On the group theoretical approach to the Quantum Theory of an interacting spin-0 particle

We point out a difficulty that arises in extending the group theoretical approach that deductively establish the quantum theory of a free particle to the case of an interacting particle. Then we develop an approach which overcomes this difficulty. The result is a theory of an interacting particle where the standard theory is characterized by specific covariance properties related to the interaction.


Introduction
A very satisfactory approach to the Quantum Theory of a free particle can be accomplished by following group theoretical methods, developed in particular by E.P. Wigner and G.W. Mackey. The key starting point is that Galilei's group G, within a non-relativistic theory, is a group of symmetry transformations for a free particle; then Wigner's theorem [1], [2] on the representation of symmetries and Mackey's imprimitivity theorem [3], [4], [5] apply. In so doing the specific Quantum Theory of a free particle is explicitly obtained [3], [6], [7], [8], [9] without invoking canonical quantization, but, instead, by means of a mathematical deduction based on symmetry principles. We recall the main mathematical tools that allow for such an approach in section 2, while the necessary formal implications of the concept of Galileian transformations in the Quantum Theory of a particle are outlined in section 3.
But the Galileian transformations are not symmetries for an interacting particle, so that an obstacle is found in extending the approach to this case. To overcome this difficulty, the approaches that in the literature extend the cited group theoretical method to the interacting case (e.g., [7], [8], [9]) adopt, among others, the following assumption.
(P) Each Galileian transformation g ∈ G is assigned a unitary operator U g which realizes the corresponding quantum transformations of states and observables according to ρ g −→U g ρU −1 g and A g −→U g AU −1 g ; the correspondence g → U g is a projective representation. Now, in section 4 we show that (P) forces the hamiltonian operator into the form H = − 1 2µ . Therefore a Quantum Theory where (P) is assumed cannot describe all empirically known interactions; in particular it excludes electromagnetic interactions. An empirically more adequate approach must abandon assumption (P).
Then in section 5 we undertake the task of establishing the Quantum Theory of an interacting particle through a group theoretical approach, but without assumption (P). The difficulties raised by the loss of symmetry are addressed through a theoretical development based on the passive interpretation of the Galileian transformations and on the concept of σ-conversion. We find that the form of the dynamical equation is constrained by the covariance properties preserved by the σ-conversion admitted by the interaction; in particular, we exactly identify which covariance properties characterize electromagnetic interaction among all possible ones.

Group theoretical key results
Here we recall the main mathematical tools of the cited group theoretical approach.

Basic achievements
Given the Hilbert space H of the Quantum Theory of a physical system, we shall make use of the following operator's structures. The main mathematical tools will be used in this work are the imprimitivity theorem of Mackey and Wigner's representation theorem.
Mackey's imprimitivity theorem is a representation theorem for imprimitivity systems relative to projective representations [4]. The following definition recalls the notion of projective representation.
Definition 2.1. Let G be a separable, locally compact group with identity element e. A correspondence U : G → U(H), g → U g , with U e = 1I, is a projective representation of G if the following conditions hold.

A projective representation with multiplier σ is called σ-representation.
A projective representation is said to be continuous if for any fixed ψ ∈ H the mapping g → U g ψ from G into H is continuous with respect to g.
Let E be the Euclidean group, i.e. the semi-direct product E = R 3 s SO(3) between the group of spatial translations R 3 and the group of spatial proper rotations SO(3); each transformation g ∈ E bi-univocally corresponds to the pair (a, R) ∈ R 3 × SO(3) such that Rx + a ≡ g(x) is the result of the transformation of the spatial point x = (x 1 , x 2 , x 3 ) by g. The general imprimitivity theorem is an advanced mathematical result, but its formulation becomes very simple when adapted to the case of the Euclidean group E, that is the case we are interested to. Then we introduce the concept of imprimitivity system and the theorem for this simple case.
holds for all (a, R) ∈ E.
In the literature different equivalent formulations of Wigner's theorem have been proved [2], [11]. Here we formulate those "Wigner's' theorems" we need for our work.

Quantum theoretical implications
Wigner's theorem has very important implications in Quantum Theory. We outline those of interest for our aims. Let g →Û g be every continuous projective representation of Galilei's group G, i.e. the group generated by E and by Galileian velocity boosts. Let T 1 ⊆ G be the abelian sub-group of spatial translations along the x 1 axis. If τ 1 (a) denotes the translation (x 1 , x 2 , x 3 ) → (x 1 + a, x 2 , x 3 ), then we have τ 1 (a)τ 1 (b) = τ 1 (a + b), so thatÛ τ 1 (a+b) = σ (τ 1 (a), τ 1 (b))Û τ 1 (a)Ûτ 1 (b) . The arbitrary phase factor of eachÛ τ 1 (a) can be chosen so that, according to Stone's theorem,Û τ 1 (a) = e −iP 1 a , whereP 1 is a self-adjoint operator called the hermitean generator of the one parameter unitary sub-group {e −iP 1 a , a ∈ R} ⊆ U(H). Now, the one-parameter abelian sub-groups T α , R α , B α of spatial translation, spatial rotations and Galileian boosts, relative to axis x α , are all additive; then the argument used for T 1 can be repeated to establish the existence of nine hermitean generatorsP α ,Ĵ α ,Ĝ α , α ∈ {1, 2, 3}, of the nine one-parameter unitary subgroups The structural properties of G as a Lie group imply the validity of the following commutation relations [9].
Another important achievement implied by Wigner's theorem is the general evolution law of Quantum Theory [5], [6] with respect to a homogeneous time:

Galileian quantum transformations
In this section we recall the concept of quantum transformation corresponding to a Galileian transformation g ∈ G according to the active interpretation.
In correspondence with each Galileian transformation g ∈ G we can conceive two mappings R (1) g and R (2) g which transform quantum states and quantum observables, respectively, according to the active interpretation of the transformation: For instance, if h ∈ G is a pure velocity boost characterized by velocity parameters (u 1 , is meant to be the observable whose measurement apparatus h A is identical to an apparatus A that measures A, also for its location, but it is endowed with a constant velocity u with respect to A. If the theory is specializaed to a localizable particle, then three commuting quantum observables Q = (Q 1 , Q 2 , Q 3 ), the position observables, exist such that R Remark 3.1. The transformation of the position at time t, i.e. of Q (t) = e iHt Qe −iHt , requires a function g t different 1 from g; in general, the following covariance relations hold.
If all transformations g ∈ G are symmetry transformations for the quantum system, further conditions can be implied in its Quantum Theory. First of all, R (1) g and R (2) g have to be bijective on S(H) and Ω(H), respectively. Furthermore, the following statement holds.
The symmetry's character of g entails that the expected values of observables do not change if both the quantum state and the observable are transformed by the same symmetry transformation; then, whenever T r(ρA) exists, the following statement holds for every g ∈ G.
4. The interacting particle problem For a free particle, the validity of (7) makes possible to apply Wigner's theorem 1, so that for every g ∈ G an operator U g exists such that R (6), these relations imply that the restriction to E of g → U g is a projective representation [5]. Then, the relation U g QU −1 g = g −1 (Q) follows from (5.ii); it entails that the spectral PV measure of Q is an imprimitivity system [5]; therefore we can apply Mackey's theorem. In so doing, to each choice of the inducing representation L in Mackey's theorem and of µ in (2), there corresponds a different theory. The simplest choice, i.e. L : as the Hilbert space of the theory, and the position operators as (Q α ψ) = x α ψ(x); moreover, by making use of Galileian invariance, valid for a free particle, it can be proved that the form of the hamiltonian operator must be H = − 1 Hence, the simplest theory 1 For instance, the pure boost h ∈ G characterized by a velocity u = (u, 0, 0), does not change the position at all; deduced from symmetry principles by the group theoretical approach is the standard Quantum Theory of a spin-0 free particle.
If the system is an interacting particle, then the group of Galileian transformations is not a group of symmetry transformations, so that conditions (6) and (7) fail to hold and we find an obstacle in extending the group theoretical approach to the interacting particle. However, in the literature several proposals can be found [5], [7], [9] where the group theoretical methods are extended to the interacting case. The aforesaid obstacles are overcome by assuming, more or less implicitly, that the following statement holds (e.g., see [5], page 201 and [7], page 236).
(P) Each Galileian transformation g is represented in the formalism of the Quantum Theory by a unitary or anti-unitary operator U g in such a way that By making use of these assumptions, the cited approaches deduce that in the Quantum Theory of a spin-0 particle undergoing an interaction homogeneous in time the hamiltonian operator H must have the following form, able to describe also interactions with electromagnetic fields.
Now we prove, instead, the following statement.

(R) Assumption (P) forces the hamiltonian of the Quantum Theory of a spin-0 particle undergoing an interaction homogeneous in time into the form
To prove (R), first we imply from (P) that g → U g is a projective representation of the Galileian group G. Then, according to section 2.2, the sub-groups T α , R α , B α can be represented by the one parameter unitary sub-groups {e −iP α a } a∈R , {e −iJ α θ } θ∈R , {e iG α u } u∈R , in such a way that the hermitean generators P α , J α , G α satisfy (2). Once defined the self-adjoint operators F α = Gα µ , it can be proved that relations (2) imply that Since the F α 's commute with each other, according to spectral theory, there is a unique PV measure P : . Then (9) easily implies that ∆ → P (∆) is an imprimitivity system for the restriction to E of g → U g ; therefore Mackey's theorem applies. In so doing, the simplest choice for H 0 , i.e. H 0 = C, leads to identify H, F α , P α ,Û g as .
Both F and P in (10) are complete systems of operators in L 2 (R 3 ), and (F, P) is an irreducible system of operators [5]. Then we can easily prove the following proposition.
≡constant for the irreducibility of (F, P); on the other hand, (P.i) together with (2.iv) and (5.ii) for By making use ofQ The comparison between (11) and (12) yields .

Quantum Theory of an interacting particle
According to the conclusion of the last section, to develop a Quantum Theory of a particle, able to describe also electromagnetic interactions, assumption (P) must be abandoned besides the symmetry's implications (6) and (7). In this section we undertake such a development. First, in sect. 5.1, we replace the active transformation A → R . This allows us to prove, without making use of (7), that a unitary or an anti-unitary operator U g exists such that S g [A] = U g AU * g , for every g ∈ G. Another problematic task is to exclude anti-unitary operators U g without the condition that g → U g is a projective representation, condition necessary in the known arguments that obtain such an exclusion [3], [5], [10]. Such a further problem is addressed and solved in sect. 5.2.
However, our correspondence U : g is yet not a projective representation, so that Mackey's theorem cannot apply. In section 5.3 we argue how g → U g can be changed into a projective representation by means of a σ-conversion. We show that the operators representing position are identifiable with the multiplication operators in the case that the interaction admits a σ-conversion that leaves unaltered the covariance properties of the position with respect to G. Therefore our approach explains such an identification by the existence of a determined covariance property related to the interaction.
In section 5.4 we determine the dynamical law for this class of interactions, whose hamiltonian operator turns out to be more general than the known hamiltonian of a particle in an electromagnetic field. However, in section 5.5 we completely characterize the electromagnetic interaction, among all possible ones, in terms of covariance properties preserved by the σconversion.

Passive interpretation of quantum transformations
To recall the meaning of the passive interpretation we make use of the Quantum Theory of a free particle, where H = L 2 (R 3 ) and the position operators Q α are the multiplication operators defined by (Q α ψ)(x) = x α ψ(x). In fact, these operators Q α represent the position observables of the particle just as referred to a pre-specified spatial reference frame Σ 0 ; if Σ g is the frame related to such a pre-fixed Σ 0 just by g, then the position operators with respect to Σ g must be g −1 (Q), in general different from Q. Such a dependence on the reference frame is not an exclusive feature of the position observables; for each g ∈ G, we can introduce a mapping with the following interpretation.
(PI) If the operator A represents a given observable with respect to the pre-fixed reference frame Σ 0 , the operator S g [A] represents that observable with respect to Σ g .
Statement (PI) expresses the passive interpretation of the transformation g. For instance, since remark 3.1 extends to the mapping S g , then the operators Q (t) transform according to , and, as a particular case, S g [Q] = g −1 (Q).
The following statements are conceptually implied by the meaning of the passive quantum transformations.
(S.1) For every g ∈ G, the mapping S g is bijective.
(S.2) Let f be any fixed real Borel function such that if A is a self-adjoint operator, then B = f (A) is a self-adjoint operator too. Since in Quantum Theory a measurement of the quantum observable f (A) can be performed by measuring A and then transforming the obtained outcome a by the purely mathematical transformation f into the outcome b = f (a) of f (A), the following equality has to hold: Conditions (S.1) and (S.2) are sufficient to show further properties of the mappings S g , according to the following propositions.
(ii) If [A, B] = 0 then C ∈ Ω(H) and two functions f a , f b exist so that A = f a (C) and

Further implications
Propositions 5.1 and 5.2 imply that for every transformation g ∈ G, the corresponding passive quantum transformation is given by A → S g [A] = U g AU * g . If the correspondence g → S g satisfied S g 1 g 2 = S g 1 • S g 2 , so that g → U g would be a projective representation, then it could be easily proved, according to [3], [5], [10], [12], that every U g must be unitary. But in presence of interaction g → U g in general is not a projective representation. Now we prove that anti-unitary U g can be excluded under the only hypothesis that the correspondence g → S g is continuous according to the following continuity notion given by Bargmann [12].
Bargmann proved that if a correspondence g → T g from G into the set of all automorphisms of Π(H) is continuous then the arbitrary phase factor of the operator U g for which T g [A] = U g AU * g can be chosen so that U g ψ is continuous with respect to g in the topology of H [12]. (16) is continuous according to def. 5.1, then every operator U g such that S g [A] = U g AU * g for all A ∈ Ω(H) is unitary. Proof. Since S e [A] = A = 1IA1I * , we can choose U e = 1I which is unitary. Hence, because of the continuity of g → U g , a maximal neighborhood K e of e must exist so that U g is unitary for all g ∈ K e ; otherwise a sequence g n → e would exist with U gn anti-unitary and so ψ | ϕ = U g n ϕ | U g n ψ , and then ψ | ϕ = lim n→∞ U g n ϕ | U g n ψ = U e ϕ | U e ψ = ϕ | ψ which cannot hold for all ψ, ϕ unless H is real. Now, such a neighborhood K e has no boundary, and since G is a connected group, K e = G; indeed, if g 0 ∈ ∂K e , two sequences g n → g 0 and h n → g 0 would exist with U gn unitary and U hn anti-unitary; therefore, the continuity of U g would imply that U g 0 should simultaneously be unitary and anti-unitary. •

σ-conversions
By switching to the passive interpretation, under a continuity condition we were able to establish the existence of a continuous correspondence U : G → U(H), g → U g , such that S g [A] = U g AU −1 g . The argument of section 4, which shows that assumption (P) is empirically inadequate, can be easily adapted to show that such a correspondence is not a σ-representation, in general, even if the passive interpretation is adopted. Without such a representation property we cannot apply Mackey's theorem to proceed with our approach. Now we address this obstacle.
The correspondence g → U g , can be converted into a σ-representation if we multiply each operator U g by a suitable unitary operator V g of H; namely, V g is a unitary operator such that the correspondence g →Û g = V g U g turns out to be a σ-representation. The transition from g → U g to g →Û g = V g U g will be called σ-conversion; the mapping V : G → U(H), g → V g that realizes the σ-conversion will be called σ-conversion mapping. If g → V g is a σ-conversion mapping for g → U g and θ : G → R is a real function, then also g → e iθ(g) V g is a σ-conversion mapping, provided that e iθ(e) = 1. In any case, V e = 1I must hold.
ButÛ g is not the unitary operator which realizes the passive transformation: S g [A] = U g AÛ −1 g . Moreover, F is not the triple Q representing the position. So, our explicit realization of the mathematical formalism of the theory is, in general, devoid of physical significance. But the approach can go on if we restrict our investigation to those interactions which admit σconversions U g →Û g = V g U g which are Q-covariant, i.e. σ-conversions that leave unaltered the covariance properties of the position operators Q, i.e. such that The proof of Prop. 4.1 can be easily adapted to show that if the interaction admits a Qcovariant σ-conversion then F = Q in the present case too. Hence, the operator which represents the position in the mathematical formalism of the theory is the multiplication operator: Q α ψ(x) = x α ψ(x). So, in the present approach the identification of the multiplication operator with the position operator is implied by the possibility that the interaction admits a σ-conversion which preserves the covariance properties of the position operators. The existence of interaction which are not Q-covariant is not excluded, of course.
However, the operatorsÛ g continue to be not the representative of the transformations of G, i.e., S g [A] =Û g AÛ −1 g does not hold. The following proposition specify howÛ g relates to the unitary operator U g that realizes the transformation g according to the passive interpretation.
Proposition 5.4. The σ-conversion mapping of a Q-covariant σ-conversion has the form g → V g = e iθ(g,Q) , for some function θ.