Photon localization revisited

In the light of Newton-Wigner-Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this example model is extracted in such a form as Quantum Field Ontology associated with Eventualization Principle, which enables us to explain the mutual relations back and forth, between quantum fields and various forms of particles in the localized form of the former.


Introduction
Extending the scope of our joint paper [24] whose essense is summarized in 1) and 2) below, we discuss in this paper the following points: 1) Starting from a specific problem of photon localization in the light of Newton-Wigner-Wightman Theorem (Sec.2), we try here to clarify the mathematical and conceptual relations among spatial points, localization processes of physical systems into restricted regions in space (and time), in contrast to the usual formulation dependent directly on the concepts of particles and their masses (in a spacetime structure given in an a priori way). In this context, Wightman's mathematical formulation of the Newton-Wigner paper plays an important role: On the basis of an imprimitivity system on the 3-dimensional space, the absence of position observables is shown to follow from the vanishing mass m = 0 of a free photon.
2) We encounter here a sharp conflict between the mathematically clearcut negative result and the actual existence of experimental devices for detecting photons in quantum optics which is impossible without the spatial localization of detected photons. Fortunately, this conflict is resolved by the presence of coupled modes of photons with material media which generates non-trivial deviations of refractive index n from 1, or equivalently generates the mass m > 0, in such typical example cases as "polaritons", as will be shown later (Sec. 3.4).
3) Through the model example of polaritons, we learn that such fundamental issues as related with mass and particles as its carriers should be viewed as something variable dependent on the contexts and situations surrounding them. Thus, we need and can elaborate on highly philosophical abstract questions like "what is a mass?" or "what are particles as mass points?", in mathematically accessible contexts. For this purpose, we certainly need to set up suitable theoretical and/or mathematical frameworks and models so that they allow us to systematically control the dynamics of our object systems coupled with their external systems. Once this coupling scheme is established, the external systems can be seen to serve as reference systems for the purpose of describing the object systems and the processes carried out by them. Such a framework and methodology are available in the form of the Tomita's integral decomposition theorem (Sec. 4.3) viewed from the standpoint of "quadrality scheme" based on "Micro-Macro duality" (Sec. 3

.2 & Sec. 4.2).
4) For instance, the delicate choice between 4-dimensional spacetime and 3-dimensional spatial setting up involved in Wightman's theorem can be naturally understood as the choice of pertinent variables to a given context. In the light of Tomita's theorem this issue is seen in such a form as the choice between central vs. subcentral decomposition measures of a relevant state. A satisfactory understanding of fundamental concepts of space (and time) coordinates and velocities is attainable in the scheme and, at the same time, crucial premise underlying such comprehension is the understanding that these concepts are never among pre-existing attributes inherent in the object system but are epigenetic properties emerging through what is to be called the "eventualization processes" as will be explained in Sec.5. These epigenetic aspects are closely related with the choices of different contexts of placing an object system and the boundary conditions specifying various different choices of subalgebras of central observables, reflected in the choices of subcentral (or central) measures appearing in Tomita's theorem (Sec. 4.3). 5) While the above explanation guarantees the naturality and genericity of the polariton picture mentioned in 2), as one of the typical explicit examples for making photons localizable, the freedom in choices of subcentral measures clarifies their speciality in the spatial homogeneity of mass generation. In fact, under such conditions that the spatial homogeneity is not required, many such forms of photon localizations are allowed as Debye shielding, various forms of dressed photons, among which cavity QED can equally be understandable. 6) Along this line of thought, it becomes also possible to compare and unify various other forms of localizations and of their "leakages" at the same time: For instance, the presence of non-vanishing mass m can be viewed as an index of timelike and spacetime-homogeneous parameter of leakage from spatial localization as exhibited by the decay rate ∝ e −mr of correlation functions in clustering limit. On the other hand, the decay width Γ in the energy spectrum can also be interpreted as a time-homogeneous parameter of leakage from chronological localizations of resonance modes (as exhibited through the decay rate ∝ e −Γ/2 of relaxation of correlations). (To be precise, it is more appropriate to regard the inverse of m and Γ as leakages.) The tunneling rate ∝ |E − V | can be interpreted as the leakage rate of spatial localization materialized by the potential barrier V .
7) The universality, naturality and the necessity of the present standpoint is verified by the above considerations in terms of subcentral measures and of the corresponding commutative algebras B. On the basis of the bidirectionality between quantum fields and particles, moreover, such a unified viewpoint will be meaningful that the microscopic quantum systems consisting of quantum fields can be controlled and designed from the macro side via the control of quantum fields. 8) To make sure of the above possibility, it would be important to recognize the constitution of the macroscopic levels in close relations with the microscopic quantum regimes. This question is answered in terms of the word, "eventualization processes", which can be mathematically described as the filtered "cones" to amplify the connections between Macro and Micro (which is analogous to the forcing method in the context of foudations of mathematics), with Micro ends given by the dynamics of quantum fields and Macro ones by the pointlike events as the apices of cones of eventualizations.

Newton-Wigner-Wightman Theorem
In 1949, Newton and Wigner [15] raised the question of localizability of single free particles. They attempted to formulate the properties of the localized states on the basis of natural requirements of relativistic covariance.
Physical quantities available in this formulation admitting direct physical meaning are restricted inevitably to the generators of Poincaré group P ↑ + = R 4 ⋊ L ↑ + (with L ↑ + the orthochronous proper Lorentz group) which is locally isomorphic to the semi-direct product H 2 (C) ⋊ SL(2, C) of the Jordan algebra H 2 (C) of hermitian (2 × 2)-matrices and SL(2, C), consisting of the energy-momentum vector P µ and of the Lorentz generators M µν (composed of angular momenta M ij and of Lorentz boosts M 0i ). The problem is then to find conditions under which "position operators" can naturally be derived from the Poincaré generators (P µ , M µν ). In [15], position operators have been shown to exist in massive cases in an essentially unique way for "elementary" systems in the sense of the irreducibility of the corresponding representations of P ↑ + so that localizability of a state can be defined in terms of such position operators. In massless cases, however, no localized states are found to exist in the above sense. That was the beginning of the story.
Wightman [25] clarified the situation by recapturing the concept of "localization" in quite a general form as follows. In place of the usual approaches with unbounded generators of position operators, he has formulated the problem in terms of their spectral resolution in the form of axioms (i)-(iii) : (i) The spectral resolution of position operators: It is defined by a family in a Hilbert space H defined for each Borel subset ∆ of R 3 , characterized by the following properties (ia), (ib), (ic): (ii) Physical interpretation of E(∆): When the system is prepared in a state ω, the expectation value ω(E(∆)) of a spectral measure E(∆) gives the probability for the system to be found in a localized region ∆; (iii) Covariance of the spectral resolution: Under a transformation (a, R) with a spatial rotation R followed by a spatial translation a, a Borel subset ∆ is transformed into R∆ + a. The corresponding unitary implementer is given in H by U (a, R), which represents (a, R) covariantly on E in such a way that Note that, in spite of the relevance of the relativistic covariance, localizability discussed above is the localization of states in space at a given time formulated in terms of spatial translations a and rotations R, respectively. To understand the reason, one should imagine the situation with the axioms (i)-(iii) replaced with those for the whole spacetime; then the CCR relations hold between 4-momenta p µ and space-time coordinates x ν , which implies the Lebesgue spectrum covering the whole R 4 for both observablesp µ and x ν . Therefore any such physical requirements as the spectrum condition or as the mass spectrum cannot be imposed on the energy-momentum spectrump µ , and hence, the concept of localizability in space-time does not make sense.
According to Mackey's theory of induced representations, Wightman's formulation can easily be seen as the condition for the family of operators {E(∆)} to constitute a system of imprimitivity ( [12]) under the action of the unitary representation U (a, R) in H of the three-dimensional Euclidean group SE(3) := R 3 ⋊ SO(3) given by the semi-direct product of the spatial translations R 3 and the rotation group SO(3). In a more algebraic form, the pair (E, U ) can also be viewed as a covariant W*-dynamical system , of the commutative algebra L ∞ (R 3 ) generated by the position operators acted on by SE(3) characterized by the covariance condition: As will be seen later, this algebraic reformulation turns out to be useful for constructing coupled systems of photon degrees of freedom with matter systems, which play the crucial roles in observing or measuring the former in the actual situations. Thus Wightman's formulation of the Newton-Wigner localizability problem is just to examine whether the Hilbert space

Corollary 2 A free photon is not localizable.
The essential mechanism causing (non-)localizability in the sense of Newton-Wigner-Wightman can be found in the structure of Wigner's little groups, the stabilizer groups of standard 4-momenta on each type of P ↑ + -orbits in p-space.
When m = 0, the little group corresponding to the residual degrees of freedom in a rest frame is the group SO(3) of spatial rotations. As a consequence, "the space of rest frames" becomes SO(1, 3)/SO(3) ∼ = R 3 . The physical meaning of this homeomorphism is just a correspondence between a rest frame r ∈ SO(1, 3)/SO(3) for registering positions and a boost k ∈ SO(1, 3) required for transforming a fixed rest frame r 0 to the chosen one r = kr 0 . The universality (or, independence for the choice the frame) of positions is recovered up to Compton wavelength h/(mc), again due to massiveness.

Remark 3
Here the coordinates of rest frames just plays the role of the order parameters (or, "sector parameters") on each P ↑ + -orbit as the space of "degenerate vacua" associated with certain of symmetry breaking, which should play the roles of position operators appearing in the imprimitivity system.
In sharp contrast, there is no rest frame for a massless particle: Its little group is isomorphic to the two-dimensional Euclidean group SE(2) = R 2 ⋊ SO(2) (locally isomorphic to C ⋊ U (1)), whose rotational generator corresponds to the helicity. Since the other two translation generators corresponding to gauge transformations span non-compact directions in distinction from the massive cases with a compact SO (3), the allowed representation (without indefinite inner product) is only the trivial one which leaves the transverse modes invariant, and hence, the little group cannot provide position operators in the massless case.
After the papers by Newton and Wigner and by Wightman, many discussions have been developed around the photon localization problem. As far as we know, the arguments seem to be divided into two opposite directions, one relying on purely dynamical bases [8] and another on pure kinematics [2], where it is almost impossible to find any meaningful agreements. Below we propose an alternative strategy based on the concept of "effective mass", which can provide a reasonable reconciliation between these conflicting ideas because of its "kinematical" nature arising from some dynamical origin.

Physical roles played by coupled external system
In spite of the above theoretical difficulty in the localizability of photons, however, it is a plain fact that almost no experiments can be performed in quantum optics where photons must be registered by localized detectors. To elaborate on this problem, we will see that it is indispensable to reexamine the behaviour of a photon in composite systems coupled with some external system such as material media constituting apparatus without which any kind of measurement processes cannot make sense. For this purpose, the above group-theoretical analysis of localizability of kinematical nature should be extended to incorporate algebraic aspects involved in the formation of a coupled dynamics between photons to be detected and the measuring devices consisting of matters. Our scheme of the localization for photons can be summarized as follows: • Photons are coupled with external system into a composite system with a coupled dynamics.
• Positive effective mass emerges in the composite system.
• Once a positive effective mass appears, Wightman's theorem itself provides the "kinematical basis" for the localization of a photon.
From our point of view, therefore, this theorem of Wightman's interpreted traditionally as a no-go theorem against the localizability becomes actually an affirmative support for it. It conveys such a strongly selective meaning (which will be discussed in detail in Sec.4) that, whenever a photon is localized, it should carry a non-zero effective mass.
In the next subsection, we explain the meaning of our scheme from a physical point of view.

How to define effective mass of a photon
As a typical example of our scheme, we focus first on a photon interacting with homogeneous medium, in the case of the monochromatic light with angular frequency ω as a classical light wave. For simplicity, we neglect here the effect of absorption, that is, the imaginary part of refractive index. When a photon interacting with matter can be treated as a single particle, it is natural to identify its velocity v with the "signal velocity" of light in medium. The relativistic total energy E of the particle should be related to v := √ v · v by its mass m eff : Since v is well known to be smaller than the light velocity c (theoretically or experimentally), m eff is positive (when the particle picture above is valid). Then we may consider m eff as the relativistic "effective (rest) mass of a photon", and identify its momentum p with Hence, as long as "an interacting photon" can be well approximated by a single particle, it should be massive, according to which its "localization problem" is resolved. The validity of this picture will be confirmed later in the next subsection. The concrete forms of energy/momentum are related to the Abraham-Minkowski controversy [1,14,4] and modified versions of Einstein/de Broglie formulae [24].
Our argument itself, however, does not depend on the energy/momentum formulae. The only essential point is that a massless particle can be made massive through some interactions. That is, while a free photon satisfies an interacting photon satisfies To sum up, an "interacting photon" can gain a positive effective mass, while a "free photon" remains massless! This is the key we have sought for. We note, however, the present argument is based on the assumption that "a photon dressed with interactions" can be viewed as a single particle. We proceed to consolidate the validity of this picture, especially the existence of particles whose effective mass is produced by the interactions, analogous to Higgs mechanism: Such a universal model for photon localization certainly exists, which is based on the concept of polariton, well known in optical and solid physics.

Polariton picture
In these areas of physics, the propagation of light in a medium is viewed as follows: By the interaction between light and matter, creation of an "excition (an excited state of polarization field above the Fermi surface)" and annihilation of a photon will be followed by annihilation of an exciton and creation of a photon, · · · , and so on. This chain of processes itself is often considered as the motion of particles called polaritons (in this case "exciton-polaritons"), which constitute particles associated with the coupled wave of the polarization wave and electromagnetic wave. The concept of polariton has been introduced to develop a microscopic theory of electromagnetic interactions in materials ( [6], [10]). Injected photons become polaritons by the interaction with matter. As exiton-phonon interaction is dissipative, the polariton picture gives a scenario of absorption. It has provided an approximation better than the scenarios without it. Moreover, the group velocity of polaritons discussed below gives another confirmation of the presence of an effective mass.
As is well known, permittivity ǫ(ω) is given by the following equality, and hence, we can determine the dispersion relation (between frequency and wave number) of polariton once the formula of permittivity is specified. In general, this dispersion relation implies branching, analogous to the Higgs mechanism. The signal pulse correponding to each branch can also be detected in many experiments, for example, in [13] cited below.
In the simple case, the permittivity is given by the transverse frequency ω T of exciton's (lattice vibration) as follows: where ǫ ∞ denotes lim ω→∞ ǫ(ω) and ǫ st = ǫ(0) (static permittivity). With a slight improvement through the wavenumber dependence of the exciton energy, the theoretical result of polariton group velocity ∂ω ∂k < c based on the above dispersion relation can explain satisfactorily experimental data of the passing time of light in materials (for example, [13]). This strongly supports the validity of the polariton picture.
From the above arguments, polaritons can be considered as a universal model of the "interacting photons in a medium" in the previous section. The positive mass of a polariton gives a solution to its "localization problem". Conversely, as the "consequence" of Wightman's theorem, it follows that "all" physically accessible photons as particles which can be localized are more or less polaritons (or similar particles) because only the interaction can give a photon its effective mass, if it does not violate particle picture.

Toward general situations
In the last subsection we have discussed that the interaction of photons with media can cause their localization by giving effective masses to them. Then a natural question arises: Is the exsistence of media a necessary condition for the emergence of effectve photon mass? The answer is no: In fact, light beams with finite transvese size have group velocities less than c.
In a recent publication [7], Giovannni et al., show experimentally that even in vacuum photons (in the optical regime) travel at the speed less than c when it is transversally structured, such as Bessel beams or Gaussian beams, by measuring a change in the arrival time of time-correlated photon pairs. They show a reduction in the velocity of photons in both a Bessel beam and a focused Gaussian one. Their work highlights that, even in free space, the invariance of the speed of light only applies to plane waves, i.e., free photons.
From our viewpoint, this result can be understood quite naturally in the light of the Newton-Wigner-Wightman theorem. As we have seen, the theorem states that every localizable elementary system (particle) with spin must be massive. It implies that photons in the real world should travel less than c, in any conditions, which makes the probability distribution of its position well-defined without contradicting with the presence of spin. Hence, transversally structured photons should become slow.
The scenario also applies to more general settings. Any kinds of boundary conditions with finite volume (like cavity), or even nanoparticles in the context of dressed photons [17], will make photons heavier and slower, even without medium!

Wightman's theorem re-interpreted as the "basis" for localization
Our general scheme of the localization for photons can be depicted as follows, whose essense can be understood in accordance with the basic formulation of "quadrality scheme" [20] underlying the Micro-Macro duality [18,19]: Localization of photons ⇑ Effective mass of photons =⇒ Change in kinematics ⇑

Dynamical interaction between photons & external system
In order to actualize the physical properties of a given system such as photons driven by an invisible microscopic dynamics, it is necessary for it to be coupled with some external measuring system through which a composite system is formed. According to this formation of coupled dynamics, the kinematics controlling the observed photons are modified and what can be actually observed is a result of this changed kinematics, realized in our case in the form of localized photons.

Tomita's theorem of integral decomposition of a state
Before going into the details of mass generation mechanisms, we examine here the theoretical framework relevant to our context. From the mathematical viewpoint, an idealized form of constructing a coupled system of the object system with an external reference one can be found conveniently in Tomita's theorem of integral decomposition of a state as follows: Theorem 4 (Tomita [5]) For a state ω of a unital C*-algebra A, the following three sets are in a 1-to-1 correspondence: 1. subcentral measures µ (pseudo-)supported by the space F A of factor states on A;

abelian von Neumann subalgebras
If µ, B and C are in the above correspondence, then the following relations hold: (ii) C = [BΩ ω ]: projection operator onto the subspace spanned by BΩ ω ; for f ∈ L ∞ (E A , µ) and A ∈ A is a *-isomorphism, satisfying the following equality for A, B ∈ A: Some vocabulary in the above need be explained: The space F A of factor states on A is the set of all the factor states ϕ whose (GNS) representations π ϕ have trivial centres: π ϕ (A) ′′ ∩ π ϕ (A) ′ = C1 Hϕ . This F A divided by the quasi-equivalence relation ≈ defined by the unitary equivalence up to multiplicity, F A / ≈ plays the role of sector-classifying space (or, sector space, for short) whose elements we call "sectors" mathematically or "pure phases" physically. Then Tomita's theorem plays a crucial role in verifying mathematically the so-called Born rule [22] postulated in quantum theory in physics.
Via the definition A(ρ) := ρ(A), ρ ∈ E A , any element A ∈ A can be expressed by a continuous function A : E A → C on the state space E A . Among measures on E A , a measure µ is called barycentric for a state ω ∈ E A if it satisfies ω = E A ρdµ(ρ) ∈ E A and is said to be subcentral if linear functionals ∆ ρdµ(ρ) and E A \∆ σdµ(σ) on A are disjoint for any Borel set ∆ ⊂ E A , having no non-vanishing intertwiners between them: i.e., T ∆ π ρ (A)dµ(ρ) = E A \∆ π σ (A)dµ(σ)T for ∀A ∈ A implies T = 0. If the abelian subalgebra B in the above theorem is equal to the centre B = Z πω (A), the measure µ is called the central measure of ω, determined uniquely by the state ω and the corresponding barycentric decomposition ω = F A ρdµ(ρ) is called the central decomposition of ω. This last concept plays crucial roles in establishing precisely the bi-directional relations between microscopic and macroscopic aspects in quantum theory, as has been exhibited by the examples of "Micro-Macro duality" (see, for instance, [18,19]).
At first sight, the distinction between central and subcentral may look too subtle, but it plays important roles in different treatments, for instance, between spatial and spacetime degrees of freedom in Wightman's theorem concerning the localizability, as mentioned already after the theorem. In this connection, we consider the problem as to how classically visible configurations of electromagnetic field can be specified in close relation with its microscopic quantum behaviour, for the purpose of which most convenient concept seems to be the coherent state and the Segal-Bargmann transform associated with it. Since coherent states are usually treated within the framework of quantum mechancs for systems with the finite degrees of freedom, the aspect commonly discussed is the so-called overcompleteness relations due to the non-orthogonality, α|β = 0, between coherent stateŝ a|α = α|α with different coherence parameters α = β.
We note that, in connection with Tomita's theorem, a composite system arises in such a form as A ⊗ C(Σ) consisting of the object system A and of the external system Σ(⊂ F A ) to which measured data are to be registered through measurement processes involving A. In this scheme, the universal reference system Σ can be viewed naturally emergent from the object system A itself just as the classifying space of its sector structure. Then, via the logical extension [21] to parametrize the object system A by its sectors in Σ, an abstract model of quantum fields ϕ : Σ → A can be created, constituting a crossed product ϕ ∈ A ⋊ U (Σ) (via the co-action of the structure group U (Σ) of Σ). Thus, the above non-orthogonality can be resolved by the effects of the classifying parameters of sectors Σ in F A . As a result, we arrive at the quantum-probabilistic realization of coherent states in such a form as the "exponential vectors" treated by Obata [16] in the context of "Fock expansions" of white noises. What is important conceptually in this framework is the analyticity due to the Segal-Bargmann transform and the associated reproducing kernel (RS) to be identified through the projection operator P in L 2 (Σ, dµ) onto its subspace HL 2 (Σ, dµ) of coherent states expressed by holomorphic functions on Σ [9], where dµ denotes the Gaussian measure.
As commented briefly above, we can find various useful relations and connections of quantum theory in terms of the concept of "quantum fields". From this viewpoint, we elaborate on its roles in attaining a transparent understanding of the mutual relations among fields, particles and mass in the next section.

From particles to fields
As we have discussed in Sec.4, the effective mass generating scenario applies to general settings. Any kinds of boundary conditions with finite volume (like cavity) will make photons heavier and slower, even without medium. This fact itself leads to a paradoxical physical question -how can the boundary condition affect a particle traveling in vacua? What is a spooky action through vacua?
Our answer is quite simple: In fact a photon is not "a particle traveling in vacua". It is just a field filling the space time, before it "becomes" a particle, or more rigorously, before it appears in a particle-like event caused via the interaction (energy-momentum exchange with external system). As we will discuss in this section, it is quite unreasonable to imagine a photon as a traveling particle unless any kinds of interaction is there.
Based on the arguments above, we discuss the limitation of particle concept in connection with a new physical interpretation of Newton-Wigner-Wightman analysis.
To begin with, we should mention that this concept involves a strong inconsistency with particle concept which seems to have been forgotten at some stage in history. In fact, the concept of a classical massless point particle with non-zero spin cannot survive special relativity with the worldline of such a particle obscured by the spin: Instead of being a purely "internal" degree of freedom, the spin causes kinematical extensivity of the particle which is exhibited in a boost transformation, as is pointed out by Bacry in [3].
The result of Newton-Wigner-Wightman analysis can be understood to show that this inconsistency cannot be eliminated by generalizing the problem in the context of quantum theory: A massless particle cannot be localized unless the spin is zero. Even in the massive case, the concept of localization is not independent of the choice of reference frames. There is no well-defined concept of "spacetime localization" as we have mentioned.
These facts are consistent with the idea that the position is not a clear cut a priori concept but an emergent property. Instead of a point particle, therefore, we should find something else having spacetime structure to accommodate events in point-like forms, which is nothing but the quantum field. In other words, the Newton-Wigner-Wightman analysis should be re-interpreted as "the existence proof of a quantum field", showing its inevitability.

From fields to particles: Principle of eventualization
This does not mean that particle-like property is artificial nor fictional. On the contrary, point-like events do take place in any kind of elementary processes of quantum measurement such as exposure on a film, photon counting, and so on.
This apparent contradiction is solved if we adopt the universality of the indeterminate processes emerging point-like events (energy-momentum exchanges) from quantum fields via formation of composite system with external systems (like media or systems giving boundary conditions), even the latter coming from the part of the degrees of freedom of quantum fields. Let us call these fundamental processes as eventualization. From our viewpoint, the most radical implication of Newton-Wigner-Wightman analysis is that we should abandon the ontology based on naïve particle picture and replace it by the one based on quantum fields with their eventualizations.
The idea of eventualization may appear to be just a palliative to avoid the contradiction between abstract theory of localization and the concrete localization phenomena, but actually, it opens the door to quite natural formulation of quantum physics. In fact, the notion of measurement process can be considered as a special kind of eventualization process with amplification. As we will discuss in a forthcoming paper [23], a glossary of "quantum paradoxes" is solved by just posing an axiom we call "eventualization principle".
Eventualization Principle: Quantum fields can effect macroscopic systems only through eventualization.
In other words, we hypothesize that the notion of "macroscopic systems"including a Schrödinger cat-can be characterized, or defined, by the collection of events, formed by perpertual eventualization.