{l}=\mathrm{Lie}(\mathcal{L})\subset\mathrm{End}(V)$.

Since we are using the $(-{++}+)$ signature convention, the minus sign in the expression $-\frac{1}{2}\omega^{\mu\nu}J_{\mu\nu}$ is necessary in order that $J_{ab}$ generate rotations in the $e_{a}$–$e_{b}$ plane from $e_{a}$ towards $e_{b}$, which is the convention we want to adopt. A detailed discussion of these issues regarding sign conventions for the generators of special orthogonal groups can be found in appendix A. Moreover, if $u\in V$ is a future-directed unit timelike vector, then $cP_{\mu}u^{\mu}$ (i.e. $cP_{0}$ in the Lorentz frame defined by $u=e_{0}$), which is minus the energy in the frame defined by $u$, is the generator of active time translations in the direction of $u$. Therefore, with our conventions, for the case of causal four-momentum $P\in V$ the energy (with respect to future-directed time directions) is positive if and only if $P$ is future-directed.

With our conventions, the commutation relations for the Poincaré generators are as follows:

	$\displaystyle[P_{\mu},P_{\nu}]$	$\displaystyle=0$		(2.4a)
	$\displaystyle[J_{\mu\nu},P_{\rho}]$	$\displaystyle=\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu}$		(2.4b)
	$\displaystyle[J_{\mu\nu},J_{\rho\sigma}]$	$\displaystyle=\eta_{\mu\rho}J_{\nu\sigma}+\text{(antisymm.)}$
		$\displaystyle=\Big{(}\eta_{\mu\rho}J_{\nu\sigma}-(\mu\leftrightarrow\nu)\Big{)}-\Big{(}\rho\leftrightarrow\sigma\Big{)}$		(2.4c)

As indicated, the abbreviation ‘antisymm.’, which we shall also use in the sequel of this paper, stands for the additional three terms that one obtains by first antisymmetrising (without a factor of $1/2$) in the first pair of indices on the left hand side, here $(\mu\nu)$, and then the ensuing combination once more in the second set of indices, here $(\rho\sigma)$, again without a factor $1/2$.

For later reference we already point out here that this Lie algebra has several convenient features, one of which being that it is perfect. This means that it equals its own derived algebra or, in other words, that each of its element is expressible as a linear combination of Lie brackets. This is easy to see directly from (2.4). Indeed, contraction of (2.4b) and (2.4) with $\eta^{\mu\rho}$ gives $(\dim V-1)P_{\nu}$ in the first and $(\dim V-2)J_{\nu\sigma}$ in the second case, showing that each basis element $P_{\nu}$ and $J_{\nu\sigma}$ is a linear combination of Lie brackets if $\dim V>2$. Being perfect implies that its first cohomology is trivial. Moreover, the second cohomology is also trivial. Being perfect and of trivial second cohomology will later allow us to conclude that symplectic actions are necessarily Poisson actions.

We employ the following sign conventions for symplectic geometry (as used by Abraham and Marsden in [22], but different to those of Arnold in [23]). Let $(\Gamma,\omega)$ be a symplectic manifold. For a smooth function $f\in C^{\infty}(\Gamma)$, we define the Hamiltonian vector field $X_{f}\in ST(\Gamma)$ ($ST$ denoting sections in the tangent bundle) corresponding to $f$ by

$$\iota_{X_{f}}\omega:=\omega(X_{f},\cdot)=\mathrm{d}f,$$

(2.5)

where $\iota$ denotes the interior product between vector fields and differential forms. The Poisson bracket of two smooth functions $f,g\in C^{\infty}(\Gamma)$ is then defined as

$$\{f,g\}:=\omega(X_{f},X_{g})=\mathrm{d}f(X_{g})=\iota_{X_{g}}\mathrm{d}f.$$

(2.6)

These conventions give the usual coordinate forms of the Hamiltonian flow equations and the Poisson bracket if the symplectic form $\omega$ takes the coordinate form (sign-opposite to that in [23])

$$\omega=\mathrm{d}q^{a}\wedge\mathrm{d}p_{a}\,.$$

(2.7)

It is important to note that $C^{\infty}(\Gamma)$ as well as $ST(\Gamma)$ are (infinite dimensional) Lie algebras with respect to the Poisson bracket and the commutator respectively, and that, with respect to these Lie structures, the map $C^{\infty}(\Gamma)\to ST(\Gamma),f\mapsto X_{f}$ is a Lie anti-homomorphism, that is,

$$X_{\{f,g\}}=-\,[X_{f},X_{g}].$$

(2.8)

The proof is simple once one recalls from (2.5) that the Lie derivative of $\omega$ with respect to any Hamiltonian vector field vanishes: $L_{X_{f}}\omega=\mathrm{d}(\iota_{X_{f}}\omega)+\iota_{X_{f}}\mathrm{d}\omega=\mathrm{d}^{2}f=0$. Therefore, $\mathrm{d}\{f,g\}=\mathrm{d}(\iota_{X_{g}}\mathrm{d}f)=L_{X_{g}}\mathrm{d}f=L_{X_{g}}(\iota_{X_{f}}\omega)=-\iota_{[X_{f},X_{g}]}\omega$.

By saying that a one-parameter group $\phi_{s}\colon\Gamma\to\Gamma$ of symplectomorphisms is generated by a function $g\in C^{\infty}(\Gamma)$, we mean that $\phi_{s}$ is the flow of the Hamiltonian vector field to $g$, i.e. that

$$\frac{\mathrm{d}}{\mathrm{d}s}\phi_{s}(\gamma)=X_{g}(\phi_{s}(\gamma))$$

(2.9)

for $\gamma\in\Gamma$, or equivalently

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}(f\circ\phi_{s})$	$\displaystyle=\Big{(}\mathrm{d}f(X_{g})\Big{)}\circ\phi_{s}$
		$\displaystyle=\{f,g\}\circ\phi_{s}$		(2.10)

for $f\in C^{\infty}(\Gamma)$. Here both sides of (2.3) are to be understood as evaluated pointwise.

A classical Poincaré-invariant system will be described by a phase space $(\Gamma,\omega)$ – i.e. a symplectic manifold – with a symplectic action

$$\Phi\colon\mathcal{P}\times\Gamma\to\Gamma,\;((\Lambda,a),\gamma)\mapsto\Phi_{(\Lambda,a)}(\gamma)$$

(2.11)

of the Poincaré group (in fact, for most of our purposes an action of $\mathcal{P}_{+}^{\uparrow}$ is enough). We will take $\Phi$ to be a ‘left’ action, i.e. to satisfy³³3We refer to [24] for a detailed discussion of left versus right actions and the corresponding sign conventions that will also play an important role in the sequel of this paper.

$$\Phi_{(\Lambda_{1},a_{1})}\circ\Phi_{(\Lambda_{2},a_{2})}=\Phi_{(\Lambda_{1}\Lambda_{2},a_{1}+\Lambda_{1}a_{2})}\;.$$

(2.12)

We will denote such systems as $(\Gamma,\omega,\Phi)$.

The left action $\Phi$ of $\mathcal{P}$ on $\Gamma$ induces vector fields $V_{\xi}$ on $\Gamma$ (the so-called ‘fundamental vector fields’), one for each $\xi$ in the Lie algebra $\mathfrak{p}$ of $\mathcal{P}$. They are given by

$$V_{\xi}(\gamma):=\left.\frac{\mathrm{d}}{\mathrm{d}s}\Phi_{\exp(s\xi)}(\gamma)\right|_{s=0}\;,$$

(2.13)

so that the map $\mathfrak{p}\to ST(\Gamma),\xi\mapsto V_{\xi}$, given by the differential of $\Phi$ with respect to its first argument and evaluated at the group identity, is clearly linear. In fact, it is straightforward to show that it is an anti-homomorphism from the Lie algebra $\mathfrak{p}$ into the Lie algebra $ST(M)$⁴⁴4Had we chosen $\Phi$ to be a right action, we would have obtained a proper Lie homomorphism; compare [24, appendix B]., i.e.

$$\bigl{[}V_{\xi_{1}},V_{\xi_{2}}\bigr{]}=-V_{[\xi_{1},\xi_{2}]}.$$

(2.14)

Moreover, a similar calculation shows [24, appendix B]

$$(\mathrm{D}\Phi_{(\Lambda,a)})\circ V_{\xi}=V_{\mathrm{Ad}_{(\Lambda,a)}(\xi)}\circ\Phi_{(\Lambda,a)}\;,$$

(2.15)

where $\mathrm{D}\Phi_{(\Lambda,a)}\colon T\Gamma\to T\Gamma$ denotes the differential of $\Phi_{(\Lambda,a)}\colon\Gamma\to\Gamma$.

As $\mathcal{P}$ acts by symplectomorphisms, we clearly have

$$L_{V_{\xi}}\omega=0\quad\text{for all}\;\xi\in\mathfrak{p}.$$

(2.16)

As $\omega$ is closed, the latter equation implies that $\iota_{V_{\xi}}\omega$ is likewise closed. Hence, by Poincaré’s lemma, locally (i.e. in a neighbourhood of each point) there exists a local function $f_{\xi}$ such that $\mathrm{d}f_{\xi}=\iota_{V_{\xi}}\omega$. This function is unique up to the addition of a $\xi$-dependent constant. Again by Poincaré’s lemma we could argue that $f_{\xi}$ existed globally if $\Gamma$ were simply connected. But, fortunately, we do not need that extra assumption.

In fact, since we are dealing with a special group, the function $f_{\xi}$ always exists globally, irrespective of $\Gamma$’s topology, so that each $V_{\xi}$ is a globally defined Hamiltonian vector field (i.e. each one-parameter group $\Phi_{\exp(s\xi)}\colon\Gamma\to\Gamma$ of symplectomorphisms is generated, in the sense of (2.3), by the corresponding function $f_{\xi}$). Moreover, the constants up to which the collection of $f_{\xi}$ is defined can be chosen in such a way that the map $\xi\mapsto f_{\xi}$ from the Lie algebra $\mathfrak{p}$ to the Lie algebra $C^{\infty}(\Gamma)$ (the Lie product of the latter being the Poisson bracket) is a Lie homomorphism, i.e.

$$\left\{f_{\xi_{1}},f_{\xi_{2}}\right\}=f_{[\xi_{1},\xi_{2}]}.$$

(2.17)

This clearly fixes the constants uniquely. Note that, according to (2.14) and (2.8), both maps, $\xi\mapsto V_{\xi}$ and $V_{\xi}\mapsto f_{\xi}$, are Lie anti-homomorphisms. Hence their combination $\xi\mapsto f_{\xi}$ is a proper Lie homomorphism (no minus sign on the right-hand side of (2.17)).

A symplectic action of a group whose generating vector fields are globally Hamiltonian and satisfy (2.17) is called a Poisson action. The statement made here is that if $\dim V>2$, any symplectic action of the Poincaré of group is always a Poisson action. This is a non-trivial statement depending crucially on properties of the groups’s Lie algebra. For example, it would fail to hold for the Galilei group (homogeneous as well as inhomogeneous) which, despite being just a contraction of the Poincaré group, behaves quite differently in that matter and, consequently, also as regards the problem of localisation [25, 5].

The underlying reason for why $f_{\xi}$ exists globally is that $\mathfrak{p}$ is perfect, as already shown above. Indeed, the proof is quite simple: Since $\xi=[\xi_{1},\xi_{2}]$ (or sums of such commutators) we have $V_{\xi}=-[V_{\xi_{1}},V_{\xi_{2}}]$ and hence $\mathrm{d}f_{\xi}=-\iota_{[V_{\xi_{1}},V_{\xi_{2}}]}\omega=-L_{V_{\xi_{1}}}(\iota_{V_{\xi_{2}}}\omega)=\mathrm{d}(\omega(V_{\xi_{1}},V_{\xi_{2}}))$, so that $f_{\xi}=\omega(V_{\xi_{1}},V_{\xi_{2}})+\text{const.}$ which is globally defined. The other statement concerning the choice of constants that guarantee (2.17) is an immediate consequence of the triviality of the second cohomology of $\mathfrak{p}$, the proof of which may, e.g., be looked up in [26, § 3.3].

Having established global existence and uniqueness of the generators $f_{\xi}$ satisfying $\omega(V_{\xi},\cdot)=\mathrm{d}f_{\xi}$, we can now deduce the transformation property of $f_{\xi}$ under the action of $\mathcal{P}$. Taking the pullback of the equation $\omega(V_{\xi},\cdot)=\mathrm{d}f_{\xi}$ with $\Phi_{(\Lambda,a)^{-1}}$ and using the invariance of $\omega$ as well as (2.15), we immediately deduce

$$\Phi_{(\Lambda,a)^{-1}}^{*}f_{\xi}:=f_{\xi}\circ\Phi_{(\Lambda,a)^{-1}}=f_{\mathrm{Ad}_{(\Lambda,a)}(\xi)}\;,$$

(2.18)

which may also be read as the invariance of the real-valued function $f\colon\mathfrak{p}\times\Gamma\to\mathbb{R}$, $(\xi,\gamma)\mapsto f_{\xi}(\gamma)$, under the combined left action of $\mathcal{P}$ on $\mathfrak{p}\times\Gamma$ given by $\mathrm{Ad}\times\Phi$. Alternatively, since $\xi\mapsto f_{\xi}$ is linear, we may regard $f$ as $\mathfrak{p}^{*}$-valued function on $\Gamma$, where $\mathfrak{p}^{*}$ denotes the vector space dual to $\mathfrak{p}$. This map is called the momentum map⁵⁵5See [22, chap. 4.2] for a general discussion on the notion of ‘momentum map’ and also [24] for an account of its use and properties restricted to the case of Poincaré-invariant systems. for the given system $(\Gamma,\omega,\Phi)$, which according to (2.18) is then $\mathrm{Ad}^{*}$-equivariant:

$$f\circ\Phi_{(\Lambda,a)}=\mathrm{Ad}^{*}_{(\Lambda,a)}\circ f\iff\mathrm{Ad}^{*}_{(\Lambda,a)}\circ f\circ\Phi_{(\Lambda,a)^{-1}}=f$$

(2.19)

The second expression is again meant to stress that the condition of equivariance is equivalent to the invariance of the function $f$ under the combined left actions in its domain and target spaces (invariance of the graph). Note that $\mathrm{Ad}^{*}$ denotes the co-adjoint representation of $\mathcal{P}$ on $\mathfrak{p}^{*}$, given by $\mathrm{Ad}^{*}_{(\Lambda,a)}:=(\mathrm{Ad}_{(\Lambda,a)^{-1}})^{\top}$ with superscript $\top$ denoting the transposed map.

Points in $\Gamma$ faithfully represent the state of the physical system whereas observables correspond to functions on $\Gamma$. In order to implement time evolution we shall employ a ‘classical Heisenberg picture’, in which the phase space point remains the same at all times, whereas the evolution will correspond to the changes of observables according to their association to different spacelike hyperplanes in spacetime. Although this is different from the (‘Schrödinger picture’) approach usually taken in classical mechanics (where the state of the system is given by a phase space point changing in ‘time’, which is an external parameter), this point of view is clearly better adapted to the Poincaré-relativistic framework, in which there simply is no absolute notion of time.

Choosing a set of ten basis vectors $(P_{\mu},J_{\mu\nu})$ for $\mathfrak{p}$ obeying (2.4) (compare appendix A), we can contract the $\mathfrak{p}^{*}$-valued momentum map with each of these basis vectors in order to obtain the corresponding ten real-valued component functions of the momentum map. By some abuse of notation we shall call these component functions by the same letters $(P_{\mu},J_{\mu\nu})$ as the Lie algebra elements themselves. Equation (2.17) now says that the map that sends the Lie algebra elements $P_{\mu}$ and $J_{\mu\nu}$ in $\mathfrak{p}$ to the corresponding component functions of the momentum map is a Lie homomorphism from $\mathfrak{p}$ to the Lie algebra $C^{\infty}(\Gamma,\mathbb{R})$ (the latter with Poisson bracket as Lie multiplication):

	$\displaystyle\{P_{\mu},P_{\nu}\}$	$\displaystyle=0$		(2.20a)
	$\displaystyle\{J_{\mu\nu},P_{\rho}\}$	$\displaystyle=\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu}$		(2.20b)
	$\displaystyle\{J_{\mu\nu},J_{\rho\sigma}\}$	$\displaystyle=\eta_{\mu\rho}J_{\nu\sigma}+\text{(antisymm.)}$		(2.20c)

The $\mathrm{Ad}^{*}$-equivariance of the momentum map can now be written down in component form if we first set $\xi=P_{\mu}$ and then $\xi=J_{\mu\nu}$. Indeed, considering (2.18) and recalling our abuse of notation in denoting the real-valued phase space functions $f_{P_{\mu}}$ and $f_{J_{\mu\nu}}$ again with the letters $P_{\mu}$ and $J_{\mu\nu}$, we can immediately read from equation (B.8) of appendix B, in which we need to replace $e_{a}$ with $P_{\mu}$ and $B_{ab}$ with $-J_{\mu\nu}$ according to (A.15) of appendix A, that

		$\displaystyle P_{\mu}\circ\Phi_{(\Lambda,a)}$	$\displaystyle=(\Lambda^{-1})^{\nu}_{\phantom{\nu}\mu}\,P_{\nu}\;,$		(2.21a)
		$\displaystyle J_{\mu\nu}\circ\Phi_{(\Lambda,a)}$	$\displaystyle=(\Lambda^{-1})^{\rho}_{\phantom{\rho}\mu}(\Lambda^{-1})^{\sigma}_{\phantom{\sigma}\nu}\,J_{\rho\sigma}+a_{\mu}(\Lambda^{-1})^{\rho}_{\phantom{\rho}\nu}\,P_{\rho}-a_{\nu}(\Lambda^{-1})^{\rho}_{\phantom{\rho}\mu}\,P_{\rho}\;.$		(2.21b)

Note that the left-hand sides of (2.21) are precisely what we need; that is, we need the composition with $\Phi_{(\Lambda,a)}$ rather than $\Phi_{(\Lambda,a)^{-1}}$ to evaluate the momenta $P_{\mu}$ and $J_{\mu\nu}$ on the actively Poincaré-displaced phase space points. Note also that if we had put the indices upstairs and had used, e.g., $P^{\mu}=\eta^{\mu\nu}P_{\nu}$ rather than $P_{\mu}$ then the right-hand side of (2.21a) would read $\Lambda^{\mu}_{\phantom{\mu}\nu}\,P^{\nu}$, and correspondingly in (2.21b). Finally recall that the last term on the right-hand side of (2.21b) just reflects the familiar transformation of angular momentum (the momentum associated to spatial rotations) under spatial translations, which is typical for the co-adjoint representation, which here gets extended to the momentum associated to boost transformations⁶⁶6One easily checks that the signs are right: translating a system whose momentum points in $y$-direction by a positive amount into the $x$-direction should enhance the angular momentum in $z$-direction. This is just what (2.21b) implies..

Given a classical Poincaré-invariant system, the Pauli–Lubański vector $W$ is the $V$-valued phase space function defined in components by

$$W_{\mu}=-\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}P^{\nu}J^{\rho\sigma}$$

(2.22)

where $\varepsilon$ denotes the volume form of Minkowski space (whose components in a positively oriented orthonormal basis are just given by the usual totally antisymmetric symbol, with $\varepsilon_{0123}=+1$). The sign convention in this definition can be understood as follows. We imagine a situation in which $P$ is timelike and future-directed (positive energy, see above), and consider the spatial components of $W$ with respect to an orthonormal basis $\{e_{0},\dots,e_{3}\}$ of $V$ with $(e_{0})^{\mu}=P^{\mu}/\sqrt{-P_{\nu}P^{\nu}}$ (‘momentum rest frame’). For those, we obtain

$$\frac{W_{a}}{\sqrt{-P_{\mu}P^{\mu}}}=-\frac{1}{2}\varepsilon_{a0\rho\sigma}J^{\rho\sigma}=\frac{1}{2}{{}^{(3)}\varepsilon}_{abc}J^{bc}$$

(2.23)

where the ${{}^{(3)}\varepsilon}_{abc}$ is the three-dimensional antisymmetric symbol / the components of the spatial volume form. Thus, since $J^{bc}=J_{bc}$ generates rotations from $e_{b}$ towards $e_{c}$, we see that $W_{a}/\sqrt{-P_{\mu}P^{\mu}}$ generates rotations ‘along the $e_{a}$ axis’ in the usual, three-dimensional sense. Thus, $W/\sqrt{-P_{\mu}P^{\mu}}$ can be interpreted as the ‘spatial spin vector’ in the momentum rest frame, which is the usual interpretation of the Pauli–Lubański vector.

Rewriting the definition of $W$ as

$$W_{\mu}=-\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}P^{\nu}J^{\rho\sigma}=\frac{1}{2}\varepsilon_{\nu\rho\sigma\mu}P^{\nu}J^{\rho\sigma}=\frac{1}{3!}\varepsilon_{\nu\rho\sigma\mu}(P^{\flat}\wedge J)^{\nu\rho\sigma},$$

(2.24)

we see that in the language of exterior algebra

$$W=(*(P^{\flat}\wedge J))^{\sharp}$$

(2.25)

where $*$ is the Hodge star operator. Here we use the standard sign conventions for the Hodge operator, i.e. the definition $\alpha\wedge*\beta=\eta(\alpha,\beta)\,\varepsilon$; see for example [27] or [24, appendix A].

We start by describing the general framework developed by Fleming in [15] and also [28] for the description of special-relativistic position observables, translated to our case of classical systems from Fleming’s quantum language. Consider a classical Poincaré-invariant system $(\Gamma,\omega,\Phi)$. By a position observable $\chi$ for this system we understand a ‘procedure’ which, given any spacelike hyperplane $\Sigma\in\mathsf{SpHP}$ in (affine) Minkowski spacetime, allows us to ‘localise’ the system on $\Sigma$. More precisely, this means that for any $\Sigma\in\mathsf{SpHP}$, we have an $M$-valued phase space function

$$\chi(\Sigma)\colon\Gamma\to M$$

(3.1)

with image contained in $\Sigma$, whose value $\chi(\Sigma)(\gamma)$ for $\gamma\in\Gamma$ is to be interpreted as the ‘$\chi$-position’ of our system in state $\gamma$ on the hyperplane $\Sigma$.

Any spacelike hyperplane $\Sigma\in\mathsf{SpHP}$ is uniquely characterised by its (timelike) future-directed unit normal $u\in V$ and its distance $\tau\in\mathbb{R}$ to the origin $o\in M$, measured along the straight line through $o$ in direction $u$. In terms of these, it has the form

$$\Sigma=\{x\in M:u_{\mu}x^{\mu}=-\tau\},$$

(3.2)

where we identified $M$ with $V$. From now on, whenever convenient, we will identify $\Sigma$ with the tuple $(u,\tau)$. The condition that the image of $\chi(\Sigma)$ be contained in $\Sigma$ then takes the form

$$u_{\mu}\chi^{\mu}(u,\tau)(\gamma)=-\tau.$$

(3.3)

We can now also spell out explicitly the left action of $\mathcal{P}$ on $\mathsf{SpHP}$ that is induced from the left action of $\mathcal{P}$ on $M$ (as already mentioned below equation (2.3)):

$$(\Lambda,a)\cdot(u,\tau)=(\Lambda u,\tau-\Lambda u\cdot a)$$

(3.4)

One easily checks that this indeed defines a left action, i.e. $(\Lambda_{1},a_{1})\cdot[(\Lambda_{2},a_{2})\cdot(u,\tau)]=(\Lambda_{1}\Lambda_{2},a_{1}+\Lambda_{1}a_{2})\cdot(u,\tau)$.

Fixing $u$ and varying $\tau$ in (3.2), we obtain the spacelike hyperplanes corresponding to different ‘instants of time’ $\tau$ in the Lorentz frame corresponding to $u$. Thus, for a fixed state $\gamma\in\Gamma$ and fixed frame $u$, the set

$$\{\chi(u,\tau)(\gamma):\tau\in\mathbb{R}\}\subset M$$

(3.5)

gives the ‘worldline’ of the $\chi$-position of the system. Following Fleming [15], who says that this is a requirement ‘easily agreed upon’, we require that this worldline should be parallel to the four-momentum⁷⁷7This assumption is natural for closed systems as we consider here. For non-closed systems, i.e. systems without local energy–momentum conservation, the four-velocity is in general not parallel to the four-momentum; see, e.g., the discussion at the beginning of section 2.6 in [29]., i.e. $\frac{\partial\chi(u,\tau)}{\partial\tau}\propto P$. Together with (3.3), this implies condition (3.8) in the definition below, which is meant to sum up all the preceding considerations.

Note that (3.8) and (3.3) imply that the four-momentum must be causal for such a position observable to exist.

In addition to the demands of the positions $\chi(\Sigma)$ being located on $\Sigma$ and of ‘worldlines’ in direction of the four-momentum, Fleming also introduces the following covariance requirement (which we, different to Fleming, do not include in the definition of a position observable):

This is indeed a sensible notion of covariance: it demands that, for any Poincaré transformation $(\Lambda,a)$, the $\chi$-position of the transformed system $\Phi_{(\Lambda,a)}(\gamma)$ on the transformed hyperplane $(\Lambda,a)\cdot\Sigma$ be the transform of the ‘original position’ $\chi(\Sigma)(\gamma)$. In terms of components, (3.9) assumes the form

$$\chi^{\mu}(\Lambda u,\tau-\Lambda u\cdot a)\circ\Phi_{(\Lambda,a)}=\Lambda^{\mu}_{\hphantom{\mu}\nu}\chi^{\nu}(u,\tau)+a^{\mu}\,,$$

(3.11)

taking into account (3.4).

The most important and widely used procedure to define special-relativistic position observables is by so-called spin supplementary conditions. Suppose we are given a causal, future-directed vector $P\in V$ and an antisymmetric 2-tensor $J\in\bigwedge^{2}V^{*}$, describing the four-momentum and the angular momentum (with respect to the origin $o\in M$) of some physical system. For any future-directed timelike vector $f\in V$, we then consider the equation

$$0=S_{\mu\nu}f^{\nu}$$

(3.12)

with $S_{\mu\nu}:=J_{\mu\nu}-x_{\mu}P_{\nu}+x_{\nu}P_{\mu}$, which we view as an equation for $x\in M$. Since $S$ is the angular momentum tensor with respect to the reference point $x$ (instead of the origin $o$ as for $J$), or the spin tensor with respect to $x$, (3.12) is called the spin supplementary condition (SSC) with respect to $f$. As is well-known (and easily verified), the set of its solutions $x$ is a line in $M$ with tangent $P$, namely

$$\{x\in M:0=S_{\mu\nu}f^{\nu}\}=\left\{x\in M:x_{\mu}=\frac{J_{\mu\rho}f^{\rho}}{f\cdot P}+\lambda P_{\mu}\;\text{with}\;\lambda\in\mathbb{R}\right\}.$$

(3.13)

This line can be given the interpretation of the ‘centre of energy’ worldline of our system with respect to the Lorentz frame defined by $f$. See [19] and references therein for further discussion on the interpretation and impact of various SSCs as regards equations of motion in General Relativity.

The idea is now to explicitly combine the SSC-based approach with Fleming’s geometric ideas, thereby introducing the two independent parameters $f$ from (3.13) and $u$ from (3.2). We define a position observable in the sense of definition 3.1 in the following way: given a classical Poincaré-invariant system $(\Gamma,\omega,\Phi)$ with causal four-momentum and a state $\gamma\in\Gamma$, we consider the SSC worldline defined by (3.12) where we now take $P_{\mu}(\gamma)$ for the four-momentum and $J_{\mu\nu}(\gamma)$ for the angular momentum tensor. We then simply define $\chi(\Sigma)(\gamma)$ to be the intersection of this worldline with the hyperplane $\Sigma=(u,\tau)$. This means that we take the $x(\lambda)$ from (3.13) and determine the parameter $\lambda$ from (3.3), i.e. from $x(\lambda)\cdot u+\tau=0$. Inserting the $\lambda=\lambda(u,\tau)$ so determined leads to

Note that for this definition to make sense, $f$ does not have to be a fixed timelike future-directed vector: it can depend on the normal $u$ (and could even depend on $\tau$), and it can also depend on phase space⁸⁸8Various choices for $f$ were given distinguished names in the literature. The main ones, different from the Newton–Wigner condition to be discussed here, are as follows. If $f$ is meant to just characterise a fixed ‘laboratory frame’, which may be preferred for any reason, like rotational symmetries in that frame, the SSC is named after Corinaldesi & Papapetrou [30]. If $f$ is proportional to the total linear momentum of the system, the SSC is named after Tulczyjew [31] and Dixon [32]. If $f$ is chosen in a somewhat self-referential way to be the four-velocity of the worldline that is to be determined by the very SSC containing that $f$, the condition is named after Frenkel [33], Mathisson [34, 35], and Pirani [36, 37].. Of course this means that according to this dependence of $f$, we will possibly be considering different worldlines for different choices of $u$.

Of course, the SSC position observable (3.14) will generally not be covariant in the sense of definition 3.2 unless $f$ is also assumed to transform appropriately. If $f$ depends on $\Sigma\in\mathsf{SpHP}$ and $\gamma\in\Gamma$ and takes values in $V$ it seems obvious that for the resulting position to be covariant $f$ itself must be a covariant function under the combined actions on its domain and target spaces. Indeed, we have

Since the vectors defining the centre of energy, the centre of inertia and the Newton–Wigner position satisfy (3.19), all of these are covariant position observables. We stress once more that for this to be true we need to take into account the action of the Poincaré group on $\mathsf{SpHP}$. This remark is particularly relevant in the Newton–Wigner case, in which $f$ is the sum of two vectors, $u$ and $P/(mc)$, the first being associated to an element of $\mathsf{SpHP}$ and the second to an element of $\Gamma$. Covariance cannot be expected to hold for non-trivial actions on $\Gamma$ alone. In the next section we will offer an insight as to why this somewhat ‘hybrid’ combination for $f$ in terms of an ‘external’ vector $u$ and an ‘internal’ vector $P/(mc)$ appears. The latter is internal, or dynamical, in the sense that it is defined entirely by the physical state of the system, i.e. a point in $\Gamma$, while the former is external, or kinematical, in the sense that it refers to the choice of $\Sigma\in\mathsf{SpHP}$, which is entirely independent of the physical system and its state.

Finally, we will need the following well-known result for SSCs with respect to different vectors $f$, which was first shown by Møller in 1949 in [38]; see also [24, theorem 17] for a recent and more geometric discussion:

For a system with timelike four-momentum, the Pauli–Lubański vector $W$ has the interpretation of being ($mc$ times) the spin vector in the momentum rest frame. We now define the spin vector in an arbitrary Lorentz frame by boosting $W/(mc)$ to the new frame:

With respect to an orthonormal basis $\{u=e_{0},\dots,e_{3}\}$ adapted to $u$, the centre of spin condition takes the form

$$s_{0}(u)=0,\quad s_{a}(u)=-\frac{1}{2}\varepsilon_{a0\rho\sigma}S^{\rho\sigma}(u)=\frac{1}{2}{{}^{(3)}\varepsilon}_{abc}S^{bc}(u),$$

(3.27)

through which it acquires an immediate interpretation: a position observable is a centre of spin if and only if, for any Lorentz frame $u$, the spin vector defined by boosting the Pauli–Lubański vector to $u$ really generates spatial rotations around the point given by the position observable.

We will now rewrite the centre of spin condition. Since $S(u)=J-(\chi(u,\tau))^{\flat}\wedge P^{\flat}$, we can rewrite the Pauli–Lubański vector as $W=\left[*\left(\frac{P^{\flat}}{mc}\wedge J\right)\right]^{\sharp}=\left[*\left(\frac{P^{\flat}}{mc}\wedge S(u)\right)\right]^{\sharp}$. Thus, the centre of spin condition takes the form

$$(B(u)^{-1})^{\top}\left[*\left(\frac{P^{\flat}}{mc}\wedge S(u)\right)\right]=*(u^{\flat}\wedge S(u)).$$

(3.28)

Since $B(u)$ is a Lorentz transformation, i.e. an isometry of $(V,\eta)$, and it maps $P/(mc)$ to $u$, this is equivalent to

$$u^{\flat}\wedge\left((B(u)^{-1})^{\top}\otimes(B(u)^{-1})^{\top}\right)(S(u))=u^{\flat}\wedge S(u).$$

(3.29)

Using the explicit form (3.24) of $B(u)$, we see that

$$\left((B(u)^{-1})^{\top}\otimes(B(u)^{-1})^{\top}\right)(S(u))=S(u)+\frac{\frac{P^{\flat}}{mc}\wedge\left(\iota_{u+\frac{P}{mc}}S(u)\right)}{1-u\cdot\frac{P}{mc}}+u^{\flat}\wedge(\ldots).$$

(3.30)

Thus, we have the following: